College Algebra and Trigonometry

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Motivating Features to Help You Succeed! Your success in college algebra and trigonometry is important to us. To guide you to that success, we have created a textbook with features that promote learning and support various learning styles. These features are highlighted below. We encourage you to examine these features and use them to successfully complete this course.

Prepare for This Section These exercises test your understanding of prerequisite skills and concepts that were covered earlier in the text. Mastery of these concepts is required for success in the following section.

Motivating Applications Large selections of contemporary applications from many different disciplines demonstrate the utility of mathematics.

Engaging Examples Examples are designed to capture your attention and help you master important concepts.

Annotated Examples Step-by-step solutions are provided for each example.

Try Exercises A reference to an exercise follows each worked example. This exercise provides you the opportunity to test your understanding by working an exercise similar to the worked example.

Solutions to Try Exercises The complete solutions to the Try Exercises can be found in the Solutions to the Try Exercises appendix, starting page S1.

Visualize the Solution When appropriate, both algebraic and graphical solutions are provided to help visualize the mathematics of the example and to create a link between the two.

Mid-Chapter Quizzes These quizzes will help you assess your understanding of the concepts studied earlier in the chapter. They provide a mini-review of the chapter material.

Chapter Test Prep This is a summary of the major concepts discussed in the chapter and will help you prepare for the chapter test. For each concept, there is a reference to a worked example illustrating how the concept is used and at least one exercise in the chapter review relating to that concept.

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COLLEGE ALGEBRA AND TRIGONOMETRY


COLLEGE ALGEBRA AND TRIGONOMETRY

Chad Ehlers/Getty Images

SEVENTH EDITION

Richard N. Aufmann Vernon C. Barker Richard D. Nation

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

College Algebra and Trigonometry, Seventh Edition Richard N. Aufmann, Vernon C. Barker, Richard D. Nation Acquisitions Editor: Gary Whalen Senior Developmental Editor: Carolyn Crockett Assistant Editor: Stefanie Beeck

© 2011, 2008 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means, graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher.

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Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10 09

CONTENTS CHAPTER P

Preliminary Concepts P.1 P.2 P.3 P.4 P.5 P.6

1

The Real Number System 2 Integer and Rational Number Exponents 17 Polynomials 32 Mid-Chapter P Quiz 39 Factoring 40 Rational Expressions 49 Complex Numbers 59

Exploring Concepts with Technology

66

Chapter P Test Prep 67 Chapter P Review Exercises 70 Chapter P Test 73

CHAPTER 1

Equations and Inequalities

75

1.1 Linear and Absolute Value Equations 76 1.2 Formulas and Applications 83 1.3 Quadratic Equations 96 Mid-Chapter 1 Quiz 109 1.4 Other Types of Equations 110 1.5 Inequalities 123 1.6 Variation and Applications 136 Exploring Concepts with Technology

144

Chapter 1 Test Prep 145 Chapter 1 Review Exercises 148 Chapter 1 Test 151 Cumulative Review Exercises 152

CHAPTER 2

Functions and Graphs

153

2.1 Two-Dimensional Coordinate System and Graphs 154 2.2 Introduction to Functions 166 2.3 Linear Functions 186 Mid-Chapter 2 Quiz 200 2.4 Quadratic Functions 200 2.5 Properties of Graphs 213 2.6 Algebra of Functions 227 2.7 Modeling Data Using Regression 237 Exploring Concepts with Technology

248

Chapter 2 Test Prep 249 Chapter 2 Review Exercises 253 Chapter 2 Test 257 Cumulative Review Exercises 258 v

vi

CONTENTS

CHAPTER 3

Polynomial and Rational Functions

259

3.1 Remainder Theorem and Factor Theorem 260 3.2 Polynomial Functions of Higher Degree 271 3.3 Zeros of Polynomial Functions 287 Mid-Chapter 3 Quiz 299 3.4 Fundamental Theorem of Algebra 299 3.5 Graphs of Rational Functions and Their Applications 307 Exploring Concepts with Technology

323


CHAPTER 4

Exponential and Logarithmic Functions

333

4.1 4.2 4.3 4.4

Inverse Functions 334 Exponential Functions and Their Applications 346 Logarithmic Functions and Their Applications 358 Properties of Logarithms and Logarithmic Scales 369 Mid-Chapter 4 Quiz 380 4.5 Exponential and Logarithmic Equations 380 4.6 Exponential Growth and Decay 390 4.7 Modeling Data with Exponential and Logarithmic Functions 404 Exploring Concepts with Technology

416


CHAPTER 5

Trigonometric Functions 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

427

Angles and Arcs 428 Right Triangle Trigonometry 442 Trigonometric Functions of Any Angle 454 Trigonometric Functions of Real Numbers 461 Mid-Chapter 5 Quiz 472 Graphs of the Sine and Cosine Functions 473 Graphs of the Other Trigonometric Functions 481 Graphing Techniques 491 Harmonic Motion—An Application of the Sine and Cosine Functions 500



505

CONTENTS

CHAPTER 6

Trigonometric Identities and Equations

515

6.1 Verification of Trigonometric Identities 516 6.2 Sum, Difference, and Cofunction Identities 522 6.3 Double- and Half-Angle Identities 532 Mid-Chapter 6 Quiz 540 6.4 Identities Involving the Sum of Trigonometric Functions 541 6.5 Inverse Trigonometric Functions 548 6.6 Trigonometric Equations 560 Exploring Concepts with Technology

572


CHAPTER 7

Applications of Trigonometry

581

7.1 Law of Sines 582 7.2 Law of Cosines and Area 592 7.3 Vectors 601 Mid-Chapter 7 Quiz 615 7.4 Trigonometric Form of Complex Numbers 616 7.5 De Moivre’s Theorem 622 Exploring Concepts with Technology

626


CHAPTER 8

Topics in Analytic Geometry 8.1 8.2 8.3 8.4

Parabolas 634 Ellipses 645 Hyperbolas 658 Rotation of Axes 670 Mid-Chapter 8 Quiz 678 8.5 Introduction to Polar Coordinates 678 8.6 Polar Equations of the Conics 691 8.7 Parametric Equations 696 Exploring Concepts with Technology


705

633

vii

viii

CONTENTS

CHAPTER 9

Systems of Equations and Inequalities

717

9.1 Systems of Linear Equations in Two Variables 718 9.2 Systems of Linear Equations in Three Variables 728 9.3 Nonlinear Systems of Equations 740 Mid-Chapter 9 Quiz 747 9.4 Partial Fractions 748 9.5 Inequalities in Two Variables and Systems of Inequalities 755 9.6 Linear Programming 762 Exploring Concepts with Technology

772


CHAPTER 10

Matrices

779

10.1 Gaussian Elimination Method 780 10.2 Algebra of Matrices 791 10.3 Inverse of a Matrix 813 Mid-Chapter 10 Quiz 823 10.4 Determinants 824 10.5 Cramer’s Rule 833 Exploring Concepts with Technology

837


CHAPTER 11

Sequences, Series, and Probability 11.1 Infinite Sequences and Summation Notation 848 11.2 Arithmetic Sequences and Series 854 11.3 Geometric Sequences and Series 860 Mid-Chapter 11 Quiz 871 11.4 Mathematical Induction 871 11.5 Binomial Theorem 878 11.6 Permutations and Combinations 883 11.7 Introduction to Probability 890 Exploring Concepts with Technology

Chapter 11 Test Prep 900 Chapter 11 Review Exercises 902 Chapter 11 Test 905 Cumulative Review Exercises 906 Solutions to the Try Exercises S1 Answers to Selected Exercises A1 Index I1

899

847

PREFACE We are proud to offer the seventh edition of College Algebra and Trigonometry. Your success in college algebra and trigonometry is important to us. To guide you to that success, we have created a textbook with features that promote learning and support various learning styles. These features are highlighted below. We encourage you to examine the features and use them to help you successfully complete this course.

Features Chapter Openers Each Chapter Opener demonstrates a contemporary application of a mathematical concept developed in that chapter.

Related Exercise References Each Chapter Opener ends with a reference to a particular exercise within the chapter that requires you to solve a problem related to that chapter opener topic.

Listing of Major Concepts A list of major concepts in each section is provided in the margin of the first page of each section. Prepare for This Section Each section (after the first section) of a chapter opens with review exercises titled Prepare for This Section. These exercises give you a chance to test your understanding of prerequisite skills and concepts before proceeding to the new topics presented in the section. ix

x

PREFACE

Thoughtfully Designed Exercise Sets We have thoroughly reviewed each exercise set to ensure a smooth progression from routine exercises to exercises that are more challenging. The exercises illustrate the many facets of topics discussed in the text. The exercise sets emphasize skill building, skill maintenance, conceptual understanding, and, as appropriate, applications. Each chapter includes a Chapter Review Exercise set and each chapter, except Chapter P, includes a Cumulative Review Exercise set.

Contemporary Applications Carefully developed mathematics is complemented by abundant, relevant, and contemporary applications, many of which feature real data, tables, graphs, and charts. Applications demonstrate the value of algebra and cover topics from a wide variety of disciplines. Besides providing motivation to study mathematics, the applications will help you develop good problem-solving skills.

PREFACE

xi

By incorporating many interactive learning techniques, including the key features outlined below, College Algebra and Trigonometry uses the proven Aufmann Interactive Method (AIM) to help you understand concepts and obtain greater mathematical fluency. The AIM consists of Annotated Examples followed by Try Exercises (and solutions) and a conceptual Question/Answer follow-up. See the samples below: Engaging Examples Examples are designed to capture your attention and help you master important concepts.

Annotated Examples Step-by-step solutions are provided for most numbered examples.

Try Exercises A reference to an exercise follows all worked examples. This exercise provides you with an opportunity to test your understanding of concepts by working an exercise related to the worked example. Solutions to Try Exercises Complete solutions to the Try Exercises can be found in the Solutions to the Try Exercises appendix on page S1.

Question/Answer In each section, we have posed at least one question that encourages you to pause and think about the concepts presented in the current discussion. To ensure that you do not miss this important information, the answer is provided as a footnote on the same page.

xii

PREFACE

Immediate Examples of Definitions and Concepts Immediate examples of many definitions and concepts are provided to enhance your understanding of new topics.

Margin Notes alert you to a point requiring special attention or are used to provide study tips.

To Review Notes in the margin will help you recognize the prerequisite skills needed to understand new concepts. These notes direct you to the appropriate page or section for review.

Calculus Connection Icons identify topics that will be revisited in a subsequent calculus course.

Visualize the Solution When appropriate, both algebraic and graphical solutions are provided to help you visualize the mathematics of an example and to create a link between the algebraic and visual components of a solution.

PREFACE

xiii

Integrating Technology Integrating Technology boxes show how technology can be used to illustrate concepts and solve many mathematical problems. Examples and exercises that require a calculator or a computer to find a solution are identified by the graphing calculator icon.

Exploring Concepts with Technology The optional Exploring Concepts with Technology feature appears after the last section in each chapter and provides you the opportunity to use a calculator or a computer to solve computationally difficult problems. In addition, you are challenged to think about pitfalls that can be produced when using technology to solve mathematical problems.

Modeling Modeling sections and exercises rely on the use of a graphing calculator or a computer. These optional sections and exercises introduce the idea of a mathematical model and help you see the relevance of mathematical concepts.

xiv

PREFACE

NEW Mid-Chapter Quizzes New to this edition, these quizzes help you assess your understanding of the concepts studied earlier in the chapter. The answers for all exercises in the Mid-Chapter Quizzes are provided in the Answers to Selected Exercises appendix along with a reference to the section in which a particular concept was presented. NEW Chapter Test Preps The Chapter Test Preps summarize the major concepts discussed in each chapter. These Test Preps help you prepare for a chapter test. For each concept there is a reference to a worked example illustrating the concept and at least one exercise in the Chapter Review Exercise set relating to that concept. Chapter Review Exercise Sets and Chapter Tests The Chapter Review Exercise sets and the Chapter Tests at the end of each chapter are designed to provide you with another opportunity to assess your understanding of the concepts presented in a chapter. The answers for all exercises in the Chapter Review Exercise sets and the Chapter Tests are provided in the Answers to Selected Exercises appendix along with a reference to the section in which the concept was presented.

PREFACE

xv

In addition to the New! Mid-Chapter Quizzes and New! Chapter Test Preps, the following changes appear in this seventh edition of College Algebra and Trigonometry: Chapter P

Preliminary Concepts P.1 This section has been reorganized. The Order of Operations Agreement has been given more prominence to ensure that students understand this important concept. P.2 Additional examples have been added to illustrate more situations with radicals and rational exponents. P.3 Another example has been added, new exercises have been added, and some of the existing exercises have been rearranged. P.4 This section has been reorganized, and new examples have been added. The exercise set has been reorganized, and new exercises have been added. P.5 New examples have been added to show operations on rational expressions.

Chapter 1

Equations and Inequalities 1.1 Two examples were added for solving first-degree equations. 1.2 This section has been reorganized, and new applications have been added. The exercise set has been changed to include new applications. 1.3 New examples showing how to solve quadratic equations were added. The exercise set has been extensively revised. 1.4 Much of this section has been rewritten and reorganized, and new application problems have been added. The exercise set has been reorganized, and many new exercises have been added. 1.5 The critical-value method of solving polynomial inequalities has been expanded, and new exercises have been added.

Chapter 2

Functions and Graphs 2.2 This section has been reorganized so that appropriate emphasis is given to the various aspects of working with functions. We introduced the connection of x-intercepts to real zeros of a function to better prepare students for a full discussion of zeros in Chapter 3. The exercise set has been reorganized, and many new exercises were added. 2.3 The introduction to slope has been expanded. New examples on finding the equation of a line were added to give students models of the various types of problems found in the exercise set. 2.5 New examples were added to illustrate various transformations. The effect was to slow the pace of this section so students could better understand these important concepts.

Chapter 3

Polynomial and Rational Functions 3.2 A new example on modeling data with a cubic function was added. This example is followed by a discussion concerning the strengths and weaknesses of modeling data from an application with cubic and quartic regression functions. Five new application exercises involving the use of cubic and quartic models were added to the exercise set. 3.5 A new example on using a rational function to solve an application was added. Two new exercises that make use of a rational function to solve an application were added. Three exercises that involve creating a rational function whose graph has given properties were added. The definition of a slant asymptote was included in this section. Several new exercises were added to the Chapter Review Exercises. A new application exercise was added to the Chapter Test.

xvi

PREFACE

Chapter 4

Exponential and Logarithmic Functions 4.1 Two new application exercises were added to the exercise set. 4.2 Two new applications were created to introduce increasing and decreasing exponential functions. Additional expository material was inserted to better explain the concept of the expression bx where x is an irrational number. 4.6 Examples and application exercises involving dates were updated or replaced. New application exercises involving the concept of a declining logistic model were added to the exercise set. 4.7 Examples and application exercises involving dates were updated or replaced. New application exercises were added to the exercise set. New exercises were added to the Chapter Review Exercises.

Chapter 5

Trigonometric Functions 5.2 New application exercises were added to the exercise set. 5.3 The reference angle evaluation procedure and the accompanying example were revised to simplify the evaluation process. Several exercises were added to the Chapter Review Exercises.

Chapter 6

Trigonometric Identities and Equations 6.6 Application examples and exercises that involve dates were updated. Several exercises were added to the Chapter Review Exercise set.

Chapter 7

Applications of Trigonometry 7.1 Three new examples were added to better illustrate the ambiguous case of the Law of Sines. New application exercises were added to the Chapter Review Exercises. A new application exercise was added to the Chapter Test.

Chapter 8

Topics in Analytic Geometry 8.4 A new art piece was included in Example 4 to better illustrate the procedure for graphing a rotated conic section with a graphing utility. 8.5 New art pieces were inserted in Example 3 to better illustrate the procedure for graphing a polar equation with a graphing utility. A new example and exercises concerning the graphs of lemniscates were added. The example concerning the transformation from rectangular to polar coordinates was revised to better illustrate the multiple representation of a point in the polar coordinate system. Several exercises were added to the exercise set. New application exercises were added to the Chapter Review Exercises. New application exercises were added to the Chapter Test.

Chapter 9

Systems of Equations and Inequalities A new chapter opener page was written to introduce some of the concepts in this chapter. 9.5 New art pieces were included to better illustrate the concept of finding the solution set of a system of inequalities by graphing. The targeted exercise heart rate formula was updated in an example and in the application exercises concerning physical fitness. 9.6 New illustrations were added to an example and two application exercises. A new application exercise on maximizing profit was added.

PREFACE

xvii

Chapter 10

Matrices 10.1 A new example on augmented matrices was added. Some new exercises were added to show different row-reduced forms and systems of equations with no solution. 10.2 A new example on finding a power of a matrix was added. A graph theory application involving multiplication of matrices was added.

Chapter 11

Sequences, Series, and Probability 11.1 A new example on finding the sum of a series was added. 11.2 Example 2 was rewritten to better illustrate that a series is the sum of the terms of a sequence. 11.5 Two examples were added that demonstrate the Binomial Theorem. New exercises were added to more gradually move from easier to more difficult applications of the Binomial Theorem. 11.7 This section has been reorganized and a new example that shows the use of the probability addition rules has been added.

SUPPLEMENTS For the Instructor Complete Solutions Manual for Aufmann/Barker/Nation’s College Algebra and Trigonometry, 7e ISBN: 0-538-73927-4 The complete solutions manual provides worked-out solutions to all of the problems in the text. (Print) *online version available; see description for Solution Builder below Text Specific DVDs for Aufmann/Barker/Nation’s College Algebra and Trigonometry Series, 7e ISBN: 0-538-79788-6 Available to adopting instructors, these DVDs, which cover all sections in the text, are hosted by Dana Mosely and captioned for the hearing-impaired. Ideal for promoting individual study and review, these comprehensive DVDs also support students in online courses or may be checked out by a student who may have missed a lecture. 12 DVDs contain over 40 hours of video. (Media) Enhanced WebAssign® (access code packaged with student edition at request of instructor; instructor access obtained by request of instructor to Cengage Learning representative) Enhanced WebAssign® allows instructors to assign, collect, grade, and record homework assignments online, minimizing workload and streamlining the grading process. EWA also gives students the ability to stay organized with assignments and have up-to-date grade information. For your convenience, the exercises available in EWA are indicated in the instructor’s edition by a blue triangle. (Online) PowerLecture ISBN: 0-538-73906-1 PowerLecture contains PowerPoint ® lecture outlines, a database of all art in the text, ExamView®, and a link to the Solution Builder. (CD) ExamView® (included on PowerLecture CD) Create, deliver, and customize tests (both print and online) in minutes with this easy-to-use assessment system. (CD) Solution Builder (included on PowerLecture CD and available online at http://academic.cengage.com/solutionbuilder/) The Solution Builder is an electronic version of the Complete Solutions Manual, providing instructors with an efficient method for creating solution sets to homework and exams that can be printed or posted. (CD and online)

xviii

PREFACE

Syllabus Creator (included on PowerLecture CD) Quickly and easily create your course syllabus with Syllabus Creator, which was created by the authors.

For the Student Study Guide with Student Solutions Manual for Aufmann/Barker/Nation’s College Algebra and Trigonometry, 7e ISBN: 0-538-73908-8 The student solutions manual reinforces student understanding and aids in test preparation with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. Includes worked solutions to the odd-numbered problems in the text. (Print) Enhanced WebAssign® (access code packaged with student edition at request of instructor) Enhanced WebAssign® allows instructors to assign, collect, grade, and record homework assignments online, minimizing workload and streamlining the grading process. EWA also gives students the ability to stay organized with assignments and have up-to-date grade information. (Online)

ACKNOWLEDGMENTS We would like to thank the wonderful team of editors, accuracy checkers, proofreaders, and solutions manual authors. Special thanks to Cindy Harvey, Helen Medley, and Christi Verity. Cindy Harvey was a very valuable asset during development and production of the manuscript. Helen Medley was the accuracy reviewer for both College Algebra and College Algebra and Trigonometry, and Christi Verity wrote the solutions for the Complete Solutions Manual and the Student Solutions Manual for College Algebra and College Algebra and Trigonometry. Both Helen and Christi have improved the accuracy of the texts and provided valuable suggestions for improving the texts. We are grateful to the users of the previous edition for their helpful suggestions on improving the text. Also, we sincerely appreciate the time, effort, and suggestions of the reviewers of this edition: Robin Anderson—Southwestern Illinois College Richard Bailey—Midlands Tech College Cecil J. Coone—Southwest Tennessee Community College Kyle Costello—Salt Lake Community College Thomas English—College of the Mainland Celeste Hernandez—Richland College Magdalen Ivanovska—New York University Skopje Rose Jenkins—Midlands Tech College Stefan C. Mancus—Embry-Riddle Aeronautical University Jamie Whittimore McGill—East Tennessee State University Zephyrinus Okonkwo—Albany State University Mike Shirazi—Germanna Community College Lalitha Subramanian—Potomac State College of West Virginia University Tan Zhang—Murray State University

CHAPTER

P

PRELIMINARY CONCEPTS

P.1 The Real Number System P.2 Integer and Rational Number Exponents P.3 Polynomials P.4 Factoring P.5 Rational Expressions

AFP/Getty Images

P.6 Complex Numbers

Albert Einstein proposed relativity theory more than 100 years ago, in 1905.

Martial Trezzini/epa/CORBIS

Relativity Is More Than 100 Years Old

The Large Hadron Collider (LHC). Atomic particles are accelerated to high speeds inside the long structure in the photo above. By studying particles moving at speeds that approach the speed of light, physicists can confirm some of the tenets of relativity theory.

Positron emission tomography (PET) scans, the temperature of Earth’s crust, smoke detectors, neon signs, carbon dating, and the warmth we receive from the sun may seem to be disparate concepts. However, they have a common theme: Albert Einstein’s Theory of Special Relativity. When Einstein was asked about his innate curiosity, he replied: The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvelous structure of reality. It is enough if one tries merely to comprehend a little of this mystery every day.

Today, relativity theory is used in conjunction with other concepts of physics to study ideas ranging from the structure of an atom to the structure of the universe. Some of Einstein’s equations require working with radical expressions, such as the expression given in Exercise 139 on page 31; other equations use rational expressions, such as the expression given in Exercise 64 on page 59.

1

2

CHAPTER P


SECTION P.1 Sets Union and Intersection of Sets Interval Notation Absolute Value and Distance Exponential Expressions Order of Operations Agreement Simplifying Variable Expressions

The Real Number System Sets Human beings share the desire to organize and classify. Ancient astronomers classified stars into groups called constellations. Modern astronomers continue to classify stars by such characteristics as color, mass, size, temperature, and distance from Earth. In mathematics it is useful to place numbers with similar characteristics into sets. The following sets of numbers are used extensively in the study of algebra. 51, 2, 3, 4, Á 6

Natural numbers

5 Á , -3, -2, - 1, 0, 1, 2, 3, Á 6

Integers

5all terminating or repeating decimals6

Rational numbers

5all nonterminating, nonrepeating decimals6

Irrational numbers

5all rational or irrational numbers6

Real numbers

If a number in decimal form terminates or repeats a block of digits, then the number is a rational number. Here are two examples of rational numbers. 0.75 is a terminating decimal. 0.245 is a repeating decimal. The bar over the 45 means that the digits 45 repeat without end. That is, 0.245 = 0.24545454 Á . p , where p and q are inteq gers and q Z 0. Examples of rational numbers written in this form are Rational numbers also can be written in the form

3 4 Note that

Math Matters Archimedes (c. 287–212 B.C.) was the first to calculate p with any degree of precision. He was able to show that 3

10 1 6 p 6 3 71 7

from which we get the approximation 3

1 22 = L p 7 7

The use of the symbol p for this quantity was introduced by Leonhard Euler (1707–1783) in 1739, approximately 2000 years after Archimedes.

27 110

-

5 2

7 1

-4 3

7 n = 7, and, in general, = n for any integer n. Therefore, all integers are rational 1 1

numbers. p , the decimal form of the rational q number can be found by dividing the numerator by the denominator. When a rational number is written in the form

3 = 0.75 4

27 = 0.245 110

In its decimal form, an irrational number neither terminates nor repeats. For example, 0.272272227 Á is a nonterminating, nonrepeating decimal and thus is an irrational number. One of the best-known irrational numbers is pi, denoted by the Greek symbol p . The number p is defined as the ratio of the circumference of a circle to its diameter. Often in applications the rational number 3.14 or the rational 22 number is used as an approximation of the irrational number p. 7 Every real number is either a rational number or an irrational number. If a real number is written in decimal form, it is a terminating decimal, a repeating decimal, or a nonterminating and nonrepeating decimal.

P.1

The relationships among the various sets of numbers are shown in Figure P.1.

Math Matters Sophie Germain (1776–1831) was born in Paris, France. Because enrollment in the university she wanted to attend was available only to men, Germain attended under the name of Antoine-August Le Blanc. Eventually her ruse was discovered, but not before she came to the attention of Pierre Lagrange, one of the best mathematicians of the time. He encouraged her work and became a mentor to her. A certain type of prime number is named after her, called a Germain prime number. It is a number p such that p and 2p + 1 are both prime. For instance, 11 is a Germain prime because 2(11) + 1 = 23 and 11 and 23 are both prime numbers. Germain primes are used in public key cryptography, a method used to send secure communications over the Internet.

3

THE REAL NUMBER SYSTEM

Positive integers (natural numbers) 7 1 103 Integers

Zero 0

−201

7

Rational numbers 3 4

0 −5

Real numbers 3 4

3.1212 −1.34 −5

3.1212 −1.34 7

Negative integers −201

−8

Irrational numbers

−5

1

0

−5

103 −201

−0.101101110... √7 π

−0.101101110... √7 π

Figure P.1

Prime numbers and composite numbers play an important role in almost every branch of mathematics. A prime number is a positive integer greater than 1 that has no positiveinteger factors1 other than itself and 1. The 10 smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Each of these numbers has only itself and 1 as factors. A composite number is a positive integer greater than 1 that is not a prime number. For example, 10 is a composite number because 10 has both 2 and 5 as factors. The 10 smallest composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.

EXAMPLE 1

Classify Real Numbers

Determine which of the following numbers are a.

integers

b. rational numbers

c. irrational numbers

d.

real numbers

e. prime numbers

f. composite numbers

-0.2,

0,

0.3,

0.71771777177771 Á ,

p,

6,

7,

41,

51

Solution a. Integers: 0, 6, 7, 41, 51 b.

Rational numbers: - 0.2, 0, 0.3, 6, 7, 41, 51

c.

Irrational numbers: 0.71771777177771. . . , p

d.

Real numbers: - 0.2, 0, 0.3, 0.71771777177771 Á , p, 6, 7, 41, 51

e.

Prime numbers: 7, 41

f.

Composite numbers: 6, 51 Try Exercise 2, page 14

Each member of a set is called an element of the set. For instance, if C = 52, 3, 56, then the elements of C are 2, 3, and 5. The notation 2 C is read “2 is an element of C.” 1

A factor of a number divides the number evenly. For instance, 3 and 7 are factors of 21; 5 is not a factor of 21.

4

CHAPTER P


Set A is a subset of set B if every element of A is also an element of B, and we write A 8 B. For instance, the set of negative integers { -1, -2, - 3, - 4, Á } is a subset of the set of integers. The set of positive integers 51, 2, 3, 4, Á 6 (the natural numbers) is also a subset of the set of integers. Question • Are the integers a subset of the rational numbers?

Note The order of the elements of a set is not important. For instance, the set of natural numbers less than 6 given at the right could have been written 53, 5, 2, 1, 46. It is customary, however, to list elements of a set in numerical order.

The empty set, or null set, is the set that contains no elements. The symbol is used to represent the empty set. The set of people who have run a 2-minute mile is the empty set. The set of natural numbers less than 6 is 51, 2, 3, 4, 56. This is an example of a finite set; all the elements of the set can be listed. The set of all natural numbers is an example of an infinite set. There is no largest natural number, so all the elements of the set of natural numbers cannot be listed. Sets are often written using set-builder notation. Set-builder notation can be used to describe almost any set, but it is especially useful when writing infinite sets. For instance, the set 52n ƒ n natural numbers6

Math Matters A fuzzy set is one in which each element is given a “degree” of membership. The concepts behind fuzzy sets are used in a wide variety of applications such as traffic lights, washing machines, and computer speech recognition programs.

is read as “the set of elements 2n such that n is a natural number.” By replacing n with each of the natural numbers, this becomes the set of positive even integers: 52, 4, 6, 8,. . .6. The set of real numbers greater than 2 is written 5x ƒ x 7 2, x real numbers6

and is read “the set of x such that x is greater than 2 and x is an element of the real numbers.” Much of the work we do in this text uses the real numbers. With this in mind, we will frequently write, for instance, 5x ƒ x 7 2, x real numbers6 in a shortened form as 5x ƒ x 7 26, where we assume that x is a real number.

EXAMPLE 2

Use Set-Builder Notation

List the four smallest elements in 5n3 ƒ n natural numbers6. Solution Because we want the four smallest elements, we choose the four smallest natural numbers. Thus n = 1, 2, 3, and 4. Therefore, the four smallest elements of the set 5n3 ƒ n natural numbers6 are 1, 8, 27, and 64. Try Exercise 6, page 14

Union and Intersection of Sets Just as operations such as addition and multiplication are performed on real numbers, operations are performed on sets. Two operations performed on sets are union and intersection. The union of two sets A and B is the set of elements that belong to A or to B, or to both A and B.

Answer • Yes.

P.1


5

Definition of the Union of Two Sets The union of two sets, written A ´ B, is the set of all elements that belong to either A or B. In set-builder notation, this is written A ´ B = 5x ƒ x A or x B6

EXAMPLE

Given A = 52, 3, 4, 56 and B = 50, 1, 2, 3, 46, find A ´ B. A ´ B = 50, 1, 2, 3, 4, 56

• Note that an element that belongs to both sets is listed only once.

The intersection of the two sets A and B is the set of elements that belong to both A and B.

Definition of the Intersection of Two Sets The intersection of two sets, written A ¨ B, is the set of all elements that are common to both A and B. In set-builder notation, this is written A ¨ B = 5x ƒ x A and x B6

EXAMPLE

Given A = 52, 3, 4, 56 and B = 50, 1, 2, 3, 46, find A ¨ B. A ¨ B = 52, 3, 46

• The intersection of two sets contains the elements common to both sets.

If the intersection of two sets is the empty set, the two sets are said to be disjoint. For example, if A = 52, 3, 46 and B = 57, 86, then A ¨ B = and A and B are disjoint sets.

EXAMPLE 3

Find the Union and Intersection of Sets

Find each intersection or union given A = 50, 2, 4, 6, 10, 126, B = 50, 3, 6, 12, 156, and C = 51, 2, 3, 4, 5, 6, 76. a.

A´C

b. B ¨ C

c. A ¨ (B ´ C)

Solution a. A ´ C = 50, 1, 2, 3, 4, 5, 6, 7, 10, 126 b. c.

d.

B ¨ C = 53, 66

d. B ´ (A ¨ C )

• The elements that belong to A or C

• The elements that belong to B and C

First determine B ´ C = 50, 1, 2, 3, 4, 5, 6, 7, 12, 156. Then A ¨ (B ´ C) = 50, 2, 4, 6, 126

• The elements that belong to A and (B ´ C )

First determine A ¨ C = 52, 4, 66. Then B ´ (A ¨ C) = 50, 2, 3, 4, 6, 12, 156

Try Exercise 16, page 14

• The elements that belong to B or (A ¨ C )

6

CHAPTER P


Interval Notation −5 −4 −3 −2 −1

0

1

2

3

4

5

2

3

4

5

2

3

4

5

2

3

4

5

2

3

4

5

Figure P.2

−5 −4 −3 −2 −1

0

1

The graph of 5x ƒ x 7 26 is shown in Figure P.2. The set is the real numbers greater than 2. The parenthesis at 2 indicates that 2 is not included in the set. Rather than write this set of real numbers using set-builder notation, we can write the set in interval notation as (2, q ). In general, the interval notation represents all real numbers between a and b, not including a and b. This is an open interval. In set-builder notation, we write 5x ƒ a 6 x 6 b6. The graph of ( -4, 2) is shown in Figure P.3.

(a, b)

Figure P.3 −5 −4 −3 −2 −1

0

1

3a, b4

represents all real numbers between a and b, including a and b. This is a closed interval. In set-builder notation, we write 5x ƒ a … x … b6. The graph of 30, 44 is shown in Figure P.4. The brackets at 0 and 4 indicate that those numbers are included in the graph.

Figure P.4

−5 −4 −3 −2 −1

0

1

represents all real numbers between a and b, not including a but including b. This is a half-open interval. In set-builder notation, we write 5x ƒ a 6 x … b6. The graph of ( -1, 34 is shown in Figure P.5.

(a, b4

Figure P.5 −5 −4 −3 −2 −1

0

1

3a, b)

represents all real numbers between a and b, including a but not including b. This is a half-open interval. In set-builder notation, we write 5x ƒ a … x 6 b6. The graph of 3 - 4, - 1) is shown in Figure P.6.

Figure P.6

Subsets of the real numbers whose graphs extend forever in one or both directions can be represented by interval notation using the infinity symbol q or the negative infinity symbol - q . a b a

represents all real numbers less than a.

(b, q )

represents all real numbers greater than b.

( - q , a4

represents all real numbers less than or equal to a.

3b, q )

b

represents all real numbers greater than or equal to b.

( - q, q)

0

EXAMPLE 4

( - q , a)

represents all real numbers.

Graph a Set Given in Interval Notation

Graph (- q , 34. Write the interval in set-builder notation. Caution It is never correct to use a bracket when using the infinity symbol. For instance, [- q , 3] is not correct. Nor is [2, q ] correct. Neither negative infinity nor positive infinity is a real number and therefore cannot be contained in an interval.

Solution The set is the real numbers less than or equal to 3. In set-builder notation, this is the set 5x ƒ x … 36. Draw a right bracket at 3, and darken the number line to the left of 3, as shown in Figure P.7. − 5 − 4 −3 −2 −1

0

1

Figure P.7


2

3

4

5

P.1

−5 −4 −3 −2 −1

0

1

2

3

4

5

2

3

4

5

Figure P.8 −5 −4 −3 −2 −1

0

1

Figure P.9


7

The set 5x ƒ x … - 26 ´ 5x ƒ x 7 36 is the set of real numbers that are either less than or equal to -2 or greater than 3. We also could write this in interval notation as (- q , -24 ´ (3, q ). The graph of the set is shown in Figure P.8. The set 5x ƒ x 7 - 46 ¨ 5x ƒ x 6 16 is the set of real numbers that are greater than -4 and less than 1. Note from Figure P.9 that this set is the interval ( - 4, 1), which can be written in set-builder notation as 5x ƒ -4 6 x 6 16.

EXAMPLE 5

Graph Intervals

Graph the following. Write a. and b. using interval notation. Write c. and d. using setbuilder notation. a. c.

5x ƒ x … - 16 ´ 5x ƒ x Ú 26

(- q , 0) ´ 31, 34

Solution a. b. c.

b. 5x ƒ x Ú - 16 ¨ 5x ƒ x 6 56 d. 3 - 1, 34 ¨ (1, 5)

− 5 − 4 −3 −2 −1

0

1

2

3

4

5

− 5 − 4 −3 −2 −1

0

1

2

3

4

5

− 5 − 4 −3 −2 −1

0

1

2

3

4

5

( - q , - 14 ´ 32, q ) 3 - 1, 5) 5x ƒ x 6 06 ´ 5x ƒ 1 … x … 36

d. The graphs of 3 -1, 34, in red, and (1, 5), in blue, are shown below. −5 −4 −3 −2 −1

0

1

2

3

4

5

Note that the intersection of the sets occurs where the graphs intersect. Although 1 3 - 1, 34, 1 > (1, 5). Therefore, 1 does not belong to the intersection of the sets. On the other hand, 3 3- 1, 34 and 3 (1, 5). Therefore, 3 belongs to the intersection of the sets. Thus we have the following. − 5 − 4 −3 −2 −1

0

1

2

3

4

5

5x ƒ 1 6 x … 36


Absolute Value and Distance − 4.25

− 52

1

−5 −4 −3 −2 −1

0

π

1

√29

2

3

4

5

2

3

4

5

Figure P.10 3 −5 −4 −3 −2 −1

3 0

1

Figure P.11

The real numbers can be represented geometrically by a coordinate axis called a real number line. Figure P.10 shows a portion of a real number line. The number associated with a point on a real number line is called the coordinate of the point. The point corresponding to zero is called the origin. Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. The absolute value of a real number a, denoted ƒ a ƒ , is the distance between a and 0 on the number line. For instance, ƒ 3 ƒ = 3 and ƒ - 3 ƒ = 3 because both 3 and - 3 are 3 units from zero. See Figure P.11.

8

CHAPTER P


In general, if a Ú 0, then ƒ a ƒ = a; however, if a 6 0, then ƒ a ƒ = - a because - a is positive when a 6 0. This leads to the following definition. Note

Definition of Absolute Value

The second part of the definition of absolute value states that if a 6 0, then ƒ a ƒ = - a.For instance, if a = - 4, then ƒ a ƒ = ƒ - 4 ƒ = - (-4) = 4.

The absolute value of the real number a is defined by ƒaƒ = e

a if a Ú 0 -a if a 6 0

EXAMPLE

ƒ5ƒ = 5

ƒ -4 ƒ = 4

EXAMPLE 6

ƒ0ƒ = 0

Simplify an Absolute Value Expression

Simplify ƒ x - 3 ƒ + ƒ x + 2 ƒ given that - 1 … x … 2. Solution Recall that ƒ a ƒ = - a when a 6 0 and ƒ a ƒ = a when a Ú 0. When - 1 … x … 2, x - 3 6 0 and x + 2 7 0. Therefore, ƒ x - 3 ƒ = - (x - 3) and ƒ x + 2 ƒ = x + 2. Thus ƒ x - 3 ƒ + ƒ x + 2 ƒ = - (x - 3) + (x + 2) = 5. Try Exercise 60, page 15

The definition of distance between two points on a real number line makes use of absolute value.

Definition of the Distance Between Points on a Real Number Line If a and b are the coordinates of two points on a real number line, the distance between the graph of a and the graph of b, denoted by d(a, b), is given by d(a, b) = ƒ a - b ƒ . EXAMPLE

Find the distance between a point whose coordinate on the real number line is -2 and a point whose coordinate is 5. d( -2, 5) = ƒ - 2 - 5 ƒ = ƒ - 7 ƒ = 7 7 −5 −4 − 3 −2 −1

0

1

2

3

4

5

Note in Figure P.12 that there are 7 units between -2 and 5. Also note that the order of the coordinates in the formula does not matter. d(5, - 2) = ƒ 5 - ( -2) ƒ = ƒ 7 ƒ = 7

Figure P.12

EXAMPLE 7

Use Absolute Value to Express the Distance Between Two Points

Express the distance between a and - 3 on the number line using absolute value notation. Solution d(a, - 3) = ƒ a - ( -3) ƒ = ƒ a + 3 ƒ Try Exercise 70, page 15

P.1

9

Exponential Expressions

Math Matters The expression 10100 is called a googol. The term was coined by the 9-year-old nephew of the American mathematician Edward Kasner. Many calculators do not provide for numbers of this magnitude, but it is no serious loss. To appreciate the magnitude of a googol, consider that if all the atoms in the known universe were counted, the number would not even be close to a googol. But if a googol is too small for you, try 10googol, which is called a googolplex. As a final note, the name of the Internet site Google.com is a takeoff on the word googol.


A compact method of writing 5 # 5 # 5 # 5 is 5 4. The expression 5 4 is written in exponential notation. Similarly, we can write 2x # 2x # 2x 3 3 3

as

a

2x 3 b 3

Exponential notation can be used to express the product of any expression that is used repeatedly as a factor.

Definition of Natural Number Exponents If b is any real number and n is a natural number, then b$''%''& is a factor n times bn = b # b # b # Á # b

where b is the base and n is the exponent. EXAMPLE

27 3 3 3 3 3 a b = # # = 4 4 4 4 64

-54 = - (5 # 5 # 5 # 5) = - 625 (-5)4 = ( -5)( - 5)( - 5)( -5) = 625

Pay close attention to the difference between - 54 (the base is 5) and ( -5)4 (the base is - 5).

EXAMPLE 8

Evaluate an Exponential Expression

Evaluate. a. (-3 4)( - 4) 2

b.

-4 4 ( - 4) 4

Solution a. (-3 4)( - 4)2 = - (3 # 3 # 3 # 3) # ( -4)( - 4) = - 81 # 16 = -1296 b.

-(4 # 4 # 4 # 4) -256 -44 = = = -1 4 ( - 4)( - 4)( -4)( - 4) 256 (-4)


Order of Operations Agreement The approximate pressure p, in pounds per square inch, on a scuba diver x feet below the water’s surface is given by p = 15 + 0.5x

10

CHAPTER P


The pressure on the diver at various depths is given below.

20 feet 40 feet

15 + 0.5(10) = 15 + 5 = 20 pounds 15 + 0.5(20) = 15 + 10 = 25 pounds 15 + 0.5(40) = 15 + 20 = 35 pounds

70 feet

15 + 0.5(70) = 15 + 35 = 50 pounds

10 feet

Note that the expression 15 + 0.5(70) has two operations, addition and multiplication. When an expression contains more than one operation, the operations must be performed in a specified order, as given by the Order of Operations Agreement.

The Order of Operations Agreement If grouping symbols are present, evaluate by first performing the operations within the grouping symbols, innermost grouping symbols first, while observing the order given in steps 1 to 3. Step 1 Evaluate exponential expressions. Step 2 Do multiplication and division as they occur from left to right. Step 3 Do addition and subtraction as they occur from left to right. EXAMPLE

5 - 7(23 - 5 2) - 16 , 2 3 = 5 - 7(23 - 25) - 16 , 23

• Begin inside the parentheses and evaluate 52 = 25.

= 5 - 7( -2) - 16 , 23

• Continue inside the parentheses and evaluate 23 - 25 = - 2.

= 5 - 7( -2) - 16 , 8 = 5 - (- 14) - 2

• Evaluate 23 = 8.

= 17

• Perform addition and subtraction from left to right.

EXAMPLE 9

• Perform multiplication and division from left to right.

Use the Order of Operations Agreement

Evaluate: 3 # 5 2 - 6( - 3 2 - 4 2) , ( -15) Solution 3 # 5 2 - 6(-3 2 - 4 2) , ( -15)

= 3 # 5 2 - 6( -9 - 16) , ( - 15) = 3 # 5 2 - 6( -25) , ( - 15)

• Begin inside the parentheses.

=

• Evaluate 5 2.

3 # 25

- 6( - 25) , ( -15)

= 75 + 150 , (- 15) = 75 + (-10) = 65 Try Exercise 80, page 15

• Simplify - 9 - 16. • Do mulipltication and division from left to right. • Do addition.

P.1

Recall Subtraction can be rewritten as addition of the opposite. Therefore, 3x2 - 4xy + 5x - y - 7 = 3x 2 + (- 4xy) + 5x + (- y) + (-7) In this form, we can see that the terms (addends) are 3x2, -4xy, 5x, - y, and -7.


11

One of the ways in which the Order of Operations Agreement is used is to evaluate variable expressions. The addends of a variable expression are called terms. The 3x 2 - 4xy + 5x - y - 7 2 terms for the expression at the right are 3x , -4xy, 5x, -y, and - 7. Observe that the sign of a term is the sign that immediately precedes it. The terms 3x 2, -4xy, 5x, and -y are variable terms. The term - 7 is a constant term. Each variable term has a numerical coefficient and a variable part. The numerical coefficient for the term 3x 2 is 3; the numerical coefficient for the term -4xy is - 4; the numerical coefficient for the term 5x is 5; and the numerical coefficient for the term -y is -1. When the numerical coefficient is 1 or - 1 (as in x and -x), the 1 is usually not written. To evaluate a variable expression, replace the variables by their given values and then use the Order of Operations Agreement to simplify the result.

EXAMPLE 10

Evaluate a Variable Expression

x3 - y3

when x = 2 and y = - 3.

a.

Evaluate

b.

Evaluate (x + 2y)2 - 4z when x = 3, y = - 2, and z = - 4.

x 2 + xy + y 2

Solution x3 - y3 a. x 2 + xy + y 2 23 - ( -3)3 22 + 2( - 3) + ( -3)2 b.

=

8 - ( - 27) 35 = = 5 4 - 6 + 9 7

(x + 2y)2 - 4z [3 + 2(- 2)]2 - 4(- 4) = [3 + ( -4)]2 - 4( -4) = ( - 1)2 - 4(- 4) = 1 - 4( -4) = 1 + 16 = 17 Try Exercise 90, page 15

Simplifying Variable Expressions Addition, multiplication, subtraction, and division are the operations of arithmetic. Addition of the two real numbers a and b is designated by a + b. If a + b = c, then c is the sum and the real numbers a and b are called terms. Multiplication of the real numbers a and b is designated by ab or a # b. If ab = c, then c is the product and the real numbers a and b are called factors of c. The number - b is referred to as the additive inverse of b. Subtraction of the real numbers a and b is designated by a - b and is defined as the sum of a and the additive inverse of b. That is, a - b = a + ( - b) If a - b = c, then c is called the difference of a and b.

12

CHAPTER P


The multiplicative inverse or reciprocal of the nonzero number b is 1>b. The division of a and b, designated by a , b with b Z 0, is defined as the product of a and the reciprocal of b. That is, 1 a , b = aa b b

provided that b Z 0

If a , b = c, then c is called the quotient of a and b. a . The real b number a is the numerator, and the nonzero real number b is the denominator of the fraction. The notation a , b is often represented by the fractional notation a>b or

Properties of Real Numbers Let a, b, and c be real numbers. Addition Properties

Multiplication Properties

Closure

a + b is a unique real number.

ab is a unique real number.

Commutative

a + b = b + a

ab = ba

Associative

(a + b) + c = a + (b + c)

(ab)c = a(bc)

Identity

There exists a unique real number 0 such that a + 0 = 0 + a = a.

There exists a unique real number 1 such that a # 1 = 1 # a = a.

Inverse

For each real number a, there is a unique real number - a such that a + ( - a) = ( -a) + a = 0.

For each nonzero real number a, there is a unique real number 1>a 1 1 such that a # = # a = 1. a a a(b + c) = ab + ac

Distributive

EXAMPLE 11

Identify Properties of Real Numbers

Identify the property of real numbers illustrated in each statement. 1 a b 11 is a real number. 5

a.

(2a)b = 2(ab)

b.

c.

4(x + 3) = 4x + 12

d. (a + 5b) + 7c = (5b + a) + 7c

e.

a

1# 2ba = 1 # a 2

f. 1 # a = a

Solution a. Associative property of multiplication b.

Closure property of multiplication

c.

Distributive property

d.

Commutative property of addition

P.1

e.

Inverse property of multiplication

f.

Identity property of multiplication


13


Note Normally, we will not show, as we did at the right, all the steps involved in the simplification of a variable expression. For instance, we will just write (6x)2 = 12x, 3(4p + 5) = 12p + 15, and 3x 2 + 9x 2 = 12x 2. It is important to know, however, that every step in the simplification process depends on one of the properties of real numbers.

We can identify which properties of real numbers have been used to rewrite an expression by closely comparing the original and final expressions and noting any changes. For instance, to simplify (6x)2, both the commutative property and associative property of multiplication are used. (6x)2 = 2(6x) • Commutative property of multiplication = (2 # 6)x

• Associative property of multiplication

= 12x To simplify 3(4p + 5), use the distributive property. 3(4p + 5) = 3(4p) + 3(5) • Distributive property = 12p + 15 Terms that have the same variable part are called like terms. The distributive property is also used to simplify an expression with like terms such as 3x 2 + 9x 2. 3x 2 + 9x 2 = (3 + 9)x 2 = 12x

• Distributive property

2

Note from this example that like terms are combined by adding the coefficients of the like terms. Question • Are the terms 2x2 and 3x like terms?

EXAMPLE 12

Simplify Variable Expressions

Simplify. a.

5 + 3(2x - 6)

b.

4x - 237 - 5(2x - 3)4

Solution a. 5 + 3(2x - 6) = 5 + 6x - 18 = 6x - 13 b.

4x - 237 - 5(2x - 3)4 = 4x - 237 - 10x + 154

• Use the distributive property. • Add the constant terms.

• Use the distributive property to remove the inner parentheses.

= 4x - 23 - 10x + 224

• Simplify.

= 4x + 20x - 44

• Use the distributive property to remove the brackets.

= 24x - 44

• Simplify.


#

Answer • No. The variable parts are not the same. The variable part of 2x2 is x x. The variable

part of 3x is x.

14

CHAPTER P


An equation is a statement of equality between two numbers or two expressions. There are four basic properties of equality that relate to equations.

Properties of Equality Let a, b, and c be real numbers. Reflexive

a = a

Symmetric

If a = b, then b = a.

Transitive

If a = b and b = c, then a = c.

Substitution

If a = b, then a may be replaced by b in any expression that involves a.

EXAMPLE 13

Identify Properties of Equality

Identify the property of equality illustrated in each statement. a.

If 3a + b = c, then c = 3a + b.

b.

5(x + y) = 5(x + y)

c.

If 4a - 1 = 7b and 7b = 5c + 2, then 4a - 1 = 5c + 2.

d. If a = 5 and b(a + c) = 72, then b(5 + c) = 72. Solution a. Symmetric

b. Reflexive

c. Transitive

d. Substitution


EXERCISE SET P.1 In Exercises 1 and 2, determine whether each number is an integer, a rational number, an irrational number, a prime number, or a real number. 1 5

1. - , 0, -44, p, 3.14, 5.05005000500005 Á , 181, 53

5 1 , , 31, -2 , 4.235653907493, 51, 0.888 Á 2. 7 2 17

In Exercises 9 to 18, perform the operations given that A { 3, 2, 1, 0, 1, 2, 3}, B {2, 0, 2, 4, 6}, C {0, 1, 2, 3, 4, 5, 6}, and D { 3, 1, 1, 3}. 9. A ´ B

10. C ´ D

11. A ¨ C

12. C ¨ D

13. B ¨ D

14. B ´ (A ¨ C)

15. D ¨ (B ´ C)

16. (A ¨ B) ´ (A ¨ C)

17. (B ´ C) ¨ (B ´ D)

18. (A ¨ C) ´ (B ¨ D)

5

In Exercises 3 to 8, list the four smallest elements of each set. 3. 52x ƒ x positive integers6

4. 5 ƒ x ƒ ƒ x integers6

5. 5 y ƒ y = 2x + 1, x natural numbers6 6. 5 y ƒ y = x2 - 1, x integers6

In Exercises 19 to 24, perform the operation, given A is any set.

7. 5z ƒ z = ƒ x ƒ , x integers6

19. A ´ A

20. A ¨ A

8. 5z ƒ z = ƒ x ƒ - x, x negative integers6

21. A ¨

22. A ´

P.1

23. If A and B are two sets and A ´ B = A, what can be said

about B? 24. If A and B are two sets and A ¨ B = B, what can be said

about B?


In Exercises 63 to 74, use absolute value notation to describe the given situation. 63. d(m, n)

64. d( p, 8)

65. The distance between x and 3

In Exercises 25 to 36, graph each set. Write sets given in interval notation in set-builder notation, and write sets given in set-builder notation in interval notation. 25. ( -2, 3)

26. 31, 54

27. 3-5, -14

28. (-3, 3)

29. 32, q )

30. (- q , 4)

31. 5x ƒ 3 6 x 6 56

32. 5x ƒ x 6 - 16

33. 5x ƒ x Ú - 26

34. 5x ƒ - 1 … x 6 56

35. 5x ƒ 0 … x … 16

36. 5x ƒ -4 6 x … 56

37. ( - q , 0) ´ 32, 44

66. The distance between a and -2 67. The distance between x and -2 is 4. 68. The distance between z and 5 is 1. 69. The distance between a and 4 is less than 5. 70. The distance between z and 5 is greater than 7. 71. The distance between x and -2 is greater than 4. 72. The distance between y and -3 is greater than 6. 73. The distance between x and 4 is greater than 0 and less than 1. 74. The distance between y and - 3 is greater than 0 and less than 0.5.

In Exercises 37 to 52, graph each set.

In Exercises 75 to 82, evaluate the expression.

38. ( -3, 1) ´ (3, 5)

75. - 53( -4)2

76. -

- 63

39. ( -4, 0) ¨ 3- 2, 54

40. (- q , 34 ¨ (2, 6)

41. (1, q ) ´ ( -2, q )

42. (- 4, q ) ´ (0, q )

77. 4 + (3 - 8)2

78. -2 # 34 - (6 - 7)6

43. (1, q ) ¨ (-2, q )

44. ( -4, q ) ¨ (0, q )

79. 28 , ( - 7 + 5)2

80. (3 - 5)2(32 - 52)

45. 3-2, 44 ¨ 34, 54

46. (- q , 14 ¨ 31, q )

81. 7 + 233(- 2)3 - 42 , 84

47. ( -2, 4) ¨ (4, 5)

48. ( - q , 1) ¨ (1, q )

In Exercises 83 to 94, evaluate the variable expression for x 3, y 2, and z 1.

50. 5x ƒ -3 … x 6 06 ´ 5x ƒ x Ú 26 51. 5x ƒ x 6 - 36 ´ 5x ƒ x 6 26 52. 5x ƒ x 6 - 36 ¨ 5x ƒ x 6 26

56. ƒ 3 ƒ - ƒ -7 ƒ

57. ƒ p + 10 ƒ 2

59. ƒ x - 4 ƒ + ƒ x + 5 ƒ , given 0 6 x 6 1 60. ƒ x + 6 ƒ + ƒ x - 2 ƒ , given 0 6 x 6 2

83. - y 3

84. -y 2

85. 2xyz

86. - 3xz

87. - 2x 2y 2

88. 2y 3z2

89. xy - z(x - y)2

In Exercises 53 to 62, write each expression without absolute value symbols. 54. - ƒ -4 ƒ 2

( -3)4

82. 5 - 433 - 6(2 # 32 - 12 , 4)4

49. 5x ƒ x 6 - 36 ´ 5x ƒ 1 6 x 6 26

53. - ƒ - 5 ƒ

15

55. ƒ 3 ƒ

# ƒ -4 ƒ

58. ƒ p - 10 ƒ 2

90. (z - 2y)2 - 3z3

91.

x2 + y2 x + y

92.

93.

3y 2z x y

94. (x - z)2(x + z)2

2xy 2z4 ( y - z)4

In Exercises 95 to 108, state the property of real numbers or the property of equality that is used. 95. (ab 2)c = a(b 2c)

61. ƒ 2x ƒ - ƒ x - 1 ƒ , given 0 6 x 6 1

96. 2x - 3y = -3y + 2x

62. ƒ x + 1 ƒ + ƒ x - 3 ƒ , given x 7 3

97. 4(2a - b) = 8a - 4b

16

CHAPTER P


98. 6 + (7 + a) = 6 + (a + 7)

123. Area of a Triangle The area of a triangle is given by

Area =

99. (3x)y = y(3x) 100. 4ab + 0 = 4ab

where b is the base of the triangle and h is its height. Find the area of a triangle whose base is 3 inches and whose height is 4 inches.

101. 1 # (4x) = 4x 102. 7(a + b) = 7(b + a)

124. Volume of a Box The volume of a

rectangular box is given by

103. x2 + 1 = x2 + 1

Volume = lwh

104. If a + b = 2, then 2 = a + b.

where l is the length, w is the width, and h is the height of the box. Find the volume of a classroom that has a length of 40 feet, a width of 30 feet, and a height of 12 feet.

105. If 2x + 1 = y and y = 3x - 2, then 2x + 1 = 3x - 2. 106. If 4x + 2y = 7 and x = 3, then 4(3) + 2y = 7. 107. 4 #

1 = 1 4

1 bh 2

125. Heart Rate The heart rate, in beats per minute, of a certain

runner during a cool-down period can be approximated by Heart rate = 65 +

108. ab + ( -ab) = 0

53 4t + 1

where t is the number of minutes after the start of cool-down. Find the runner’s heart rate after 10 minutes. Round to the nearest natural number.

109. Is division of real numbers an associative operation? Give a

reason for your answer. 110. Is subtraction of real numbers a commutative operation? Give

a reason for your answer. 111. Which of the properties of real numbers are satisfied by the

integers? 112. Which of the properties of real numbers are satisfied by the

rational numbers? In Exercises 113 to 122, simplify the variable expression. 113. 2 + 3(2x - 5) 114. 4 + 2(2a - 3) 115. 5 - 3(4x - 2y) 116. 7 - 2(5n - 8m) 117. 3(2a - 4b) - 4(a - 3b) 118. 5(4r - 7t) - 2(10r + 3t)

126.

Body Mass Index According to the National Institutes of Health, body mass index (BMI) is a measure of body fat based on height and weight that applies to both adult men and women, with values between 18.5 and 24.9 considered 705w healthy. BMI is calculated as BMI = , where w is the h2 person’s weight in pounds and h is the person’s height in inches. Find the BMI for a person who weighs 160 pounds and is 5 feet 10 inches tall. Round to the nearest natural number.

127. Physics The height, in feet, of a ball t seconds after it is

thrown upward is given by height = - 16t 2 + 80t + 4

119. 5a - 233 - 2(4a + 3)4

Find the height of the ball 2 seconds after it has been released.

120. 6 + 332x - 4(3x - 2)4

128. Chemistry Salt is being added to water in such a way that

1 3 121. (5a + 2) - (3a - 5) 4 2 122. -

3 2 (2x + 3) + (3x - 7) 5 4

the concentration of salt, in grams per liter, is given by 50t concentration = , where t is the time in minutes after t + 1 the introduction of the salt. Find the concentration of salt after 24 minutes.

P.2

SECTION P.2 Integer Exponents Scientific Notation Rational Exponents and Radicals Simplifying Radical Expressions

INTEGER AND RATIONAL NUMBER EXPONENTS

17

Integer and Rational Number Exponents PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A1.

#

43 [P.1] 45

PS1. Simplify: 22 23 [P.1]

PS2. Simplify:

PS3. Simplify: (23)2 [P.1]

PS4. Simplify: 3.14(105) [P.1]

PS5. True or false: 34 32 = 96 [P.1]

PS6. True or false: (3 + 4)2 = 32 + 42 [P.1]

#

Integer Exponents

b is a factor n times

$''%''& Recall that if n is a natural number, then b = b # b # b # Á # b. We can extend the definition of exponent to all integers. We begin with the case of zero as an exponent. n

Definition of b0 For any nonzero real number b, b0 = 1. EXAMPLE

Note

3 0 a b = 1 4

30 = 1

Note that -70 = - (70) = - 1.

- 70 = - 1

(a2 + 1)0 = 1

Now we extend the definition to include negative integers.

Definition of bn If b Z 0 and n is a natural number, then b-n =

1 1 and -n = b n. bn b

EXAMPLE

3-2 =

1 1 = 9 32

EXAMPLE 1

1 = 43 = 64 4-3

7 7 5-2 = 2 = 25 7-1 5

Evaluate an Exponential Expression

Evaluate. a. b. c.

(-24)( - 3)2 (-4)-3 (-2)-5 -p0 (continued)

18

CHAPTER P


Solution a. (-24)( -3)2 = - (2 # 2 # 2 # 2)( - 3)( - 3) = - (16)(9) = - 144 b. c.

(- 4)-3 (-2)

-5

=

(- 2)( -2)( -2)( - 2)( - 2) -32 1 = = ( -4)( - 4)( - 4) -64 2

- p0 = - (p0) = -1 Try Exercise 10, page 29

When working with exponential expressions containing variables, we must ensure that a value of the variable does not result in an undefined expression. Take, for instance, 1 x-2 = 2 . Because division by zero is not allowed, for the expression x-2, we must assume x that x Z 0. Therefore, to avoid problems with undefined expressions, we will use the following restriction agreement.

Restriction Agreement a are all undefined 0 expressions. Therefore, all values of variables in this text are restricted to avoid any one of these expressions. The expressions 00, 0 n (where n is a negative integer), and

EXAMPLE

In the expression

x0y-3 , x Z 0, y Z 0, and z Z 4. z - 4

In the expression

(a - 1)0 , a Z 1 and b Z - 2. b + 2

Exponential expressions containing variables are simplified using the following properties of exponents.

Properties of Exponents If m, n, and p are integers and a and b are real numbers, then Product

b m # b n = b m+n

Quotient

bm = b m - n, bn

Power

(b m)n = b mn

b Z 0

(a mb n) p = a m pb n p a

am p a mp b = , bn b np

b Z 0

P.2


19

EXAMPLE

a4 # a # a3 = a 4 + 1 + 3 = a 8

• Add the exponents of the like bases. Recall that a = a1.

(x4y 3)(xy 5z 2) = x4 + 1y 3 + 5z 2 = x 5y8z 2

• Add the exponents of the like bases.

a7b a5 = a7 - 2b 1 - 5 = a 5b-4 = 4 2 5 ab b

• Subtract the exponents of the like bases.

#

#

(uv 3) 5 = u 1 5v 3 5 = u 5v 15 a

5

3

1#3 5#3

• Multiply the exponents.

3 15

15

2x 2 x 2x 8x b = 1 # 3 4 # 3 = 3 12 = 4 5y 5 y 5y 125y 12

Question • Can the exponential expression

• Multiply the exponents.

x 5y 3 be simplified using the properties of exponents?

Integrating Technology c

Exponential expressions such as a b can be confusing. The generally accepted meaning c c of a b is a(b ). However, some graphing calculators do not evaluate exponential expressions in this way. Enter 2^3^4 in a graphing calculator. If the result is approximately 4 2.42 * 10 24, then the calculator evaluated 2(3 ). If the result is 4096, then the calculator evaluated (23)4. To ensure that you calculate the value you intend, we strongly urge you to use parentheses. For instance, entering 2^(3^4) will produce 2.42 * 10 24 and entering (2^3)^4 will produce 4096.

To simplify an expression involving exponents, write the expression in a form in which each base occurs at most once and no powers of powers or negative exponents occur.

EXAMPLE 2

Simplify Exponential Expressions

Simplify. a.

b. (3x 2yz -4 ) 3

(5x 2y)(- 4x 3y 5)

Solution a. (5x 2y)(- 4x 3y 5) = 35( - 4)4x 2 + 3y 1 + 5 = - 20x 5y 6 b.

#

#

#

#

(3x 2yz-4) 3 = 31 3 x 2 3 y 1 3 z -4 3 27x 6y 3 = 3 3 x 6y 3 z-12 = z 12

c.

-12x 5y 18x 2y 6

d.

a

4p 2q 6pq

b 4

-2

• Multiply the coefficients. Multiply the variables by adding the exponents of the like bases. • Use the power property of exponents.

(continued)

Answer • No. The bases are not the same.

20

CHAPTER P


c.

d.

-12x 5y 18x 2y6

a

4p2q 6pq

b 4

= -

2 5-2 1-6 y x 3

= -

2 3 -5 x y 3

= -

2x3 3y5

-2

= a = =

- 12 2 = - . Divide the variables by 18 3 subtracting the exponents of the like bases.

• Simplify

2p2 - 1q1 - 4 -2 2pq-3 -2 b = a b 3 3

21(-2)p1(-2)q-3(-2) 31(-2)

=

2-2p-2q6 3-2

9q6

• Use the quotient property of exponents. • Use the power property of exponents. • Write the answer in simplest form.

4p2


Scientific Notation The exponent theorems provide a compact method of writing very large or very small numbers. The method is called scientific notation. A number written in scientific notation has the form a # 10n, where n is an integer and 1 … a 6 10. The following procedure is used to change a number from its decimal form to scientific notation. For numbers greater than 10, move the decimal point to the position to the right of the first digit. The exponent n will equal the number of places the decimal point has been moved. For example, 7,430,000 = 7.43 * 106 6 places

For numbers less than 1, move the decimal point to the right of the first nonzero digit. The exponent n will be negative, and its absolute value will equal the number of places the decimal point has been moved. For example, 0.00000078 = 7.8 * 10-7 m

7 places

To change a number from scientific notation to its decimal form, reverse the procedure. That is, if the exponent is positive, move the decimal point to the right the same number of places as the exponent. For example, 3.5 * 105 = 350,000

m

5 places

If the exponent is negative, move the decimal point to the left the same number of places as the absolute value of the exponent. For example, 2.51 * 10-8 = 0.0000000251 m

Approximately 3.1 * 106 orchid seeds weigh 1 ounce. I Computer scientists measure an operation in nanoseconds. 1 nanosecond = 1 * 10-9 second I If a spaceship traveled at 25,000 mph, it would require approximately 2.7 * 109 years to travel from one end of the universe to the other. I

m

Math Matters

8 places

Most calculators display very large and very small numbers in scientific notation. The number 450,0002 is displayed as 2.025 E 11 . This means 450,0002 = 2.025 * 1011.

P.2

EXAMPLE 3


21

Simplify an Expression Using Scientific Notation

The Andromeda galaxy is approximately 1.4 * 1019 miles from Earth. If a spacecraft could travel 2.8 * 1012 miles in 1 year (about one-half the speed of light), how many years would it take for the spacecraft to reach the Andromeda galaxy? Solution To find the time, divide the distance by the speed. t =

1.4 * 1019 1.4 * 1019 - 12 = 0.5 * 107 = 5.0 * 106 = 2.8 2.8 * 1012

It would take 5.0 * 106 (or 5,000,000) years for the spacecraft to reach the Andromeda galaxy. Try Exercise 52, page 29

Rational Exponents and Radicals To this point, the expression bn has been defined for real numbers b and integers n. Now we wish to extend the definition of exponents to include rational numbers so that expres> sions such as 21 2 will be meaningful. Not just any definition will do. We want a definition of rational exponents for which the properties of integer exponents are true. The following example shows the direction we can take to accomplish our goal. If the product property for exponential expressions is to hold for rational exponents, then for rational numbers p and q, b pb q = b p + q. For example, 91>2 # 91>2

must equal

91>2+1>2 = 91 = 9

Thus 91>2 must be a square root of 9. That is, 91>2 = 3. The example suggests that b1>n can be defined in terms of roots according to the following definition.

Definition of b1/n If n is an even positive integer and b Ú 0, then b1>n is the nonnegative real number such that (b1>n)n = b. If n is an odd positive integer, then b1>n is the real number such that (b1>n)n = b. EXAMPLE

251>2 = 5 because 52 = 25. (-64)1>3 = - 4 because ( - 4)3 = - 64. 161>2 = 4 because 42 = 16. -161>2 = - (161>2) = - 4. (-16)1>2 is not a real number. (-32)1>5 = - 2 because (-2)5 = - 32. If n is an even positive integer and b 6 0, then b1>n is a complex number. Complex numbers are discussed in Section P.6.

22

CHAPTER P


To define expressions such as 82>3, we will extend our definition of exponents even further. Because we want the power property (b p) q = b pq to be true for rational exponents also, we must have (b 1>n) m = b m>n. With this in mind, we make the following definition.

Definition of b m/n For all positive integers m and n such that m>n is in simplest form, and for all real numbers b for which b1>n is a real number, b m>n = (b1>n ) m = (b m )1>n Because bm>n is defined as (b1>n)m and as (bm)1>n, we can evaluate expressions such as 8 in more than one way. For example, because 81>3 is a real number, 84>3 can be evaluated in either of the following ways. 4>3

84>3 = (81>3)4 = 24 = 16 84>3 = (84)1>3 = 40961>3 = 16 Of the two methods, the bm>n = (b1>n)m method is usually easier to apply, provided you can evaluate b1>n.

EXAMPLE 4

Evaluate a Number with a Rational Exponent

Simplify. a.

642>3

b.

32 - 3>5

c.

a

16 - 3>4 b 81

Solution a. 642>3 = (641>3)2 = 42 = 16 1 1 = 3 8 2 1>4 3 81 3 3 27 = ca b d = a b = 16 2 8

b.

32 - 3>5 = (321>5) - 3 = 2 - 3 =

c.

a

81 3>4 16 - 3>4 = a b b 81 16


The following exponent properties were stated earlier, but they are restated here to remind you that they have now been extended to apply to rational exponents.

Properties of Rational Exponents If p, q, and r represent rational numbers and a and b are positive real numbers, then Product

b p # bq = b p + q

Quotient

bp = b p-q bq

Power

(b p ) q = b pq a

ap r a pr b = bq b qr

(a pb q ) r = a prb qr b-p =

1 bp

P.2


23

Recall that an exponential expression is in simplest form when no powers of powers or negative exponents occur and each base occurs at most once.

EXAMPLE 5

Simplify Exponential Expressions

Simpify. a.

(2x1>3y3>5)2 (9x3y3>2)1>2

(a3>4b1>2)2 (a2>3b3>4)3

b.

Solution a. (2x1>3y 3>5)2 (9x 3y 3>2)1>2 = (2 2x 2>3y 6>5) (91>2x 3>2y 3>4)

• Use the power property.

= (4x2>3y6>5)(3x3>2y3>4) 2

3

6

3

4

9

24

15

= 12x 3 + 2y 5 + 4 = 12x 6 + 6y 20 + 20 = 12x

13>6 39>20

(a b ) (a2>3b3>4)3 3>4 1>2 2

b.

=

y

a3>2b

• Use the power property.

a2b9>4 3

9

= a 2 - 2b1 - 4 3 4 4 9 = a2 - 2b 4 - 4 = a-1>2b-5>4 =

• Add the exponents on like bases.

• Subtract the exponents on like bases.

1 1>2 5>4

a b


Simplifying Radical Expressions

Math Matters The formula for kinetic energy (energy of motion) that is used in Einstein’s Theory of Relativity involves a radical, K.Er = mc 2

1 v2 P A1 - c 2

n

Radicals, expressed by the notation 1b, are also used to denote roots. The number b is the radicand, and the positive integer n is the index of the radical. n

Definition of 1b

- 1

Q

where m is the mass of the object at rest, v is the speed of the object, and c is the speed of light.

If n is a positive integer and b is a real number such that b1>n is a real number, n then 1b = b1>n.

2

If the index n equals 2, then the radical 1b is written as simply 1b, and it is referred to as the principal square root of b, or simply the square root of b. The symbol 1b is reserved to represent the nonnegative square root of b. To represent the negative square root of b, write - 1b. For example, 125 = 5, whereas - 125 = - 5. n

Definition of (1b)m n

For all positive integers n, all integers m, and all real numbers b such that 1b is a n n real number, ( 1b)m = 2bm = bm>n.

24

CHAPTER P


n

When 1b is a real number, the equations m

mm mn bm bm>n = 2

and

mm mn m bm>n = ( 1b)m

can be used to write exponential expressions such as bm>n in radical form. Use the denominator n as the index of the radical and the numerator m as the power of the radicand or as the power of the radical. For example, 3 (5xy)2>3 = (1 5xy)2 =

2 25x y 3

2 2

• Use the denominator 3 as the index of the radical and the numerator 2 as the power of the radical.

The equations n

bm>n = 1bm

and

n

bm>n = ( 1b)m

also can be used to write radical expressions in exponential form. For example, 2(2ab)3 = (2ab)3>2

• Use the index 2 as the denominator of the power and the exponent 3 as the numerator of the power.

n

The definition of ( 1b)m often can be used to evaluate radical expressions. For instance, 3

( 18)4 = 84>3 = (81>3)4 = 24 = 16 Care must be exercised when simplifying even roots (square roots, fourth roots, sixth roots, and so on) of variable expressions. Consider 2x2 when x = 5 and when x = - 5. Case 1

If x = 5, then 2x2 = 252 = 125 = 5 = x.

Case 2

If x = - 5, then 2x2 = 2( - 5)2 = 125 = 5 = - x.

These two cases suggest that 2x2 = e Absolute Value See pages 7–8.

x, if x Ú 0 - x, if x 6 0

Recalling the definition of absolute value, we can write this more compactly as 2x2 = ƒ x ƒ . Simplifying odd roots of a variable expression does not require using the absolute 3 3 value symbol. Consider 2 x when x = 5 and when x = - 5. Case 1

3 3 3 3 3 If x = 5, then 2 x = 2 5 = 1 125 = 5 = x.

Case 2

3 3 3 3 If x = - 5, then 2 x = 2 ( -5)3 = 1 -125 = - 5 = x.

3 3 Thus 2 x = x. Although we have illustrated this principle only for square roots and cube roots, the same reasoning can be applied to other cases. The general result is given below.

n

Definition of 1bn If n is an even natural number and b is a real number, then n

2bn = ƒ b ƒ If n is an odd natural number and b is a real number, then n

2bn = b EXAMPLE 4 2 16z4 = 2 ƒ z ƒ

5 2 32a5 = 2a

P.2


25

Because radicals are defined in terms of rational powers, the properties of radicals are similar to those of exponential expressions.

Properties of Radicals If m and n are natural numbers and a and b are positive real numbers, then Product

1a # 1b = 1ab

Quotient

1a n a = n Ab 1b

Index

31a = 2a

n

n

n

n

m n

mn

A radical is in simplest form if it meets all of the following criteria. 1. The radicand contains only powers less than the index. ( 2x5 does not satisfy this requirement because 5, the exponent, is greater than 2, the index.) 9 3 2. The index of the radical is as small as possible. ( 2 x does not satisfy this 9 3 3 3>9 1>3 requirement because 2x = x = x = 1x.)

3. The denominator has been rationalized. That is, no radicals occur in the denominator. (1> 12 does not satisfy this requirement.) 4 4. No fractions occur under the radical sign. ( 2 2>x3 does not satisfy this requirement.)

Radical expressions are simplified by using the properties of radicals. Here are some examples.

EXAMPLE 6

Simplify Radical Expressions

Simplify. a.

4 2 32x3y4

b.

3 2 162x4y6

Solution 4 4 5 3 4 4 a. 2 32x3y4 = 2 2xy = 2 (24y4) # (2x3) 4 4 4# 4 = 2 2 y 22x3

= 2 ƒ y ƒ 22x 4

b.

3

3 3 2 162x4x6 = 2 (2 # 34)x4y6

= =

2(3xy ) # (2 # 3x) 3 3 2 (3xy2)3 # 2 6x 3

2 3

2 3

= 3xy 16x Try Exercise 84, page 30

• Factor and group factors that can be written as a power of the index. • Use the product property of radicals. n

• Recall that for n even, 2bn = ƒ b ƒ . • Factor and group factors that can be written as a power of the index. • Use the product property of radicals. n

• Recall that for n odd, 1bn = b.

26

CHAPTER P


Like radicals have the same radicand and the same index. For instance, 3

325xy2

and

3

-4 25xy2

are like radicals. Addition and subtraction of like radicals are accomplished by using the distributive property. For example, 413x - 913x = (4 - 9)13x = - 5 13x 3 2 3 2 3 2 3 2 3 2 22 y + 42 y - 2 y = (2 + 4 - 1) 2 y = 52 y

The sum 213 + 6 15 cannot be simplified further because the radicands are not the 3 4 same. The sum 31 x + 51 x cannot be simplified because the indices are not the same. Sometimes it is possible to simplify radical expressions that do not appear to be like radicals by simplifying each radical expression.

EXAMPLE 7

Combine Radical Expressions

3 3 Simplify: 5x2 16x4 - 2 128x7

Solution 3 3 5x2 16x4 - 2 128x7 3

3

= 5x224x4 - 2 27x7 3

= 5x22 x

3 3

# 23 2x -

3

22 x

6 6

# 13 2x

= 5x(2x 12x) - 22x2 # 12x 3 3 = 10x2 1 2x - 4x2 1 2x 3

3

• Factor. • Group factors that can be written as a power of the index. • Use the product property of radicals. • Simplify.

2 3

= 6x 12x Try Exercise 92, page 30

Multiplication of radical expressions is accomplished by using the distributive property. For instance, 15(120 - 3115) = 15 ( 120) - 15 (3 115) = 1100 - 3175 = 10 - 3 # 513

• Use the distributive property. • Multiply the radicals. • Simplify.

= 10 - 1513 Finding the product of more complicated radical expressions may require repeated use of the distributive property.

EXAMPLE 8

Multiply Radical Expressions

Perform the indicated operation. a. (5 16 - 7)(3 16 + 2) b. (3 - 1x - 7)2, x Ú 7

P.2


27

Solution a. (5 16 - 7)(3 16 + 2) = 5 16(3 16 + 2) - 7(3 16 + 2)

= (15 # 6

+ 1016) - (21 16 + 14)

= 90 + 10 16 - 2116 - 14

• Use the distributive property. • Use the distributive property. • Simplify.

= 76 - 11 16 (3 - 1x - 7)2

b.

= (3 - 1x - 7)(3 - 1x - 7) = 9 - 31x - 7 - 31x - 7 + ( 1x - 7)2

• Use the distributive property.

= 9 - 61x - 7 + (x - 7)

• ( 1x - 7)2 = x - 7, since x Ú 7.

= 2 - 61x - 7 + x Try Exercise 102, page 30

To rationalize the denominator of a fraction means to write the fraction in an equivalent form that does not involve any radicals in the denominator. This is accomplished by multiplying the numerator and denominator of the radical expression by an expression that will cause the radicand in the denominator to be a perfect root of the index. 5 5 # 13 5 13 5 13 = = = 3 13 13 13 232

2 3 1 7

5 4 5 2 x

=

# 23 7 3

2

2

3 1 7 272

=

5

3

=

# 24 x 4

3

4 5 2 x 2x3

EXAMPLE 9

2 272 3 3 2 7

3

=

2 249 7

4

=

2 • Recall that 13 means 1 3. Multiply numerator and denominator by 13 so that the radicand is a perfect root of the index of the radical.

5 2x3 4 8 2 x

4

=

5 2x3 x2

3 2 • Multiply numerator and denominator by 2 7 so that the radicand is a perfect root of the index of the radical. 4 3 • Multiply numerator and denominator by 2 x so that the radicand is a perfect root of the index of the radical.

Rationalize the Denominator

Rationalize the denominator. a.

5 3

1a

b.

3 ,y 7 0 A 32y

Solution 3 3 3 5 5 # 2a 2 5 2a2 5 2a2 a. = = = 3 3 3 2 3 3 a 1 a 1 a 2 a 2 a

3 # 3 2 3 3 • Use 1 a 2a = 2 a = a.

(continued)

28

CHAPTER P


b.

3

A 32y

=

16y 13 13 13 # 12y = = = 8y 132y 412y 412y 12y


To rationalize the denominator of a fractional expression such as 1 1m + 1n we use the conjugate of 1m + 1n, which is 1m - 1n. The product of these conjugate pairs does not involve a radical. ( 1m + 1n)( 1m - 1n) = m - n

EXAMPLE 10

Rationalize the Denominator

Rationalize the denominator. a.

3 + 215 1 - 415

b.

2 + 41x ,x 7 0 3 - 51x

Solution 3 + 2 15 # 1 + 415 3 + 215 a. = 1 - 415 1 - 415 1 + 415 =

3(1 + 415) + 2 15(1 + 4 15)

12 - (4 15)2 3 + 12 15 + 215 + 8 # 5 = 1 - 16 # 5 43 + 1415 = -89 43 + 1415 = 89 b.

2 + 41x 2 + 4 1x # 3 + 51x = 3 - 51x 3 - 51x 3 + 51x =

• Multiply numerator and denominator by the conjugate of the denominator.

2(3 + 5 1x) + 4 1x(3 + 5 1x)

32 - (5 1x)2 6 + 101x + 121x + 20x = 9 - 25x 6 + 221x + 20x = 9 - 25x Try Exercise 116, page 30

• Simplify.

• Multiply numerator and denominator by the conjugate of the denominator.

P.2


29

EXERCISE SET P.2 In Exercises 1 to 12, evaluate each expression. 1. -53

2. (- 5)3

3. a b

4. -60

5. 4-2

6. 3-4

7.

10.

1 2 4

8.

-5 -2

2 3

1 3

6

x 4

12.

In Exercises 13 to 38, write the exponential expression in simplest form. 13. 2x-4

15.

14. 3y-2

5

16.

39. 2,011,000,000,000

40. 49,100,000,000

41. 0.000000000562

42. 0.000000402

In Exercises 43 to 46, change the number from scientific notation to decimal notation.

-3

0

11. - 2x0

2-3

2-3

9.

-3

In Exercises 39 to 42, write the number in scientific notation. 0

43. 3.14 * 107

44. 4.03 * 109

45. - 2.3 * 10-6

46. 6.14 * 10-8

In Exercises 47 to 54, perform the indicated operation and write the answer in scientific notation. 47. (3 * 1012)(9 * 10-5)

8 -5

49.

17. (x3y 2)(xy5)

18. (uv6)(u2v)

51.

19. ( -2ab4)(- 3a2b5)

20. (9xy 2)( -2x 2y5)

z

-6

21. (- 4x-3y)(7x5y-2)

23.

25.

27.

6a4

22. (- 6x4y)(7x-3y-5)

24.

8a8 12x3y4

26.

18x5y 2 36a-2b3

28.

4

3ab

x

5v4w-3 10v8 -48ab10 -32a b

4 3

31. (x-2y)2(xy)-2

32. (x-1y 2)-3(x2y-4)-3

33. a

35.

3a 2b 3

b 4

2

6a4b

( -4x 2y 3)2

37. a

(2xy 2)3 a-2b a3b

b -4

2

3 2 3

34. a

36.

6 * 108 (3.2 * 10-11)(2.7 * 1018) 1.2 * 10

-5

(4.0 * 10-9)(8.4 * 105) (3.0 * 10-6)(1.4 * 1018)

52.

54.

2.5 * 108 5 * 1010 (6.9 * 1027)(8.2 * 10-13) 4.1 * 1015 (7.2 * 108)(3.9 * 10-7) (2.6 * 10-10)(1.8 * 10-8)

In Exercises 55 to 76, evaluate each exponential expression.

16x4

30. (2a b ) (- 4a b )

2 2

50.

12x3

29. (-2m n )( -3mn ) 3 2

53.

9 * 10-3

48. (8.9 * 10-5)(3.4 * 10-6)

4 2

2ab 2c3 5ab 2

b

(- 2ab4)3 x-3y-4 x-2y

b

56. - 163>2

57. - 642>3

58. 1254>3

59. 9-3>2

60. 32-4>5

61. a b

1>2

62. a

16 3>2 b 25

63. a b

-4>3

64. a

8 -2>3 b 27

65. (4a2>3b1>2)(2a1>3b 3>2)

66. (6a3>5b1>4)(- 3a1>5b 3>4)

67. ( -3x 2>3)(4x1>4)

68. ( -5x1>3)(- 4x1>2)

69. (81x8y12)1>4

70. (27x 3y6)2>3

4 9 1 8

3

(- 3a2b 3)2

38. a

55. 43>2

-2

71.

16z3>5 12z1>5

72.

6a 2>3 9a1>3

30

CHAPTER P


73. (2x 2>3y1>2)(3x1>6y1>3)

75.

9a3>4b

74.

76.

3a2>3b2

In Exercises 105 to 126, simplify each expression by rationalizing the denominator. Write the result in simplest form. Assume x>0 and y>0.

x1>3y5>6 x

2>3 1>6

y

12x1>6y1>4

105.

16x3>4y1>2

In Exercises 77 to 86, simplify each radical expression. 77. 145

107.

78. 175

3

3

79. 124

80. 1135

3

109.

82. 1 -250

83. 224x 2y3

84. 218x 2y5

3

12 5

108.

A 18 3

110.

3

12

3x 13 7

A 40 2 3

14

111.

4 3

28x

2

112.

2 4

14y

3

85. 216a3y7

86. 254m2n7 113.

In Exercises 87 to 94, simplify each radical and then combine like radicals. 87. 2 132 - 3198 4

3

91. 4 232y4 + 3y1108y 3

3

3

3

3

115.

88. 5132 + 21108

89. - 8148 + 21243 3

106.

3

81. 1 -135

4

2

3

93. x28x3y4 - 4y 264x6y

90. 2140 - 31135 3

116.

15 - 2 -7 312 - 5

118.

6 - 312 5 - 12

119.

613 - 11 413 - 7

120.

217 + 8 1217 - 6

121.

2 + 1x 3 - 21x

122.

4 - 21x 5 + 31x

123.

x - 15 x + 215

124.

x + 317 x + 217

95. (15 + 3)(15 + 4) 96. (17 + 2)(17 - 5)

6 215 + 2

2

3 + 215 5 - 315

92. -3x254x4 + 2216x7

In Exercises 95 to 104, find the indicated product and express each result in simplest form.

13 + 4

114.

117. 3

94. 42a5b - a2 1ab

3

97. (12 - 3)(12 + 3) 98. (2 17 + 3)(2 17 - 3)

125.

3 15 + 1x

126.

5 1y - 13

99. (3 1z - 2)(4 1z + 3) 100. (4 1a - 1b)(31a + 21b) 101. (1x + 2)

In Exercises 127 and 128, rationalize the numerator, a technique that is occasionally used in calculus. 127.

102. (3 15y - 4)2 103. (1x - 3 + 2)2 104. (12x + 1 - 3)2

129.

14 + h - 2 h

128.

19 + h - 3 h

Weight of an Orchid Seed An orchid seed weighs

approximately 3.2 * 10-8 ounce. If a package of seeds contains 1 ounce of orchid seeds, how many seeds are in the package?

P.2

130. Biology The weight of one E. coli baterium is approximately

670 femtograms, where 1 femtogram 1 1015 gram. If one E. coli baterium can divide into two bacteria every 20 minutes, then after 24 hours there would be (assuming all bacteria survived) approximately 4.7 * 1021 bacteria. What is the weight, in grams, of these bacteria?


31

137. Oceanography The percent P of light that will pass to a

depth d, in meters, at a certain place in the ocean is given by P = 102 - (d>40). Find, to the nearest percent, the amount of light that will pass to a depth of a. 10 meters and b. 25 meters below the surface of the ocean. 138. Learning Theory In a psychology experiment, students were

131.

given a nine-digit number to memorize. The percent P of students who remembered the number t minutes after it was read to them can be given by P = 90 - 3t 2>3. What percent of the students remembered the number after 1 hour?

Doppler Effect Astronomers can approximate the dis-

tance to a galaxy by measuring its red shift, which is a shift in the wavelength of light due to the velocity of the galaxy. This is similar to the way the sound of a siren coming toward you seems to have a higher pitch than the sound of the siren moving away from you. A formula for red shift is lr - ls , where lr and ls are wavelengths of a certain frequency ls of light. Calculate the red shift for a galaxy for which lr = 5.13 * 10-7 meter and ls = 5.06 * 10-7 meter.

139.

Relativity Theory A moving object has energy, called kinetic energy, because of its motion. The Theory of Relativity, mentioned on page 1, uses the following formula for kinetic energy.

K.Er = mc 2

132. Laser Wavelength The wavelength of a certain helium-neon

laser is 800 nanometers. (1 nanometer is 1 * 10-9 meter.) The 1 . frequency, in cycles per second, of this wave is wavelength What is the frequency of this laser?

133.

134.

Astronomy The sun is approximately 1.44 * 10 meters from Earth. If light travels 3 * 108 meters per second, how many minutes does it take light from the sun to reach Earth?

is 9.3 * 107 miles. This distance is called the astronomical unit (AU). Jupiter is 5.2 AU from the sun. Find the distance in miles from Jupiter to the sun. Jupiter 5.2 AU = ? mi

Earth 1 AU = 9.3 × 10 7 mi

v2 P A1 - 2 c

- 1

Q

When the speed of an object is much less than the speed of light (3.0 * 108 meters per second) the formula K.En =

11

Astronomical Unit Earth’s mean distance from the sun

1

1 2 mv 2

is used. In each formula, v is the velocity of the object in meters per second, m is its rest mass in kilograms, and c is the speed of light given previously. In a. through e., calculate the percent error for each of the given velocities. The formula for percent error is % error =

ƒ K.Er - K.En ƒ * 100 K.Er

a. v = 30 meters per second (speed of a speeding car on an

expressway) b. v = 240 meters per second (speed of a commercial jet) c. v = 3.0 * 107 meters per second (10% of the speed of

light) Sun

d. v = 1.5 * 108 meters per second (50% of the speed of light) 135.

Mass of an Atom One gram of hydrogen contains

6.023 * 1023 atoms. Find the mass of one hydrogen atom.

e. v = 2.7 * 108 meters per second (90% of the speed of

light) 136. Drug Potency The amount A (in milligrams) of digoxin,

a drug taken by cardiac patients, remaining in the blood t hours after a patient takes a 2-milligram dose is given by A = 2(10-0.0078t ). a. How much digoxin remains in the blood of a patient

4 hours after taking a 2-milligram dose? b. Suppose that a patient takes a 2-milligram dose of digoxin

at 1:00 P.M. and another 2-milligram dose at 5:00 P.M. How much digoxin remains in the patient’s blood at 6:00 P.M.?

f. Use your answers from a. through e. to explain why the for-

mula for kinetic energy given by K.En is adequate for most of our common experiences involving motion (walking, running, bicycling, driving, flying, and so on). g. According to the Theory of Relativity, a particle (such as an

electron or a spacecraft) cannot reach the speed of light. Explain why the equation for K.Er suggests such a conclusion.

32

CHAPTER P


SECTION P.3

Polynomials

Operations on Polynomials Applications of Polynomials

PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A2.

PS1. Simplify: - 3(2a - 4b) [P.1] PS2. Simplify: 5 - 2(2x - 7) [P.1] PS3. Simplify: 2x 2 + 3x - 5 + x 2 - 6x - 1 [P.1] PS4. Simplify: 4x 2 - 6x - 1 - 5x 2 + x [P.1] PS5. True or false: 4 - 3x - 2x 2 = 2x 2 - 3x + 4 [P.1] 3

12 + 15 12 + 15 PS6. True or false: = 18 [P.1] = 4 4

Operations on Polynomials A monomial is a constant, a variable, or the product of a constant and one or more variables, with the variables having only nonnegative integer exponents. -8 A number

z A variable

7y The product of a constant and one variable

- 12a 2bc3 The product of a constant and several variables

The expression 3x-2 is not a monomial because it is the product of a constant and a variable with a negative integer exponent. The constant multiplying the variables is called the numerical coefficient or coefficient. For 7y, the coefficient is 7; for -12a 2bc3, the coefficient is - 12. The coefficient of z is 1 because z = 1 # z. Similarly, the coefficient of -x is -1 because - x = - 1 # x. The degree of a monomial is the sum of the exponents of the variables. The degree of a nonzero constant is 0. The constant zero has no degree. 7y Degree is 1 because y = y1.

-12a 2bc 3

-8

Degree is 2 + 1 + 3 = 6.

Degree is 0.

A polynomial is the sum of a finite number of monomials. Each monomial is called a term of the polynomial. The degree of a polynomial is the greatest of the degrees of the terms. See Table P.1. Terms See page 11.

Table P.1

Note The sign of a term is the sign that precedes the term.

Terms and Degree of a Polynomial

Polynomial

Terms

Degree

5x 4 - 6x 3 + 5x 2 - 7x - 8

5x 4, - 6x 3, 5x 2, - 7x, - 8

4

- 3xy 2, - 8xy, 6x

3

- 3xy - 8xy + 6x 2

Terms that have exactly the same variables raised to the same powers are called like terms. For example, 14x 2 and -x 2 are like terms. 7x 2y and 5yx 2 are like terms; the order of the variables is not important. The terms 6xy 2 and 6x 2y are not like terms; the exponents on the variables are different.

P.3

POLYNOMIALS

33

A polynomial is said to be in simplest form if all its like terms have been combined. For example, the simplified form of 4x 2 + 3x + 5x - x 2 is 3x 2 + 8x. A binomial is a simplified polynomial with two terms; 3x 4 - 7, 2xy - y 2, and x + 1 are binomials. A trinomial is a simplified polynomial with three terms; 3x 2 + 6x - 1, 2x 2 - 3xy + 7y 2, and x + y + 2 are trinomials. A nonzero constant, such as 5, is a constant polynomial.

Definition of the Standard Form of a Polynomial The standard form of a polynomial of degree n in the variable x is an x n + an-1x n-1 + + a2x 2 + a1x + a0 where an Z 0 and n is a nonnegative integer. The coefficient an is the leading coefficient, and a0 is the constant term. EXAMPLE

Polynomial

Standard Form

6x - 7 + 2x 3 4z - 2z + 3z - 9 3

-2z + 4z + 3z - 9

4

y - 3y + 1 - 2y - y 5

3

EXAMPLE 1

Leading Coefficient

2x 3 + 6x - 7 4

2

3

y - 3y - y - 2y + 1 5

3

2

2 -2

1

Identify Terms Related to a Polynomial

Write the polynomial 6x 3 - x + 5 - 2x 4 in standard form. Identify the degree, terms, constant term, leading coefficient, and coefficients of the polynomial. Solution A polynomial is in standard form when the terms are written in decreasing powers of the variable. The standard form of the polynomial is -2x 4 + 6x 3 - x + 5. In this form, the degree is 4; the terms are -2x 4, 6x 3, - x, and 5; the constant term is 5. The leading coefficient is - 2; the coefficients are -2, 6, -1, and 5. Try Exercise 12, page 37

To add polynomials, add the coefficients of the like terms.

EXAMPLE 2

Add Polynomials

Add: (3x3 - 2x2 - 6) + (4x2 - 6x - 7) Solution (3x3 - 2x2 - 6) + (4x2 - 6x - 7)

= 3x3 + ( - 2x2 + 4x2) + ( - 6x) + 3( - 6) + ( -7)4 = 3x3 + 2x2 - 6x - 13


The additive inverse of the polynomial 3x - 7 is - (3x - 7) = - 3x + 7


Question • What is the additive inverse of 3x 2 - 8x + 7?

To subtract a polynomial, we add its additive inverse. For example, (2x - 5) - (3x - 7) = (2x - 5) + ( - 3x + 7)

= 32x + ( - 3x)4 + 3( - 5) + 74

= -x + 2 The distributive property is used to multiply polynomials. For instance, (2x 2 - 5x + 3)(3x + 4) = (2x 2 - 5x + 3)(3x) + (2x 2 - 5x + 3)4 = (6x 3 - 15x 2 + 9x) + (8x 2 - 20x + 12) = 6x 3 - 7x 2 - 11x + 12 Although we could always multiply polynomials using the preceding procedure, we frequently use a vertical format. Here is the same product as shown previously using that format. 2x 2 - 5x + 3 3x + 4 8x 2 - 20x + 12 = (2x 2 - 5x + 3)4 = (2x 2 - 5x + 3)(3x) 6x 3 - 15x 2 + 9x 6x 3 - 7x 2 - 11x + 12

EXAMPLE 3

Multiply Polynomials

Multiply: (2x - 5)(x 3 - 4x + 2) Solution Note in the following solution how like terms are placed in columns. - 4x + 2

x3

2x - 5 - 5x + 20x - 10 - 8x 2 + 4x 2x 4 2x 4 - 5x 3 - 8x 2 + 24x - 10 3


If the terms of the binomials (a + b) and (c + d) are labeled as shown below, then the product of the two binomials can be computed mentally by the FOIL method. Last First

+

m b) m

(am

#

(cm m

m

CHAPTER P

Outer

First

+

m = ac d) m

34

Inner Outer Answer • The additive inverse is -3x 2 + 8x - 7.

+

ad

Last

Inner

+

bc

+

bd

P.3

POLYNOMIALS

35

In the following illustration, we find the product of (7x - 2) and (5x + 4) by the FOIL method. First

Outer

Inner

Last

(7x - 2)(5x + 4) = (7x)(5x) + (7x)(4) + ( -2)(5x) + ( -2)(4)

EXAMPLE 4

=

35x2

=

35x + 18x - 8

+

-

28x

10x

-

8

2

Multiply Binomials

Multiply. a.

(4x + 5)(3x - 7)

b.

(2x - 3y)(4x - 5y)

Solution a. (4x + 5)(3x - 7) = (4x)(3x) - (4x)7 + 5(3x) - 5(7) = 12x 2 - 28x + 15x - 35 = 12x 2 - 13x - 35 b.

(2x - 3y)(4x - 5y) = (2x)(4x) - (2x)(5y) - (3y)(4x) + (3y)(5y) = 8x 2 - 10xy - 12xy + 15y 2 = 8x 2 - 22xy + 15y 2 Try Exercise 50, page 38

Certain products occur so frequently in algebra that they deserve special attention. See Table P.2. Table P.2

Special Product Formulas

Special Form (Sum)(Difference)

Formula(s) (x + y)(x - y) = x 2 - y 2 (x + y)2 = x 2 + 2xy + y 2

2

(Binomial)

(x - y)2 = x 2 - 2xy + y 2

The variables x and y in these special product formulas can be replaced by other algebraic expressions, as shown in Example 5.

EXAMPLE 5

Use the Special Product Formulas

Find each special product. a.

(7x + 10)(7x - 10)

b.

(2y2 + 11z)2

Solution a. (7x + 10)(7x - 10) = (7x)2 - (10)2 = 49x2 - 100 b.

(2y 2 + 11z)2 = (2y 2)2 + 23(2y 2)(11z)4 + (11z)2 = 4y 4 + 44y 2z + 121z 2 Try Exercise 56, page 38

36

CHAPTER P


Many application problems require you to evaluate polynomials. To evaluate a polynomial, substitute the given value or values for the variable or variables and then perform the indicated operations using the Order of Operations Agreement.

EXAMPLE 6

Evaluate a Polynomial

Evaluate the polynomial 2x 3 - 6x 2 + 7 for x = - 4. Solution 2x 3 - 6x 2 + 7 2(-4)3 - 6(-4)2 + 7 = 2( - 64) - 6(16) + 7

• Substitute -4 for x. Evaluate the powers.

= - 128 - 96 + 7

• Perform the multiplications.

= -217

• Perform the additions and subtractions.


Applications of Polynomials EXAMPLE 7

Solve an Application

The number of singles tennis matches that can be played among n tennis players is given 1 1 by the polynomial n 2 - n. Find the number of singles tennis matches that can be 2 2 played among four tennis players. Solution 1 1 2 n - n 2 2 1 1 1 1 2 (4) - (4) = (16) - (4) = 8 - 2 = 6 2 2 2 2

• Substitute 4 for n. Then simplify.

Therefore, four tennis players can play a total of six singles matches. See Figure P.13. Figure P.13

Four tennis players can play a total of six singles matches.


EXAMPLE 8


A scientist determines that the average time in seconds that it takes a particular computer to determine whether an n-digit natural number is prime or composite is given by 0.002n2 + 0.002n + 0.009,

20 … n … 40

P.3

Math Matters The procedure used by the computer to determine whether a number is prime or composite is a polynomial time algorithm, because the time required can be estimated using a polynomial. The procedure used to factor a number is an exponential time algorithm. In the field of computational complexity, it is important to distinguish between polynomial time algorithms and exponential time algorithms. Example 8 illustrates that the polynomial time algorithm can be run in about 2 seconds, whereas the exponential time algorithm requires about 44 minutes!

POLYNOMIALS

37

The average time in seconds that it takes the computer to factor an n-digit number is given by 0.00032(1.7)n,

20 … n … 40

Estimate the average time it takes the computer to a.

determine whether a 30-digit number is prime or composite

b.

factor a 30-digit number

Solution a. 0.002n2 + 0.002n + 0.009 0.002(30)2 + 0.002(30) + 0.009 = 1.8 + 0.06 + 0.009 = 1.869 L 2 seconds b.

0.00032(1.7)n 0.00032(1.7)30 L 0.00032(8,193,465.726) L 2600 seconds Try Exercise 84, page 39

EXERCISE SET P.3 In Exercises 1 to 10, match the descriptions, labeled A to J, with the appropriate examples. A. x 3y xy 1 C. x 2 xy y 2 2 E. 8x 3 1 G. 8 I. 8x 4 15x 3 7

B. 7x 2 5x 11 D. 4xy F. 3 4x 2 H. 3x 5 4x 2 7x 11 J. 0

1. A monomial of degree 2

13. x 3 - 1 14. 4x 2 - 2x + 7 15. 2x 4 + 3x 3 + 5 + 4x 2 16. 3x 2 - 5x 3 + 7x - 1

In Exercises 17 to 22, determine the degree of the given polynomial. 17. 3xy 2 - 2xy + 7x

18. x 3 + 3x 2y + 3xy 2 + y 3

19. 4x 2y 2 - 5x 3y 2 + 17xy 3

20. -9x 5y + 10xy 4 - 11x 2y 2

21. xy

22. 5x 2y - y 4 + 6xy

2. A binomial of degree 3 3. A polynomial of degree 5 4. A binomial with a leading coefficient of - 4 5. A zero-degree polynomial 6. A fourth-degree polynomial that has a third-degree term

In Exercises 23 to 40, perform the indicated operation and simplify if possible by combining like terms. Write the result in standard form.

7. A trinomial with integer coefficients

23. (3x 2 + 4x + 5) + (2x 2 + 7x - 2)

8. A trinomial in x and y

24. (5y 2 - 7y + 3) + (2y 2 + 8y + 1)

9. A polynomial with no degree

25. (4w 3 - 2w + 7) + (5w 3 + 8w 2 - 1)

10. A fourth-degree binomial

In Exercises 11 to 16, for each polynomial, determine its a. standard form, b. degree, c. coefficients, d. leading coefficient, and e. terms.

26. (5x 4 - 3x 2 + 9) + (3x 3 - 2x 2 - 7x + 3) 27. (r 2 - 2r - 5) - (3r 2 - 5r + 7)

11. 2x + x 2 - 7

28. (7s 2 - 4s + 11) - (- 2s 2 + 11s - 9)

12. -3x 2 - 11 - 12x 4

29. (u 3 - 3u 2 - 4u + 8) - (u 3 - 2u + 4)

38

CHAPTER P


30. (5v 4 - 3v 2 + 9) - (6v 4 + 11v 2 - 10)

65. (r + s)(r 2 - rs + s 2)

31. (4x - 5)(2x 2 + 7x - 8)

66. (r - s)(r 2 + rs + s 2)

32. (5x - 7)(3x 2 - 8x - 5)

67. (3c - 2)(4c + 1)(5c - 2)

33. (3x 2 - 5x + 6)(3x - 1)

68. (4d - 5)(2d - 1)(3d - 4)

34. (3x - 4)(x - 6x - 9) 2

In Exercises 69 to 76, evaluate the given polynomial for the indicated value of the variable.

35. (2x + 6)(5x 3 - 6x 2 + 4) 36. (2x 3 - 7x - 1)(6x - 3)

69. x2 + 7x - 1, for x = 3

37. (x 3 - 4x 2 + 9x - 6)(2x + 5)

70. x2 - 8x + 2, for x = 4

38. (3x 3 + 4x 2 - x + 7)(3x - 2)

71. -x2 + 5x - 3, for x = - 2

39. (3x 2 - 2x + 5)(2x 2 - 5x + 2)

72. - x2 - 5x + 4, for x = - 5

40. (2y 3 - 3y + 4)(2y 2 - 5y + 7)

73. 3x3 - 2x2 - x + 3, for x = - 1

In Exercises 41 to 54, use the FOIL method to find the indicated product.

74. 5x3 - x2 + 5x - 3, for x = - 1

41. (2x + 4)(5x + 1)

42. (5x - 3)(2x + 7)

75. 1 - x5, for x = - 2

43. (y + 2)(y + 1)

44. (y + 5)(y + 3)

76. 1 - x3 - x5, for x = 2

45. (4z - 3)(z - 4)

46. (5z - 6)(z - 1)

77. Recreation The air resistance (in pounds) on a cyclist riding

47. (a + 6)(a - 3)

48. (a - 10)(a + 4)

49. (5x - 11y)(2x - 7y)

50. (3a - 5b)(4a - 7b)

a bicycle in an upright position can be given by 0.016v 2, where v is the speed of the cyclist in miles per hour (mph). Find the air resistance on a cyclist when

51. (9x + 5y)(2x + 5y)

52. (3x - 7z)(5x - 7z)

53. (3p + 5q)(2p - 7q)

54. (2r - 11s)(5r + 8s)

In Exercises 55 to 62, use the special product formulas to perform the indicated operation.

a. v = 10 mph

b. v = 15 mph

78. Highway Engineering On an expressway, the recommended

safe distance between cars in feet is given by 0.015v 2 + v + 10, where v is the speed of the car in miles per hour. Find the safe distance when a. v = 30 mph

b. v = 55 mph

55. (3x + 5)(3x - 5)

56. (4x - 3y)(4x + 3y)

57. (3x - y)

58. (6x + 7y)

below) is given by pr 2h, where r is the radius of the base and h is the height of the cylinder. Find the volume when

59. (4w + z)2

60. (3x - 5y2)2

a. r = 3 inches,

2

2

2

2

2

61. 3(x + 5) + y4 3(x + 5) - y4 62. 3(x - 2y) + 74 3(x - 2y) - 74

In Exercises 63 to 68, perform the indicated operation or operations and simplify. 63. (4d - 1)2 - (2d - 3)2 64. (5c - 8)2 - (2c - 5)2

79. Geometry The volume of a right circular cylinder (as shown

r

h = 8 inches

h

b. r = 5 centimeters, h = 12 centimeters

80. Automotive Engineering The fuel efficiency (in miles per

gallon of gas) of a car is given by -0.02v 2 + 1.5v + 2, where v is the speed of the car in miles per hour. Find the fuel efficiency when a. v = 45 mph

b. v = 60 mph

MID-CHAPTER P QUIZ

81. Psychology Based on data from one experiment, the reac-

tion time, in hundredths of a second, of a person to visual stimulus varies according to age and is given by the expression 0.005x 2 - 0.32x + 12, where x is the age of the person. Find the reaction time to the stimulus for a person who is a. x = 20 years old

b. x = 50 years old

82. Committee Membership The number of committees con-

sisting of exactly 3 people that can be formed from a group of n people is given by the polynomial 1 3 1 1 n - n2 + n 6 2 3

39

where 1000 … n … 10,000. Using this polynomial, estimate the time it takes this computer to calculate 4000! and 8000!. 86. Air Velocity of a Cough The velocity, in meters per sec-

ond, of the air that is expelled during a cough is given by velocity = 6r 2 - 10r 3, where r is the radius of the trachea in centimeters. a. Find the velocity as a polynomial in standard form. b. Find the velocity of the air in a cough when the radius of

the trachea is 0.35 cm. Round to the nearest hundredth. 87. Sports The height, in feet, of a baseball released by a pitcher

Find the number of committees consisting of exactly 3 people that can be formed from a group of 21 people.

t seconds after it is released is given by (ignoring air resistance)

83. Chess Matches Find the number of chess matches that can be

For the pitch to be a strike, it must be at least 2 feet high and no more than 5 feet high when it crosses home plate. If it takes 0.5 second for the ball to reach home plate, will the ball be high enough to be a strike?

played among the members of a group of 150 people. Use the formula from Example 7.

Height = - 16t 2 + 4.7881t + 6

84. Computer Science A computer scientist determines that the

time in seconds it takes a particular computer to calculate n digits of p is given by the polynomial 4.3 * 10-6n2 - 2.1 * 10-4n

6 ft

where 1000 … n … 10,000. Estimate the time it takes the computer to calculate p to a. 1000 digits

b. 5000 digits

60 ft Not to scale

c. 10,000 digits

85. Computer Science If n is a positive integer, then n!, which is

read “n factorial,” is given by n(n - 1)(n - 2) Á 2 # 1

For example, 4! = 4 # 3 # 2 # 1 = 24. A computer scientist determines that each time a program is run on a particular computer, the time in seconds required to compute n! is given by the polynomial 1.9 * 10-6n2 - 3.9 * 10-3n

88. Medicine The temperature, in degrees Fahrenheit, of a patient

after receiving a certain medication is given by Temperature = 0.0002t 3 - 0.0114t 2 + 0.0158t + 104 where t is the number of minutes after receiving the medication. a. What was the patient’s temperature just before the medica-

tion was given? b. What was the patient’s temperature 25 minutes after the

medication was given?

MID-CHAPTER P QUIZ 1. Evaluate 2x3 - 4(3xy - z 2) for x = -2, y = 3, and z = -4. 2. Simplify: 5 - 2[3x - 5(2x - 3) + 1] 3. Simplify:

24x-3y4 2 -3

6x y

4. Simplify: (3a-1>2b3>4)2(- 2a2>3b5>6)3

3 5. Simplify: 2 16a4b9c8

6. Simplify:

2 3 - 215

7. Multiply: (3x - 4y)(2x + 5y) 8. Multiply: (2a + 7)2 9. Multiply: (2x - 3)(4x 2 5x - 7)

40

CHAPTER P


SECTION P.4 Greatest Common Factor Factoring Trinomials Special Factoring Factor by Grouping General Factoring

Factoring PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A2.

PS1. Simplify:

6x3 [P.2] 2x

PS2. Simplify: ( -12x 4)3x 2 [P.2]

PS3. Express x 6 as a power of a. x 2 and b. x 3. [P.2] In Exercises PS4 to PS6, replace the question mark to make a true statement.

#

PS4. 6a3b4 ? = 18a3b 7 [P.2]

PS5. - 3(5a - ?) = - 15a + 21 [P.1]

PS6. 2x(3x - ?) = 6x2 - 2x [P.1]

Writing a polynomial as a product of polynomials is called factoring. Factoring is an important procedure that is often used to simplify fractional expressions and to solve equations. In this section, we consider only the factorization of polynomials that have integer coefficients. Also, we are concerned only with factoring over the integers. That is, we search only for polynomial factors that have integer coefficients.

Greatest Common Factor The first step in the factorization of any polynomial is to use the distributive property to factor out the greatest common factor (GCF) of the terms of the polynomial. Given two or more exponential expressions with the same prime number base or the same variable base, the GCF is the exponential expression with the smallest exponent. For example, 23 is the GCF of 23, 25, and 28

and

a is the GCF of a4 and a

The GCF of two or more monomials is the product of the GCFs of all the common bases. For example, to find the GCF of 27a3b4 and 18b3c, factor the coefficients into prime factors and then write each common base with its smallest exponent. 27a3b4 = 33 # a3 # b4

18b3c = 2 # 32 # b3 # c

The only common bases are 3 and b. The product of these common bases with their smallest exponents is 32b3. The GCF of 27a3b4 and 18b3c is 9b3. The expressions 3x(2x + 5) and 4(2x + 5) have a common binomial factor, which is 2x + 5. Thus the GCF of 3x(2x + 5) and 4(2x + 5) is 2x + 5.

EXAMPLE 1

Factor Out the Greatest Common Factor

Factor out the GCF. a.

12x3y4 - 24x2y5 + 18xy6

b. (6x - 5)(4x + 3) - (4x + 3)(3x - 7)

Solution a. 12x3y4 - 24x2y5 + 18xy6 = (6xy4)2x2 - (6xy4)4xy + (6xy4)3y 2 = 6xy4(2x2 - 4xy + 3y2)

• The GCF is 6xy4. • Factor out the GCF.

P.4

b.

FACTORING

41

(6x - 5)(4x + 3) - (4x + 3)(3x - 7) = (4x + 3)3(6x - 5) - (3x - 7)4

• The common binomial factor is 4x + 3.

= (4x + 3)(3x + 2) Try Exercise 6, page 48

Factoring Trinomials

The FOIL method See pages 34–35.

Some trinomials of the form x2 + bx + c can be factored by a trial procedure. This method makes use of the FOIL method in reverse. For example, consider the following products. (x + 3)(x + 5) = x2 + 5x + 3x + (3)(5) = x2 + 8x + 15 (x - 2)(x - 7) = x2 - 7x - 2x + ( -2)( - 7) = x2 - 9x + 14 (x + 4)(x - 9) = x2 - 9x + 4x + (4)(-9) = x2 - 5x - 36 m

m

The coefficient of x is the sum of the r constant terms of the binomials. The constant term of the trinomial is the product r of the constant terms of the binomials. Question • Is 共x 2兲共x 7兲 the correct factorization of x2 5x 14?

Points to Remember to Factor x 2 bx c 1. The constant term c of the trinomial is the product of the constant terms of the binomials. 2. The coefficient b in the trinomial is the sum of the constant terms of the binomials. 3. If the constant term c of the trinomial is positive, the constant terms of the binomials have the same sign as the coefficient b in the trinomial. 4. If the constant term c of the trinomial is negative, the constant terms of the binomials have opposite signs.

EXAMPLE 2

Factor a Trinomial

Factor. a.

Note In b. the last term of the trinomial contains y 2, so the last term of each binomial factor has a y.

x2 + 7x - 18

b.

x2 + 7xy + 10y2

Solution a. Find two integers whose product is 18 and whose sum is 7. The integers are 2 and 9: 2(9) 18, 2 9 7. x 2 + 7x - 18 = (x - 2)(x + 9) b.

Find two integers whose product is 10 and whose sum is 7. The integers are 2 and 5: 2(5) 10, 2 5 7. x2 + 7xy + 10y2 = (x + 2y)(x + 5y) Try Exercise 12, page 48

Answer • No. (x - 2)(x + 7) = x2 + 5x - 14.

42

CHAPTER P


Sometimes it is impossible to factor a polynomial into the product of two polynomials having integer coefficients. Such polynomials are said to be nonfactorable over the integers. For example, x2 + 3x + 7 is nonfactorable over the integers because there are no integers whose product is 7 and whose sum or difference is 3. The trial method sometimes can be used to factor trinomials of the form ax2 + bx + c, which do not have a leading coefficient of 1. We use the factors of a and c to form trial binomial factors. Factoring trinomials of this type may require testing many factors. To reduce the number of trial factors, make use of the following points.

Points to Remember to Factor ax 2 bx c, a>0 1. If the constant term of the trinomial is positive, the constant terms of the binomials have the same sign as the coefficient b in the trinomial. 2. If the constant term of the trinomial is negative, the constant terms of the binomials have opposite signs. 3. If the terms of the trinomial do not have a common factor, then neither binomial will have a common factor.

EXAMPLE 3

Factor a Trinomial of the Form ax 2 bx c

Factor: 6x2 - 11x + 4 Solution Because the constant term of the trinomial is positive and the coefficient of the x term is negative, the constant terms of the binomials will both be negative. We start by finding factors of the first term and factors of the constant term.

Factors of 6x 2

Factors of 4 (both negative)

x, 6x

- 1, - 4

2x, 3x

- 2, - 2

Use these factors to write trial factors. Use the FOIL method to see whether any of the trial factors produce the correct middle term. If the terms of a trinomial do not have a common factor, then a binomial factor cannot have a common factor (point 3). Such trial factors need not be checked. Trial Factors

Middle Term

(x - 1)(6x - 4)

Common factor

(x - 4)(6x - 1)

-1x - 24x = - 25x

(x - 2)(6x - 2)

Common factor

• 6x and 2 have a common factor.

(2x - 1)(3x - 4)

-8x - 3x = - 11x

• This is the correct middle term.

Thus 6x2 - 11x + 4 = (2x - 1)(3x - 4). Try Exercise 18, page 48

• 6x and 4 have a common factor. • This is not the correct middle term.

P.4

FACTORING

43

If you have difficulty factoring a trinomial, you may wish to use the following theorem. It will indicate whether the trinomial is factorable over the integers.

Factorization Theorem The trinomial ax2 + bx + c, with integer coefficients a, b, and c, can be factored as the product of two binomials with integer coefficients if and only if b2 - 4ac is a perfect square.

EXAMPLE 4

Apply the Factorization Theorem

Determine whether each trinomial is factorable over the integers. a.

4x2 + 8x - 7

b.

6x2 - 5x - 4

Solution a. The coefficients of 4x2 + 8x - 7 are a = 4, b = 8, and c = - 7. Applying the factorization theorem yields b2 - 4ac = 82 - 4(4)(-7) = 176 Because 176 is not a perfect square, the trinomial is nonfactorable over the integers. b.

The coefficients of 6x2 - 5x - 4 are a = 6, b = - 5, and c = - 4. Thus b2 - 4ac = ( -5)2 - 4(6)(-4) = 121 Because 121 is a perfect square, the trinomial is factorable over the integers. Using the methods we have developed, we find 6x2 - 5x - 4 = (3x - 4)(2x + 1) Try Exercise 24, page 48

Special Factoring The product of a term and itself is called a perfect square. The exponents on variables of perfect squares are always even numbers. The square root of a perfect square is one of the two equal factors of the perfect square. To find the square root of a perfect square variable term, divide the exponent by 2. For the examples in Table P.3, assume that the variables represent positive numbers. Table P.3

Term 7 y 2x3 xn

Perfect Squares and Square Roots

7#7 = y#y = 2x3 # 2x3 = xn # xn =

Perfect Square

Square Root

49

149 = 7

2

y

2y2 = y

4x6

24x6 = 2x3

x 2n

2x2n = xn

The factors of the difference of two perfect squares are the sum and difference of the square roots of the perfect squares.

44

CHAPTER P


Factors of the Difference of Two Perfect Squares a2 - b2 = (a + b)(a - b)

The difference of two perfect squares always factors over the integers. However, the sum of squares does not factor over the integers. For instance, a2 + b2 does not factor over the integers. As another example, x2 + 4 is the sum of squares and does not factor over the integers. There are no integers whose product is 4 and whose sum is 0.

EXAMPLE 5

Factor the Difference of Squares

Factor. a.

49x2 - 144

b. a4 - 81

Solution a. 49x2 - 144 = (7x)2 - 122 = (7x + 12)(7x - 12) b.

a4 - 81 = (a2)2 - (9)2 = (a2 + 9)(a2 - 9) = (a + 3)(a - 3)(a2 + 9)

• Write as the difference of squares. • The binomial factors are the sum and the difference of the square roots of the squares. • Write as the difference of squares. • The binomial factors are the sum and the difference of the square roots of the squares. • a2 - 9 is the difference of squares. Factor as (a + 3)(a - 3). The sum of squares, a 2 + 9, does not factor over the integers.


A perfect-square trinomial is a trinomial that is the square of a binomial. For example, x2 + 6x + 9 is a perfect-square trinomial because (x + 3)2 = x2 + 6x + 9 Every perfect-square trinomial can be factored by the trial method, but it generally is faster to factor perfect-square trinomials by using the following factoring formulas.

Factors of a Perfect-Square Trinomial a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

EXAMPLE 6

Factor a Perfect-Square Trinomial

Factor: 16m2 - 40mn + 25n2 Solution Because 16m2 = (4m)2 and 25n2 = (5n)2, try factoring 16m2 - 40mn + 25n2 as the square of a binomial. 16m2 - 40mn + 25n2 ⱨ (4m - 5n)2

P.4

Caution It is important to check the proposed factorization. For instance, consider x 2 + 13x + 36. Because x 2 is the square of x and 36 is the square of 6, it is tempting to factor, using the perfect-square trinomial formulas, as x 2 + 13x + 36 ⱨ (x + 6)2. Note that (x + 6) 2 = x 2 + 12x + 36, which is not the original trinomial. The correct factorization is x 2 + 13x + 36 = (x + 4)(x + 9).

= 16m2 - 20mn - 20mn + 25n2 = 16m2 - 40mn + 25n2 The factorization checks. Therefore, 16m2 - 40mn + 25n2 = (4m - 5n)2. Try Exercise 46, page 49

The product of the same three terms is called a perfect cube. The exponents on variables of perfect cubes are always divisible by 3. The cube root of a perfect cube is one of the three equal factors of the perfect cube. To find the cube root of a perfect cube variable term, divide the exponent by 3. See Table P.4. Perfect Cubes and Cube Roots

Term

5#5#5 =

5

a + b = (a + b)(a - ab + b ) 2

2

3x2 # 3x2 # 3x2 =

3x2

Pay attention to the pattern of the signs when factoring the sum or the difference of two perfect cubes.

3

z#z#z =

z

Study tip

3

45

Check: (4m - 5n)2 = (4m - 5n)(4m - 5n)

Table P.4

Same signs

FACTORING

xn # xn # xn =

xn

Perfect Cube

Cube Root

125

3 1 125 = 5

z3

3 3 2 z = z

27x6

3 2 27x6 = 3x 2

x3n

3 3n 2 x = xn

The following factoring formulas are used to factor the sum or difference of two perfect cubes.

Opposite signs Same signs

Factors of the Sum or Difference of Two Perfect Cubes a3 + b3 = (a + b)(a2 - ab + b2) a3 - b3 = (a - b)(a2 + ab + b2)

a3 - b3 = (a - b)(a2 + ab + b2) Opposite signs

EXAMPLE 7

Factor the Sum or Difference of Cubes

Factor. a.

8a3 + b3

b.

a3 - 64

Solution a. 8a3 + b3 = (2a)3 + b3

• Recognize the sum-of-cubes form.

= (2a + b)(4a2 - 2ab + b2) b.

a3 - 64 = a3 - 43

• Factor. • Recognize the difference-of-cubes form.

= (a - 4)(a + 4a + 16) 2

• Factor.


Certain trinomials can be expressed as quadratic trinomials by making suitable variable substitutions. A trinomial is quadratic in form if it can be written as au2 + bu + c

46

CHAPTER P


If we let x2 = u, the trinomial x4 + 5x2 + 6 can be written as shown at the right. The trinomial is quadratic in form.

x4 + 5x2 + 6 = (x2)2 + 5(x2) + 6 = u2 + 5u + 6

If we let xy = u, the trinomial 2x2y2 + 3xy - 9 can be written as shown at the right. The trinomial is quadratic in form.

2x2y2 + 3xy - 9 = 2(xy)2 + 3(xy) - 9 = 2u2 + 3u - 9

When a trinomial that is quadratic in form is factored, the variable part of the first term in each binomial factor will be u. For example, because x4 + 5x2 + 6 is quadratic in form when x2 = u, the first term in each binomial factor will be x2. x4 + 5x2 + 6 = (x2)2 + 5(x2) + 6 = (x2 + 2)(x2 + 3) The trinomial x2y2 - 2xy - 15 is quadratic in form when xy = u. The first term in each binomial factor will be xy. x2y2 - 2xy - 15 = (xy)2 - 2(xy) - 15 = (xy + 3)(xy - 5)

EXAMPLE 8

Factor a Polynomial That Is Quadratic in Form

Factor. a.

6x 2y 2 - xy - 12

b. x4 + 5x 2 - 36

Solution a. 6x 2y 2 - xy - 12 = 6u2 - u - 12 = (3u + 4)(2u - 3) = (3xy + 4)(2xy - 3) b.

x4 + 5x2 - 36 = u2 + 5u - 36 = (u - 4)(u + 9) = (x2 - 4)(x2 + 9) = (x - 2)(x + 2)(x2 + 9)

• The trinomial is quadratic in form when xy u. Then x 2y 2 = u 2. • Factor. • Replace u with xy.

• The trinomial is quadratic in form when x 2 = u. Then x4 = u 2. • Factor. • Replace u with x 2. • Factor the difference of squares. The sum of squares does not factor.


Factor by Grouping Note - a + b = - (a - b). Thus - 4y + 14 = - (4y - 14).

Some polynomials can be factored by grouping. Pairs of terms that have a common factor are first grouped together. The process makes repeated use of the distributive property, as shown in the following factorization of 6y3 - 21y2 - 4y + 14. 6y3 - 21y2 - 4y + 14 = (6y3 - 21y2) - (4y - 14)

• Group the first two terms and the last two terms.

P.4

FACTORING

47

= 3y2(2y - 7) - 2(2y - 7)

• Factor out the GCF from each of the groups.

= (2y - 7)(3y - 2)

• Factor out the common binomial factor.

2

When you factor by grouping, some experimentation may be necessary to find a grouping that fits the form of one of the special factoring formulas.

EXAMPLE 9

Factor by Grouping

Factor by grouping. a.

a2 + 10ab + 25b2 - c2

b.

p2 + p - q - q2

Solution a. a2 + 10ab + 25b2 - c2

b.

= (a2 + 10ab + 25b2) - c2

• Group the terms of the perfectsquare trinomial.

= (a + 5b)2 - c2 = 3(a + 5b) + c43(a + 5b) - c4 = (a + 5b + c)(a + 5b - c)

• Factor the trinomial.

p + p - q - q = p2 - q2 + p - q 2

• Factor the difference of squares. • Simplify.

2

= ( p - q ) + ( p - q) = ( p + q)( p - q) + ( p - q) = ( p - q)( p + q + 1) 2

2

• Rearrange the terms. • Regroup. • Factor the difference of squares. • Factor out the common factor (p q).


General Factoring A general factoring strategy for polynomials is shown below.

General Factoring Strategy 1. Factor out the GCF of all terms. 2. Try to factor a binomial as a.

the difference of two squares

b. the sum or difference of two cubes 3. Try to factor a trinomial a.

as a perfect-square trinomial

b. using the trial method 4. Try to factor a polynomial with more than three terms by grouping. 5. After each factorization, examine the new factors to see whether they can be factored.

48

CHAPTER P


EXAMPLE 10

Factor Using the General Factoring Strategy

Factor: 2vx6 + 14vx3 - 16v Solution 2vx6 + 14vx3 - 16v = 2v(x6 + 7x3 - 8)

• The GCF is 2v.

= 2v(u + 7u - 8)

• x6 + 7x3 - 8 is quadratic in form. Let u = x3. Then u2 = x6.

= 2v(u + 8)(u - 1)

• Factor.

= 2v(x + 8)(x - 1)

• Replace u with x3. x3 + 8 is the sum of cubes. x3 - 1 is the difference of cubes.

2

3

3

= 2v(x + 2)(x2 - 2x + 4)(x - 1)(x2 + x + 1)

• Factor the sum and difference of cubes.


EXERCISE SET P.4 In Exercises 1 to 8, factor out the GCF from each polynomial.

In Exercises 23 to 28, use the factorization theorem to determine whether each trinomial is factorable over the integers.

1. 5x + 20

2. 8x + 12x - 40

3. -15x2 - 12x

4. -6y2 - 54y

23. 8x 2 + 26x + 15

24. 16x 2 + 8x - 35

5. 10x2y + 6xy - 14xy2

6. 6a3b2 - 12a2b + 72ab3

25. 4x 2 - 5x + 6

26. 6x 2 + 8x - 3

27. 6x 2 - 14x + 5

28. 10x 2 - 4x - 5

2

7. (x - 3)(a + b) + (x - 3)(a + 2b) 8. (x - 4)(2a - b) + (x + 4)(2a - b)

In Exercises 9 to 22, factor each trinomial over the integers. 9. x 2 + 7x + 12

In Exercises 29 to 42, factor each difference of squares over the integers. 29. x 2 - 9

30. x2 - 64

31. 4a2 - 49

32. 81b2 - 16c2

33. 1 - 100x2

34. 1 - 121y2

35. (x + 1)2 - 4

36. (5x + 3)2 - 9

37. 6x 2 - 216

38. - 2z 3 + 2z

10. x2 + 9x + 20

11. a - 10a - 24

12. b + 12b - 28

13. x + 6x + 5

14. x + 11x + 18

15. 6x2 + 25x + 4

16. 8a2 - 26a + 15

17. 51x2 - 5x - 4

18. 57y 2 + y - 6

19. 6x 2 + xy - 40y 2

20. 8x 2 + 10xy - 25y 2

39. x 4 - 625

40. y 4 - 1

21. 6x 2 + 23x + 15

22. 9x 2 + 10x + 1

41. x 5 - 81x

42. 3xy 6 - 48xy 2

2

2

2

2

P.5

In Exercises 43 to 50, factor each perfect-square trinomial. 43. x 2 + 10x + 25

44. y 2 + 6y + 9

45. a - 14a + 49

46. b - 24b + 144

47. 4x2 + 12x + 9

48. 25y2 + 40y + 16

49. z4 + 4z2w2 + 4w4

50. 9x4 - 30x 2y 2 + 25y 4

2

RATIONAL EXPRESSIONS

49

71. 6w 3 + 4w 2 - 15w - 10 72. 10z3 - 15z2 - 4z + 6

2

In Exercises 73 to 92, use the general factoring strategy to completely factor each polynomial. If the polynomial does not factor, then state that it is nonfactorable over the integers.

In Exercises 51 to 58, factor each sum or difference of cubes over the integers. 51. x3 - 8

52. b3 + 64

53. 8x3 - 27y3

54. 64u3 - 27v3

55. 8 - x6

56. 1 + y12

57. (x - 2)3 - 1

58. ( y + 3)3 + 8

In Exercises 59 to 66, factor over the integers the polynomials that are quadratic in form. 59. x4 - x 2 - 6

60. y4 + 3y2 + 2

61. x 2y 2 + 4xy - 5

62. x 2y 2 - 8xy + 12

63. 4x 5 - 4x 3 - 8x

64. z4 + 3z2 - 4

65. z4 + z 2 - 20

66. x 4 - 13x 2 + 36

75. 16x4 - 1

76. 81y4 - 16

77. 12ax 2 - 23axy + 10ay 2

78. 6ax 2 - 19axy - 20ay 2

79. 3bx3 + 4bx2 - 3bx - 4b

80. 2x6 - 2

81. 72bx2 + 24bxy + 2by2

82. 64y 3 - 16y 2z + yz 2

83. (w - 5)3 + 8

84. 5xy + 20y - 15x - 60

85. x2 + 6xy + 9y2 - 1

86. 4y 2 - 4yz + z 2 - 9

87. 8x2 + 3x - 4

88. 16x2 + 81

89. 5x(2x - 5)2 - (2x - 5)3

90. 6x(3x + 1)3 - (3x + 1)4

91. 4x2 + 2x - y - y2

92. a2 + a + b - b2

93. x 2 + kx + 16

67. 3x 3 + x 2 + 6x + 2

68. 18w 3 + 15w 2 + 12w + 10

69. ax 2 - ax + bx - b

70. a 2y 2 - ay 3 + ac - cy

Simplifying Rational Expressions Operations on Rational Expressions Determining the LCD of Rational Expressions Complex Fractions Application of Rational Expressions

74. 4bx 3 + 32b

In Exercises 93 and 94, find all positive values of k such that the trinomial is a perfect-square trinomial.

In Exercises 67 to 72, factor over the integers by grouping.

SECTION P.5

73. 18x 2 - 2

94. 36x 2 + kxy + 100y 2

In Exercises 95 and 96, find k such that the trinomial is a perfect-square trinomial. 95. x 2 + 16x + k

96. x 2 - 14xy + ky 2

Rational Expressions PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A3.

PS1. Simplify: 1 +

1 2 -

1 3

[P.1]

PS2.

w -1 y -1 Simplify: a b a b [P.2] x z

PS3. What is the common binomial factor of x2 + 2x - 3 and x2 + 7x + 12? [P.4]

50

CHAPTER P


In Exercises PS4 to PS6, factor completely over the integers.

PS4. (2x - 3)(3x + 2) - (2x - 3)(x + 2) [P.4] PS5. x2 - 5x - 6 [P.4] PS6. x3 - 64 [P.4]

Math Matters Evidence from work left by early Egyptians more than 3600 years ago shows that they used, with one exception, unit fractions— that is, fractions whose numerators are 1. The one exception was 2兾3. A unit fraction was represented by placing an oval over the symbol for the number in the denominator. For instance, 1兾4 = .

A rational expression is a fraction in which the numerator and denominator are polynomials. For example, the expressions below are rational expressions. 3 x + 1

x 2 - 4x - 21 x2 - 9

and

The domain of a rational expression is the set of all real numbers that can be used as replacements for the variable. Any value of the variable that causes division by zero is excluded from the domain of the rational expression. For example, the domain of x + 3 , x2 - 5x

x Z 0, x Z 5

is the set of all real numbers except 0 and 5. Both 0 and 5 are excluded values because the denominator x2 - 5x equals zero when x = 0 and also when x = 5. Sometimes the excluded values are specified to the right of a rational expression, as shown here. However, a rational expression is meaningful only for those real numbers that are not excluded values, regardless of whether the excluded values are specifically stated. Question • What value of x must be excluded from the domain of

x-2 ? x + 1

Rational expressions have properties similar to the properties of rational numbers.

Properties of Rational Expressions For all rational expressions

P R and , where Q Z 0 and S Z 0, Q S

Equality

R P = Q S

Equivalent expressions

PR P = , Q QR

Sign

-

if and only if PS = QR R Z 0

P -P P = = Q Q -Q

Answer • When x = - 1, x + 1 = 0. Therefore, -1 must be excluded from the domain. When

x - 2 2 - 2 0 = = 0. The value of the numerator can equal is x + 1 2 + 1 3 zero; the value denominator cannot equal zero. x = 2, the value of

P.5


51

Simplifying Rational Expressions To simplify a rational expression, factor the numerator and denominator. Then use the equivalent expressions property to eliminate factors common to both the numerator and the denominator. A rational expression is simplified when 1 is the only common factor of both the numerator and the denominator.

EXAMPLE 1 Simplify:

Simplify a Rational Expression

7 + 20x - 3x 2 2x 2 - 11x - 21

Solution (7 - x)(1 + 3x) 7 + 20x - 3x 2 = 2 (x - 7)(2x + 3) 2x - 11x - 21 - (x - 7)(1 + 3x) = (x - 7)(2x + 3) - (x - 7)(1 + 3x) = (x - 7)(2x + 3) - (1 + 3x) = 2x + 3 3x + 1 3 = ,x 2x + 3 2

• Factor. • Use (7 x) (x 7). • x 7.


Operations on Rational Expressions Arithmetic operations are defined on rational expressions in the same way as they are on rational numbers.

Definitions of Arithmetic Operations for Rational Expressions For all rational expressions Addition Subtraction Multiplication Division

P R R , , and , where Q Z 0 and S Z 0, Q Q S

R P + R P + = Q Q Q R P - R P = Q Q Q PR P#R = Q S QS P R P#S PS , = = , R Z 0 Q S Q R QR

Factoring and the equivalent expressions property of rational expressions are used in the multiplication and division of rational expressions.

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CHAPTER P


EXAMPLE 2 Multiply:

Multiply Rational Expressions

x 2 - 11x + 28 4 - x2 # x 2 + 2x - 8 x 2 - 5x - 14

Solution 2 4 - x2 # x 2 - 11x + 28 2 x + 2x - 8 x - 5x - 14

• Factor.

=

(2 - x)(2 + x) # (x - 4)(x - 7) (x - 2)(x + 4) (x + 2)(x - 7)

=

-(x - 2)(2 + x) # (x - 4)(x - 7) (x - 2)(x + 4) (x + 2)(x - 7)

• 2 x (x 2).

=

-(x - 2)(2 + x)(x - 4)(x - 7) (x - 2)(x + 4)(x + 2)(x - 7)

• Simplify.

=

-(x - 4) x - 4 = x + 4 x + 4


EXAMPLE 3 Divide:

Divide Rational Expressions

x 2 + 7x + 12 x 2 + 6x + 9 , x 3 + 27 x 3 - 3x 2 + 9x

Solution x 2 + 7x + 12 x 2 + 6x + 9 , 3 3 x + 27 x - 3x 2 + 9x Factors of the Sum or Difference of Two Perfect Cubes See page 45.

= = = =

(x + 3)2 (x + 3)(x - 3x + 9) 2

,

(x + 4)(x + 3) x(x 2 - 3x + 9)

- 3x + 9) (x + 4)(x + 3) (x + 3)(x - 3x + 9) 2 2 (x + 3) x(x - 3x + 9) (x + 3)2 2

# x(x

• Factor.

2

(x + 3)(x 2 - 3x + 9)(x + 4)(x + 3)

• Multiply by the reciprocal. • Simplify.

x x + 4


Addition of rational expressions with a common denominator is accomplished by writing the sum of the numerators over the common denominator. For example, 5x x 5x + x 6x 6x x + = = = # = 18 18 18 18 6 3 3 If the rational expressions do not have a common denominator, then they can be written as equivalent expressions that have a common denominator by multiplying the numerator and

P.5


53

denominator of each of the rational expressions by the required polynomials. The following procedure can be used to determine the least common denominator (LCD) of rational expressions. It is similar to the process used to find the LCD of rational numbers.

Determining the LCD of Rational Expressions 1. Factor each denominator completely and express repeated factors using exponential notation. 2. Identify the largest power of each factor in any single factorization. The LCD is the product of each factor raised to its largest power. For example, the rational expressions 1 x + 3

and

5 2x - 1

have an LCD of (x + 3)(2x - 1). The rational expressions 5x (x + 5)(x - 7)3

and

7 x(x + 5)2(x - 7)

have an LCD of x(x + 5)2(x - 7)3.

EXAMPLE 4

Add and Subtract Rational Expressions

Perform the indicated operation and then simplify, if possible. a.

x + 2 2x + 1 + x - 3 x + 5

b.

23x - 16 39x + 36 - 2 x - 3x - 10 x - 7x + 10 2

Solution a. The LCD is (x - 3)(x + 5). Write equivalent fractions in terms of the LCD, and then add. x + 2 2x + 1 # x + 5 x + 2#x - 3 2x + 1 + = + x - 3 x + 5 x - 3 x + 5 x + 5 x - 3

b.

=

x2 - x - 6 2x 2 + 11x + 5 + (x - 3)(x + 5) (x - 3)(x + 5)

=

(2x 2 + 11x + 5) + (x 2 - x - 6) (x - 3)(x + 5)

• Add.

=

3x 2 + 10x - 1 (x - 3)(x + 5)

• Simplify.

Factor the denominators: x 2 - 3x - 10 = (x - 5)(x + 2) x 2 - 7x + 10 = (x - 5)(x - 2) The LCD is (x - 5)(x + 2)(x - 2). Write equivalent fractions in terms of the LCD, and then subtract. (continued)

54

CHAPTER P


23x - 16 39x + 36 - 2 x - 3x - 10 x - 7x + 10 23x - 16 # x + 2 39x + 36 # x - 2 = (x - 5)(x + 2) x - 2 (x - 5)(x - 2) x + 2 2

=

23x2 + 30x - 32 39x2 - 42x - 72 (x - 5)(x + 2)(x - 2) (x - 5)(x + 2)(x - 2)

=

(39x2 - 42x - 72) - (23x2 + 30x - 32) (x - 5)(x + 2)(x - 2)

=

8(2x2 - 9x - 5) 16x2 - 72x - 40 = (x - 5)(x + 2)(x - 2) (x - 5)(x + 2)(x - 2)

=

8(2x + 1) 8(2x + 1)(x - 5) = (x - 5)(x + 2)(x - 2) (x + 2)(x - 2)


EXAMPLE 5

Simplify:

Use the Order of Operations Agreement with Rational Expressions

x2 + 5x + 4 x + 4 x + 3 , 2 x - 2 x - 1 x + 4x - 5

Solution The Order of Operations Agreement requires that division be completed before subtraction. To divide fractions, multiply by the reciprocal as shown below. x + 4 x2 + 5x + 4 x + 3 , 2 x - 2 x - 1 x + 4x - 5 =

x + 4 # x 2 + 4x - 5 x + 3 x - 2 x - 1 x 2 + 5x + 4

• Multiply by the reciprocal.

=

x + 4 # (x - 1)(x + 5) x + 3 x - 2 x - 1 (x + 1)(x + 4)

• Factor the trinomials.

=

(x + 4)(x - 1)(x + 5) x + 3 x - 2 (x - 1)(x + 1)(x + 4)

• Multiply.

=

x + 5 x + 3 x - 2 x + 1

• Simplify.

=

x + 5#x - 2 x + 3#x + 1 x - 2 x + 1 x + 1 x - 2

• Subtract. The LCD is (x 2)(x 1).

=

(x 2 + 4x + 3) - (x 2 + 3x - 10) (x - 2)(x + 1)

=

x + 13 (x - 2)(x + 1)


P.5


55

Complex Fractions A complex fraction is a fraction whose numerator or denominator contains one or more fractions. Simplify complex fractions using one of the following methods.

Methods for Simplifying Complex Fractions Method 1: Multiply by 1 in the form

LCD . LCD

1. Determine the LCD of all fractions in the complex fraction. 2. Multiply both the numerator and the denominator of the complex fraction by the LCD. 3. If possible, simplify the resulting rational expression. Method 2: Multiply the numerator by the reciprocal of the denominator. 1. Simplify the numerator to a single fraction and the denominator to a single fraction. 2. Using the definition for dividing fractions, multiply the numerator by the reciprocal of the denominator. 3. If possible, simplify the resulting rational expression.

EXAMPLE 6

Simplify Complex Fractions

Simplify.

a.

2 + x - 2 3x x - 5 x

1 x 2 - 5

b. 4 -

2x x - 2 2 x

Solution a. Simplify the numerator to a single fraction and the denominator to a single fraction. 1 # (x - 2) 2#x 1 2 + # + # (x - 2) x x (x - 2) x x - 2 = 3x 3x - 2 2 x - 5 x - 5 x - 5

• Simplify the numerator and denominator.

2x + (x - 2) 3x - 2 x(x - 2) x(x - 2) = = 3x - 2 3x - 2 x - 5 x - 5 =

3x - 2 x(x - 2)

=

x - 5 x(x - 2)

#

x - 5 3x - 2

• Multiply the numerator by the reciprocal of the denominator.

(continued)

56

CHAPTER P


b.

4 -

2x 2x #x = 4 x - 2 x - 2 x 2 2 x x 2x 2 = 4 2x - (x - 2) = 4 -

• Multiply the numerator and denominator by the LCD of all the fractions.

2x 2 x + 2

• Simplify.

=

2x 2 4#x + 2 1 x + 2 x + 2

=

2x 2 4x + 8 x + 2 x + 2

=

- 2x 2 + 4x + 8 x + 2

• Subtract. The LCD is x 2.


EXAMPLE 7

Simplify a Fraction

Simplify the fraction

c-1 . a-1 + b-1

Solution The fraction written without negative exponents becomes 1 c c-1 = -1 -1 1 1 a + b + a b 1# abc c = 1 1 a + b abc a b ab = bc + ac

• Use x-n =

1 . xn

• Multiply the numerator and denominator by abc, which is the LCD of the fraction in the numerator and the fraction in the denominator.


Application of Rational Expressions EXAMPLE 8


The average speed for a round trip is given by the complex fraction 2 1 1 + v1 v2 where v1 is the average speed on the way to your destination and v2 is the average speed on your return trip. Find the average speed for a round trip if v1 = 50 mph and v2 = 40 mph.

P.5


57

Solution Evaluate the complex fraction with v1 = 50 and v2 = 40. 2 2 2 = = 1 1 1 1#4 1 1#5 + + + # v1 v2 50 40 50 4 40 # 5 =

2 4 5 + 200 200

= 2#

=

• Substitute the given values for v1 and v2. Then simplify the denominator.

2 9 200

400 4 200 = = 44 9 9 9

The average speed for the round trip is 44

4 mph. 9

Try Exercise 64, page 59 Question • In Example 8, why is the average speed for the round trip not the average of v1 and v2?

Answer • Because you were traveling more slowly on the return trip, the return trip took longer

than the trip to your destination. More time was spent traveling at the slower speed. Thus the average speed is less than the average of v1 and v2.

EXERCISE SET P.5 In Exercises 1 to 10, simplify each rational expression. x - x - 20 3x - 15 2

1.

3.

5.

7.

8.

9.

10.

2.

x 3 - 9x x + x - 6x 3

2

a3 + 8

4.

6.

a2 - 4 x2 + 3x - 40 -x + 3x + 10 2

2x3 - 6x2 + 5x - 15 9-x

2

4y3 - 8y2 + 7y - 14 -y - 5y + 14 2

x3 - x2 + x x3 + 1

2x - 5x - 12 2

2x + 5x + 3 2

x 3 + 125 2x - 50x 3

y3 - 27 -y2 + 11y - 24

In Exercises 11 to 40, simplify each expression. 11. a12. a

13. a 14. a

15.

16.

17.

ba

4a 3b

2

12x2y 4

5z

6b a4

b a-

b

25x2z3 15y2

b

-1 2p 2 b a b 5q2 3q2

6p2

4r 2s 3t 3

-1

b

a

6rs3 5t 2

b

x2 + x # 3x2 + 19x + 28 2x + 3 x2 + 5x + 4 x2 - 16 x + 7x + 12 2

#x

2

- 4x - 21 x2 - 4x

3x - 15 # 2x2 + 16x + 30 6x + 9 2x2 - 50

58

18.

19.

20.

21.

22.

CHAPTER P

y3 - 8


y2 + 3y

#

y2 + y - 6 y3 + 2y2 + 4y 12y2 + 28y + 15 6y + 35y + 25 2

z2 - 81 z - 16 a2 + 9 a - 64 2

z + 5z - 36

,

a3 - 3a2 + 9a - 27

4x - 9y 2

2

,

24.

-2s + 3t 2s + 5t + 4t 4t

25.

5y - 7 2y - 3 y + 4 y + 4

26.

6x - 5 3x - 8 x-3 x-3

27.

7x x + x-5 x+3

28.

2x 5x + 3x + 1 x - 7

29.

4z 5z + 2z - 3 z - 5

32.

33.

3x2 - xy - 2y2 2x + xy - 3y 2

x2 - 9

36.

p p p + 2 + , 2 p + 5 p - 4 p - p - 12

2 a2 - 3a + 2

+

1

+

x - 9 2

3 a2 - 1

39. a1 +

1 2 b a3 - b x x

40. a4 -

2 1 b a4 + b z z

-

1 x - 16 2

5 a2 + 3a - 10

x - 2 y 43. y - x

1 x 41. 1 1 x

2 a 42. 3 5 + a

2 x - 3 44. 1 4 + 1 2 + x

1 x + 2 45. 3 1 + 3 1 + x

3 +

1 b - 2 47. 1 1 b + 3

3 -

1

5 -

1 -

+

3m - 5n

- 1

h

r

48. r -

r +

1 3

1

x2 49. 1 1 + x

m2 + mn - 2n2

2 1 2 # 3x + 11x - 4 + x 3x - 1 x - 5

(x + h)2

46.

1 +

x2 + 7x + 12

m - n

38.

x + 7x + 12

+

4 +

3x - 1

m2 - mn - 6n2

2

1 2

In Exercises 41 to 58, simplify each complex fraction.

3y - 1 2y - 5 3y + 1 y-3 -

q + 1 2q q + 5 , q - 3 q - 3 q - 3

37.

a + 5a - 24

6x2 + 13xy + 6y2

x

35.

2

p + 5 2p - 7 + r r

31.

3y + 11y - 20 2

3 # y2 - 1 2 y y + 1 y + 4

2

23.

30.

2y2 - y - 3

z2 - z - 20

,

2

,

34.

m 51. 2 1 - m 1 -m

50.

1 1 1 + a b

x + h + 1 x x + h x + 1 52. h

P.6

3y - 2 2 y y - 1 54. y y - 1

1 x - 4 x x + 1 53. x x + 1 1 x + 3 x 55. x + x - 1 x

x + 2

57.

v1 = 180 mph and v2 = 110 mph. b. Simplify the complex fraction. 64. Relativity Theory Using Einstein’s Theory of Relativity, the

“sum” of the two speeds v1 and v2 is given by the complex fraction

+

v1 + v2 v1v2 1 + 2 c where c is the speed of light.

2y2 + 11y + 15

x2 + 3x - 10 x2 + x - 6 x2 - x - 30

58.

2x2 - 15x + 18

y2 - 4y - 21

a. Evaluate this complex fraction with v1 = 1.2 * 108 mph,

6y2 + 11y - 10

v2 = 2.4 * 108 mph, and c = 6.7 * 108 mph.

3y - 23y + 14 2

b. Simplify the complex fraction.

In Exercises 59 to 62, simplify each algebraic fraction. Write all answers with positive exponents. 59.

61.

a-1 + b-1 a - b

60.

a-1b - ab-1

65. Find the rational expression in simplest form that represents

the sum of the reciprocals of the consecutive integers x and x + 1.

e-2 - f -1 ef


62. (a + b-2)-1

a2 + b 2

the positive difference between the reciprocals of the consecutive even integers x and x + 2.

63. Average Speed According to Example 8, the average speed


for a round trip in which the average speed on the way to your destination is v1 and the average speed on your return is v2 is given by the complex fraction

the sum of the reciprocals of the consecutive even integers x - 2, x, and x + 2.

2 1 1 + v1 v2

SECTION P.6 Introduction to Complex Numbers Addition and Subtraction of Complex Numbers Multiplication of Complex Numbers Division of Complex Numbers Powers of i

59

a. Find the average speed for a round trip by helicopter with

1 2 x + 1 x - 1 56. 1 x + x - 1 2x2 - x - 1

2 - 1 3 + 3

COMPLEX NUMBERS


the sum of the reciprocals of the squares of the consecutive even integers x - 2, x, and x + 2.

Complex Numbers PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A3. In Exercises PS1 to PS5, simplify the expression.

PS1. (2 - 3x)(4 - 5x) [P.3] PS2. (2 - 5x)2 [P.3] PS3. 196 [P.2] PS4. (2 + 315)(3 - 415) [P.2] PS5.

5 + 12 [P.2] 3 - 12

PS6. Which of the following polynomials, if any, does not factor over the integers? [P.4]

a. 81 - x2

b.

9 + z2

60

CHAPTER P


Math Matters It may seem strange to just invent new numbers, but that is how mathematics evolves. For instance, negative numbers were not an accepted part of mathematics until well into the thirteenth century. In fact, these numbers often were referred to as “fictitious numbers.” In the seventeenth century, René Descartes called square roots of negative numbers “imaginary numbers,” an unfortunate choice of words, and started using the letter i to denote these numbers. These numbers were subjected to the same skepticism as negative numbers. It is important to understand that these numbers are not imaginary in the dictionary sense of the word. This misleading word is similar to the situation of negative numbers being called fictitious. If you think of a number line, then the numbers to the right of zero are positive numbers and the numbers to the left of zero are negative numbers. One way to think of an imaginary number is to visualize it as up or down from zero.

Math Matters The imaginary unit i is important in the field of electrical engineering. However, because the letter i is used by engineers as the symbol for electric current, these engineers use j for the complex unit.

Introduction to Complex Numbers Recall that 19 = 3 because 32 = 9. Now consider the expression 1 - 9. To find 1- 9, we need to find a number c such that c 2 = - 9. However, the square of any real number c (except zero) is a positive number. Consequently, we must expand our concept of number to include numbers whose squares are negative numbers. Around the seventeenth century, a new number, called an imaginary number, was defined so that a negative number would have a square root. The letter i was chosen to represent the number whose square is - 1.

Definition of i The imaginary unit, designated by the letter i, is the number such that i 2 = -1.

The principal square root of a negative number is defined in terms of i.

Definition of an Imaginary Number If a is a positive real number, then 1 - a = i1a. The number i1a is called an imaginary number. EXAMPLE

1- 36 = i136 = 6i

1 -18 = i118 = 3i12

1 - 23 = i123

1 -1 = i11 = i

It is customary to write i in front of a radical sign, as we did for i 123, to avoid confusing 1ai with 1ai.

Definition of a Complex Number A complex number is a number of the form a + bi, where a and b are real numbers and i = 1 - 1. The number a is the real part of a + bi, and b is the imaginary part. EXAMPLE

- 3 + 5i

• Real part: -3; imaginary part: 5

2 - 6i

• Real part: 2; imaginary part: - 6

5

• Real part: 5; imaginary part: 0

7i

• Real part: 0; imaginary part: 7

Note from these examples that a real number is a complex number whose imaginary part is zero, and an imaginary number is a complex number whose real part is zero and whose imaginary part is not zero.

P.6

COMPLEX NUMBERS

61

Question • What are the real part and imaginary part of 3 - 5i? Complex numbers a + bi Real numbers a + 0i (b = 0)

Imaginary numbers 0 + bi (a = 0, b ≠ 0)

Note from the diagram at the left that the set of real numbers is a subset of the complex numbers and the set of imaginary numbers is a separate subset of the complex numbers. The set of real numbers and the set of imaginary numbers are disjoint sets. Example 1 illustrates how to write a complex number in the standard form a + bi.

EXAMPLE 1

Write a Complex Number in Standard Form

Write 7 + 1- 45 in the form a + bi.

7 + 1 - 45 = 7 + i 145 = 7 + i 19 # 15 = 7 + 3i 15

Solution


Addition and Subtraction of Complex Numbers All the standard arithmetic operations that are applied to real numbers can be applied to complex numbers.

Definition of Addition and Subtraction of Complex Numbers If a + bi and c + di are complex numbers, then Addition

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtraction

(a + bi) - (c + di) = (a - c) + (b - d)i

Basically, these rules say that to add two complex numbers, add the real parts and add the imaginary parts. To subtract two complex numbers, subtract the real parts and subtract the imaginary parts.

EXAMPLE 2 Simplify.

Add or Subtract Complex Numbers

a. (7 - 2i) + ( -2 + 4i)

b. ( -9 + 4i) - (2 - 6i)

Solution a. (7 - 2i) + ( - 2 + 4i) = (7 + ( - 2)) + ( - 2 + 4)i = 5 + 2i b.

(-9 + 4i) - (2 - 6i) = ( -9 - 2) + (4 - ( - 6))i = - 11 + 10i Try Exercise 18, page 65

Answer • Real part: 3; imaginary part: 5.

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CHAPTER P


Multiplication of Complex Numbers When multiplying complex numbers, the term i 2 is frequently a part of the product. Recall that i 2 = - 1. Therefore, 3i(5i) = 15i 2 = 15(- 1) = -15 -2i(6i) = - 12i 2 = - 12( - 1) = 12 4i(3 - 2i) = 12i - 8i 2 = 12i - 8( - 1) = 8 + 12i Caution Recall that the definition of the product of radical expressions requires that the radicand be a positive number. Therefore, when multiplying expressions containing negative radicands, we first must rewrite the expression using i and a positive radicand.

When multiplying square roots of negative numbers, first rewrite the radical expressions using i. For instance, 1-6 # 1 -24 = i 16 # i 124

= i 2 1144 = - 1 # 12

• 1- 6 = i 16, 1- 24 = i 124

= -12 Note from this example that it would have been incorrect to multiply the radicands of the two radical expressions. To illustrate: 1 - 6 # 1 -24 Z 1( -6)( - 24)

Question • What is the product of 1- 2 and 1- 8?

To multiply two complex numbers, we use the following definition.

Definition of Multiplication of Complex Numbers If a + bi and c + di are complex numbers, then (a + bi)(c + di) = (ac - bd) + (ad + bc)i

Because every complex number can be written as a sum of two terms, it is natural to perform multiplication on complex numbers in a manner consistent with the operation defined on binomials and the definition i 2 = - 1. By using this analogy, you can multiply complex numbers without memorizing the definition.

EXAMPLE 3

Multiply Complex Numbers

Multiply. a.

3i(2 - 5i)

b.

(3 - 4i)(2 + 5i)

Solution a. 3i(2 - 5i) = 6i - 15i 2 = 6i - 15(- 1) = 15 + 6i

#

#

• Replace i 2 with 1. • Write in standard form.

#

Answer • 1 -2 1 -8 = i 12 i 18 = i2 116 = - 1 4 = - 4.

P.6

Integrating Technology Some graphing calculators can be used to perform operations on complex numbers. Here are some typical screens for a TI-83/TI-83 Plus/TI-84 Plus graphing calculator. Press MODE . Use the down arrow key to highlight a + bi. Normal Sci Eng Float 0 12 34 56 7 8 9 Radian Degree Func Par Pol Seq Connected Dot Sequential Simul Real a+bi re^θi Full Horiz

b.

(3 - 4i)(2 + 5i) = 6 + 15i - 8i - 20i 2 = 6 + 15i - 8i - 20(- 1)

COMPLEX NUMBERS

63

• Replace i 2 with 1.

= 6 + 15i - 8i + 20

• Simplify.

= 26 + 7i

• Write in standard form.


Division of Complex Numbers Recall that the number

3

is not in simplest form because there is a radical expression in 12 3 the denominator. Similarly, is not in simplest form because i = 1 - 1. To write this i expression in simplest form, multiply the numerator and denominator by i. 3i 3i 3#i = 2 = = -3i i i -1 i Here is another example.

Press ENTER 2nd [QUIT]. The following screen shows two examples of computations on complex numbers. To enter an i, use 2nd [i], which is located above the decimal point key. (3 –4 i)(2 + 5 i ) 2 6+ 7 i (16 –11i)/(5 + 2 i ) 2 –3 i

3i - 6(- 1) 3 3 - 6i 3 - 6i # i 3i - 6i 2 3i + 6 = = = = = -3 - i 2 2i 2i i 2( -1) -2 2 2i Recall that to simplify the quotient

2 + 13

, we multiply the numerator and denomina5 + 213 tor by the conjugate of 5 + 2 13, which is 5 - 213. In a similar manner, to find the quotient of two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The complex numbers a + bi and a - bi are called complex conjugates or conjugates of each other. The conjugate of the complex number z is denoted by z. For instance, 2 + 5i = 2 - 5i

and

3 - 4i = 3 + 4i

Consider the product of a complex number and its conjugate. For instance, (2 + 5i)(2 - 5i) = 4 - 10i + 10i - 25i 2 = 4 - 25( -1) = 4 + 25 = 29 Note that the product is a real number. This is always true.

Product of Complex Conjugates The product of a complex number and its conjugate is a real number. That is, (a + bi)(a - bi) = a2 + b2. EXAMPLE

(5 + 3i)(5 - 3i) = 52 + 32 = 25 + 9 = 34

The next example shows how the quotient of two complex numbers is determined by using conjugates.

64

CHAPTER P


EXAMPLE 4 Simplify:

Divide Complex Numbers

16 - 11i 5 + 2i

Solution 16 - 11i 16 - 11i # 5 - 2i = 5 + 2i 5 + 2i 5 - 2i 80 - 32i - 55i + 22i 2 = 52 + 22 80 - 32i - 55i + 22( -1) = 25 + 4 80 - 87i - 22 = 29 58 - 87i = 29 29(2 - 3i) = 2 - 3i = 29

• Multiply the numerator and denominator by the conjugate of the denominator.


Powers of i The following powers of i illustrate a pattern: i1 i2 i3 i4

= = = =

i5 i6 i7 i8

i -1 i 2 # i = ( - 1)i = - i i 2 # i 2 = ( - 1)( -1) = 1

= = = =

i4 # i = 1 # i = i i 4 # i 2 = 1(- 1) = - 1 i 4 # i 3 = 1(- i) = - i (i 4)2 = 12 = 1

Because i 4 = 1, (i 4)n = 1n = 1 for any integer n. Thus it is possible to evaluate powers of i by factoring out powers of i 4, as shown in the following. i 27 = (i 4)6 # i 3 = 16 # i 3 = 1 # ( - i) = - i

The following theorem can also be used to evaluate powers of i.

Powers of i If n is a positive integer, then i n = i r, where r is the remainder of the division of n by 4.

EXAMPLE 5

Evaluate a Power of i

Evaluate: i 153 Solution Use the powers of i theorem. i 153 = i1 = i Try Exercise 60, page 65

• Remainder of 153 , 4 is 1.

P.6

COMPLEX NUMBERS

65

EXERCISE SET P.6 In Exercises 1 to 10, write the complex number in standard form. 1. 1-81

39.

6 + 3i i

40.

4 - 8i 4i

41.

1 7 + 2i

42.

5 3 + 4i

43.

2i 1 + i

44.

5i 2 - 3i

45.

5 - i 4 + 5i

46.

4 + i 3 + 5i

47.

3 + 2i 3 - 2i

48.

8 - i 2 + 3i

49.

- 7 + 26i 4 + 3i

50.

- 4 - 39i 5 - 2i

2. 1- 64

3. 1-98

4. 1- 27

5. 116 + 1-81

6. 125 + 1 -9

7. 5 + 1-49

8. 6 - 1 -1

9. 8 - 1-18

10. 11 + 1 - 48

In Exercises 11 to 36, simplify and write the complex number in standard form. 11. (5 + 2i) + (6 - 7i)

12. (4 - 8i) + (5 + 3i)

13. ( -2 - 4i) - (5 - 8i)

14. (3 - 5i) - (8 - 2i)

15. (1 - 3i) + (7 - 2i)

16. (2 - 6i) + (4 - 7i)

17. ( -3 - 5i) - (7 - 5i)

18. (5 - 3i) - (2 + 9i)

19. 8i - (2 - 8i)

20. 3 - (4 - 5i)

21. 5i # 8i

22. (- 3i)(2i)

23. 1-50 # 1-2

51. (3 - 5i)2

52. (2 + 4i)2

53. (1 + 2i)3

54. (2 - i)3

In Exercises 55 to 62, evaluate the power of i. 55. i 15

56. i 66

24. 1- 12 # 1- 27

57. - i 40

58. - i 51

25. 3(2 + 5i) - 2(3 - 2i)

26. 3i(2 + 5i) + 2i(3 - 4i)

59.

27. (4 + 2i)(3 - 4i)

28. (6 + 5i)(2 - 5i)

29. ( -3 - 4i)(2 + 7i)

30. (-5 - i)(2 + 3i)

31. (4 - 5i)(4 + 5i)

32. (3 + 7i)(3 - 7i)

33. (3 + 1- 4)(2 - 1 -9) 34. (5 + 21-16)(1 - 1 -25)

61. i -34

36. (5 - 31 -48)(2 - 41- 27)

In Exercises 37 to 54, write each expression as a complex number in standard form. -8 38. 2i

60.

1 i 83

62. i -52

In Exercises 63 to 68, evaluate

b

2b

2

4ac

for the 2a given values of a, b, and c. Write your answer as a complex number in standard form. 63. a = 3, b = - 3, c = 3

35. (3 + 21-18)(2 + 21- 50)

6 37. i

1 i 25

64. a = 2, b = 4, c = 4 65. a = 2, b = 6, c = 6 66. a = 2, b = 1, c = 3 67. a = 4, b = - 4, c = 2 68. a = 3, b = - 2, c = 4

66

CHAPTER P



Can You Trust Your Calculator? You may think that your calculator always produces correct results in a predictable manner. However, the following experiment may change your opinion. First note that the algebraic expression p + 3p(1 - p) is equal to the expression 4p - 3p2

Integrating Technology To perform the iterations at the right with a TI graphing calculator, first store 0.05 in p and then store p + 3p(1 - p) in p, as shown below. 0.05->p .05 p+3p(1–p)->p .1925

Each time you press ENTER , the expression p + 3p(1 - p) will be evaluated with p equal to the previous result. 0.05->p .05 p+3p(1–p)->p .1925 .65883125 1.33314915207 7.366232839E-4

Use a graphing calculator to evaluate both expressions with p = 0.05. You should find that both expressions equal 0.1925. So far we do not observe any unexpected results. Now replace p in each expression with the current value of that expression (0.1925 in this case). This is called feedback because we are feeding our output back into each expression as input. Each new evaluation is referred to as an iteration. This time each expression takes on the value 0.65883125. Still no surprises. Continue the feedback process. That is, replace p in each expression with the current value of that expression. Now each expression takes on the value 1.33314915207, as shown in the following table. The iterations were performed on a TI-85 calculator. Iteration

p 3p(1 p)

4p 3p 2

1

0.1925

0.1925

2

0.65883125

0.65883125

3

1.33314915207

1.33314915207

The following table shows that if we continue this feedback process on a calculator, the expressions p + 3p(1 - p) and 4p - 3p2 will start to take on different values beginning with the fourth iteration. By the 37th iteration, the values do not even agree to two decimal places. Iteration

p 3p(1 p)

4p 3p 2

4

7.366232839E-4

7.366232838E-4

5

0.002944865294

0.002944865294

6

0.011753444481

0.0117534448

7

0.046599347553

0.046599347547

20

1.12135618652

1.12135608405

30

0.947163304835

0.947033128433

37

0.285727963839

0.300943417861

1. Use a calculator to find the first 20 iterations of p + 3p(1 - p) and 4p - 3p2, with the initial value of p = 0.5. 2. Write a report on chaos and fractals. Include information on the “butterfly effect.” An excellent source is Chaos and Fractals, New Frontiers of Science by Heinz-Otto Peitgen, Hartmut Jurgens, and Dietmar Saupe (New York: Springer-Verlag, 1992). 3. Equations of the form pn + 1 = pn + rpn(1 - pn) are called Verhulst population models. Write a report on Verhulst population models.

CHAPTER P TEST PREP

67

CHAPTER P TEST PREP The following test prep table summarizes essential concepts in this chapter. The references given in the right-hand column list Examples and Exercises that can be used to test your understanding of a concept.

P.1 The Real Number System The following sets of numbers are used extensively in algebra: Natural numbers {1, 2, 3, 4, Á } Integers { Á , -3, -2, -1, 0, 1, 2, 3, Á } Rational numbers {all terminating and repeating decimals} Irrational numbers {all nonterminating, nonrepeating decimals} Real numbers {all rational or irrational numbers}

See Example 1, page 3, and then try Exercises 1 and 2, page 70.

Set-builder notation is a method of writing sets that has the form {variable | condition on the variable}.

See Example 2, page 4, and then try Exercise 5, page 70.

The union of two sets A and B is the set of all elements that belong to either A or B. The intersection of two sets A and B is the set of all elements that belong to both A and B.


Sets of real numbers can be written in interval notation. Page 6 shows the various forms of interval notation.

See Examples 4 and 5, pages 6 and 7, and then try Exercises 9 and 12, page 70.

a if a Ú 0 - a if a 6 0


The distance d(a, b) between two points a and b on a real number line is given by d(a, b) = ƒ a - b ƒ .


The absolute value of a real number a is given by ƒ a ƒ = e

f

If b is any real number and n is a natural number, then bn = b # b # b # Á

# b.

b is a factor n times


The Order of Operations Agreement specifies the order in which operations must be performed. See page 10.


To evaluate a variable expression, replace the variables with their given values. Then use the Order of Operations Agreement to simplify the result.


The properties of real numbers are used to simplify variable expressions. See page 12.


Four properties of equality are symmetric, reflexive, transitive, and substitution.


P.2 Integer and Rational Number Exponents If b Z 0, then b0 = 1. If b Z 0 and n is a natural number, then b - n

1 1 = n and - n = bn. b b


68

CHAPTER P


If n is an even positive integer and b Ú 0, then b1>n is the nonnegative real number such that (b1>n)n = b. If n is an odd positive integer, then b1>n is the real number such that (b1>n)n = b. For all positive integers m and n such that m>n is in simplest form, and for all real number b for which b1>n is a real number, bm>n = (b1>n)m = (bm)1>n.


Properties of Rational Exponents If p, q, and r are rational numbers and a and b are positive real numbers, then


Product Quotient Power

b p # bq = b p + q bp = bp-q bq (b p )q = b pq ap r a pr a q b = qr b b

See Example 5, page 23, and then try Exercises 55 and 57, page 71. (a pb q)r = a prb qr 1 b-p = p b

A number written in scientific notation has the form a * 10n, where 1 … a 6 10 and n is an integer.


Properties of Radicals If m and n are natural numbers, and a and b are positive real numbers, then


Product

1a # 1b = 1ab

Quotient

1a n a = n Ab 1b

Index

21b = 2b

n

n

n

n

m n

mn

A radical expression is in simplest form if it meets the criteria listed on page 25.

See Example 7, page 26, and then try Exercise 63, page 71. See Example 8, page 26, and then try Exercises 65 and 68, page 71.

To rationalize the denominator of a fraction means to write the fraction as an equivalent fraction that does not involve any radicals in the denominator.

See Examples 9 and 10, pages 27 and 28, and then try Exercises 70 and 71, pages 71 and 72.

P.3 Polynomials The standard form of a polynomial of degree n is an expression of the form an x + an - 1x n

n-1

+ Á + a1x + a0


where n is a natural number and an Z 0. The leading coefficient is an, and a0 is the constant term. The properties of real numbers are used to perform operations on polynomials.

See Example 2, page 33, and then try Exercise 75, page 72. See Example 3, page 34, and then try Exercise 78, page 72. See Example 4, page 35, and then try Exercise 80, page 72.

CHAPTER P TEST PREP

Special product formulas are as follows.


Special Form

Formula(s)

(Sum)(Difference)

(x + y)(x - y) = x 2 - y 2

(Binomial)2

(x + y)2 = x 2 + 2xy + y 2 (x - y)2 = x 2 - 2xy + y 2

P.4 Factoring The greatest common factor (GCF) of a polynomial is the product of the GCF of the coefficients of the polynomial and the monomial of greatest degree that is a factor of each term of the polynomial.


Some trinomials of the form ax2 + bx + c can be factored over the integers as the product of two binomials.

See Example 2, page 41, and then try Exercise 88, page 72. See Example 3, page 42, and then try Exercise 90, page 72.

Some special factoring formulas are as follows.

See Example 5, page 44, and then try Exercise 93, page 72. See Example 6, page 44, and then try Exercise 94, page 72. See Example 7, page 45, and then try Exercise 98, page 72.

Special Form

Formula(s)

Difference of squares

x - y = (x + y)(x - y)

Perfect-square trinomials

x2 + 2xy + y2 = (x + y)2 x 2 - 2xy + y 2 = (x - y)2

Sum of cubes

x3 + y 3 = (x + y)(x 2 - xy + y 2)

Difference of cubes

x3 - y3 = (x - y)(x 2 + xy + y 2)

2

2

A polynomial that can be written as au2 + bu + c is said to be quadratic in form. A strategy that is similar to that of factoring a quadratic trinomial can be used to factor some of these polynomials.


Factoring by grouping may be helpful for polynomials with four or more terms.


Use the general factoring strategy given on page 47 to factor a polynomial.


P.5 Rational Expressions A rational expression is a fraction in which the numerator and denominator are polynomials. A rational expression is in simplest form when 1 is the only common factor of the numerator and denominator.


Operations on Rational Expressions • To multiply rational expressions, multiply numerators and multiply denominators. • To divide rational expressions, invert the divisor and then multiply the rational expressions. • To add or subtract rational expressions, write each expression in terms of a common denominator. Then perform the indicated operation.

See Example 2, page 52, and then try Exercise 105, page 72. See Example 3, page 52, and then try Exercise 106, page 72. See Example 4, page 53, and then try Exercises 107 and 108, page 72.

69

70

CHAPTER P


A complex fraction is a fraction whose numerator or denominator contains one or more fractions. There are two basic methods for simplifying a complex fraction. Method 1: Multiply both the numerator and the denominator by the least common denominator of all fractions in the complex fraction. Method 2: Simplify the numerator to a single fraction and the denominator to a single fraction. Multiply the numerator by the reciprocal of the denominator.

See Example 6, page 55, then try Exercises 109 and 110, page 72.

P.6 Complex Numbers The imaginary unit, designated by the letter i, is the number such that i 2 = - 1. If a is a positive real number, then 1- a = i1a. The number i 1a is called an imaginary number. A complex number is one of the form a + bi , where a and b are real numbers and i is the imaginary unit. The real part of the complex number is a; the imaginary part of the complex number is b.


Operations on Complex Numbers • To add or subtract two complex numbers, add or subtract the real parts and add or subtract the imaginary parts. • To multiply two complex numbers, use the FOIL method (first, outer, inner, last) and the fact that i2 = - 1. • To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

See Example 2, page 61, and then try Exercises 113 and 114, page 72. See Example 3, page 62, and then try Exercise 116, page 72. See Example 4, page 64, and then try Exercise 120, page 72.

Powers of i If n is a positive integer, then i n = i r, where r is the remainder when n is divided by 4.


CHAPTER P REVIEW EXERCISES In Exercises 1 to 4, classify each number as one or more of the following: integer, rational number, irrational number, real number, prime number, composite number. 1. 3

2. 2 7

1 3. 2

In Exercises 9 and 10, graph each interval and write the interval in set-builder notation. 10. (- 1, q )

9. [-3, 2)

4. 0.5

In Exercises 5 and 6, list the four smallest elements of the set. 5. {y ƒ y = x 2, x integers}

In Exercises 11 and 12, graph each set and write the set in interval notation. 11. {x ƒ -4 6 x … 2}

12. {x ƒ x … - 1} ´ {x ƒ x 7 3}

6. {y ƒ y = 2x + 1, x natural numbers}

In Exercises 13 to 18, write each expression without absolute value symbols. In Exercises 7 and 8, use A {1, 5, 7} and B {2, 3, 5, 11} to find the indicated intersection or union. 7. A ´ B

8. A ¨ B

13. ƒ 7 ƒ

14. ƒ 2 - p ƒ

15. ƒ 4 - p ƒ

17. ƒ x - 2 ƒ + ƒ x + 1 ƒ , -1 6 x 6 2

16. ƒ - 11 ƒ

CHAPTER P REVIEW EXERCISES

18. ƒ 2x + 3 ƒ - ƒ x - 4 ƒ , - 3 … x … - 2 19. If -3 and 7 are the coordinates of two points on the real number

line, find the distance between the two points. 20. If a = 4 and b = - 1 are the coordinates of two points on the

real number line, find d(a, b).

In Exercises 41 and 42, write each number in scientific notation. 41. 620,000

42. 0.0000017

In Exercises 43 and 44, change each number from scientific notation to decimal form. 44. 4.31 * 10-7

43. 3.5 * 104

In Exercises 21 to 24, evaluate the expression. 21. -44

22. -42(- 3)2

23. -5 # 32 + 455 - 23-6 - (- 4)46 24. 6 - 2 c4 -

( -5)2 - 29 - 22

d

In Exercises 25 and 26, evaluate the variable expressions for x 2, y 3, and z 5. 25. -3x 3 - 4xy - z 2

26. 2x - 3y(4z - x3)

In Exercises 27 to 34, identify the real number property or property of equality that is illustrated.

In Exercises 45 to 48, evaluate each exponential expression. 45. 251>2

46. - 27 2>3

47. 36 - 1>2

48.

29. (6c)d = 6(cd)

3 81 - 1>4

In Exercises 49 to 58, simplify the expression. 12a5b

49. ( -4x3y2)(6x4y3)

50.

51. ( -3x - 2y3) - 3

52. a

53. ( -4x - 3y2) - 2(8x - 2y - 3)2

54.

55. (x - 1>2)(x3>4)

56.

27. 5(x + 3) = 5x + 15 28. a(3 + b) = a(b + 3)

18a3b6

57. a

8x5/4

32. 1x = x 33. If 7 = x, then x = 7.

In Exercises 35 and 36, simplify the variable expression. 35. 8 - 3(2x - 5)

z

-4

6a - 3b

(4x - 3y4) - 2 a2>3b - 3>4 a5>6b2

58. a

x 2y x1/2y

b -3

1>2

59. 248a2b7

60. 212a3b

3

3

61. 2 - 135x2y7

62. 2 - 250xy6

3

38. -

40.

3

64. 3x 216x5y10 - 4y 2 22x8y4

In Exercises 37 to 40, simplify the exponential expression.

2

-2

63. b28a4b3 + 2a218a2b5

36. 5x - 3[7 - 2(6x - 7) - 3x]

39.

2>3

b 6

In Exercises 59 to 72, simplify each radical expression. Assume that the variables are positive real numbers.

34. If 3x + 4 = y and y = 5z, then 3x + 4 = 5z.

37. -2 - 5

x1/2

b

2a2b - 4

( - 2x4y - 5) - 3

30. 12 + 3 is a real number. 31. 7 + 0 = 7

x-4 y

65. (3 + 2 15)(7 - 315)

66. (5 12 - 7)(3 12 + 6)

67. (4 - 217)2

68. (2 - 31x)2

1 p0

-3

71

69.

6 18

70.

9 3

19x

72

71.

CHAPTER P


3 + 217

72.

9 - 327

99. 4x4 - x 2 - 4x 2y 2 + y 2

5 21x - 3

100. 2a 3 + a 2b - 2ab2 - b3

73. Write the polynomial 4x - 7x + 5 - x in standard form. 2

3

Identify the degree, the leading coefficient, and the constant term.

101. 24a2b2 - 14ab3 - 90b4 102. 3x5y2 - 9x3y2 - 12xy2

74. Evaluate the polynomial 3x - 4x + 2x - 1 when x = - 2. 3

2

In Exercises 75 to 82, perform the indicated operation and express each result as a polynomial in standard form. 75. (2a 2 + 3a - 7) + (-3a 2 - 5a + 6) 76. (5b2 - 11) - (3b2 - 8b - 3)

103.

6x2 - 19x + 10

104.

2x + 3x - 20 2

4x 3 - 25x 8x 4 + 125x

In Exercises 105 to 108, perform the indicated operation and simplify, if possible.

77. (3x - 2)(2x 2 + 4x - 9)

105.

78. (4y - 5)(3y - 2y - 8) 3

In Exercises 103 and 104, simplify each rational expression.

2

79. (3x - 4)(x + 2)

106.

10x2 + 13x - 3 # 6x2 + 5x + 1 6x2 - 13x - 5

15x2 + 11x - 12

107.

x x2 - 9

82. (4x - 5y)(4x + 5y) 108.

,

25x2 - 9

80. (5x + 1)(2x - 7) 81. (2x + 5)2

10x2 + 3x - 1

+

3x2 + 13x + 12 10x2 + 11x + 3

2x x2 + x - 12

3x x2 + 7x + 12

-

x 2x2 + 5x - 3

In Exercises 83 to 86, factor out the GCF. 83. 12x3y4 + 10x2y3 - 34xy2

In Exercises 109 and 110, simplify each complex fraction.

84. 24a4b3 + 12a3b4 - 18a2b5

1 x - 5 109. 2 3 x - 5 2 +

85. (2x + 7)(3x - y) - (3x + 2)(3x - y) 86. (5x + 2)(3a - 4) - (3a - 4)(2x - 6)

In Exercises 87 to 102, factor the polynomial over the integers. 87. x 2 + 7x - 18

88. x 2 - 2x - 15

89. 2x 2 + 11x + 12

90. 3x 2 - 4x - 15

93. 9x - 100

94. 25x - 30xy + 9y

95. x - 5x - 6

96. x + 2x - 3

97. x - 27

98. 3x + 192

4

3

2

4

1 +

112. 2 - 1-18

113. (2 - 3i) + (4 + 2i)

114. (4 + 7i) - (6 - 3i)

115. 2i(3 - 4i)

116. (4 - 3i)(2 + 7i)

117. (3 + i)2

118. i345

2

2

3

4 x

In Exercises 113 to 120, perform the indicated operation and write the answer in simplest form.

92. - 2a4b3 - 2a3b3 + 12a 2b3 2

3

2 +

In Exercises 111 and 112, write the complex number in standard form. 111. 5 + 1- 64

91. 6x 3y 2 - 12x 2y 2 - 144xy 2

2

1

110.

119.

4 - 6i 2i

120.

2 - 5i 3 + 4i

CHAPTER P TEST

CHAPTER P TEST 1. For real numbers a, b, and c, identify the property that is illus-

trated by (a + b)c = ac + bc. 2. Graph {x ƒ -3 … x 6 4} and write the set in interval notation. 3. Given -1 6 x 6 4, simplify ƒ x + 1 ƒ - ƒ x - 5 ƒ . 0 -2 2

2 -1 -2

17. Factor: 7x 2 + 34x - 5 18. Factor: 3ax - 12bx - 2a + 8b 19. Factor: 16x 4 - 2xy 3 20. Factor: x4 - 15x 2 - 16

4. Simplify: ( -2x y ) (- 3x y ) 5. Simplify:

21. Simplify:

(2a-1bc-2)2 -1

-1

-2 3

22. Simplify:

x1>3y-3>4 x

23. Multiply:

-1>2 3>2

y

3

3

8. Simplify: 3x281xy4 - 2y23x4y 24. Simplify: 9. Simplify: (2 13 - 4)(5 13 + 2) 10. Simplify: (2 - 1x + 4)2 11. Simplify:

25 - x 2

(3 b)(2 ac )

6. Write 0.00137 in scientific notation. 7. Simplify:

x 2 - 2x - 15

x 4

2 + 15 13. Simplify: 4 - 2 15 14. Subtract: (3x - 2x - 5) - (2x + 4x - 7) 3

2

2

x 2 + x - 20

2x 2 + 3x - 2 x 2 - 3x

2 x 2 - 5x + 6

x 2 + 2x - 8 ,

2x 2 - 7x + 3 x 3 - 3x 2

x x +

1 2

26. Write 7 + 1-20 in standard form.

In Exercises 27 to 30, write the complex number in simplest form. 27. (4 - 3i) - (2 - 5i)

15. Multiply: (3a + 7b)(2a - 9b) 16. Multiply: (2x + 5)(3x2 - 6x - 2)

-

x 2 - 3x - 4 # x 2 + 3x - 10

25. Simplify: x -

3 12. Simplify: 2 - 317

22x3

x x2 + x - 6

29.

3 + 4i 5 - i

28. (2 + 5i)(1 - 4i) 30. i 97

73


CHAPTER

1

EQUATIONS AND INEQUALITIES a F b

1.1 Linear and Absolute Value Equations 1.2 Formulas and Applications 1.3 Quadratic Equations 1.4 Other Types of Equations 1.5 Inequalities

a

1.6 Variation and Applications

Pegaz/Alamy

b

Cheops was the second king of the Fourth Egyptian Dynasty and ruled from 2551–2578 B.C. By some accounts, he was the first Egyptian king to be mummified.

The Golden Mean There is a common theme among the following: a stage for the 2000 Olympic Games in Sydney, Australia; the giant pyramid of Cheops in Egypt; the Parthenon in Athens, Greece; and the Mona Lisa, which hangs in the Louvre in Paris, France. All of these are related through a number called the golden mean, symbolized by f. The golden mean is thought to be an aesthetically pleasing ratio—thus its popularity in art and architecture. Exercises 82 and 83 on page 122 give a method for calculating the golden mean. Generally, this number manifests itself as a ratio of sides of geometric figures. The photo above shows how the golden mean is represented in the structure of the Cheops pyramid.

75

76

CHAPTER 1

EQUATIONS AND INEQUALITIES

SECTION 1.1 Linear Equations Contradictions, Conditional Equations, and Identities Absolute Value Equations Applications of Linear Equations

Linear and Absolute Value Equations Linear Equations An equation is a statement about the equality of two expressions. If either of the expressions contains a variable, the equation may be a true statement for some values of the variable and a false statement for other values. For example, the equation 2x + 1 = 7 is a true statement for x = 3, but it is false for any number except 3. The number 3 is said to satisfy the equation 2x + 1 = 7 because substituting 3 for x produces 2(3) + 1 = 7, which is a true statement. To solve an equation means to find all values of the variable that satisfy the equation. The values that satisfy an equation are called solutions or roots of the equation. For instance, 2 is a solution of x + 3 = 5. Equivalent equations are equations that have exactly the same solution or solutions. The process of solving an equation is often accomplished by producing a sequence of equivalent equations until we arrive at an equation or equations of the form Variable = Constant To produce these equivalent equations, apply the properties of real numbers and the following two properties of equality.

Addition and Subtraction Property of Equality Adding the same expression to each side of an equation or subtracting the same expression from each side of an equation produces an equivalent equation. EXAMPLE

Begin with the equation 2x - 7 = 11. Replacing x with 9 shows that 9 is a solution of the equation. Now add 7 to each side of the equation. The resulting equation is 2x = 18, and the solution of the new equation is still 9.

Multiplication and Division Property of Equality Multiplying or dividing each side of an equation by the same nonzero expression produces an equivalent equation. EXAMPLE

2 x = 8. Replacing x with 12 shows that 12 is a solution of 3 3 the equation. Now multiply each side of the equation by . The resulting equation is 2 x = 12, and the solution of the new equation is still 12. Begin with the equation

Many applications can be modeled by linear equations in one variable.

1.1

LINEAR AND ABSOLUTE VALUE EQUATIONS

77

Definition of a Linear Equation A linear equation, or first-degree equation, in the single variable x is an equation that can be written in the form ax + b = 0 where a and b are real numbers, with a Z 0.

Linear equations are solved by applying the properties of real numbers and the properties of equality.

EXAMPLE 1

Solve a Linear Equation in One Variable

Solve: 3x - 5 = 7x - 11 Study tip You should check a proposed solution by substituting it back into the original equation. 3x - 5 = 7x - 11 3 3 3a b - 5 ⱨ 7a b - 11 2 2 9 21 - 5ⱨ - 11 2 2 -

1 1 = 2 2

Solution 3x - 5 3x - 7x - 5 -4x - 5 - 4x - 5 + 5 -4x -4x -4

= = = = =

7x - 11 7x - 7x - 11 - 11 - 11 + 5 -6 -6 = -4

x =

3 2

• Subtract 7x from each side of the equation. • Add 5 to each side of the equation.

• Divide each side of the equation by 4. • The equation is now in the form Variable Constant.

As shown to the left,

3 3 satisfies the original equation. The solution is . 2 2


When an equation contains parentheses, use the distributive property to remove the parentheses.

EXAMPLE 2

Solve a Linear Equation in One Variable

Solve: 8 - 5(2x - 7) = 3(16 - 5x) + 5 Solution 8 - 5(2x - 7) 8 - 10x + 35 -10x + 43 -10x + 15x + 43

= = = =

3(16 - 5x) + 5 48 - 15x + 5 - 15x + 53 - 15x + 15x + 53

• Use the distributive property. • Simplify. • Add 15x to each side of the equation. (continued)

78

CHAPTER 1


5x + 43 = 53 5x + 43 - 43 = 53 - 43

• Subtract 43 from each side of the equation.

5x = 10 10 5x = 5 5 x = 2

• Divide each side of the equation by 5. • Check in the original equation.

The solution is 2. Try Exercise 8, page 81

If an equation involves fractions, it is helpful to multiply each side of the equation by the least common denominator (LCD) of all denominators to produce an equivalent equation that does not contain fractions.

EXAMPLE 3 Solve:

Solve by Clearing Fractions

x 36 2 x + 10 - = 3 5 5

Solution x 36 2 x + 10 = 3 5 5 x 36 2 15a x + 10 - b = 15a b 3 5 5

• Multiply each side of the equation by 15, the LCD of all denominators.

10x + 150 - 3x = 108 7x + 150 = 108

• Simplify.

7x + 150 - 150 = 108 - 150 7x = - 42

• Subtract 150 from each side.

- 42 7x = 7 7 x = -6

• Divide each side by 7. • Check in the original equation.

The solution is - 6. Try Exercise 14, page 81

Contradictions, Conditional Equations, and Identities An equation that has no solutions is called a contradiction. The equation x = x + 1 is a contradiction. No number is equal to itself increased by 1. An equation that is true for some values of the variable but not true for other values of the variable is called a conditional equation. For example, x + 2 = 8 is a conditional equation because it is true for x = 6 and false for any number not equal to 6.

1.1


79

An identity is an equation that is true for all values of the variable for which all terms of the equation are defined. Examples of identities include the equations x + x = 2x and 4(x + 3) - 1 = 4x + 11.

EXAMPLE 4

Classify Equations

Classify each equation as a contradiction, a conditional equation, or an identity. a.

x + 1 = x + 4

b.

4x + 3 = x - 9

c.

5(3x - 2) - 7(x - 4) = 8x + 18

Solution a. Subtract x from both sides of x + 1 = x + 4 to produce the equivalent equation 1 = 4. Because 1 = 4 is a false statement, the original equation x + 1 = x + 4 has no solutions. It is a contradiction. b.

Solve using the procedures that produce equivalent equations. 4x + 3 3x + 3 3x x

= = = =

x - 9 -9 - 12 -4

• Subtract x from each side. • Subtract 3 from each side. • Divide each side by 3.

Check to confirm that -4 is a solution. The equation 4x + 3 = x - 9 is true for x = - 4, but it is not true for any other values of x. Thus 4x + 3 = x - 9 is a conditional equation. c.

Simplify the left side of the equation to show that it is identical to the right side. 5(3x - 2) - 7(x - 4) = 8x + 18 15x - 10 - 7x + 28 = 8x + 18 8x + 18 = 8x + 18 The original equation 5(3x - 2) - 7(x - 4) = 8x + 18 is true for all real numbers x. The equation is an identity. Try Exercise 24, page 82

Question • Dividing each side of x = 4x by x produces 1 = 4. Are the equations x = 4x and

1 = 4 equivalent equations?

Absolute Value Equations −5 −4 −3 −2 −1

0

x

1

3

Figure 1.1

2

3

4

5

The absolute value of a real number x is the distance between the number x and the number 0 on the real number line. Thus the solutions of ƒ x ƒ = 3 are all real numbers that are 3 units from 0. Therefore, the solutions of ƒ x ƒ = 3 are x = 3 or x = - 3. See Figure 1.1. The following property is used to solve absolute value equations. Answer • No. The real number 0 is a solution of x = 4x, but 0 is not a solution of 1 = 4.

80

CHAPTER 1


A Property of Absolute Value Equations For any variable expression E and any nonnegative real number k, ƒEƒ = k

Note Some absolute value equations have no solutions. For example, ƒ x + 2 ƒ = - 5 is false for all values of x. Because an absolute value is always nonnegative, the equation is never true.

if and only if

E = k or

E = -k

EXAMPLE

If ƒ x ƒ = 5, then x = 5 or x = - 5. 3 3 3 If ƒ x ƒ = , then x = or x = - . 2 2 2 If ƒ x ƒ = 0, then x = 0.

EXAMPLE 5

Solve an Absolute Value Equation

Solve: ƒ 2x - 5 ƒ = 21 Solution ƒ 2x - 5 ƒ = 21 implies 2x - 5 = 21 or 2x - 5 = - 21. Solving each of these linear equations produces 2x - 5 = 21 2x = 26 x = 13

or

2x - 5 = - 21 2x = - 16 x = -8

The solutions are -8 and 13. Try Exercise 38, page 82

Applications of Linear Equations Linear equations often can be used to model real-world data.

EXAMPLE 6

Table 1.1 Average U.S. Movie Theater Ticket Price

Movie Theater Ticket Prices

Movie theater ticket prices have been increasing steadily in recent years (see Table 1.1). An equation that models the average U.S. movie theater ticket price p, in dollars, is given by p = 0.211t + 5.998

2003

6.03

where t is the number of years after 2003. (This means that t = 0 corresponds to 2003.) Use this equation to predict the year in which the average U.S. movie theater ticket price will reach $7.50.

2004

6.21

Solution

2005

6.41

2006

6.55

2007

6.88

2008

7.08

Year

Price (in dollars)

Source: National Association of Theatre Owners, http://www. natoonline.org/statisticstickets.htm.

p 7.50 1.502 t

= = = L

0.211t + 5.998 0.211t + 5.998 0.211t 7.1

• Substitute 7.50 for p. • Solve for t.

Our equation predicts that the average U.S. movie theater ticket price will reach $7.50 about 7.1 years after 2003, which is 2010. Try Exercise 50, page 82

1.1

EXAMPLE 7


81

Driving Time

Alicia is driving along a highway that passes through Centerville (see Figure 1.2). Her distance d, in miles, from Centerville is given by the equation

Highway

d = ƒ 135 - 60t ƒ Starting point

Centerville

where t is the time in hours since the start of her trip and 0 … t … 5. Determine when Alicia will be exactly 15 miles from Centerville.

Figure 1.2

Solution Substitute 15 for d. d = ƒ 135 - 60t ƒ 15 = ƒ 135 - 60t ƒ 15 = 135 - 60t

or

- 15 = 135 - 60t

- 120 = - 60t

- 150 = - 60t

2 = t

5 = t 2

• Solve for t.

Alicia will be exactly 15 miles from Centerville after she has driven for 2 hours and after 1 she has driven for 2 hours. 2 Try Exercise 52, page 82

EXERCISE SET 1.1 In Exercises 1 to 22, solve each equation and check your solution.

13.

2 1 x - 5 = x - 3 3 2

14.

1 1 19 x + 7 - x = 2 4 2

1. 2x + 10 = 40 2. - 3y + 20 = 2 3. 5x + 2 = 2x - 10

15. 0.2x + 0.4 = 3.6

4. 4x - 11 = 7x + 20

16. 0.04x - 0.2 = 0.07

5. 2(x - 3) - 5 = 4(x - 5)

17. x + 0.08(60) = 0.20(60 + x)

6. 5(x - 4) - 7 = - 2(x - 3)

18. 6(t + 1.5) = 12t

7. 3x + 5(1 - 2x) = 4 - 3(x + 1)

19. 53x - (4x - 5)4 = 3 - 2x

8. 6 - 2(4x + 1) = 3x - 2(2x + 5)

20. 633y - 2(y - 1)4 - 2 + 7y = 0

9. 4(2r - 17) + 5(3r - 8) = 0 10. 6(5s - 11) - 12(2s + 5) = 0 11.

3 1 2 x + = 4 2 3

12.

x 1 - 5 = 4 2

21.

6x + 7 40 - 3x = 5 8

22.

12 + x 5x - 7 = + 2 -4 3

82

CHAPTER 1


In Exercises 23 to 32, classify each equation as a contradiction, a conditional equation, or an identity. 23. - 3(x - 5) = - 3x + 15 24. 2x +

6x + 1 1 = 3 3

49. Biology The male magnificent frigatebird inflates a red pouch

under his neck to attract females. Along with the inflated pouch, the bird makes a drumming-like sound whose frequency F, in hertz, is related to the volume V, in cubic centimeters, of the pouch by the equation F = - 5.5V + 5400.

25. 2x + 7 = 3(x - 1) 26. 432x - 5(x - 3)4 = 6

4x + 8 = x + 8 4

Dreamstime LLC

27.

28. 33x - (4x - 1)4 = - 3(2x - 5) 29. 33x - 2(x - 5)4 - 1 = - 3x + 29 30. 433(x - 5) + 74 = 12x - 32

50. Health According to one formula for lean body mass (LBM, in

31. 2x - 8 = - x + 9

kilograms) given by R. Hume, the mass of the body minus fat is LBM = 0.3281W + 0.3393H - 29.5336

32. ƒ 3(x - 4) + 7 ƒ = ƒ 3x - 5 ƒ

In Exercises 33 to 48, solve each absolute value equation for x. 33. ƒ x ƒ = 4

34. ƒ x ƒ = 7

35. ƒ x - 5 ƒ = 2

36. ƒ x - 8 ƒ = 3

37. ƒ 2x - 5 ƒ = 11

38. ƒ 2x - 3 ƒ = 21

39. ƒ 2x + 6 ƒ = 10

40. ƒ 2x + 14 ƒ = 60

41. `

x - 4 ` = 8 2

42. `

x + 3 ` = 6 4

Use the equation to estimate the volume of the pouch when the frequency of the sound is 550 hertz. Round to the nearest cubic centimeter.

where W is a person’s weight in kilograms and H is the person’s height in centimeters. If a person is 175 centimeters tall, what should that person weigh to have an LBM of 55 kilograms? Round to the nearest kilogram. 51. Travel Ruben is driving along a highway that passes through

Barstow. His distance d, in miles, from Barstow is given by the equation d = ƒ 210 - 50t ƒ , where t is the time, in hours, since the start of his trip and 0 … t … 6. When will Ruben be exactly 60 miles from Barstow? 52. Automobile Gas Mileage The gas mileage m, in miles per

gallon, obtained during a long trip is given by m = -

1 ƒ s - 55 ƒ + 25 2

where s is the speed of Kate’s automobile in miles per hour and 40 … s … 70. At what constant speed can Kate drive to obtain a gas mileage of exactly 22 miles per gallon?

43. ƒ 2x + 5 ƒ = - 8 44. ƒ 4x - 1 ƒ = - 17 45. 2 ƒ x + 3 ƒ + 4 = 34 46. 3 ƒ x - 5 ƒ - 16 = 2 47. ƒ 2x - a ƒ = b

(b 7 0)

48. 3 ƒ x - d ƒ = c

(c 7 0)

53. Office Carpeting The cost to install new carpet in an office

is determined by a $550 fixed fee plus a fee of $45 per square yard of floor space to be covered. How many square yards of floor space can be carpeted at a cost of $3800? Round to the nearest square yard. 54. Wholesale Price A retailer determines the retail price of a

coat by first computing 175% of the wholesale price of the coat and then adding a markup of $8.00. What is the wholesale price of a coat that has a retail price of $156.75?

1.2

55. Computer Science If p% of a file remains to be downloaded

using a cable modem, then p = 100 -

30 t N

FORMULAS AND APPLICATIONS

83

defined by the linear equations shown below, where a is your age in years and the heart rate is in beats per minute.1

where N is the size of the file in megabytes and t is the number of seconds since the download began. In how many minutes will 25% of a 110-megabyte file remain to be downloaded? Round to the nearest tenth of a minute.

Maximum exercise heart rate = 0.85(220 – a) Minimum exercise heart rate = 0.65(220 – a)

56. Aviation The number of miles that remain to be flown by a

commercial jet traveling from Boston to Los Angeles can be approximated by the equation Miles remaining = 2650 - 475t where t is the number of hours since leaving Boston. In how many hours will the plane be 1000 miles from Los Angeles? Round to the nearest tenth of an hour.

57.

To benefit from an aerobic exercise program, many experts recommend that you exercise three to five times a week for 20 minutes to an hour. It is also important that your heart rate be in the training zone, which is

58.

Exercise Heart Rate Find the maximum exercise heart

rate and the minimum exercise heart rate for a person who is 25 years of age. (Round to the nearest beat per minute.) Maximum Exercise Heart Rate How old is a person

who has a maximum exercise heart rate of 153 beats per minute?

SECTION 1.2

Formulas and Applications

Formulas Applications


PS1. The sum of two numbers is 32. If one of the numbers is represented by x, then the

1 expression 32 - x represents the other number. Evaluate 32 - x for x = 8 . [P.1] 2 PS2. Evaluate

1 2 4 and h = . [P.1] bh for b = 2 3 5

PS3. What property has been applied to rewrite 2l + 2w as 2(l + w)? [P.1] PS4. What property has been applied to rewrite a b bh as

1 2

PS5. Add:

1 2 x + x [P.1] 5 3

PS6. Simplify:

1

1 [P.5] 1 1 + a b

“The Heart of the Matter,” American Health, September 1995.

1 (bh)? [P.1] 2

84

CHAPTER 1


Formulas A formula is an equation that expresses known relationships between two or more variables. Table 1.2 lists several formulas from geometry that are used in this text. The variable P represents perimeter, C represents circumference of a circle, A represents area, S represents surface area of an enclosed solid, and V represents volume.

Table 1.2

Formulas from Geometry

Rectangle P

2l

A

lw

Square

2w

P

4s

P

A

s2

A

1 bh 2

c

h

s

w

V

lwh

lw

hl

S

V

r r2 1 3

b

c

Circle C

d

A

r2

Parallelogram

2 r

P

2b

A

bh

2s

r

a

h

h2 r2

r 2h

Sphere Sphere S

V

4 r2 4 3

s

b

b

Righ t Circ u la r Cone

Rectangular Solid 2 wh

a

s

l

S

Triangle

r3

Right Circular Cylinder S

2 rh

V

r 2h

2 r2

Frustum of a Cone S

V

R 1 3

r

h2

h r2

rR

R r2

r2 R2

R2

r h

h

r l

w

h

r

r

h R

It is often necessary to solve a formula for a specified variable. Begin the process by isolating all terms that contain the specified variable on one side of the equation and all terms that do not contain the specified variable on the other side.

EXAMPLE 1

Note In Example 1a, the solution P - 2w also can be written as l = 2 P l = - w. 2

Solve a Formula for a Specified Variable

a.

Solve 2l + 2w = P for l.

b.

Solve S = 2(wh + lw + hl) for h.

Solution a. 2l + 2w = P 2l = P - 2w P - 2w l = 2

• Subtract 2w from each side to isolate the 2l term. • Divide each side by 2.

1.2


85

S = 2(wh + lw + hl )

b.

S = 2wh + 2lw + 2hl S - 2lw = 2wh + 2hl S - 2lw = 2h(w + l ) S - 2lw = h 2(w + l )

• Isolate the terms that involve the variable h on the right side. • Factor 2h from the right side. • Divide each side by 2(w + l ).


Question • If ax + b = c, does x =

c - b? a

Formulas are often used to compare the performances of athletes. Here is an example of a formula that is used in professional football.

EXAMPLE 2

Calculate a Quarterback Rating

The National Football League uses the following formula to rate quarterbacks. QB rating =

100 30.05(C - 30) + 0.25(Y - 3) + 0.2T + (2.375 - 0.25I )4 6

In this formula, C is the percentage of pass completions, Y is the average number of yards gained per pass attempt, T is the percentage of touchdown passes, and I is the percentage of interceptions. During the 2008 season, Philip Rivers, the quarterback of the San Diego Chargers, completed 65.3% of his passes. He averaged 8.39 yards per pass attempt, 7.1% of his passes were for touchdowns, and 2.3% of his passes were intercepted. Determine Rivers’s quarterback rating for the 2008 season. Solution Because C is defined as a percentage, C = 65.3. We are also given Y = 8.39, T = 7.1, and I = 2.3. Substitute these values into the rating formula. QB rating 100 30.05(65.3 - 30) + 0.25(8.39 - 3) + 0.2(7.1) + (2.375 - 0.25(2.3))4 6 = 105.5 =

Philip Rivers’s quarterback rating for the 2008 season was 105.5. Try Exercise 14, page 93

Answer • No. x =

c - b , provided a Z 0. a

86

CHAPTER 1


Applications Linear equations emerge in a variety of application problems. In solving such problems, it generally helps to apply specific techniques in a series of small steps. The following general strategies should prove helpful in the remaining portion of this section.

Strategies for Solving Application Problems 1. Read the problem carefully. If necessary, reread the problem several times. 2. When appropriate, draw a sketch and label parts of the drawing with the specific information given in the problem. 3. Determine the unknown quantities, and label them with variables. Write down any equation that relates the variables. 4. Use the information from step 3, along with a known formula or some additional information given in the problem, to write an equation. 5. Solve the equation obtained in step 4, and check to see whether the results satisfy all the conditions of the original problem.

EXAMPLE 3

Dimensions of a Painting

w

Solution 1. Read the problem carefully. l

l

2.

Draw a rectangle. See Figure 1.3.

3.

Label the rectangle. We have used w for its width and l for its length. The problem states that the length is 24 centimeters more than the width. Thus l and w are related by the equation l = w + 24

4. w

Figure 1.3

The perimeter of a rectangle is given by the formula P = 2l + 2w. To produce an equation that involves only constants and a single variable (say, w), substitute 260 for P and w + 24 for l. P = 2l + 2w 260 = 2(w + 24) + 2w

5.

Solve for w. 260 260 212 w

= = = =

2w + 48 + 2w 4w + 48 4w 53

The length is 24 centimeters more than the width. Thus l = 53 + 24 = 77.

Gianni Dagli Orti/CORBIS

One of the best known paintings is the Mona Lisa by Leonardo da Vinci. It is on display at the Musée du Louvre, in Paris. The length (or height) of this rectangular-shaped painting is 24 centimeters more than its width. The perimeter of the painting is 260 centimeters. Find the width and length of the painting.

1.2


87

A check verifies that 77 is 24 more than 53 and that twice the length (77) plus twice the width (53) gives the perimeter (260). The width of the painting is 53 centimeters, and its length is 77 centimeters. Try Exercise 18, page 93

Similar triangles are ones for which the measures of corresponding angles are equal. The triangles below are similar.

D

e A

b

B

C

f

c E

F

a

d ∠A = ∠D

∠B = ∠E

∠C = ∠F

An important relationship among the sides of similar triangles is that the ratios of corresponding sides are equal. Thus, for the triangles above, d a = e b

b e = c f

a d = c f

This fact is used in many applications.

EXAMPLE 4

A Problem Involving Similar Triangles

A person 6 feet tall is in the shadow of a building 40 feet tall and is walking directly away from the building. When the person is 30 feet from the building, the tip of the person’s shadow is at the same point as the tip of the shadow of the building. How much farther must the person walk to be just out of the shadow of the building? Round to the nearest tenth of a foot. Solution Let x be the distance the person has to walk. Draw a picture of the situation using similar triangles. A

40 ft F 6 ft B

30 ft

D

x

C

(continued)

88

CHAPTER 1


Triangles ABC and FDC are similar triangles. Therefore, the ratios of the lengths of the corresponding sides are equal. Using this fact, we can write an equation. x 30 + x = 40 6 Now solve the equation. x 30 + x = 40 6 120a

x 30 + x b = 120a b 40 6

3(30 + x) = 20x

• Multiply each side by 120, the LCD of 40 and 6. • Solve for x.

90 + 3x = 20x 90 = 17x 5.3 L x The person must walk an additional 5.3 feet. Try Exercise 24, page 94

Many business applications can be solved by using the equation Profit = revenue - cost

EXAMPLE 5

A Business Application

It costs a tennis shoe manufacturer $26.55 to produce a pair of tennis shoes that sells for $49.95. How many pairs of tennis shoes must the manufacturer sell to make a profit of $14,274.00? Solution The profit is equal to the revenue minus the cost. If x equals the number of pairs of tennis shoes to be sold, then the revenue will be $49.95x and the cost will be $26.55x. Therefore, Profit 14,274.00 14,274.00 610

= = = =

revenue - cost 49.95x - 26.55x 23.40x x

The manufacturer must sell 610 pairs of tennis shoes to make the desired profit. Try Exercise 26, page 94

Simple interest problems can be solved by using the formula I = Prt, where I is the interest, P is the principal, r is the simple interest rate per period, and t is the number of periods.

EXAMPLE 6

An Investment Problem

An accountant invests part of a $6000 bonus in a 5% simple interest account and invests the remainder of the money at 8.5% simple interest. Together the investments earn $370 per year. Find the amount invested at each rate.

1.2


89

Solution Let x be the amount invested at 5%. The remainder of the money is $6000 - x, which is the amount invested at 8.5%. Using the simple interest formula I = Prt with t = 1 year yields Interest at 5% = x # 0.05 = 0.05x

Interest at 8.5% = (6000 - x) # (0.085) = 510 - 0.085x

The interest earned on the two accounts equals $370. 0.05x + (510 - 0.085x) = 370 - 0.035x + 510 = 370 -0.035x = - 140 x = 4000 The accountant invested $4000 at 5% and the remaining $2000 at 8.5%. Try Exercise 32, page 94

Many uniform motion problems can be solved by using the formula d = rt, where d is the distance traveled, r is the rate of speed, and t is the time.

EXAMPLE 7

A Uniform Motion Problem

A runner runs a course at a constant speed of 6 mph. One hour after the runner begins, a cyclist starts on the same course at a constant speed of 15 mph. How long after the runner starts does the cyclist overtake the runner? Solution If we represent the time the runner has spent on the course by t, then the time the cyclist takes to overtake the runner is t - 1. The following table organizes the information and helps us determine how to write the distance each person travels. Rate r

Time t

Distance d

Runner

6

t

=

6t

Cyclist

15

t - 1

=

15(t - 1)

Figure 1.4 indicates that the runner and the cyclist cover the same distance. Thus 6t = 15(t - 1) 6t = 15t - 15 - 9t = - 15

d = 6t

t = 1 d = 15(t − 1)

Figure 1.4

2 3

2 The cyclist overtakes the runner 1 hours after the runner starts. 3 Try Exercise 36, page 94

90

CHAPTER 1


Percent mixture problems involve combining solutions or alloys that have different concentrations of a common substance. Percent mixture problems can be solved by using the formula pA = Q, where p is the percent of concentration (in decimal form), A is the amount of the solution or alloy, and Q is the quantity of a substance in the solution or alloy. For example, in 4 liters of a 25% acid solution, p is the percent of acid (0.25 as a decimal), A is the amount of solution (4 liters), and Q is the amount of acid in the solution, which equals (0.25)(4) liters = 1 liter.

EXAMPLE 8

A Percent Mixture Problem

A chemist mixes an 11% hydrochloric acid solution with a 6% hydrochloric acid solution. How many milliliters of each solution should the chemist use to make a 600-milliliter solution that is 8% hydrochloric acid? Solution Let x be the number of milliliters of the 11% solution. Because the solution after mixing will have a total of 600 milliliters of fluid, 600 - x is the number of milliliters of the 6% solution. See Figure 1.5. Solutions before mixing

Solution after mixing

11

%

6%

x ml

added to

8%

600 – x ml

yields

600 ml

Figure 1.5

Because all the hydrochloric acid in the solution after mixing comes from either the 11% solution or the 6% solution, the number of milliliters of hydrochloric acid in the 11% solution added to the number of milliliters of hydrochloric acid in the 6% solution must equal the number of milliliters of hydrochloric acid in the 8% solution. a

ml of acid in ml of acid in ml of acid in b b + a b = a 6% solution 8% solution 11% solution 0.11x

+ 0.06(600 - x) 0.11x + 36 - 0.06x 0.05x + 36 0.05x x

= = = = =

0.08(600) 48 48 12 240

The chemist should use 240 milliliters of the 11% solution and 360 milliliters of the 6% solution to make a 600-milliliter solution that is 8% hydrochloric acid. Try Exercise 44, page 95

Value mixture problems involve combining two or more ingredients that have different prices into a single blend. The solution of a value mixture problem is based on the equation

1.2


91

V = CA, where V is the value of the ingredient, C is the unit cost of the ingredient, and A is the amount of the ingredient. For instance, if the cost C of tea is $4.30 per pound, then 5 pounds (the amount A) of tea has a value V = (4.30)(5) = 21.50, or $21.50. The solution of a value mixture problem is based on the sum of the values of all ingredients taken separately equaling the value of the mixture.

EXAMPLE 9

A Value Mixture Problem

How many ounces of pure silver costing $10.50 per ounce must be mixed with 60 ounces of a silver alloy that costs $7.35 per ounce to produce a silver alloy that costs $9.00 per ounce? Solution Let x be the number of ounces of pure silver being added. The value of the silver added is 10.50x. The value of the 60 ounces of the existing alloy is 7.35(60). Mixing the x ounces of the pure silver to the 60 ounces of the existing alloy yields an alloy that contains (x + 60) ounces. The value of the new alloy is 9.00(x + 60). a

Value of Value of Value of b + a b = a b new alloy pure silver existing alloy 10.50x

+

7.35(60) 10.5x + 441 1.5x + 441 1.5x x

= = = = =

9.00(x + 60) 9x + 540 540 99 66

66 ounces of pure silver must be added. Try Exercise 54, page 95

To solve a work problem, use the equation Rate of work * time worked = part of task completed For example, if a painter can paint a wall in 15 minutes, then the painter can paint

1 of the 15

1 of the wall each minute. In general, if a 15 1 task can be completed in x minutes, then the rate of work is of the task each minute. x wall in 1 minute. The painter’s rate of work is

EXAMPLE 10

A Work Problem

Pump A can fill a pool in 6 hours, and pump B can fill the same pool in 3 hours. How long will it take to fill the pool if both pumps are used? Solution 1 Because pump A fills the pool in 6 hours, represents the part of the pool filled by 6 1 pump A in 1 hour. Because pump B fills the pool in 3 hours, represents the part of 3 the pool filled by pump B in 1 hour. (continued)

92

CHAPTER 1


Let t equal the number of hours to fill the pool using both pumps. Then t#

t 1 = 6 6

• Part of the pool filled by pump A

t#

1 t = 3 3

• Part of the pool filled by pump B

a

Part filled Part filled 1 filled b + a b = a b by pump A by pump B pool t 6

+

t 3

=

1

Multiplying each side of the equation by 6 produces t + 2t = 6 3t = 6 t = 2 2 1 2 , or , of the pool in 2 hours and pump B fills of the pool 6 3 3 in 2 hours, so 2 hours is the time required to fill the pool if both pumps are used. Check: Pump A fills


EXERCISE SET 1.2 In Exercises 1 to 10, solve the formula for the specified variable. 1. V =

1 pr 2 h; h (geometry) 3

2. P = S - Sdt; t 3. I = Prt; t

5. F =

6. A =

d2

; m1

(geometry)

7. an = a1 + (n - 1)d; d

(mathematics)

8. y - y1 = m(x - x1 ); x

(mathematics)

9. S =

12.

Quarterback Rating During the 2008 season, Peyton Manning, the quarterback of the Indianapolis Colts, completed 66.8% of his passes. He averaged 7.21 yards per pass attempt, 4.9% of his passes were for touchdowns, and 2.2% of his passes were intercepted. Determine Manning’s quarterback rating for the 2008 season. Round to the nearest tenth. (Hint: See Example 2, page 85.)

(business)

1 h(b1 + b2); b1 2

a1 ; r (mathematics) 1 - r

(chemistry)

Quarterback Rating During the 2008 season, Drew Brees, the quarterback of the New Orleans Saints, completed 65.0% of his passes. He averaged 7.98 yards per pass attempt, 5.4% of his passes were for touchdowns, and 2.7% of his passes were intercepted. Determine Brees’s quarterback rating for the 2008 season. (Hint: See Example 2, page 85.)

(business)

(physics)

P1V1 P2V2 = ; V2 T1 T2

11.

(business)

4. A = P + Prt; P

Gm1 m2

10.

The simplified measure of gobbledygook (SMOG) readability formula is often used to estimate the reading grade level required if a person is to fully understand the written material being assessed. The formula is given by SMOG reading grade level 1w 3

1.2

where w is the number of words that have three or more syllables in a sample of 30 sentences. Use this information in Exercises 13 and 14.

14.

A sample of 30 sentences from Alice’s Adventures in Wonderland, by Lewis Carroll, shows a total of 42 words that have three or more syllables. Use the SMOG reading grade level formula to estimate the reading grade level required to fully understand this novel. Round the reading grade level to the nearest tenth.

A sample of 30 sentences from A Tale of Two Cities, by Charles Dickens, shows a total of 105 words that have three or more syllables. Use the SMOG reading grade level formula to estimate the reading grade level required to fully understand this novel. Round the reading grade level to the nearest tenth.

Another reading level formula is the Gunning-Fog Index. Here is the formula.

18. Geometry The width of a rectangle is 1 meter more than half

the length of the rectangle. If the perimeter of the rectangle is 110 meters, find the width and the length.

of the two longer sides of the triangle is three times as long as the shortest side. Find the length of each side of the triangle. 20. Geometry A triangle has a perimeter of 161 miles. The length

of each of the two smaller sides of the triangle is two-thirds the length of the longest side. Find the length of each side of the triangle. 21. Height of a Tree One way to approximate the height of a tree

is to measure its shadow and then measure the shadow of a known height. Use similar triangles and the diagram below to estimate the height of the tree.

h

Gunning-Fog Index 0.4(A P) where A is the average number of words per sentence and P is the percentage of words that have three or more syllables. The Gunning-Fog Index is defined as the minimum grade level required if a person is to easily understand the text on the first reading. Use this information in Exercises 15 and 16. 15.

16.

93

19. Geometry A triangle has a perimeter of 84 centimeters. Each Syndicated Features Limited/Heritage/ The Image Works

13.


In a large sample of sentences from the novel The Red Badge of Courage, by Stephen Crane, the average number of words per sentence is 14.8 and the percentage of words with three or more syllables is 15.1. Use the Gunning-Fog Index formula to estimate the reading grade level required to easily understand this novel. Round the grade level to the nearest tenth. In a large sample of sentences from the novel Emma, by Jane Austen, the average number of words per sentence is 18.8 and the percentage of words with three or more syllables is 14.2. Use the Gunning-Fog Index formula to estimate the reading grade level required to easily understand this novel. Round the grade level to the nearest tenth.

6 ft

10 ft

22. Height of a Building A building casts a shadow 50 feet long.

A rod 4 feet tall placed near the building casts a shadow 3 inches long. Find the height of the building. 23. Shadow Length A building 50 feet tall casts a shadow 20 feet

long. A person 6 feet tall is walking directly away from the building toward the edge of the building’s shadow. How far from the building will the person be when the person’s shadow just begins to emerge from that of the building?

50 ft

In Exercises 17 to 60, solve by using the strategies for solving application problems (see page 86). 17. Geometry The length of a rectangle is 3 feet less than twice

the width of the rectangle. If the perimeter of the rectangle is 174 feet, find the width and the length.

4 ft

6 ft x ft

20 ft

94

CHAPTER 1


24. Shadow Length A person 6 feet tall is standing at the base of

a lamppost that is 25 feet tall and then begins to walk away from the lamppost. When the person is 10 feet from the lamppost, what is the length of the person’s shadow? Round to the nearest tenth of a foot.

7%. The amount of interest earned for 1 year was $405. How much was invested in each account? 33. Investment An investment of $2500 is made at an annual

simple interest rate of 5.5%. How much additional money must be invested at an annual simple interest rate of 8% so that the total interest earned is 7% of the total investment? 34. Investment An investment of $4600 is made at an annual

simple interest rate of 6.8%. How much additional money must be invested at an annual simple interest rate of 9% so that the total interest earned is 8% of the total investment?

25 ft

35. Uniform Motion Running at an average rate of 6 meters per 6 ft 10 ft

x

25. Business It costs a manufacturer $8.95 to produce sunglasses

that sell for $29.99. How many pairs of sunglasses must the manufacturer sell to make a profit of $17,884? 26. Business It costs a restaurant owner 18 cents per glass for

orange juice, which sells for 75 cents per glass. How many glasses of orange juice must the restaurant owner sell to make a profit of $2337? 27. Determine Individual Prices A book and a bookmark together

second, a sprinter ran to the end of a track. The sprinter then jogged back to the starting point at an average rate of 2 meters per second. The total time for the sprint and the jog back was 2 minutes 40 seconds. Find the length of the track. 36. Uniform Motion A motorboat left a harbor and traveled to an

island at an average rate of 15 knots. The average speed on the return trip was 10 knots. If the total trip took 7.5 hours, how many nautical miles is the harbor from the island? 37. Uniform Motion A plane leaves an airport traveling at an

average speed of 240 kilometers per hour. How long will it take a second plane traveling the same route at an average speed of 600 kilometers per hour to catch up with the first plane if it leaves 3 hours later?

sell for $10.10. If the price of the book is $10.00 more than the price of the bookmark, find the price of the book and the price of the bookmark.

240 km/h

28. Share an Expense Three people decide to share the cost of a

yacht. By bringing in an additional partner, they can reduce the cost to each person by $4000. What is the total cost of the yacht? 29. Business The price of a computer fell 20% this year. If the

computer now costs $750, how much did it cost last year? 30. Business The price of a magazine subscription rose 4% this

year. If the subscription now costs $26, how much did it cost last year?

600 km/h

Airport

38. Uniform Motion A plane leaves Chicago headed for Los

Angeles at 540 mph. One hour later, a second plane leaves Los Angeles headed for Chicago at 660 mph. If the air route from Chicago to Los Angeles is 1800 miles, how long will it take for the first plane to pass the second plane? How far from Chicago will they be at that time?

31. Investment An investment adviser invested $14,000 in two

accounts. One investment earned 8% annual simple interest, and the other investment earned 6.5% annual simple interest. The amount of interest earned for 1 year was $1024. How much was invested in each account? 32. Investment A total of $7500 is deposited into two simple inter-

est accounts. In one account the annual simple interest rate is 5%, and in the second account the annual simple interest rate is

1800

Los Angeles

mi

Chicago

1.2


95

39. Speed of Sound in Air Two seconds after firing a rifle at a

48. Metallurgy How much 14-karat gold should be melted with

target, the shooter hears the impact of the bullet. Sound travels at 1100 feet per second and the bullet at 1865 feet per second. Determine the distance to the target (to the nearest foot).

4 ounces of pure gold to produce 18-karat gold? (Hint: See Exercise 47.) 49. Tea Mixture A tea merchant wants to make 20 pounds of a

40. Speed of Sound in Water Sound travels through sea water

4.62 times as fast as through air. The sound of an exploding mine on the surface of the water and partially submerged reaches a ship through the water 4 seconds before it reaches the ship through the air. How far is the ship from the explosion (to the nearest foot)? Use 1100 feet per second as the speed of sound through air.

blended tea costing $5.60 per pound. The blend is made using a $6.50-per-pound grade of tea and a $4.25-per-pound grade of tea. How many pounds of each grade of tea should be used? 50. Gold Alloy How many ounces of pure gold that costs $850 per

ounce must be mixed with 25 ounces of a gold alloy that costs $500 per ounce to make a new alloy that costs $725 per ounce?

41. Uniform Motion A car traveling at 80 kilometers per hour is

51. Trail Mix A grocery mixes some dried cranberries that cost $6

passed by a second car going in the same direction at a constant speed. After 30 seconds, the two cars are 500 meters apart. Find the speed of the second car.

per pound with some granola that costs $3 per pound. How many pounds of each should be used to make a 25-pound mixture that costs $3.84 per pound?

42. Uniform Motion Marlene rides her bicycle to her friend Jon’s

52. Coffee Mixture A coffee shop decides to blend a coffee that

house and returns home by the same route. Marlene rides her bike at constant speeds of 6 mph on level ground, 4 mph when going uphill, and 12 mph when going downhill. If her total time riding was 1 hour, how far is it to Jon’s house? (Hint: Let d1 be the distance traveled on level ground and let d2 be the distance traveled on the hill. Then the distance between the two houses is d1 + d2. Write an equation for the total time. For instance, the time spent traveling to Jon’s house on level d1 ground is .) 6

sells for $12 per pound with a coffee that sells for $9 per pound to produce a blend that will sell for $10 per pound. How much of each should be used to yield 20 pounds of the new blend?

43. Metallurgy How many grams of pure silver must a silversmith

mix with a 45% silver alloy to produce 200 grams of a 50% alloy? 44. Chemistry How many liters of a 40% sulfuric acid solution

53. Coffee Mixture The vendor of a coffee cart mixes coffee

beans that cost $8 per pound with coffee beans that cost $4 per pound. How many pounds of each should be used to make a 50-pound blend that sells for $5.50 per pound? 54. Silver Alloy A jeweler wants to make a silver alloy to be used

to make necklaces. How many ounces of a silver alloy that costs $6.50 per ounce should be mixed with one that costs $8.00 per ounce to make a new 20-ounce alloy that costs $7.40 per ounce? 55. Install Electrical Wires An electrician can install the electric

should be mixed with 4 liters of a 24% sulfuric acid solution to produce a 30% solution?

wires in a house in 14 hours. A second electrician requires 18 hours. How long would it take both electricians, working together, to install the wires?

45. Chemistry How many liters of water should be evaporated

56. Print a Report Printer A can print a report in 3 hours. Printer

from 160 liters of a 12% saline solution so that the solution that remains is a 20% saline solution? 46. Automotive A radiator contains 6 liters of a 25% antifreeze

solution. How much should be drained and replaced with pure antifreeze to produce a 33% antifreeze solution?

B can print the same report in 4 hours. How long would it take both printers, working together, to print the report? 57. Painting A painter can paint a kitchen in 10 hours. An appren-

tice can paint the same kitchen in 15 hours. If they worked together, how long would it take them to paint the kitchen?

47. Metallurgy How much pure gold should be melted with

58. Sports A snowmaking machine at a ski resort can produce

15 grams of 14-karat gold to produce 18-karat gold? (Hint: A karat is a measure of the purity of gold in an alloy. Pure gold measures 24 karats. An alloy that measures x karats is 18 x 3 gold. For example, 18-karat gold is = gold.) 24 24 4

enough snow for a beginner’s ski trail in 16 hours. With a typical natural snowfall, it takes 24 hours to deposit enough snow to open the beginner’s ski trail. If the snowmaking machine is run during a typical natural snowfall, how long will it take to deposit enough snow to open the beginner’s trail?

96

CHAPTER 1


59. Road Construction A new machine that deposits cement

60. Masonry A mason can lay the bricks in a sidewalk in

for a road requires 12 hours to complete a one-half mile section of road. An older machine requires 16 hours to pave the same amount of road. After depositing cement for 4 hours, the new machine develops a mechanical problem and quits working. The older machine is brought into place and continues the job. How long does it take the older machine to complete the job?

12 hours. The mason’s apprentice requires 16 hours to do the same job. After working together for 4 hours, the mason leaves for another job, and the apprentice continues working. How long will it take the apprentice to complete the job?

SECTION 1.3 Solving Quadratic Equations by Factoring Solving Quadratic Equations by Taking Square Roots Solving Quadratic Equations by Completing the Square Solving Quadratic Equations by Using the Quadratic Formula The Discriminant of a Quadratic Equation Applications of Quadratic Equations

Quadratic Equations PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A5.

PS1. Factor: x 2 - x - 42 [P.4] PS2. Factor: 6x 2 - x - 15 [P.4] PS3. Write 3 + 1- 16 in a + bi form. [P.6] PS4. If a = - 3, b = - 2, and c = 5, evaluate PS5. If a = 2, b = - 3, and c = 1, evaluate

-b - 2b 2 - 4ac [P.1/P.2] 2a

-b + 2b 2 - 4ac [P.1/P.2] 2a

PS6. If x = 3 - i, evaluate x 2 - 6x + 10. [P.6]

Solving Quadratic Equations by Factoring In Section 1.1 you solved linear equations. In this section you will learn to solve a type of equation that is referred to as a quadratic equation.

Math Matters The term quadratic is derived from the Latin word quadrare, which means “to make square.” Because the area of a square that measures x units on each side is x 2, we refer to equations that can be written in the form ax 2 + bx + c = 0 as equations that are “quadratic in x.”

Definition of a Quadratic Equation A quadratic equation in x is an equation that can be written in the standard quadratic form ax 2 + bx + c = 0 where a, b, and c are real numbers and a Z 0.

Several methods can be used to solve a quadratic equation. For instance, if you can factor ax 2 + bx + c into linear factors, then ax 2 + bx + c = 0 can be solved by applying the following property.

The Zero Product Principle If A and B are algebraic expressions such that AB = 0, then A = 0 or B = 0.

1.3

QUADRATIC EQUATIONS

97

The zero product principle states that if the product of two factors is zero, then at least one of the factors must be zero. In Example 1, the zero product principle is used to solve a quadratic equation.

EXAMPLE 1

Solve by Factoring

Solve each quadratic equation by factoring. a.

x 2 + 2x - 15 = 0

2x 2 - 5x = 12

b.

Solution a. x 2 + 2x - 15 = 0 (x - 3)(x + 5) = 0 x - 3 = 0 x = 3

or

• Factor.

x + 5 = 0

• Set each factor equal to zero.

x = -5

• Solve each linear equation.

A check shows that 3 and - 5 are both solutions of x 2 + 2x - 15 = 0. b.

2x 2 - 5x = 12 2x - 5x - 12 = 0 (x - 4)(2x + 3) = 0 or 2x + 3 = 0 x - 4 = 0 2

x = 4

• Write in standard quadratic form. • Factor. • Set each factor equal to zero.

2x = - 3

• Solve each linear equation.

3 x = 2 A check shows that 4 and -

3 are both solutions of 2x 2 - 5x = 12. 2


Some quadratic equations have a solution that is called a double root. For instance, consider x 2 - 8x + 16 = 0. Solving this equation by factoring, we have x 2 - 8x + 16 = 0 (x - 4)(x - 4) = 0 x - 4 = 0 x = 4

or

• Factor.

x - 4 = 0 x = 4

• Set each factor equal to zero. • Solve each linear equation.

The only solution of x 2 - 8x + 16 = 0 is 4. In this situation, the single solution 4 is called a double solution or double root because it was produced by solving the two identical equations x - 4 = 0, both of which have 4 as a solution.

Solving Quadratic Equations by Taking Square Roots Recall that 2x 2 = ƒ x ƒ . This principle can be used to solve some quadratic equations by taking the square root of each side of the equation.

98

CHAPTER 1


In the following example, we use this idea to solve x 2 = 25. x 2 = 25

Square Roots of Variable Expressions See page 23. Absolute Value Equations See page 80.

2x 2 = 125 ƒxƒ = 5

• Take the square root of each side.

x = - 5 or x = 5 The solutions are -5 and 5.

• Solve the absolute value equation.

• Use the fact that 2x 2 = ƒ x ƒ and 125 = 5.

We will refer to the preceding method of solving a quadratic equation as the square root procedure.

The Square Root Procedure If x 2 = c, then x = 1c or x = - 1c , which can also be written as x = 1c . EXAMPLE

If x 2 = 9, then x = 19 = 3 or x = - 19 = - 3. This can be written as x = 3. If x 2 = 7, then x = 17 or x = - 17 . This can be written as x = 17 . If x 2 = - 4, then x = 1 -4 = 2i or x = - 1- 4 = - 2i. This can be written as x = 2i.

EXAMPLE 2

Solve by Using the Square Root Procedure

Use the square root procedure to solve each equation. a.

3x 2 + 12 = 0

Solution a. 3x 2 + 12 3x 2 x2 x

= = = =

b.

(x + 1)2 = 48

0 - 12 -4 1 -4

• Solve for x 2. • Take the square root of each side of the equation and insert a plus-or-minus sign in front of the radical.

x = - 2i or x = 2i The solutions are -2i and 2i. b.

(x + 1)2 = 48 x + 1 = 148 x + 1 = 413 x = - 1 413

• Take the square root of each side of the equation and insert a plus-or-minus sign in front of the radical. • Simplify.

x = - 1 + 4 13 or x = -1 - 413 The solutions are - 1 + 413 and -1 - 413. Try Exercise 28, page 106

1.3

QUADRATIC EQUATIONS

99

Solving Quadratic Equations by Completing the Square Consider two binomial squares and their perfect-square trinomial products. Square of a Binomial

Perfect-Square Trinomial

(x + 5)

=

x 2 + 10x + 25

(x - 3)

=

x 2 - 6x + 9

2 2

Math Matters Mathematicians have studied quadratic equations for centuries. Many of the initial quadratic equations they studied resulted from attempts to solve geometry problems. One of the most famous, which dates from around 500 B.C., concerns “squaring a circle.” The question was, Is it possible to construct a square whose area is the same as the area of a given circle? For these early mathematicians, to construct meant to draw with only a straightedge and a compass. It was approximately 2300 years later when mathematicians proved that such a construction is impossible.

In each of the preceding perfect-square trinomials, the coefficient of x 2 is 1 and the constant term is the square of half the coefficient of the x term. a

x 2 + 10x + 25, x 2 - 6x + 9,

a

2 1# 10b = 25 2

2 1# ( -6) b = 9 2

Adding to a binomial of the form x 2 + bx the constant term that makes the binomial a perfect-square trinomial is called completing the square. For example, to complete the square of x 2 + 8x, add a

1# 2 8b = 16 2

to produce the perfect-square trinomial x 2 + 8x + 16. Completing the square is a powerful procedure that can be used to solve any quadratic equation. For instance, to solve x 2 - 6x + 13 = 0, first isolate the variable terms on one side of the equation and the constant term on the other side. x 2 - 6x = - 13 x 2 - 6x + 9 = - 13 + 9 (x - 3)2 x - 3 x - 3 x

= = = =

-4 1 -4 2i 3 2i

• Subtract 13 from each side of the equation. 2 1 • Complete the square by adding c (- 6) d = 9 2 to each side of the equation.

• Factor and solve by the square root procedure.

The solutions of x 2 - 6x + 13 = 0 are 3 - 2i and 3 + 2i. You can check these solutions by substituting each solution into the original equation. For instance, the following check shows that 3 - 2i does satisfy the original equation. x 2 - 6x + 13 = 0 (3 - 2i)2 - 6(3 - 2i) + 13 ⱨ 0 9 - 12i + 4i 2 - 18 + 12i + 13 ⱨ 0 4 + 4( -1) ⱨ 0 0 = 0

EXAMPLE 3

• Substitute 3 - 2i for x. • Simplify. • The left side equals the right side, so 3 - 2i checks.

Solve by Completing the Square

Solve x 2 = 2x + 6 by completing the square. (continued)

100

CHAPTER 1


Math Matters Ancient mathematicians thought of “completing the square” in a geometric manner. For instance, to complete the square of x 2 + 8x, draw a square that measures x units on each side and add four rectangles that measure 1 unit by x units to the right side of and the bottom of the square. x+4 x

Solution x 2 = 2x + 6 x 2 - 2x = 6 x 2 - 2x + 1 = 6 + 1 (x - 1)

2

• Isolate the constant term. • Complete the square.

= 7

• Factor and simplify.

x - 1 = 17

• Apply the square root procedure.

x = 1 17

• Solve for x.

The exact solutions of x 2 = 2x + 6 are 1 - 17 and 1 + 17. A calculator can be used to show that 1 - 17 L - 1.646 and 1 + 17 L 3.646. The decimals -1.646 and 3.646 are approximate solutions of x 2 = 2x + 6. Try Exercise 46, page 106

x x+4

Each of the rectangles has an area of x square units, so the total area of the figure is x 2 + 8x. To make this figure a complete square, we must add 16 squares that measure 1 unit by 1 unit, as shown below. x+4 x x x+4

Completing the square by adding the square of half the coefficient of the x term requires that the coefficient of the x 2 term be 1. If the coefficient of the x2 term is not 1, then first multiply each term on each side of the equation by the reciprocal of the coefficient of x2 to produce a coefficient of 1 for the x 2 term.

EXAMPLE 4

Solve by Completing the Square

Solve 2x 2 + 8x - 1 = 0 by completing the square. Solution 2x 2 + 8x - 1 = 0 2x 2 + 8x = 1 1 1 (2x 2 + 8x) = (1) 2 2

• Isolate the constant term. • Multiply both sides of the equation by the reciprocal of the coefficient of x 2.

1 2 1 x 2 + 4x + 4 = + 4 2 9 (x + 2)2 = 2 9 x + 2 = A2 x 2 + 4x =

This figure is a complete square whose area is (x + 4)2 = x 2 + 8x + 16

x = -2 3

• Complete the square. • Factor and simplify. • Apply the square root procedure.

1

A2

12 2 - 4 312 x = 2 x = -2 3

The solutions are

• Solve for x. • Simplify.

- 4 - 312 -4 + 312 and . 2 2


1.3

101

Solving Quadratic Equations by Using the Quadratic Formula

The Granger Collection

Math Matters

Completing the square for ax 2 + bx + c = 0 (a Z 0) produces a formula for x in terms of the coefficients a, b, and c. The formula is known as the quadratic formula, and it can be used to solve any quadratic equation.

The Quadratic Formula If ax2 + bx + c = 0, a Z 0, then

Evariste Galois (1811–1832)

x =

The quadratic formula provides the solutions to the general quadratic equation ax 2 + bx + c = 0 Formulas also have been developed to solve the general cubic

ax 2 + bx + c = 0 (a Z 0) ax 2 + bx = - c

2

b c x2 + x = a a

and the general quartic ax4 + bx3 + cx 2 + dx + e = 0 However, the French mathematician Evariste Galois, shown above, proved that there are no formulas that can be used to solve “by radicals” general equations of degree 5 or larger.

- b 2b 2 - 4ac 2a

Proof We assume a is a positive real number. If a were a negative real number, then we could multiply each side of the equation by -1 to make it positive.

ax + bx + cx + d = 0 3

QUADRATIC EQUATIONS

x2 +

b b 2 c b 2 x + a b = a b a a 2a 2a ax +

b2 c b 2 b = 2 a 2a 4a

b 2 b2 4a # c b = 2 a 2a 4a 4a b b 2 - 4ac = x + 2a A 4a 2

ax +

Shortly after completion of his remarkable proof, Galois was shot in a duel. It has been reported that as Galois lay dying, he asked his brother, Alfred, to “Take care of my work. Make it known. Important.” When Alfred broke into tears, Evariste said, “Don’t cry, Alfred. I need all my courage to die at twenty.” (Source: Whom the Gods Love, by Leopold Infeld, The National Council of Teachers of Mathematics, 1978, p. 299.)

x +

b 2b 2 - 4ac = 2a 2a x = x =

2b 2 - 4ac b 2a 2a

• Given. • Isolate the constant term. • Multiply each term on each side of 1 the equation by . a • Complete the square. • Factor the left side. Simplify the powers on the right side. • Use a common denominator to simplify the right side. • Apply the square root procedure. • Because a 7 0, 24a 2 = 2a. • Add -

b to each side. 2a

- b 2b 2 - 4ac 2a

As a general rule, you should first try to solve quadratic equations by factoring. If the factoring process proves difficult, then solve by using the quadratic formula.

EXAMPLE 5

Solve by Using the Quadratic Formula

Use the quadratic formula to solve each of the following. a.

x 2 = 3x + 5

b.

4x 2 - 4x + 3 = 0 (continued)

102

CHAPTER 1


Solution a.

x 2 = 3x + 5

x 2 - 3x - 5 = 0 x =

- b 2b 2 - 4ac 2a

• Use the quadratic formula.

x =

- ( - 3) 2( - 3)2 - 4(1)(- 5) 2(1)

• a = 1, b = -3, c = -5.

3 129 2

= x =

3 + 129 3 - 129 or 2 2

The solutions are b.

• Write the equation in standard form.

3 + 129 3 - 129 and . 2 2

4x 2 - 4x + 3 = 0

• The equation is in standard form.

x =

- b 2b2 - 4ac 2a


x =

- ( -4) 2( -4)2 - 4(4)(3) 2(4)

• a = 4, b = - 4, c = 3.

=

4 1 -32 4 116 - 48 = 2(4) 8

=

4 4i12 8

x = The solutions are

4 - 4i12 1 12 = i 8 2 2

or

x =

1 12 4 + 4i12 = + i 8 2 2

1 1 12 12 i and + i. 2 2 2 2

Try Exercise 58, page 107 Question • Can the quadratic formula be used to solve any quadratic equation ax 2 + bx + c = 0

with real coefficients and a Z 0?

The Discriminant of a Quadratic Equation The solutions of ax 2 + bx + c = 0, a Z 0, are given by x =

-b 2b 2 - 4ac 2a

Answer • Yes. However, it is sometimes easier to find the solutions by factoring, by the square

root procedure, or by completing the square.

1.3

QUADRATIC EQUATIONS

103

The expression under the radical, b 2 - 4ac, is called the discriminant of the equation ax 2 + bx + c = 0. If b 2 - 4ac Ú 0, then 2b 2 - 4ac is a real number. If b 2 - 4ac 6 0, then 2b 2 - 4ac is not a real number. Thus the sign of the discriminant can be used to determine whether the solutions of a quadratic equation are real numbers.

The Discriminant and the Solutions of a Quadratic Equation The equation ax 2 + bx + c = 0, with real coefficients and a Z 0, has as its discriminant b 2 - 4ac. If b 2 - 4ac 7 0, then ax 2 + bx + c = 0 has two distinct real solutions. If b 2 - 4ac = 0, then ax 2 + bx + c = 0 has one real solution. The solution is a double solution.

Complex Conjugates See page 63.

If b 2 - 4ac 6 0, then ax 2 + bx + c = 0 has two distinct nonreal complex solutions. The solutions are conjugates of each other.

EXAMPLE 6

Use the Discriminant to Determine the Number of Real Solutions

For each equation, determine the discriminant and state the number of real solutions. a.

2x 2 - 5x + 1 = 0

b.

3x 2 + 6x + 7 = 0

c.

x 2 + 6x + 9 = 0

Solution a. The discriminant of 2x 2 - 5x + 1 = 0 is b 2 - 4ac = ( - 5)2 - 4(2)(1) = 17. Because the discriminant is positive, 2x 2 - 5x + 1 = 0 has two distinct real solutions. b.

The discriminant of 3x 2 + 6x + 7 = 0 is b 2 - 4ac = 6 2 - 4(3)(7) = -48. Because the discriminant is negative, 3x 2 + 6x + 7 = 0 has no real solutions.

c.

The discriminant of x 2 + 6x + 9 = 0 is b 2 - 4ac = 6 2 - 4(1)(9) = 0. Because the discriminant is 0, x 2 + 6x + 9 = 0 has one real solution. Try Exercise 72, page 107

Applications of Quadratic Equations A right triangle contains one 90 angle. The side opposite the 90 angle is called the hypotenuse. The other two sides are called legs. The lengths of the sides of a right triangle are related by a theorem known as the Pythagorean Theorem. The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs. This theorem is often used to solve applications that involve right triangles.

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The Pythagorean Theorem If a and b denote the lengths of the legs of a right triangle and c the length of the hypotenuse, then c 2 = a 2 + b 2 .

Hypotenuse c Right angle

b Leg

a Leg

EXAMPLE 7

Determine the Dimensions of a Television Screen

A television screen measures 60 inches diagonally, and its aspect ratio is 16 to 9. This means that the ratio of the width of the screen to the height of the screen is 16 to 9. Find the width and height of the screen.

Note 9x

Many movies are designed to be shown on a screen that has a 16-to-9 aspect ratio.

60 inches

16x

A 60-inch television screen with a 16:9 aspect ratio.

Solution Let 16x represent the width of the screen and let 9x represent the height of the screen. Applying the Pythagorean Theorem gives us (16x)2 + (9x)2 = 602 256x 2 + 81x 2 = 3600

• Solve for x.

337x = 3600 3600 x2 = 337 2

x =

3600 L 3.268 inches A 337

• Apply the square root procedure. The plus-or-minus sign is not used in this application because we know x is positive.

The height of the screen is about 9(3.268) L 29.4 inches, and the width of the screen is about 16(3.268) L 52.3 inches. Try Exercise 82, page 107

EXAMPLE 8

Determine the Dimensions of a Candy Bar

A company makes rectangular solid candy bars that measure 5 inches by 2 inches by 0.5 inch. Due to difficult financial times, the company has decided to keep the price of the candy bar fixed and reduce the volume of the bar by 20%. What should the

1.3

QUADRATIC EQUATIONS

105

dimensions be for the new candy bar if the company keeps the height at 0.5 inch and makes the length of the candy bar 3 inches longer than the width?

Integrating Technology In many application problems, it is helpful to use a calculator to estimate the solutions of a quadratic equation by applying the quadratic formula. For instance, the following figure shows the use of a graphing calculator to estimate the solutions of w 2 + 3w - 8 = 0.

0.5 in.

w+3 w

Solution The volume of a rectangular solid is given by V = lwh. The original candy bar had a volume of 5 # 2 # 0.5 = 5 cubic inches. The new candy bar will have a volume of 80%(5) = 0.80(5) = 4 cubic inches. Let w represent the width and w + 3 represent the length of the new candy bar. For the new candy bar we have

√(32−4*1*(-8)))/2 (-3+√ 1.701562119 √(32−4*1*(-8)))/2 (-3−√ -4.701562119

lwh = V (w + 3)(w)(0.5) = 4 (w + 3)(w) = 8 w 2 + 3w = 8

• Substitute in the volume formula. • Multiply each side by 2. • Simplify the left side.

w 2 + 3w - 8 = 0 - (3) 2(3)2 - 4(1)(-8) w = 2(1) =

• Write in ax 2 + bx + c = 0 form. • Use the quadratic formula.

- 3 141 2

L 1.7

or

- 4.7

We can disregard the negative value because the width must be positive. The width of the new candy bar, to the nearest tenth of an inch, should be 1.7 inches. The length should be 3 inches longer, which is 4.7 inches. Try Exercise 94, page 108

Quadratic equations are often used to determine the height (position) of an object that has been dropped or projected. For instance, the position equation s = - 16t 2 + v0t + s0 can be used to estimate the height of a projected object near the surface of Earth at a given time t in seconds. In this equation, v0 is the initial velocity of the object in feet per second and s0 is the initial height of the object in feet.

EXAMPLE 9

Determine the Time of Descent

Dreamstime LLC

A ball is thrown downward with an initial velocity of 5 feet per second from the Golden Gate Bridge, which is 220 feet above the water. How long will it take for the ball to hit the water? Round your answer to the nearest hundreth of a second. Solution The distance s, in feet, of the ball above the water after t seconds is given by the position equation s = -16t 2 - 5t + 220. We have replaced v0 with -5 because the ball is thrown downward. (If the ball had been thrown upward, we would use v0 = 5.) To determine the time it takes the ball to hit the water, substitute 0 for s in the equation (continued)

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s = - 16t 2 - 5t + 220 and solve for t. In the following work, we have solved by using the quadratic formula. 0 = - 16t 2 - 5t + 220 t = =

- (-5) 2( - 5)2 - 4( -16)(220) 2( -16)


5 114,105 -32

L - 3.87

or

3.56

• Use a calculator to estimate t.

Because the time must be positive, we disregard the negative value. The ball will hit the water in about 3.56 seconds. Try Exercise 96, page 108

EXERCISE SET 1.3 In Exercises 1 to 10, solve each quadratic equation by factoring and applying the zero product principle.

In Exercises 33 to 50, solve each equation by completing the square.

1. x 2 - 2x - 15 = 0

2. x 2 + 3x - 10 = 0

33. x 2 - 2x - 15 = 0

34. x 2 + 2x - 8 = 0

3. 2x 2 - x = 1

4. 2x 2 + 5x = 3

35. 2x 2 - 5x - 12 = 0

36. 3x 2 - 5x - 2 = 0

5. 8x 2 + 189x - 72 = 0

6. 12x 2 - 41x + 24 = 0

37. x 2 + 6x + 1 = 0

38. x 2 + 8x - 10 = 0

7. 3x 2 - 7x = 0

8. 5x 2 = - 8x

39. x 2 + 3x - 1 = 0

9. (x - 5)2 - 9 = 0

10. (3x + 4)2 - 16 = 0

In Exercises 11 to 32, use the square root procedure to solve the equation.

40. x 2 + 7x - 2 = 0 41. 2x 2 + 4x - 1 = 0 42. 2x 2 + 10x - 3 = 0

11. x 2 = 81

12. x 2 = 225

13. y 2 = 24

14. y 2 = 48

15. z 2 = - 16

16. z 2 = - 100

45. x 2 + 4x + 5 = 0

17. (x - 5)2 = 36

18. (x + 4)2 = 121

46. x 2 - 6x + 10 = 0

19. (x + 2)2 = 27

20. (x - 3)2 = 8

47. 4x 2 + 4x + 2 = 0

21. (z - 4)2 + 25 = 0

22. (z + 1)2 + 64 = 0

48. 9x 2 + 12x + 5 = 0

23. 2(x + 3)2 - 18 = 0

24. 5(x - 2)2 - 45 = 0

49. 3x 2 + 2x + 1 = 0

25. (y - 6)2 - 4 = 14

26. (y + 2)2 + 5 = 15

50. 4x 2 - 4x = - 15

27. 5(x + 6)2 + 60 = 0

28. (x + 2)2 + 28 = 0

29. 2(x + 4)2 = 9

30. 3(x - 2)2 = 20

51. x 2 - 2x = 15

52. x 2 - 5x = 24

31. 4(x - 2)2 + 15 = 0

32. 6(x + 5)2 + 21 = 0

53. 12x 2 - 11x - 15 = 0

54. 10x 2 + 19x - 15 = 0

43. 3x 2 - 8x = - 1 44. 4x 2 = 13 - 4x

In Exercises 51 to 70, solve by using the quadratic formula.

1.3

QUADRATIC EQUATIONS

107

83. Dimensions of a Television Screen A television screen

55. x 2 - 2x = 2

56. x 2 + 4x - 1 = 0

57. x = - x + 1

58. 2x + 4x = 1

measures 54 inches diagonally, and its aspect ratio is 4 to 3. Find the width and the height of the screen.

59. 4x 2 = 41 - 8x

60. 2x = 9 - 3x 2

84. Publishing Costs The cost, in dollars, of publishing x books is

2

61.

2

1 2 3 x + x - 1 = 0 2 4

C(x) = 40,000 + 20x + 0.0001x 2. How many books can be published for $250,000? 85. Sports The height of a kicked football during a field goal attempt

19 5x = 0 62. x + 2 8 2

63. x 2 + 6x + 13 = 0

64. x 2 = 2x - 26

65. 2x 2 = 2x - 13

66. 9x 2 - 24x + 20 = 0

67. x 2 + 2x + 29 = 0

68. x 2 + 6x + 21 = 0

69. 4x 2 + 4x + 13 = 0

70. 9x 2 = 12x - 49

In Exercises 71 to 80, determine the discriminant of the quadratic equation and then state the number of real solutions of the equation. Do not solve the equation.

can be approximated by h = - 0.0114x 2 + 1.732x, where h is the height, in feet, of the football when it is x feet from the kicker. To clear the goalpost the football must be at least 10 feet above the ground. What is the maximum number of yards the kicker can be from the goalpost so that the football clears it? Round to the nearest tenth of a yard. 86. Revenue The demand for a certain product is given by

p = 26 - 0.01x, where x is the number of units sold per month and p is the price, in dollars, at which each item is sold. The monthly revenue is given by R = xp. What number of items sold produces a monthly revenue of $16,500? 87. Profit A company has determined that the profit, in dollars, it

71. 2x 2 - 5x - 7 = 0

72. x 2 + 3x - 11 = 0

can expect from the manufacture and sale of x tennis racquets is given by

73. 3x 2 - 2x + 10 = 0

74. x 2 + 3x + 3 = 0

P = - 0.01x 2 + 168x - 120,000

75. x 2 - 20x + 100 = 0

76. 4x 2 + 12x + 9 = 0

How many racquets should the company manufacture and sell to earn a profit of $518,000?

77. 24x 2 = - 10x + 21

78. 32x 2 - 44x = - 15

88. Quadratic Growth A plant’s ability to create food through the

79. 12x 2 + 15x = - 7

80. 8x 2 = 5x - 3

81. Geometry The length of each side of an equilateral triangle is

31 centimeters. Find the altitude of the triangle. Round to the nearest tenth of a centimeter.

process of photosynthesis depends on the surface area of its leaves. A biologist has determined that the surface area A of a maple leaf can be closely approximated by the formula A = 0.72(1.28)h 2, where h is the height of the leaf in inches. 1.28 h

82.

Dimensions of a Baseball Diamond How far, to the

nearest tenth of a foot, is it from home plate to second base on a baseball diamond? (Hint: The bases in a baseball diamond form a square that measures 90 feet on each side.)

h

Second base

90 ft First base

x 90 ft

a. Find the surface area of a maple leaf with a height of 7 inches.

Round to the nearest tenth of a square inch. Home plate

b. Find the height of a maple leaf with an area of 92 square

inches. Round to the nearest tenth of an inch.

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89. Dimensions of an Animal Enclosure A veterinarian wishes

94. Dimensions of a Candy Bar A company makes rectangular

to use 132 feet of chain-link fencing to enclose a rectangular region and subdivide the region into two smaller rectangular regions, as shown in the following figure. If the total enclosed area is 576 square feet, find the dimensions of the enclosed region.

solid candy bars that measure 5 inches by 2 inches by 0.5 inch. Due to difficult financial times, the company has decided to keep the price of the candy bar fixed and reduce the volume of the bar by 20%. What should be the dimensions, to the nearest tenth of an inch, of the new candy bar if the company decides to keep the height at 0.5 inch and to make the length of the new candy bar 2.5 times as long as its width?

w l 2.5 w

0.5 in.

90. Construction of a Box A square piece of cardboard is formed

into a box by cutting out 3-inch squares from each of the corners and folding up the sides, as shown in the following figure. If the volume of the box needs to be 126.75 cubic inches, what size square piece of cardboard is needed? 3 in. 3 in.

w

95. Height of a Rocket A model rocket is launched upward with

an initial velocity of 220 feet per second. The height, in feet, of the rocket t seconds after the launch is given by the equation h = - 16t 2 + 220t. How many seconds after the launch will the rocket be 350 feet above the ground? Round to the nearest tenth of a second. 96. Baseball The height h, in feet, of a baseball above the ground

x 3 in.

x

91. Population Density of a City The population density D (in

people per square mile) of a city is related to the horizontal distance x, in miles, from the center of the city by the equation D = - 45x 2 + 190x + 200, 0 6 x 6 5. At what distances from the center of the city does the population density equal 250 people per square mile? Round each result to the nearest tenth of a mile.

t seconds after it is hit is given by h = - 16t 2 + 52t + 4.5. Use this equation to determine the number of seconds, to the nearest tenth of a second, from the time the ball is hit until the ball touches the ground. 97. Baseball Two equations can be used to track the position of

a baseball t seconds after it is hit. For instance, suppose h = - 16t 2 + 50t + 4.5 gives the height, in feet, of a baseball t seconds after it is hit and s = 103.9t gives the horizontal distance, in feet, of the ball from home plate t seconds after it is hit. (See the following figure.) Use these equations to determine whether this particular baseball will clear a 10-foot fence positioned 360 feet from home plate.

92. Traffic Control Traffic engineers install “flow lights” at the

entrances of freeways to control the number of cars entering the freeway during times of heavy traffic. For a particular freeway entrance, the number of cars N waiting to enter the freeway during the morning hours can be approximated by N = - 5t 2 + 80t - 280, where t is the time of the day and 6 … t … 10.5. According to this model, when will there be 35 cars waiting to enter the freeway? 93.

h

s

Daredevil Motorcycle Jump In March 2000, Doug

Danger made a successful motorcycle jump over an L1011 jumbo jet. The horizontal distance of his jump was 160 feet, and his height, in feet, during the jump was approximated by h = - 16t 2 + 25.3t + 20, t Ú 0. He left the takeoff ramp at a height of 20 feet, and he landed on the landing ramp at a height of about 17 feet. How long, to the nearest tenth of a second, was he in the air?

98. Basketball Michael Jordan was known for his hang time,

which is the amount of time a player is in the air when making a jump toward the basket. An equation that approximates the height h, in inches, of one of Jordan’s jumps is given by h = - 16t 2 + 26.6t, where t is time in seconds. Use this equation to determine Michael Jordan’s hang time, to the nearest tenth of a second, for this jump.

MID-CHAPTER 1 QUIZ

99. Number of Handshakes If everyone in a group of n people

101.

shakes hands with everyone other than themselves, then the total number of handshakes h is given by

109

Centenarians According to data provided by the U.S.

The total number of handshakes that are exchanged by a group of people is 36. How many people are in the group?

Census Bureau, the number N, in thousands, of centenarians (a person whose age is 100 years or older) who will be living in the United States during a year from 2010 to 2050 can be approximated by N = 0.3453x 2 - 9.417x + 164.1, where x is the number of years after the beginning of 2000. Use this equation to determine in what year will there be 200,000 centenarians living in the United States.

Data Storage The projected data storage requirements

102. Automotive Engineering The number of feet N that a car

h =

100.

1 n(n - 1) 2

D, in petabytes, for the U.S. National Archives and Records Administration (NARA) can be modeled by D = 1.525x 2 - 21.35x + 72.225, 7 … x … 22, where x = 7 corresponds to the year 2007. Use this model to predict the year in which the storage requirement for the NARA will first exceed 100 petabytes. Note: 1 petabyte = 250 L 1.13 * 1015 bytes, or one-half of all information stored in academic libraries. (Source: U.S. National Archives and Records Administration, as reported in Technology Review, July 2005.)

Plethora of Petabytes

Projected accumulated NARA electronic records holdings (in petabytes)

By 2022, NARA is expected to be responsible for 347 petabytes of electronic records.

needs to stop on a certain road surface is given by the equation N = - 0.015v 2 + 3v, 0 … v … 90, where v is the speed of the car in miles per hour when the driver applies the brakes. What is the maximum speed, to the nearest mile per hour, that a motorist can be traveling and stop the car within 100 feet? 103.

Orbital Debris The amount of space debris is increas-

ing. The number N, in thousands, of objects greater than 10 cm in diameter in low-Earth orbits (orbits 200 km to 2000 km above Earth) can be approximated by the equation N = 0.004t 2 + 0.103t + 8.242, where t = 0 corresponds to the year 2000. (Source: http://orbitaldebris.jsc.nase.gov.) a. Use the equation to estimate the number of objects of

400

space debris we can expect in 2010. Round to the nearest hundred.

300

b. According to this model, in what year will the number of

objects of space debris first exceed 15,000? 200

100

0 ’07 ’08 ’09 ’10 ’11 ’12 ’13 ’14 ’15 ’16 ’17 ’18 ’19 ’20 ’21 ’22 SOURCE: U.S. NATIONAL ARCHIVES AND RECORDS ADMINISTRATION

MID-CHAPTER 1 QUIZ 1. Solve: 6 - 4(2x + 1) = 5(3 - 2x)

2. Solve:

1 x 3 2 x - = + 3 4 6 2

3. Solve by factoring: x 2 - 5x = 6 4. Solve by completing the square: x 2 + 4x - 2 = 0 5. Solve by using the quadratic formula: x 2 - 6x + 12 = 0

6. A runner runs a course at a constant speed of 8 miles per hour.

One hour later, a cyclist begins the same course at a constant speed of 16 miles per hour. How long after the runner starts does the cyclist overtake the runner? 7. A pharmacist mixes a 9% acetic acid solution with a 4% acetic

solution. How many milliliters of each solution should the pharmacist use to make a 500-milliliter solution that is 6% acetic acid? 8. A mason can complete a wall in 10 hours, but an apprentice

mason requires 15 hours to do the same job. How long will it take to build the wall with both people working?

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SECTION 1.4 Polynomial Equations Rational Equations Radical Equations Rational Exponent Equations Equations That Are Quadratic in Form Applications of Other Types of Equations

Other Types of Equations PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A6.

PS1. Factor: x3 - 16x [P.4] PS2. Factor: x4 - 36x2 [P.4] PS3. Evaluate: 82/3 [P.2] PS4. Evaluate: 163/2 [P.2] PS5. Multiply: (1 + 1x - 5)2, x 7 5 [P.2/P.3] PS6. Multiply: (2 - 1x + 3)2, x 7 - 3 [P.2/P.3]

Polynomial Equations Some polynomial equations that are neither linear nor quadratic can be solved by the various techniques presented in this section. For instance, the third-degree equation, or cubic equation, in Example 1 can be solved by factoring the polynomial and using the zero product principle.

EXAMPLE 1

Solve a Polynomial Equation

Solve: x3 + 3x2 - 4x - 12 = 0 Solution x3 + 3x2 - 4x - 12 = 0 (x3 + 3x2) - (4x + 12) = 0

• Factor by grouping.

x2(x + 3) - 4(x + 3) = 0 (x + 3)(x2 - 4) = 0 (x + 3)(x + 2)(x - 2) = 0 x + 3 = 0 or x + 2 = 0 or x = -3 x = -2

• Use the zero product principle.

x - 2 = 0 x = 2

The solutions are 3, 2, and 2. Try Exercise 8, page 120

Rational Equations Rational Expressions. See page 50.

A rational equation is one that involves rational expressions. The following two equations are rational equations. 2x x + 4 - 5 = x + 3 x - 1

x + 1 x 4 + = 2x - 3 x - 1 x2 - 1

When solving a rational equation, be aware of the domain of the equation, which is the intersection of the domains of rational expressions. For the first equation above, 3 and 1 are

1.4

OTHER TYPES OF EQUATIONS

111

excluded as possible values of x and are not in the domain. For the second equation, -1, 1, 3 and are excluded as possible values of x and are not in the domain. 2 3x 9 The domain is important, as shown by trying to solve . We begin + 2 = x - 3 x - 3 by noting that 3 is not in the domain of the rational expressions and then multiplying each side of the equation by x - 3. 3x 9 + 2 = x - 3 x - 3

Note Just because an equation can be written does not mean that there is a solution, as the equation at the right illustrates. Recall from Section 1.1 that equations with no solution are called contradictions. A simple example of a contradiction is x = x + 1. This equation has no solution.

(x - 3)a

• 3 is not in the domain.

3x 9 + 2b = (x - 3) a b x - 3 x - 3

• Multiply each side by x 3, the LCD of the denominators.

9 + 2(x - 3) = 3x

• Solve for x.

9 + 2x - 6 = 3x 3 = x However, the proposed solution, 3, is not in the domain, and replacing x with 3 in the original equation would require division by zero, which is not allowed. Therefore, the equation has no solution.

EXAMPLE 2

Solve a Rational Equation

Solve. a.

2x + 1 -2 + 3 = x + 4 x + 4

c.

x + 1 x - 1 2x + = x - 3 x + 4 x - 3

b.

3x +

- 4x + 12 4 = x - 2 x - 2

Solution -2 2x + 1 + 3 = x + 4 x + 4

a. (x + 4) a

-2 2x + 1 + 3b = (x + 4) a b x + 4 x + 4

(2x + 1) + 3(x + 4) 5x + 13 5x x

= = = =

-2 -2 - 15 -3

• 4 is not in the domain. • Multiply each side by x 4, the LCD of the denominators. • Solve for x.

3 checks as a solution. The solution is 3. b.

3x + (x - 2) a3x +

- 4x + 12 4 = x - 2 x - 2

-4x + 12 4 b = (x - 2) a b x - 2 x - 2

3x(x - 2) + 4 = - 4x + 12 3x2 - 6x + 4 = - 4x + 12

• 2 is not in the domain. • Multiply each side by x 2. • Solve for x. (continued)

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CHAPTER 1


3x 2 - 2x - 8 = 0

c.

• Factor and use the zero product principle. (3x + 4)(x - 2) = 0 3x + 4 = 0 or x - 2 = 0 4 x = x = 2 3 4 - checks as a solution; 2 is not in the domain and does not check as a solution. 3 4 The solution is - . 3 x + 1 x - 1 2x • 3 and 4 are not + = in the domain. x - 3 x + 4 x - 3 2x x + 1 x - 1 (x - 3)(x + 4)a + b = (x - 3)(x + 4) a b • Multiply each side by the x - 3 x + 4 x - 3

2x(x + 4) + (x + 2x2 + 8x + x2 2x2 (2x + 2x + 1 = 0

or

= = = =

(x + 4)(x - 1) x2 + 3x - 4 0 0

• Factor and use the zero product principle.

x + 1 = 0

1 x = 2 -

1)(x - 3) - 2x - 3 + 3x + 1 1)(x + 1)

LCD of the denominators. • Simplify.

x = -1

1 1 and 1 check as solutions. The solutions are - and -1. 2 2


Radical Equations Some equations that involve radical expressions can be solved by using the following principle.

The Power Principle If P and Q are algebraic expressions and n is a positive integer, then every solution of P = Q is a solution of P n = Q n.

EXAMPLE 3

Solve a Radical Equation

Use the power principle to solve 1x + 4 = 3. Solution 1x + 4 = 3 (1x + 4)2 = 32 x + 4 = 9 x = 5

• Square each side of the equation. (Apply the power principle with n = 2.)

1.4

Check: 1x + 4 15 + 4 19 3

= 3 ⱨ3 ⱨ3 = 3


113

• Substitute 5 for x. • 5 checks.

The solution is 5. Try Exercise 28, page 120

Some care must be taken when using the power principle because the equation P n = Qn may have more solutions than the original equation P = Q. As an example, consider x = 3. The only solution is the real number 3. Square each side of the equation to produce x 2 = 9, and you get both 3 and -3 as solutions. The - 3 is called an extraneous solution because it is not a solution of the original equation x = 3.

Definition of an Extraneous Solution Any solution of P n = Qn that is not a solution of P = Q is called an extraneous solution. Extraneous solutions may be introduced whenever each side of an equation is raised to an even power.

EXAMPLE 4

Solve Radical Equations

Solve. a.

212x - 1 - x = 1

b.

1x + 1 - 12x - 5 = 1

Solution a. 212x - 1 - x = 1 212x - 1 = x + 1 (2 12x - 1)2 = (x + 1)2 4(2x - 1) = x + 2x + 1 8x - 4 = x 2 + 2x + 1 0 = x 2 - 6x + 5 2

0 = (x - 1)(x - 5) x - 1 = 0 or x - 5 = 0 x = 1 x = 5

• Isolate the radical. • Square each side. • Simplify and solve for x.

• Factor.

Check: 212x - 1 - x = 1 2 12(1) - 1 - 1 ⱨ 1 211 - 1 ⱨ 1

2 - 1ⱨ1

212x - 1 - x = 1 212(5) - 1 - 5 ⱨ 1 219 - 5 ⱨ 1 6 - 5ⱨ1

1 = 1

1 = 1

1 and 5 check as solutions. The solutions are 1 and 5. b.

1x + 1 - 12x - 5 = 1 1x + 1 = 1 + 12x - 5 ( 1x + 1)2 = (1 + 12x - 5)2

• Isolate one of the radicals. • Square each side. (continued)

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CHAPTER 1


x + 1 = 1 + 2 12x - 5 + (2x - 5) • There is still a radical expression. Isolate the remaining radical.

-x + 5 ( -x + 5)2 x 2 - 10x + 25 x 2 - 10x + 25 x 2 - 18x + 45

= = = = =

212x - 5 (2 12x - 5)2 4(2x - 5) 8x - 20 0

• Square each side.

• Write the equation in standard form. • Factor.

(x - 3)(x - 15) = 0 x - 3 = 0 or x - 15 = 0 x = 3 x = 15

Note In the check at the right, 15 is an example of an extraneous solution. Squaring both sides of the equation created the extraneous solution.

Check:

1x + 1 - 12x 13 + 1 - 12(3) 14 2

- 5 - 5 11 - 1

= 1 ⱨ1 ⱨ1 = 1

1x + 1 - 12x - 5 115 + 1 - 12(15) - 5 116 - 125 4 - 5

= 1 ⱨ1 ⱨ1 Z 1

3 checks as a solution, but 15 does not. The solution is 3. Try Exercise 30, page 120

Rational Exponent Equations n

n

Definition of 2 bn See page 24.

n

Recall that 2bn = ƒ b ƒ when n is a positive even integer and 2bn = b (the absolute value sign is not necessary) when n is a positive odd integer. These results can be restated using rational exponents. (b n)1>n = ƒ b ƒ , n is a positive even integer (b n)1>n = b, n is a positive odd integer For instance, (x 2)1/2 = ƒ x ƒ (n is an even integer) and (x 3)1/3 = x (n is an odd integer). It is important to remember this when solving equations that involve a variable with a rational exponent. Here is an example that shows the details. x 2>3 (x ) [(x 2)1>3]3 x2 (x 2)1>2 2 1>3

= = = = =

16 16 163 163 (163)1>2

3 1>2 ƒ x ƒ = (16 ) 1>2 ƒ x ƒ = 4096 x = 64

• Rewrite x 2>3 as (x 2)1>3. • Cube each side of the equation. • To take the square root, raise each side of the equation to the 1>2 power. • (x 2)1>2 = ƒ x ƒ • Use the fact that if ƒ x ƒ = a (a 7 0), then x = a.

Here is a check. x 2>3 (-64)2>3 [(-64)1>3]2 3- 442 16

= ⱨ ⱨ ⱨ

16 16 • Replace x with 64. 16 16 = 16 • The solution checks.

The solutions are 64 and 64.

x 2>3 (64)2>3 [(64)1>3]2 3442 16

= ⱨ ⱨ ⱨ

16 16 • Replace x with 64. 16 16 = 16 • The solution checks.

1.4


115

Although we could use this procedure every time we solve an equation containing a variable with a rational exponent, we will rely on a shortcut that recognizes the need for the absolute value symbol when the numerator of the rational exponent is an even integer. Here is the solution of x 2>3 = 16, using this shortcut. x 2>3 = 16 (x 2>3)3>2 = 163>2 ƒ x ƒ = 64

• Raise each side of the equation to the 3>2 (the reciprocal of 2>3) power. • Because the numerator in the exponent of x 2>3 is an even number, the absolute value sign is necessary.

x = 64 The solutions are 64 and 64. Now consider x3>4 = 8. We solve this equation as x 3>4 = 8 (x3>4)4>3 = 84>3 x = 16

• Raise each side of the equation to the 4>3 (the reciprocal of 3>4) power. • Because the numerator in the exponent of x 3>4 is an odd number, the absolute value sign is not necessary.

The solution is 16.

EXAMPLE 5

Solve an Equation That Involves a Variable with a Rational Exponent

Solve. a.

2x4>5 - 47 = 115

5x3>4 + 4 = 44

b.

Solution a. 2x4>5 - 47 = 115 2x4>5 = 162 4>5

x (x )

4>5 5>4

• Add 47 to each side.

= 81 = 815>4

• Divide each side by 2.

• Raise each side of the equation to the 5>4 (the reciprocal of 4>5) power.

ƒ x ƒ = 243

• Because the numerator in the exponent of x 4>5 is an even number, use absolute value.

x = 243 The solutions are -243 and 243. b.

5x3>4 + 4 = 44 5x3>4 = 40 x3>4 = 8 3>4 4>3

(x )

= 8

4>3

x = 16

• Subtract 4 from each side. • Divide each side by 5.

• Raise each side of the equation to the 4>3 (the reciprocal of 3>4) power. • Because the numerator in the exponent of x 3>4 is an odd number, do not use absolute value.

Substituting 16 into 5x 3>4 + 4 = 44, we can verify that the solution is 16. Try Exercise 50, page 120

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CHAPTER 1


Equations That Are Quadratic in Form The equation 4x 4 - 25x 2 + 36 = 0 is said to be quadratic in form, which means that it can be written in the form au 2 + bu + c = 0,

a Z 0

where u is an algebraic expression involving x. For example, if we make the substitution u = x 2 (which implies that u 2 = x 4), then our original equation can be written as 4u2 - 25u + 36 = 0 This quadratic equation can be solved for u, and then, using the relationship u = x 2, we can find the solutions of the original equation.

EXAMPLE 6

Solve an Equation That Is Quadratic in Form

Solve: 4x 4 - 25x 2 + 36 = 0 Solution Make the substitutions u = x 2 and u 2 = x 4 to produce the quadratic equation 4u2 - 25u + 36 = 0. Factor the quadratic polynomial on the left side of the equation. (4u - 9)(u - 4) = 0 or u - 4 = 0 4u - 9 = 0 9 u = u = 4 4 Substitute x 2 for u to produce x2 =

The solutions are - 2, -

9 4

or

x2 = 4

x =

9 A4

x = 14

x =

3 2

x = 2

• Check in the original equation.

3 3 , , and 2. 2 2


The following table shows equations that are quadratic in form. Each equation is accompanied by an appropriate substitution that will enable it to be written in the form au2 + bu + c = 0. Equations That Are Quadratic in Form

Substitution

au2 bu c 0 Form

x4 - 8x 2 + 15 = 0

u = x2

u2 - 8u + 15 = 0

x 6 + x 3 - 12 = 0

u = x3

u 2 + u - 12 = 0

1/4

u = x

u2 - 9u + 20 = 0

2x 2/3 + 7x 1/3 - 4 = 0

u = x1/3

2u2 + 7u - 4 = 0

15x-2 + 7x-1 - 2 = 0

u = x-1

15u2 + 7u - 2 = 0

Original Equation

1/2

x

- 9x

1/4

+ 20 = 0

1.4

EXAMPLE 7


117

Solve an Equation That Is Quadratic in Form

Solve: 3x 2>3 - 5x1>3 - 2 = 0 Solution Substituting u for x1>3 gives 3u2 - 5u - 2 = 0 (3u + 1)(u - 2) = 0 or u - 2 3u + 1 = 0 1 u u = 3 1 x1/3 x1/3 = 3 1 x x = 27 A check will verify that both -

• Factor.

= 0 = 2 = 2

• Replace u with x1>3.

= 8

• Cube each side.

1 and 8 are solutions. 27


It is possible to solve equations that are quadratic in form without making a formal substitution. For example, to solve x4 + 5x 2 - 36 = 0, factor the equation and apply the zero product principle. x 4 + 5x 2 - 36 = 0 (x + 9)(x 2 - 4) = 0 or x2 + 9 = 0 2

x2 - 4 = 0

x2 = - 9 x = 3i

x2 = 4 x = 2

Applications of Other Types of Equations EXAMPLE 8

Solve a Uniform Motion Problem

Two buses are transporting a football team to a game that is 120 miles away. The second bus travels at an average speed that is 10 mph faster than the first bus and arrives 1 hour sooner than the first bus. Find the average speed of each bus. Solution Let r be the rate of the first bus. Then r + 10 is the rate of the second bus. Solving the d uniform motion equation d = rt for time gives t = . Thus r Time for first bus =

120 distance = r rate of first bus

Time for second bus =

distance 120 = rate of second bus r + 10

(continued)

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CHAPTER 1


Time for second bus = time for first bus - 1 120 120 - 1 = r r + 10 120 120 r(r + 10)a - 1b b = r(r + 10) a r r + 10

120r = (r + 10) # 120 - r(r + 10) 120r = 120r + 1200 - r2 - 10r

• Multiply each side by the LCD r(r + 10).

r2 + 10r - 1200 = 0

• Write the quadratic equation in standard form.

(r + 40)(r - 30) = 0

• Factor.

Applying the zero product principle, r = - 40 or r = 30. A negative average speed is not possible. The rate of the first bus is 30 miles per hour. The rate of the second bus is 40 miles per hour. Try Exercise 70, page 121

EXAMPLE 9

Solve a Work Problem

A small pipe takes 12 minutes longer than a larger pipe to empty a tank. Working together, they can empty the tank in 1.75 minutes. How long would it take the smaller pipe to empty the tank if the larger pipe is closed? Solution Let t be the time it takes the smaller pipe to empty the tank. Then t - 12 is the time for the larger pipe to empty the tank. Both pipes are open for 1.75 minutes. Therefore, 1.75 1.75 is the portion of the tank emptied by the smaller pipe and is the portion t t - 12 of the tank emptied by the larger pipe. Working together, they empty one tank. Thus 1.75 1.75 + = 1. Solve this equation for t. t t - 12 1.75 1.75 + = 1 t t - 12 t(t - 12)a

1.75 1.75 + b = t(t - 12) # 1 t t - 12

1.75(t - 12) + 1.75t = t 2 - 12t 1.75t - 21 + 1.75t = t 2 - 12t 0 = t 2 - 15.5t + 21

• Multiply each side by the LCD t(t - 12).

• Write the quadratic equation in standard form.

Using the quadratic formula, the solutions of the above equation are t = 1.5 and t = 14. Substituting t = 1.5 into the time for the larger pipe would give a negative time (1.5 - 12 = - 10.5), so that answer is not possible. The time for the smaller pipe to empty the tank with the larger pipe closed is 14 minutes. Try Exercise 72, page 121

1.4

EXAMPLE 10


119

Solve an Application Involving Radicals

A cabin is located in a meadow at the end of a straight driveway 2 kilometers long. A post office is 5 kilometers from the driveway along a straight road. (See the diagram below.) A woman walks 2 kilometers per hour through the meadow to point P and then 5 kilometers per hour along the road to the post office. If it takes the woman 2.25 hours to reach the post office, what is the distance x of point P from the end of the driveway? Round to the nearest tenth of a kilometer.

Cabin

4 + x2

2 km

Post office P x

5−x 5 km

Solution distance Recall that distance = rate * time. Therefore, time = . Using this equation, rate we have 24 + x2 distance from cabin to P = rate of walking in meadow 2 distance from P to post office 5 - x = Time to walk from P to post office = rate of walking on road 5

Time to walk from cabin to P =

The sum of these two times equals the total time (2.25 hours). Thus 5 - x 24 + x2 + = 2.25 2 5 Solve the equation. 5 - x 24 + x 2 + = 2.25 2 5 10a

5 - x 24 + x 2 + b = 10(2.25) 2 5

524 + x 2 + 2(5 - x) = 22.5 524 + x 2 + 10 - 2x = 22.5 524 + x 2 = 12.5 + 2x (5 24 + x 2)2 = (12.5 + 2x)2

• Clear the denominators. • Simplify. • Isolate the radical. • Square each side.

25(4 + x 2) = 4x 2 + 50x + 156.25 100 + 25x 2 = 4x 2 + 50x + 156.25 21x 2 - 50x - 56.25 = 0

• Write in standard form. (continued)

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CHAPTER 1


Using the quadratic formula to solve the last equation, we have x L - 0.8 and x L 3.2. Because x cannot be negative, point P is 3.2 kilometers from the end of the driveway. Try Exercise 84, page 122

EXERCISE SET 1.4 In Exercises 1 to 12, solve each polynomial equation by factoring and using the principle of zero products. 1. x3 - 25x = 0

2. x3 - x = 0

3. x3 - 2x 2 - x + 2 = 0

4. 4x 3 + 4x 2 - 9x - 9 = 0

5. x 3 - 3x 2 - 5x + 15 = 0

6. x 3 - 4x 2 - 2x + 8 = 0

In Exercises 27 to 42, solve the radical equation. 27. 1x - 4 - 6 = 0 28. 110 - x = 4 29. 13x - 5 - 1x + 2 = 1 30. 1x + 7 - 2 = 1x - 9

7. 3x3 + 2x 2 - 27x - 18 = 0

31. 12x + 11 - 12x - 5 = 2

8. 4x 3 + 5x 2 - 16x - 20 = 0 9. x3 - 8 = 0

32. 1x + 7 + 1x - 5 = 6

10. x3 + 8 = 0

33. 1x - 4 + 1x + 1 = 1

11. x4 - 2x 3 + 27x - 54 = 0

34. 12x - 9 + 12x + 6 = 3

12. x + 3x - 8x - 24 = 0 4

3

35. 19x - 20 = x

In Exercises 13 to 26, solve the rational equation. 13.

7x + 18 5 -2= x + 4 x+4

15. 2 +

9 3r = r - 3 r - 3

3 5 17. = x + 2 2x - 7 19. x -

2x + 3 2x + 9 = x + 3 x + 3

5 3 4 = 21. x-3 x-2 x-3

14.

x + 4 -2 + 3 = x - 2 x - 2

16.

t 4 + 3 = t - 4 t - 4

4 7 18. = y + 2 y - 4 20. 2x +

4 7 5 + = 22. x - 1 x + 7 x - 1

x x + 2 x - 12 = 23. x + 1 x - 1 x + 1 24.

2x + 1 x - 4 10x + 13 = x - 3 x + 5 x + 5

25.

4 - 3x 3x + 2 4x - 5 + = 2x + 1 x + 2 2x + 1

5x + 3 x - 1 2x + 3 = 26. 3x - 2 x - 3 x - 3

3 -7x + 10 = x -1 x-1

36. x = 112x - 35 37. 12x - 1 - 1x - 1 = 1 38. 16 - x + 15x + 6 = 6 39. 1-7x + 2 + x = 2 40. 1- 9x - 9 + x = 1 3

41. 2x 3 - 2x - 13 = x - 1 3

42. 2x 3 - 5x - 17 = x - 1

In Exercises 43 to 52, solve each equation containing a rational exponent on the variable. 43. x1>3 = 2

44. x1>2 = 5

45. x 2>5 = 9

46. x4>3 = 81

47. x 3>2 = 27

48. x 3>4 = 125

49. 3x 2>3 - 16 = 59

50. 4x4>5 - 27 = 37

51. 4x3>4 - 31 = 77

52. 4x4>5 - 54 = 270

1.4

In Exercises 53 to 68, find all real solutions of each equation by first rewriting each equation as a quadratic equation. 53. x4 - 9x 2 + 14 = 0

54. x4 - 10x 2 + 9 = 0

55. 2x4 - 11x 2 + 12 = 0

56. 6x4 - 7x 2 + 2 = 0

57. x6 + x 3 - 6 = 0

58. 6x6 + x 3 - 15 = 0

59. x1>2 - 3x1>4 + 2 = 0

60. 2x1>2 - 5x1>4 - 3 = 0

61. 3x 2>3 - 11x 1>3 - 4 = 0

62. 6x 2>3 - 7x 1>3 - 20 = 0

63. x4 + 8x 2 - 9 = 0

64. 4x4 + 7x 2 - 36 = 0

65. x 2>5 - x1>5 - 2 = 0

66. 2x 2>5 - x1>5 = 6

67. 9x - 521x + 64 = 0

68. 8x - 38 1x + 9 = 0


121

74. Parallel Processing Parallel processing uses two or more

computers, working together, to solve a single problem. Using parallel processing, two computers can solve a problem in 12 minutes. If, working alone, one computer can solve a problem in 7 minutes less than the time needed by the second computer, how long would it take the faster computer working alone to solve the problem? In Exercises 75 and 76, the depth s from the opening of a well to the water below can be determined by measuring the total time between the instant you drop a stone and the moment you hear it hit the water. The time, in seconds, it takes the stone to hit the water is given by 1s /4, where s is measured in feet. The time, also in seconds, required for the sound of the impact to travel up to your ears is given by s /1100. Thus the total time T, in seconds, between the instant you drop the stone and the moment you hear its impact is T

s 1s 4 1100

69. Boating A small fishing boat heads to a point 24 miles down-

river and then returns. The river’s current moves at 3 miles per hour. If the trip up and back takes 6 hours and the boat keeps a constant speed relative to the water, what is the speed of the boat? (Hint: If v is the speed of the boat, then its speed downriver is (v + 3) miles per hour and its speed upriver is (v - 3) miles per hour.)

s Time of fall =

s 4

Time for sound s = to travel up 1100

70. Running Maureen can run at a rate that is 2 miles per hour

faster than her friend Hector’s rate. While training for a mini marathon, Maureen gives Hector a half-hour head start and then begins chasing Hector on the same route. If Maureen passes Hector 12 miles from the starting point, how fast is each running? 71. Fence Construction A worker can build a fence in 8 hours.

75. Time of Fall One of the world’s deepest water wells is 7320

Working together, the worker and an assistant can build the fence in 5 hours. How long should it take the assistant, working alone, to build the fence?

feet deep. Find the time between the instant you drop a stone and the time you hear it hit the water if the surface of the water is 7100 feet below the opening of the well. Round your answer to the nearest tenth of a second.

72. Roof Repair A roofer and an assistant can repair a roof

together in 6 hours. Working alone, the assistant can repair the roof in 14 hours. If both the roofer and the assistant work together for 2 hours and then the assistant is left alone to finish the job, how much longer should the assistant need to finish the repairs?

76. Depth of a Well Find the depth from the opening of a well to

the water level if the time between the instant you drop a stone and the moment you hear its impact is 3 seconds. Round your answer to the nearest foot. 77. Radius of a Cone A conical funnel has a height h of 4 inches

73. Painting a Room An experienced painter and an apprentice

can paint a room in 6 hours. Working alone, it takes the apprentice 5 hours less than twice the time needed by the experienced painter to paint the room. How long does it take the experienced painter to paint the room?

and a lateral surface area L of 15p square inches. Find the radius r of the cone. (Hint: Use the formula L = pr 2r 2 + h2.) 78. Diameter of a Cone As flour is poured onto a table, it forms

a right circular cone whose height is one-third the diameter of

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CHAPTER 1


the base. What is the diameter of the base when the cone has a volume of 192 cubic inches? Round to the nearest tenth of an inch.

B

C

2

79. Precious Metals A solid silver sphere has a diameter of 8 mil-

limeters, and a second silver sphere has a diameter of 12 millimeters. The spheres are melted down and recast to form a single cube. What is the length s of each edge of the cube? Round your answer to the nearest tenth of a millimeter.

A

84.

80. Pendulum The period T of a pendulum is the time it takes the

pendulum to complete one swing from left to right and back. For a pendulum near the surface of Earth, L T = 2p A 32 where T is measured in seconds and L is the length of the pendulum in feet. Find the length of a pendulum that has a period of 4 seconds. Round to the nearest tenth of a foot.

1

E

1

F

D

Providing Power A power station is on one side of a river that is 1 mile wide, and a factory is 6 miles downstream on the other side of the river. The cost is $0.125 million per mile to run power lines over land and $0.2 million per mile to run power lines under water. How far over land should the power line be run if the total cost of the project is to be $1 million? Round to the nearest tenth of a mile. See the diagram below. (Hint: Cost for a segment equals cost per mile times the number of miles.)

Power station

81. Distance to the Horizon On a ship, the distance d that you

can see to the horizon is given by d = 11.5h, where h is the height of your eye measured in feet above sea level and d is measured in miles. How high is the eye level of a navigator who can see 14 miles to the horizon? Round to the nearest foot.

x

P

6−x 1 mi

1 + (6 − x)2 6 mi

As mentioned in the chapter opener, the golden mean, F, occurs in many situations. The exact value of F is 1 15 . Exercises 82 and 83 involve the golden mean. 2 82. The golden mean can be found by dividing a line segment into

two parts so that the ratio of the length of the longer part to the length of the shorter part equals the ratio of the length of the whole segment to the length of the longer part. Use the accompanying diagram to write an equation that represents this relationship. Solve the equation. Show that the positive solution of the equation makes the ratios described in the first sentence equal to f.

85.

Factory

Triathlon Training To prepare for a triathlon, a person swims across a river to point P and then runs along a path as shown in the diagram below.

16 − x

P x

Run Swim 16 + x 2

1 x

1−x

4 km

16 km

83. Here is a method of constructing a golden rectangle, a rectangle

in which the ratio of the length to the width is f. Begin with a square whose sides are 2 units. (You can use any length.) From the midpoint of one side, draw a line segment to an opposite vertex. Using a compass, create an arc that intersects an extension of the base of the square. Now complete the rectangle. AD (See the following diagram.) Show that = f. AB

The person swims at 7 kilometers per hour and runs at 22 kilometers per hour. For what distance x is the total time for swimming and running 2 hours? Round to the nearest tenth of a kilometer. (Hint: Time swimming + time running = 2 hours, distance and = time.) rate

1.5

SECTION 1.5 Properties of Inequalities Compound Inequalities Absolute Value Inequalities Polynomial Inequalities Rational Inequalities Applications of Inequalities

INEQUALITIES

123

Inequalities PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A6.

PS1. Find: 5x ƒ x 7 26 ¨ 5x ƒ x 7 56 [P.1] PS2. Evaluate 3x 2 - 2x + 5 for x = - 3. [P.1] PS3. Evaluate

x + 3 for x = 7. [P.1/P.5] x - 2

PS4. Factor: 10x 2 + 9x - 9 [P.4] PS5. For what value of x is

x - 3 undefined? [P.1/P.5] 2x - 7

PS6. Solve: 2x 2 - 11x + 15 = 0 [1.3]

Properties of Inequalities

Math Matters Another property of inequalities, called the transitive property, states that for real numbers a, b, and c, if a 7 b and b 7 c, then a 7 c. We say that the relationship “is greater than” is a transitive relationship. Not all relationships are transitive relationships. For instance, consider the game of scissors, paper, rock. In this game, scissors beats paper and paper beats rock, but scissors does not beat rock!

In Section P.1 we used inequalities to describe the order of real numbers and to represent subsets of real numbers. In this section we consider inequalities that involve a variable. In particular, we consider how to determine which real numbers make an inequality a true statement. The solution set of an inequality is the set of all real numbers for which the inequality is a true statement. For instance, the solution set of x + 1 7 4 is the set of all real numbers greater than 3. Two inequalities are equivalent inequalities if they have the same solution set. We can solve many inequalities by producing simpler but equivalent inequalities until the solutions are readily apparent. To produce these simpler but equivalent inequalities, we often apply the following properties.

Properties of Inequalities Let a, b, and c be real numbers. 1. Addition–Subtraction Property If the same real number is added to or subtracted from each side of an inequality, the resulting inequality is equivalent to the original inequality. a 6 b and a + c 6 b + c are equivalent inequalities. 2. Multiplication–Division Property a. Multiplying or dividing each side of an inequality by the same positive real number produces an equivalent inequality. If c 7 0, then a 6 b and ac 6 bc are equivalent inequalities. b.

Multiplying or dividing each side of an inequality by the same negative real number produces an equivalent inequality provided the direction of the inequality symbol is reversed. If c 6 0, then a 6 b and ac 7 bc are equivalent inequalities.

124

CHAPTER 1


EXAMPLE

Property 1

Adding or subtracting the same number to (from) each side of an inequality produces an equivalent inequality. x - 4 6 7 x - 4 + 4 6 7 + 4 x 6 11

x + 3 7 5 x + 3 - 3 7 5 - 3 x72

Property 2a Multiplying or dividing each side of an inequality by the same positive number produces an equivalent inequality. 2 x 7 -4 3 3 3#2 x 7 ( - 4) 2 3 2 x 7 -6

3x 6 12 3x 12 6 3 3 x 6 4

Property 2b Multiplying or dividing each side of an inequality by the same negative number produces an equivalent inequality provided the direction of the inequality symbol is reversed. -2x 6 6

-

- 2x 6 7 -2 -2 x 7 -3

-

3 x73 4

3 4 4 a- xb 6 a- b3 3 4 3 x 6 -4

Note the difference between Property 2a and Property 2b. Property 2a states that an equivalent inequality is produced when each side of a given inequality is multiplied (divided) by the same positive real number and the inequality symbol is not changed. By contrast, Property 2b states that when each side of a given inequality is multiplied (divided) by a negative real number, we must reverse the direction of the inequality symbol to produce an equivalent inequality. For instance, multiplying both sides of - b 6 4 by -1 produces the equivalent inequality b 7 - 4. (We multiplied both sides of the first inequality by - 1, and we changed the “less than” symbol to a “greater than” symbol.)

EXAMPLE 1

Solve Linear Inequalities

Solve each of the following inequalities. a. Study tip Solutions of inequalities can be stated using set-builder notation or interval notation. For instance, the solutions of 2x + 1 6 7 can be written in set-builder notation as 5x ƒ x 6 36 or in interval notation as (- q , 3).

2x + 1 6 7

Solution a. 2x + 1 6 7 2x 6 6 x 6 3

b.

- 3x - 2 … 10

• Add -1 to each side and keep the inequality symbol as is. • Divide each side by 2 and keep the inequality symbol as is.

The inequality 2x + 1 6 7 is true for all real numbers less than 3. In set-builder notation, the solution set is given by 5x ƒ x 6 36. In interval notation, the solution set is (- q , 3). See the following figure. −1

0

1

2

3

4

5

6

7

1.5

b.

125

-3x - 2 … 10 -3x … 12 x Ú -4

Interval Notation See page 6.

INEQUALITIES

• Add 2 to each side and keep the inequality symbol as is. • Divide each side by -3 and reverse the direction of the inequality symbol.

The inequality -3x - 2 … 10 is true for all real numbers greater than or equal to -4. In set-builder notation, the solution set is given by 5x ƒ x Ú - 46. In interval notation, the solution set is 3 -4, q ). See the following figure. − 6 − 5 − 4 −3

−2

−1

0

1

2


Compound Inequalities A compound inequality is formed by joining two inequalities with the connective word and or or. The inequalities shown below are compound inequalities. x + 1 7 3 x + 3 7 5

and or

2x - 11 6 7 x - 1 6 9

The solution set of a compound inequality with the connective word or is the union of the solution sets of the two inequalities. The solution set of a compound inequality with the connective word and is the intersection of the solution sets of the two inequalities.

EXAMPLE 2

Solve Compound Inequalities

Solve each compound inequality. Write each solution in set-builder notation. a.

2x 6 10 or x + 1 7 9

Solution a. 2x 6 10 x 6 5

or

b.

x + 3 7 4 and 2x + 1 7 15

x + 1 7 9 x 7 8 5x ƒ x 7 86

5x ƒ x 6 56 5x ƒ x 6 56 ´ 5x ƒ x 7 86 = 5x ƒ x 6 5 or x 7 86

b.

x + 3 7 4

and

• Solve each inequality. • Write each solution as a set. • Write the union of the solution sets.

2x + 1 7 15

x 7 1

2x 7 14

x 7 7 5x ƒ x 7 16 5x ƒ x 7 76 5x ƒ x 7 16 ¨ 5x ƒ x 7 76 = 5x ƒ x 7 76

• Solve each inequality. • Write each solution as a set. • Write the intersection of the solution sets.

Try Exercise 10, page 133 Question • What is the solution set of the compound inequality x 7 1 or x 6 3? Answer • The solution is the set of all real numbers. Using interval notation, the solution set is

written as ( - q , q ).

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CHAPTER 1


The inequality given by

Note

12 6 x + 5 6 19

We reserve the notation a 6 b 6 c to mean a 6 b and b 6 c. Thus the solution set of 2 7 x 7 5 is the empty set, because there are no numbers less than 2 and greater than 5.

is equivalent to the compound inequality 12 6 x + 5 and x + 5 6 19. You can solve 12 6 x + 5 6 19 by either of the following methods. Method 1 Find the intersection of the solution sets of the inequalities 12 6 x + 5 and x + 5 6 19. 12 6 x + 5

and

x + 5 6 19 x 6 14

7 6 x

The solution set is 5x ƒ x 7 76 ¨ 5x ƒ x 6 146 = 5x ƒ 7 6 x 6 146.

Note The compound inequality a 6 b and b 6 c can be written in the compact form a 6 b 6 c. However, the compound inequality a 6 b or b 7 c cannot be expressed in a compact form.

Method 2 Subtract 5 from each of the three parts of the inequality. 12 6 x + 5 6 19 12 - 5 6 x + 5 - 5 6 19 - 5 x 6 14 7 6

The solution set is 5x ƒ 7 6 x 6 146.

Absolute Value Inequalities −5 −4 −3 −2 −1

0

1

2

3

4

5

2

3

4

5

ƒ x - 1ƒ 6 3

Figure 1.6 −5 −4 −3 −2 −1

0

1

ƒ x - 1ƒ 7 3

Figure 1.7

The solution set of the absolute value inequality ƒ x - 1 ƒ 6 3 is the set of all real numbers whose distance from 1 is less than 3. Therefore, the solution set consists of all numbers between - 2 and 4. See Figure 1.6. In interval notation, the solution set is ( -2, 4). The solution set of the absolute value inequality ƒ x - 1 ƒ 7 3 is the set of all real numbers whose distance from 1 is greater than 3. Therefore, the solution set consists of all real numbers less than -2 or greater than 4. See Figure 1.7. In interval notation, the solution set is (- q , -2) ´ (4, q ). The following properties are used to solve absolute value inequalities.

Properties of Absolute Value Inequalities Note

For any variable expression E and any nonnegative real number k,

Some inequalities have a solution set that consists of all real numbers. For example, ƒ x + 9 ƒ Ú 0 is true for all values of x. Because an absolute value is always nonnegative, the inequality is always true.

ƒEƒ … k ƒEƒ Ú k

if and only if if and only if

-k … E … k E … - k or E Ú k

These properties also hold true when the 6 symbol is substituted for the symbol and when the 7 symbol is substituted for the symbol. EXAMPLE

If ƒ x ƒ 6 5, then - 5 6 x 6 5. If ƒ x ƒ 7 7, then x 6 - 7 or x 7 7.

In Example 3, we use the preceding properties to solve absolute value inequalities.

EXAMPLE 3

Solve Absolute Value Inequalities

Solve each of the following inequalities. a.

ƒ 2 - 3x ƒ 6 7

b.

ƒ 4x - 3 ƒ Ú 5

1.5

INEQUALITIES

127

Solution a. ƒ 2 - 3x ƒ 6 7 if and only if - 7 6 2 - 3x 6 7. Solve this compound inequality. -7 6 2 - 3x 6 7 -9 6

5 − 3 −4 −3

−2 −1

0

1

2

3

3 7

4

5 a - , 3b 3

- 3x x

6 5

• Subtract 2 from each of the three parts of the inequality.

5 7 3

1

• Multiply each part of the inequality by - and reverse 3 the inequality symbols.

5 In interval notation, the solution set is given by a- , 3b. See Figure 1.8. 3

Figure 1.8

b.

ƒ 4x - 3 ƒ Ú 5 implies 4x - 3 … - 5 or 4x - 3 Ú 5. Solving each of these inequalities produces 4x - 3 … - 5

1 − 2 −4 −3

−2 −1

0

1

2

1 a - q , - d ´ [2, q ) 2

Figure 1.9

3

4

or

4x - 3 Ú 5

4x … - 2

4x Ú 8

1 2

x Ú 2

x … -

1 The solution set is a- q , - d ´ 32, q ). See Figure 1.9. 2 Try Exercise 18, page 133

Polynomial Inequalities Any value of x that causes a polynomial in x to equal zero is called a zero of the polynomial. For example, - 4 and 1 are both zeros of the polynomial x 2 + 3x - 4 because (-4)2 + 3( -4) - 4 = 0 and 12 + 3 # 1 - 4 = 0.

Sign Property of Polynomials Polynomials in x have the following property: for all values of x between two consecutive real zeros, all values of the polynomial are positive or all values of the polynomial are negative.

In our work with inequalities that involve polynomials, the real zeros of the polynomial are also referred to as critical values of the inequality. On a number line, the critical values of an inequality separate the real numbers that make the inequality true from those that make it false. For instance, to solve the inequality x 2 + 3x - 4 6 0, we begin by solving the equation x 2 + 3x - 4 = 0 to find the real zeros of the polynomial. x 2 + 3x - 4 = 0 (x + 4)(x - 1) = 0 or x - 1 = 0 x + 4 = 0 x = -4 x = 1

128

CHAPTER 1

−5 −4 −3 −2 −1


0

1

2

3

4

5

Figure 1.10

The real zeros are - 4 and 1. They are the critical values of the inequality x 2 + 3x - 4 6 0, and they separate the real number line into three intervals, as shown in Figure 1.10. To determine the intervals in which x 2 + 3x - 4 is less than 0, pick a number called a test value from each of the three intervals and then determine whether x 2 + 3x - 4 is less than 0 for each of these test values. For example, in the interval ( - q , - 4), pick a test value of -5. Then x 2 + 3x - 4 = ( - 5)2 + 3(- 5) - 4 = 6 Because 6 is not less than 0, by the sign property of polynomials, no number in the interval (- q , - 4) makes x 2 + 3x - 4 less than 0. Now pick a test value from the interval ( -4, 1)—say, 0. When x = 0, x 2 + 3x - 4 = 0 2 + 3(0) - 4 = - 4 Because -4 is less than 0, by the sign property of polynomials, all numbers in the interval (- 4, 1) make x 2 + 3x - 4 less than 0. If we pick a test value of 2 from the interval (1, q ), then x 2 + 3x - 4 = (2)2 + 3(2) - 4 = 6 Because 6 is not less than 0, by the sign property of polynomials, no number in the interval (1, q ) makes x 2 + 3x - 4 less than 0. The following table is a summary of our work. Interval ( - q , - 4)

−5 −4 − 3 −2 −1

0

1

Figure 1.11

2

3

4

5

?

x 2 3x 4m2. Try Exercise 32, page 143

Joint and Combined Variations Some variations involve more than two variables.

Definition of Joint Variation The variable z varies jointly as the variables x and y if and only if z = kxy where k is a constant.

EXAMPLE 5

Solve a Joint Variation

The cost of insulating the ceiling of a house varies jointly as the thickness of the insulation and the area of the ceiling. It costs $175 to insulate a 2100-square-foot ceiling with insulation that is 4 inches thick. Find the cost of insulating a 2400-square-foot ceiling with insulation that is 6 inches thick.

1.6

VARIATION AND APPLICATIONS

141

Solution Because the cost C varies jointly as the area A of the ceiling and the thickness T of the insulation, we know C = kAT. Using the fact that C = 175 when A = 2100 and T = 4 gives us 175 = k(2100)(4)

which implies

Consequently, the specific formula for C is C =

k =

175 1 = (2100)(4) 48

1 AT. Now, when A = 2400 and 48

T = 6, we have C =

1 (2400)(6) = 300 48

The cost of insulating the 2400-square-foot ceiling with 6-inch insulation is $300. Try Exercise 34, page 143

Combined variations involve more than one type of variation.

EXAMPLE 6

Solve a Combined Variation

The weight that a horizontal beam with a rectangular cross section can safely support varies jointly as the width and the square of the depth of the cross section and inversely as the length of the beam. See Figure 1.21. If a 10-foot-long 4- by 4-inch beam safely supports a load of 256 pounds, what load L can be safely supported by a beam made of the same material and with a width w of 4 inches, a depth d of 6 inches, and a length l of 16 feet? Solution The general variation equation is L = k # w

L d

wd 2 . Using the given data yields l

256 = k #

4(42) 10

Solving for k produces k = 40, so the specific formula for L is l

L = 40 #

wd 2 l

Substituting 4 for w, 6 for d, and 16 for l gives L = 40 #

Figure 1.21

4(6 2) = 360 pounds 16


EXERCISE SET 1.6 In Exercises 1 to 12, write an equation that represents the relationship between the given variables. Use k as the variation constant.

3. y varies inversely as x. 4. p is inversely proportional to q.

1. d varies directly as t.

5. m varies jointly as n and p.

2. r varies directly as the square of s.

6. t varies jointly as r and the cube of s.

142

CHAPTER 1


22. Hooke’s Law Hooke’s Law states that the distance a spring

7. V varies jointly as l, w, and h. 8. u varies directly as v and inversely as the square of w. 9. A is directly proportional to the square of s.

stretches varies directly as the weight on the spring. A weight of 80 pounds stretches a spring 6 inches. How far will a weight of 100 pounds stretch the spring?

10. A varies jointly as h and the square of r. 11. F varies jointly as m1 and m2 and inversely as the square

of d. 12. T varies jointly as t and r and the square of a.

6 in.

In Exercises 13 to 20, write the equation that expresses the relationship between the variables, and then use the given data to solve for the variation constant. 13. y varies directly as x, and y = 64 when x = 48. 14. m is directly proportional to n, and m = 92 when n = 23. 15. r is directly proportional to the square of t, and r = 144 when

t = 108.

16. C varies directly as r, and C = 94.2 when r = 15. 17. T varies jointly as r and the square of s, and T = 210 when

r = 30 and s = 5.

80 lb

23. Semester Hours vs. Quarter Hours A student plans to trans-

fer from a college that uses the quarter system to a college that uses the semester system.The number of semester hours a student receives credit for is directly proportional to the number of quarter hours the student has earned. A student with 51 quarter hours is given credit for 34 semester hours. How many semester hours credit should a student receive after completing 93 quarter hours? 24. Pressure and Depth The pressure a liquid exerts at a given

18. u varies directly as v and inversely as the square root of w, and

u = 0.04 when v = 8 and w = 0.04. 19. V varies jointly as l, w, and h, and V = 240 when l = 8,

point on a submarine is directly proportional to the depth of the point below the surface of the liquid. If the pressure at a depth of 3 feet is 187.5 pounds per square foot, find the pressure at a depth of 7 feet.

w = 6, and h = 5.

25. Amount of Juice Contained in a Grapefruit The amount of 20. t varies directly as the cube of r and inversely as the square root

of s, and t = 10 when r = 5 and s = 0.09. 21. Charles’s Law Charles’s Law states that the volume V occu-

pied by a gas (at a constant pressure) is directly proportional to its absolute temperature T. An experiment with a balloon shows that the volume of the balloon is 0.85 liter at 270 K (absolute temperature).1 What will the volume of the balloon be when its temperature is 324 K? Gas expands and the balloon inflates

juice in a grapefruit is directly proportional to the cube of its diameter. A grapefruit with a 4-inch diameter contains 6 fluid ounces of juice. How much juice is contained in a grapefruit with a 5-inch diameter? Round to the nearest tenth of a fluid ounce. 26. Motorcycle Jump The range of a projectile is directly propor-

tional to the square of its velocity. If a motorcyclist can make a jump of 140 feet by coming off a ramp at 60 mph, find the distance the motorcyclist could expect to jump if the speed coming off the ramp were increased to 65 mph. Round to the nearest tenth of a foot. 27. Period of a Pendulum The period T of a pendulum (the time

it takes the pendulum to make one complete oscillation) varies directly as the square root of its length L. A pendulum 3 feet long has a period of 1.8 seconds. Ice water 270 K 1

Hot water 324 K

Absolute temperature is measured on the Kelvin scale. A unit (called a kelvin) on the Kelvin scale is the same measure as a degree on the Celsius scale; however, 0 on the Kelvin scale corresponds to -273°C.

a. Find the period of a pendulum that is 10 feet long. Round

to the nearest tenth of a second. b. What is the length of a pendulum that beats seconds (that

is, has a 2-second period)? Round to the nearest tenth of a foot.

1.6

VARIATION AND APPLICATIONS

143

28. Area of a Projected Picture The area of a projected picture

34. Safe Load The load L that a horizontal beam can safely sup-

on a movie screen varies directly as the square of the distance from the projector to the screen. If a distance of 20 feet produces a picture with an area of 64 square feet, what distance produces a picture with an area of 100 square feet?

port varies jointly as the width w and the square of the depth d. If a beam with a width of 2 inches and a depth of 6 inches safely supports up to 200 pounds, how many pounds can a beam of the same length that has width 4 inches and depth 4 inches be expected to support? Round to the nearest pound. Assume the two beams are made of the same material. 35. Ideal Gas Law The Ideal Gas Law states that the volume V of

a gas varies jointly as the number of moles of gas n and the absolute temperature T and inversely as the pressure P. What happens to V when n is tripled and P is reduced by a factor of one-half? 4 ft 3 ft

36. Maximum Load The maximum load a cylindrical column

of circular cross section can support varies directly as the fourth power of the diameter and inversely as the square of the height. If a column 2 feet in diameter and 10 feet high supports up to 6 tons, how many tons can a column 3 feet in diameter and 14 feet high support? Round to the nearest tenth of a ton. Assume the two columns are made of the same material.

2 ft 1 ft 0

29. Speed of a Bicycle Gear The speed of a bicycle gear, in rev-

olutions per minute, is inversely proportional to the number of teeth on the gear. If a gear with 64 teeth has a speed of 30 revolutions per minute, what will be the speed of a gear with 48 teeth? 30. Vibration of a Guitar String The frequency of vibration of a

guitar string under constant tension varies inversely as the length of the string. A guitar string with a length of 20 inches has a frequency of 144 vibrations per second. Find the frequency of a guitar string with a length of 18 inches. Assume the tension is the same for both strings. 31. Jet Engine Noise The sound intensity of a jet engine, meas-

ured in watts per meter squared (W>m2), is inversely proportional to the square of the distance between the engine and an airport ramp worker. For a certain jet, the sound intensity measures 0.5 W>m2 at a distance of 7 meters from the ramp worker. What is the sound intensity for a ramp worker 10 meters from the jet? 32. Illumination The illumination a source of light provides is

inversely proportional to the square of the distance from the source. If the illumination at a distance of 10 feet from the source is 50 footcandles, what is the illumination at a distance of 15 feet from the source? Round to the nearest tenth of a footcandle. 33. Volume Relationships The volume V of a right circular cone

varies jointly as the square of the radius r and the height h. Tell what happens to V when

37.

Earned Run Average A pitcher’s earned run average (ERA) is directly proportional to the number of earned runs the pitcher has allowed and is inversely proportional to the number of innings the pitcher has pitched. During the 2002 season, Randy Johnson of the Arizona Diamondbacks had an ERA of 2.32. He allowed 67 earned runs in 260 innings. During the same season, Tom Glavine of the Atlanta Braves allowed 74 earned runs in 224.2 innings. What was Glavine’s ERA for the 2002 season? Round to the nearest hundredth. (Source: MLB.com)

38. Safe Load The load L a horizontal beam can safely support

varies jointly as the width w and the square of the depth d and inversely as the length l. If a 12-foot beam with a width of 4 inches and a depth of 8 inches safely supports 800 pounds, how many pounds can a 16-foot beam that has a width of 3.5 inches and a depth of 6 inches be expected to support? Round to the nearest pound. Assume the two beams are made of the same material. 39. Force, Speed, and Radius Relationships The force needed

to keep a car from skidding on a curve varies jointly as the weight of the car and the square of its speed and inversely as the radius of the curve. It takes 2800 pounds of force to keep an 1800-pound car from skidding on a curve with a radius of 425 feet at 45 mph. What force is needed to keep the same car from skidding when it takes a similar curve with a radius of 450 feet at 55 mph? Round to the nearest 10 pounds.

a. r is tripled b. h is tripled c. both r and h are tripled

40. Stiffness of a Beam A cylindrical log is to be cut so that it

will yield a beam that has a rectangular cross section of depth d and width w. The stiffness of a beam of given length is

144

CHAPTER 1


directly proportional to the width and the cube of the depth. The diameter of the log is 18 inches. What depth will yield the “stiffest” beam: d = 10 inches, d = 12 inches, d = 14 inches, or d = 16 inches?

18 in.

distance d between the planet and the Sun. The Earth, which averages 93 million miles from the Sun, completes one revolution in 365 days. Find the average distance from the Sun to Mars if Mars completes one revolution about the Sun in 686 days. Round to the nearest million miles.

d

Earth Sun

w

One revolution every 365 days Average distance from the Sun: 93 million miles

Mars One revolution every 686 days

41. Kepler’s Third Law Kepler’s Third Law states that the time

T needed for a planet to make one complete revolution about 3 the Sun is directly proportional to the power of the average 2


Use a Graphing Calculator to Solve Equations Most graphing calculators can be used to solve equations. The following example shows how to solve an equation using the solve( feature that is available on a TI83/TI-83 Plus/TI-84 Plus graphing calculator. The calculator display that follows indicates that the solution of 2x - 17 = 0 is 8.5. Expression that is set equal to 0 Variable for which you wish to solve Initial guess at a solution

solve(2X−17,X,8) 8.5

Actual solution

The calculator display above was produced by the following keystrokes. Press 2nd [catalog] S (scroll down to solve( ) ENTER . Now enter 2 X,T, ,n 17 , X,T, ,n , 8 ) ENTER . In this example, the 8.5 represents the solution of 2x - 17 = 0, which is close to our initial guess of 8. Because 2x - 17 = 0 has only one solution, we are finished. Note that the solve( feature can be used only to solve equations of the form Expression = 0 Also, you are required to indicate the variable you wish to solve for, and you must enter an initial guess. In the preceding display, we entered X as the variable and 8 as our initial guess. The solve( feature can be used only to find real solutions. Also, the solve( feature finds only one solution each time the solution procedure is applied. If you know that an

CHAPTER 1 TEST PREP

solve(2X2−X−15,X,2) 3 solve(2X2−X−15,X,-1) -2.5

145

equation has two real solutions, then you need to apply the solution procedure twice. Each time you must enter an initial guess that is close to the solution you are trying to find. The calculator display to the left indicates that the solutions of 2x 2 - x - 15 = 0 are 3 and -2.5. To find these solutions, we first used the solve( feature with an initial guess of 2, and we then used the solve( feature with an initial guess of - 1. The chapters that follow will illustrate additional techniques and calculator procedures that can be used to solve equations.

CHAPTER 1 TEST PREP The following test prep table summarizes essential concepts in this chapter. The references given in the right-hand column list Examples and Exercises that can be used to test your understanding of a concept.

1.1 Linear and Absolute Value Equations Linear or first-degree equation A linear or first-degree equation in a single variable is one for which all of the variable expressions have degree one. To solve a linear equation, apply the properties of real numbers and the properties of equality to produce equivalent equations until an equation in the form variable = constant is reached.


Clearing fractions When solving an equation containing fractions, it is helpful to clear the equation of fractions by multiplying each side of the equation by the LCD of all denominators.


Linear absolute value equation A linear absolute value equation in the variable x is one that can be written in the form ƒ ax + b ƒ = c.


1.2 Formulas and Applications Formula A formula is an equation that expresses known relationships between two or more variables. Applications Some of the applications of linear equations include • Geometry • Business • Investment • Uniform motion • Percent mixture problems • Value mixture problems • Work problems


See Examples 3 and 4, pages 86 and 87, and then try Exercises 58 and 60, page 148. See Example 5, page 88, and then try Exercise 63, page 149. See Example 6, page 88, and then try Exercise 64, page 149. See Example 7, page 89, and then try Exercise 65, page 149. See Example 8, page 90, and then try Exercise 67, page 149. See Example 9, page 91, and then try Exercise 69, page 149. See Example 10, page 91, and then try Exercise 71, page 149.

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CHAPTER 1


1.3 Quadratic Equations Quadratic equation A quadratic equation in the variable x is one that can be written in the form ax 2 + bx + c = 0, where a Z 0. Some ways in which a quadratic equation can be solved include • Factoring and using the zero product principle • Using the square root procedure • Completing the square • Using the quadratic formula

See Example 1, page 97, and then try Exercise 14, page 148. See Example 2, page 98, and then try Exercise 15, page 148. See Example 4, page 100, and then try Exercise 18, page 148. See Example 5, page 101, and then try Exercise 20, page 148.

Discriminant The discriminant of ax 2 + bx + c = 0, where a, b, and c are real numbers and a Z 0, is the value of the expression b2 - 4ac. If b 2 - 4ac 7 0, then the quadratic equation has two distinct real solutions. If b 2 - 4ac = 0, then the quadratic equation has one real solution. If b 2 - 4ac 6 0, then the quadratic equation has two distinct nonreal complex solutions.


Pythagorean Theorem The Pythagorean Theorem states that if a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then a 2 + b 2 = c 2.


Applications Quadratic equations can be applied to a variety of situations.


1.4 Other Types of Equations Polynomial equations Some polynomial equations of a degree greater than two can be solved by factoring.


Rational equations A rational equation is one that contains rational expressions. These equations can be solved by multiplying each side of the equation by the LCD of the denominators of the rational expressions.


Radical equations A radical equation is one that involves one or more radical expressions.


Equations with a rational exponent An equation of the form ax p>q + b = c can be solved by isolating x p>q and then raising each side of the equation to the q>p power.


Equations that are quadratic in form An equation that is quadratic in form—that is, an equation that can be written as au2 + bu + c = 0—can be solved using any of the techniques used to solve a quadratic equation.


CHAPTER 1 TEST PREP

Applications


1.5 Inequalities Linear or first-degree inequality A linear or first-degree inequality in a single variable is one for which all variable expressions have degree one. To solve a linear inequality, apply the properties of real numbers and the properties of inequalities. (See page 123.)


Compound inequality A compound inequality is formed by joining two inequalities with the connective word and or or.


Absolute value inequality An absolute value inequality can be solved by rewriting it as a compound inequality.


Polynomial inequality A polynomial inequality can be solved by using the sign property of polynomials. (See page 127.)


Rational inequality A rational inequality can be solved by using the critical value method. (See page 130.)


Applications


1.6 Variation Direct variation The variable y varies directly as the variable x, or y is directly proportional to x, if and only if y = kx, where k is a constant.


Direct variation as the nth power The variable y varies directly as the nth power of the variable x if and only if y = kx n, where k is a constant.


Inverse variation The variable y varies inversely as the variable x, or y k is inversely proportional to x, if and only if y = , where k is a constant. x


Inverse variation as the nth power The variable y varies inversely as the nth power of the variable x, or y is inversely proportional to the k nth power of x, if and onlyif y = n , where k is a constant. x


Joint variation The variable z varies jointly as the variables x and y if and only if z = kxy, where k is a constant.


147

148

CHAPTER 1


CHAPTER 1 REVIEW EXERCISES In Exercises 1 to 20, solve each equation.

In Exercises 23 to 40, solve each equation.

1. 4 - 5x = 3x + 14

23. 3x 3 - 5x 2 = 0

2. 7 - 5(1 - 2x) = 3(2x + 1)

24. 2x 3 - 8x = 0

3.

4.

4x 4x - 1 1 = 3 6 2 2x - 1 3 3x = 4 8 2

25. 2x 3 + 3x 2 - 8x - 12 = 0 26. 3x 3 - 2x 2 - 3x + 2 = 0 27.

1 x + = 5 x + 2 4

28.

y - 1 2 - 1 = y + 1 y

5. ƒ x - 3 ƒ = 2 6. ƒ x + 5 ƒ = 4 7. ƒ 2x + 1 ƒ = 5

29. 3x +

8. ƒ 3x - 7 ƒ = 8 9. V = pr 2h, for h 10. P =

A , for t 1 + rt

11. A =

h (b + b 2), for b1 2 1

30.

2 4x - 1 = x - 2 x - 2

x + 1 2x - 1 3x + 5 + = x+3 x-2 x+3

31. 12x + 6 - 1 = 3 32. 15x - 1 + 3 = 1 33. 1 -2x - 7 + 2x = - 7

12. P = 2(l + w), for w

34. 1 -8x - 2 + 4x = - 1

13. x 2 - 5x + 6 = 0

35. 13x + 4 + 1x - 3 = 5

14. 6x 2 + x - 12 = 0

36. 12x + 2 - 1x + 2 = 1

15. (x - 2)2 = 50

37. x 5>4 - 32 = 0

16. 2(x + 4)2 + 18 = 0

38. 2x 2>3 - 5 = 13

17. x 2 - 6x - 1 = 0

39. 6x 4 - 23x 2 + 20 = 0

18. 4x 2 - 4x - 1 = 0

40. 3x + 161x - 12 = 0

19. 3x 2 - x - 1 = 0 20. x 2 - x + 1 = 0

In Exercises 21 and 22, use the discriminant to determine whether the equation has real number solutions or nonreal complex number solutions.

In Exercises 41 to 56, solve each inequality. Write the answer using interval notation. 41. - 3x + 4 Ú - 2 42. - 2x + 7 … 5x + 1

21. 2x 2 + 4x = 5

43. 3x + 1 7 7 or 3x + 2 6 - 7

22. x 2 + 4x + 7 = 0

44. 5x - 4 … 6 and 4x + 1 7 - 7

CHAPTER 1 REVIEW EXERCISES

45. 61 …

9 C + 32 … 95 5

46. 30 6

5 (F - 32) 6 65 9

149

60. Shadow Length A person 5 feet 6 inches tall is walking away

from a lamppost that is 22 feet tall. What is the length of the person’s shadow at a point 12 feet from the lamppost? See the diagram below.

47. ƒ 3x - 4 ƒ 6 2 48. ƒ 2x - 3 ƒ Ú 1 49. 0 6 ƒ x - 2 ƒ 6 1

22 ft

50. 0 6 ƒ x - a ƒ 6 b (b 7 0) 5.5 ft

51. x 2 + x - 6 Ú 0 12 ft

52. x 3 + 2x 2 - 16x - 32 6 0 53.

x + 3 7 0 x - 4

54.

x(x - 5) … 0 x + 7

61. Diameter of a Cone As sand is poured from a chute, it forms

a right circular cone whose height is one-fourth the diameter of the base. What is the diameter of the base when the cone has a volume of 144 cubic feet? Round to the nearest foot. 62. Individual Price A calculator and a battery together sell for $21.

The price of the calculator is $20 more than the price of the battery. Find the price of the calculator and the price of the battery.

2x … 10 55. 3 - x 56.

x

63. Maintenance Cost Eighteen owners share the maintenance

x Ú 1 5 - x

57. Rectangular Region The length of a rectangle is 9 feet less

than twice the width of the rectangle. The perimeter of the rectangle is 54 feet. Find the width and the length. 58. Rectangular Region The perimeter of a rectangle is 40 inches

and its area is 96 square inches. Find the length and the width of the rectangle.

cost of a condominium complex. If six more units are sold, the maintenance cost will be reduced by $12 per month for each of the present owners. What is the total monthly maintenance cost for the condominium complex? 64. Investment A total of $5500 was deposited into two simple

interest accounts. On one account the annual simple interest rate is 4%, and on the second account the annual simple interest rate is 6%. The amount of interest earned for 1 year was $295. How much was invested in each account?

59. Height of a Tree The height of a tree is estimated by using its

shadow and the known height of a pole as shown in the figure below. Find the height of the tree.

65. Distance to an Island A motorboat left a harbor and traveled

to an island at an average rate of 8 knots. The average speed on the return trip was 6 knots. If the total trip took 7 hours, how many nautical miles is it from the harbor to the island? 66. Running Inez can run at a rate that is 2 miles per hour faster

than Olivia’s rate. One day, Inez gave Olivia a 25-minute head start on a run. If Inez passes Olivia 5 miles from the starting point, how fast is each running? h

67. Chemistry A chemist mixes a 5% salt solution with an 11% salt

solution. How many milliliters of each should be used to make 600 milliliters of a 7% salt solution?

9 ft

68. Pharmacy How many milliliters of pure water should a phar15 ft

6 ft

macist add to 40 milliliters of a 5% salt solution to produce a 2% salt solution?

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CHAPTER 1


69. Alloys How many ounces of a gold alloy that costs $460 per

75. Sports In an Olympic 10-meter diving competition, the height

ounce must be mixed with 25 ounces of a gold alloy that costs $220 per ounce to make a mixture that costs $310 per ounce?

h, in meters, of a diver above the water t seconds after leaving the board can be given by h = - 4.9t 2 + 7.5t + 10. In how many seconds will the diver be 5 meters above the water? Round to the nearest tenth of a second.

70. Blends A grocer makes a snack mixture of raisins and nuts by

combining raisins that cost $2.50 per pound and nuts that cost $4.50 per pound. How many pounds of each should be mixed to make 20 pounds of this snack that costs $3.25 per pound?

76. Fair Coin If a fair coin is tossed 100 times, we would expect

heads to occur about 50 times. But how many heads would suggest that a coin is not fair? An inequality used by statisticians to x - 50 answer this question is ` ` 6 1.96, where x is the actual 5 number of heads that occurred in 100 tosses of a coin. What range of heads would suggest that the coin is a fair coin?

71. Construction of a Wall A mason can build a wall in 9 hours

less than an apprentice. Together they can build the wall in 6 hours. How long would it take the apprentice, working alone, to build the wall? 72. Parallel Processing One computer can solve a problem 5 min-

utes faster than a second computer. Working together, the computers can solve the problem in 6 minutes. How long does it take the faster computer working alone to solve the problem?

77.

height (the mean is the sum of all the measurements divided by the number of measurements) of women in the United States, the height of every woman would have to be measured and then the mean height calculated—an impossible task. Instead, researchers find a representative sample of women and find the mean height of the sample. Because the entire population of women is not used, there is a possibility that the calculated mean height is not the true mean height. For 63.8 - m one study, researchers used the formula ` ` 6 1.645, 0.45 where m is the true meanheight, in inches, of all women, to be 90% sure of the range of values for the true mean height. Using this inequality, what is the range of mean heights of women in the United States? Round to the nearest tenth of an inch. (Source: Based on data from the National Center for Health Statistics)

73. Dogs on a Beach Two dogs start, at the same time, from points

C and D on a beach and run toward their owner, who is positioned at point X. If the dogs run at the same rate and reach their owner at the same instant, what is the distance AX? See the diagram below. Note: Angle A and Angle B are right angles.

C D

78.

60 yd 40 yd A

B

X 100 yd

74. Constructing a Box A square piece of cardboard is formed

into a box by cutting an 8-centimeter square from each corner and folding up the sides. If the volume of the box is to be 80,000 cubic centimeters, what size square piece of cardboard is needed? (Hint: Volume of a box is V = lwh.) 8 cm 8 cm

Mean Height If a researcher wanted to know the mean

Mean Waist Size If a researcher wanted to know the

mean waist size (see Exercise 77 for the definition of mean) of men in the United States, the waist size of every man would have to be measured and then the mean waist size calculated. Instead, researchers find a representative sample of men and find the mean waist size of the sample. Because the entire population of men is not used, there is a possibility that the calculated mean waist size is not the true mean waist size. 39 - m For one study, researchers used the formula ` ` 6 1.96, 0.53 where m is the true mean waist size, in inches, of all men to be 95% sure of the range of values for the true mean waist size. Using this inequality, what is the range of mean waist sizes of men in the United States? Round to the nearest tenth of an inch. (Source: Based on data from the National Center for Health Statistics) 79. Basketball Dimensions A basketball

is to have a circumference of 29.5 to 30.0 inches. Find the acceptable range of diameters for the basketball. Round results to the nearest hundredth of an inch.

CHAPTER 1 TEST

80. Population Density The population density D, in people per

square mile, of a city is related to the horizontal distance x, in miles, from the center of the city by the equation D = - 45x 2 + 190x + 200,

0 6 x 6 5

151

players can be sold when the price is $150, how many players could be sold if the price is $125? 84. Magnetism The repulsive force between the north poles of two

Describe the region of the city in which the population density exceeds 300 people per square mile. Round critical values to the nearest tenth of a mile.

magnets is inversely proportional to the square of the distance between the poles. If the repulsive force is 40 pounds when the distance between the poles is 2 inches, what is the repulsive force when the distance between the two poles is 4 inches?

81. Physics Force F is directly proportional to acceleration a. If a

85. Acceleration The acceleration due to gravity on the surface

force of 10 pounds produces an acceleration of 2 feet per second squared, what acceleration will be produced by a 15-pound force?

of a planetary body is directly proportional to the mass of the body and inversely proportional to the square of its radius. If the acceleration due to gravity is 9.8 meters per second squared on Earth, whose radius is 6,370,000 meters and whose mass is 5.98 * 1026 grams, find the acceleration due to gravity on the moon, whose radius is 1,740,000 meters and whose mass is 7.46 * 1024 grams. Round to the nearest hundredth of a meter per second squared.

82. Physics The distance an object will fall on the moon is

directly proportional to the square of the time it falls. If an object falls 10.6 feet in 2 seconds, how far would an object fall in 3 seconds? 83. Business The number of MP3 players a company can sell is

inversely proportional to the price of the player. If 5000 MP3

CHAPTER 1 TEST 1.

1 x 3 2x + = 3 2 2 4

2. Solve: ƒ 2x + 5 ƒ = 13 3. Solve ax - c = c(x - d) for x.

14. a. Solve the compound inequality:

2x - 5 … 11

- 3x + 2 7 14

Write the solution set using set-builder notation. b. Solve the compound inequality: 2x - 1 6 9

4. Solve 6x 2 - 13x - 8 = 0 by factoring and applying the zero

product principle.

or

and

-3x + 1 … 7

Write the solution set using interval notation.

5. Solve 2x 2 - 8x + 1 = 0 by completing the square.

15. Solve ƒ 3x - 4 ƒ 7 5. Write the answer in interval notation.

6. Solve x 2 + 13 = 4x by using the quadratic formula.

16. Solve x 2 - 5x - 6 6 0. Write the answer using interval notation.

7. Determine the discriminant of 2x 2 + 3x + 1 = 0 and state

17. Solve:

the number of real solutions of the equation. 8. Solve: 1x - 2 - 1 = 13 - x 9. Solve: 13x + 1 - 1x - 1 = 2 10. Solve: 3x 4>5 - 7 = 41

x 2 + x - 12 Ú 0 x + 1 Write the solution set using interval notation.

18. Automotive A radiator contains 6 liters of a 20% antifreeze

solution. How much should be drained and replaced with pure antifreeze to produce a 50% antifreeze solution? 19. Paving A worker can cover a parking lot with asphalt in 10 hours.

3 3 5 - = 11. Solve: x + 2 4 x + 2

With the help of an assistant, the work can be done in 6 hours. How long would it take the assistant, working alone, to cover the parking lot with asphalt?

12. Solve: 2x 3 + x 2 - 8x - 4 = 0 20. Shadow Length Geraldine is 6 feet tall and walking away 13. Solve: x 3 - 64 = 0

from a lamppost that is 20 feet tall. What is the length of

152

CHAPTER 1


Geraldine’s shadow when she is 10 feet from the lamppost? Round to the nearest tenth of a foot.

on a run. Assuming that each runs at a constant rate and Zoey passes Tessa 15 miles from the starting point, what is Zoey’s rate?

21. Mixtures A market offers prepackaged meatloaf that is made

by combining ground beef that costs $3.45 per pound with ground sausage that costs $2.70 per pound. How many pounds of each should be used to make 50 pounds of a meatloaf mixture that costs $3.15 per pound? 22. Rockets A toy rocket is launched from a platform that is 4 feet

above the ground. The height h, in feet, of the rocket t seconds after launch is given by h = -16t 2 + 160t + 4. How many seconds after launch will the rocket be 100 feet above the ground? Round to the nearest tenth of a second. 23. Running Zoey can run at a rate that is 4 miles per hour faster

than Tessa’s rate. One day, Zoey gave Tessa a 1-hour head start

24. Pass Completions One part of the NFL quarterback rating

formula requires that 0 6 0.05p - 1.5 6 2.375, where p% is the percent of completed passes. What is the range of p used in the formula? 25. Astronomy A meteorite approaching the moon has a veloc-

ity that varies inversely as the square root of its distance from the center of the moon. If the meteorite has a velocity of 4 miles per second at 3000 miles from the center of the moon, find the velocity of the meteorite when it is 2500 miles from the center of the moon. Round to the nearest tenth of a mile per second.

CUMULATIVE REVIEW EXERCISES 1. Evaluate: 4 + 3(- 5) 2. Write 0.00017 in scientific notation. 3. Perform the indicated operations and simplify:

(3x - 5)2 - (x + 4)(x - 4) 4. Factor: 8x + 19x - 15

7x - 3 - 5 x - 4

18. Business The revenue R, in dollars, earned by selling x inkjet

printers is given by R = 200x - 0.004x 2. The cost C, in dollars, of manufacturing x inkjet printers is given by the equation C = 65x + 320,000. How many printers should be manufactured and sold to earn a profit of at least $600,000?

6. Simplify: a2>3 # a1>4 7. Simplify: (2 + 5i)(2 - 5i) 8. Solve: 2(3x - 4) + 5 = 17

19. Course Grade An average score of at least 80, but less than

9. Solve 2x 2 - 4x = 3 by using the quadratic formula. 10. Solve: ƒ 2x - 6 ƒ = 4 11. Solve: x = 3 + 19 - x

90, in a history class receives a B grade. Rebecca has scores of 86, 72, and 94 on three tests. Find the range of scores she could receive on the fourth test that would give her a B grade for the course. Assume that the highest test score she can receive is 100. 20. Ticketing Speeding Drivers A highway patrol department

estimates that the cost of ticketing p percent of the speeders who travel on a freeway is given by

12. Solve: x - 36x = 0 3

13. Solve: 2x 4 - 11x 2 + 15 = 0

C =

14. Solve the compound inequality:

3x - 1 7 2

17. Dimensions of a Field A fence built around the border of a

rectangular field measures a total of 200 feet. If the length of the fence is 16 feet longer than the width, what are the dimensions of the fence?

2

5. Simplify:

x - 2 Ú 4. Write the solution set using set-builder 2x - 3 notation.

16. Solve

or

-3x + 5 Ú 8

Write the solution set using set-builder notation. 15. Solve ƒ x - 6 ƒ Ú 2. Write the solution set using interval notation.

600p , 0 6 p 6 100 100 - p

where C is in thousands of dollars. If the highway patrol department plans to fund its program to ticket speeding drivers with $100,000 to $180,000, what is the range of the percent of speeders the department can expect to ticket? Round your percents to the nearest 0.1%.

CHAPTER

2

FUNCTIONS AND GRAPHS

2.1 Two-Dimensional Coordinate System and Graphs 2.2 Introduction to Functions 2.3 Linear Functions 2.4 Quadratic Functions 2.5 Properties of Graphs

NASA/Johnson Space Center

2.6 Algebra of Functions 2.7 Modeling Data Using Regression

Astronauts experiencing microgravity. A typical training session may consist of 40 to 60 microgravity maneuvers, each lasting about 18 seconds. The zero gravity scenes in the movie Apollo 13 were produced using these microgravity maneuvers.

Altitude (in meters)

Functions as Models 9500 9000 8500 8000

Microgravity begins here Microgravity ends here 10 0 Time (in seconds)

20

To prepare astronauts for the experience of zero gravity (technically, microgravity) in space, the National Aeronautics and Space Administration (NASA) uses a specially designed jet. A pilot accelerates the jet upward to an altitude of approximately 9000 meters and then reduces power. At that time, the plane continues upward, noses over, and begins to descend until the pilot increases power. The maneuver is then repeated. The figure to the left shows one maneuver. During the climb and the point at which the pilot increases power, the force on the astronauts is approximately twice what they experience on Earth. During the time of reduced power (about 15 to 20 seconds), the plane is in free fall and the astronauts experience microgravity. The sudden changes in gravity effects have a tendency to make astronauts sick. Because of this, the plane has been dubbed the Vomit Comet. A parabola, one of the topics of this chapter, can approximate the height of the jet. Using an equation of the parabola, the time during which the astronauts experience microgravity in one maneuver can be determined. See Exercise 47, page 210. 153

154

CHAPTER 2


SECTION 2.1 Cartesian Coordinate Systems Distance and Midpoint Formulas Graph of an Equation Intercepts Circles, Their Equations, and Their Graphs

Note Abscissa comes from the same root word as scissors. An open pair of scissors looks like an x.

Math Matters The concepts of analytic geometry developed over an extended period, culminating in 1637 with the publication of two works: Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences by René Descartes (1596–1650) and Introduction to Plane and Solid Loci by Pierre de Fermat. Each of these works was an attempt to integrate the study of geometry with the study of algebra. Of the two mathematicians, Descartes is usually given most of the credit for developing analytic geometry. In fact, Descartes became so famous in La Haye, the city in which he was born, that it was renamed La Haye-Descartes.

Two-Dimensional Coordinate System and Graphs Cartesian Coordinate Systems Each point on a coordinate axis is associated with a number called its coordinate. Each point on a flat, two-dimensional surface, called a coordinate plane or xy-plane, is associated with an ordered pair of numbers called coordinates of the point. Ordered pairs are denoted by (a, b), where the real number a is the x-coordinate or abscissa and the real number b is the y-coordinate or ordinate. The coordinates of a point are determined by the point’s position relative to a horizontal coordinate axis called the x-axis and a vertical coordinate axis called the y-axis. The axes intersect at the point (0, 0), called the origin. In Figure 2.1, the axes are labeled such that positive numbers appear to the right of the origin on the x-axis and above the origin on the y-axis. The four regions formed by the axes are called quadrants and are numbered counterclockwise. This two-dimensional coordinate system is referred to as a Cartesian coordinate system in honor of René Descartes.

y Quadrant II Horizontal axis −4

4 2

−2 −2

Quadrant III

−4

y 4

Quadrant I

4

Quadrant IV

Figure 2.1

(4, 3)

(0, 1)

(3, 1)

2

Vertical axis 2 Origin

(1, 3)

(−3, 1) x

−4

−2

2 −2

4

x

(3, −2)

(−2, −3) −4

Figure 2.2

To plot a point P(a, b) means to draw a dot at its location in the coordinate plane. In Figure 2.2, we have plotted the points (4, 3), ( - 3, 1), ( -2, -3), (3, - 2), (0, 1), (1, 3), and (3, 1). The order in which the coordinates of an ordered pair are listed is important. Figure 2.2 shows that (1, 3) and (3, 1) do not denote the same point. Data often are displayed in visual form as a set of points called a scatter diagram or scatter plot. For instance, the scatter diagram in Figure 2.3 shows the current and projected revenue of Web-filtering software vendors. (Web-filtering software allows businesses to control which Internet sites are available to employees while at work.) The point whose coordinates are approximately (2005, 520) means that in 2005 approximately $520 million in revenue were generated by companies that supplied this software. The line segments that connect the points in Figure 2.3 help illustrate trends.

2.1

TWO-DIMENSIONAL COORDINATE SYSTEM AND GRAPHS

155

Revenue from Web-filtering software (in millions of dollars)

R

Note The notation (a, b) was used earlier to denote an interval on a one-dimensional number line. In this section, (a, b) denotes an ordered pair in a two-dimensional plane. This should not cause confusion in future sections because as each mathematical topic is introduced, it will be clear whether a one-dimensional or a two-dimensional coordinate system is involved.

1000 800 600 400 200 0 2003 2004 2005 2006 2007 2008 2009 Year

t

Figure 2.3 Source: IDC, 2005. Question • According to the data in Figure 2.3, will the revenue from Web-filtering software in

2009 be more or less than twice the revenue in 2003?

In some instances, it is important to know when two ordered pairs are equal.

Definition of the Equality of Ordered Pairs y

The ordered pairs (a, b) and (c, d) are equal if and only if a = c and b = d. (1, 2)

EXAMPLE

2

If (3, y) = (x, - 2), then x = 3 and y = - 2.

−2

2

4

x

Distance and Midpoint Formulas

5 −2

The Cartesian coordinate system makes it possible to combine the concepts of algebra and geometry into a branch of mathematics called analytic geometry. The distance between two points on a horizontal line is the absolute value of the difference between the x-coordinates of the two points. The distance between two points on a vertical line is the absolute value of the difference between the y-coordinates of the two points. For example, as shown in Figure 2.4, the distance d between the points with coordinates (1, 2) and (1, -3) is d = ƒ 2 - ( -3) ƒ = 5. If two points are not on a horizontal or vertical line, then a distance formula for the distance between the two points can be developed as follows. The distance between the points P1(x1, y1) and P2(x2, y2) in Figure 2.5 is the length of the hypotenuse of a right triangle whose sides are horizontal and vertical line segments that measure ƒ x2 - x1 ƒ and ƒ y2 - y1 ƒ , respectively. Applying the Pythagorean Theorem to this triangle produces

(1, −3)

Figure 2.4

Pythagorean Theorem See pages 103–104. y P1(x1, y1) y1 d

| y2 – y1|

d 2 = ƒ x2 - x1 ƒ 2 + ƒ y2 - y1 ƒ 2 d = 2 ƒ x2 - x1 ƒ 2 + ƒ y2 - y1 ƒ 2

y2

P2(x2, y2) x2

x1

| x2 – x1|

x

= 2(x2 - x1)2 + (y2 - y1)2

• Use the square root procedure. Because d is nonnegative, the negative root is not listed. • ƒ x2 - x1 ƒ 2 = (x2 - x1)2 and 2 2 ƒ y2 - y1 ƒ = (y2 - y1)

Thus we have established the following theorem. Figure 2.5 Answer • More. The revenue in 2003 was approximately $350 million. The projected revenue in

2009 is approximately $925 million, which is more than twice $350 million.

156

CHAPTER 2


Distance Formula The distance d(P1, P2) between the points P1(x1, y1) and P2(x2, y2) is d(P1, P2) = 2(x2 - x1)2 + (y2 - y1)2 EXAMPLE

y 8

The distance between P1( -3, 4) and P2(7, 2) is given by d(P1, P2) = 2(x2 - x1) + (y 2 - y 1) 2

= 2[7 - ( - 3)]2 + (2 - 4)2

P2(x2, y2)

4

d(P1, P2) = 2 26

2

2

= 1104 = 2126 L 10.2 y

P1(−3, 4)

P2(7, 2)

= 210 + ( -2) 2

6

2

−4

−2

4

2

6

8 x

The midpoint M of a line segment is the point on the line segment that is equidistant from the endpoints P1(x1, y1) and P2(x2, y2) of the segment. See Figure 2.6.

M(x, y)

Midpoint Formula P1(x1, y1)

The midpoint M of the line segment from P1(x1, y1) to P2(x2, y2) is given by a

x

Figure 2.6

x1 + x2 y1 + y2 , b 2 2

EXAMPLE

The midpoint of the line segment between P1(-2, 6) and P2(3, 4) is given by x1 + x2 y1 + y2 , b 2 2 (-2) + 3 6 + 4 1 = a , b = a , 5b 2 2 2

M = a

y 8

M=

6

1

( 2 , 5)

P1(−2, 6) 4 P2(3, 4) 2 −4

−2

2

4

6 x

The midpoint formula states that the x-coordinate of the midpoint of a line segment is the average of the x-coordinates of the endpoints of the line segment and that the y-coordinate of the midpoint of a line segment is the average of the y-coordinates of the endpoints of the line segment.

EXAMPLE 1

Find the Midpoint and Length of a Line Segment

Find the midpoint and the length of the line segment connecting the points whose coordinates are P1(- 4, 3) and P2(4, - 2). Solution x1 + x2 y1 + y2 , b 2 2 -4 + 4 3 + ( -2) 1 = a , b = a0, b 2 2 2

Midpoint = a

2.1


157

d(P1, P2) = 2(x2 - x1)2 + ( y2 - y1)2 = 2(4 - ( -4))2 + ( - 2 - 3)2 = 2(8)2 + ( -5)2 = 164 + 25 = 189 Try Exercise 6, page 164

Graph of an Equation The equations below are equations in two variables. y = 3x 3 - 4x + 2

y

2

−2

2

4

x

−2

Definition of the Graph of an Equation

Consider y = 2x - 1. Substituting various values of x into the equation and solving for y produces some of the ordered pairs that satisfy the equation. It is convenient to record the results in a table similar to the one shown below. The graph of the ordered pairs is shown in Figure 2.7.

Figure 2.7 y 4

y 2x 1

y

2

2( -2) - 1

5

( -2,- 5)

1

2(- 1) - 1

3

( -1, - 3)

0

2(0) - 1

1

(0, - 1)

1

2(1) - 1

1

(1, 1)

2

2(2) - 1

3

(2, 3)

x

2

−2

x x + 1

The graph of an equation in the two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation.

−4

−4

y =

The solution of an equation in two variables is an ordered pair (x, y) whose coordinates satisfy the equation. For instance, the ordered pairs (3, 4), (4, - 3), and (0, 5) are some of the solutions of x 2 + y 2 = 25. Generally, there are an infinite number of solutions of an equation in two variables. These solutions can be displayed in a graph.

4

−4

x 2 + y 2 = 25

2

4

x

−4

(x, y)

Choosing some noninteger values of x produces more ordered pairs to graph, such as 3 5 a- ,-4 b and a , 4 b, as shown in Figure 2.8. Using still other values of x would add 2 2 even more ordered pairs to graph. The result would be so many dots that the graph would appear as the straight line shown in Figure 2.9, which is the graph of y = 2x - 1.

Figure 2.8 y

4 2

EXAMPLE 2 −4

−2

2

−4

4

x

Draw a Graph by Plotting Points

Graph: -x 2 + y = 1 Solution Solve the equation for y. y = x2 + 1

Figure 2.9

(continued)

158

CHAPTER 2


Select values of x and use the equation to calculate y. Choose enough values of x so that an accurate graph can be drawn. Plot the points and draw a curve through them. See Figure 2.10.

y (−2, 5)

(2, 5) 4

(−1, 2)

(1, 2) (0, 1)

−4

−2

2

4

x

Figure 2.10

x

y x2 1

y

(x, y)

2

2

(- 2) + 1

5

( - 2, 5)

1

(- 1) + 1

2

(- 1, 2)

0

(0) + 1

1

(0, 1)

1

(1)2 + 1

2

(1, 2)

2

(2) + 1

5

(2, 5)

2 2

2


Math Matters

Integrating Technology

Maria Agnesi (1718–1799) wrote Foundations of Analysis for the Use of Italian Youth, one of the most successful textbooks of the eighteenth century. The French Academy authorized a translation into French in 1749, noting that “there is no other book, in any language, which would enable a reader to penetrate as deeply, or as rapidly, into the fundamental concepts of analysis.” A curve that Agnesi discusses in her text is given by the equation y =

a3 x + a2 2

Unfortunately, due to a translation error from Italian to English, the curve became known as the “witch of Agnesi.” y

a

y=

a3 x2 + a2

x

Some graphing calculators, such as the TI-83/TI-83 Plus/TI-84 Plus, have a TABLE feature that allows you to create a table similar to the one shown in Example 2. Enter the equation to be graphed, the first value for x, and the increment (the difference between successive values of x). For instance, entering y1 = x 2 + 1, an initial value of x of - 2, and an increment of 1 yields a display similar to the one in Figure 2.11. Changing the initial value to - 6 and the increment to 2 gives the table in Figure 2.12. Plot1 Plot2 Plot3 \Y 1 = X2+1 \Y2 = TABLE SETUP \Y3 = TblStart=-2 \Y4 = ΔTbl=1 \Y5 = Indpnt: Auto Ask \Y6 = Depend: Auto Ask \Y7 =

X -2 -1 0 1 2 3 4 X=-2

Y1 5 2 1 2 5 10 17

Figure 2.11

TABLE SETUP TblStart=-6 ΔTbl=2 Indpnt: Auto Ask Depend: Auto Ask

X -6 -4 -2 0 2 4 6 X=-6

Y1 37 17 5 1 5 17 37

Figure 2.12

With some calculators, you can scroll through the table by using the up- or downarrow keys. In this way, you can determine many more ordered pairs of the graph.

2.1

EXAMPLE 3

5

(−2, 4) (−1, 3)

(5, 3)

(0, 2)

(4, 2) (1, 1)

−2

159

Graph by Plotting Points

Graph: y = ƒ x - 2 ƒ Solution This equation is already solved for y, so start by choosing an x value and using the equation to determine the corresponding y value. For example, if x = - 3, then y = ƒ (-3) - 2 ƒ = ƒ - 5 ƒ = 5. Continuing in this manner produces the following table.

y (−3, 5)


5

-2 4

-1 3

0 2

1 1

2 0

3 1

4 2

5 3

Now plot the points listed in the table. Connecting the points forms a V shape, as shown in Figure 2.13.

(3, 1)

(2, 0)

-3 5

When x is y is

x


Figure 2.13

EXAMPLE 4

Graph by Plotting Points

Graph: y 2 = x Solution Solve the equation for y. y2 = x y = 1x y 4 2

(16, 4) (4, 2) (1, 1)

(0, 0) −2 −4

Choose several x values, and use the equation to determine the corresponding y values.

(9, 3)

(1, −1) (4, −2)

8

12

16 x

When x is

0

1

4

9

16

y is

0

1

2

3

4

Plot the points as shown in Figure 2.14. The graph is a parabola. Try Exercise 32, page 164

(9, −3) (16, −4)

Figure 2.14

Integrating Technology A graphing calculator or computer graphing software can be used to draw the graphs in Examples 3 and 4. These graphing utilities graph a curve in much the same way as you would, by selecting values of x and calculating the corresponding values of y. A curve is then drawn through the points. If you use a graphing utility to graph y = ƒ x - 2 ƒ , you will need to use the absolute value function that is built into the utility. The equation you enter will look similar to Y1=abs(X–2). To graph the equation in Example 4, you will enter two equations. The equations you enter will be similar to Y1 1 (X) Y2 1 (X) The graph of the first equation will be the top half of the parabola; the graph of the second equation will be the bottom half.

160

CHAPTER 2


Intercepts On a graph, any point that has an x- or a y-coordinate of zero is called an intercept of the graph, because it is at this point that the graph intersects the x- or the y-axis.

Definitions of x-Intercepts and y-Intercepts If (x1, 0) satisfies an equation in two variables, then the point whose coordinates are (x1, 0) is called an x-intercept of the graph of the equation. If (0, y1) satisfies an equation in two variables, then the point whose coordinates are (0, y1) is called a y-intercept of the graph of the equation.

To find the x-intercepts of the graph of an equation, let y = 0 and solve the equation for x. To find the y-intercepts of the graph of an equation, let x = 0 and solve the equation for y.

EXAMPLE 5

Find x- and y-Intercepts

Find the x- and y-intercepts of the graph of y = x 2 - 2x - 3. Algebraic Solution

Visualize the Solution

To find the y-intercept, let x = 0 and solve for y.

The graph of y = x 2 - 2x - 3 is shown below. Observe that the graph intersects the x-axis at (- 1, 0) and (3, 0), the x-intercepts. The graph also intersects the y-axis at (0, -3), the y-intercept.

y = 02 - 2(0) - 3 = - 3 To find the x-intercepts, let y = 0 and solve for x. 0 = x 2 - 2x - 3 0 = (x - 3)(x + 1) (x - 3) = 0 or (x + 1) = 0 x = 3 or x = -1

y (4, 5) 4

Because y = - 3 when x = 0, (0, -3) is a y-intercept. Because x = 3 or - 1 when y = 0, (3, 0) and (-1, 0) are x-intercepts. Figure 2.15 confirms that these three points are intercepts.

2 (−1, 0)

(3, 0)

−2

(0, −3)

2

4

(2, −3) −4

(1, −4)

Figure 2.15


x

2.1

161


Integrating Technology In Example 5, it was possible to find the x-intercepts by solving a quadratic equation. In some instances, however, solving an equation to find the intercepts may be very difficult. In these cases, a graphing calculator can be used to estimate the x-intercepts. The x-intercepts of the graph of y = x 3 + x + 4 can be estimated using the ZERO feature of a TI-83/TI-83 Plus/TI-84 Plus calculator. The keystrokes and some sample screens for this procedure are shown below. Press 2nd CALC to access the CALCULATE menu. The y-coordinate of an x-intercept is zero. Therefore, select 2: zero. Press ENTER .

Press Y= . Now enter X^3+X+4. Press ZOOM and select the standard viewing window. Press ENTER .

10

-2) = - 8( - 8 … - 2) + ( -8)2( - 8 7 - 2)

−10

= - 8(1) + 64(0) = - 8

• When x = - 8, the value assigned to - 8 … - 2 is 1; the value assigned to - 8 7 - 2 is 0.

Y1=X*(X ◊ -2)+X2*(X>-2) = 2(2 … - 2) + 22(2 7 - 2) = 2(0) + 4(1) = 4

• When x = 2, the value assigned to 2 … - 2 is 0; the value assigned to 2 7 - 2 is 1.

In a similar manner, for any value x … - 2, the value assigned to (X ◊ -2) is 1 and the value assigned to (X>-2) is 0. Thus Y1=X*1+X2*0=X on that interval. This means that only the f (x) = x piece of the function is graphed. When x 7 - 2, the value assigned to (X ◊ -2) is 0 and the value assigned to (X>-2) is 1. Thus Y1=X*0+X2*1=X2 on that interval. This means that only the f (x) = x 2 piece of the function is graphed on that interval.

2

1. Graph: f (x) = e

x 2, -x,

x 6 2 x Ú 2

2. Graph: f (x) = e

x 2 - x, - x + 4,

x 6 2 x Ú 2

3. Graph: f(x) = e

-x 2 + 1, x 6 0 x 2 - 1, x Ú 0

4. Graph: f (x) = e

x 3 - 4x, x 2 - x + 2,

x 6 1 x Ú 1

Note that pressing 2ND TEST will display the inequality menu.


2.1 Two-Dimensional Coordinate System and Graphs Distance Formula The distance d between two points P1(x1, y1) and P2(x2, y2) is d = 2(x2 - x1)2 + (y2 - y1)2.


Midpoint Formula The coordinates of the midpoint of the line segment x1 + x2 y1 + y2 from P1(x1, y1) to P2(x2, y2) are a , b. 2 2


Graph of an Equation The graph of an equation in the two variables x and y is the graph of all ordered pairs that satisfy the equation.

See Examples 2 and 3, pages 157 and 159, and then try Exercise 7, page 253.

250

CHAPTER 2


x-Intercepts and y-Intercepts If (x1, 0) satisfies an equation in two variables, then the point P(x1, 0) is an x-intercept of the graph of the equation. If (0, y1) satisfies an equation in two variables, then the point P(0, y1) is a y-intercept of the graph of the equation.


Equation of a Circle The standard form of the equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r 2.


2.2 Introduction to Functions Definition of a Function A function is a set of ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinates.


Evaluate a Function To evaluate a function, replace the independent variable with a number in the domain of the function and then simplify the resulting numerical expression.


Piecewise-Defined Function A piecewise-defined function is represented by more than one expression.


Domain and Range of a Function The domain of a function is the set of all See Example 4, page 170, and then try first coordinates of the ordered pairs of the function. The range of a function Exercise 26, page 254. See Example 6, is the set of all second coordinates of the ordered pairs of the function. page 172, and then try Exercise 29, page 254. Graph a Function The graph of a function is the graph of all ordered pairs of the function.


Zero of a Function A value a in the domain of a function f for which f (a) = 0 is called a zero of the function.


Greatest Integer Function (Floor Function) The value of the greatest integer function at the real number x is the greatest integer that is less than or equal to x.


2.3 Linear Functions Slope of a Line If P1(x1, y1) and P2(x2, y2) are two points on a line, then y2 - y1 the slope m of the line between the two points is given by m = , x2 - x1 x1 Z x2. If x1 = x2, the line is vertical and the slope is undefined.


Slope–Intercept Form of the Equation of a Line The equation f (x) = mx + b is called the slope–intercept form of a linear function because the graph of the function is a straight line. The slope is m, and the y-intercept is (0, b).


General Form of a Linear Equation in Two Variables An equation of the form Ax + By = C, where A, B, and C are real numbers and both A and B are not zero, is called the general form of a linear equation in two variables.


Point–Slope Form The equation y - y1 = m(x - x1) is called the point–slope form of the equation of a line. This equation is frequently used to find the equation of a line.


CHAPTER 2 TEST PREP

251

Parallel Lines If m1 and m2 are the slopes of two lines in the plane, then the graphs of the lines are parallel if and only if m1 = m2. That is, parallel lines have the same slope. Vertical lines are parallel.

See Example 6a, page 192, and then try Exercise 50, page 254.

Perpendicular Lines If m1 and m2 are the slopes of two lines in the plane, 1 then the graphs of the lines are perpendicular if and only if m1 = - . m2 That is, the slopes of perpendicular lines are negative reciprocals of each other. A vertical line is perpendicular to a horizontal line.

See Example 6b, page 192, and then try Exercise 52, page 254.

Applications


2.4 Quadratic Functions Quadratic Function A quadratic function f can be represented by the equation f (x) = ax 2 + bx + c, where a, b, and c are real numbers and a Z 0. Every quadratic function given by f (x) = ax 2 + bx + c, a Z 0, can be written in standard form as f (x) = a(x - h)2 + k. The graph of f is a parabola with vertex (h, k).


Parabola The graph of a quadratic function given by f (x) = ax 2 + bx + c, a Z 0, is a parabola. The coordinates of the vertex of the parabola are b b b a - , f a- bb . The equation of the axis of symmetry is x = - . 2a 2a 2a The parabola opens up when a 7 0 and opens down when a 6 0.


Minimum or Maximum of a Quadratic Function If a 7 0, then the b b graph of f (x) = ax 2 + bx + c opens up and the vertex a- , f a- bb 2a 2a b is the lowest point on the graph; f a- b is the minimum value of the 2a function. If a 6 0, then the graph of f (x) = ax 2 + bx + c opens b b down and the vertex a- , f a- bb is the highest point on the graph; 2a 2a b f a- b is the maximum value of the function. 2a


Applications of Quadratic Functions

See Examples 5 through 8, pages 206–209, and then try Exercises 67 through 69, page 255.

2.5 Properties of Graphs Symmetry of a Graph with Respect to • the x-axis The graph of an equation is symmetric with respect to the x-axis if the replacement of y with –y leaves the equation unaltered. • the y-axis The graph of an equation is symmetric with respect to the y-axis if the replacement of x with –x leaves the equation unaltered. • the origin The graph of an equation is symmetric with respect to the origin if the replacement of x with –x and the replacement of y with –y leaves the equation unaltered.

See Examples 1 and 2, pages 214 and 215, and then try Exercises 72, 73, and 77, page 255.

252

CHAPTER 2


Even and Odd Functions The function f is an even function if f (-x) = f (x) for all x in the domain of the function. The function f is an odd function if f (-x) = - f (x) for all x in the domain of the function.


Vertical Translation of a Graph If f is a function and c is a positive constant, then the graph of • y = f (x) + c is a vertical shift c units upward of the graph of y = f (x). • y = f (x) - c is a vertical shift c units downward of the graph of y = f (x).


Horizontal Translation of a Graph If f is a function and c is a positive constant, then the graph of • y = f (x + c) is a horizontal shift c units to the left of the graph of y = f (x). • y = f (x - c) is a horizontal shift c units to the right of the graph of y = f (x).


Reflections of a Graph The graph of • y = - f (x) is the graph of y = f (x) reflected across the x-axis. • y = f (-x) is the graph of y = f (x) reflected across the y-axis.


Vertical Stretching and Compressing of a Graph Assume that f is a function and c is a positive constant. Then • if c 7 1, the graph of y = c # f (x) is the graph of y = f (x) stretched vertically away from the x-axis by a factor of c. • if 0 6 c 6 1, the graph of y = c # f (x) is the graph of y = f (x) compressed vertically toward the x-axis by a factor of c.


Horizontal Stretching and Compressing of a Graph Assume that f is a function and c is a positive constant. Then • if c 7 1, the graph of y = f (c # x) is the graph of y = f (x) compressed 1 horizontally toward the y-axis by a factor of . c # • if 0 6 c 6 1, the graph of y = f (c x) is the graph of y = f (x) stretched 1 horizontally away from the y-axis by a factor of . c


2.6 Algebra of Functions Operations on Functions If f and g are functions with domains Df and Dg, then • ( f + g)(x) = f (x) + g(x) Domain: Df ¨ Dg • ( f - g)(x) = f (x) - g(x) Domain: Df ¨ Dg • ( f # g)(x) = f (x) # g(x) Domain: Df ¨ Dg f f (x) • a b(x) = Domain: Df ¨ Dg, g(x) Z 0 g g(x)


Difference Quotient For a given function f, the expression f (x + h) - f (x) , h Z 0, is called the difference quotient. h



Composition of Functions Let f and g be two functions such that g(x) is in the domain of f for all x in the domain of g. Then the composition of the two functions, denoted by f ⴰ g, is the function whose value at x is given by (f ⴰ g)(x) = f [g(x)].

253


2.7 Modeling Data Using Regression Linear Regression Linear regression is a method of fitting a linear function to data.


Quadratic Regression Quadratic regression is a method of fitting a quadratic function to data.


CHAPTER 2 REVIEW EXERCISES In Exercises 1 and 2, find the distance between the points whose coordinates are given. 1. (-3, 2)

2. (5, - 4)

(7, 11)

(- 3, -8)

In Exercises 3 and 4, find the midpoint of the line segment with the given endpoints. 3. (2, 8) ( -3, 12)

4. ( -4, 7)

(8, -11)

In Exercises 5 to 8, graph each equation by plotting points. 5. 2x - y = - 2

6. 2x + y = 4

7. y = ƒ x - 2 ƒ + 1

8. y = - ƒ 2x ƒ

17. x - y = 4

18. x + y 2 = 4

19. ƒ x ƒ + ƒ y ƒ = 4

20. ƒ x ƒ + y = 4

21. If f (x) = 3x 2 + 4x - 5, find a. f (1)

b. f ( -3)

c. f (t)

d. f (x + h)

e. 3f (t)

f. f (3t)

2

In Exercises 9 to 12, find the x- and y-intercepts of the graph of each equation. Use the intercepts and some additional points as needed to draw the graph of the equation. 10. ƒ x - y ƒ = 4

9. x = y 2 - 1

12. x = ƒ y - 1 ƒ + 1

11. 3x + 4y = 12

In Exercises 13 and 14, determine the center and radius of the circle with the given equation. 13. (x - 3) + ( y + 4) = 81 2

In Exercises 17 to 20, determine whether the equation defines y as a function of x.

2

14. x 2 + y 2 + 10x + 4y + 20 = 0

In Exercises 15 and 16, find the equation in standard form of the circle that satisfies the given conditions. 15. Center C = (2, -3), radius r = 5 16. Center C = (-5, 1), passing through (3, 1)

22. If g(x) = 264 - x 2, find a. g(3)

b. g( -5)

c. g(8)

d. g(- x)

e. 2g(t)

f. g(2t)

23. Let f be a piecewise-defined function given by

f (x) = e

3x + 2, x 6 0 x2 - 3, x Ú 0

Find each of the following. a. f (3)

b. f (- 2)

c. f (0)

24. Let f be a piecewise-defined function given by

x + 4, f (x) = c x2 + 1, x - 7,

x 6 -3 -3 … x 6 5 x Ú 5

Find each of the following. a. f (0)

b. f ( - 3)

c. f (5)

254

CHAPTER 2


In Exercises 25 to 28, determine the domain of the function represented by the given equation. 25. f (x) = - 2x + 3

26. f (x) = 16 - x

27. f (x) = 225 - x 2

28. f (x) =

2

3 x 2 - 2x - 15

29. Find the values of a in the domain of f (x) = x 2 + 2x - 4 for

which f (a) = - 1.

30. Find the value of a in the domain of f (x) =

f (a) = 2.

4 for which x + 1

In Exercises 31 and 32, graph the given equation. 31. f (x) = ƒ x - 1 ƒ - 1

48. Find the equation of the line that passes through the points with

coordinates (-4, -6) and (8, 15). 49. Find the slope–intercept form of the equation of the line that

passes through the point with coordinates (3, -5) and is parallel 2 to the graph of y = x - 1. 3 50. Find the slope–intercept form of the equation of the line that

passes through the point with coordinates (-1, -5) and is parallel to the graph of 2x - 5y = 2. 51. Find the slope–intercept form of the equation of the line that

passes through the point with coordinates (3, -1) and is per3 pendicular to the graph of y = - x - 2. 2

32. f (x) = 4 - 1x 52. Find the slope–intercept form of the equation of the line that

In Exercises 33 and 34, find the zero or zeros of the given function. 33. f (x) = 2x + 6

34. f (x) = x 2 - 4x - 12

In Exercises 35 and 36, find each function value. 35. Let g(x) = 冀2x冁 . a. g(p)

b. g a- b

2 3

b. f (0.5)

c. f (-p)

In Exercises 37 to 40, find the slope of the line between the points with the given coordinates. 37. (- 3, 6); (4, -1)

38. (- 5, 2); (-5, 4)

39. (4, -2); (-3, -2)

40. (6, -3); (-4, -1)

41. Graph f (x) = -

3 x + 2 using the slope and y-intercept. 4

42. Graph f (x) = 2 - x using the slope and y-intercept. 43. Graph 3x - 4y = 8.

44. Graph 2x + 3y = 9

45. Find the equation of the line that passes through the point with

2 coordinates (-3, 2) and whose slope is - . 3 46. Find the equation of the line that passes through the point with

coordinates (1, -4) and whose slope is -2. 47. Find the equation of the line that passes through the points with

coordinates (-2, 3) and (1, 6).

53. Sports The speed of a professional golfer’s swing and the

speed of the ball as it leaves the club are important factors in the distance the golf ball travels. The table below shows five measurements of clubhead speed and ball speed, each in miles per hour.

c. g( -2)

36. Let f (x) = 冀 1 - x冁 . a. f ( 12)

passes through the point with coordinates (2, 6) and is perpendicular to the graph of 2x - 5y = 10.

Measurement

Clubhead Speed (mph)

Ball Speed (mph)

1

106

155

2

108

159

3

114

165

4

116

171

5

118

175

Use measurements 1 and 5 to find a linear function that could be used to determine ball speed for a given clubhead speed. 54. Food Science Newer heating elements allow an oven to

reach a normal baking temperature (350°F) more quickly. The table below shows the time, in minutes, since an oven was turned on and the temperature of the oven.

Measurement

Time (min)

Temperature (°F)

1

0

75

2

2

122

3

4

182

4

6

255

5

8

300

6

10

350


Use measurements 2 and 6 to find a linear function that could be used to determine the temperature of the oven as a function of time.

255

In Exercises 70 and 71, sketch a graph that is symmetric to the given graph with respect to the a. x-axis, b. y-axis, and c. origin. 70.

71.

y 6

In Exercises 55 to 60, use the method of completing the square to write each quadratic equation in its standard form.

y 4

4

2

2

55. f (x) = x + 6x + 10 2

−6

−4

−2

56. f (x) = 2x 2 + 4x + 5

2

−2

4

6 x

−4

−2

−4

57. f (x) = - x - 8x + 3 2

2

4

−2 −4

−6

58. f (x) = 4x 2 - 6x + 1

In Exercises 72 to 79, determine whether the graph of each equation is symmetric with respect to the a. x-axis, b. y-axis, and c. origin.

59. f (x) = - 3x 2 + 4x - 5 60. f (x) = x 2 - 6x + 9

In Exercises 61 to 64, find the vertex of the graph of the quadratic function. 61. f(x) = 3x 2 - 6x + 11

62. h(x) = 4x 2 - 10

63. k(x) = - 6x 2 + 60x + 11

64. m(x) = 14 - 8x - x 2

72. y = x 2 - 7

73. x = y 2 + 3

74. y = x 3 - 4x

75. y 2 = x 2 + 4

76.

x2 32

+

y2 42

77. xy = 8

= 1

78. ƒ y ƒ = ƒ x ƒ

79. ƒ x + y ƒ = 4

In Exercises 65 and 66, find the requested value. 65. The maximum value of f (x) = - x 2 + 6x - 3 66. The minimum value of g(x) = 2x 2 + 3x - 4

In Exercises 80 to 85, sketch the graph of g. a. Find the domain and the range of g. b. State whether g is even, odd, or neither. 80. g(x) = - x 2 + 4

81. g(x) = - 2x - 4

82. g(x) = ƒ x - 2 ƒ + ƒ x + 2 ƒ

83. g(x) = 216 - x 2

84. g(x) = x 3 - x

85. g(x) = 2 冀x 冁

67. Height of a Ball A ball is thrown vertically upward with an

initial velocity of 50 feet per second. The height h, in feet, of the ball t seconds after it is released is given by the equation h(t) = - 16t 2 + 50t + 4 . What is the maximum height reached by the ball? 68. Delivery Cost A freight company has determined that its cost,

in dollars, per delivery of x parcels is

In Exercises 86 to 91, use the graph of f shown below to sketch a graph of g.

C(x) = 1050 + 0.5x

y 6

The price it charges to send a parcel is $13.00 per parcel. Determine a. the revenue function b. the profit function

2 −6

−4

2

−2

4

6 x

−2

c. the minimum number of parcels the company must ship to

−4

break even

−6

69. Agriculture A farmer wishes to enclose a rectangular region

bordering a river using 700 feet of fencing. What is the maximum area that can be enclosed with the fencing?

y = f (x)

4

86. g(x) = f (x) - 2

87. g(x) = f (x + 3)

x

256

CHAPTER 2


88. g(x) = f (x - 1) - 3

89. g(x) = f (x + 2) - 1

90. g(x) = f (-x)

91. g(x) = - f (x)

In Exercises 92 to 95, use the graph of f shown below to sketch a graph of g.

101. If f(x) = 2x 2 + 7 and g(x) = ƒ x - 1 ƒ , find

102.

y 6 4

y = f(x)

a. ( f ⴰ g)( -5)

b. ( g ⴰ f )( -5)

c. ( f ⴰ g)(x)

d. ( g ⴰ f )(x)

Sports A soccer coach examined the relationship between the speed, in meters per second, of a soccer player’s foot when it strikes the ball and the initial speed, in meters per second, of the ball. The table below shows the values obtained by the coach.

2 −6

−4

2

−2

4

6 x

Foot Speed (m/s)

Initial Ball Speed (m/s)

5

12

8

13

11

18

14

22

17

26

20

28

−2 −4 −6

92. g(x) = 2f (x)

1 93. g(x) = f (x) 2

94. g(x) = f (2x)

1 95. g(x) = f a xb 2

a. Find a linear regression equation for these data.

96. Let f (x) = x 2 + x - 2 and g(x) = 3x + 1. Find each of the

b. Using the regression model, what is the expected initial

following. a. (f + g)(2)

f b. a b(- 1) g

c. (f - g)(x)

d. (f # g)(x)

speed of a ball that is struck with a foot speed of 12 meters per second? Round to the nearest meter per second. 103.

97. If f (x) = 4x 2 - 3x - 1, find the difference quotient

f (x + h) - f (x) h

Water Escaping a Ruptured Can

98. If g(x) = x3 - x, find the difference quotient

g(x + h) - g(x) h 99. Ball Rolling on a Ramp The distance traveled by a ball

rolling down a ramp is given by s(t) = 3t2, where t is the time in seconds after the ball is released and s(t) is measured in feet. Evaluate the average velocity of the ball for each of the following time intervals.

a. 32, 44

b. 32, 34

c. 32, 2.54

d. 32, 2.014

e. What appears to be the average velocity of the ball for the

time interval 32, 2 + ¢t4 as ¢t approaches 0?

100. If f (x) = x 2 + 4x and g(x) = x - 8, find a. ( f ⴰ g)(3)

b. ( g ⴰ f )( - 3)

c. ( f ⴰ g)(x)

d. ( g ⴰ f )(x)

Physics The rate at which water will escape from the bottom of a ruptured can depends on a number of factors, including the height of the water, the size of the hole, and the diameter of the can. The table below shows the height h (in millimeters) of water in a can after t seconds.

Time (t)

Height (h)

Time (t)

Height (h)

180

0

93

60

163

10

81

70

147

20

70

80

133

30

60

90

118

40

50

100

105

50

48

110

a. Find the quadratic regression model for these data. b. On the basis of this model, will the can ever empty? c.

Explain why there seems to be a contradiction between the model and reality, in that we know that the can will eventually run out of water.

257

CHAPTER 2 TEST

CHAPTER 2 TEST 1. Find the midpoint and the length of the line segment with end-

points (- 2, 3) and (4, -1).

In Exercises 14 to 18, sketch the graph of g given the graph of f below.

2. Determine the x- and y-intercepts of the equation x = 2y 2 - 4.

y 6

Then graph the equation. 3. Graph the equation y = ƒ x + 2 ƒ + 1. 4. Find the center and radius of the circle that has the general

form x 2 - 4x + y 2 + 2y - 4 = 0.

2 − 10

−8

−6

−4

4

6

8

x

−2 −4

f(x) = - 2x 2 - 16

−6

6. Find the elements a in the domain of f (x) = x 2 + 6x - 17 for

14. g(x) = 2f (x)

7. Find the slope of the line that passes through the points with

15. g(x) = f a xb

coordinates (5, -2) and (-1, 3).

2

−2

5. Determine the domain of the function

which f (a) = - 1.

y = f(x)

4

1 2

16. g(x) = - f (x) 8. Find the slope–intercept form of the equation of the line that

passes through the point with coordinates (5, -3) and whose slope is -2. 9. Find the slope–intercept form of the equation of the line that

passes through the point with coordinates (4, -2) and is perpendicular to the graph of 3x - 2y = 4. 10. Write the equation of the parabola f (x) = x 2 + 6x - 2 in

standard form. What are the coordinates of the vertex, and what is the equation of the axis of symmetry? 11. Find the maximum or minimum value of the function

f (x) = x 2 - 4x - 8. State whether this value is a maximum or a minimum.

17. g(x) = f (x - 1) + 3 18. g(x) = f ( -x) 19. Let f (x) = x 2 - x + 2 and g(x) = 2x - 1. Find a. (f - g)(x)

b. (f # g)(- 2)

c. (f ⴰ g)(3)

d. (g ⴰ f)(x)

20. Find the difference quotient of the function f (x) = x 2 + 1. 21. Dog Run A homeowner has 80 feet of fencing to make a rec-

tangular dog run alongside a house as shown below.

12. Classify each of the following as an even function, an odd

function, or neither. a. f(x) = x4 - x 2 b. f (x) = x3 - x c. f (x) = x - 1 13. Classify the graph of each equation as being symmetric with

respect to the x-axis, the y-axis, or the origin. a. y = x + 1 2

b. y = 2x3 + 3x c. y = 3x2 - 2

x

y

What dimensions x and y of the rectangle will produce the maximum area?

258

CHAPTER 2


22. Ball Rolling on a Ramp The distance traveled by a ball

rolling down a ramp is given by s(t) = 5t 2, where t is the time in seconds after the ball is released and s(t) is measured in feet. Evaluate the average velocity of the ball for each of the following time intervals.

a. 32, 34

b. 32, 2.54

c. 32, 2.014

23. Calorie Content The table to the right shows the percentage

of water and the number of calories in various canned soups to which 100 grams of water are added. a. Find the equation of the linear regression line for these data. b. Using the linear model from part a., find the expected num-

ber of calories in a soup that is 89% water. Round to the nearest calorie.

CUMULATIVE REVIEW EXERCISES 1. What property of real numbers is demonstrated by the equation

3(a + b) = 3(b + a)?

15. Find the distance between the points P1( -2, - 4) and P2(2, - 3). 16. Given G(x) = 2x3 - 4x - 7, find G( -2).

2 6 2. Which of the numbers - 3, - , , 0, 116, and 12 are not 3 p rational numbers? In Exercises 3 to 8, simplify the expression. 3. 3 + 4(2x - 9) 5.

7.

24a4b3 4 5

18a b

x2 + 6x - 27 x2 - 9

4. (- 4xy2)3(- 2x2 y4)

P1(2, - 3) and P2( -2, - 1). 18. Chemistry How many ounces of pure water must be added to

60 ounces of an 8% salt solution to make a 3% salt solution? 19. Tennis The path of a tennis ball during a serve is given by

6. (2x + 3)(3x - 7)

8.

4 2 2x - 1 x - 1

In Exercises 9 to 14, solve for x. 9. 6 - 2(2x - 4) = 14

17. Find the equation of the line that passes through the points

10. x2 - x - 1 = 0

11. (2x - 1)(x + 3) = 4

12. 3x + 2y = 15

13. x4 - x2 - 2 = 0

14. 3x - 1 6 5x + 7

h(x) = - 0.002x 2 - 0.03x + 8, where h(x) is the height of the ball in feet x feet from the server. For a serve to be legal in tennis, the ball must be at least 3 feet high when it is 39 feet from the server, and it must land in a spot that is less than 60 feet from the server. Does the path of the ball satisfy the conditions of a legal serve? 20. Medicine A patient with a fever is given a medication to

reduce the fever. The equation T = - 0.04t + 104 models the patient’s temperature T, in degrees Fahrenheit, t minutes after taking the medication. What is the rate, in degrees Fahrenheit per minute, at which the patient’s temperature is decreasing?

CHAPTER

3

POLYNOMIAL AND RATIONAL FUNCTIONS

3.1 Remainder Theorem and Factor Theorem 3.2 Polynomial Functions of Higher Degree

Shutterstock

3.3 Zeros of Polynomial Functions 3.4 Fundamental Theorem of Algebra 3.5 Graphs of Rational Functions and Their Applications

Applications of Polynomial Functions and Rational Functions In this chapter, you will study polynomial functions and rational functions. A polynomial function is a function defined by a polynomial. For instance, f (x) = x 3 + 4x 2 - x + 1 is a polynomial function. It is a third-degree, or cubic, polynomial function because the largest exponent of the variable x is 3. A rational function is a function defined by the quotient of two polynomials. For instance, f (x) =

7x 2x + 5 2

is a rational function. Polynomial and rational functions have many practical applications. In Exercise 67, page 285, a cubic polynomial function is used to model the power generated by a wind turbine at various wind speeds. In Exercise 74, page 322, a rational function is used to model the amount of medication in the bloodstream of a patient t hours after an injection.

259

260

CHAPTER 3


SECTION 3.1

Remainder Theorem and Factor Theorem

Division of Polynomials Synthetic Division Remainder Theorem Factor Theorem Reduced Polynomials

If P is a polynomial function, then the values of x for which P(x) is equal to 0 are called the zeros of P. For instance, -1 is a zero of P(x) = 2x3 - x + 1 because P( -1) = 2( - 1)3 - ( -1) + 1 = -2 + 1 + 1 = 0 Question • Is 0 a zero of P(x) = 2x3 - x + 1?

Much of this chapter concerns finding the zeros of polynomial functions. Sometimes the zeros of a polynomial function are determined by dividing one polynomial by another.

Division of Polynomials Recall A fraction bar acts as a grouping symbol. Division of a polynomial by a monomial is an application of the distributive property.

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. For instance, 16x 3 8x 2 12x 16x 3 - 8x 2 + 12x = + 4x 4x 4x 4x = 4x2 - 2x + 3

• Divide each term in the numerator by the denominator. • Simplify.

To divide a polynomial by a binomial, we use a method similar to that used to divide natural numbers. For instance, consider (6x3 - 16x2 + 23x - 5) , (3x - 2). 3x - 2 冄 6x 3 - 16x 2 + 23x - 5 2x 2 3x - 2 冄 6x 3 - 16x 2 + 23x - 5 6x 3 - 4x 2 - 12x 2 + 23x

6x 3 = 2x 2. 3x • Multiply: 2x 2(3x - 2) = 6x 3 - 4x 2

• Think

• Subtract and bring down the next term, 23x.

2x 2 - 4x 3x - 2 冄 6x - 16x 2 + 23x - 5 6x 3 - 4x 2 - 12x 2 + 23x - 12x 2 + 8x 3

15x - 5

m

• Subtract and bring down the next term, -5.

+ 5 - 5

Quotient

- 5 - 10

• Think

m m

Divisor

2x 2 - 4x 3x - 2 冄 6x - 16x 2 + 23x 6x 3 - 4x 2 - 12x 2 + 23x - 12x 2 + 8x 15x 15x 3

-12x 2 = - 4x. 3x • Multiply: - 4x(3x - 2) = - 12x 2 + 8x

• Think

5

Dividend

15x = 5. 3x • Multiply: 5(3x - 2) = 15x - 10 • Subtract to produce the remainder, 5.

Answer • No. P(0) = 2(0)3 - 0 + 1 = 1. Because P(0) Z 0, we know that 0 is not a zero of P.

3.1

Dividend Quotient $''%''& $''''%''''& 5 6x 3 - 16x 2 + 23x - 5 = 2x 2 - 4x + 5 + 3x - 2 3x - 2

Remainder Divisor

m m

Divisor

In every division, the dividend is equal to the product of the divisor and quotient, plus the remainder. That is,

# (2x 2 - 4x + # Quotient

5) +

$''%''&

6x 3 - 16x 2 + 23x - 5 = (3x - 2) = Divisor Dividend

5

#

$'%'&

$''''%''''&

20 written as a mixed number is 3 2 2 6 . Recall, however, that 6 3 3 2 means 6 + , which is in the form 3 remainder . quotient + divisor

261

The division process ends when the expression in the bottom row is of lesser degree than the divisor. The expression in the bottom row is the remainder, and the polynomial in the top row is the quotient. Thus (6x 3 - 16x 2 + 23x - 5) , (3x - 2) = 2x 2 - 4x + 5 with a remainder of 5. Although there is nothing wrong with writing the answer as we did above, it is more common to write the answer as the quotient plus the remainder divided by the divisor. (See the note at the left.) Using this method, we write

$'%'&

Note

REMAINDER THEOREM AND FACTOR THEOREM

+ Remainder

The preceding polynomial division concepts are summarized by the following theorem.

Division Algorithm for Polynomials Let P(x) and D(x) be polynomials, with D(x) of lower degree than P(x) and D(x) of degree 1 or more. Then there exist unique polynomials Q(x) and R(x) such that P(x) = D(x) # Q(x) + R(x)

where R(x) is either 0 or of degree less than the degree of D(x). The polynomial P(x) is called the dividend, D(x) is the divisor, Q(x) is the quotient, and R(x) is the remainder.

Before dividing polynomials, make sure that each polynomial is written in descending order. In some cases, it is helpful to insert a 0 in the dividend for a missing term (one whose coefficient is 0) so that like terms align in the same column. This is demonstrated in Example 1. Question • What is the first step you should perform to find the quotient of

(2x + 1 + x2) , (x - 1)?

EXAMPLE 1 Divide:

Divide Polynomials

- 5x 2 - 8x + x 4 + 3 x - 3

Solution Write the numerator in descending order. Then divide. - 5x 2 - 8x + x 4 + 3 x 4 - 5x 2 - 8x + 3 = x - 3 x - 3 Answer • Write the dividend in descending order as x 2 + 2x + 1.

(continued)


x 3 + 3x 2 + 4x + 4 x - 3 冄 x 4 + 0x 3 - 5x 2 - 8x + 3 x 4 - 3x 3 3x 3 - 5x 2 3x 3 - 9x 2

• Inserting 0x3 for the missing term helps align like terms in the same column.

4x 2 - 8x 4x 2 - 12x 4x + 3 4x - 12 15 15 -5x - 8x + x + 3 = x 3 + 3x 2 + 4x + 4 + . x - 3 x - 3 2

Thus

4


Synthetic Division A procedure called synthetic division can expedite the division process. To apply the synthetic division procedure, the divisor must be a polynomial of the form x - c, where c is a constant. In the synthetic division procedure, the variables that occur in the polynomials are not listed. To understand how synthetic division is performed, examine the following long division on the left and the related synthetic division on the right. Long Division

Synthetic Division Coefficients of the quotient

4

-5 8

2 6

- 10 16

3 8

Coefficients of the quotient

First row Second row

6

Third row Remainder

m

Remainder

4

m

+ 2x - 6x 8x - 10 8x - 16 6

2

m

+ 3x + 8 + 2x - 10

m

4x 2 x - 2 冄 4x 3 - 5x 2 4x 3 - 8x 2 3x 2 3x 2

$'%''&

CHAPTER 3

m

262

In the long division, the dividend is 4x 3 - 5x 2 + 2x - 10 and the divisor is x - 2. Because the divisor is of the form x - c, with c = 2, the division can be performed by the synthetic division procedure. Observe that in the above synthetic division: 1. The constant c is listed as the first number in the first row, followed by the coefficients of the dividend. 2. The first number in the third row is the leading coefficient of the dividend. 3. Each number in the second row is determined by computing the product of c and the number in the third row of the preceding column. 4. Each of the numbers in the third row, other than the first number, is determined by adding the numbers directly above it.

3.1


263

The following explanation illustrates the steps used to find the quotient and remainder of (2x 3 - 8x + 7) , (x + 3) using synthetic division. The divisor x + 3 is written in x - c form as x - ( -3), which indicates that c = - 3. The dividend 2x 3 - 8x + 7 is missing an x 2 term. If we insert 0x 2 for the missing term, the dividend becomes 2x 3 + 0x 2 - 8x + 7. Coefficients of the dividend $''%''&

-3 2

0

-8

7

m 2 -3 2

• Write the constant c, -3, followed by the coefficients of the dividend. Bring down the first coefficient in the first row, 2, as the first number of the third row.

0

-8

7

• Multiply c times the first number in the third row, 2, to produce the first number of the second row, -6. Add the 0 and the -6 to produce the next number of the third row, -6.

2 -3 2 2

m

-8 7 18 10 -8 7 18 -30 10 -23 m

0 -6 -6 0 -6 -6

m

-3 2

$''%''& Coefficients of the quotient

• Multiply c times the second number in the third row, -6, to produce the next number of the second row, 18. Add the -8 and the 18 to produce the next number of the third row, 10. • Multiply c times the third number in the third row, 10, to produce the next number of the second row, -30. Add the 7 and the -30 to produce the last number of the third row, -23.

m

-6

m

m

m

-6

2

Remainder

The last number in the bottom row, -23, is the remainder. The other numbers in the bottom row are the coefficients of the quotient. The quotient of a synthetic division always has a degree that is one less than the degree of the dividend. Thus the quotient in this example is 2x 2 - 6x + 10. The results of the synthetic division can be expressed in fractional form as -23 2x3 - 8x + 7 = 2x2 - 6x + 10 + x + 3 x + 3

Note

- 23 2x 2 - 6x + 10 + x + 3 can also be written as 23 2x 2 - 6x + 10 x + 3

or as 2x 3 - 8x + 7 = (x + 3)(2x 2 - 6x + 10) - 23 In Example 2, we illustrate the compact form of synthetic division, obtained by condensing the process explained here.

EXAMPLE 2

Use Synthetic Division to Divide Polynomials

Use synthetic division to divide x4 - 4x2 + 7x + 15 by x + 4. Solution Because the divisor is x + 4, we perform synthetic division with c = - 4. -4

1

0 -4

-4 16

7 -48

15 164

1

-4

12

- 41

179 (continued)

264

CHAPTER 3


The quotient is x 3 - 4x 2 + 12x - 41, and the remainder is 179. x 4 - 4x 2 + 7x + 15 179 = x 3 - 4x 2 + 12x - 41 + x + 4 x + 4 Try Exercise 12, page 268


p r g mS Y D I V DEGREE? 4 D I VI D E N D C OE F ?1 ?0 ?-4 ?7 ?1 5

Figure 3.1

A TI-83/TI-83 Plus/TI-84 Plus synthetic division program called SYDIV is available on the Internet at http://www.cengage.com/math/aufmann/algtrig7e. The program prompts you to enter the degree of the dividend, the coefficients of the dividend, and the constant c from the divisor x - c. For instance, to perform the synthetic division in Example 2, enter 4 for the degree of the dividend, followed by the coefficients 1, 0, -4, 7, and 15. See Figure 3.1. Press ENTER followed by -4 to produce the display in Figure 3.2. Press ENTER to produce the display in Figure 3.3. Press ENTER again to produce the display in Figure 3.4.

C? -4

R E M A IN D E R

C O E F O F Q U O T IE N T 1 -4 12 -41

179 Q U IT ? P R E S S 1 NEW C? PRESS 2

Figure 3.3

Figure 3.2

Figure 3.4

Remainder Theorem The following theorem shows that synthetic division can be used to determine the value P(c) for a given polynomial P( x) and constant c.

Remainder Theorem If a polynomial P(x) is divided by x - c, then the remainder equals P(c).

Proof of the Remainder Theorem

Because the degree of the remainder must be less than the degree of the divisor (x - c), we know that the remainder must be a constant. If we call the constant remainder r, then by the division algorithm we have P(x) = (x - c) # Q(x) + r

Setting x = c produces

P(c) = (c - c) # Q(c) + r P(c) = 0 + r P(c) = r

N

3.1


265

The following example shows that the remainder of P( x) = x 2 + 9x - 16 divided by x - 3 is the same as P(3). x + 12 x - 3 冄 x 2 + 9x - 16 x 2 - 3x

Let x = 3 and P(x) = x 2 + 9x - 16. Then P(3) = 3 2 + 9(3) - 16 = 9 + 27 - 16

12x - 16 12x - 36

= 20 c

c

20

The remainder of P(x) divided by x - 3 is equal to P(3). In Example 3, we use synthetic division and the Remainder Theorem to evaluate a polynomial function.

EXAMPLE 3

Use the Remainder Theorem to Evaluate a Polynomial Function

Let P(x) = 2x3 + 3x2 + 2x - 2. Use the Remainder Theorem to find P(c) 1 for c = - 2 and c = . 2 Algebraic Solution


1 Perform synthetic division with c = - 2 and c = and examine the 2 remainders.

1 The points ( - 2, -10) and a , 0b 2 are on the graph of P.

-2

2

y

-2 -8

3 2 -4 2

10

2 - 1 4 -10 The remainder is -10. Therefore, P(-2) = - 10. 1 2

2

2

3

2

-2

1

2

2

4

4

0

5 −4

−2

( 12 , 0) 2

4

x

−5 (−2, −10)

−10

P(x) = 2x3 + 3x2 + 2x − 2

1 The remainder is 0. Therefore, Pa b = 0. 2 Try Exercise 26, page 268

Using the Remainder Theorem to evaluate a polynomial function is often faster than evaluating the polynomial function by direct substitution. For instance, evaluating P(x) = x5 - 10x4 + 35x3 - 50x2 + 24x by substituting 7 for x requires the following work. P(7) = (7)5 - 10(7)4 + 35(7)3 - 50(7)2 + 24(7) = 16,807 - 10(2401) + 35(343) - 50(49) + 24(7) = 16,807 - 24,010 + 12,005 - 2450 + 168 = 2520

266

CHAPTER 3


Caution

Using the Remainder Theorem to evaluate P(7) requires only the following work.

Because P has a constant term of 0, we must include 0 as the last number in the first row of the synthetic division at the right.

7

1

- 10

35

-50

24

0

7

-21

98

336

2520

-3

14

1

48 360

2520 — P(7)

Factor Theorem 1 1 Note from Example 3 that P a b = 0. Recall that is a zero of P because P(x) = 0 when 2 2 1 x = . 2 The following theorem shows the important relationship between a zero of a given polynomial function and a factor of the polynomial.

Factor Theorem A polynomial P(x) has a factor (x - c) if and only if P(c) = 0. That is, (x - c) is a factor of P(x) if and only if c is a zero of P.

Proof of the Factor Theorem

If (x - c) is a factor of P(x), then P(x) = (x - c) # Q(x) and P(c) = (c - c) # Q(c) = 0 # Q(c) = 0 Conversely, if P(c) = 0, then, by the Remainder Theorem, R(x) = 0 and P(x) = (x - c) # Q(x) + 0 = (x - c) # Q(x) This result indicates that (x - c) is a factor of P(x).

EXAMPLE 4

N

Apply the Factor Theorem

Use synthetic division and the Factor Theorem to determine whether (x + 5) or (x - 2) is a factor of P(x) = x4 + x3 - 21x2 - x + 20. Solution -5

1

1 -5

- 21 20

-1 5

20 -20

1

-4

-1

4

0

The remainder of 0 indicates that (x + 5) is a factor of P(x).

3.1

2

1

1


1

-21

-1

20

2

6

-30

-62

3

-15

-31

- 42

267

The remainder of - 42 indicates that (x - 2) is not a factor of P(x). Try Exercise 36, page 268 Question • Is - 5 a zero of the function P as given in Example 4?

Here is a summary of the important role played by the remainder in the division of a polynomial by (x - c).

Remainder of a Polynomial Division In the division of the polynomial P(x) by (x - c), the remainder is equal to P(c). 0 if and only if (x - c) is a factor of P(x). 0 if and only if c is a zero of P. If c is a real number, then the remainder of P(x) , (x - c) is 0 if and only if (c, 0) is an x-intercept of the graph of P.

Reduced Polynomials In Example 4 we showed that (x + 5) is a factor of P(x) = x4 + x 3 - 21x 2 - x + 20 and that the quotient of P(x) divided by (x + 5) is x 3 - 4x 2 - x + 4. Thus P(x) = (x + 5)(x 3 - 4x 2 - x + 4) The quotient x 3 - 4x 2 - x + 4 is called a reduced polynomial, or a depressed polynomial, of P(x) because it is a factor of P(x) and its degree is 1 less than the degree of P(x). Reduced polynomials play an important role in Sections 3.3 and 3.4.

EXAMPLE 5

Find a Reduced Polynomial

Verify that (x - 3) is a factor of P(x) = 2x 3 - 3x 2 - 4x - 15, and write P(x) as the product of (x - 3) and the reduced polynomial Q(x). Solution 2

-4 9

3

5

-15 15 0

This 0 indicates that (x - 3) is a factor of P(x).

m

2

-3 6 m

3

Coefficients of the reduced polynomial Q(x) (continued)

Answer • Yes. Because (x + 5) is a factor of P(x), the Factor Theorem states that P(- 5) = 0.

Thus - 5 is a zero of P.

268

CHAPTER 3


Thus (x - 3) and the reduced polynomial 2x 2 + 3x + 5 are both factors of P. That is, P(x) = 2x3 - 3x2 - 4x - 15 = (x - 3)(2x2 + 3x + 5) Try Exercise 54, page 269

EXERCISE SET 3.1 In Exercises 1 to 10, use long division to divide the first polynomial by the second. 1. 5x 3 + 6x 2 - 17x + 20,

3. x4 - 5x 2 + 3x - 1,

5. x 2 + x 3 - 2x - 5,

x - 3

6. 4x + 3x + x - 5, 3

2x 2 - x + 1

x 2 - 2x + 2 2x 2 + 2x - 3 x2 + 1

In Exercises 11 to 24, use synthetic division to divide the first polynomial by the second. 11. 4x - 5x + 6x - 7,

x - 2

12. 5x + 6x - 8x + 1,

x - 5

2

3

2

13. 4x 3 - 2x + 3, 14. 6x 3 - 4x 2 + 17,

c = 2

26. P(x) = 2x 3 - x 2 + 3x - 1,

10. x5 + 3x4 - 2x 3 - 7x 2 - x + 4,

3

x - 4

25. P(x) = 3x 3 + x 2 + x - 5,

x - 2

9. 2x5 - x 3 + 5x 2 - 9x + 6,

x + 3

In Exercises 25 to 34, use synthetic division and the Remainder Theorem to find P(c).

7. 2x4 + 5x 3 - 6x 2 + 4x + 3, 8. 3x3 + x 2 - 5x + 2,

x + 1

24. 2x5 - 3x4 - 5x 2 - 10,

x - 2 x - 1

2

23. x6 + x - 10,

x + 4

4. x4 - 5x3 + x - 4,

x - 2

22. -x7 - x5 - x 3 - x - 5,

x + 3

2. 6x 3 + 15x 2 - 8x + 2,

21. x8 + x6 + x4 + x 2 + 4,

c = 3

27. P(x) = 4x4 - 6x 2 + 5,

c = -2

28. P(x) = 6x 3 - x 2 + 4x,

c = -3

29. P(x) = - 2x 3 - 2x 2 - x - 20,

c = 10

30. P(x) = - x 3 + 3x 2 + 5x + 30,

c = 8

31. P(x) = - x4 + 1, 32. P(x) = x5 - 1,

c = 3 c = 1

33. P(x) = x4 - 10x 3 + 2,

c = 3

34. P(x) = x5 + 20x 2 - 1,

c = -4

x + 1 In Exercises 35 to 44, use synthetic division and the Factor Theorem to determine whether the given binomial is a factor of P(x).

x + 3

15. x5 - 10x 3 + 5x - 1, 16. 6x4 - 2x 3 - 3x 2 - x,

x - 4 x - 5

35. P(x) = x3 + 2x 2 - 5x - 6,

x - 2

36. P(x) = x 3 + 4x 2 - 27x - 90,

x + 6

5

17. x - 1,

x - 1

37. P(x) = 2x 3 + x 2 - 3x - 1,

18. x + 1,

x + 1

38. P(x) = 3x 3 + 4x 2 - 27x - 36,

4

19. 8x 3 - 4x 2 + 6x - 3, 20. 12x + 5x + 5x + 6, 3

2

x -

1 2

3 x + 4

x + 1

39. P(x) = x4 - 25x 2 + 144,

x + 3

40. P(x) = x4 - 25x 2 + 144,

x - 3

x - 4

41. P(x) = x5 + 2x4 - 22x 3 - 50x 2 - 75x,

x - 5

3.1

42. P(x) = 9x4 - 6x 3 - 23x 2 - 4x + 4,

x + 1

2

44. P(x) = 10x4 + 9x 3 - 4x 2 + 9x + 6,

ways the bride can select her bridesmaids if she chooses from n = 7 girlfriends.

x +

b. Evaluate P(n) for n = 7 by substituting 7 for n. How does

this result compare with the result obtained in a.?

1 2

59. House of Cards The number of cards C needed to build a

In Exercises 45 to 52, use synthetic division to show that c is a zero of P. 45. P(x) = 3x 3 - 8x 2 - 10x + 28, 46. P(x) = 4x 3 - 10x 2 - 8x + 6, 47. P(x) = x4 - 1,

c = 1

48. P(x) = x 3 + 8,

c = -2

c = 2 c = 3

49. P(x) = 3x4 + 8x 3 + 10x 2 + 2x - 20, 50. P(x) = x4 - 2x 2 - 100x - 75,

c = -2

c = 5

51. P(x) = 2x 3 - 18x 2 - 50x + 66,

c = 11

52. P(x) = 2x4 - 34x 3 + 70x 2 - 153x + 45,

c = 15

In Exercises 53 to 56, verify that the given binomial is a factor of P(x), and write P(x) as the product of the binomial and its reduced polynomial Q(x). 53. P(x) = x 3 + x 2 + x - 14,

house of cards with r rows (levels) is given by the function C(r) = 1.5r 2 + 0.5r.

Topham/The Image Works

3

269

a. Use the Remainder Theorem to determine the number of

1 x 4

43. P(x) = 16x - 8x + 9x + 14x + 4, 4


x - 2

Use the Remainder Theorem to determine the number of cards needed to build a house of cards with a. r = 8 rows b. r = 20 rows

54. P(x) = x + 5x + 3x - 5x - 4,

x + 1

60. Display of Soda Cans The number of soda cans S needed to

55. P(x) = x4 - x 3 - 9x 2 - 11x - 4,

x - 4

build a square pyramid display with n levels is given by the function

4

3

2

56. P(x) = 2x5 - x4 - 7x 3 + x 2 + 7x - 10,

x - 2

S(n) =

1 3 1 1 n + n2 + n 3 2 6

57. Selection of Cards The number of ways you can select three

cards from a stack of n cards, in which the order of selection is important, is given by P(n) = n3 - 3n2 + 2n,

Level 1 12 = 1 soda can

n Ú 3

a. Use the Remainder Theorem to determine the number of ways

you can select three cards from a stack of n = 8 cards. b. Evaluate P(n) for n = 8 by substituting 8 for n. How does

this result compare with the result obtained in a.? 58. Selection of Bridesmaids A bride-to-be has many girlfriends,

but she has decided to have only five bridesmaids, including the maid of honor. The number of different ways n girlfriends can be chosen and assigned a position, such as maid of honor, first bridesmaid, second bridesmaid, and so on, is given by the polynomial function P(n) = n5 - 10n4 + 35n3 - 50n2 + 24n,

n Ú 5

Level 2 22 = 4 soda cans Level 3 32 = 9 soda cans Level 4 42 = 16 soda cans

A square pyramid display with n2 soda cans in level n

270

CHAPTER 3


Use the Remainder Theorem to determine the number of soda cans needed to build a square pyramid display with a. n = 6 levels b. n = 12 levels

64. Volume of a Box A rectangular box has a volume of

V(x) = x 3 + 10x 2 + 31x + 30 cubic inches. The height of the box is x + 2 inches. The width of the box is x + 3 inches. Find the length of the box in terms of x.

61. Election of Class Officers The number of ways a class of x+2

n students can elect a president, a vice president, a secretary, and a treasurer is given by P(n) = n4 - 6n3 + 11n2 - 6n, where n Ú 4. Use the Remainder Theorem to determine the number of ways the class can elect officers if the class consists of a. n = 12 students

x+3 ?

65. Use synthetic division to divide each of the following polyno-

b. n = 24 students

mials by x - 1.

62. Volume of a Solid The volume, in cubic inches, of the fol-

lowing solid is given by V(x) = x3 + 3x2.

2 x+2 1 x

x 3 - 1,

x 5 - 1,

x7 - 1

Use the pattern suggested by these quotients to write the quotient of (x9 - 1) , (x - 1).

In Exercises 66 to 69, determine the value of k so that the divisor is a factor of the dividend. 66. (x3 - x 2 - 14x + k) , (x - 2) 67. (2x 3 + x 2 - 25x + k) , (x - 3)

x+1

68. (3x3 + 14x 2 + kx - 6) , (x + 2)

Use the Remainder Theorem to determine the volume of the solid if a. x = 7 inches

69. (x4 + 3x 3 - 8x 2 + kx + 16) , (x + 4) 70. Use the Factor Theorem to show that for any positive integer n

b. x = 11 inches

P(x) = x n - 1 63. Volume of a Solid The volume, in cubic inches, of the fol-

lowing solid is given by V(x) = x3 + x2 + 10x - 8.

has x - 1 as a factor. 71. Find the remainder of

5x 48 + 6x10 - 5x + 7 divided by x - 1.

2 x−2

x+2

x−2

72. Find the remainder of

18x 80 - 6x 50 + 4x 20 - 2

x

x+1

Use the Remainder Theorem to determine the volume of the solid if a. x = 6 inches b. x = 9 inches

divided by x + 1. 73. Determine whether i is a zero of

P(x) = x3 - 3x 2 + x - 3 74. Determine whether - 2i is a zero of

P(x) = x4 - 2x3 + x2 - 8x - 12

3.2

SECTION 3.2 Far-Left and Far-Right Behavior Maximum and Minimum Values Real Zeros of a Polynomial Function Intermediate Value Theorem Real Zeros, x-Intercepts, and Factors of a Polynomial Function Even and Odd Powers of (x - c) Theorem Procedure for Graphing Polynomial Functions Cubic and Quartic Regression Models

POLYNOMIAL FUNCTIONS OF HIGHER DEGREE

271

Polynomial Functions of Higher Degree PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A19.

PS1. Find the minimum value of P(x) = x2 - 4x + 6. [2.4] PS2. Find the maximum value of P(x) = - 2x 2 - x + 1. [2.4] PS3. Find the interval on which P(x) = x2 + 2x + 7 is increasing. [2.4] PS4. Find the interval on which P(x) = - 2x2 + 4x + 5 is decreasing. [2.4] PS5. Factor: x4 - 5x2 + 4 [P.4] PS6. Find the x-intercepts of the graph of P(x) = 6x2 - x - 2. [2.4]

Table 3.1 summarizes information developed in Chapter 2 about graphs of polynomial functions of degree 0, 1, or 2. Table 3.1

Polynomial Function P(x)

Graph

P(x) = a (degree 0)

Horizontal line through (0, a)

P(x) = ax + b (degree 1), a Z 0

Line with y-intercept (0, b) and slope a

P(x) = ax2 + bx + c (degree 2), a Z 0

Parabola with vertex a -

b b , Pa - b b 2a 2a

In this section, we will focus on polynomial functions of degree 3 or higher. These functions can be graphed by the technique of plotting points; however, some additional knowledge about polynomial functions will make graphing easier. All polynomial functions have graphs that are smooth continuous curves. The terms smooth and continuous are defined rigorously in calculus, but for the present, a smooth curve is a curve that does not have sharp corners, like the graph shown in Figure 3.5a. A continuous curve does not have a break or hole, like the graph shown in Figure 3.5b. Note

y

y

The general form of a polynomial is given by an x n + an - 1 x n - 1 + Á + a0 In this text, the coefficients an , an - 1, . . . , a0 are all real numbers unless specifically stated otherwise.

Hole Break

Sharp corner x

a. Continuous, but not smooth

x

b. Not continuous Figure 3.5

272

CHAPTER 3


Far-Left and Far-Right Behavior Note The leading term of a polynomial function in x is the nonzero term that contains the largest power of x. The leading coefficient of a polynomial function is the coefficient of the leading term.

Table 3.2

The graph of a polynomial function may have several up and down fluctuations; however, the graph of every polynomial function eventually will increase or decrease without bound as n ƒ x ƒ becomes larger. The leading term an x is said to be the dominant term of the polynon n-1 mial function P(x) = an x + an - 1x + Á + a1x + a0 because, as ƒ x ƒ becomes larger, n the absolute value of an x will be much larger than the absolute value of any of the other terms. Because of this condition, you can determine the far-left and far-right behavior of the polynomial by examining the leading coefficient an and the degree n of the polynomial. Table 3.2 shows the far-left and far-right behavior of a polynomial function P with leading term an x n.

The Leading Term Test

The far-left and far-right behavior of the graph of the polynomial function P (x ) an x n an1x n1 can be determined by examining its leading term an x n. n Is Even an>0

y

a1x a0

n Is Odd

If an 7 0 and n is even, then the graph of P goes up to the far left and up to the far right. As x → − ∞, P(x) → ∞

Á

If an 7 0 and n is odd, then the graph of P goes down to the far left and up to the far right. y

As x → ∞, P(x) → ∞

As x → ∞, P(x) → ∞

x

x

As x → − ∞, P(x) → − ∞

an or N repeatedly to select an x-value that is to the left of the relative maximum point. Press ENTER . A left bound is displayed in the bottom left corner. 4. Press N repeatedly to select an x-value that is to the right of the relative maximum point. Press ENTER . A right bound is displayed in the bottom left corner. 5. The word Guess? is now displayed in the bottom left corner. Press > repeatedly to move to a point near the maximum point. Press ENTER . Answer • Yes, the absolute minimum y5 also satisfies the requirements of a relative minimum.

3.2


275

6. The cursor appears on the relative maximum point, and the coordinates of the relative maximum point are displayed. In this example, the y value 6.312608 is the approximate relative maximum of the function P. (Note: If your window settings, bounds, or guess are different from those shown here, then your final results may differ slightly from the final results shown in step 6.) Plot1 Plot2 Plot3 \Y 1 = .3X^3+-2.8X^2+6.4X+2 \Y 2 = WINDOW \Y 3 = Xmin = 0 \Y 4 = Xmax = 8 \Y 5 = Xscl = 1 \Y 6 = Ymin = -4 \Y 7 = Ymax = 10 Yscl = 1 Xres = 1

10

Y1=.3X^3+-2.8X^2+6.4X+2

CALCULATE 1 : value 2: zero 3: minimum 4: maximum 5: intersect 6: dy/dx 7: ∫f(x)dx

Step 1

0

−4

Step 2

10

10

Y1=.3X^3+-2.8X^2+6.4X+2

0

−4

10

8 0

Guess? X=1.7021277 −4

Step 4

Step 3

Y1=.3X^3+-2.8X^2+6.4X+2

8 0

Right Bound? X=2.5531915 Y=5.0809358

8

Left Bound? X=.85106383 Y=5.6036716

8

Maximum X=1.5086448 Y=6.312608

Y=6.26079

−4

Step 5

Step 6

The following example illustrates the role a maximum may play in an application.

EXAMPLE 2


A rectangular piece of cardboard measures 12 inches by 16 inches. An open box is formed by cutting squares that measure x inches by x inches from each of the corners of the cardboard and folding up the sides, as shown below.

12 in. x

x x

12 − 2x 16 − 2x

16 in.

a. b.

Express the volume V of the box as a function of x. Determine (to the nearest tenth of an inch) the x value that maximizes the volume.

Solution a. The height, width, and length of the open box are x, 12 - 2x, and 16 - 2x. The volume is given by V(x) = x(12 - 2x)(16 - 2x) V(x) = 4x3 - 56x2 + 192x

(continued)

276

CHAPTER 3


250

b.

Use a graphing utility to graph y = V(x). The graph is shown in Figure 3.12. Note that we are interested only in the part of the graph for which 0 6 x 6 6. This is because the length of each side of the box must be positive. In other words, x 7 0,

10

−2

−100

y = 4x3 − 56x2 + 192x Figure 3.12

Integrating Technology A TI graphing calculator program is available that simulates the construction of a box by cutting out squares from each corner of a rectangular piece of cardboard. This program, CUTOUT, can be found at the online study center at http://www.cengage.com/math/ aufmann/algtrig7e.

12 - 2x 7 0, x 6 6

and

16 - 2x 7 0 x 6 8

The domain of V is the intersection of the solution sets of the three inequalities. Thus the domain is 5x ƒ 0 6 x 6 66. 250 Now use a graphing utility to find that V attains its maximum of about 194.06736 cubic inches when x L 2.3 inches. See Figure 3.13. 6

0 Maximum

X=2.262967

Y=194.06736

−40

y = 4x3 − 56x2 + 192x, 0 x 6 Figure 3.13


Real Zeros of a Polynomial Function Sometimes the real zeros of a polynomial function can be determined by using the factoring procedures developed in previous chapters. We illustrate this concept in the next example.

EXAMPLE 3

Factor to Find the Real Zeros of a Polynomial Function

Factor to find the three real zeros of P(x) = x3 + 3x2 - 4x. Algebraic Solution


P can be factored as shown below. P(x) = x + 3x - 4x = x(x 2 + 3x - 4) = x(x - 1)(x + 4) 3

The graph of P has x-intercepts at (0, 0), (1, 0), and ( - 4, 0).

2

16

• Factor out the common factor x. • Factor the trinomial x + 3x - 4. 2

The real zeros of P(x) are x = 0, x = 1, and x = - 4. −4.7

(−4, 0

(0, 0

(1, 0

−4

P(x) = x3 + 3x2 − 4x


Intermediate Value Theorem The following theorem states an important property of polynomial functions.

4.7

3.2


277

Intermediate Value Theorem If P is a polynomial function and P(a) Z P(b) for a 6 b, then P takes on every value between P(a) and P(b) in the interval 3a, b4. The Intermediate Value Theorem is often used to verify the existence of a zero of a polynomial function in an interval. The essential idea is to find two values a and b such that the polynomial function is positive at one of the values and negative at the other. Then you can conclude by the Intermediate Value Theorem that the function has a zero between a and b. Stated in geometric terms, if the points (a, P(a)) and (b, P(b)) are on opposite sides of the x-axis, then the graph of the polynomial function P must cross the x-axis at least once between a and b. See Figure 3.14. y P(b)

y

P (b, P(b))

P(a)

(a, P(a))

P(a)

P(x) = 0 a

P

P(x) = 0 x

b

a P(b)

(a, P(a))

P(a) 6 0, P(b) 7 0

b

x

(b, P(b))

P(b) 6 0, P(a) 7 0 Figure 3.14

EXAMPLE 4

Apply the Intermediate Value Theorem

Use the Intermediate Value Theorem to verify that P(x) = x3 - x - 2 has a real zero between 1 and 2. Algebraic Solution


Use substitution or synthetic division to evaluate P(1) and P(2).

The graph of P crosses the x-axis between x = 1 and x = 2. Thus P has a real zero between 1 and 2.

P(x) = x 3 - x - 2 P(1) = (1)3 - (1) - 2

y

= 1 - 1 - 2 = -2

4

(2 , 4 )

3

and P(x) = x 3 - x - 2 P(2) = (2)3 - (2) - 2 = 8 - 2 - 2 = 4 Because P(1) and P(2) have opposite signs, we know by the Intermediate Value Theorem that the polynomial function P has at least one real zero between 1 and 2.


2 1 2

−2 −1 −2

(1 , −2 )

P(x) = x3 − x − 2

x

278

CHAPTER 3


Real Zeros, x-Intercepts, and Factors of a Polynomial Function The following theorem summarizes important relationships among the real zeros of a polynomial function, the x-intercepts of its graph, and its factors; this theorem can be written in the form (x - c), where c is a real number.

Polynomial Functions, Real Zeros, Graphs, and Factors (x c) If P is a polynomial function and c is a real number, then all of the following statements are equivalent in the following sense: If any one statement is true, then they are all true, and if any one statement is false, then they are all false. y

(x - c) is a factor of P. x = c is a real solution of P(x) = 0.

6

x = c is a real zero of P.

4 (−2, 0) −4

(1, 0)

2

2

−2

(3, 0) 4

(c, 0) is an x-intercept of the graph of y = P(x). x

−4

Sometimes it is possible to make use of the preceding theorem and a graph of a polynomial function to find factors of the function. For example, the graph of

−6 −8

S(x) = x 3 - 2x 2 - 5x + 6

S(x) = x3 − 2x2 − 5x + 6

is shown in Figure 3.15. The x-intercepts are (- 2, 0), (1, 0), and (3, 0). Hence -2, 1, and 3 are zeros of S, and 3x - ( -2)4, (x - 1), and (x - 3) are all factors of S.

Figure 3.15

Even and Odd Powers of (x - c) Theorem Use a graphing utility to graph P(x) = (x + 3)(x - 4)2. Compare your graph with Figure 3.16. Examine the graph near the x-intercepts ( - 3, 0) and (4, 0). Observe that the graph of P

60

−5

6 −20

y = (x + 3)(x − 4)2

crosses the x-axis at ( -3, 0). intersects but does not cross the x-axis at (4, 0). The following theorem can be used to determine at which x-intercepts the graph of a polynomial function will cross the x-axis and at which x-intercepts the graph will intersect but not cross the x-axis.

Figure 3.16

Even and Odd Powers of (x c) Theorem If c is a real number and the polynomial function P has (x - c) as a factor exactly k times, then the graph of P will intersect but not cross the x-axis at (c, 0), provided k is an even positive integer. cross the x-axis at (c, 0), provided k is an odd positive integer.

3.2

EXAMPLE 5


279

Apply the Even and Odd Powers of (x c) Theorem

Determine where the graph of P(x) = (x + 3)(x - 2)2(x - 4)3 crosses the x-axis and where the graph intersects but does not cross the x-axis. Solution The exponents of the factors (x + 3) and (x - 4) are odd integers. Therefore, the graph of P will cross the x-axis at the x-intercepts ( -3, 0) and (4, 0). The exponent of the factor (x - 2) is an even integer. Therefore, the graph of P will intersect but not cross the x-axis at (2, 0). Use a graphing utility to check these results. Try Exercise 34, page 283

Procedure for Graphing Polynomial Functions You may find that you can sketch the graph of a polynomial function just by plotting several points; however, the following procedure will help you sketch the graph of many polynomial functions in an efficient manner.

Procedure for Graphing Polynomial Functions P(x) = an x n + an - 1 x n - 1 + Á + a1 x + a0, an Z 0 To graph P, follow these steps. 1. Determine the far-left and far-right behavior Examine the leading coefficient an x n to determine the far-left and far-right behavior of the graph. 2. Find the y-intercept Determine the y-intercept by evaluating P(0).

Factoring of Polynomials See Section P.4.

3. Find the x-intercept or x-intercepts and determine the behavior of the graph near the x-intercept or x-intercepts If possible, find the x-intercepts by factoring. If (x - c), where c is a real number, is a factor of P, then (c, 0) is an x-intercept of the graph. Use the Even and Odd Powers of (x - c) Theorem to determine where the graph crosses the x-axis and where the graph intersects but does not cross the x-axis. 4. Find additional points on the graph Find a few additional points (in addition to the intercepts). 5. Check for symmetry a.

The graph of an even function is symmetric with respect to the y-axis.

b.

The graph of an odd function is symmetric with respect to the origin.

6. Sketch the graph Use all the information previously obtained to sketch the graph of the polynomial function. The graph should be a smooth continuous curve that passes through the points determined in steps 2 through 4. The graph should have a maximum of n - 1 turning points.

280

CHAPTER 3


EXAMPLE 6

Graph a Polynomial Function

Sketch the graph of P(x) = x3 - 4x2 + 4x. Solution 1. Determine the far-left and far-right behavior The leading term is 1x 3. Because the leading coefficient, 1, is positive and the degree of the polynomial, 3, is odd, the graph of P goes down to the far left and up to the far right. 2. Find the y-intercept

P(0) = 03 - 4(0)2 + 4(0) = 0. The y-intercept is (0, 0).

3. Find the x-intercept or x-intercepts and determine the behavior of the graph near the x-intercept or x-intercepts Try to factor x 3 - 4x 2 + 4x. x 3 - 4x 2 + 4x = x(x 2 - 4x + 4) = x(x - 2)(x - 2) = x(x - 2)2 Because (x - 2) is a factor of P, the point (2, 0) is an x-intercept of the graph of P. Because x is a factor of P (think of x as x - 0), the point (0, 0) is an x-intercept of the graph of P. Applying the Even and Odd Powers of (x - c) Theorem allows us to determine that the graph of P crosses the x-axis at (0, 0) and intersects but does not cross the x-axis at (2, 0).

y 8

4

−4

−2

4. Find additional points on the graph

2

4 x

x

P(x)

-1

-9

0.5

1.125

1

1

3

3

−8

P(x) = x3 − 4x2 + 4x Figure 3.17

5. Check for symmetry The function P is neither an even nor an odd function, so the graph of P is not symmetric to either the y-axis or the origin. 6. Sketch the graph

See Figure 3.17.


Cubic and Quartic Regression Models In Section 2.7, we used linear and quadratic functions to model several data sets. In many applications, data can be modeled more closely using cubic and quartic functions.

EXAMPLE 7

Model an Application with a Cubic Function

The table on the next page lists the number of U.S. indoor movie screens from 1992 to 2007. Find the cubic regression function that models the data and use the function to predict the number of indoor movie screens in 2013.

3.2

Note

During recent years, many cinema sites have been replaced with multiscreen complexes, or multiplexes. Thus there is an upward trend in the number of movie screens and a downward trend in the number of cinema sites. In Exercise 69, on page 285, you will model this downward trend in the number of cinema sites.


Number of U.S. Indoor Movie Screens

Year

Number of Screens

Year

Number of Screens

1992

24,344

2000

35,567

1993

24,789

2001

34,490

1994

25,830

2002

35,170

1995

26,995

2003

35,361

1996

28,905

2004

36,012

1997

31,050

2005

37,092

1998

33,418

2006

37,776

1999

36,448

2007

38,159

Source: National Association of Theatre Owners.

Solution 1. Construct a scatter diagram for these data Enter the data and construct a scatter diagram as explained on pages 237–243. We have used x 1 to represent 1992 and x 16 to represent 2007. The upward, downward, upward trends shown in the scatter diagram suggest that we may be able to closely model the data with a cubic or a quartic function.

45,000

1

CubicReg y=ax3+bx2+cx+d a=.9201794078 b=-88.15295025 c=2218.870794 d=20906.0467 R2=.9430217124

f(x) = 0.9201794078x 3 - 88.15295025x 2 + 2218.870794x + 20906.0467 The value of the coefficient of determination will not be displayed unless the DiagnosticOn command is enabled. The DiagnosticOn command is in the CATALOG menu.

24 24,000

2. Find the regression equation To find the cubic regression function on a TI-83/TI-83 Plus/TI-84 Plus calculator, select 6: CubicReg in the STAT CALC menu. The cubic model is

Recall

281

3. Examine the coefficient of determination The coefficient of determination, R 2, is 0.9430217124, which is relatively close to 1. Thus the cubic function provides a fairly good model of the data, as shown by the graph at the right.

45,000

1

24 24,000

4. Use the model to make a prediction Use the value command, in the CALCULATE menu, to predict the number of movie screens in 2013 (represented by x 22). Press 2ND (CALC) ENTER 22 ENTER to produce the screen at the right. The cubic model predicts about 36,853 indoor movie screens in 2013. Try Exercise 68, page 285

45,000

Y1=.9201794077973X^3+-8_ X

1 X=22 24,000

Y=36853.247 24

282

CHAPTER 3


Cubic and quartic regression models often yield poor predictions. For instance, you may have noticed that in Example 7 the cubic model predicts there will be fewer movie screens in 2013 than in 2007. This scenario is unlikely because the number of screens has been increasing every year since 2001. A quartic regression model will always model an application at least as well as a cubic regression model. To illustrate this, find the quartic regression model for the data in Example 7. This can be accomplished by choosing 7: QuartReg in the STAT CALC menu (keystrokes STAT N 7 ENTER ). QuarticReg y=ax4+bx3+…+e a=2.732416448 b=-91.98197984 c=948.9942645 d=-1988.269846 ↓e=25444.98077

QuarticReg y=ax4+bx3+…+e b=-91.98197984 c=948.9942645 d=-1988.269846 e=25444.98077 R2=.9721649195 ↓

R2 is the coefficient of determination.

The coefficients of the quartic regression model

The quartic model is f(x) = 2.732416448x4 - 91.98197984x3 + 948.9942645x2 1988.269846x + 25444.98077 The coefficient of determination (R 2 0.9721649195) and the graph of the quartic model shown on the left below both confirm that this quartic model does provide a better fit to the data in Example 7 than does the cubic model. 45,000

45,000 Y2=2.7324164483655X^4+-_

24

1 24,000

X=22 …….... Y=101677.09

1

24

24,000

A graph of the quartic regression model and the data

The quartic model predicts about 101,677 indoor movie screens in 2013.

If you use the quartic model to predict the number of indoor movie screens in 2013, you get 101,677 screens, as shown by the graph on the right above. This huge increase (from 38,159 in 2007) is highly unlikely.

EXERCISE SET 3.2 In Exercises 1 to 8, examine the leading term and determine the far-left and far-right behavior of the graph of the polynomial function.

3. P(x) = 5x 5 - 4x 3 - 17x 2 + 2 4. P(x) = - 6x 4 - 3x 3 + 5x 2 - 2x + 5

1. P(x) = 3x 4 - 2x 2 - 7x + 1

5. P(x) = 2 - 3x - 4x 2

2. P(x) = - 2x 3 - 6x 2 + 5x - 1

6. P(x) = - 16 + x 4

3.2

7. P(x) =

1 3 (x + 5x 2 - 2) 2


15. P(x) = x4 - 4x 3 - 2x 2 + 12x - 5 16. P(x) = x4 - 10x 2 + 9

1 8. P(x) = - (x 4 + 3x 2 - 2x + 6) 4 9. The following graph is the graph of a third-degree (cubic) poly-

nomial function. What does the far-left and far-right behavior of the graph say about the leading coefficient a? 5

7.5

(3)

17. P(x) = x3 - 2x 2 - 15x

(3) (3)

19. P(x) = x4 - 13x 2 + 36

(4)

20. P(x) = 4x4 - 37x 2 + 9

(4)

21. P(x) = x5 - 5x3 + 4x −5

(5)

22. P(x) = x5 - 25x3 + 144x

P(x) = ax3 + bx2 + cx + d

10. The following graph is the graph of a fourth-degree (quartic)

polynomial function. What does the far-left and far-right behavior of the graph say about the leading coefficient a?

(5)

In Exercises 23 to 32, use the Intermediate Value Theorem to verify that P has a zero between a and b. 23. P(x) = 2x 3 + 3x 2 - 23x - 42; 24. P(x) = 4x 3 - x 2 - 6x + 1;

5

25. P(x) = 3x 3 + 7x 2 + 3x + 7; 5

−5

(3)

In Exercises 17 to 22, find the real zeros of each polynomial function by factoring. The number in parentheses to the right of each polynomial indicates the number of real zeros of the given polynomial function.

18. P(x) = x 3 - 6x 2 + 8x

−7.5

a = 3, b = 4

a = 0, b = 1 a = - 3, b = - 2

26. P(x) = 2x 3 - 21x 2 - 2x + 25;

a = 1, b = 2

27. P(x) = 4x4 + 7x 3 - 11x 2 + 7x - 15;

a = 1, b = 1

−5

P(x) = ax4 + bx3 + cx2 + dx + e

In Exercises 11 to 16, use a graphing utility to graph each polynomial. Use the maximum and minimum features of the graphing utility to estimate, to the nearest tenth, the coordinates of the points where P(x) has a relative maximum or a relative minimum. For each point, indicate whether the y value is a relative maximum or a relative minimum. The number in parentheses to the right of the polynomial is the total number of relative maxima and minima. 11. P(x) = x 3 + x 2 - 9x - 9

(2)

12. P(x) = x 3 + 4x 2 - 4x - 16

(2)

13. P(x) = x 3 - 3x 2 - 24x + 3

(2)

14. P(x) = - 2x 3 - 3x 2 + 12x + 1

28. P(x) = 5x 3 - 16x 2 - 20x + 64; 29. P(x) = x4 - x2 - x - 4; 30. P(x) = x3 - x - 8;

a = 3, b = 3

1 2

1 2

a = 1.7, b = 1.8

a = 2.1, b = 2.2

31. P(x) = - x4 + x3 + 5x - 1;

a = 0.1, b = 0.2

32. P(x) = - x3 - 2x2 + x - 3;

a = - 2.8, b = - 2.7

In Exercises 33 to 40, determine the x-intercepts of the graph of P. For each x-intercept, use the Even and Odd Powers of (x c) Theorem to determine whether the graph of P crosses the x-axis or intersects but does not cross the x-axis. 33. P(x) = (x - 1)(x + 1)(x - 3)

(2)

283

34. P(x) = (x + 2)(x - 6)2

284

CHAPTER 3


35. P(x) = - (x - 3)2(x - 7)5 36. P(x) = (x + 2) (x - 6) 3

55. P(x) = x 5 - x 4 - 5x 3 + x 2 + 8x + 4

(In factored form, P(x) = (x + 1)3(x - 2)2.)

10

56. P(x) = 2x 5 - 3x 4 - 4x 3 + 3x 2 + 2x

37. P(x) = (2x - 3) (x - 1) 4

15

38. P(x) = (5x + 10)6(x - 2.7)5 39. P(x) = x 3 - 6x 2 + 9x 40. P(x) = x + 3x + 4x 4

3

In Exercises 57 to 62, use translation, reflection, or both concepts to explain how the graph of P can be used to produce the graph of Q. 57. P(x) = x 3 + x;

Q(x) = x 3 + x + 2

2

58. P(x) = x 4;

Q(x) = x 4 - 3

59. P(x) = x 4;

Q(x) = (x - 1)4

41. P(x) = x 3 - x 2 - 2x

60. P(x) = x 3;

Q(x) = (x + 3)3

42. P(x) = x 3 + 2x 2 - 3x

61. P(x) = x 5;

Q(x) = - (x - 2)5 + 3

43. P(x) = - x3 - 2x2 + 5x + 6

62. P(x) = x 6;

Q(x) = (x + 4)6 - 5

In Exercises 41 to 56, sketch the graph of the polynomial function. Do not use a graphing calculator.

(In factored form, P(x) = - (x + 3)(x + 1)(x - 2).) 63.

44. P(x) = - x - 3x + x + 3 3

2

(In factored form, P(x) = - (x + 3)(x + 1)(x - 1).)

Medication Level Pseudoephedrine hydrochloride is an allergy medication. The function

L(t) = 0.03t 4 + 0.4t 3 - 7.3t 2 + 23.1t

45. P(x) = x 4 - 4x 3 + 2x 2 + 4x - 3

where 0 … t … 5, models the level of pseudoephedrine hydrochloride, in milligrams, in the bloodstream of a patient t hours after 30 milligrams of the medication have been taken.

46. P(x) = x 4 - 6x 3 + 8x 2

a. Use a graphing utility and the function L(t) to determine

(In factored form, P(x) = (x + 1)(x - 1)2(x - 3).)

the maximum level of pseudoephedrine hydrochloride in the patient’s bloodstream. Round your result to the nearest 0.01 milligram.

47. P(x) = x 3 + 6x 2 + 5x - 12

(In factored form, P(x) = (x - 1)(x + 3)(x + 4).)

Pseudoephedrine hydrochloride in the bloodstream (in milligrams)

48. P(x) = - x 3 + 4x 2 + x - 4 49. P(x) = - x + 7x - 6 3

50. P(x) = x 3 - 6x 2 + 9x

(In factored form, P(x) = x(x - 3)2.)

51. P(x) = - x 3 + 4x 2 - 4x

(In factored form, P(x) = - x(x - 2)2.)

52. P(x) = - x 4 + 2x 3 + 3x 2 - 4x - 4

3 1 54. P(x) = x 4 + x 3 - 2x 2 - x + 2 2 1 (In factored form, P(x) = (x - 1)2(x + 1)(x + 3).) 2

16 12 8 4 1

2

3

4

5

t

Time (in hours)

b. At what time t, to the nearest minute, is this maximum level

(In factored form, P(x) = - (x - 2)2(x + 1)2.)

53. P(x) = - x 4 + 3x 3 + x 2 - 3x

L 20

of pseudoephedrine hydrochloride reached? 64.

Profit A software company produces a computer game.

The company has determined that its profit P, in dollars, from the manufacture and sale of x games is given by P(x) = - 0.000001x 3 + 96x - 98,000 where 0 6 x … 9000.

3.2

a. What is the maximum profit, to the nearest thousand dollars,


67. Wind Turbine Power The power P, in watts, generated by a

the company can expect from the sale of its game?

particular wind turbine with winds blowing at v meters per second is given by the cubic polynomial function

b. How many games, to the nearest unit, does the company

P(v) = 4.95v 3

need to produce and sell to obtain the maximum profit?

a. Find the power generated, to the nearest 10 watts, when the

Construction of a Box A company constructs boxes from

65.

285

wind speed is 8 meters per second.

rectangular pieces of cardboard that measure 10 inches by 15 inches. An open box is formed by cutting squares that measure x inches by x inches from each corner of the cardboard and folding up the sides, as shown in the following figure.

b. What wind speed, in meters per second, is required to gen-

erate 10,000 watts? Round to the nearest tenth. c. If the wind speed is doubled, what effect does this have on

the power generated by the turbine? d. If the wind speed is tripled, what effect does this have on the

10 in.

power generated by the turbine?

x x

x x

68.

15 in.

Vehicle Sales The following table shows the approximate sales of U.S. sport/cross utility vehicles from 1988 to 2006.

a. Express the volume V of the box as a function of x.

U.S. Sport/Cross Utility Vehicle Sales b. Determine (to the nearest hundredth of an inch) the x value

that maximizes the volume of the box. 66.

Maximizing Volume A closed box is to be constructed from a rectangular sheet of cardboard that measures 18 inches by 42 inches. The box is made by cutting rectangles that measure x inches by 2x inches from two of the corners and by cutting two squares that measure x inches by x inches from the top and from the bottom of the rectangle, as shown in the following figure. What value of x (to the nearest thousandth of an inch) will produce a box with maximum volume?

x x

x

42 in. x x

18 in.

Year

Sales (millions)

Year

Sales (millions)

1988

1.0

1998

2.8

1989

1.0

1999

3.2

1990

0.9

2000

3.5

1991

0.9

2001

3.9

1992

1.1

2002

4.1

1993

1.4

2003

4.5

1994

1.5

2004

4.7

1995

1.8

2005

4.6

1996

2.2

2006

4.5

1997

2.4

Source: The World Almanac and Book of Facts, 2008.

a. Find the cubic regression function that models the data. Use

x 1 to represent 1988 and x 19 to represent 2006. b. Use the cubic regression function to predict the sales of

these vehicles in 2011 (x 24). Round to the nearest tenth of a million vehicles. 69.

Number of Cinema Sites The number of U.S. cin-

ema sites from 1996 to 2007 are given in the table on the next page.

286

CHAPTER 3


Fuel Efficiency The fuel efficiency, in miles per gallon, for

71.

Number of U.S. Cinema Sites

Year

Number of Cinema Sites

Year


1996

7215

2002

5712

1997

6903

2003

5700

1998

6894

2004

5629

1999

7031

2005

2000

6550

2001

5813

a midsize car at various speeds, in miles per hour, is given in the following table.

Fuel Efficiency of a Midsize Car

5713

Speed (mph)

Fuel Efficiency (mpg)

Speed (mph)

Fuel Efficiency (mpg)

2006

5543

0

0.0

40

30.2

2007

5545

5

11.1

45

30.6

10

17.2

50

31.7

15

22.4

55

30.8

20

26.2

60

29.5

25

27.1

65

28.2

30

28.3

70

26.3

35

29.4

75

24.1


a. Find a cubic and a quartic model for the data. Use x 1 to

represent 1996 and x 12 to represent 2007.

b. Do you think either of the models will accurately predict the

number of cinema sites in 2011? Box Office Grosses The total movie theater box

70.

office grosses in the United States and Canada, for the years 1994 to 2007, are given in the following table.

a. Find a cubic and a quartic model for the data. Let x repre-

sent the speed of the car in miles per hour. Box Office Grosses

b. Use the cubic and the quartic models to predict the fuel effi-

Year

Gross (billions of dollars)

Year

Gross (billions of dollars)

1994

5.184

2001

8.125

1995

5.269

2002

9.272

1996

5.817

2003

9.165

1997

6.216

2004

9.215

1998

6.760

2005

8.832

1999

7.314

2006

9.138

2000

7.468

2007

9.629


ciency, in miles per gallon, of the car traveling at a speed of 80 miles per hour. Round to the nearest tenth. c. Which of the fuel efficiency values from b. is the more real-

istic value? 72.

Use a graph of P(x) = 4x 4 - 12x 3 + 13x 2 - 12x + 9 to determine between which two consecutive integers P has a real zero.

73. The point (2, 0) is on the graph of P. What point must be on the

graph of P(x - 3)? 74. The point (3, 5) is on the graph of P. What point must be on the

graph of P(x + 1) - 2? a. Find a quartic regression function for the data. Use x 1 to

represent 1994 and x 14 to represent 2007.

b. Graph the quartic regression function and the scatter dia-

gram of the data in the same window. c. Use the quartic regression function to predict the box

office gross for 2010. Round to the nearest tenth of a billion dollars.

75.

Explain how to use the graph of y = x3 to produce the graph of P(x) = (x - 2)3 + 1.

76.

Consider the following conjecture. Let P be a polynomial function. If a and b are real numbers such that a 6 b, P(a) 7 0, and P(b) 7 0, then P(x) does not have a real zero between a and b. Is this conjecture true or false? Support your answer.

3.3

SECTION 3.3

ZEROS OF POLYNOMIAL FUNCTIONS

287

Zeros of Polynomial Functions

Multiple Zeros of a Polynomial Function Rational Zero Theorem Upper and Lower Bounds for Real Zeros Descartes’ Rule of Signs Zeros of a Polynomial Function Applications of Polynomial Functions


PS1. Find the zeros of P(x) = 6x 2 - 25x + 14. [1.3, 2.4] PS2. Use synthetic division to divide 2x3 + 3x 2 + 4x - 7 by x + 2. [3.1] PS3. Use synthetic division to divide 3x4 - 21x 2 - 3x - 5 by x - 3. [3.1] PS4. List all natural numbers that are factors of 12. [P.1] PS5. List all integers that are factors of 27. [P.1] PS6. Given P(x) = 4x3 - 3x 2 - 2x + 5, find P( -x). [2.5]

Multiple Zeros of a Polynomial Function Recall that if P is a polynomial function then the values of x for which P(x) is equal to 0 are called the zeros of P or the roots of the equation P(x) = 0. A zero of a polynomial function may be a multiple zero. For example, P(x) = x2 + 6x + 9 can be expressed in factored form as (x + 3)(x + 3). Setting each factor equal to zero yields x = - 3 in both cases. Thus P(x) = x2 + 6x + 9 has a zero of - 3 that occurs twice. The following definition will be most useful when we are discussing multiple zeros. y

Definition of Multiple Zeros of a Polynomial Function 4000

If a polynomial function P has (x - r) as a factor exactly k times, then r is a zero of multiplicity k of the polynomial function P.

2000

EXAMPLE −4

−2

2

4

P(x) = (x − 5)2(x + 2)3(x + 4) Figure 3.18

x

The graph of the polynomial function P(x) = (x - 5)2(x + 2)3(x + 4) is shown in Figure 3.18. This polynomial function has 5 as a zero of multiplicity 2. -2 as a zero of multiplicity 3. -4 as a zero of multiplicity 1.

A zero of multiplicity 1 is generally referred to as a simple zero. When searching for the zeros of a polynomial function, it is important that we know how many zeros to expect. This question is answered completely in Section 3.4. For the work in this section, the following result is valuable.

288

CHAPTER 3


Number of Zeros of a Polynomial Function A polynomial function P of degree n has at most n zeros, where each zero of multiplicity k is counted k times.

Rational Zero Theorem The rational zeros of polynomial functions with integer coefficients can be found with the aid of the following theorem.

Rational Zero Theorem Study tip The Rational Zero Theorem is one of the most important theorems of this chapter. It enables us to narrow the search for rational zeros to a finite list.

If P(x) = an x n + an - 1x n - 1 + Á + a1x + a0 has integer coefficients (an Z 0) p and is a rational zero (in simplest form) of P, then q p is a factor of the constant term a0. q is a factor of the leading coefficient an.

The Rational Zero Theorem often is used to make a list of all possible rational zeros p of a polynomial function. The list consists of all rational numbers of the form , where p q is an integer factor of the constant term a0 and q is an integer factor of the leading coefficient an.

EXAMPLE 1

Apply the Rational Zero Theorem

Use the Rational Zero Theorem to list all possible rational zeros of P(x) = 4x4 + x 3 - 40x 2 + 38x + 12 Solution List all integers p that are factors of 12 and all integers q that are factors of 4. Caution The Rational Zero Theorem gives the possible rational zeros of a polynomial function. That is, if P has a rational zero, then it must be one indicated by the theorem. However, P may not have any rational zeros. In the case of the polynomial function in Example 1, 1 the only rational zeros are - and 4 2. The remaining rational numbers in the list are not zeros of P.

p: q:

1, 2, 3, 4, 6, 12 1, 2, 4

Form all possible rational numbers using 1, 2, 3, 4, 6, and 12 for the numerator and 1, 2, and 4 for the denominator. By the Rational Zero Theorem, the possible rational zeros are 1 3 3 1 1, , , 2, 3, , , 4, 6, 12 2 4 2 4 It is not necessary to list a factor that is already listed in reduced form. For 3 6 example, is not listed because it is equal to . 4 2 Try Exercise 12, page 296

3.3


289

Question • If P(x) = an x n + an - 1x n - 1 + Á + a1x + a0 has integer coefficients and a leading

coefficient of an = 1, must all the rational zeros of P be integers?

Upper and Lower Bounds for Real Zeros A real number b is called an upper bound of the zeros of the polynomial function P if no zero is greater than b. A real number b is called a lower bound of the zeros of P if no zero is less than b. The following theorem is often used to find positive upper bounds and negative lower bounds for the real zeros of a polynomial function.

Upper- and Lower-Bound Theorem Let P be a polynomial function with real coefficients. Use synthetic division to divide P by x - b, where b is a nonzero real number. Upper bound

a. If b 7 0 and the leading coefficient of P is positive, then b is an upper bound for the real zeros of P provided none of the numbers in the bottom row of the synthetic division are negative. b. If b 7 0 and the leading coefficient of P is negative, then b is an upper bound for the real zeros of P provided none of the numbers in the bottom row of the synthetic division are positive.

Lower bound If b 6 0 and the numbers in the bottom row of the synthetic division alternate in sign (the number zero can be considered positive or negative as needed to produce an alternating sign pattern), then b is a lower bound for the real zeros of P.

Upper and lower bounds are not unique. For example, if b is an upper bound for the real zeros of P, then any number greater than b is also an upper bound. Likewise, if a is a lower bound for the real zeros of P, then any number less than a is also a lower bound.

EXAMPLE 2

Find Upper and Lower Bounds

According to the Upper- and Lower-Bound Theorem, what is the smallest positive integer that is an upper bound and the largest negative integer that is a lower bound of the real zeros of P(x) = 2x 3 + 7x 2 - 4x - 14? (continued)

p , where p is q p p = p. an integer factor of a0 and q is an integer factor of an. Thus q = 1 and = q 1

Answer • Yes. By the Rational Zero Theorem, the rational zeros of P are of the form

290

CHAPTER 3


Study tip When you check for bounds, you do not need to limit your choices to the possible zeros given by the Rational Zero Theorem. For instance, in Example 2 the integer 4 is a lower bound; however, 4 is not one of the possible zeros of P as given by the Rational Zero Theorem.

Integrating Technology Note in Figure 3.19 that the zeros of P(x) = 2x 3 + 7x 2 - 4x - 14 are between -4 (a lower bound) and 2 (an upper bound). You can use the Upper- and Lower-Bound Theorem to help determine Xmin and Xmax for the viewing window of a graphing utility. This will ensure that all the x-intercepts of the polynomial function will be shown. 20

3

−5

−20

Solution To find the smallest positive-integer upper bound, use synthetic division with 1, 2, . . . , as test values. 1 2 7

-4

- 14

2

9

5

2 9

5

-9

2

2

2

7

-4

-14

4

22

36

11

18

22

• No negative numbers

According to the Upper- and Lower-Bound Theorem, 2 is the smallest positive-integer upper bound. Now find the largest negative-integer lower bound. -1 2

7

-4

- 14

-2

-5

9

2 -3 2

5 7 -6

-9 -4 -3

-5 - 14 21

2

1

-7

7

-2

-4

7

-4

- 14

-4

-6

20

2 2

3 7 -8

-10 -4 4

6 - 14 0

2

-1

0

- 14

2

• Alternating signs

According to the Upper- and Lower-Bound Theorem, - 4 is the largest negative-integer lower bound. Try Exercise 20, page 296

Descartes’ Rule of Signs Descartes’ Rule of Signs is another theorem that is often used to obtain information about the zeros of a polynomial function. In Descartes’ Rule of Signs, the number of variations in sign of the coefficients of P(x) or P( -x) refers to sign changes of the coefficients from positive to negative or from negative to positive that we find when we examine successive terms of the function. The terms are assumed to appear in order of descending powers of x. For example, the polynomial function

P(x) = 2x3 + 7x2 − 4x − 14

P(x) = + 3x4 - 5x3 - 7x2 + x - 7

Figure 3.19

1

2

3

has three variations in sign. The polynomial function P( -x) = + 3( -x)4 - 5( - x)3 - 7( -x)2 + ( -x) - 7 = + 3x4 + 5x3 - 7x2 - x - 7 1

has one variation in sign. Terms that have a coefficient of 0 are not counted as variations in sign and may be ignored. For example, P(x) = - x5 + 4x2 + 1 1

has one variation in sign.

3.3

Math Matters


291

Descartes’ Rule of Signs Let P be a polynomial function with real coefficients and with the terms arranged in order of decreasing powers of x.

The Granger Collection

The number of positive real zeros of P is equal to the number of variations in sign of P(x) or to that number decreased by an even integer.

Descartes’ Rule of Signs first appeared in his La Géométrie (1673). Although a proof of Descartes’ Rule of Signs is beyond the scope of this course, we can see that a polynomial function with no variations in sign cannot have a positive zero. For instance, consider P(x) = x 3 + x 2 + x + 1. Each term of P is positive for any positive value of x. Thus P is never zero for x 7 0.

The number of negative real zeros of P is equal to the number of variations in sign of P(- x) or to that number decreased by an even integer.

EXAMPLE 3

Apply Descartes’ Rule of Signs

Use Descartes’ Rule of Signs to determine both the number of possible positive and the number of possible negative real zeros of each polynomial function. a.

P(x) = x 4 - 5x 3 + 5x 2 + 5x - 6

Solution a.

b. P(x) = 2x 5 + 3x 3 + 5x 2 + 8x + 7

P(x) = + x 4 - 5x 3 + 5x 2 + 5x - 6 1

2

3

There are three variations in sign. By Descartes’ Rule of Signs, there are either three or one positive real zeros. Now examine the variations in sign of P( -x). P( -x) = x 4 + 5x 3 + 5x 2 - 5x - 6 1

There is one variation in sign of P( - x). By Descartes’ Rule of Signs, there is one negative real zero. b.

P(x) = 2x5 + 3x3 + 5x2 + 8x + 7 has no variations in sign, so there are no positive real zeros. P( -x) = - 2x5 - 3x3 + 5x2 - 8x + 7 1

2

3

P(-x) has three variations in sign, so there are either three or one negative real zeros. Try Exercise 30, page 296 Question • If P(x) = ax2 + bx + c has two variations in sign, must P have two positive real zeros?

In applying Descartes’ Rule of Signs, we count each zero of multiplicity k as k zeros. For instance, P(x) = x2 - 10x + 25 has two variations in sign. Thus, by Descartes’ Rule of Signs, P must have either two or no positive real zeros. Factoring P produces (x - 5)2, from which we can observe that 5 is a positive zero of multiplicity 2.

Answer • No. According to Descartes’ Rule of Signs, P will have either two positive real zeros or

no positive real zeros.

292

CHAPTER 3


Zeros of a Polynomial Function Guidelines for Finding the Zeros of a Polynomial Function with Integer Coefficients 1. Gather general information Determine the degree n of the polynomial function. The number of distinct zeros of the polynomial function is at most n. Apply Descartes’ Rule of Signs to find the possible number of positive zeros and possible number of negative zeros. 2. Check suspects Apply the Rational Zero Theorem to list rational numbers that are possible zeros. Use synthetic division to test numbers in your list. If you find an upper or a lower bound, then eliminate from your list any number that is greater than the upper bound or less than the lower bound. Each time a zero is found, you obtain a

3. Work with the reduced polynomials reduced polynomial.

If a reduced polynomial is of degree 2, find its zeros either by factoring or by applying the quadratic formula. If the degree of a reduced polynomial is 3 or greater, repeat the preceding steps for this polynomial.

Example 4 illustrates the procedure discussed in the preceding guidelines.

EXAMPLE 4 Integrating Technology If you have a graphing utility, you can produce a graph similar to the one below. By looking at the x-intercepts of the graph, you can reject as possible zeros some of the values suggested by the Rational Zero Theorem. This will reduce the amount of work that is necessary to find the zeros of the polynomial function.

Find the Zeros of a Polynomial Function

Find the zeros of P(x) = 3x4 + 23x3 + 56x2 + 52x + 16. Solution 1. Gather general information The degree of P is 4. Thus the number of zeros of P is at most 4. By Descartes’ Rule of Signs, there are no positive zeros and there are four, two, or no negative zeros. 2. Check suspects zeros of P are p : q

By the Rational Zero Theorem, the possible negative rational 1 2 4 8 16 -1, - 2, -4, - 8, -16, - , - , - , - , 3 3 3 3 3

Use synthetic division to test the possible rational zeros. The following work shows that -4 is a zero of P. -4 3

−4.5

0.5

−20

P(x) = 3x4 + 23x3 + 56x2 + 52x + 16

3

23

56

52

16

-12

- 44

- 48

- 16

11

12

4

0

m

30

Coefficients of the first reduced polynomial

3.3


293

3. Work with the reduced polynomials Because - 4 is a zero, (x + 4) and the first reduced polynomial (3x3 + 11x2 + 12x + 4) are both factors of P. Thus P(x) = (x + 4)(3x3 + 11x2 + 12x + 4) All remaining zeros of P must be zeros of 3x3 + 11x2 + 12x + 4. The Rational Zero Theorem indicates that the only possible negative rational zeros of 3x3 + 11x2 + 12x + 4 are p : q

1 2 4 -1, - 2, - 4, - , - , 3 3 3

Synthetic division is again used to test possible zeros. 11

12

4

-6

- 10

-4

5

2

0

3

m

-2 3

Coefficients of the second reduced polynomial

Because - 2 is a zero, (x + 2) is also a factor of P. Thus P(x) = (x + 4)(x + 2)(3x 2 + 5x + 2) The remaining zeros of P must be zeros of 3x 2 + 5x + 2. 3x 2 + 5x + 2 = 0 (3x + 2)(x + 1) = 0 x = -

2 3

and

x = -1

2 The zeros of P(x) = 3x4 + 23x 3 + 56x 2 + 52x + 16 are -4, -2, - , and -1. 3 Try Exercise 44, page 296

Applications of Polynomial Functions In the following example we use an upper bound to eliminate several of the possible zeros that are given by the Rational Zero Theorem.

EXAMPLE 5


Glasses can be stacked to form a triangular pyramid. The total number of glasses in one of these pyramids is given by

Level 1 Level 2

T =

Level 3 Level 4 Level 5 Level 6

1 3 (k + 3k 2 + 2k) 6

where k is the number of levels in the pyramid. If 220 glasses are used to form a triangular pyramid, how many levels are in the pyramid? Solution 1 3 (k + 3k 2 + 2k) for k. Multiplying each side of the 6 equation by 6 produces 1320 = k 3 + 3k 2 + 2k, which can be written as We need to solve 220 =

(continued)

294

CHAPTER 3


k 3 + 3k 2 + 2k - 1320 = 0. The number 1320 has many natural number divisors, but we can eliminate many of these by showing that 12 is an upper bound. 12

2

- 1320

12

180

2184

1

15

182

864

m

3

1

No number in the bottom row is negative. Thus 12 is an upper bound.

The only natural number divisors of 1320 that are less than 12 are 1, 2, 3, 4, 5, 6, 8, 10, and 11. The following synthetic division shows that 10 is a zero of k 3 + 3k 2 + 2k - 1320.

Note

10

The reduced polynomial k 2 + 13k + 132 has zeros of - 13 i 2359 . These zeros 2 are not solutions of this application because the number of levels must be a natural number. k =

1

3 10

2 130

-1320 1320

1

13

132

0

The pyramid has 10 levels. There is no need to seek additional solutions, because the number of levels is uniquely determined by the number of glasses. Try Exercise 76, page 298

The procedures developed in this section will not find all solutions of every polynomial equation. However, a graphing utility can be used to estimate the real solutions of any polynomial equation. In Example 6 we utilize a graphing utility to solve an application.

EXAMPLE 6 4 in.

x x

Use a Graphing Utility to Solve an Application

A carbon dioxide cartridge for a paintball rifle has the shape of a right circular cylinder with a hemisphere at each end. The cylinder is 4 inches long, and the volume of the cartridge is 2p cubic inches (approximately 6.3 cubic inches). In the figure at the left, the common interior radius of the cylinder and the hemispheres is denoted by x. Use a graphing utility to estimate, to the nearest hundredth of an inch, the length of the radius x. Solution The volume of the cartridge is equal to the volume of the two hemispheres plus the volume of the cylinder. Recall that the volume of a sphere of radius x is given by 4 3 1 4 px . Therefore, the volume of a hemisphere is a px 3b . The volume of a right 3 2 3 circular cylinder is px 2h, where x is the radius of the base and h is the height of the cylinder. Thus the volume V of the cartridge is given by V = =

1 4 1 4 3 a px b + a px 3b + px 2h 2 3 2 3 4 3 px + px 2h 3

Replacing V with 2p and h with 4 yields 2p = 2 =

4 3 px + 4px 2 3 4 3 x + 4x 2 3

3 = 2x 3 + 6x 2

• Divide each side by p. 3 • Multiply each side by . 2

3.3


295

Here are two methods that can be used to solve 3 = 2x 3 + 6x 2

(1)

for x with the aid of a graphing utility. 1. Intersection method Use a graphing utility to graph y = 2x3 + 6x2 and y = 3 on the same screen, with x 7 0. The x-coordinate of the point of intersection of the two graphs is the desired solution. The graphs intersect at x L 0.64 inch. See the following figures. 4

4

y=3 y = 2x3 + 6x2 0

1.5


y=3 y = 2x3 + 6x2 0

−1

Intersection X=.64178353 Y=3

1.5

−1

The length of the radius is approximately 0.64 inch. 2. Intercept method Rewrite Equation (1) as 2x 3 + 6x 2 - 3 = 0. Graph y = 2x 3 + 6x 2 - 3 with x 7 0. Use a graphing utility to find the x-intercept of the graph. This method also shows that x L 0.64 inch. 2

2

0

1.5 y=

2x3 +

6x2 −

3


0

−2

1.5

Zero X=.64178353 Y=0 −2

The length of the radius is approximately 0.64 inch. Try Exercise 80, page 298

EXERCISE SET 3.3 In Exercises 1 to 8, find the zeros of the polynomial function and state the multiplicity of each zero. 1. P(x) = (x - 3)2(x + 5)

7. P(x) = (3x - 5)2(2x - 7) 8. P(x) = (5x - 2)(x + 3)4

2. P(x) = (x + 4)3(x - 1)2 3. P(x) = x 2(3x + 5)2 4. P(x) = x 3(2x + 1)(3x - 12)2

In Exercises 9 to 18, use the Rational Zero Theorem to list possible rational zeros for each polynomial function. 9. P(x) = x 3 + 3x 2 - 6x - 8

5. P(x) = (x 2 - 4)(x + 3)2

10. P(x) = x 3 - 19x - 30

6. P(x) = (x + 4)3(x 2 - 9)2

11. P(x) = 2x 3 + x 2 - 25x + 12

296

CHAPTER 3


12. P(x) = 3x 3 + 11x 2 - 6x - 8

38. P(x) = x4 - 1

13. P(x) = 6x4 + 23x 3 + 19x 2 - 8x - 4

39. P(x) = 10x6 - 9x5 - 14x4 - 8x3 - 18x2 + x + 6

14. P(x) = 2x 3 + 9x 2 - 2x - 9

40. P(x) = 2x6 - 5x5 - 26x4 + 76x3 - 60x2 - 255x + 700

15. P(x) = 4x4 - 12x 3 - 3x 2 + 12x - 7

41. P(x) = 12x7 - 112x6 + 421x5 - 840x4 + 1038x3

16. P(x) = x 5 - x4 - 7x 3 + 7x 2 - 12x - 12 17. P(x) = x - 32 5

42. P(x) = x7 + 2x5 + 3x3 + x

In Exercises 43 to 66, find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity.

18. P(x) = x4 - 1

In Exercises 19 to 28, find the smallest positive integer and the largest negative integer that, by the Upper- and Lower-Bound Theorem, are upper and lower bounds for the real zeros of each polynomial function. 19. P(x) = x 3 + 3x 2 - 6x - 6 20. P(x) = x 3 - 19x - 28 21. P(x) = 2x + x - 25x + 10 3

- 938x2 + 629x - 210

43. P(x) = x 3 + 3x 2 - 6x - 8 44. P(x) = x 3 - 19x - 30 45. P(x) = 2x 3 + x 2 - 25x + 12 46. P(x) = 3x 3 + 11x 2 - 6x - 8 47. P(x) = 6x4 + 23x 3 + 19x 2 - 8x - 4

2

22. P(x) = 3x + 11x - 6x - 9 3

48. P(x) = 2x 3 + 9x 2 - 2x - 9

2

23. P(x) = 6x + 23x + 19x - 8x - 4 4

3

24. P(x) = - 2x - 9x + 2x + 9 3

50. P(x) = 3x 3 - x 2 - 6x + 2

2

25. P(x) = - 4x4 + 12x 3 + 3x 2 - 12x + 7 26. P(x) = x5 - x4 - 7x 3 + 7x 2 - 12x - 12 27. P(x) = x 5 - 32

51. P(x) = x 3 - 8x 2 + 8x + 24 52. P(x) = x 3 - 7x 2 - 7x + 69 53. P(x) = 2x4 - 19x 3 + 51x 2 - 31x + 5 54. P(x) = 4x4 - 35x 3 + 71x 2 - 4x - 6

28. P(x) = x4 - 1

55. P(x) = 3x6 - 10x 5 - 29x4 + 34x 3 + 50x 2 - 24x - 24

In Exercises 29 to 42, use Descartes’ Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. 29. P(x) = x 3 + 3x 2 - 6x - 8 30. P(x) = x 3 - 19x - 30 31. P(x) = 2x + x - 25x + 12 3

49. P(x) = 2x4 - 9x 3 - 2x 2 + 27x - 12

2

2

56. P(x) = 2x4 + 3x 3 - 4x 2 - 3x + 2 57. P(x) = x 3 - 3x - 2 58. P(x) = 3x4 - 4x 3 - 11x 2 + 16x - 4 59. P(x) = x4 - 5x 2 - 2x 60. P(x) = x 3 - 2x + 1

32. P(x) = 3x 3 + 11x 2 - 6x - 8

61. P(x) = x4 + x 3 - 3x 2 - 5x - 2

33. P(x) = 6x4 + 23x 3 + 19x 2 - 8x - 4

62. P(x) = 6x4 - 17x 3 - 11x 2 + 42x

34. P(x) = 2x 3 + 9x 2 - 2x - 9

63. P(x) = 2x4 - 17x 3 + 4x 2 + 35x - 24

35. P(x) = 4x4 - 12x 3 - 3x 2 + 12x - 7

64. P(x) = x 5 + 5x4 + 10x 3 + 10x 2 + 5x + 1

36. P(x) = x5 - x4 - 7x 3 + 7x 2 - 12x - 12

65. P(x) = x 3 - 16x

37. P(x) = x5 - 32

66. P(x) = x 3 - 4x2 - 3x

3.3


297

67. Find the Dimensions A cube measures n inches on each

a. Use the function in the preceding column to determine the

edge. If a slice 2 inches thick is cut from one face of the cube, the resulting solid has a volume of 567 cubic inches. Find n.

maximum number of pieces that can be produced by five straight cuts. b.

68. Find the Dimensions A cube

measures n inches on each edge. If a slice 1 inch thick is cut from Second cut one face of the cube and then a slice 3 inches thick is cut from another face of the cube as shown, the resulting solid has a volume of 1560 cubic inches. Find the dimensions of the original cube.

n

n

72.

n First cut

What is the fewest number of straight cuts that are needed to produce 64 pieces?

Inscribed Quadrilateral Isaac Newton discovered that if a quadrilateral with sides of lengths a, b, c, and x is inscribed in a semicircle with diameter x, then the lengths of the sides are related by the following equation.

x3 - (a 2 + b2 + c2)x - 2abc = 0 b

69. Dimensions of a Solid For what value of x will the volume

of the following solid be 112 cubic inches? a

c x

2 x+2

Given a = 6, b = 5, and c = 4, find x. Round to the nearest hundredth.

1 x

73. Cannonball Stacks Cannonballs can be stacked to form a

x+1

pyramid with a square base. The total number of cannonballs T in one of these square pyramids is

70. Dimensions of a Box The length of a rectangular box is 1 inch

more than twice the height of the box, and the width is 3 inches more than the height. If the volume of the box is 126 cubic inches, find the dimensions of the box.

T =

1 (2n3 + 3n2 + n) 6

where n is the number of rows (levels). If 140 cannonballs are used to form a square pyramid, how many rows are in the pyramid? x+3

x

74.

2x + 1

71. Pieces and Cuts One straight cut through a thick piece of

cheese produces two pieces. Two straight cuts can produce a maximum of four pieces. Three straight cuts can produce a maximum of eight pieces.

P(x) = - 0.02x 3 + 0.01x 2 + 1.2x - 1.1 where P is the profit in millions of dollars and x is the amount, in hundred-thousands of dollars, spent on advertising. Determine the minimum amount, rounded to the nearest thousand dollars, the company needs to spend on advertising if it is to earn a profit of $2 million.

Cut 2 Cut 1

Profit (in millions of dollars)

Cut 3

You might be inclined to think that every additional cut doubles the previous number of pieces. However, for four straight cuts, you get a maximum of 15 pieces. The maximum number of pieces P that can be produced by n straight cuts is given by P(n) =

n3 + 5n + 6 6

Advertising Expenses A company manufactures digital cameras. The company estimates that the profit from camera sales is

P 3 2 1 −1

1

2

3

4

5

6

7

8

x

Advertising expenses (in hundred-thousands of dollars)

75.

Cost Cutting A nutrition bar in the shape of a rectangular solid measures 0.75 inch by 1 inch by 5 inches.

298

CHAPTER 3


To reduce costs, the manufacturer has decided to decrease each of the dimensions of the nutrition bar by x inches. What value of x, rounded to the nearest thousandth of an inch, will produce a new nutrition bar with a volume that is 0.75 cubic inch less than the present bar’s volume?

80.

Propane Tank Dimensions A propane tank has the shape of a circular cylinder with a hemisphere at each end. The cylinder is 6 feet long and the volume of the tank is 9p cubic feet. Find, to the nearest thousandth of a foot, the length of the radius x. 6 ft

x x 5

5−x 0.75

0.75 − x

1

1−x

Original

New

76. Selection of Cards The number of ways one can select three

cards from a group of n cards (the order of the selection matters), where n Ú 3, is given by P(n) = n3 - 3n2 + 2n. For a certain card trick, a magician has determined that there are exactly 504 ways to choose three cards from a given group. How many cards are in the group? Dimensions of a Box

A rectangular box is square on two ends and has length plus girth of 81 inches. (The girth of a box is the w shortest distance “around” l the box.) Determine the posw sible lengths l of the box if its volume is 4900 cubic inches. Round approximate values to the nearest hundredth of an inch. In the above figure, assume that w 6 l. 78.

Medication Level Pseudoephedrine hydrochloride is an

allergy medication. The polynomial function L(t) = 0.03t + 0.4t - 7.3t + 23.1t 4

3

2

where 0 … t … 5, models the level of pseudoephedrine hydrochloride, in milligrams, in the bloodstream of a patient t hours after 30 milligrams of the medication have been taken. At what times, to the nearest minute, does the level of pseudoephedrine hydrochloride in the bloodstream reach 12 milligrams? 79.

Cauchy’s Bound Theorem Let P(x) = anxn + an-1x n-1 + Á + a1 + a0 be a polynomial function with complex coefficients. The absolute value of each zero of P is less than maximum of ( ƒ an-1 ƒ , ƒ an-2 ƒ , Á , ƒ a1 ƒ , ƒ a0 ƒ ) + 1b B = a ƒ an ƒ 81. P(x) = 2x3 - 5x2 - 28x + 15, zeros: 3,

83. P(x) = x4 - 2x3 + 9x 2 + 2x - 10, zeros: 1 + 3i, 1 - 3i, 1, 1 84. P(x) = x4 - 4x3 + 14x2 - 4x + 13,

zeros: 2 + 3i, 2 - 3i, i, i

Weight and Height of Giraffes A veterinarian at a wild animal park has determined that the average weight w, in pounds, of an adult male giraffe is closely approximated by the function

w = 8.3h3 - 307.5h2 + 3914h - 15,230 where h is the giraffe’s height in feet, and 15 … h … 18. Use the above function to estimate the height of a giraffe that weighs 3150 pounds. Round to the nearest tenth of a foot.

1 ,5 2

82. P(x) = x3 - 5x2 + 2x + 8, zeros: 1, 2, 4

Bettmann/CORBIS

77.

The mathematician Augustin Louis Cauchy proved a theorem that can be used to quickly establish a bound B for all the absolute values of the zeros (real and complex) of a given polynomial function. In Exercises 81 to 84, a polynomial function and its zeros are given. For each polynomial, apply Cauchy’s Bound Theorem (shown below) to determine the bound B for the polynomial and verify that the absolute value of each of the given zeros is less than B. (Hint: ƒ a bi ƒ 2a 2 b2)

Augustin Louis Cauchy (1789–1857)

3.4

FUNDAMENTAL THEOREM OF ALGEBRA

299

MID-CHAPTER 3 QUIZ 1. Use the Remainder Theorem to find P(5) for the function

6. Use the Rational Zero Theorem to list all possible rational

P(x) = 3x 3 + 7x 2 - 2x - 5.

zeros of P(x) = 3x3 + 7x2 - 18x + 8.

2. Use the Factor Theorem to determine whether (x 5) is a factor

of P(x) = x 3 - 6x 2 + 3x + 10.

Use a graphing utility to estimate, to the nearest hundredth, the coordinates of the point where the function P(x) = - x 3 + 10x 2 - 27x + 25 has a relative minimum.

7.

3. Determine the far-left and the far-right behavior of the graph of

P(x) = 4x 3 - 3x 2 - 6x - 1.

8. Find the zeros of P(x) = 2x4 - 19x 3 + 57x 2 - 64x + 20. If

4. Find the zeros of P(x) = 3x - 2x - 27x + 18 3

2

a zero is a multiple zero, state its multiplicity.

5. Use the Intermediate Value Theorem to verify that

P(x) = 2x4 - 5x 3 + x 2 - 20x - 28 has a zero between 3 and 4.

SECTION 3.4 Fundamental Theorem of Algebra Number of Zeros of a Polynomial Function Conjugate Pair Theorem Finding a Polynomial Function with Given Zeros

Fundamental Theorem of Algebra PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A21.

PS1. What is the conjugate of 3 - 2i? [P.6] PS2. What is the conjugate of 2 + i15? [P.6] PS3. Multiply: (x - 1)(x - 3)(x - 4) [P.3] PS4. Multiply: 3x - (2 + i)43x - (2 - i)4 [P.3/P.6] PS5. Solve: x 2 + 9 = 0 [1.3] PS6. Solve: x 2 - x + 5 = 0 [1.3]

Fundamental Theorem of Algebra The German mathematician Carl Friedrich Gauss (1777–1855) was the first to prove that every polynomial function has at least one complex zero. This concept is so basic to the study of algebra that it is called the Fundamental Theorem of Algebra. The proof of the Fundamental Theorem is beyond the scope of this text; however, it is important to understand the theorem and its consequences. As you consider each of the following theorems, keep in mind that the terms complex coefficients and complex zeros include real coefficients and real zeros because the set of real numbers is a subset of the set of complex numbers.

Fundamental Theorem of Algebra If P is a polynomial function of degree n Ú 1 with complex coefficients, then P has at least one complex zero.

300

CHAPTER 3


Number of Zeros of a Polynomial Function

Math Matters

Let P be a polynomial function of degree n Ú 1 with complex coefficients. The Fundamental Theorem implies that P has a complex zero—say, c1. The Factor Theorem implies that

Bettmann/CORBIS

P(x) = (x - c1)Q(x)

Carl Friedrich Gauss (1777–1855) has often been referred to as the Prince of Mathematics. His work covered topics in algebra, calculus, analysis, probability, number theory, non-Euclidean geometry, astronomy, and physics, to name but a few. The following quote by Eric Temple Bell gives credence to the fact that Gauss was one of the greatest mathematicians of all time. “Archimedes, Newton, and Gauss, these three, are in a class by themselves among the great mathematicians, and it is not for ordinary mortals to attempt to range them in order of merit.”*

where Q(x) is a polynomial of degree one less than the degree of P. Recall that the polynomial Q(x) is called a reduced polynomial. Assuming that the degree of Q(x) is 1 or more, the Fundamental Theorem implies that it also must have a zero. A continuation of this reasoning process leads to the following theorem.

Linear Factor Theorem If P is a polynomial function of degree n Ú 1 with leading coefficient an Z 0, P(x) = a n x n + a n - 1x n - 1 + Á + a1x1 + a0 then P has exactly n linear factors P(x) = an(x - c1)(x - c2) Á (x - cn) where c1, c2, . . . , cn are complex numbers.

The following theorem follows directly from the Linear Factor Theorem.

Number of Zeros of a Polynomial Function Theorem If P is a polynomial function of degree n Ú 1, then P has exactly n complex zeros, provided each zero is counted according to its multiplicity.

*Men of Mathematics, by E. T. Bell, New York, Simon and Schuster, 1937.

The Linear Factor Theorem and the Number of Zeros of a Polynomial Function Theorem are referred to as existence theorems. They state that an nth-degree polynomial will have n linear factors and n complex zeros, but they do not provide any information on how to determine the linear factors or the zeros. In Example 1, we use previously developed methods to actually find the linear factors and zeros of some polynomial functions.

EXAMPLE 1

Find the Zeros and Linear Factors of a Polynomial Function

Find all the zeros of each of the following polynomial functions, and write each function as a product of its leading coefficient and its linear factors. a.

P(x) = x4 - 4x3 + 8x2 - 16x + 16

b.

S(x) = 2x4 + x3 + 39x2 + 136x - 78

Solution a. By the Linear Factor Theorem, P will have four linear factors and thus four zeros. The possible rational zeros are 1, 2, 4, 8, and 16. Use synthetic division to show that 2 is a zero of multiplicity 2.

3.4

2

1

1


-4

8

-16

16

2

-4

8

-16

-2

4

-8

0

2

1

-2

4

-8

2

0

8

0

4

0

1

301

The final reduced polynomial is x + 4. Solve x + 4 = 0 to find the remaining zeros. 2

2

x2 + 4 x2 x x

Concepts Involving Complex Numbers See Section P.6.

= = = =

0 -4 1-4 2i

The four zeros of P are 2, 2, - 2i, and 2i. The leading coefficient of P is 1. Thus the linear factored form of P is P(x) = 1(x - 2)(x - 2)(x - ( -2i))(x - 2i) or, after simplifying, P(x) = (x - 2)2(x + 2i)(x - 2i) b.

By the Linear Factor Theorem, S will have four linear factors and thus four zeros. The possible rational zeros are 3 13 39 1 , 1, , 2, 3, 6, , 13, 26, , 39, and 78 2 2 2 2 1 are zeros of S. 2 1 - 78 2 -5

Use synthetic division to show that - 3 and -3

2

2

1

39

136

-6

15

-162

78

-5

54

- 26

0

2

2

54

-26

1 -2

26

-4

52

0

The final reduced polynomial is 2x 2 - 4x + 52. The remaining zeros can be found by using the quadratic formula to solve 2x 2 - 4x + 52 = 0 x 2 - 2x + 26 = 0 - ( -2) 2( - 2)2 - 4(1)(26) x = 2 2 1 - 100 = 2 2 10i = 1 5i = 2


1 The four zeros are - 3, , 1 + 5i, and 1 - 5i. The leading coefficient of S is 2. 2 Thus the linear factored form of S is S(x) = 23x - ( -3)4ax -

1 b 3x - (1 + 5i)43x - (1 - 5i)4 2

or, after simplifying, S(x) = 2(x + 3) ax Try Exercise 2, page 305

1 b (x - 1 - 5i)(x - 1 + 5i) 2

302

CHAPTER 3


Conjugate Pair Theorem You may have noticed that the complex zeros of the polynomial function in Example 1 were complex conjugates. The following theorem shows that this is not a coincidence.

Conjugate Pair Theorem Complex Conjugates See page 63.

If a + bi (b Z 0) is a complex zero of a polynomial function with real coefficients, then the conjugate a - bi is also a complex zero of the polynomial function.

EXAMPLE 2

Use the Conjugate Pair Theorem to Find Zeros

Find all the zeros of P(x) = x 4 - 4x 3 + 14x 2 - 36x + 45, given that 2 + i is a zero. Solution Because the coefficients are real numbers and 2 + i is a zero, the Conjugate Pair Theorem implies that 2 - i also must be a zero. Using synthetic division with 2 + i and 2 - i, we have 2 + i

2 - i y

1

-4 2 + i

14 -5

-36 18 + 9i

45 - 45

1

-2 + i

9

- 18 + 9i

0

1

-2 + i 2 - i

9 0

-18 + 9i 18 - 9i

1 200

0

9

0

• The coefficients of the reduced polynomial

• The coefficients of the next reduced polynomial

The resulting reduced polynomial is x2 + 9, which has 3i and -3i as zeros. Therefore, the four zeros of x4 - 4x3 + 14x2 - 36x + 45 are 2 + i, 2 - i, 3i, and -3i. Try Exercise 18, page 305

−4

−2

2

4

x

P(x) = x4 − 4x3 + 14x2 − 36x + 45 Figure 3.20

A graph of P(x) = x4 - 4x 3 + 14x 2 - 36x + 45 is shown in Figure 3.20. Because the polynomial in Example 2 is a fourth-degree polynomial and because we have verified that P has four nonreal solutions, it comes as no surprise that the graph does not intersect the x-axis. When performing synthetic division with complex numbers, it is helpful to write the coefficients of the given polynomial as complex coefficients. For instance, - 10 can be written as -10 + 0i. This technique is illustrated in the next example.

EXAMPLE 3

Apply the Conjugate Pair Theorem

Find all the zeros of P(x) = x 5 - 10x 4 + 65x 3 - 184x 2 + 274x - 204, given that 3 - 5i is a zero.

3.4

Integrating Technology Many graphing calculators can be used to do computations with complex numbers. The following TI-83/TI-83 Plus/TI-84 Plus screen display shows that the product of 3 - 5i and -7 - 5i is -46 + 20i. The i symbol is located above the decimal point key.


303

Solution Because the coefficients are real numbers and 3 - 5i is a zero, 3 + 5i also must be a zero. Use synthetic division to produce 3 - 5i 1

3 + 5i 1

1

-10 + 0i

65 + 0i

- 184 + 0i

274 + 0i

- 204

3 - 5i

-46 + 20i

157 - 35i

-256 + 30i

204

- 7 - 5i

19 + 20i

-27 - 35i

18 + 30i

0

3 + 5i

- 12 - 20i

21 + 35i

- 18 - 30i

-4

-6

7

0

(3 – 5i) (-7 – 5i) -46 + 20i

Descartes’ Rule of Signs can be used to show that the reduced polynomial x3 - 4x2 + 7x - 6 has three or one positive zeros and no negative zeros. Using the Rational Zero Theorem, we have p = 1, 2, 3, 6 q Use synthetic division to determine that 2 is a zero. 2

1

1

-4

7

-6

2

-4

6

-2

3

0

Use the quadratic formula to solve x 2 - 2x + 3 = 0. x =

- ( -2) 2( - 2)2 - 4(1)(3) 2 1-8 2 2i 12 = = = 1 i 12 2(1) 2 2

The zeros of P(x) = x 5 - 10x 4 + 65x 3 - 184x 2 + 274x - 204 are 3 - 5i, 3 + 5i, 2, 1 + 12i, and 1 - 12 i. Try Exercise 22, page 305 Question • Is it possible for a third-degree polynomial function with real coefficients to have two

real zeros and one nonreal complex zero?

Recall that the real zeros of a polynomial function P are the x-coordinates of the xintercepts of the graph of P. This important connection between the real zeros of a polynomial function and the x-intercepts of the graph of the polynomial function is the basis for using a graphing utility to solve equations. Careful analysis of the graph of a polynomial function and your knowledge of the properties of polynomial functions can be used to solve many polynomial equations. Answer • No. Because the coefficients of the polynomial are real numbers, the nonreal complex

zeros of the polynomial function must occur as conjugate pairs.

304

CHAPTER 3


EXAMPLE 4

Solve a Polynomial Equation

Solve: x4 - 5x 3 + 4x 2 + 3x + 9 = 0 Solution Let P(x) = x 4 - 5x 3 + 4x 2 + 3x + 9. The real zeros of P are the real solutions of the equation. Use a graphing utility to graph P. See Figure 3.21. From the graph, it appears that (3, 0) is an x-intercept and the only x-intercept. Because the graph of P intersects but does not cross the x-axis at (3, 0), we know that 3 is a multiple zero of P with an even multiplicity.

100

5

−3

3 1

-5 3

4 -6

3 -6

9 -9

1

-2

-2

-3

0

−10

P(x) = x4 − 5x3 + 4x2 + 3x + 9 Figure 3.21

• Coefficients of P

• The remainder is 0. Thus 3 is a zero.

By the Number of Zeros Theorem, there are three more zeros of P. Use synthetic division to show that 3 is also a zero of the reduced polynomial x 3 - 2x 2 - 2x - 3. 3 1

-2 3

-2 3

-3 3

1

1

1

0

• Coefficients of reduced polynomial

• The remainder is 0. Thus 3 is a zero of multiplicity 2.

We now have 3 as a double root of the original equation, and from the last line of the preceding synthetic division, the remaining solutions must be solutions of x 2 + x + 1 = 0. Use the quadratic formula to solve this equation. x =

- 1 212 - 4(1)(1) -1 1 - 3 - 1 i13 = = 2(1) 2 2

1 13 The solutions of x4 - 5x 3 + 4x 2 + 3x + 9 = 0 are 3, 3, - + i, and 2 2 13 1 i. - 2 2 Try Exercise 30, page 306

Finding a Polynomial Function with Given Zeros Many of the problems in this section and in Section 3.3 dealt with the process of finding the zeros of a given polynomial function. Example 5 considers the reverse process: finding a polynomial function when the zeros are given.

EXAMPLE 5

Determine a Polynomial Function Given Its Zeros

Find each polynomial function. a.

A polynomial function of degree 3 that has 1, 2, and - 3 as zeros

b.

A polynomial function of degree 4 that has real coefficients and zeros 2i and 3 - 7i

3.4


305

Solution a. Because 1, 2, and -3 are zeros, (x - 1), (x - 2), and (x + 3) are factors. The product of these factors produces a polynomial function with the indicated zeros. P(x) = (x - 1)(x - 2)(x + 3) = (x2 - 3x + 2)(x + 3) = x3 - 7x + 6 By the Conjugate Pair Theorem, the polynomial function also must have - 2i and 3 + 7i as zeros. The product of the factors x - 2i, x - ( -2i), x - (3 - 7i), and x - (3 + 7i) produces the desired polynomial function.

b. y

P(x) = x 3 − 7x + 6

P(x) = (x - 2i)(x + 2i)3x - (3 - 7i)43x - (3 + 7i)4 = (x 2 + 4)(x 2 - 6x + 58) = x 4 - 6x 3 + 62x 2 - 24x + 232

10

Try Exercise 48, page 306 −3

1 −10

2

x

S(x) = 2x 3 − 14x + 12

Figure 3.22

A polynomial function that has a given set of zeros is not unique. For example, P(x) = x 3 - 7x + 6 has zeros 1, 2, and - 3, but so does any nonzero multiple of P, such as S(x) = 2x 3 - 14x + 12. This concept is illustrated in Figure 3.22. The graphs of the two polynomial functions are different, but they have the same zeros.

EXERCISE SET 3.4 In Exercises 1 to 16, find all the zeros of the polynomial function and write the polynomial as a product of its leading coefficient and its linear factors. (Hint: First determine the rational zeros.) 1. P(x) = x4 + x 3 - 2x 2 + 4x - 24 2. P(x) = x 3 - 3x 2 + 7x - 5

13. P(x) = 2x4 - x 3 - 2x 2 + 13x - 6 14. P(x) = 4x4 - 4x 3 + 13x 2 - 12x + 3 15. P(x) = 3x5 + 2x4 + 10x 3 + 6x 2 - 25x - 20 16. P(x) = 2x6 - 11x5 + 5x4 + 60x 3 - 62x 2 - 64x + 40

3. P(x) = 2x4 - x 3 - 4x 2 + 10x - 4 4. P(x) = x 3 - 13x 2 + 65x - 125 5. P(x) = x5 - 9x4 + 34x 3 - 58x 2 + 45x - 13 6. P(x) = x4 - 4x 3 + 53x 2 - 196x + 196 7. P(x) = 2x4 - x 3 - 15x 2 + 23x + 15 8. P(x) = 3x4 - 17x 3 - 39x 2 + 337x + 116 9. P(x) = 2x4 - 14x 3 + 33x 2 - 46x + 40 10. P(x) = 3x4 - 10x 3 + 15x 2 + 20x - 8 11. P(x) = 2x 3 - 9x 2 + 18x - 20 12. P(x) = 3x4 - 19x 3 + 59x 2 - 79x + 36

In Exercises 17 to 28, use the given zero to find the remaining zeros of each polynomial function. 17. P(x) = 2x 3 - 5x 2 + 6x - 2;

1 + i

18. P(x) = 3x 3 - 29x 2 + 92x + 34; 19. P(x) = x 3 + 3x 2 + x + 3;

5 + 3i

-i

20. P(x) = x4 - 6x 3 + 71x 2 - 146x + 530; 21. P(x) = x4 - 4x 3 + 14x 2 - 4x + 13;

2 + 7i

2 - 3i

22. P(x) = x 5 - 6x4 + 22x 3 - 64x 2 + 117x - 90; 23. P(x) = x4 - 4x 3 + 19x 2 - 30x + 50; 24. P(x) = x 5 - x4 - 4x 3 - 4x 2 - 5x - 3;

1 + 3i i

3i

306

CHAPTER 3


25. P(x) = x 5 - 3x4 + 7x 3 - 13x 2 + 12x - 4; 26. P(x) = x4 - 8x 3 + 18x 2 - 8x + 17;

In Exercises 47 to 52, find a polynomial function P, with real coefficients, that has the indicated zeros and satisfies the given conditions.

-2i

i

27. P(x) = x4 - 17x 3 + 112x 2 - 333x + 377;

5 + 2i

28. P(x) = 2x 5 - 8x4 + 61x 3 - 99x 2 + 12x + 182;

1 - 5i

In Exercises 29 to 36, use a graph and your knowledge of the zeros of polynomial functions to determine the exact values of all the solutions of each equation. 29. 2x 3 - x 2 + x - 6 = 0 30. 4x 3 + 3x 2 + 16x + 12 = 0 31. 24x 3 - 62x 2 - 7x + 30 = 0 32. 12x 3 - 52x 2 + 27x + 28 = 0

47. Zeros: 2 - 5i, - 4; degree 3 48. Zeros: 3 + 2i, 7; degree 3 49. Zeros: 4 + 3i, 5 - i; degree 4 50. Zeros: i, 3 - 5i; degree 4 51. Zeros: - 2, 1, 3, 1 + 4i, 1 - 4i; degree 5 52. Zeros: - 5, 3 (multiplicity 2), 2 + i, 2 - i; degree 5

In Exercises 53 to 56, find a polynomial function P with real coefficients that has the indicated zeros and satisfies the given conditions. 53. Zeros: -1, 2, 3; degree 3; P(1) = 12 [Hint: First find a third-

degree polynomial function T(x) with real coefficients that has - 1, 2, and 3 as zeros. Now evaluate T(1). If T(1) = P(1) = 12, then T(x) is the desired polynomial function. If T(1) Z 12, then 12 you need to multiply T(x) by to produce the polynomial T(1) function that has the given zeros and whose graph passes 12 through (1, 12). That is, P(x) = T(x).] T(1)

33. x - 4x + 5x - 4x + 4 = 0 4

3

2

34. x4 + 4x 3 + 8x 2 + 16x + 16 = 0 35. x4 + 4x 3 - 2x 2 - 12x + 9 = 0 36. x4 + 3x 3 - 6x 2 - 28x - 24 = 0

54. Zeros: 3i, 2; degree 3; P(3) = 27 (See the hint in Exercise 53.)

In Exercises 37 to 46, find a polynomial function of lowest degree with integer coefficients that has the given zeros.

55. Zeros: 3, -5, 2 + i; degree 4; P(1) = 48 (See the hint in

Exercise 53.)

37. 4, -3, 2 38. - 1, 1, - 5 39. 3, 2i, - 2i 40. 0, i, -i 41. 3 + i, 3 - i, 2 + 5i, 2 - 5i 42. 2 + 3i, 2 - 3i, - 5, 2 43. 6 + 5i, 6 - 5i, 2, 3, 5

1 44. , 4 - i, 4 + i 2

1 , 1 - i; degree 3; P(4) = 140 (See the hint in 2 Exercise 53.)

56. Zeros:

57.

Conjugate Pair Theorem Verify that the function P(x) = x 3 - x 2 - ix 2 - 9x + 9 + 9i has 1 + i as a zero and that its conjugate 1 - i is not a zero. Explain why this does not contradict the Conjugate Pair Theorem.

58. Polynomial Function Give an example of a polynomial func-

tion that has the given properties, or explain why no such polynomial function exists. a. A polynomial function of degree 3 that has one rational

zero and two irrational zeros b. A polynomial function of degree 4 that has four irrational

zeros 3 45. , 2 + 7i, 2 - 7i 4 46.

1 1 , - , i, -i 4 5

c. A polynomial function of degree 3, with real coefficients,

that has no real zeros d. A polynomial function of degree 4, with real coefficients,

that has no real zeros

3.5

SECTION 3.5

GRAPHS OF RATIONAL FUNCTIONS AND THEIR APPLICATIONS

307

Graphs of Rational Functions and Their Applications

Vertical and Horizontal Asymptotes Sign Property of Rational Functions General Graphing Procedure Slant Asymptotes Graphing Rational Functions That Have a Common Factor Applications of Rational Functions


PS1. Simplify:

x2 - 9 [P.5] x 2 - 2x - 15

PS2. Evaluate

x + 4 for x = - 1. [P.1] x - 2x - 5

PS3. Evaluate

2x 2 + 4x - 5 for x = - 3. [P.1] x + 6

2

PS4. For what values of x does the denominator of

x2 - x - 5 equal zero? [1.4] 2x 3 + x 2 - 15x

PS5. Determine the degree of the numerator and the degree of the denominator of

x 3 + 3x 2 - 5 . [P.3] x2 - 4

PS6. Write

R(x) x 3 + 2x 2 - x - 11 in Q(x) + 2 form. [3.1] 2 x - 2x x - 2x

Vertical and Horizontal Asymptotes If P(x) and Q(x) are polynomials, then the function F given by F(x) =

P(x) Q(x)

is called a rational function. The domain of F is the set of all real numbers except those for which Q(x) = 0. For example, let F(x) =

Setting the denominator equal to zero, we have

y

2x 3 + x 2 - 15x = 0 x(2x - 5)(x + 3) = 0

6

y=1

3

The denominator is 0 for x = 0, x = −6

−3

3 −3 −6

x2 - x - 5 2x 3 + x 2 - 15x

x=2

6

x

5 , and x = - 3. Thus the domain of F is the set of 2

5 , and -3. 2 x + 1 The graph of G(x) = is given in Figure 3.23. The graph shows that G has the x - 2 following properties: all real numbers except 0,

G(x) = x + 1 x−2

The graph has an x-intercept at ( -1, 0) and a y-intercept at a 0, -

Figure 3.23

The graph does not exist when x = 2.

1 b. 2

308

CHAPTER 3


Note the behavior of the graph as x takes on values that are close to 2 but less than 2. Mathematically, we say that “x approaches 2 from the left.” x G(x)

1.9

1.95

1.99

1.995

1.999

-29

-59

-299

-599

-2999

From this table and the graph, it appears that as x approaches 2 from the left, the functional values G(x) decrease without bound. In this case, we say that “G(x) approaches negative infinity.” Now observe the behavior of the graph as x takes on values that are close to 2 but greater than 2. Mathematically, we say that “x approaches 2 from the right.” x

2.1

2.05

2.01

2.005

2.001

G(x)

31

61

301

601

3001

From this table and the graph, it appears that as x approaches 2 from the right, the functional values G(x) increase without bound. In this case, we say that “G(x) approaches positive infinity.” Now consider the values of G(x) as x increases without bound. The following table gives values of G(x) for selected values of x. x G(x) y

y f

1000

5000

10,000

50,000

100,000

1.00301

1.00060

1.00030

1.00006

1.00003

As x increases without bound, the values of G(x) become closer to 1.

f

Now let the values of x decrease without bound. The following table gives the values of G(x) for selected values of x. x

a

a

x

x G(x)

a. f (x) → ∞ as x → a+

c. f (x) → −∞ as x → a+

a

d. f (x) → −∞ as x → a− Figure 3.24

-50,000

-100,000

0.997006

0.999400

0.999700

0.999940

0.999970

f(x) : q

f f

x

-10,000

When we are discussing functional values that increase or decrease without bound, it is convenient to use mathematical notation. The notation

y

a

-5000

As x decreases without bound, the values of G(x) become closer to 1.

b. f (x) → ∞ as x → a−

y

-1000

x

as

x : a+

means that the functional values f (x) increase without bound as x approaches a from the right. Recall that the symbol q does not represent a real number but is used merely to describe the concept of a variable taking on larger and larger values without bound. See Figure 3.24a. The notation f (x) : q

as

x : a-

means that the function values f (x) increase without bound as x approaches a from the left. See Figure 3.24b.

3.5


309

The notation f(x) : - q

as

x : a+

means that the functional values f(x) decrease without bound as x approaches a from the right. See Figure 3.24c. The notation f (x) : - q

as

x : a-

means that the functional values f (x) decrease without bound as x approaches a from the left. See Figure 3.24d. Each graph in Figure 3.24 approaches a vertical line through (a, 0) as x : a + or a - . The line is said to be a vertical asymptote of the graph.

Definition of a Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function F provided F(x) : q

F(x) : - q

or

as x approaches a from either the left or right.

In Figure 3.23 on page 307, the line x = 2 is a vertical asymptote of the graph of G. Note that the graph of G in Figure 3.23 also approaches the horizontal line y = 1 as x : q and as x : - q . The line y = 1 is a horizontal asymptote of the graph of G.

Definition of a Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function F provided F(x) : b as

x: q

or

x: -q

Figure 3.25 illustrates some of the ways in which the graph of a rational function may approach its horizontal asymptote. It is common practice to display the asymptotes of the graph of a rational function by using dashed lines. Although a rational function may have several vertical asymptotes, it can have at most one horizontal asymptote. The graph may intersect its horizontal asymptote. y

y

y=b

y=b

y f

y=b

f

f x

x

Figure 3.25 f (x) : b as x : q

x

310

CHAPTER 3


Geometrically, a line is an asymptote of a curve if the distance between the line and a point P(x, y) on the curve approaches zero as the distance between the origin and the point P increases without bound. Vertical asymptotes of the graph of a rational function can be found by using the following theorem.

Theorem on Vertical Asymptotes If the real number a is a zero of the denominator Q(x), then the graph of F(x) = P(x)>Q(x), where P(x) and Q(x) have no common factors, has the vertical asymptote x = a.

EXAMPLE 1

Find the vertical asymptotes of each rational function.

y

a.

f(x) =

x3 x + 1

b.

g(x) =

x x2 - x - 6

2 1 −3

−2

1

2

3

x

−1 −2

2

Solution a. To find the vertical asymptotes, determine the real zeros of the denominator. The denominator x 2 + 1 has no real zeros, so the graph of f has no vertical asymptotes. See Figure 3.26.

x3 f(x) = 2 x +1 Figure 3.26 y

Vertical asymptote: x = −2

Find the Vertical Asymptotes of a Rational Function

b.

4 2

The denominator x 2 - x - 6 = (x - 3)(x + 2) has zeros of 3 and - 2. The numerator has no common factors with the denominator, so x = 3 and x = - 2 are both vertical asymptotes of the graph of g, as shown in Figure 3.27. Try Exercise 10, page 320

4

−2 −2 −4

x g(x) = 2 x −x−6

x

Vertical asymptote: x=3

Question • Can a graph of a rational function cross its vertical asymptote?

If the denominator of a rational function is written as a product of linear factors, then the following theorem can be used to determine the manner in which the graph of the rational function approaches its vertical asymptotes.

Figure 3.27

Behavior Near a Vertical Asymptote Theorem Let F be a function defined by a rational expression in simplest form. If (x - a)n is the largest power of (x - a) that is a factor of the denominator of F, then the graph of F will have a vertical asymptote at x = a, and its graph will approach the asymptote in the manner shown on page 311.

Answer • No. If x = a is a vertical asymptote of a rational function R, then R(a) is undefined.

3.5


311

1. If n is odd, F will approach q on one side of the vertical asymptote and - q on the other side of the vertical asymptote. F

F a x

a

x

or

2. If n is even, F will approach q on both sides of the vertical asymptote or F will approach - q on both sides of the vertical asymptote. F a a

x

x

or F

Here is a specific example of how we can use the Behavior Near a Vertical Asymptote Theorem. Consider the rational function F(x) = y 5

5

10 x

−5

F(x) =

x (x − 2)(x − 5)2

Figure 3.28

As x : 2-, F : As x : 2+, F : As x : 5-, F : As x : 5+, F :

- q. q. q. q.

x (x - 2)(x - 5)2

The exponent of (x - 2) is 1. Because 1 is an odd number, the graph of F will approach q on one side of the vertical asymptote x = 2 and it will approach - q on the other side. The exponent of (x - 5) is 2. Because 2 is an even number, the graph of F will approach q on both sides of the vertical asymptote x = 5 or it will approach - q on both sides of x = 5. We can examine the factors of F to determine the exact manner in which F approaches its vertical asymptotes. For instance, as x : 2 + , the factor (x - 2) approaches 0 through positive values and the factor (x - 5)2 approaches ( - 3)2 = 9 through positive values. Thus, as x : 2 + , the denominator of F approaches 0 # 9 = 0 through positive values and the numerator of F approaches 2 through positive values. From this analysis, we see that as x : 2 + the quotient of x and (x - 2)(x - 5)2 will be positive and that it will become larger and larger as x gets closer and closer to 2. That is, as x : 2 + , F : q . We could use a similar analysis to determine the behavior of F as x : 2 - ; however, we have already determined that F approaches q on one side of x = 2 and - q on the other side. Thus we can conclude that as x : 2 - , F : - q . See Figure 3.28. As x : 5 + , the factor (x - 2) approaches 3 through positive values and the factor (x - 5)2 approaches (0)2 = 0 through positive values. Thus, as x : 5 + , the denominator of F approaches 3 # 0 = 0 through positive values and the numerator of F approaches 5 through positive values. From this analysis, we see that as x : 5 + the quotient of x and (x - 2)(x - 5)2 will be positive and that it will become larger and larger as x gets closer and closer to 5. That is, as x : 5 + , F : q . We have already determined that F approaches q on both sides of x = 5 or F approaches - q on both sides of x = 5. Thus we can conclude that as x : 5 - , F : q . See Figure 3.28. The following theorem indicates that a horizontal asymptote can be determined by examining the leading terms of the numerator and the denominator of a rational function.

312

CHAPTER 3


Theorem on Horizontal Asymptotes F(x) =

Let

an x n + an - 1x n - 1 + Á + a1x + a0 bm x m + bm - 1x m - 1 + Á + b1x + b0

be a rational function with numerator of degree n and denominator of degree m. 1. If n 6 m, then the x-axis, which is the line given by y = 0, is the horizontal asymptote of the graph of F.

2. If n = m, then the line given by y = an>bm is the horizontal asymptote of the graph of F. 3. If n 7 m, then the graph of F has no horizontal asymptote. y

EXAMPLE 2

2

Find the Horizontal Asymptote of a Rational Function

Find the horizontal asymptote of each rational function. −2

2 −1

4

x

Horizontal asymptote: x-axis

f (x) =

a.

x2 + 1

Figure 3.29 y

1

−1

g(x) =

4x2 + 1 3x2

Figure 3.30

b.

g(x) =

4x2 + 1 3x2

c.

h(x) =

x3 + 1 x - 2

b.

The numerator 4x 2 + 1 and the denominator 3x 2 of g are both of degree 2. By the 4 Theorem on Horizontal Asymptotes, the line y = is the horizontal asymptote of g. 3 See the graph of g in Figure 3.30.

c.

The degree of the numerator x 3 + 1 is larger than the degree of the denominator x - 2, so by the Theorem on Horizontal Asymptotes, the graph of h has no horizontal asymptotes.

Horizontal 4 asymptote: y = 3

2

2x + 3 x2 + 1

Solution a. The degree of the numerator 2x + 3 is less than the degree of the denominator x 2 + 1. By the Theorem on Horizontal Asymptotes, the x-axis is the horizontal asymptote of f. See the graph of f in Figure 3.29.

2x + 3

−2

f(x) =

x


The proof of the Theorem on Horizontal Asymptotes uses the technique employed in the following verification. To verify that y =

5x 2 + 4 3x + 8x + 7 2

5 , divide the numerator and the denominator by the 3 2 largest power of the variable x (x in this case). has a horizontal asymptote of y =

5x 2 + 4 4 5 + 2 2 x x y = = , 8 7 3x 2 + 8x + 7 3 + + 2 x x x2

x Z 0

3.5


313

4 8 7 As x increases without bound or decreases without bound, the fractions 2 , , and 2 x x x approach zero. Thus y: Hence the line given by y =

5 5 + 0 = 3 + 0 + 0 3

as

x: q

5 is a horizontal asymptote of the graph. 3

Sign Property of Rational Functions y

Vertical asymptote: x = −3

The zeros and vertical asymptotes of a rational function F divide the x-axis into intervals. In each interval, F(x) is positive for all x in the interval or F(x) is negative for all x in the interval. For example, consider the rational function

4 2

−3

−1 −2 −4

g(x) =

x

1 Vertical asymptote: x=1

x+1 x2 + 2x − 3

Figure 3.31

g(x) =

x + 1 x + 2x - 3 2

which has vertical asymptotes of x = - 3 and x = 1 and a zero of - 1. These three numbers divide the x-axis into the four intervals ( - q , -3), ( - 3, - 1), ( - 1, 1), and (1, q ). Note in Figure 3.31 that the graph of g is negative for all x such that x 6 - 3, positive for all x such that - 3 6 x 6 - 1, negative for all x such that -1 6 x 6 1, and positive for all x such that x 7 1.

General Graphing Procedure If F(x) = P(x)>Q(x), where P(x) and Q(x) are polynomials that have no common factors, then the following general procedure offers useful guidelines for graphing F.

General Procedure for Graphing Rational Functions That Have No Common Factors 1. Asymptotes Find the real zeros of the denominator Q(x). For each zero a, draw the dashed line x = a. Each line is a vertical asymptote of the graph of F. Also graph any horizontal asymptotes. 2. Intercepts Find the real zeros of the numerator P(x). For each real zero c, plot the point (c, 0). Each such point is an x-intercept of the graph of F. For each x-intercept, use the even and odd powers of (x - c) to determine whether the graph crosses the x-axis at the intercept or intersects but does not cross the x-axis. Also evaluate F(0). Plot (0, F(0)), the y-intercept of the graph of F. 3. Symmetry Use the tests for symmetry to determine whether the graph of the function has symmetry with respect to the y-axis or symmetry with respect to the origin. 4. Additional points Plot some points that lie in the intervals between and beyond the vertical asymptotes and the x-intercepts. 5. Behavior near asymptotes If x = a is a vertical asymptote, determine whether F(x) : q or F(x) : - q as x : a- and as x : a + . 6. Sketch the graph Use all the information obtained in steps 1 through 5 to sketch the graph of F.

314

CHAPTER 3


EXAMPLE 3

Graph a Rational Function

Sketch a graph of f(x) =

2x 2 - 18 . x2 + 3

Solution 1. Asymptotes The denominator x 2 + 3 has no real zeros, so the graph of f has no vertical asymptotes. The numerator and denominator both are of degree 2. The leading coefficients are 2 and 1, respectively. By the Theorem on Horizontal 2 Asymptotes, the graph of f has a horizontal asymptote of y = = 2. 1 2.

Intercepts The zeros of the numerator occur when 2x 2 - 18 = 0 or, solving for x, when x = - 3 and x = 3. Therefore, the x-intercepts are ( -3, 0) and (3, 0). The factored numerator is 2(x + 3)(x - 3). Each linear factor has an exponent of 1, an odd number. Thus the graph crosses the x-axis at its x-intercepts. To find the y-intercept, evaluate f when x = 0. This gives y = - 6. Therefore, the y-intercept is (0, - 6).

3.

Symmetry Below we show that f( -x) = f(x), which means that f is an even function and therefore its graph is symmetric with respect to the y-axis. f( -x) =

4.

5.

2( -x)2 - 18 ( -x)2 + 3

=

2x 2 - 18 = f(x) x2 + 3

Additional points The intervals determined by the x-intercepts are x 6 - 3, -3 6 x 6 3, and x 7 3. Generally, it is necessary to determine points in all intervals. However, because f is an even function, its graph is symmetric with respect to the y-axis. The following table lists a few points for x 7 0. Symmetry can be used to locate corresponding points for x 6 0. x

1

f (x)

-4

2 -

10 L - 1.43 7

6 18 L 1.38 13

Behavior near asymptotes As x increases or decreases without bound, f(x) approaches the horizontal asymptote y = 2. To determine whether the graph of f intersects the horizontal asymptote at any point, solve the equation f(x) = 2. There are no solutions of f(x) = 2 because 2x 2 - 18 = 2 x2 + 3

implies

2x 2 - 18 = 2x 2 + 6

implies

-18 = 6

This is not possible. Thus the graph of f does not intersect the horizontal asymptote but approaches it from below as x increases or decreases without bound. 6.

Sketch the graph Use the summary in Table 3.3 to sketch the graph. The completed graph is shown in Figure 3.32.

3.5


y

Table 3.3

Vertical asymptote

None

Horizontal asymptote

y=2

315

Horizontal asymptote: y = 2

2 −6

x-intercepts

Crosses at (- 3, 0) and (3, 0)

y-intercept

(0, -6)

Additional points

(1, -4), (2, -1.43), (6, 1.38)

x

6

−6

f (x) =

2x2 − 18 x2 + 3

Figure 3.32


EXAMPLE 4

Graph a Rational Function

Sketch a graph of h(x) =

x2 + 1 . x2 + x - 2

Solution 1. Asymptotes The denominator x2 + x - 2 = (x + 2)(x - 1) has zeros - 2 and 1; because there are no common factors of the numerator and the denominator, the lines x = - 2 and x = 1 are vertical asymptotes. The numerator and denominator both are of degree 2. The leading coefficients of the numerator and denominator are both 1. Thus h has the horizontal asymptote 1 y = = 1. 1 2.

Intercepts The numerator x 2 + 1 has no real zeros, so the graph of h has no x-intercepts. Because h(0) = - 0.5, h has the y-intercept (0, - 0.5).

3.

Symmetry By applying the tests for symmetry, we can determine that the graph of h is not symmetric with respect to the origin or with respect to the y-axis.

4.

Additional points The intervals determined by the vertical asymptotes are (- q , - 2), ( - 2, 1), and (1, q ). Plot a few points from each interval. x

-5

-3

-1

0.5

2

3

4

h(x)

13 9

5 2

-1

-1

5 4

1

17 18

The graph of h will intersect the horizontal asymptote y = 1 exactly once. This can be determined by solving the equation h(x) = 1. x2 + 1 = x2 + x - 2 x2 + 1 = 1 = 3 =

1 x2 + x - 2 x - 2 x

• Multiply both sides by x 2 + x - 2. (continued)

316

CHAPTER 3


The only solution is x = 3. Therefore, the graph of h intersects the horizontal asymptote at (3, 1). 5.

Behavior near asymptotes The manner in which h approaches its vertical asymptote x = - 2 as x : - 2- can be determined by examining the numerator and the factors of the denominators as x : -2-. For instance, as x : -2-, the numerator x 2 + 1 approaches (- 2)2 + 1 = 5 through positive values. As x : - 2 - , the (x + 2) factor in the denominator approaches 0 through negative values and the (x - 1) factor in the denominator approaches - 3 through negative values. Thus, as x : -2- , the denominator of h approaches 0 # ( - 3) = 0 through positive values. From this analysis, we see that as x : - 2 - , the quotient of the numerator and the denominator will be positive, and it will become larger and larger as x : -2 - . That is, h(x) : q as x : -2 We could use a similiar analysis to determine the behavior of h as x : - 2 + . However, the Behavior Near a Vertical Asymptote Theorem indicates that h approaches q on one side of the vertical asymptote x = - 2, and h approaches - q on the other side. Thus we know that h(x) : - q as x : - 2 + A similar analysis can be used to show that h(x) : - q as x : 1 h(x) : q as x : 1 +

6.

Sketch the graph Figure 3.33.

Use the summary in Table 3.4 to sketch the graph. See

Table 3.4 y

Vertical asymptote

x = - 2, x = 1

Horizontal asymptote

y = 1

x-intercepts

None

y-intercept

(0, -0.5)

Additional points

( -5, 1.4), ( - 3, 2.5), ( -1, -1), (0.5, -1), (2, 1.25), (3, 1), (4, 0.94)

Horizontal asymptote: y=1

2 (3, 1) 2

−4 Vertical asymptote: x = −2

6

x


−2

h(x) =

4

x2 + 1 x2 + x − 2 Figure 3.33


Slant Asymptotes Some rational functions have an asymptote that is neither vertical nor horizontal but slanted.

Definition of a Slant Asymptote The line given by y = mx + b, m Z 0, is a slant asymptote of the graph of a function F provided F(x) : mx + b as x : q or x : - q .

3.5


317

The following theorem can be used to determine which rational functions have a slant asymptote.

Theorem on Slant Asymptotes The rational function given by F(x) = P(x)>Q(x), where P(x) and Q(x) have no common factors, has a slant asymptote if the degree of P(x) is one greater than the degree of Q(x).

To find the slant asymptote, divide P(x) by Q(x) and write F(x) in the form F(x) =

r(x) P(x) = (mx + b) + Q(x) Q(x)

where the degree of r(x) is less than the degree of Q(x). Because r(x) : 0 as Q(x)

x : q

we know that F(x) : mx + b as x : q . The line represented by y = mx + b is the slant asymptote of the graph of F.

EXAMPLE 5

Find the Slant Asymptote of a Rational Function

Find the slant asymptote of f(x) =

2x 3 + 5x 2 + 1 . x2 + x + 3

Solution Because the degree of the numerator 2x 3 + 5x 2 + 1 is exactly one larger than the degree of the denominator x 2 + x + 3 and f is in simplest form, f has a slant asymptote. To find the asymptote, divide 2x 3 + 5x 2 + 1 by x 2 + x + 3. x + x + 3 冄 2x + 5x 2x 3 + 2x 2 3x 2 3x 2 2

y

10 Slant asymptote: y = 2x + 3 −6

5

−4

2

+ + +

2x 0x 6x 6x 3x

+ 3 + 1 + 1 + 9

-9x - 8 2

−5 −10

f (x) =

3

2x3 + 5x2 + 1 x2 + x + 3

Figure 3.34

4

x

Therefore, f(x) =

- 9x - 8 2x 3 + 5x 2 + 1 = 2x + 3 + 2 2 x + x + 3 x + x + 3

and the line given by y = 2x + 3 is the slant asymptote for the graph of f. Figure 3.34 shows the graph of f and its slant asymptote. Try Exercise 44, page 321

The function f in Example 5 does not have a vertical asymptote because the denominator x 2 + x + 3 does not have any real zeros. However, the function g(x) =

2x 2 - 4x + 5 3 - x

318

CHAPTER 3


y


10

−2

2

has both a slant asymptote and a vertical asymptote. The vertical asymptote is x = 3, and the slant asymptote is y = - 2x - 2. Figure 3.35 shows the graph of g and its asymptotes.

x 6 8 Slant asymptote: y = −2x − 2

4

−10 −20

g(x) =

2x2 − 4x + 5 3−x

Graphing Rational Functions That Have a Common Factor If a rational function has a numerator and denominator that have a common factor, then you should reduce the rational function to simplest form before you apply the general procedure for sketching the graph of a rational function.

EXAMPLE 6

Graph a Rational Function That Has a Common Factor

Sketch the graph of f(x) =

Figure 3.35

x 2 - 3x - 4 . x 2 - 6x + 8

Solution Factor the numerator and denominator to obtain y

f (x) =

6

(x + 1)(x - 4) x 2 - 3x - 4 = , 2 (x - 2)(x - 4) x - 6x + 8

Thus for all x values other than x = 4, the graph of f is the same as the graph of

4 (4, 2.5) 2

–4

–2

G(x) = 2

–2 –4

f (x) =

x2 − 3x − 4 x2 − 6x + 8

4

6

x Z 2, x Z 4

x

x + 1 x - 2

Figure 3.23 on page 307 shows a graph of G. The graph of f will be the same as this graph, except that it will have an open circle at (4, 2.5) to indicate that it is undefined at x = 4. See the graph of f in Figure 3.36. The height of the open circle was found by x + 1 evaluating the resulting reduced rational function G(x) = at x = 4. x - 2 Try Exercise 62, page 321

Figure 3.36 Question • Does F(x) =

x2 - x - 6 x2 - 9

have a vertical asymptote at x = 3?

Applications of Rational Functions EXAMPLE 7

Determine the Average Speed for a Trip

Jordan averages 30 miles per hour during 12 miles of city driving. For the remainder of the trip, she drives on a highway at a constant rate of 60 miles per hour. Her average speed for the entire trip is given by s(x) =

12 + x 1 2 + x 5 60

where x is the number of miles she drives on the highway. Answer • No. F(x) =

x2 - x - 6 x - 9 2

=

(x - 3)(x + 2) 5 x + 2 = , x Z 3. As x : 3, F(x) : . (x - 3)(x + 3) x + 3 6

3.5

a.

How far will she need to drive on the highway to bring her average speed for the entire trip up to 50 miles per hour?

b.

Determine the horizontal asymptote of the graph of s, and explain the meaning of the horizontal asymptote in the context of this application.

Algebraic Solution a.


12 + x = 50 1 2 + x 5 60 12 + x = 50a

a.

• Set s(x) equal to 50.

The following graph shows that s(x) = 50 when x = 48. 80

1 2 + xb 5 60

12 + x = 20 + x -

319


• Multiply each side by a

5 x 6

Y1=(12+X)/(2/5+1/60X)

1 2 + xb. 5 60

• Simplify.

5 x = 20 - 12 6

0 X=48 0

• Solve for x.

Y=50

120

Figure 3.37

1 x = 8 6 x = 48 Jordan needs to drive 48 miles at 60 miles per hour to bring her average speed up to 50 miles per hour. See Figure 3.37. b.

The numerator and denominator of s are both of degree 1. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1/60. Thus the graph of s has a horizontal asymptote of y =

b.

The graphs of s and y = 60 are shown in the same window. 80

1 = 60 1 a b 60

The horizontal asymptote of the graph of s is the line y = 60. As Jordan continues to drive at 60 miles per hour, her average speed for the entire trip will approach 60 miles per hour. See Figure 3.38.

0

120 0

Figure 3.38


EXAMPLE 8


A cylindrical soft drink can is to be constructed so that it will have a volume of 21.6 cubic inches. See Figure 3.39. a.

Write the total surface area A of the can as a function of r, where r is the radius of the can in inches.

b.

Use a graphing utility to estimate the value of r (to the nearest tenth of an inch) that produces the minimum surface area.

r

Figure 3.39 (continued)

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CHAPTER 3


Integrating Technology A Web applet is available to explore the relationship between the radius of a cylinder with a given volume and the surface area of the cylinder. This applet, CYLINDER, can be found at http://www.cengage.com/math/ aufmann/algtrig7e.

Solution a. The formula for the volume of a cylinder is V = pr 2h, where r is the radius and h is the height. Because we are given that the volume is 21.6 cubic inches, we have 21.6 = pr 2h 21.6 = h pr 2

• Solve for h.

The surface area of the cylinder is given by A = 2pr 2 + 2prh A = 2pr 2 + 2pr a

21.6 b pr 2

100

A = 2pr 2 + A = Minimum 0 X=1.5092371 Y=42.935551 0

4

2πx3 + 43.2 x

y=

• Substitute for h.

2(21.6) r

• Simplify.

2pr 3 + 43.2 r

(1)

Use Equation (1) with y = A and x = r and a graphing utility to determine that A is a minimum when r L 1.5 inches. See Figure 3.40.

b.


Figure 3.40

EXERCISE SET 3.5 In Exercises 1 to 8, determine the domain of the rational function. 1 1. F(x) = x 3. F(x) =

5. F(x) =

7. F(x) =

2 2. F(x) = x - 3

x - 3 2

4. F(x) =

x + 1 2

2x - 1 2x - 15x + 18 2

2x 2 x - 4x - 12x 3

2

6. F(x) =

8. F(x) =

x + 4 3

x 2 - 25

13. F(x) =

4x - 27x + 18 3x 2 x - 5 2

4x - 25x + 6x 3

2

15. F(x) =

16. F(x) =

17. F(x) =

4x 2 + 1 x2 + x + 1 3x 3 - 27x 2 + 5x - 11 x5 - 2x 3 + 7 15,000x 3 + 500x - 2000 700 + 500x 3

18. F(x) = 6000a 1 -

In Exercises 9 to 14, find all vertical asymptotes of each rational function. 9. F(x) =

11. F(x) =

2x - 1 x 2 + 3x x 2 + 11 6x - 5x - 4 2

10. F(x) =

12. F(x) =

3x 2 + 5

19. F(x) =

x3 - 8

20. F(x) =

25 (x + 5)2

4x 2 - 11x + 6 4 - x +

x2 - 4 3x - 5

14. F(x) =

5x x - 81 4

In Exercises 15 to 20, find the horizontal asymptote of each rational function.

3x - 2 2

5x 2 - 3

1 2 x 3

(2x - 3)(3x + 4) (1 - 2x)(3 - 5x)

b

3.5

In Exercises 21 to 42, determine the vertical and horizontal asymptotes and sketch the graph of the rational function F. Label all intercepts and asymptotes. 21. F(x) =

1 x + 4

22. F(x) =

1 x - 2

23. F(x) =

-4 x - 3

24. F(x) =

-3 x + 2

25. F(x) =

27. F(x) =

29. F(x) =

31. F(x) =

33. F(x) =

35. F(x) =

37. F(x) =

39. F(x) =

41. F(x) =

4 x

26. F(x) =

x x + 4 x + 4 2 - x 1 x - 9 2

1 x 2 + 2x - 3 x2 x + 4x + 4 2

10 x2 + 2 2x 2 - 2 x2 - 9 x2 + x + 4 x 2 + 2x - 1

28. F(x) =

30. F(x) =

32. F(x) =

34. F(x) =

36. F(x) =

38. F(x) =

40. F(x) =

42. F(x) =

-4 x


47. F(x) =

48. F(x) =

x - 4

2x 2 x - 1 2

x2 x 2 - 6x + 9 6x 2 - 5 2x 2 + 6 2x 2 - 14

50. F(x) =

x2 + 10 2x

51. F(x) =

x 2 - 3x - 4 x + 3

52. F(x) =

x 2 - 4x - 5 2x + 5

53. F(x) =

2x 2 + 5x + 3 x - 4

54. F(x) =

4x 2 - 9 x + 3

55. F(x) =

x2 - x x + 2

56. F(x) =

x2 + x x - 1

57. F(x) =

44. F(x) =

45. F(x) =

46. F(x) =

3x 2 + 5x - 1 x + 4

x - 4 2

58. F(x) =

59. F(x) =

x2 + x x + 1

60. F(x) =

61. F(x) =

2x 3 + 4x 2 2x + 4

62. F(x) =

63. F(x) =

65. F(x) = 67.

43. F(x) =

x3 + 1

- 2x 3 + 6x 2x - 6x 2

x 2 - 3x - 10 x 2 + 4x + 4

64. F(x) =

66. F(x) =

x 2 - x - 12 x 2 - 2x - 8 x 3 + 3x 2 x(x + 3)(x - 1) 2x 2 + x - 3 x 2 - 2x + 1

9-V battery A

x 2 - 3x + 5

4000 + 20x + 0.0001x 2 x

x 2 - 3x x - 3

Variable resistor

Ammeter

x2

3x 2

Electrical Current A variable resistor, an ammeter, and a 9-volt battery are connected as shown in the following diagram.

x 3 - 2x 2 + 3x + 4

x3 - 1

x3 - 1

In Exercises 59 to 66, sketch the graph of the rational function F.

x 2 - 6x + 5

In Exercises 43 to 48, find the slant asymptote of each rational function.

x3 - 1

x2 - 4 x

1 x 2 - 2x - 8

- x4 - 2x 3 - 3x 2 + 4x - 1

49. F(x) =

-2 2

-4x 2 + 15x + 18 x - 5

In Exercises 49 to 58, determine the vertical and slant asymptotes and sketch the graph of the rational function F.

x x - 2 x + 3 1 - x

321

The internal resistance of the ammeter is 4.5 ohms. The current I, in amperes, through the ammeter is given by 9 I(x) = x + 4.5 where x is the resistance, in ohms, provided by the variable resistor.

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CHAPTER 3


a. Find the current through the ammeter when the variable

71.

resistor has a resistance of 3 ohms.

the salt in a tank of seawater is given by

b. Determine the resistance of the variable resistor when the

c. Determine the horizontal asymptote of the graph of I, and explain the meaning of the horizontal asymptote in the context of this application.

r(x) =

b. Find the cost of removing 80% of the salt. c. Sketch the graph of C. 72.

C(x) = 0.0006x 2 + 9x + 401,000

30 + x 3 1 + x 4 70

The average cost per cell phone is C(x) =

a. How far will you need to drive on the highway to bring your

100,000 phones are produced.

b. Determine the horizontal asymptote of the graph of r, and

b. What is the minimum average cost per cell phone? How

write a sentence that explains the meaning of the horizontal asymptote in the context of this application.

producing x golf balls is given by

C(x) 0.0006x 2 + 9x + 401,000 = x x

a. Find the average cost per cell phone when 1000, 10,000, and

average speed up to 60 miles per hour?

Average Cost of Golf Balls The cost, in dollars, of

Production Costs The cost, in dollars, of producing x

cell phones is given by

where x is the number of miles you have driven on the highway.

69.

0 … p 6 100

a. Find the cost of removing 40% of the salt.

Average Speed During the first 30 miles of city driving,

you average 40 miles per hour. For the remainder of the trip, you drive on a highway at a constant rate of 70 miles per hour. Your average speed for the entire trip is given by

2000p , 100 - p

C( p) =

current through the ammeter is 0.24 amperes.

68.

Desalinization The cost C, in dollars, to remove p% of

many cell phones should be produced to minimize the average cost per phone? 73.

A Sales Model A music company expects that the monthly

sales S, in thousands, of a new music CD it has produced will be closely approximated by

C(x) = 0.43x + 76,000 The average cost per golf ball is given by

S(t) =

0.43x + 76,000 C(x) = C(x) = x x

150t 1.5t 2 + 80

where t is the number of months after the CD is released. a. Find the monthly sales the company expects for t = 2, 4, and

a. Find the average cost per golf ball of producing 1000,

10 months. Round to the nearest 100 CDs.

10,000, and 100,000 golf balls.

b. Use S to predict the month in which sales are expected to

b. What is the equation of the horizontal asymptote of the graph

reach a maximum.

of C? Explain the significance of the horizontal asymptote as it relates to this application.

c. What does the company expect the monthly sales will

approach as the years go by? 70.

Average Cost of DVD Players The cost, in dollars, of

producing x DVD players is given by

74.

C(x) = 0.001x 2 + 54x + 175,000

Medication Model The rational function

M(t) =

The average cost per DVD player is given by

0.5t + 400 0.04t 2 + 1

models the number of milligrams of medication in the bloodstream of a patient t hours after 400 milligrams of the medication have been injected into the patient’s bloodstream.

C(x) 0.001x 2 + 54x + 175,000 C(x) = = x x

a. Find M(5) and M(10). Round to the nearest milligram.

a. Find the average cost per DVD player of producing 1000,

10,000, and 100,000 DVD players.

b. What will M approach as t : q ?

b. What is the minimum average cost per DVD player? How

many DVD players should be produced to minimize the average cost per DVD player?

75.

Minimizing Surface Area A cylindrical soft drink can is

to be made so that it will have a volume of 354 milliliters.

EXPLORING CONCEPTS WITH TECHNOLOGY

If r is the radius of the can in centimeters, then the total surface area A in square centimeters of the can is given by the rational function A(r) =

One resistor has a resistance of R1 ohms, and the other has a resistance of R2 ohms. The total resistance for the circuit, measured in ohms, is given by the formula

2pr 3 + 708 r

RT =

r

323

R1R2 R1 + R2

Assume that R1 has a fixed resistance of 10 ohms. a. Compute RT for R2 = 2 ohms and for R2 = 20 ohms. b. Find R2 when RT = 6 ohms. c.

What happens to RT as R2 : q ?

77. Determine the point at which the graph of

F(x) =

a. Graph A and use the graph to estimate (to the nearest tenth

of a centimeter) the value of r that produces the minimum value of A.

2x 2 + 3x + 4 x 2 + 4x + 7

intersects its horizontal asymptote.

b. Does the graph of A have a slant asymptote? c.

Explain the meaning of the following statement as it applies to the graph of A. As r : q , A : 2p r 2.

76. Resistors in Parallel The electronic circuit below shows two

resistors connected in parallel. R1 R2

In Exercises 78 to 80, create a rational function whose graph has the given characteristics. 78. Has a vertical asymptote at x = 2, has a horizontal asymptote

at y = 5, and intersects the x-axis at (4, 0)

79. Is symmetric to the y-axis, has vertical asymptotes at x = 3

and x = - 3, has a horizontal asymptote at y = 2, and passes through the origin

80. Has a vertical asymptote at x = 5, has y = x - 3 as a slant

asymptote, and intersects the x-axis at (4, 0)


Finding Zeros of a Polynomial Using Mathematica Computer algebra systems (CAS) are computer programs that are used to solve equations, graph functions, simplify algebraic expressions, and help us perform many other mathematical tasks. In this exploration, we will demonstrate how to use one of these programs, Mathematica, to find the zeros of a polynomial function. Recall that a zero of a function P is a number x for which P(x) = 0. The idea behind finding a zero of a polynomial function by using a CAS is to solve the polynomial equation P(x) = 0 for x. Two commands in Mathematica that can be used to solve an equation are Solve and NSolve. (Mathematica is sensitive about syntax, or the way in which an expression is typed.You must use uppercase and lowercase letters as we indicate.) Solve will attempt to find an exact solution of the equation; NSolve will attempt to find approximate solutions. Here are some examples. To find the exact values of the zeros of P(x) = x 3 + 5x 2 + 11x + 15, input the following. (Note: The two equals signs are necessary.) Solve[x^3+5x^2+11x+15==0] (continued)

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CHAPTER 3


Press Enter . The result should be {{x–>-3}, {x–>-1–2 I}, {x–>-1+2 I}} Thus the three zeros of P are -3, - 1 - 2i, and - 1 + 2i. To find the approximate values of the zeros of P(x) = x 4 - 3x 3 + 4x 2 + x - 4, input the following. NSolve[x^4–3x^3+4x^2+x–4==0] Press Enter . The result should be {{x–>-0.821746}, {x–>1.2326}, {x–>1.29457–1.50771 I}, {x–>1.29457+1.50771 I}} The four zeros are (approximately) -0.821746, 1.2326, 1.29457 - 1.50771i, and 1.29457 + 1.50771i. Not all polynomial equations can be solved exactly. This means that Solve will not always give solutions with Mathematica. Consider the two examples below. Input Output

NSolve[x^5–3x^3+2x^2–5==0] {{x–>-1.80492}, {x–>-1.12491}, {x–>0.620319–1.03589 I}, {x–>0.620319+1.03589 I}, {x–>1.68919}}

These are the approximate zeros of the polynomial. Input Output

Solve[x^5–3x^3+2x^2–5==0] {ToRules[Roots[2x2–3x3+x5==5]]}

In this case, no exact solution could be found. In general, there are no formulas (like the quadratic formula) that yield exact solutions for fifth-degree (or higher) polynomial equations. Use Mathematica (or another CAS) to find the zeros of each of the following polynomial functions. 1. P(x) = x 4 - 3x 3 + x - 5

2. P(x) = 3x 3 - 4x 2 + x - 3

3. P(x) = 4x 5 - 3x 3 + 2x 2 - x + 2

4. P(x) = - 3x 4 - 6x 3 + 2x - 8


3.1 Remainder Theorem and Factor Theorem Synthetic division Synthetic division is a procedure that can be used to expedite the division of a polynomial by a binomial of the form x – c.


Remainder Theorem If a polynomial P(x) is divided by x – c, then the remainder equals P(c).


Factor Theorem A polynomial P(x) has a factor (x – c) if and only if P(c) = 0.


CHAPTER 3 TEST PREP

325

3.2 Polynomial Functions of Higher Degree Leading Term Test The far-left and far-right behavior of the graph of a polynomial function P can be determined by examining its leading term, an x n. • If an 7 0 and n is even, then the graph of P goes up to the far left and up to the far right.


• If an 7 0 and n is odd, then the graph of P goes down to the far left and up to the far right. • If an 6 0 and n is even, then the graph of P goes down to the far left and down to the far right. • If an 6 0 and n is odd, then the graph of P goes up to the far left and down to the far right. Definition of Relative Minimum and Relative Maximum If there is an open interval I containing c on which • f(c) … f(x) for all x in I, then f(c) is a relative minimum of f.

See the Integrating Technology feature, page 274, and then try Exercises 15 and 16, page 329.

• f(c) Ú f(x) for all x in I, then f(c) is a relative maximum of f. Intermediate Value Theorem If P is a polynomial function and P(a) Z P(b) for a 6 b, then P takes on every value between P(a) and P(b) in the interval [a, b]. The following statement is a special case of the Intermediate Value Theorem. If P(a) and P(b) have opposite signs, then you can conclude by the Intermediate Value Theorem that P has a zero between a and b.


Even and Odd Powers of (x c) Theorem If c is a real number and the polynomial function P has (x – c) as a factor exactly k times, then the graph of P will intersect but not cross the x-axis at (c, 0), provided k is an even positive integer, and the graph of P will cross the x-axis at (c, 0), provided k is an odd positive integer.


Procedure for Graphing Polynomial Functions To graph a polynomial function P


1. Examine the leading coefficient of P to determine the far-left and far-right behavior of the graph. 2. Find the y-intercept by evaluating P(0). 3. Find the x-intercept(s). If (x – c), where c is a real number, is a factor of P, then (c, 0) is an x-intercept of the graph. Use the Even and Odd Powers of (x – c) Theorem to determine where the graph crosses the x-axis and where the graph intersects but does not cross the x-axis. 4. Find additional points on the graph. 5. Check for symmetry with respect to the y-axis and with respect to the origin. 6. Use all of the information obtained to sketch the graph. The graph should be a smooth, continuous curve that passes through the points determined in steps 2 through 4. The graph should have a maximum of n – 1 turning points.

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CHAPTER 3


3.3 Zeros of Polynomial Functions Rational Zero Theorem If P(x) = anxn + an - 1x n - 1 + Á + a1x + a0 p has integer coefficients (an Z 0) and is a rational zero (in simplest form) q of P, then p is a factor of the constant term a0 and q is a factor of the leading coefficient an.


Descartes’ Rule of Signs Let P be a polynomial function with real coefficients and with the terms arranged in order of decreasing powers of x. • The number of positive real zeros of P is equal to the number of variations in sign of P(x) or to that number decreased by an even integer.


• The number of negative real zeros of P is equal to the number of variations in sign of P(–x) or to that number decreased by an even integer. Guidelines for Finding the Zeros of a Polynomial Function with Integer Coefficients 1. Determine the degree of the function. The number of distinct zeros of the polynomial function is at most n. Apply Descartes’ Rule of Signs to find the possible number of positive zeros and the possible number of negative zeros.


2. Apply the Rational Zero Theorem to list rational numbers that are possible zeros. Use synthetic division to test numbers in your list. If you find an upper bound or lower bound, then eliminate from your list any number that is greater than the upper bound or less than the lower bound. 3. Work with the reduced polynomial. • If the reduced polynomial is of degree 2, find its zeros either by factoring or by applying the quadratic formula. • If the degree of the reduced polynomial is 3 or greater, repeat the preceding steps for this reduced polynomial.

3.4 Fundamental Theorem of Algebra Fundamental Theorem of Algebra If P is a polynomial function of degree n Ú 1 with complex coefficients, then P has at least one complex zero. The Fundamental Theorem of Algebra can be used to establish the following theorem. • Linear Factor Theorem If P is a polynomial function of degree n Ú 1 with leading coefficient an Z 0, then P has exactly n linear factors and can be written as P(x) = an(x - c1)(x - c2) Á (x - cn) where c1, c2, Á , cn are complex numbers. The Linear Factor Theorem can be used to establish the following theorem.


CHAPTER 3 TEST PREP

327

• Number of Zeros of a Polynomial Function Theorem If P is a polynomial function of degree n Ú 1, then P has exactly n complex zeros, provided each zero is counted according to its multiplicity. Conjugate Pair Theorem If a + bi (b Z 0) is a complex zero of a polynomial function with real coefficients, then the conjugate a - bi is also a complex zero of the polynomial function.


Finding a Polynomial Function with Given Zeros If c1, c2, Á , cn are given as zeros, then the product (x - c1)(x - c2) Á (x - cn) yields a polynomial function that has the given zeros.

See Example 5 page 304, and then try Exercises 49 and 50, page 329.

3.5 Graphs of Rational Functions and Their Applications Vertical Asymptotes • Definition of a Vertical Asymptote The line given by x = a is a vertical asymptote of the graph of a function F, provided F(x) : q or F(x) : - q as x approaches a from either the left or the right.


• Theorem on Vertical Asymptotes If the real number a is a zero of the denominator Q(x), then the graph of F(x) = P(x)兾Q(x), where P(x) and Q(x) have no common factors, has the vertical asymptote x = a. Horizontal Asymptotes • Definition of a Horizontal Asymptote The line given by y = b is a horizontal asymptote of the graph of a function F, provided F(x) : b as x : q or x : - q .


• Theorem on Horizontal Asymptotes See the Theorem on Horizontal Asymptotes on page 312. The method used to determine the horizontal asymptote of a rational function depends upon the relationship between the degree of the numerator and the degree of the denominator of the rational function. Slant Asymptotes • Definition of a Slant Asymptote The line given by y = mx + b, m Z 0, is a slant asymptote of the graph of a function F, provided F(x) : mx + b as x : q or x : - q .


• Theorem on Slant Asymptotes The rational function F(x) = P(x)>Q(x), where P(x) and Q(x) have no common factors, has a slant asymptote if the degree of P(x) is one greater than the degree of Q(x). The equation of the asymptote can be determined by setting y equal to the quotient of P(x) divided by Q(x). General Procedure for Graphing Rational Functions That Have No Common Factors To graph a rational function F 1. Find the real zeros of the denominator. For each real zero a, the vertical line x = a will be a vertical asymptote. Use the Theorem on Horizontal Asymptotes and the Theorem on Slant Asymptotes to determine whether F has a horizontal asymptote or a slant asymptote. Use dashed lines to graph all asymptotes.

See Examples 3 and 4, pages 314 and 315, and then try Exercises 60, 63, and 65, page 330.

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CHAPTER 3


2. Find the real zeros of the numerator. For each real zero c, plot (c, 0). These are the x-intercepts. The y-intercept of the graph is the point (0, F(0)), provided F(0) is a real number. 3. Use the tests for symmetry to determine whether the graph has symmetry with respect to the y-axis or with respect to the origin. 4. Find and plot additional points that lie in the intervals between and beyond the vertical asymptotes and the x-intercepts. 5. Determine the behavior of the graph near asymptotes. 6. Use all of the information obtained in steps 1 through 5 to sketch the graph. General Procedure for Graphing Rational Functions That Have a Common Linear Factor To graph a rational function F that has a numerator and a denominator with (x – a) as a common factor


1. Reduce the rational function to simplest form. Then use the general procedure for graphing rational functions that have no common factors. 2. If the reduced rational function does not have (x – a) as a factor of the denominator, then the graph produced in step 1 is the graph of F, provided you place an open circle on the graph at x = a. The height of the open circle can be determined by evaluating the reduced rational function at x = a. If (x - a) is a factor of the denominator of the reduced rational function, then the graph produced in step 1 is the graph of F and it will have a vertical asymptote at x = a.

CHAPTER 3 REVIEW EXERCISES In Exercises 1 and 2, use synthetic division to divide the first polynomial by the second. 1. 4x 3 - 11x 2 + 5x - 2,

x - 3

2. x4 + 9x 3 + 6x 2 - 65x - 63,

In Exercises 7 to 10, use synthetic division to show that c is a zero of the given polynomial function. 7. P(x) = x 3 + 2x 2 - 26x + 33,

x + 7

c = 3

8. P(x) = 2x 4 + 8x 3 - 8x 2 - 31x + 4, 9. P(x) = x 5 - x 4 - 2x 2 + x + 1,

In Exercises 3 to 6, use the Remainder Theorem to find P(c). 3. P(x) = x 3 + 2x 2 - 5x + 1, 4. P(x) = - 4x 3 - 10x + 8,

c = 4

c = -1

5. P(x) = 6x 4 - 12x 2 + 8x + 1,

c = 1

c =

1 2

In Exercises 11 and 12, use the Factor Theorem to determine whether the given binomial is a factor of P.

c = -2

6. P(x) = 5x 5 - 8x 4 + 2x 3 - 6x 2 - 9,

10. P(x) = 2x 3 + 3x 2 - 8x + 3,

c = -4

c = 3

11. P(x) = x 3 - 11x 2 + 39x - 45,

(x - 5)

12. P(x) = 2x 4 - 11x 3 + 11x 2 - 33x + 15,

(x + 2)


In Exercises 13 and 14, determine the far-left and the far-right behavior of the graph of the function. 13. P(x) = - 2x 3 - 5x 2 + 6x - 3

329

In Exercises 33 to 36, use Descartes’ Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. 33. P(x) = x 3 + 3x 2 + x + 3

14. P(x) = - x 4 + 3x 3 - 2x 2 + x - 5

34. P(x) = x4 - 6x 3 - 5x 2 + 74x - 120

In Exercises 15 and 16, use the maximum and minimum features of a graphing utility to estimate, to the nearest thousandth, the x and y coordinates of the points where P has a relative maximum or a relative minimum.

35. P(x) = x4 - x - 1 36. P(x) = x 5 - 4x4 + 2x 3 - x 2 + x - 8

15. P(x) = 2x 3 - x 2 - 3x + 1

In Exercises 37 to 42, find the zeros of the polynomial function.

16. P(x) = x - 2x + x + 1 4

2

37. P(x) = x 3 + 6x 2 + 3x - 10

In Exercises 17 and 18, use the Intermediate Value Theorem to verify that P has a zero between a and b.

38. P(x) = x 3 - 10x 2 + 31x - 30

17. P(x) = 3x3 - 7x2 - 3x + 7; a = 2, b = 3

39. P(x) = 6x 4 + 35x 3 + 72x 2 + 60x + 16

18. P(x) = 3x4 - 5x3 - 6x2 - 10x - 24; a = - 2, b = - 1

40. P(x) = 2x 4 + 7x 3 + 5x 2 + 7x + 3 41. P(x) = x 4 - 4x 3 + 6x 2 - 4x + 1

In Exercises 19 and 20, determine the x-intercepts of the graph of P. For each x-intercept, use the Even and Odd Powers of (x c) Theorem to determine whether the graph of P crosses the x-axis or intersects but does not cross the x-axis. 19. P(x) = (x + 3)(x - 5)2

42. P(x) = 2x 3 - 7x 2 + 22x + 13

In Exercises 43 and 44, find all the zeros of P and write P as a product of its leading coefficient and its linear factors. 43. P(x) = 2x 4 - 9x 3 + 22x 2 - 29x + 10

20. P(x) = (x - 4)4(x + 1)

44. P(x) = x 4 - 6x 3 + 21x 2 - 46x + 30

In Exercises 21 to 26, graph the polynomial function. In Exercises 45 and 46, use the given zero to find the remaining zeros of each polynomial function.

21. P(x) = x 3 - x 22. P(x) = - x 3 - x 2 + 8x + 12 23. P(x) = x - 6

24. P(x) = x - x

4

5

25. P(x) = x - 10x + 9 4

2

26. P(x) = x - 5x 5

27. P(x) = x - 7x - 6 28. P(x) = 2x 3 + 3x 2 - 29x - 30 29. P(x) = 15x - 91x + 4x + 12 3

2

30. P(x) = x4 - 12x 3 + 52x 2 - 96x + 64 31. P(x) = x 3 + x 2 - x - 1 32. P(x) = 6x5 + 3x - 2

1 - 2i

46. P(x) = x4 - x 3 - 17x 2 + 55x - 50;

2 + i

3

In Exercises 27 to 32, use the Rational Zero Theorem to list all possible rational zeros for each polynomial function. 3

45. P(x) = x4 - 4x 3 + 6x 2 - 4x - 15;

In Exercises 47 to 50, find the requested polynomial function. 47. Find a third-degree polynomial function with integer

coefficients and zeros of 4, - 3, and

1 . 2

48. Find a fourth-degree polynomial function with zeros of 2, -3,

i, and - i.

49. Find a fourth-degree polynomial function with real coeffi-

cients that has zeros of 1, 2, and 5i. 50. Find a fourth-degree polynomial function with real coefficients

that has - 2 as a zero of multiplicity 2 and has 1 + 3i as a zero.

330

CHAPTER 3


In Exercises 51 and 52, determine the domain of the rational function. 51. F(x) =

2

x

52. F(x) =

x + 7 2

68. Food Temperature The temperature F, in degrees Fahrenheit,

of a dessert placed in a freezer for t hours is given by the rational function

3x + 2x - 5 2

6x2 - 25x + 4

F(t) =

4x - 1 x - x - 12x 3

2

54. f (x) =

x - 9

69.

56. f (x) =

U. S. Motor Vehicle Thefts, in thousands

x2 - 2

Year

Thefts

Year

Thefts

2x 3 - x - 4

1992

1611

2000

1160

1993

1563

2001

1228

1994

1539

2002

1247

1995

1472

2003

1261

1996

1394

2004

1238

1997

1354

2005

1236

1998

1243

2006

1193

1999

1152

2007

1096

In Exercises 57 and 58, determine the slant asymptote for the graph of each rational function. 57. f (x) =

2x 3 - 3x 2 - x + 5 x2 - x + 1

58. f (x) =

3x 2 + 7 x - 2

In Exercises 59 to 66, graph each rational function. 59. f (x) =

61. f (x) =

63. f (x) =

65. f (x) =

3x - 2 x

60. f (x) =

12x - 24

62. f (x) =

x2 - 4 2x - 4x + 6 3

x -4 2

3x 2 - 6 x2 - 9

64. f (x) =

66. f (x) =

Motor Vehicle Thefts The following table lists the number

of motor vehicle thefts in the United States for each year from 1992 to 2007.

In Exercises 55 and 56, determine the horizontal asymptote for the graph of each rational function. 3x 2 - x 55. f (x) = 1 2 x + 5 2

t Ú 0

b. What temperature will the dessert approach as t : q ?

3x 2 - 5 2

,

a. Find the temperature of the dessert after it has been in the freezer for 1 hour.

In Exercises 53 and 54, determine the vertical asymptotes for the graph of each rational function. 53. f (x) =

60 t 2 + 2t + 1

x + 4 x - 2

Source: Federal Bureau of Investigation.

a. Find a cubic regression model for the data. Use x = 1 to rep-

resent 1992 and x = 16 to represent 2007.

4x 2 x2 + 1

b. Use the cubic model to predict the number of motor vehicle

thefts for the year 2012. Round to the nearest thousand.

x x - 1 3

c. Do you think the above prediction is reliable?

-x 3 + 6 x2

67. Average Cost of Skateboards The cost, in dollars, of pro-

ducing x skateboards is given by C(x) = 5.75x + 34,200 The average cost per skateboard is given by 5.75x + 34,200 C(x) C (x) = = x x

70.

Physiology One of Poiseuille’s laws states that the resist-

ance R encountered by blood flowing through a blood vessel is given by R(r) = C

L r4

where C is a positive constant determined by the viscosity of the blood, L is the length of the blood vessel, and r is its radius. r

a. Find the average cost per skateboard, to the nearest cent, of

producing 5000 and 50,000 skateboards.

L

b. What is the equation of the horizontal asymptote of the graph

a. Explain the meaning of R(r) : q as r : 0.

of C? Explain the significance of the horizontal asymptote as it relates to this application.

b. Explain the meaning of R(r) : 0 as r : q .

CHAPTER 3 TEST

331

CHAPTER 3 TEST 1. Use synthetic division to divide

15. Graph: P(x) = x3 - 6x2 + 9x + 1

(3x + 5x + 4x - 1) , (x + 2) 3

2

2. Use the Remainder Theorem to find P(-2) if

16. Graph: f (x) =

P(x) = - 3x + 7x + 2x - 5 3

x2 - 1 x - 2x - 3 2

2

3. Use the Factor Theorem to show that x - 1 is a factor of

x 4 - 4x 3 + 7x 2 - 6x + 2 4. Determine the far-left and far-right behavior of the graph of

P(x) = - 3x 3 + 2x 2 - 5x + 2. 5. Find the real zeros of P(x) = 3x 3 + 7x 2 - 6x. 6. Use the Intermediate Value Theorem to verify that

P(x) = 2x 3 - 3x 2 - x + 1 has a zero between 1 and 2. 7. Find the zeros of

P(x) = (x 2 - 4)2(2x - 3)(x + 1)3 and state the multiplicity of each. 8. Use the Rational Zero Theorem to list the possible rational

zeros of P(x) = 6x 3 - 3x 2 + 2x - 3

17. Graph: f (x) =

2x 2 + 2x + 1 x + 1

Burglaries The following table lists the number of

18.

burglaries in the United States for each year from 1992 to 2007. U. S. Burglaries, in thousands

Year

Burglaries

Year

Burglaries

1992

2980

2000

2051

1993

2835

2001

2117

1994

2713

2002

2151

1995

2594

2003

2155

1996

2506

2004

2144

1997

2461

2005

2155

1998

2333

2006

2184

1999

2101

2007

2179

Source: Federal Bureau of Investigation.

9. Use Descartes’ Rule of Signs to state the number of possible

positive and negative real zeros of P(x) = x4 - 3x 3 + 2x 2 - 5x + 1 10. Find the zeros of P(x) = 2x 3 - 3x 2 - 11x + 6. 11. Given that 2 + 3i is a zero of

P(x) = 6x4 - 5x 3 + 12x 2 + 207x + 130

a. Find a quartic regression model for the data. Use x = 1 to

represent 1992 and x = 16 to represent 2007.

b. Use the quartic model to predict the number of burglaries in

2010. Round to the nearest ten thousand. 19. Typing Speed The rational function

find the remaining zeros. w(t) =

12. Find all the zeros of

P(x) = x 5 - 6x4 + 14x 3 - 14x 2 + 5x 13. Find a polynomial function of smallest degree that has real

coefficients and zeros 1 + i, 3, and 0. 14. Find the vertical asymptotes and the horizontal asymptotes of

the graph of f (x) =

3x 2 - 2x + 1 x 2 - 5x + 6

70t + 120 , t + 40

t Ú 0

models Rene’s typing speed, in words per minute, after t hours of typing lessons. a. Find w(1), w(10), and w(20). Round to the nearest word per

minute. b. How many hours of typing lessons will be needed before

Rene can expect to type at 60 words per minute? c. What will Rene’s typing speed approach as t : q ?

332 20.

CHAPTER 3


Maximizing Volume You are to construct an open box

from a rectangular sheet of cardboard that measures 18 inches by 25 inches. To assemble the box, you make the four cuts shown in the figure below and then fold on the dashed lines. What value of x (to the nearest 0.01 inch) will produce a box with maximum volume? What is the maximum volume (to the nearest 0.1 cubic inch)? 25 in. cut x

x

cut

x

x

x

x

Fold on dashed lines

18 in.

cut

CUMULATIVE REVIEW EXERCISES 1. Write

3 + 4i in a + bi form. 1 - 2i

12. Determine the far-right behavior of the graph of

P(x) = - 3x4 - x2 + 7x - 6.

2. Use the quadratic formula to solve x2 - x - 1 = 0. 3. Solve: 12x + 5 - 1x - 1 = 2

13. Determine the relative maximum of the polynomial function

P(x) = - 3x3 - x2 + 4x - 1. Round to the nearest tenthousandth. 14. Use the Rational Zero Theorem to list all possible rational

4. Solve: ƒ x - 3 ƒ … 11

zeros of P(x) = 3x 4 - 4x 3 - 11x 2 + 16x - 4.

5. Find the distance between the points (2, 5) and (7, -11). 6. Explain how to use the graph of y = x to produce the graph 2

15. Use Descartes’ Rule of Signs to state the number of possible

positive and negative real zeros of P(x) = x 3 + x 2 + 2x + 4

of y = (x - 2) + 4. 2

7. Find the difference quotient for the function

P(x) = x - 2x - 3.

16. Find all zeros of P(x) = x 3 + x + 10.

2

8. Given f(x) = 2x 2 + 5x - 3 and g(x) = 4x - 7,

find ( f ⴰ g)(x).

17. Find a polynomial function of smallest degree that has real

coefficients and - 2 and 3 + i as zeros. 18. Write P(x) = x 3 - 2x 2 + 9x - 18 as a product of linear factors.

9. Given f(x) = x 3 - 2x + 7 and g(x) = x 2 - 3x - 4,

find ( f - g)(x).

10. Use synthetic division to divide (4x 4 - 2x 2 - 4x - 5) by

(x + 2).

19. Determine the vertical and horizontal asymptotes of the

graph of F(x) =

4x 2 x2 + x - 6

.

20. Find the equation of the slant asymptote for the graph of

11. Use the Remainder Theorem to find P(3) for

P(x) = 2x - 3x + 4x - 6. 4

2

F(x) =

x 3 + 4x 2 + 1 x2 + 4

.

CHAPTER

4

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

4.1 Inverse Functions 4.2 Exponential Functions and Their Applications 4.3 Logarithmic Functions and Their Applications

Lori Adamski Peek/Getty Images

4.4 Properties of Logarithms and Logarithmic Scales 4.5 Exponential and Logarithmic Equations 4.6 Exponential Growth and Decay 4.7 Modeling Data with Exponential and Logarithmic Functions

Applications of Exponential and Logarithmic Functions

Logarithmic functions can be used to scale very large (or very small) numbers so that they are easier to comprehend. In Exercise 67, page 379, a logarithmic function is used to determine the Richter scale magnitude of an earthquake. The photo to the right shows some of the damage caused by the San Francisco–Oakland earthquake, which struck during the pregame warm-up for the third game of the 1989 World Series. It was the first powerful earthquake in the United States to be broadcast live by a major television network. This earthquake measured 7.1 on the Richter scale and was responsible for 67 deaths. 333

AP Photo/Paul Sakuma

Exponential and logarithmic functions are often used to model data and make predictions. For instance in Exercise 19, page 412, an exponential function is used to model the price of a lift ticket at a ski resort for recent years. The function is also used to predict the price of a lift ticket in 2014.

334

CHAPTER 4


SECTION 4.1 Introduction to Inverse Functions Graphs of Inverse Functions Composition of a Function and Its Inverse Finding an Inverse Function

Inverse Functions Introduction to Inverse Functions Consider the “doubling function” f (x) = 2x that doubles every input. Some of the ordered pairs of this function are 5 10 e ( -4, -8), (-1.5, -3), (1, 2), a , b , (7, 14) f 3 3 Now consider the “halving function” g(x) =

1 x that takes one-half of every input. Some of 2

the ordered pairs of this function are e ( - 8, - 4), (- 3, -1.5), (2, 1), a

10 5 , b , (14, 7) f 3 3

Observe that the ordered pairs of g are the ordered pairs of f with the order of the coordinates reversed. The following two examples illustrate this concept.

Ordered pair: (5, 10)

f (a) = 2(a) = 2a Ordered pair: (a, 2a)

1 (2a) = a 2 Ordered pair: (2a, a) g(2a) =

m m

In this section, our primary interest is finding the inverse of a function; however, we can also find the inverse of a relation. Recall that a relation r is any set of ordered pairs. The inverse of r is the set of ordered pairs formed by reversing the order of the coordinates of the ordered pairs in r.

1 (10) = 5 2 Ordered pair: (10, 5) g(10) =

m m

Note

f (5) = 2(5) = 10

The function g is said to be the inverse function of f.

Definition of an Inverse Function If the ordered pairs of a function g are the ordered pairs of a function f with the order of the coordinates reversed, then g is the inverse function of f.

Consider a function f and its inverse function g. Because the ordered pairs of g are the ordered pairs of f with the order of the coordinates reversed, the domain of the inverse function g is the range of f, and the range of g is the domain of f. Not all functions have an inverse that is a function. Consider, for instance, the “square function” S(x) = x2. Some of the ordered pairs of S are 5( -3, 9), (-1, 1), (0, 0), (1, 1), (3, 9), (5, 25)6 If we reverse the coordinates of the ordered pairs, we have 5(9, -3), (1, - 1), (0, 0), (1, 1), (9, 3), (25, 5)6

4.1

(3, 9)

8 6

S(x) = x

2

335

This set of ordered pairs is not a function because there are ordered pairs, for instance (9, -3) and (9, 3), with the same first coordinate and different second coordinates. In this case, S has an inverse relation but not an inverse function. A graph of S is shown in Figure 4.1. Note that x = - 3 and x = 3 produce the same value of y. Thus the graph of S fails the horizontal line test; therefore, S is not a one-to-one function. This observation is used in the following theorem.

y 10 (−3, 9)

INVERSE FUNCTIONS

4 2

−4

−2

2

4

Figure 4.1

x

Condition for an Inverse Function A function f has an inverse function if and only if f is a one-to-one function.

Recall that increasing functions and decreasing functions are one-to-one functions. Thus we can state the following theorem. Horizontal Line Test See page 175.

Alternative Condition for an Inverse Function If f is an increasing function or a decreasing function, then f has an inverse function.

Question • Which of the functions graphed below has an inverse function? y

y

y

g

h

f x

Caution f -1(x) does not mean f(x) = 2x, f -1(x) = 1 1 = . f (x) 2x

1 . For f (x)

1 x, but 2

x

x

If a function g is the inverse of a function f, we usually denote the inverse function by f -1 rather than g. For the doubling and halving functions f and g discussed on page 334, we write f (x) = 2x

f -1(x) =

1 x 2

Graphs of Inverse Functions Because the coordinates of the ordered pairs of the inverse of a function f are the ordered pairs of f with the order of the coordinates reversed, we can use them to create a graph of f -1. Answer • The graph of f is the graph of an increasing function. Therefore, f is a one-to-one func-

tion and has an inverse function. The graph of h is the graph of a decreasing function. Therefore, h is a one-to-one function and has an inverse function. The graph of g is not the graph of a one-to-one function. g does not have an inverse function.

336

CHAPTER 4


EXAMPLE 1

Sketch the graph of f -1 given that f is the function shown in Figure 4.2.

y f 6 4 2 (−1, 0.5) −4

−2

Solution Because the graph of f passes through ( -1, 0.5), (0, 1), (1, 2), and (2, 4), the graph of f -1 must pass through (0.5, -1), (1, 0), (2, 1), and (4, 2). Plot the points and then draw a smooth curve through the points, as shown in Figure 4.3.

(2, 4) (1, 2) (0, 1) 2

4

Sketch the Graph of the Inverse of a Function

6

x y

−2

f y=x

6

−4

4 (1, 2) (0, 1) 2 (−1, 0.5)

Figure 4.2

−4

(2, 4)

f −1 (4, 2) (2, 1)

−2

2 4 (1, 0) (0.5, −1)

−2

x

6

−4

Figure 4.3

Try Exercise 10, page 342 Question • If f is a one-to-one function and f (4) = 5, what is f

-1

(5)?

The graph from the solution to Example 1 is shown again in Figure 4.4. Note that the graph of f -1 is symmetric to the graph of f with respect to the graph of y = x. If the graph were folded along the dashed line, the graph of f would lie on top of the graph of f -1. This is a characteristic of all graphs of functions and their inverses. In Figure 4.5, although S does not have an inverse that is a function, the graph of the inverse relation S -1 is symmetric to S with respect to the graph of y = x. y f y=x

6 4 (1, 2) (0, 1) 2 (−1, 0.5) −4

−2 −2

(2, 4)

f

2 4 (1, 0) (0.5, −1)

6

Answer • Because (4, 5) is an ordered pair of

(5) = 4.

2

(4, 2) (2, 1)

Figure 4.4

f

y=x

4 S

−4

-1

y

−1

x

−4

−2

2

4

x

−2 −4

S −1

Figure 4.5

f, (5, 4) must be an ordered pair of f -1. Therefore,

4.1

INVERSE FUNCTIONS

337

Composition of a Function and Its Inverse Observe the effect of forming the composition of f (x) = 2x and g(x) = f (x) = 2x Study tip If we think of a function as a machine, then the composition of inverse functions property can be represented as shown below. Take any input x for f. Use the output of f as the input for f -1. The result is the original input, x.

g(x) =

1 f 3g(x)4 = 2 c x d 2 f 3g(x)4 = x

• Replace x with g(x).

g3 f(x)4 =

1 x. 2

1 x 2 1 32x4 2

• Replace x with f (x).

g3 f(x)4 = x

This property of the composition of inverse functions always holds true. When taking the composition of inverse functions, the inverse function reverses the effect of the original function. For the two functions above, f doubles a number, and g halves a number. If you double a number and then take one-half of the result, you are back to the original number.

x

Composition of Inverse Functions Property If f is a one-to-one function, then f -1 is the inverse function of f if and only if

f (x)

( f ⴰ f -1)(x) = f 3 f -1(x)4 = x

f function x f

−1

function

and

( f -1 ⴰ f )(x) = f -13 f(x)4 = x

EXAMPLE 2

for all x in the domain of f -1

for all x in the domain of f.

Use the Composition of Inverse Functions Property

Use composition of functions to show that f -1(x) = 3x - 6 is the inverse function 1 of f(x) = x + 2. 3 Solution We must show that f 3 f -1(x)4 = x and f -13 f (x)4 = x. f(x) =

1 x + 2 3

f 3 f -1(x)4 =

1 33x - 64 + 2 3 f 3 f -1 (x)4 = x

f -1(x) = 3x - 6 1 f -13 f (x)4 = 3 c x + 2 d - 6 3 f -13 f(x)4 = x


Integrating Technology In the standard viewing window of a calculator, the distance between two tick marks on the x-axis is not equal to the distance between two tick marks on the y-axis. As a result, the graph of y = x does not appear to bisect the first and third quadrants. (continued)

338

CHAPTER 4


See Figure 4.6. This anomaly is important if a graphing calculator is being used to check whether two functions are inverses of one another. Because the graph of y = x does not appear to bisect the first and third quadrants, the graphs of f and f -1 will not 1 appear to be symmetric about the graph of y = x. The graphs of f (x) = x + 2 and 3 f -1(x) = 3x - 6 from Example 2 are shown in Figure 4.7. Notice that the graphs do not appear to be quite symmetric about the graph of y = x. 10

−10

10

Distances between tick marks are not equal.

10

− 10

10

−10

y = x in the standard viewing window

− 10

f , f −1, and y = x in the standard viewing window

Figure 4.6

To get a better view of a function and its inverse, it is necessary to use the SQUARE viewing window, as in Figure 4.8. In this window, the distance between two tick marks on the x-axis is equal to the distance between two tick marks on the y-axis.

Figure 4.7 10 Distances are equal. − 15

15

− 10

f , f −1, and y = x in a square viewing window Figure 4.8

Finding an Inverse Function If a one-to-one function f is defined by an equation, then we can use the following method to find the equation for f -1.

Study tip

Steps for Finding the Inverse of a Function

If the ordered pairs of f are given by (x, y), then the ordered pairs of f -1 are given by (y, x). That is, x and y are interchanged. This is the reason for Step 2 at the right.

To find the equation of the inverse f -1 of the one-to-one function f, follow these steps. 1. Substitute y for f(x). 2. Interchange x and y. 3. Solve, if possible, for y in terms of x. 4. Substitute f -1(x) for y.

4.1

EXAMPLE 3

INVERSE FUNCTIONS

339

Find the Inverse of a Function

Find the inverse of f (x) = 3x + 8. Solution f(x) y x x - 8 x - 8 3 1 8 x 3 3

= = = =

3x + 8 3x + 8 3y + 8 3y

• Replace f (x) with y. • Interchange x and y. • Solve for y.

= y = f -1(x)

The inverse function is given by f -1(x) =

• Replace y with f -1.

1 8 x - . 3 3


In the next example, we find the inverse of a rational function.

EXAMPLE 4

Find the Inverse of a Function

Find the inverse of f(x) =

2x + 1 , x Z 0. x

Solution f(x) = y = x = xy = xy - 2y = y(x - 2) = y = f -1(x) =

2x + 1 x 2x + 1 x 2y + 1 y 2y + 1 1 1 1 x - 2 1 ,x Z 2 x - 2

• Replace f (x) with y. • Interchange x and y. • Solve for y. • Factor the left side.

• Replace y with f -1.


The graph of f (x) = x 2 + 4x + 3 is shown in Figure 4.9a on the next page. The function f is not a one-to-one function and therefore does not have an inverse function.

340

CHAPTER 4


However, the function given by G(x) = x 2 + 4x + 3, shown in Figure 4.9b, for which the domain is restricted to 5x ƒ x Ú - 26, is a one-to-one function and has an inverse function G -1. This is shown in Example 5. y

f

−4

y

4

4

2

2

−2

2

4

x

−2

2

−2

−2

−4

−4

Figure 4.9a

EXAMPLE 5

−4

G

4

x

Figure 4.9b

Find the Inverse of a Function with a Restricted Domain

Find the inverse of G(x) = x 2 + 4x + 3, where the domain of G is 5x ƒ x Ú - 26. Solution

Recall The range of a function f is the domain of f -1, and the domain of f is the range of f -1.

G(x) = x 2 + 4x + 3 y = x 2 + 4x + 3

• Replace G(x) with y.

x = y + 4y + 3

• Interchange x and y.

x = ( y + 4y + 4) - 4 + 3

• Solve for y by completing the square of y 2 + 4y.

x = ( y + 2)2 - 1

• Factor.

2

2

x + 1 = ( y + 2)

2

y

−4

G

4

y=x

2

−1

−2

G 2

−2 −4

Figure 4.10

4

x

1x + 1 = 2( y + 2)2 1x + 1 = y + 2

• Add 1 to each side of the equation. • Take the square root of each side of the equation. • Recall that if a 2 = b, then

a = 1b.

1x + 1 - 2 = y

Because the domain of G is 5x ƒ x Ú - 26, the range of G -1 is 5y ƒ y Ú - 26. This means that we must choose the positive value of 1x + 1. Thus G -1(x) = 1x + 1 - 2. See Figure 4.10. Try Exercise 44, page 343

In Example 6, we use an inverse function to determine the wholesale price of a gold bracelet for which we know the retail price.

4.1

EXAMPLE 6

INVERSE FUNCTIONS

341


A merchant uses the function S(x) =

4 x + 100 3

to determine the retail selling price S, in dollars, of a gold bracelet for which she has paid a wholesale price of x dollars. a.

The merchant paid a wholesale price of $672 for a gold bracelet. Use S to determine the retail selling price of this bracelet.

b.

Find S -1 and use it to determine the merchant’s wholesale price for a gold bracelet that retails at $1596.

Solution a.

b.

4 (672) + 100 = 896 + 100 = 996 3 The merchant charges $996 for a bracelet that has a wholesale price of $672. S(672) =

To find S -1, begin by substituting y for S(x). S(x) =

4 x + 100 3

y =

4 x + 100 3

• Replace S(x) with y.

x =

4 y + 100 3

• Interchange x and y.

4 y 3

• Solve for y.

x - 100 =

3 (x - 100) = y 4 3 x - 75 = y 4 Using inverse notation, the above equation can be written as S -1(x) =

3 x - 75 4

Substitute 1596 for x to determine the wholesale price. S -1(1596) =

3 (1596) - 75 4

= 1197 - 75 = 1122 A gold bracelet that the merchant retails at $1596 has a wholesale price of $1122. Try Exercise 52, page 344

342

CHAPTER 4


Integrating Technology Some graphing utilities can be used to draw the graph of the inverse of a function without the user having to find the inverse function. For instance, Figure 4.11 shows the graph of f(x) = 0.1x 3 - 4. The graphs of f and f -1 are both shown in Figure 4.12, along with the graph of y = x. Note that the graph of f -1 is the reflection of the graph of f with respect to the graph of y = x. The display shown in Figure 4.12 was produced on a TI-83/TI-83 Plus/TI-84 Plus graphing calculator by using the DrawInv command, which is in the DRAW menu. 10

10 y=x f −1

− 15

− 15

15

15

f(x) = 0.1x3 − 4

f(x) = 0.1x3 − 4

− 10

− 10

Figure 4.11

Figure 4.12

EXERCISE SET 4.1 In Exercises 1 to 4, assume that the given function has an inverse function. 1. Given f (3) = 7,

11.

2. Given g(-3) = 5,

find f -1(7).

12.

y

find g-1(5).

8

8

(−6, 4) 4

4

(2, 6)

3. Given h ( -3) = - 4,

4. Given f

find h( -4).

5. If 3 is in the domain of f

-1

-1

(7) = 0,

−8

-1

(5)

b.

, find f 3 f -1(3)4.

−8

13.

8. The range of the inverse function f

-1

-1

is the

of f.

is the

−8

of f.

10.

y

8

8 (8, 6)

4

4

15.

(2, 3)

−4

4

4 x

x

4

8

x

4

8

x

y 8

8

8

−8

14.

4

4 −4

(−5, −6)

(0, −3)

−8

−4

−4

−4

−8

−8

16.

y

(0, 6)

−8

8

−4

In Exercises 9 to 16, draw the graph of the inverse relation. Is the inverse relation a function? y

x 8 (6, −5)

y

f -1(2)

7. The domain of the inverse function f

9.

4 −4

f (2) = 7, find the following.

a. f

−4

find f (0).

6. If f is a one-to-one function and f (0) = 5, f (1) = 2, and

(0, 3)

(−3, 0)

(− 4, 0) -1

y

y

8

8

4

4

(2, 3)

(−4, 0) −8 −4 (−8, −2) −4 −8

4

8

x

−8

−4

4 −4 −8

(6, −3)

8

x

−8

−4

4

8

x

−8

−4

−4

−4

−8

−8

4.1

In Exercises 17 to 26, use composition of functions to determine whether f and g are inverses of one another.

37. f (x) =

2x , x - 1

x Z 1

17. f (x) = 4x; g(x) =

x 4

38. f (x) =

x , x - 2

x Z 2

18. f (x) = 3x; g(x) =

1 3x

39. f (x) =

x - 1 , x + 1

x Z -1

40. f (x) =

2x - 1 , x + 3

19. f (x) = 4x - 1; g(x) =

20. f (x) =

1 1 x + 4 4

3 1 x - ; g(x) = 2x + 3 2 2

1 1 21. f (x) = - x - ; g(x) = - 2x + 1 2 2 22. f (x) = 3x + 2; g(x) =

2 1 x 3 3

5 5 23. f (x) = ; g(x) = + 3 x - 3 x 24. f (x) =

2x x ; g(x) = x - 1 x - 2 3

25. f (x) = x + 2; g(x) = 1x - 2 3

3

26. f (x) = (x + 5)3; g(x) = 1x - 5

41. f (x) = x 2 + 1,

x Ú 0

42. f (x) = x 2 - 4,

x Ú 0

43. f (x) = 1x - 2 ,

x Ú 2

44. f (x) = 14 - x ,

x … 4

45. f (x) = x 2 + 4x ,

x Ú -2

46. f (x) = x 2 - 6x ,

x … 3

47. f (x) = x 2 + 4x - 1 ,

x … -2

48. f (x) = x 2 - 6x + 1 ,

x Ú 3

49. Fahrenheit to Celsius The function

In Exercises 27 to 30, find the inverse of the function. If the function does not have an inverse function, write “no inverse function.” 28. 5(-5, 4), ( -2, 3), (0, 1), (3, 2), (7, 11)6

343

x Z -3

f (x) =

27. 5(-3, 1), (-2, 2), (1, 5), (4, -7)6

INVERSE FUNCTIONS

5 (x - 32) 9

is used to convert x degrees Fahrenheit to an equivalent Celsius temperature. Find f -1 and explain how it is used. 50. Retail Sales A clothing merchant uses the function

3 x + 18 2 to determine the retail selling price S, in dollars, of a winter coat for which she has paid a wholesale price of x dollars. S(x) =

29. 5(0, 1), (1, 2), (2, 4), (3, 8), (4, 16)6 30. 5(1, 0), (10, 1), (100, 2), (1000, 3), (10,000, 4)6

a. The merchant paid a wholesale price of $96 for a winter

In Exercises 31 to 48, find f the domain of f 1(x).

coat. Use S to determine the retail selling price she will charge for this coat.

1

(x). State any restrictions on

b. Find S -1 and use it to determine the merchant’s wholesale

31. f (x) = 2x + 4 32. f (x) = 4x - 8 33. f (x) = 3x - 7 34. f (x) = - 3x - 8 35. f (x) = - 2x + 5 36. f (x) = - x + 3

price for a coat that retails at $399. 51.

Fashion The function

s(x) = 2x + 24 can be used to convert a U.S. women’s shoe size into an Italian women’s shoe size. Determine the function s-1 (x) that can be used to convert an Italian women’s shoe size to its equivalent U.S. shoe size.

344 52.

53.

CHAPTER 4


Fashion The function K(x) = 1.3x - 4.7 converts a men’s shoe size in the United States to the equivalent shoe size in the United Kingdom. Determine the function K -1(x) that can be used to convert a U.K. men’s shoe size to its equivalent U.S. shoe size. Catering A catering service uses the function

c(x) =

300 + 12x x

57. The Birthday Problem A famous problem called the birth-

day problem goes like this: Suppose there is a randomly selected group of n people in a room. What is the probability that at least two of the people have a birthday on the same day of the year? It may surprise you that for a group of 23 people, the probability that at least two of the people share a birthday is about 50.7%. The following graph can be used to estimate shared birthday probabilities for 1 … n … 60. p

to determine the amount, in dollars, it charges per person for a sit-down dinner, where x is the number of people in attendance.

1.0 Probability that at least two people in the group share the same birthday

a. Find c(30) and explain what it represents. -1

b. Find c . c. Use c-1 to determine how many people attended a dinner

for which the cost per person was $15.00. 54.

p(n)

0.9

Landscaping A landscaping company uses the function

600 + 140x c(x) = x

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

to determine the amount, in dollars, it charges per tree to deliver and plant x palm trees.

0

10

40

50

60

n

b. Find c-1.

a. Use the graph of p to estimate p(10) and p(30).

c. Use c-1 to determine how many palm trees were delivered

b. Consider the function p with 1 … n … 60, as shown in the

and planted if the cost per tree was $160.

graph. Explain how you can tell that p has an inverse that is a function.

Compensation The monthly earnings E(s), in dollars, of

Grading A professor uses the function defined by the following table to determine the grade a student receives on a test. Does this grading function have an inverse function? Explain your answer. Grading Scale

Score

Grade

90–100

A

80–89

B

70–79

C

60–69

D

0–59

F

c.

Write a sentence that explains the meaning of p-1(0.223) in the context of this application.

58. Medication Level The function L shown in the following

graph models the level of pseudoephedrine hydrochloride, in milligrams, in the bloodstream of a patient t hours after 30 milligrams of the medication have been administered. Pseudoephedrine hydrochloride in the bloodstream (in milligrams)

a software sales executive are given by E(s) = 0.05s + 2500, where s is the value, in dollars, of the software sold by the executive during the month. Find E -1(s) and explain how the executive could use this function. 56.

30

Number of people in the group

a. Find c(5) and explain what it represents.

55.

20

L

L(t) = 0.03t 4 + 0.4t 3 − 7.3t 2 + 23.1t

20 16 12 8 4

0

1

2 3 4 Time (in hours)

t

4.1

a. Use the graph of L to estimate two different values of t

b. Use f

for which the pseudoephedrine hydrochloride levels are the same.

secret codes. Secret codes are often used to send messages over the Internet. By devising a code that is difficult to break, the sender hopes to prevent the messages from being read by an unauthorized person. In practice, complicated one-to-one functions and their inverses are used to encode and decode messages. The following procedure uses the simple function f (x) = 2x - 1 to illustrate the basic concepts that are involved. Assign to each letter of the alphabet, and a blank space, a two-digit numerical value, as shown below.

10 11 12 13 14 15 16

H I J K L M N

17 18 19 20 21 22 23

O P Q R S T U

24 25 26 27 28 29 30

V W X Y Z m

A B C D E F G

31 32 33 34 35 36

Note: A blank space is represented by the numerical value 36. Using these numerical values, the message MEET YOU AT NOON would be represented by 22 14 14 29 36 34 24 30 36 10 29 36 23 24 24 23 Let f (x) = 2x - 1 define a coding function. The above message can be encoded by finding f (22), f (14), f (14), f (29), f (36), f (34), f (24), Á , f (23), which yields 43 27 27 57 71 67 47 59 71 19 57 71 45 47 47 45 The inverse of f , which is x + 1 f -1(x) = 2 is used by the receiver of the message to decode the message. For instance, f -1(43) =

43 + 1 = 22 2

(x) to decode the message

c.

Explain why it is important to use a one-to-one function to encode a message.

60. Cryptography A friend is using the letter–number correspon-

dence in Exercise 59 and the coding function g(x) = 2x + 3. Your friend sends you the coded message 59 31 39 73 31 75 61 37 31 75 29 23 71 Use g-1(x) to decode this message. In Exercises 61 to 66, answer the question without finding the equation of the linear function. 61.

Suppose that f is a linear function, f (2) = 7, and f (5) = 12. If f (4) = c, then is c less than 7, between 7 and 12, or greater than 12? Explain your answer.

62.

Suppose that f is a linear function, f (1) = 13, and f (4) = 9. If f (3) = c, then is c less than 9, between 9 and 13, or greater than 13? Explain your answer.

63. Suppose that f is a linear function, f (2) = 3, and f (5) = 9.

Between which two numbers is f -1(6)?

64. Suppose that f is a linear function, f (5) = - 1, and f (9) = - 3.

Between which two numbers is f -1( -2)?

Only one-to-one functions have inverses that are functions. In Exercises 65 to 68, determine whether the given function is a one-to-one function. 65. f (x) = x 2 + 1 66. v(t) = 1 16 + t 67. F(x) = ƒ x ƒ + x 68. T(x) = ƒ x 2 - 6 ƒ ,

x Ú 0

69. Consider the linear function f (x) = mx + b, m Z 0. The

graph of f has a slope of m and a y-intercept of (0, b). What are the slope and y-intercept of the graph of f -1? 70. Find the inverse of f (x) = ax 2 + bx + c, a Z 0, x Ú -

which represents M, and f -1(27) =

345

49 33 47 45 27 71 33 47 43 27

b. Does L have an inverse that is a function? Explain. 59. Cryptology Cryptology is the study of making and breaking

-1

INVERSE FUNCTIONS

27 + 1 = 14 2

b . 2a

71. Use a graph of f (x) = - x + 3 to explain why f is its own

inverse.

which represents E. a. Use the above coding procedure to encode the message

DO YOUR HOMEWORK.

72. Use a graph of f (x) = 216 - x 2, with 0 … x … 4, to explain

why f is its own inverse.

346

CHAPTER 4


SECTION 4.2 Exponential Functions Graphs of Exponential Functions Natural Exponential Function

Exponential Functions and Their Applications PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A26.

PS1. Evaluate: 23 [P.2] PS2. Evaluate: 3-4 [P.2] PS3. Evaluate:

22 + 2-2 [P.2/P.5] 2

PS4. Evaluate:

32 - 3-2 [P.2/P.5] 2

PS5. Evaluate f(x) = 10x for x = - 1, 0, 1, and 2. [P.2] x

PS6. Evaluate f(x) = a b for x = - 1, 0, 1, and 2. [P.2]

1 2

Exponential Functions

Daily parking fee

$40

When an airport parking facility opened in 1968, it charged $0.75 for all day parking. Since then it has doubled its daily parking fee every 8 years as shown in the following table.

$30 $20

Table 4.1 $10

Year

1968

1976

1984

1992

2000

2008

Daily parking fee

$0.75

$1.50

$3.00

$6.00

$12.00

$24.00

1968 1976 1984 1992 2000 2008 Year

In Figure 4.13, we have plotted the data in the above table and modeled the upward trend in the parking fee by a smooth curve. This model is based on an exponential function, which is one of the major topics of this chapter. The effectiveness of a drug, which is used for sedation during a surgical procedure, depends on the concentration of the drug in the patient. Through natural body chemistry, the amount of this drug in the body decreases over time. The graph in Figure 4.14 models this decrease. This model is another example of an exponential model.

Figure 4.13

Concentration (in mg>L)

600

400

Definition of an Exponential Function 200

The exponential function with base b is defined by f(x) = b x 1

2 3 4 Time (in hours)

5

where b 7 0, b Z 1, and x is a real number.

Figure 4.14

The base b of f(x) = b x is required to be positive. If the base were a negative number, the value of the function would be a complex number for some values of x. For instance, if

4.2

EXPONENTIAL FUNCTIONS AND THEIR APPLICATIONS

347

1 1 , then f a b = ( - 4)1>2 = 2i. To avoid complex number values of a 2 2 function, the base of any exponential function must be a positive number. Also, b is defined such that b Z 1 because f(x) = 1x = 1 is a constant function. You may have noticed that in the definition of an exponential function the exponent x is a real number. We have already worked with expressions of the form bx, where b 0 and x is a rational number. For instance, b = - 4 and x =

23 = 2 # 2 # 2 = 8

3 272>3 = ( 127)2 = 32 = 9 5

320.4 = 322>5 = ( 132)2 = 22 = 4 To extend the meaning of bx to real numbers, we need to give meaning to bx when x is an irrational number. For example, what is the meaning of 5p? To completely answer this question requires concepts from calculus. However, for our purposes, we can think of 5p as the unique real number that is approached by 5x as x takes on ever closer rational number approximations of p. For instance, each successive number in the following list is a closer approximation of 5p than the number to its left. 53, 53.1, 53.14, 53.142, 53.1416, 53.14159, 53.141593, 53.1415927, 53.14159265, Á A calculator can be used to show that 5p L 156.9925453. A computer algebra system, such as Mathematica, can produce even closer decimal approximations of 5p by using closer rational-number approximations of p. For instance, if you use 3.1415926535897932385 as your approximation of p, then Mathematica produces 156.9925453088659076 as an approximation of 5p. In a similar manner, we can think of 713 as the number that is approached by ever closer rational-number approximations of 13. For instance, each successive number in the following list is a closer approximation of 713 than the number to its left. 71, 71.7, 71.73, 71.732, 71.7321, 71.73205, 71.732051, 71.7320508, 71.73205081, Á A calculator can be used to show that 713 L 29.0906043. It can be shown that the properties of rational-number exponents, as stated in Section P.2, hold for real exponents.

EXAMPLE 1

Evaluate an Exponential Function

Evaluate f (x) = 3x at x = 2, x = - 4, and x = p. Solution f(2) = 32 = 9 1 1 = 81 34 f(p) = 3p L 33.1415927 L 31.54428

f(-4) = 3-4 =

• Evaluate with the aid of a calculator.


Graphs of Exponential Functions The graph of f (x) = 2x is shown in Figure 4.15 on page 348. The coordinates of some of the points on the curve are given in Table 4.2.

348

CHAPTER 4


Table 4.2 y

f(x) = 2

x

x

8 6 The graph approaches the negative x-axis, but it does not intersect the axis.

y f (x) 2x

-2

f ( - 2) = 2-2 =

1 4

1 a -2, b 4

-1

f ( - 1) = 2-1 =

1 2

1 a -1, b 2

0

f (0) = 20 = 1

(0, 1)

1

f (1) = 2 = 2

(1, 2)

2

f (2) = 2 = 4

(2, 4)

3

f (3) = 2 = 8

(3, 8)

4 2

− 4 −3 −2 −1

1

2

3

4

(x, y)

x

Figure 4.15

1 2 3

Note the following properties of the graph of the exponential function f(x) = 2x. The y-intercept is (0, 1). The graph passes through (1, 2). As x decreases without bound (that is, as x : - q ), f(x) : 0. The graph is a smooth, continuous increasing curve. Now consider the graph of an exponential function for which the base is between 0 x 1 and 1. The graph of f(x) = a b is shown in Figure 4.16. The coordinates of some of the 2 points on the curve are given in Table 4.3. Table 4.3 y

x

x

1 y f (x) a b 2

(x, y)

-3

1 -3 f ( - 3) = a b = 8 2

( -3, 8)

-2

1 f ( - 2) = a b 2

= 4

( -2, 4)

-1

1 -1 f ( - 1) = a b = 2 2

( -1, 2)

8 6 The graph approaches the positive x-axis, but it does not intersect the axis.

4 f(x) =

1 x 2

()

2

−4 −3 −2 −1

1

2

3 4

x

Figure 4.16

-2

0

0

1 f (0) = a b = 1 2

1

1 1 f (1) = a b = 2 2

2

1 1 f (2) = a b = 2 4 x

1 Note the following properties of the graph of f(x) = a b . 2 The y-intercept is (0, 1). 1 The graph passes through a 1, b . 2

(0, 1)

1

1 a1, b 2

2

1 a2, b 4

4.2

349


As x increases without bound (that is, as x : q ), f(x) : 0. The graph is a smooth, continuous decreasing curve. The basic properties of exponential functions are provided in the following summary.

Properties of f (x) bx For positive real numbers b, b Z 1, the exponential function defined by f (x) = b x has the following properties: The function f is a one-to-one function. It has the set of real numbers as its domain and the set of positive real numbers as its range. The graph of f is a smooth, continuous curve with a y-intercept of (0, 1), and the graph passes through (1, b). If b 7 1, f is an increasing function and the graph of f is asymptotic to the negative x-axis. [As x : q , f(x) : q , and as x : - q , f (x) : 0.] See Figure 4.17a. If 0 6 b 6 1, f is a decreasing function and the graph of f is asymptotic to the positive x-axis. [As x : - q , f(x) : q, and as x : q, f(x) : 0.] See Figure 4.17b. y

y

(− 2, b ) −2

(2, b ) 2

(1, b ) 1

(−2, b ) (−1, b ) (−3, b ) −2

−1

−3

−2

−1

(1, b ) (2, b ) 1

(0, 1)

(0, 1) 2

a. f (x) = b x, b > 1

(−1, b ) x

−2

2

(3, b ) 3

x

2

b. f (x) = b x, 0 < b < 1 Figure 4.17

x

Question • What is the x-intercept of the graph of f (x) = a b ?

1 3

EXAMPLE 2

Graph an Exponential Function

3 x Graph: g(x) = a b 4 Solution 3 Because the base is less than 1, we know that the graph of g is a decreasing function 4 that is asymptotic to the positive x-axis. The y-intercept of the graph is the point (0, 1), (continued) Answer • The graph does not have an x-intercept. As x increases without bound, the graph

approaches, but does not intersect, the x-axis.

350

CHAPTER 4


3 and the graph passes through a 1, b . Plot a few additional points (see Table 4.4), and 4 then draw a smooth curve through the points, as in Figure 4.18. Table 4.4

x

3 x y g (x) a b 4

-3

64 3 -3 a b = 4 27 -2

3 a b 4

-2

=

(x, y)

16 9

3 -1 4 a b = 4 3

-1

a -2,

16 b 9

a2,

y 4 g(x) = 2 −4

3 27 a b = 4 64

−2

2

() 3 4

4

x

x

−2

9 b 16

−4

27 a 3, b 64

3

3

64 b 27

4 a -1, b 3

3 2 9 a b = 4 16

2

a -3,

Figure 4.18


Consider the functions F(x) = 2x - 3 and G(x) = 2x - 3. You can construct the graphs of these functions by plotting points; however, it is easier to construct their graphs by using translations of the graph of f(x) = 2x, as shown in Example 3.

EXAMPLE 3

Use a Translation to Produce a Graph

a.

Explain how to use the graph of f (x) = 2x to produce the graph of F(x) = 2x - 3.

b.

Explain how to use the graph of f (x) = 2x to produce the graph of G(x) = 2x - 3.

Solution a. F(x) = 2x - 3 = f(x) - 3. The graph of F is a vertical translation of f down 3 units, as shown in Figure 4.19. b.

G(x) = 2x - 3 = f(x - 3). The graph of G is a horizontal translation of f to the right 3 units, as shown in Figure 4.20.

f (x) = 2 x

y

y

6

6

4

4 f (x) = 2 x

2

2 G(x) = 2 x − 3

−4

−2

2

4

6

x

−4

−2

2

F(x) = 2 x − 3

Figure 4.19


Figure 4.20

4

6

x

4.2


351

The graphs of some functions can be constructed by stretching, compressing, or reflecting the graph of an exponential function.

EXAMPLE 4

Use Stretching or Reflecting Procedures to Produce a Graph

a.

Explain how to use the graph of f (x) = 2x to produce the graph of M(x) = 2(2x).

b.

Explain how to use the graph of f(x) = 2x to produce the graph of N(x) = 2-x.

Solution a. M(x) = 2(2x) = 2 f(x). The graph of M is a vertical stretching of f away from the x-axis by a factor of 2, as shown in Figure 4.21. (Note: If (x, y) is a point on the graph of f (x) = 2x, then (x, 2y) is a point on the graph of M.)

Math Matters

b.

N(x) = 2-x = f( - x). The graph of N is the graph of f reflected across the y-axis, as shown in Figure 4.22. (Note: If (x, y) is a point on the graph of f(x) = 2x, then (-x, y) is a point on the graph of N.) y

y

Bettmann/CORBIS

8 6 f(x) = 2

Source: wikiquote.org/wiki/leonhard_ Euler.

6

−x

x

f(x) = 2 x

N(x) = 2

4

4

2

2

x

M(x) = 2(2 )

Leonhard Euler (1707–1783)

Some mathematicians consider Euler to be the greatest mathematician of all time. He certainly was the most prolific writer of mathematics of all time. He made substantial contributions in the areas of number theory, geometry, calculus, differential equations, differential geometry, topology, complex variables, and analysis, to name but a few. Euler was the first to introduce many of the mathematical notations that we use today. For instance, he introduced the symbol i for the square root of -1, the symbol p for pi, the functional notation f (x), and the letter e for the base of the natural exponential function. Euler’s computational skills were truly amazing. The mathematician François Arago remarked, “Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.”

8

−4

−2

2

4

x

−4

Figure 4.21

−2

2

4

x

Figure 4.22


Natural Exponential Function The irrational number p is often used in applications that involve circles. Another irrational number, denoted by the letter e, is useful in many applications that involve growth or decay.

Definition of e The letter e represents the number that a1 +

1 n b n

approaches as n increases without bound.

The letter e was chosen in honor of the Swiss mathematician Leonhard Euler. He was able n 1 to compute the value of e to several decimal places by evaluating a1 + b for large n values of n, as shown in Table 4.5 on page 352.

352

CHAPTER 4


Table 4.5

Value of n

1 n b n

Value of a1

1

2

10

2.59374246

100

2.704813829

1000

2.716923932

10,000

2.718145927

100,000

2.718268237

1,000,000

2.718280469

10,000,000

2.718281693

The value of e accurate to eight decimal places is 2.71828183. The base of an exponential function can be any positive real number other than 1. The number 10 is a convenient base to use for some situations, but we will see that the number e is often the best base to use in real-life applications. The exponential function with e as the base is known as the natural exponential function.

Definition of the Natural Exponential Function For all real numbers x, the function defined by f(x) = e x is called the natural exponential function.

A calculator can be used to evaluate e x for specific values of x. For instance, e 2 L 7.389056,

Integrating Technology The graph of f (x) = e x below was produced on a TI-83/TI-83 Plus/ TI-84 Plus graphing calculator by entering e x in the Y = menu. Plot1 Plot2 Plot3 \Y 1 = e^(X ) \Y 2 = 6 \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 = − 4.7

f (x) = e

4.7 −1

e-1.4 L 0.246597

On a TI-83/TI-83 Plus/TI-84 Plus calculator, the e x function is located above the LN key. To graph f(x) = e x, use a calculator to find the range values for a few domain values. The range values in Table 4.6 have been rounded to the nearest tenth. Table 4.6

x f(x) e

x

e 3.5 L 33.115452, and

x

-2

-1

0

1

2

0.1

0.4

1.0

2.7

7.4

Plot the points given in Table 4.6, and then connect the points with a smooth curve. Because e 7 1, we know that the graph is an increasing function. To the far left, the graph will approach the x-axis. The y-intercept is (0, 1). See Figure 4.23. Note in Figure 4.24 how the graph of f(x) = e x compares with the graphs of g(x) = 2x and h(x) = 3x. You may have anticipated that the graph of f (x) = e x would lie between the two other graphs because e is between 2 and 3.

4.2


353

y h (x) = 3 x

y

g(x) = 2 x

f(x) = e x

8

3

6

f(x) = e x

4 1 2

−4

−3

−2 −1

1

2

3

4

x

−1

1

Figure 4.23

2

x

Figure 4.24

Many applications can be modeled effectively by functions that involve an exponential function. For instance, in Example 5 we use a function that involves an exponential function to model the temperature of a cup of coffee.

EXAMPLE 5

Use a Mathematical Model

A cup of coffee is heated to 160°F and placed in a room that maintains a temperature of 70°F. The temperature T of the coffee, in degrees Fahrenheit, after t minutes is given by T = 70 + 90e-0.0485t a. b.

Find the temperature of the coffee, to the nearest degree, 20 minutes after it is placed in the room. Use a graphing utility to determine when the temperature of the coffee will reach 90°F.

Solution a. T = 70 + 90e-0.0485t # = 70 + 90e-0.0485 (20) L 70 + 34.1 L 104.1

• Substitute 20 for t.

After 20 minutes the temperature of the coffee is about 104°F. Note In Example 5b, we use a graphing utility to solve the equation 90 = 70 + 90e-0.0485t. Analytic methods of solving this type of equation without the use of a graphing utility will be developed in Section 4.5.

b.

Graph T = 70 + 90e-0.0485t and T = 90. See the following figure. 170

0

Intersection X=31.011905 Y=90

− 40

Xscl = 5

45

Yscl = 20

The graphs intersect near (31.01, 90). It takes the coffee about 31 minutes to cool to 90°F. Try Exercise 48, page 356

354

CHAPTER 4


EXAMPLE 6

Use a Mathematical Model

The weekly revenue R, in dollars, from the sale of a product varies with time according to the function R(x) =

1760 8 + 14e-0.03x

where x is the number of weeks that have passed since the product was put on the market. What will the weekly revenue approach as time goes by? Solution Method 1 Use a graphing utility to graph R(x), and use the TRACE feature to see what happens to the revenue as the time increases. The graph on the right shows that as the weeks go by, the weekly revenue will increase and approach $220 per week.

280

Y1=1760/(8+14e^(-.03X))

0

400

X=400

Method 2 Write the revenue function in the following form. R(x) =

1760 14 8 + 0.03x e

− 40

Xscl = 100

• 14e- 0.03x =

Y=219.99763 Yscl = 100

14 0.03x

e

As x increases without bound, e0.03x increases without bound, and the fraction 1760 14 approaches 0. Therefore, as x : q , R(x) : = 220. Both methods 0.03x 8 + 0 e indicate that, as the number of weeks increases, the revenue approaches $220 per week. Try Exercise 54, page 357

EXERCISE SET 4.2 In Exercises 1 to 8, evaluate the exponential function for the given x-values. 1. f (x) = 3x; x = 0 and x = 4 2. f (x) = 5x; x = 3 and x = - 2

x

7. j(x) = a b ; x = - 2 and x = 4

1 2

x

8. j(x) = a b ; x = - 1 and x = 5

1 4

3. g(x) = 10x; x = - 2 and x = 3 4. g(x) = 4x; x = 0 and x = - 1 x

5. h(x) = a b ; x = 2 and x = - 3

3 2

x

6. h(x) = a b ; x = - 1 and x = 3

2 5

In Exercises 9 to 14, use a calculator to evaluate the exponential function for the given x-value. Round to the nearest hundredth. 9. f (x) = 2x; x = 3.2

10. f (x) = 3x; x = - 1.5

11. g(x) = ex; x = 2.2

12. g(x) = ex; x = - 1.3

13. h(x) = 5x; x = 12

14. h(x) = 0.5x; x = p

4.2

In Exercises 15 and 16, examine the four functions and the graphs labeled a, b, c, and d. For each graph, determine which function has been graphed. g(x) = 1 + 5-x

15. f (x) = 5x

h(x) = 5x + 3 a.

26. f (x) = 4x, F(x) = 4x - 3 b.

y

8

8

27. f (x) = 10x, F(x) = 10x-2

4

4

28. f (x) = 6x, F(x) = 6x + 5

−4

4

c.

x

−4

d.

y

4

8

4

4

4

x

x-2

1 k(x) = 3 a b 4

4

4

d.

x

y

4

4

x

19. f (x) = 10

−4

4

20. f (x) = 6

x

21. f (x) = a b

x

22. f (x) = a b

x

2 24. f (x) = a b 3

3 2

1 23. f (x) = a b 3

x

x

1 2 ca b d 2 3

x

37. f (x) = 0.5x, F (x) = 3 + 0.5-x

x

In Exercises 39 to 46, use a graphing utility to graph each function. If the function has a horizontal asymptote, state the equation of the horizontal asymptote. 39. f (x) =

3x + 3-x 2

40. f (x) = 4 # 3-x

41. f (x) =

e x - e-x 2

42. f (x) =

18. f (x) = 4x x

1 3

38. f (x) = 0.5x, F (x) = 3(0.5x + 2 ) - 1

In Exercises 17 to 24, sketch the graph of each function. 17. f (x) = 3x

x

36. f (x) = 2x, F (x) = - (2 - x ) 4

8

4

1 3

35. f (x) = 2x, F (x) = - (2x-4)

−4

8

−4

x

5 2

34. f (x) = e x, F(x) = e x-3 + 1

y

4

y

x

33. f (x) = e x, F(x) = e-x + 2

8

x

-x

31. f (x) = a b , F(x) = 2c a b d

2 3

b.

4

3 2

32. f (x) = a b , F(x) =

8

c.

x

x

y

3 2

5 2

-x

1 g(x) = a b 4

1 h(x) = a b 4

x

29. f (x) = a b , F(x) = a b

30. f (x) = a b , F(x) = - c a b d

−4

x

1 16. f(x) = a b 4

−4

x

y

8

−4

a.

In Exercises 25 to 38, explain how to use the graph of the first function f to produce the graph of the second function F. 25. f (x) = 3x, F(x) = 3x + 2

k(x) = 5x + 3

y

355


5 2

x

x

43. f (x) = - e(x - 4) 45. f (x) =

x Ú 0

e x + e-x 2

44. f (x) = 0.5e-x

10 1 + 0.4e

2

-0.5x

,

46. f (x) =

x Ú 0

10 1 + 1.5e-0.5x

,

356


E. Coli Infection Escherichia coli (E. coli) is a bacterium that can reproduce at an exponential rate. The E. coli reproduce by dividing. A small number of E. coli bacteria in the large intestine of a human can trigger a serious infection within a few hours. Consider a particular E. coli infection that starts with 100 E. coli bacteria. Each bacterium splits into two parts every half hour. Assuming none of the bacteria die, the size of the E. coli population after t hours is given by P(t) = 100 # 22t, where 0 … t … 16.

b. What will the monthly income from the product approach

as the time increases without bound? 51.

Photochromatic Eyeglass Lenses Photochromatic eye-

glass lenses contain molecules of silver chloride or silver halide. These molecules are transparent in the absence of ultraviolet (UV) rays. UV rays are normally absent in artificial lighting. However, when the lenses are exposed to UV rays, as in direct sunlight, the molecules take on a new molecular structure, which causes the lenses to darken. The number of molecules affected varies with the intensity of the UV rays. The intensity of UV rays is measured using a scale called the UV index. On this scale, a value near 0 indicates a low UV intensity and a value near 10 indicates a high UV intensity. For the photochromatic lenses shown below, the function P(x) = (0.9)x models the transparency P of the lenses as a function of the UV index x.

Charles O’Rear/CORBIS

47.

CHAPTER 4

a. Find P(3) and P(6).

C PHOTOCHROMATIC PHOTOCHROMATIC

b. Use a graphing utility to find the time, to the nearest tenth

of an hour, it takes for the E. coli population to number 1 billion.

C PHOTOCHROMATIC PHOTOCHROMATIC

48.

UV index, 0 Lens transparency, 100%

UV index, 5 Lens transparency, 59.0%

Medication in the Bloodstream The exponential function

A(t) = 200e-0.014t C PHOTOCHROMATIC PHOTOCHROMATIC

gives the amount of medication, in milligrams, in a patient’s bloodstream t minutes after the medication has been injected into the patient’s bloodstream.

UV index, 9 Lens transparency, 38.7%

a. Find the transparency of these lenses, to the nearest tenth of

a. Find the amount of medication, to the nearest milligram, in

a percent, when they are exposed to light rays with a UV index of 3.5.

the patient’s bloodstream after 45 minutes. b. Use a graphing utility to determine how long it will take, to

b. What is the UV index of light rays that cause these pho-

the nearest minute, for the amount of medication in the patient’s bloodstream to reach 50 milligrams.

tochromatic lenses to have a transparency of 45%? Round to the nearest tenth.

49. Demand for a Product The demand d for a specific product,

in items per month, is given by

52.

percentage of a certain radiation that can penetrate x millimeters of lead shielding is given by I(x) = 100e-1.5x.

d(p) = 880e-0.18p

a. What percentage of radiation, to the nearest tenth of a percent,

where p is the price, in dollars, of the product.

will penetrate a lead shield that is 1 millimeter thick?

a. What will be the monthly demand, to the nearest unit, when

the price of the product is $10 and when the price is $18?

b. How many millimeters of lead shielding are required so that

less than 0.05% of the radiation penetrates the shielding? Round to the nearest millimeter.

b. What will happen to the demand as the price increases with-

out bound? 50. Sales The monthly income I, in dollars, from a new product is

given by I(t) = 8600 - 5500e-0.005t where t is the time, in months, since the product was first put on the market. a. What was the monthly income after the 10th month and

after the 100th month?

Radiation Lead shielding is used to contain radiation. The

53.

The Pay It Forward Model In the movie Pay It Forward,

Trevor McKinney, played by Haley Joel Osment, is given a school assignment to “think of an idea to change the world— and then put it into action.” In response to this assignment, Trevor develops a pay it forward project. In this project, anyone who benefits from another person’s good deed must do a good deed for three additional people. Each of these three people is then obligated to do a good deed for another three people, and so on.

4.2

55.

357

A Temperature Model A cup of coffee is heated to

180F and placed in a room that maintains a temperature of 65F. The temperature of the coffee after t minutes is given by T(t) = 65 + 115e-0.042t.

David James/Getty Images

The following diagram shows the number of people who have been a beneficiary of a good deed after one round and after two rounds of this project.


a. Find the temperature, to the nearest degree, of the coffee

10 minutes after it is placed in the room. b. Use a graphing utility to determine when, to the nearest

tenth of a minute, the temperature of the coffee will reach 100F. 56.

Three beneficiaries after one round

Intensity of Light The percent I(x) of the original intensity of light striking the surface of a lake that is available x feet below the surface of the lake is given by the equation I(x) = 100e-0.95x. a. What percentage of the light, to the nearest tenth of a percent,

is available 2 feet below the surface of the lake? A total of 12 beneficiaries after two rounds (3 + 9 = 12)

b. At what depth, to the nearest hundredth of a foot, is the

intensity of the light one-half the intensity at the surface?

A mathematical model for the number of pay-it-forward 3n + 1 - 3 beneficiaries after n rounds is given by B(n) = . Use 2 this model to determine

57.

Musical Scales Starting on the left side of a standard

88-key piano, the frequency, in vibrations per second, of the nth note is given by f (n) = (27.5)2(n-1)/12.

a. the number of beneficiaries after 5 rounds and after

Symphony #9

by L. von Beethoven

10 rounds. Assume that no person is a beneficiary of more than one good deed. b. how many rounds are required to produce at least 2 million

beneficiaries. 54.

Fish Population The number of bass in a lake is given by

P(t) =

3600

Middle C

D

E

1 + 7e-0.05t

where t is the number of months that have passed since the lake was stocked with bass.

a. Using this formula, determine the frequency, to the nearest

hundredth of a vibration per second, of middle C, key number 40 on an 88-key piano. b. Is the difference in frequency between middle C (key num-

ber 40) and D (key number 42) the same as the difference in frequency between D (key number 42) and E (key number 44)? Explain.

In Exercises 58 and 59, verify that the given function is odd or even as requested. a. How many bass were in the lake immediately after it was

stocked?

58. Verify that f (x) =

e x + e-x is an even function. 2

59. Verify that f (x) =

e x - e-x is an odd function. 2

b. How many bass were in the lake 1 year after the lake was

stocked? Round to the nearest bass. c. What will happen to the bass population as t increases with-

out bound?

358

CHAPTER 4


In Exercises 60 and 61, draw the graphs as indicated.

65. f (x) = 2e x - e-x

64. f (x) = 21 - e x

60. Graph g(x) = 10 , and then sketch the graph of g reflected x

across the line given by y = x.

Average Height Explain why the graph of

66.

61. Graph f (x) = e x, and then sketch the graph of f reflected

f (x) =

across the line given by y = x.

can be produced by plotting the average height of g(x) = ex and h(x) = e-x for each value of x.

In Exercises 62 to 65, determine the domain of the given function. Write the domain using interval notation. 62. f (x) =

e x - e-x e x + e-x

63. f (x) =

SECTION 4.3 Logarithmic Functions Graphs of Logarithmic Functions Domains of Logarithmic Functions Common and Natural Logarithms

e x + e-x 2

e ƒ xƒ 1 + ex

Logarithmic Functions and Their Applications PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A27.

PS1. If 2x = 16, determine the value of x. [4.2] PS2. If 3 - x =

1 , determine the value of x. [4.2] 27

PS3. If x4 = 625, determine the value of x. [4.2] PS4. Find the inverse of f (x) =

2x . [4.1] x + 3

PS5. State the domain of g(x) = 1x - 2. [2.2] PS6. If the range of h(x) is the set of all positive real numbers, then what is the domain

of h-1(x)? [4.1]

Logarithmic Functions Every exponential function of the form g(x) = b x is a one-to-one function and therefore has an inverse function. Sometimes we can determine the inverse of a function represented by an equation by interchanging the variables of its equation and then solving for the dependent variable. If we attempt to use this procedure for g(x) = b x, we obtain g(x) = b x y = bx x = by

• Interchange the variables.

None of our previous methods can be used to solve the equation x = b y for the exponent y. Thus we need to develop a new procedure. One method would be to merely write y = the power of b that produces x Although this would work, it is not concise. We need a compact notation to represent “y is the power of b that produces x.” This more compact notation is given in the following definition.

4.3

Math Matters

LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS

359

Definition of a Logarithm and a Logarithmic Function

Logarithms were developed by John Napier (1550–1617) as a means of simplifying the calculations of astronomers. One of his ideas was to devise a method by which the product of two numbers could be determined by performing an addition.

If x 7 0 and b is a positive constant (b Z 1), then y = logb x

if and only if

by = x

The notation logb x is read “the logarithm (or log) base b of x.” The function defined by f (x) = logb x is a logarithmic function with base b. This function is the inverse of the exponential function g(x) = b x.

It is essential to remember that f (x) = logb x is the inverse function of g(x) = b x. Because these functions are inverses and because functions that are inverses have the property that f ( g(x)) = x and g( f(x)) = x, we have the following important relationships.

Composition of Logarithmic and Exponential Functions Let g(x) = b x and f (x) = logb x (x 7 0, b 7 0, b Z 1). Then g( f (x)) = b log b x = x

f( g(x)) = logb b x = x

and

As an example of these relationships, let g(x) = 2x and f(x) = log2 x. Then 2log2 x = x

log2 2x = x

and

The equations y = logb x

and

by = x

are different ways of expressing the same concept.

Definition of Exponential Form and Logarithmic Form The exponential form of y = logb x is b y = x. The logarithmic form of b y = x is y = logb x.

These concepts are illustrated in the next two examples.

EXAMPLE 1

Change from Logarithmic to Exponential Form

Write each equation in its exponential form. 3 = log 2 8

b.

2 = log10(x + 5)

Study tip

a.

c.

The notation logb x replaces the phrase “the power of b that produces x.” For instance, “3 is the power of 2 that produces 8” is abbreviated 3 = log2 8. In your work with logarithms, remember that a logarithm is an exponent.

Solution Use the definition y = logb x if and only if b y = x.

log e x = 4

d.

log b b 3 = 3

Logarithms are exponents. a.

3 = log 2 8

if and only if

23 = 8

Base (continued)

360

CHAPTER 4


b.

2 = log10(x + 5) if and only if 102 = x + 5.

c.

log e x = 4 if and only if e 4 = x.

d.

log b b3 = 3 if and only if b 3 = b 3. Try Exercise 4, page 366

EXAMPLE 2

Change from Exponential to Logarithmic Form

Write each equation in its logarithmic form. a.

32 = 9

b.

53 = x

c.

ab = c

d.

blogb 5 = 5

Solution The logarithmic form of b y = x is y = log b x. Exponent a.

32 = 9

if and only if

2 = log 3 9

Base b.

53 = x if and only if 3 = log 5 x.

c.

ab = c if and only if b = log a c.

d.

blogb 5 = 5 if and only if log b 5 = log b 5. Try Exercise 14, page 366

The definition of a logarithm and the definition of an inverse function can be used to establish many properties of logarithms. For instance, log b b = 1 because b = b1. log b 1 = 0 because 1 = b0. log b(b x ) = x because b x = b x. blogb x = x because f(x) = log b x and g(x) = b x are inverse functions. Thus g3 f (x)4 = x. We will refer to the preceding properties as the basic logarithmic properties.

Basic Logarithmic Properties 1.

log b b = 1

EXAMPLE 3

2.

log b 1 = 0

3.

log b(b x ) = x

4.

Apply the Basic Logarithmic Properties

Evaluate each of the following logarithms. a.

log 8 1

b.

log 5 5

c.

log 2(24 )

d.

3log 3 7

blogb x = x

4.3


361

Solution a. By property 2, log8 1 = 0. b.

By property 1, log 5 5 = 1.

c.

By property 3, log 2(24) = 4.

d. By property 4, 3log 3 7 = 7. Try Exercise 32, page 366

Some logarithms can be evaluated just by remembering that a logarithm is an exponent. For instance, log5 25 equals 2 because the base 5 raised to the second power equals 25. log10 100 = 2 because 102 = 100. log 4 64 = 3 because 43 = 64. log 7

1 1 1 = - 2 because 7 -2 = 2 = . 49 49 7

Question • What is the value of log5 625?

Graphs of Logarithmic Functions y

g(x) = 2 x

8 6

y=x

4

Table 4.7

2 −4

−2

Because f (x) = log b x is the inverse function of g(x) = b x, the graph of f is a reflection of the graph of g across the line given by y = x. The graph of g(x) = 2x is shown in Figure 4.25. Table 4.7 shows some of the ordered pairs of the graph of g.

f(x) = log2 x 2

4

6

8 x

x

-3

-2

-1

0

1

2

3

g(x) 2 x

1 8

1 4

1 2

1

2

4

8

−2 −4

Figure 4.25

The graph of the inverse of g, which is f(x) = log 2 x, is also shown in Figure 4.25. Some of the ordered pairs of f are shown in Table 4.8. Note that if (x, y) is a point on the graph of g, then ( y, x) is a point on the graph of f. Also notice that the graph of f is a reflection of the graph of g across the line given by y = x. Table 4.8

x

1 8

1 4

1 2

1

2

4

8

f(x) log2 x

-3

-2

-1

0

1

2

3

The graph of a logarithmic function can be drawn by first rewriting the function in its exponential form. This procedure is illustrated in Example 4.

Answer • log 5 625 = 4 because 54 = 625.

362

CHAPTER 4


EXAMPLE 4

Graph a Logarithmic Function

Graph f (x) = log3 x. Solution To graph f (x) = log3 x, consider the equivalent exponential equation x = 3 y. Because this equation is solved for x, choose values of y and calculate the corresponding values of x, as shown in Table 4.9.

y 3 2

Table 4.9

1 −1

2

4

6

8

10 x

−2 −3 f(x) = log3 x

Figure 4.26

x 3y

1 9

1 3

1

3

9

y

-2

-1

0

1

2

Now plot the ordered pairs and connect the points with a smooth curve, as shown in Figure 4.26. Try Exercise 44, page 366

y 4 2

2

4

x

We can use a similar procedure to draw the graph of a logarithmic function with a fractional base. For instance, consider y = log2>3 x. Rewriting this in exponential form gives us 2 y a b = x. Choose values of y and calculate the corresponding x values. See Table 4.10. 3 Plot the points corresponding to the ordered pairs (x, y), and then draw a smooth curve through the points, as shown in Figure 4.27.

−2

Table 4.10 −4

y = log2>3 x Figure 4.27

2 y x a b 3

2 -2 9 a b = 3 4

2 -1 3 a b = 3 2

2 0 a b = 1 3

2 1 2 a b = 3 3

2 2 4 a b = 3 9

y

-2

-1

0

1

2

Properties of f (x) log b x For all positive real numbers b, b Z 1, the function f (x) = log b x has the following properties. The domain of f consists of the set of positive real numbers, and its range consists of the set of all real numbers. The graph of f has an x-intercept of (1, 0) and passes through (b, 1). If b 7 1, f is an increasing function and its graph is asymptotic to the negative y-axis. [As x : q , f(x) : q , and as x : 0 from the right, f (x) : - q .] See Figure 4.28a on page 363. If 0 6 b 6 1, f is a decreasing function and its graph is asymptotic to the positive y-axis. [As x : q , f (x) : - q , and as x : 0 from the right, f (x) : q .] See Figure 4.28b on page 363.

4.3

y 3


y 3

3

(b , 3) (b 2, 2)

(b 1, 1) b −1

(b, 1)

(1, 0) 1

b

b2

b3

x

(1, 0)

b −2 (b −1, −1)

(b −1, −1)

a. f (x) = logb x, b > 1

363

(b −2 , − 2 )

b −3

x

(b − 3, −3)

b. f(x) = logb x, 0 < b < 1 Figure 4.28

Domains of Logarithmic Functions The function f(x) = log b x has as its domain the set of positive real numbers. The function f (x) = log b( g(x)) has as its domain the set of all x for which g(x) 7 0. To determine the domain of a function such as f (x) = log b( g(x)), we must determine the values of x that make g(x) positive. This process is illustrated in Example 5.

EXAMPLE 5

Find the Domain of a Logarithmic Function

Find the domain of each of the following logarithmic functions. a.

f(x) = log6 (x - 3)

b.

F(x) = log 2 ƒ x + 2 ƒ

c.

R(x) = log 5 a

x b 8 - x

Solution a. Solving (x - 3) 7 0 for x gives us x 7 3. The domain of f consists of all real numbers greater than 3. In interval notation, the domain is (3, q ). b.

c.

The solution set of ƒ x + 2 ƒ 7 0 consists of all real numbers x except x = - 2. The domain of F consists of all real numbers x Z - 2. In interval notation, the domain is ( - q , -2) ´ ( -2, q ). x b 7 0 yields the set of all real numbers x between 0 and 8. 8 - x The domain of R is all real numbers x such that 0 6 x 6 8. In interval notation, the domain is (0, 8).

Solving a


Some logarithmic functions can be graphed by using horizontal or vertical translations of a previously drawn graph.

364

CHAPTER 4


EXAMPLE 6 Graph.

Use Translations to Graph Logarithmic Functions

a.

f (x) = log4(x + 3)

b.

f (x) = log4 x + 3

Solution a. The graph of f(x) = log4(x + 3) can be obtained by shifting the graph of g(x) = log4 x to the left 3 units. See Figure 4.29. Note that the domain of f consists of all real numbers x greater than - 3 because x + 3 7 0 for x 7 - 3. The graph of f is asymptotic to the vertical line x = - 3. b.

The graph of f(x) = log4 x + 3 can be obtained by shifting the graph of g(x) = log4 x upward 3 units. See Figure 4.30. y

y

4

4

2

−4

−2

f (x) = log4(x + 3)

2 −2

4

f (x) = log4 x + 3

2

6

x

−4

−2

g (x) = log4 x

2 −2

4

6

x

g (x) = log4 x

−4

−4

Figure 4.29

Figure 4.30


Common and Natural Logarithms Two of the most frequently used logarithmic functions are common logarithms, which have base 10, and natural logarithms, which have base e (the base of the natural exponential function).

Definition of Common and Natural Logarithms The function defined by f(x) = log10 x is called the common logarithmic function. It is customarily written as f(x) = log x, without stating the base. The function defined by f(x) = log e x is called the natural logarithmic function. It is customarily written as f(x) = ln x.

Most scientific or graphing calculators have a LOG key for evaluating common logarithms and an LN key to evaluate natural logarithms. For instance, using a graphing calculator, log 24 L 1.3802112 and

ln 81 L 4.3944492

The graphs of f (x) = log x and f (x) = ln x can be drawn using the same techniques we used to draw the graphs in the preceding examples. However, these graphs also can be produced

4.3


365

with a graphing calculator by entering log x and ln x into the Y = menu. See Figure 4.31 and Figure 4.32. Plot1 Plot2 Plot3 \Y 1 = log(X) \Y2 = ln(X) \ WINDOW \ Xmin=0 \ Xmax=9.4 \ Xscl=1 \ Ymin=-3 Ymax=3 Yscl=1 Xres=1

3 f (x) = ln x

9.4

0 f (x) = log x

−3

Figure 4.31

Figure 4.32

Observe that each graph passes through (1, 0). Also note that as x : 0 from the right, the functional values f(x) : - q . Thus the y-axis is a vertical asymptote for each of the graphs. The domain of both f (x) = log x and f (x) = ln x is the set of positive real numbers. Each of these functions has a range consisting of the set of real numbers. Many applications can be modeled by logarithmic functions.

Jan Halaska/Index Stock Imagery/ Jupiter Images

EXAMPLE 7

Math Matters Although logarithms were originally developed to assist with computations, logarithmic functions have a much broader use today. They are often used in such disciplines as geology, acoustics, chemistry, physics, and economics, to name a few.

Applied Physiology

In the study The Pace of Life, M. H. Bornstein and H. G. Bornstein (Nature, Vol. 259, pp. 557–558, 1976) reported that as the population of a city increases, the average walking speed of a pedestrian also increases. An approximate relation between the average pedestrian walking speed s, in miles per hour, and the population x, in thousands, of a city is given by the function s(x) = 0.37 ln x + 0.05 a.

Determine the average walking speed, to the nearest tenth of a mile per hour, in San Francisco, which has a population of 765,000, and in Round Rock, Texas, which has a population of 86,000.

b.

Estimate the population of a city for which the average pedestrian walking speed is 3.1 miles per hour. Round to the nearest hundred-thousand.

Solution a. The population of San Francisco, in thousands, is 765. s(x) = 0.37 ln x + 0.05 s(765) = 0.37 ln 765 + 0.05 L 2.5

• Substitute 765 for x. • Use a calculator to evaluate.

The average walking speed in San Francisco is about 2.5 miles per hour. The population of Round Rock, in thousands, is 86. s(x) = 0.37 ln x + 0.05 s(86) = 0.37 ln 86 + 0.05 L 1.7

• Substitute 86 for x. • Use a calculator to evaluate.

The average walking speed in Round Rock is about 1.7 miles per hour. (continued)

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CHAPTER 4


Graph s(x) = 0.37 ln x + 0.05 and s = 3.1 in the same viewing window.

b.

5

0

10,000

Intersection X=3801.8507 Y=3.1 −1.5

Xscl = 1000 Yscl = 1

The x value of the intersection point represents the population in thousands. The function indicates that a city with an average pedestrian walking speed of 3.1 miles per hour should have a population of about 3,800,000. Try Exercise 86, page 367

EXERCISE SET 4.3 In Exercises 1 to 12, write each equation in its exponential form. 1. 1 = log 10

2. 4 = log 10,000

3. 2 = log8 64

4. 3 = log4 64

5. 0 = log7 x

1 6. -4 = log3 81

7. ln x = 4 9. ln 1 = 0 11. 2 = log(3x + 1)

13. 3 = 9 15. 4-2 =

1 243

29. ln e 3

8. ln e = 2 10. ln x = - 3 12.

1 x + 1 = ln a 2 b 3 x

14. 5 = 125 3

1 16

27. log3

8 27

26. log3>2

25. log4 16

28. logb 1 30. logb b

2

In Exercises 13 to 24, write each equation in its logarithmic form. Assume y>0 and b>0. 2

In Exercises 25 to 42, evaluate each logarithm. Do not use a calculator.

31. log

1 100

32. log10(106 )

34. log0.3

35. 4 log 1000

36. log5 125 2

37. 2 log7 2401

38. 3 log11 161,051

5

3

39. log3 19

40. log6 136 3

16. 100 = 1

100 9

33. log0.5 16

41. 5 log13 1169

7

42. 2 log7 1343

In Exercises 43 to 50, graph each function by using its exponential form.

17. bx = y

18. 2x = y

19. y = e x

20. 51 = 5

21. 100 = 102

22. 2-4 =

23. e2 = x + 5

24. 3x = 47

1 16

43. f (x) = log 4 x

44. f (x) = log6 x

45. f (x) = log12 x

46. f (x) = log 8 x

47. f (x) = log1>2 x

48. f (x) = log1>4 x

49. f (x) = log 5>2 x

50. f (x) = log 7>3 x

4.3

In Exercises 51 to 64, find the domain of the function. Write the domain using interval notation. 51. f (x) = log 5(x - 3)

52. k(x) = log4(5 - x)

53. k(x) = log 2>3(11 - x)

54. H(x) = log1>4(x 2 + 1)

55. P(x) = ln(x 2 - 4)

56. J(x) = ln a


74. f (x) = ln x + 3

57. h(x) = ln a

x2 b x - 4

g(x) = ln(x - 3)

h(x) = ln(3 - x)

k(x) = - ln( -x)

y

a.

y

b.

4

x - 3 b x

4

−4

4

x

−4

−4

58. R(x) = ln(x4 - x 2)

c.

4

x

4

x

−4

y

y

d.

4

59. N(x) = log 2(x 3 - x)

367

4

60. s(x) = log 7(x 2 + 7x + 10) x

4

61. g(x) = log 12x - 11

62. m(x) = log ƒ 4x - 8 ƒ

63. t(x) = 2 ln(3x - 7)

64. v(x) = ln(x - 4)2

−4

−4

−4

In Exercises 75 to 84, use a graphing utility to graph the function. In Exercises 65 to 72, use translations of the graphs in Exercises 43 to 50 to produce the graph of the given function. 66. f (x) = log6(x + 3)

65. f (x) = log 4(x - 3)

75. f (x) = - 2 ln x

76. f (x) = - log x

77. f (x) = ƒ ln x ƒ

78. f (x) = ln ƒ x ƒ

3

67. f (x) = log12 x + 2

68. f (x) = log 8 x - 4

69. f (x) = 3 + log1>2 x

70. f (x) = 2 + log1>4 x

71. f (x) = 1 + log 5>2(x - 4)

72. f (x) = log 7>3(x - 3) - 1

79. f (x) = log 1x

80. f (x) = ln 1x

81. f (x) = log(x + 10)

82. f (x) = ln(x + 3)

83. f (x) = 3 log ƒ 2x + 10 ƒ

84. f (x) =

85.

In Exercises 73 and 74, examine the four functions and the graphs labeled a, b, c, and d. For each graph, determine which function has been graphed. 73. f (x) = log5(x - 2)

h(x) = log5(- x) y

a.

b.

4

−4

k(x) = - log5(x + 3)

a. What interest rate, to the nearest tenth of a percent, will the

x

average typing speed S, in words per minute, of a student who has been typing for t months.

4

x

which a person must invest to receive an interest rate of at least 3%? 86. Average Typing Speed The following function models the

y

d.

4 −4

b. What is the minimum number of complete months during 4

4

−4

bank pay on a money market account with a term of 9 months?

y

−4

y

c.

r(t) = 0.69607 + 0.60781 ln t

g(x) = 2 + log5 x

x

−4

Money Market Rates The function

gives the annual interest rate r, as a percent, a bank will pay on its money market accounts, where t is the term (the time the money is invested) in months.

4

4

1 ln ƒ x - 4 ƒ 2

−4

S(t) = 5 + 29 ln(t + 1), 0 … t … 16 4

−4

x

a. What was the student’s average typing speed, to the nearest

word per minute, when the student first started to type? What was the student’s average typing speed, to the nearest word per minute, after 3 months?

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CHAPTER 4

b.

87.


Use a graph of S to determine how long, to the nearest tenth of a month, it will take the student to achieve an average typing speed of 65 words per minute.

Brightness x, relative to a first-magnitude star

Apparent magnitude M(x)

Advertising Costs and Sales The function

1

1

N(x) = 2750 + 180 ln a

1 2.51

2

1 1 L 6.31 2.512

3

1 1 L 15.85 2.513

4

1 1 L 39.82 2.514

5

1 1 L 100 2.515

6

x + 1b 1000

models the relationship between the dollar amount x spent on advertising a product and the number of units N that a company can sell. a. Find the number of units that will be sold with advertising

expenditures of $20,000, $40,000, and $60,000. b. How many units will be sold if the company does not pay

to advertise the product? Medicine In

anesthesiology it is necessary to accurately estimate the body surface area of a patient. One formula for estimating body surface area (BSA) was developed by Edith Boyd (University of Minnesota Press, 1935). Her formula for the BSA (in square meters) of a patient of height H (in centimeters) and weight W (in grams) is

#

The following logarithmic function gives the apparent magnitude M(x) of a star as a function of its brightness x. M(x) = - 2.51 log x + 1, 0 6 x … 1 a. Use M(x) to find the apparent magnitude of a star that is

1 as bright as a first-magnitude star. Round to the nearest 10 hundredth.

#

1 as bright 400 as a first-magnitude star. Round to the nearest hundredth.

BSA 0.0003207 H 0.3 W (0.7285 - 0.0188 log W)

b. Find the apparent magnitude of a star that is

In Exercises 88 and 89, use Boyd’s formula to estimate the body surface area of a patient with the given weight and height. Round to the nearest hundredth of a square meter.

c. Which star appears brighter: a star with an apparent mag-

nitude of 12 or a star with an apparent magnitude of 15?

88. W = 110 pounds (49,895.2 grams)

H = 5 feet 4 inches (162.56 centimeters)

89. W = 180 pounds (81,646.6 grams)

H = 6 feet 1 inch (185.42 centimeters)

90.

Astronomy Astronomers measure the apparent brightness of a star by a unit called the apparent magnitude. This unit was created in the second century B.C. when the Greek astronomer Hipparchus classified the relative brightness of several stars. In his list, he assigned the number 1 to the stars that appeared to be the brightest (Sirius, Vega, and Deneb). They are first-magnitude stars. Hipparchus assigned the number 2 to all the stars in the Big Dipper. They are secondmagnitude stars. The following table shows the relationship between a star’s brightness relative to a first-magnitude star and the star’s apparent magnitude. Notice from the table that a first-magnitude star appears to be about 2.51 times as bright as a second-magnitude star.

d. Is M(x) an increasing function or a decreasing function? Number of Digits in b X An engineer has determined

91.

that the number of digits N in the expansion of b x, where both b and x are positive integers, is N = int(x log b) + 1, where int(x log b) denotes the greatest integer of x log b. (Note: See pages 175–176 for information on the greatest integer function.) a. Because 210 = 1024, we know that 210 has four digits. Use

the equation N = int(x log b) + 1 to verify this result.

b. Find the number of digits in 3200. c. Find the number of digits in 74005. d.

The largest known prime number as of August 23, 2008 was 243,112,609 - 1. Find the number of digits in this prime number. (Hint: Because 243,112,609 is not a power of 10, both 243,112,609 and 243,112,609 - 1 have the same number of digits.)

4.4

92.

Number of Digits in 9(9

PROPERTIES OF LOGARITHMS AND LOGARITHMIC SCALES

9)

A science teacher has offered 10 points extra credit to any student who will write out all 9 the digits in the expansion of 9(9 ).

94.

a. Use the formula from Exercise 91 to determine the number

of digits in this number. b. Assume that you can write 1000 digits per page and that

95. The functions f (x) =

In Exercises 93 and 94, use a graphing utility to determine the relationship between f and g. 93.

96.

-x

e - e and 2 g(x) = ln(x + 2x 2 + 1 ) on the same screen. Use a square viewing window. What appears to be the relationship between f and g? Use a graphing utility to graph f (x) =

SECTION 4.4 Properties of Logarithms Change-of-Base Formula Logarithmic Scales

ex + e-x for x Ú 0 2 and g(x) = ln(x + 2x 2 - 1 ) for x Ú 1 on the same screen. Use a square viewing window. What appears to be the relationship between f and g? Use a graphing utility to graph f (x) =

1 1 + x ex - e -x and g(x) = ln are e x + e -x 2 1 - x inverse functions. The domain of f is the set of all real numbers. The domain of g is 5x ƒ - 1 6 x 6 16. Use this information to determine the range of f and the range of g.

500 pages of paper are in a ream of paper. How many reams of paper, to the nearest tenth of a ream, are required to write 9 out the expansion of 9(9 ) ? Assume that you write on only one side of each page.

x

369

Use a graph of f (x) =

2 to determine the domain e x + e -x

and range of f.

Properties of Logarithms and Logarithmic Scales PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A28. In Exercises PS1 to PS6, use a calculator to compare the values of the given expressions.

PS1. log 3 + log 2; log 6 [4.3] PS2. ln 8 - ln 3; ln a b [4.3]

8 3

PS3. 3 log 4; log(43) [4.3] PS4. 2 ln 5; ln(52) [4.3] PS5. ln 5;

log 5 [4.3] log e

PS6. log 8;

ln 8 [4.3] ln 10

Properties of Logarithms In Section 4.3 we introduced the following basic properties of logarithms. log b b = 1

and

log b 1 = 0

Also, because exponential functions and logarithmic functions are inverses of each other, we observed the relationships log b(b x ) = x

and

blogb x = x

370

CHAPTER 4


Caution Pay close attention to these properties. Note that

We can use the properties of exponents to establish the following additional logarithmic properties.

logb(MN) Z logb M # logb N

Properties of Logarithms

and logb

In the following properties, b, M, and N are positive real numbers (b Z 1).

logb M M Z N logb N

log b(MN) = log b M + log b N M = log b M - log b N log b N

Product property

Also,

Quotient property

logb(M + N) Z logb M + logb N

Power property

In fact, the expression logb(M + N) cannot be expanded.

Logarithm-of-each-side property One-to-one property

log b(M p) = p log b M M = N implies log b M = log b N log b M = log b N implies M = N

Here is a proof of the product property. Proof

Let r = logb M and s = logb N. These equations can be written in exponential form as M = br

and

N = bs

Now consider the product MN. MN = brbs

• Substitute for M and N.

r+s

MN = b logb MN = r + s logb MN = logb M + logb N

• Product property of exponents • Write in logarithmic form. • Substitute for r and s.

N

The last equation is our desired result.

The quotient property and the power property can be proved in a similar manner. See Exercises 87 and 88 on page 380. The properties of logarithms are often used to rewrite logarithmic expressions in an equivalent form. The process of using the product or quotient rules to rewrite a single logarithm as the sum or difference of two or more logarithms, or using the power property to rewrite logb(M p ) in its equivalent form p logb M, is called expanding the logarithmic expression. We illustrate this process in Example 1.

EXAMPLE 1

Expand Logarithmic Expressions

Use the properties of logarithms to expand the following logarithmic expressions. Assume all variable expressions represent positive real numbers. When possible, evaluate logarithmic expressions. a.

log 5(xy 2)

b.

ln a

e1y z3

b

Solution a. log 5(xy 2) = log 5 x + log 5 y 2 = log 5 x + 2 log 5 y

• Product property • Power property

4.4

b.

ln a

e 1y z3


b = ln(e 1y) - ln z 3 = ln e + ln 1y = ln e + ln y1>2 1 = ln e + ln y 2 1 = 1 + ln y 2

371

• Quotient property

- ln z 3 - ln z3

• Product property

- 3 ln z

• Power property

3 ln z

• Evaluate ln e.

• Write 1y as y1>2.


The properties of logarithms are also used to condense expressions that involve the sum or difference of logarithms into a single logarithm. For instance, we can use the product property to rewrite logb M + logb N as logb(MN), and the quotient property to rewrite M logb M - logb N as logb . Before applying the product or quotient properties, use the N power property to write all expressions of the form p logb M in their equivalent logb M p form. See Example 2. Question • Does log 2 + log 5 = 1?

EXAMPLE 2

Condense Logarithmic Expressions

Use the properties of logarithms to rewrite each expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers. a.

2 ln x +

1 ln(x + 4) 2

b.

log5(x 2 - 4) + 3 log5 y - log5(x - 2)2

Solution a.

b.

2 ln x +

1 ln(x + 4) = ln x 2 + ln(x + 4)1>2 2 = ln3x 2 (x + 4)1>24 = ln3x 2 1(x + 4)4

• Power property • Product property • Rewriting (x + 4)1>2 as 1x + 4 is an optional step.

log5(x 2 - 4) + 3 log5 y - log5(x - 2)2 = log5(x 2 - 4) + log5 y 3 - log5(x - 2)2 = 3log5(x 2 - 4) + log5 y 34 - log5(x - 2)2 = log53(x 2 - 4) y 34 - log5(x - 2)2 (x 2 - 4) y 3 = log5 B R (x - 2)2

• Power property • Order of Operations Agreement • Product property • Quotient property

(x + 2)(x - 2) y 3 = log5 B R (x - 2)2

• Factor.

(x + 2) y 3 = log5 B R x - 2

• Simplify.


#

Answer • Yes. By the product property, log 2 + log 5 = log(2 5) = log 10 = 1.

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CHAPTER 4


Change-of-Base Formula Recall that to determine the value of y in log3 81 = y, we ask the question, “What power of 3 is equal to 81?” Because 34 = 81, we have log3 81 = 4. Now suppose that we need to determine the value of log3 50. In this case, we need to find the power of 3 that produces 50. Because 33 = 27 and 34 = 81, the value we are seeking is somewhere between 3 and 4. The following procedure can be used to produce an estimate of log3 50. The exponential form of log3 50 = y is 3 y = 50. Applying logarithmic properties gives us 3 y = 50 ln 3 y = ln 50

• Logarithm-of-each-side property

y ln 3 = ln 50

• Power property

ln 50 L 3.56088 y = ln 3

• Solve for y.

Thus log3 50 L 3.56088. In the preceding procedure we could just as well have used logarithms of any base and arrived at the same value. Thus any logarithm can be expressed in terms of logarithms of any base we wish. This general result is summarized in the following formula.

Change-of-Base Formula If x, a, and b are positive real numbers with a Z 1 and b Z 1, then log b x =

log a x log a b

Because most calculators use only common logarithms (a = 10) or natural logarithms (a = e), the change-of-base formula is used most often in the following form. If x and b are positive real numbers and b Z 1, then log b x =

EXAMPLE 3

log x ln x = log b ln b

Use the Change-of-Base Formula

Evaluate each logarithm. Round to the nearest ten-thousandth. a. Study tip If common logarithms had been used for the calculations in Example 3, the final results would have been the same. log 3 18 = log12 400 =

log 18 L 2.6309 log 3 log 400 L 2.4111 log 12

log 3 18

b.

log12 400

Solution To approximate these logarithms, we may use the change-of-base formula with a = 10 or a = e. For this example, we choose to use the change-of-base formula with a = e. That is, we will evaluate these logarithms by using the LN key on a scientific or graphing calculator. a.

log3 18 =

ln 18 L 2.6309 ln 3


b.

log12 400 =

ln 400 L 2.4111 ln 12

4.4

y 4 2

−2

2

4

x


373

The change-of-base formula and a graphing calculator can be used to graph logarithmic functions that have a base other than 10 or e. For instance, to graph f (x) = log3(2x + 3), we rewrite the function in terms of base 10 or base e. Using base 10 logarithms, we have log(2x + 3) . The graph is shown in Figure 4.33. f(x) = log3(2x + 3) = log 3

−2 −4

EXAMPLE 4 f (x) = log3(2x + 3)

Figure 4.33

Use the Change-of-Base Formula to Graph a Logarithmic Function

Graph f (x) = log2 ƒ x - 3 ƒ . Solution Rewrite f using the change-of-base formula. We will use the natural logarithm function; however, the common logarithm function could be used instead. f(x) = log2 ƒ x - 3 ƒ =

ln ƒ x - 3 ƒ ln 2

ln ƒ x - 3 ƒ into Y 1. The graph is shown at the ln 2 right. Note that the domain of f(x) = log 2 ƒ x - 3 ƒ is all real numbers except 3 because ƒ x - 3 ƒ = 0 when x = 3 and ƒ x - 3 ƒ is positive for all other values of x.

4

Enter

f (x) = log2|x − 3| − 2.7

6.7

−4


Logarithmic Scales Logarithmic functions are often used to scale very large (or very small) numbers into numbers that are easier to comprehend. For instance, the Richter scale magnitude of an earthquake uses a logarithmic function to convert the intensity of the earthquake’s shock waves I into a number M, which for most earthquakes is in the range of 0 to 10. The intensity I of an earthquake is often given in terms of the constant I0, where I0 is the intensity of the smallest earthquake (called a zero-level earthquake) that can be measured on a seismograph near the earthquake’s epicenter. The following formula is used to compute the Richter scale magnitude of an earthquake.

Math Matters The Richter scale was created by the seismologist Charles F. Richter in 1935. Notice that a tenfold increase in the intensity level of an earthquake increases the Richter scale magnitude of the earthquake by only 1.

Richter Scale Magnitude of an Earthquake An earthquake with an intensity of I has a Richter scale magnitude of I M = log a b I0 where I0 is the measure of the intensity of a zero-level earthquake.

374

CHAPTER 4


EXAMPLE 5

Determine the Magnitude of an Earthquake

Find the Richter scale magnitude (to the nearest tenth) of the 2008 Panama–Costa Rica earthquake that had an intensity of I = 1,584,893I0. Study tip Notice in Example 5 that we did not need to know the value of I0 to determine the Richter scale magnitude of the quake.

Solution 1,584,893I0 I M = log a b = log a b = log(1,584,893) L 6.2 I0 I0 The 2008 Panama–Costa Rica earthquake had a Richter scale magnitude of 6.2. Try Exercise 68, page 379

If you know the Richter scale magnitude of an earthquake, you can determine the intensity of the earthquake.

EXAMPLE 6

Determine the Intensity of an Earthquake

Find the intensity of the July 2008, Greater Los Angeles Area earthquake, which measured 5.4 on the Richter scale. Solution I log a b = 5.4 I0 I = 105.4 I0

• Write in exponential form.

I = 105.4I0 I L 251,189I0

• Solve for I.

The 2008 Greater Los Angeles Area earthquake had an intensity that was approximately 251,000 times the intensity of a zero-level earthquake. Try Exercise 70, page 379

In Example 7 we use the Richter scale magnitudes of two earthquakes to compare the intensities of the earthquakes.

EXAMPLE 7

Compare Intensities of Earthquakes

The 1960 Chile earthquake had a Richter scale magnitude of 9.5. The 1989 San Francisco earthquake had a Richter scale magnitude of 7.1. Compare the intensities of the earthquakes. Study tip The results of Example 7 show that if an earthquake has a Richter scale magnitude of M 1 and a smaller earthquake has a Richter scale magnitude of M 2 , then the larger earthquake is 10 M1 - M2 times as intense as the smaller earthquake.

Solution Let I1 be the intensity of the Chilean earthquake, and let I2 be the intensity of the San Francisco earthquake. Then I1 log a b = 9.5 I0 I1 = 109.5 I0 I1 = 109.5I0

and

I2 log a b = 7.1 I0 I2 = 107.1 I0 I2 = 107.1I0

4.4


375

To compare the intensities of the earthquakes, we compute the ratio I1>I2 . I1 109.5I0 109.5 = 7.1 = 7.1 = 109.5 - 7.1 = 102.4 L 251 I2 10 I0 10 The earthquake in Chile was approximately 251 times as intense as the San Francisco earthquake. Try Exercise 72, page 379

Seismologists generally determine the Richter scale magnitude of an earthquake by examining a seismogram. See Figure 4.34. Arrival of first s-wave Arrival of first p-wave

Amplitude = 23 mm

24 s Time between s-wave and p-wave

Figure 4.34

The magnitude of an earthquake cannot be determined just by examining the amplitude of a seismogram because this amplitude decreases as the distance between the epicenter of the earthquake and the observation station increases. To account for the distance between the epicenter and the observation station, a seismologist examines a seismogram for small waves called p-waves and larger waves called s-waves. The Richter scale magnitude M of an earthquake is a function of both the amplitude A of the s-waves and the difference in time t between the occurrence of the s-waves and the occurrence of the p-waves. In the 1950s, Charles Richter developed the following formula to determine the magnitude of an earthquake from the data in a seismogram.

Amplitude–Time–Difference Formula The Richter scale magnitude M of an earthquake is given by M = log A + 3 log 8t - 2.92 where A is the amplitude, in millimeters, of the s-waves on a seismogram and t is the difference in time, in seconds, between the s-waves and the p-waves.

EXAMPLE 8

Determine the Magnitude of an Earthquake from Its Seismogram

Find the Richter scale magnitude of the earthquake that produced the seismogram in Figure 4.34. (continued)

376

CHAPTER 4


Solution M = log A + 3 log 8t - 2.92

= log 23 + 3 log38 # 244 - 2.92

Note

• Substitute 23 for A and 24 for t.

L 1.36173 + 6.84990 - 2.92

The Richter scale magnitude is usually rounded to the nearest tenth.

L 5.3 The earthquake had a magnitude of about 5.3 on the Richter scale. Try Exercise 76, page 379

Logarithmic scales are also used in chemistry. One example concerns the pH of a liquid, which is a measure of the liquid’s acidity or alkalinity. (You may have tested the pH of the water in a swimming pool or an aquarium.) Pure water, which is considered neutral, has a pH of 7.0. The pH scale ranges from 0 to 14, with 0 corresponding to the most acidic solutions and 14 to the most alkaline. Lemon juice has a pH of about 2, whereas household ammonia measures about 11. Specifically, the pH of a solution is a function of the hydronium-ion concentration of the solution. Because the hydronium-ion concentration of a solution can be very small (with values such as 0.00000001 mole per liter), pH uses a logarithmic scale.

Definition of the pH of a Solution

The pH of a solution with a hydronium-ion concentration of 3H + 4 mole per liter is given by pH = - log3H + 4

EXAMPLE 9

Find the pH of a Solution

Find the pH of each liquid. Round to the nearest tenth. a. b. c.

Orange juice with 3H + 4 = 2.8 * 10-4 mole per liter Milk with 3H + 4 = 3.97 * 10-7 mole per liter

Rainwater with 3H + 4 = 6.31 * 10-5 mole per liter

d. A baking soda solution with 3H + 4 = 3.98 * 10-9 mole per liter

Math Matters The pH scale was created by the Danish biochemist Søren Sørensen in 1909 to measure the acidity of water used in the brewing of beer. pH is an abbreviation for pondus hydrogenii, which translates as “potential hydrogen.”

Solution a. pH = - log3H + 4 = - log(2.8 * 10-4 ) L 3.6 The orange juice has a pH of 3.6. b. c. d.

pH = - log3H + 4 = - log(3.97 * 10-7 ) L 6.4 The milk has a pH of 6.4. pH = - log3H + 4 = - log(6.31 * 10-5 ) L 4.2 The rainwater has a pH of 4.2. pH = - log3H + 4 = - log(3.98 * 10-9 ) L 8.4 The baking soda solution has a pH of 8.4.


4.4


377

Figure 4.35 illustrates the pH scale, along with the corresponding hydronium-ion concentrations. A solution on the left half of the scale, with a pH of less than 7, is an acid, and a solution on the right half of the scale is an alkaline solution, or a base. Because the scale is logarithmic, a solution with a pH of 5 is 10 times more acidic than a solution with a pH of 6. From Example 9, we see that the orange juice, milk, and rainwater are acids whereas the baking soda solution is a base. Neutral Acidic pH

0

[H+ ] 100

Basic

1

2

3

4

5

6

7

10−1

10−2

10−3

10−4

10−5

10−6

10−7

8

9

10

11

12

13

14

10−8

10−9

10−10

10−11

10−12

10−13

10−14

+

pH = −log[H ]

Figure 4.35

EXAMPLE 10

Find the Hydronium-Ion Concentration

A sample of blood has a pH of 7.3. Find the hydronium-ion concentration of the blood. Solution

pH = - log3H + 4 7.3 = - log3H + 4 -7.3 = log3H + 4 10-7.3 = 3H + 4 5.0 * 10

-8

• Substitute 7.3 for pH. • Multiply both sides by -1. • Change to exponential form.

L 3H 4 +

The hydronium-ion concentration of the blood is about 5.0 * 10-8 mole per liter. Try Exercise 80, page 379

EXERCISE SET 4.4 In Exercises 1 to 16, expand the given logarithmic expression. Assume all variable expressions represent positive real numbers. When possible, evaluate logarithmic expressions. Do not use a calculator. 1. log b(xyz)

3. ln

x

z3 1xy

4. log 5

z4

5. log2

2. ln

1x y

3

xy

7. log 7

1xz

3

8. ln 2x 2 1y

y2

9. ln (e 2 z) 11. log4 a

10. ln (x 1>2 y 2>3)

3 1z

16y

3

b

2

z4

6. log b (x1y ) 3

12. log5 a

13. log 2x1z

14. ln a

15. ln ( 2z 1e )

16. ln c

3

1xz4 b 125

3 2 2 x

z2

x 2 1z y-3

b d

378

CHAPTER 4


In Exercises 17 to 32, write each expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.

41. log12 17

42. log13 5.5

43. logp e

44. logp 115

17. log(x + 5) + 2 log x 18. 3 log2 t -

1 log2 u + 4 log2 v 3

19. ln(x 2 - y 2 ) - ln(x - y) 20.

1 log8(x + 5) - 3 log8 y 2

21. 3 log x +

1 log y + log(x + 1) 3

In Exercises 45 to 52, use a graphing utility and the change-of-base formula to graph the logarithmic function. 45. f (x) = log 4 x

46. g(x) = log8(5 - x)

47. g(x) = log 8(x - 3)

48. t(x) = log 9(5 - x)

49. h(x) = log 3(x - 3)2

50. J(x) = log12( -x)

51. F(x) = - log5 ƒ x - 2 ƒ

52. n(x) = log 2 1x - 8

In Exercises 53 to 62, determine whether the statement is true or false for all x >0, y >0. If it is false, write an example that disproves the statement.

y 22. ln(xz) - ln(x 1y ) + 2 ln z

53. logb(x + y) = logb x + logb y

23. log(xy 2) - log z

54. logb(xy) = logb x # logb y

24. ln(y1>2z) - ln z1>2 25. 2(log6 x + log6 y 2) - log6 (x + 2)

1 26. log3 x - log3 y + 2 log3(x + 2) 2 27. 2 ln(x + 4) - ln x - ln(x 2 - 3)

55. logb(xy) = logb x + logb y 56. logb x # logb y = logb x + logb y 57. logb x - logb y = logb(x - y), 58. logb

28. log(3x) - (2 log x - log y)

1 29. ln(2x + 5) - ln y - 2 ln z + ln w 2 30. logb x + logb(y + 3) + logb(y + 2) - logb(y 2 + 5y + 6) 31. ln(x - 9) - 2 ln(x - 3) + 3 ln y 2

32. logb(x 2 + 7x + 12) - 2 logb(x + 4)

In Exercises 33 to 44, use the change-of-base formula to approximate the logarithm accurate to the nearest tenthousandth.

59.

x 7 y

logb x x = y logb y

logb x = logb x - logb y logb y

60. logb(x n) = n logb x 61. (logb x)n = n logb x 62. logb 1x =

1 logb x 2

In Exercises 63 and 64, evaluate the given expression without using a calculator. 63. log3 5 # log5 7 # log7 9

33. log 7 20

34. log 5 37

64. log5 20 # log20 60 # log60 100 # log100 125

35. log11 8

36. log 50 22

65. Which is larger, 500501 or 506500? These numbers are too large

1 37. log6 3

7 38. log3 8

39. log9 117

40. log 4 17

for most calculators to handle. (They each have 1353 digits!) (Hint: Compare the logarithms of each number.) 66. Which is smaller,

1 50

300

or

1 151233

? See the hint in Exercise 65.

4.4

67.

68.

Earthquake Magnitude The Baja California earthquake of November 20, 2008, had an intensity of I = 101,400I0. What did this earthquake measure on the Richter scale?


76. Earthquake Magnitude Find the Richter scale magnitude of

the earthquake that produced the seismogram in the following figure.

Earthquake Magnitude The Colombia earthquake of

s-wave

1906 had an intensity of I = 398,107,000I0 . What did this earthquake measure on the Richter scale? 69.

Earthquake Intensity The Coalinga, California, earth-

Earthquake Intensity The earthquake that occurred just south of Concepción, Chile, in 1960 had a Richter scale magnitude of 9.5. Find the intensity of this earthquake.

71. Comparison of Earthquakes Compare the intensity of an

earthquake that measures 5.0 on the Richter scale to the intensity of an earthquake that measures 3.0 on the Richter scale by finding the ratio of the larger intensity to the smaller intensity. 72.

73.

74.

A = 26 mm

p-wave

quake of 1983 had a Richter scale magnitude of 6.5. Find the intensity of this earthquake. 70.

379

Comparison of Earthquakes How many times as great was the intensity of the 1960 earthquake in Chile, which measured 9.5 on the Richter scale, than the San Francisco earthquake of 1906, which measured 8.3 on the Richter scale? Comparison of Earthquakes On March 2, 1933, an earthquake of magnitude 8.9 on the Richter scale struck Japan. In October 1989, an earthquake of magnitude 7.1 on the Richter scale struck the San Francisco Bay Area. Compare the intensity of the larger earthquake to the intensity of the smaller earthquake by finding the ratio of the larger intensity to the smaller intensity. Comparison of Earthquakes An earthquake that occurred in China in 1978 measured 8.2 on the Richter scale. In 1988, an earthquake in California measured 6.9 on the Richter scale. Compare the intensity of the larger earthquake to the intensity of the smaller earthquake by finding the ratio of the larger intensity to the smaller intensity.

t = 17 s

77. pH Milk of magnesia has a hydronium-ion concentration of

about 3.97 * 10-11 mole per liter. Determine the pH of milk of magnesia and state whether milk of magnesia is an acid or a base.

78. pH Vinegar has a hydronium-ion concentration of 1.26 * 10 - 3

mole per liter. Determine the pH of vinegar and state whether vinegar is an acid or a base. 79. Hydronium-Ion Concentration A morphine solution has a

pH of 9.5. Determine the hydronium-ion concentration of the morphine solution. 80. Hydronium-Ion Concentration A rainstorm in New York City

produced rainwater with a pH of 5.6. Determine the hydroniumion concentration of the rainwater. Decibel Level The range of sound intensities that

the human ear can detect is so large that a special decibel scale (named after Alexander Graham Bell) is used to measure and compare sound intensities. The decibel level (dB) of a sound is given by I dB(I ) = 10 log a b I0 where I0 is the intensity of sound that is barely audible to the human ear. Use the decibel level formula to work Exercises 81 to 84. 81. Find the decibel level for the following sounds. Round to the

nearest tenth of a decibel. Sound

75. Earthquake Magnitude Find the Richter scale magnitude of

a. Automobile traffic

the earthquake that produced the seismogram in the following figure.

b. Quiet conversation c. Fender guitar d. Jet engine

s-wave p-wave

t = 31 s

A = 18 mm

Intensity

I = 1.58 * 108 # I0 I = 10,800 # I0

I = 3.16 * 1011 # I0

I = 1.58 * 1015 # I0

82. A team in Arizona installed in a Ford Bronco a 48,000-watt

sound system that it claims can output 175-decibel sound. The human pain threshold for sound is 125 decibels. How many times as great is the intensity of the sound from the Bronco than the human pain threshold for sound?

380

CHAPTER 4


83. How many times as great is the intensity of a sound that mea-

intermediate step if a map is to start with a scale of 1:1,000,000, and proceed through five intermediate steps to end with a scale of 1:500,000.

sures 120 decibels than a sound that measures 110 decibels? 84. If the intensity of a sound is doubled, what is the increase in the

decibel level? (Hint: Find dB(2I) - dB(I).)

86. Animated Maps Use the equation in Exercise 85 to deter-

mine the scales for each stage of an animated map zoom that goes from a scale of 1:250,000 to a scale of 1:100,000 in four steps (following the initial scale).

85. Animated Maps A software company that creates interactive

maps for websites has designed an animated zooming feature such that when a user selects the zoom-in option the map appears to expand on a location. This is accomplished by displaying several intermediate maps to give the illusion of motion. The company has determined that zooming in on a location is more informative and pleasing to observe when the scale of each step of the animation is determined using the equation Sn = S0 # 10

87. Prove the quotient property of logarithms

logb

n N (log Sf - log S0)

M = logb M - logb N N

(Hint: See the proof of the product property of logarithms on page 370.)

where Sn represents the scale of the current step n (n = 0 corresponds to the initial scale), S0 is the starting scale of the map, Sf is the final scale, and N is the number of steps in the animation following the initial scale. (If the initial scale of the map is 1:200, then S0 = 200.) Determine the scales to be used at each

88. Prove the power property of logarithms

logb (M p) = p logb M See the hint given in Exercise 87.

MID-CHAPTER 4 QUIZ 1. Use composition of functions to verify that

f (x) =

500 + 120x x

and

g(x) =

500 x - 120

6. Expand ln a

xy 3 e2

b . Assume x and y are positive real numbers.

7. Write log3 x4 - 2 log3 z + log3 (xy 2) as a single logarithm with a

are inverses of each other.

coefficient of 1. Assume all variables are positive real numbers.

24x + 5 2. Find the inverse of f (x) = , x Z 4. State any restrictions x - 4 -1 on the domain of f (x).

8. Use the change-of-base formula to evaluate log8 411. Round to

3. Evaluate f (x) = e x, for x = - 2.4. Round to the nearest ten-

9. What is the Richter scale magnitude of an earthquake with an

the nearest ten-thousandth.

intensity of 789,251I0? Round to the nearest tenth.

thousandth. 4. Write ln x = 6 in exponential form.

10. How many times as great is the intensity of an earthquake that

measures 7.9 on the Richter scale than the intensity of an earthquake that measures 5.1 on the Richter scale?

5. Graph f (x) = log3(x + 3).

SECTION 4.5

Exponential and Logarithmic Equations

Solving Exponential Equations Solving Logarithmic Equations


PS1. Use the definition of a logarithm to write the exponential equation 36 = 729 in

logarithmic form. [4.3] PS2. Use the definition of a logarithm to write the logarithmic equation log5 625 = 4

in exponential form. [4.3]

4.5

EXPONENTIAL AND LOGARITHMIC EQUATIONS

381

PS3. Use the definition of a logarithm to write the exponential equation a x + 2 = b in

logarithmic form. [4.3] PS4. Solve for x: 4a = 7bx + 2cx [1.2] PS5. Solve for x: 165 = PS6. Solve for x: A =

300 [1.4] 1 + 12x

100 + x [1.4] 100 - x

Solving Exponential Equations If a variable appears in the exponent of a term of an equation, such as in 2x + 1 = 32, then the equation is called an exponential equation. Example 1 uses the following Equality of Exponents Theorem to solve 2x + 1 = 32.

Equality of Exponents Theorem If bx = b y, then x = y, provided b 7 0 and b Z 1.

EXAMPLE 1

Solve an Exponential Equation

Use the Equality of Exponents Theorem to solve 2x + 1 = 32. Solution 2x + 1 = 32 2x + 1 = 25 x + 1 = 5 x = 4

• Write each side as a power of 2. • Equate the exponents. • Solve for x.

Check: Let x = 4. Then 2x + 1 = 24 + 1 = 25 = 32 Try Exercise 2, page 386

Integrating Technology A graphing utility can also be used to find the solutions of an equation of the form f(x) = g(x). Either of the following two methods can be employed. Intersection Method Graph y1 = f(x) and y2 = g(x) on the same screen. The solutions of f(x) = g(x) are the x-coordinates of the points of intersection of the graphs. (continued)

382

CHAPTER 4


Intercept Method The solutions of f(x) = g(x) are the x-coordinates of the x-intercepts of the graph of y = f(x) - g(x). Figure 4.36 and Figure 4.37 illustrate the graphical methods for solving 2x + 1 = 32. 60

8 y1 = 2x + 1− 32

y2 = 32

9.4

0 y 1 = 2x + 1 0

Intersection X=4

x-intercept 9.4

Y=32

Zero X=4

Y=0

−8

−15

Intercept method

Intersection method Figure 4.36

Figure 4.37

In Example 1, we were able to write both sides of the equation as a power of the same base. If you find it difficult to write both sides of an exponential equation in terms of the same base, then try the procedure of taking the logarithm of each side of the equation. This procedure is used in Example 2.

EXAMPLE 2


Solve: 5x = 40 Visualize the Solution

Algebraic Solution 5 = 40 log(5x) = log 40 x log 5 = log 40 log 40 x = log 5 x

x L 2.3

• Take the logarithm of each side. • Power property

Intersection Method The solution of 5x = 40 is the x-coordinate of the point of intersection of y = 5x and y = 40.

• Exact solution 60

• Decimal approximation

To the nearest tenth, the solution is 2.3. y = 40 y = 5x 0

Intersection X=2.2920297 Y=40

4.7

−15


An alternative approach to solving the equation in Example 2 is to rewrite the exponential equation in logarithmic form: 5x = 40 is equivalent to the logarithmic equation log 40 log 5 40 = x. Using the change-of-base formula, we find that x = log 5 40 = . log 5 In Example 3, however, we must take logarithms of both sides to reach a solution.

4.5

EXAMPLE 3

383



Solve: 32x - 1 = 5x + 2 Visualize the Solution

Algebraic Solution 2x - 1

x+2

3 = 5 2x - 1 = ln 5x + 2 ln 3 (2x - 1) ln 3 2x ln 3 - ln 3 2x ln 3 - x ln 5 x(2 ln 3 - ln 5)

• Take the natural logarithm of each side.

= = = =

(x + 2) ln 5 x ln 5 + 2 ln 5 2 ln 5 + ln 3 2 ln 5 + ln 3 2 ln 5 + ln 3 x = 2 ln 3 - ln 5 x L 7.3

Intercept Method The solution of 32x - 1 = 5x + 2 is the x-coordinate of the x-intercept of y = 32x - 1 - 5x + 2.

• Power property

400,000

• Distributive property

y = 3 2x − 1− 5 x + 2

• Solve for x. • Factor.

−6

12

• Exact solution Zero X=7.3453319 Y=0

• Decimal approximation

− 400,000

To the nearest tenth, the solution is 7.3. Try Exercise 18, page 386

In Example 4, we solve an exponential equation that has two solutions.

EXAMPLE 4 Solve:

Solve an Exponential Equation Involving b x b x

2x + 2-x = 3 2

Algebraic Solution


Multiplying each side by 2 produces -x

2 + 2 = 6 22x + 20 = 6(2x) x

(2x)2 - 6(2x) + 1 = 0 (u)2 - 6(u) + 1 = 0

• Multiply each side by 2x to clear negative exponents. • Write in quadratic form. • Substitute u for 2x.

Intersection Method The solutions 2x + 2-x of = 3 are the x-coordinates 2 of the points of intersection of 2x + 2-x and y = 3. y = 2

By the quadratic formula,

5

6 136 - 4 6 4 12 u = = = 3 212 2 2 2x = 3 212 • Replace u with 2x. log 2x = log(3 212) • Take the common logarithm of each side.

x log 2 = log(3 212) log(3 212) x = L 2.54 log 2 The approximate solutions are 2.54 and 2.54. Try Exercise 42, page 387

• Power property • Solve for x.

y=3

(2.54, 3)

y=

(−2.54, 3) −4

2 x + 2 −x 2 4

−1

384

CHAPTER 4


Solving Logarithmic Equations Equations that involve logarithms are called logarithmic equations. The properties of logarithms, along with the definition of a logarithm, are often used to find the solutions of a logarithmic equation.

EXAMPLE 5

Solve a Logarithmic Equation

Solve: log(3x - 5) = 2 Solution log(3x - 5) = 2 3x - 5 = 102 3x = 105

• Definition of a logarithm • Solve for x.

x = 35 Check:

log33(35) - 54 = log 100 = 2


EXAMPLE 6


Solve: log 2x - log(x - 3) = 1 Solution log 2x - log(x - 3) = 1 log

2x = 1 x - 3

• Quotient property

2x = 101 x - 3

• Definition of a logarithm

2x = 10x - 30 - 8x = - 30 x = Check the solution by substituting

• Multiply each side by x 3. • Solve for x.

15 4 15 into the original equation. 4


In Example 7 we use the one-to-one property of logarithms to find the solution of a logarithmic equation.

4.5

EXAMPLE 7


385


Solve: ln(3x + 8) = ln(2x + 2) + ln(x - 2) Algebraic Solution


ln(3x + 8) = ln(2x + 2) + ln(x - 2)

The graph of

ln(3x + 8) = ln3(2x + 2)(x - 2)4

• Product property

y = ln(3x + 8) - ln(2x + 2) - ln(x - 2)

ln(3x + 8) = ln(2x - 2x - 4) 2

3x + 8 = 2x 2 - 2x - 4

x = -

3 2

• One-to-one property of logarithms

0 = 2x 2 - 5x - 12

• Subtract 3x 8 from each side.

0 = (2x + 3)(x - 4)

• Factor.

or

x = 4

has only one x-intercept. Thus there is only one real solution. y

2

• Solve for x.

x=4

3 A check will show that 4 is a solution but that - is not a solution. 2

2

4

x


Question • Why does x = -

EXAMPLE 8

3 not check in Example 7? 2

Velocity of a Sky Diver Experiencing Air Resistance

During the free fall portion of a jump, the time t, in seconds, required for a sky diver to reach a velocity v, in feet, per second is given by t = -

v 175 ln a1 b , 0 … v 6 175 32 175

a.

Determine the velocity of the diver after 5 seconds.

b.

The graph of t has a vertical asymptote at v = 175. Explain the meaning of the vertical asymptote in the context of this example.

Solution a. Substitute 5 for t and solve for v. t = -

v 175 ln a1 b 32 175

5 = -

175 v ln a1 b 32 175

• Replace t with 5. (continued)

3 7 7 , the original equation becomes ln a b = ln( - 1) + ln a - b . This cannot 2 2 2 be true because the function f(x) = ln x is not defined for negative values of x.

Answer • If x = -

386

CHAPTER 4


a-

32 v b5 = ln a1 b 175 175 -

• Multiply each side by -

v 32 = ln a1 b 35 175

e-32/35 = 1 -

32 . 175

• Simplify.

v 175

• Write in exponential form.

v 175 v = 175(1 - e-32/35)

e-32/35 - 1 = -

• Subtract 1 from each side. • Multiply each side by -175.

v L 104.86 After 5 seconds the velocity of the sky diver will be about 104.9 feet per second. See Figure 4.38.

Note

t v = 175

Time (in seconds)

20 15 10 5

If air resistance is not considered, then the time in seconds required for a sky diver to reach a given velocity (in feet per second) is v . The function in Example 8 t = 32 is a more realistic model of the time required to reach a given velocity during the free fall of a sky diver who is experiencing air resistance.

104.9 50

100

150

v

Velocity (in feet per second)

t = − 175 ln(1 − v ) 32 175 Figure 4.38

The vertical asymptote v = 175 indicates that the velocity of the sky diver approaches, but never reaches or exceeds, 175 feet per second. In Figure 4.38, note that as v : 175 from the left, t : q .

b.


EXERCISE SET 4.5 In Exercises 1 to 48, use algebraic procedures to find the exact solution or solutions of the equation. 1. 2x = 64 3. 49x =

2. 3x = 243

1 343

5. 25x + 3 =

1 8

2 x 8 7. a b = 5 125

4. 9x =

1 243

6. 34x - 7 =

1 9

2 x 25 8. a b = 5 4

9. 5x = 70

10. 6 x = 50

11. 3 - x = 120

12. 7 - x = 63

13. 102x + 3 = 315

14. 106 - x = 550

15. e x = 10

16. e x + 1 = 20

17. 21 - x = 3x + 1

18. 3 x - 2 = 4 2x + 1

19. 22x - 3 = 5 - x - 1

20. 53x = 3x + 4

21. log(4x - 18) = 1

22. log(x 2 + 19) = 2

4.5

23. ln(x 2 - 12) = ln x


55. ln(2x + 4) +

24. log(2x 2 + 3x) = log(10x + 30)

1 x = -3 2

57. 2 x + 1 = x 2 - 1

25. log 2 x + log 2(x - 4) = 2

387

56. 2 ln(3 - x) + 3x = 4 58. ln x = - x 2 + 4

59. Population Growth The population P of a city grows expo-

nentially according to the function

26. log 3 x + log 3(x + 6) = 3

P(t) = 8500(1.1)t,

0 … t … 8

27. log(5x - 1) = 2 + log(x - 2)

where t is measured in years.

28. 1 + log(3x - 1) = log(2x + 1)

a. Find the population at time t = 0 and at time t = 2.

29. ln(1 - x) + ln(3 - x) = ln 8

b. When, to the nearest year, will the population reach 15,000? 60. Physical Fitness After a race, a runner’s pulse rate R, in beats

30. log(4 - x) = log(x + 8) + log(2x + 13)

per minute, decreases according to the function

1 31. log 2x - 17 = 2

32. log(x ) = (log x)

33. log(log x) = 1

34. ln(ln x) = 2

3

3

2

0 … t … 15

where t is measured in minutes. a. Find the runner’s pulse rate at the end of the race and 1 minute

after the end of the race. b. How long, to the nearest minute, after the end of the race

35. ln(e3x) = 6 36. ln x =

R(t) = 145e - 0.092t,

will the runner’s pulse rate be 80 beats per minute?

1 5 1 ln a2x + b + ln 2 2 2 2

61. Rate of Cooling A can of soda at 79°F is placed in a refrig-

erator that maintains a constant temperature of 36°F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by

37. log7(5x) - log7 3 = log7(2x + 1)

T(t) = 36 + 43e - 0.058t

38. log4 x + log4(x - 2) = log4 15

a. Find the temperature, to the nearest degree, of the soda 39. eln(x - 1) = 4 41.

10 x - 10 - x = 20 2

43.

45.

40. 10log(2x + 7) = 8 42.

10 x + 10 - x = 8 2

10 + 10 = 5 10 x - 10 - x

44.

10 - 10 1 = 10 x + 10 - x 2

ex + e - x = 15 2

46.

ex - e - x = 15 2

10 minutes after it is placed in the refrigerator. b. When, to the nearest minute, will the temperature of the

soda be 45°F? 62. Medicine During surgery, a patient’s circulatory system

x

-x

1 47. x = 4 e - e-x

x

-x

ex + e - x 48. x = 3 e - e-x

requires at least 50 milligrams of an anesthetic. The amount of anesthetic present t hours after 80 milligrams of anesthetic is administered is given by T(t) = 80(0.727)t a. How much, to the nearest milligram, of the anesthetic is

present in the patient’s circulatory system 30 minutes after the anesthetic is administered? b. How long, to the nearest minute, can the operation last if

the patient does not receive additional anesthetic? In Exercises 49 to 58, use a graphing utility to approximate the solution or solutions of the equation to the nearest hundredth. 49. 2 - x + 3 = x + 1

50. 3x - 2 = - 2x - 1

51. e 3 - 2x - 2x = 1

52. 2e x + 2 + 3x = 2

53. 3 log 2(x - 1) = - x + 3

54. 2 log 3(2 - 3x) = 2x - 1

In 1938, the biologist Ludwig von Bertalanffy developed the equation

Bertalanffy’s Equation

L m (m L0)erx which models the length L, in centimeters, of a fish as it grows under optimal conditions for a period of x years. In Bertalanffy’s equation, m represents the maximum

388

CHAPTER 4


and

length, in centimeters, the fish is expected to attain; L0 is the length, in centimeters, of the fish at birth; and r is a constant related to the growth rate of the fish species. Use Bertalanffy’s equation to predict the age of the fish described in Exercises 63 and 64.

h2(x) = 568.2 - 161.5 ln x,

6.1 … x … 16.47

approximate the height, in meters, of the Eiffel Tower x meters to the right of the center line, shown by the y-axis in the following figure.

63. A barracuda has a length of 114 centimeters. Use Bertalanffy’s

equation to predict, to the nearest tenth of a year, the age of the barracuda. Assume m = 198 centimeters, L 0 = 0.9 centimeter, and r = 0.23.

y Third stage 276.13 m

300

h2

64. A haddock has a length of 21 centimeters. Use Bertalanffy’s

200

equation to predict, to the nearest tenth of a year, the age of the haddock. Assume m = 94 centimeters, L0 = 0.6 centimeters, and r = 0.21. Second stage 115.73 m

65. Typing Speed The following function models the average

typing speed S, in words per minute, for a student who has been typing for t months.

100

S(t) = 5 + 29 ln(t + 1), 0 … t … 9

h1

First stage 57.63 m

Use S to determine how long it takes the student to achieve an average typing speed of 65 words per minute. Round to the nearest tenth of a month. −50

66. Walking Speed An approximate relation between the average

pedestrian walking speed s, in miles per hour, and the population x, in thousands, of a city is given by the formula

50

x

The graph of h1 models the shape of the tower from ground level up to the second stage in the figure, and the graph of h2 models the shape of the tower from the second stage up to the third stage. Determine the horizontal distance across the Eiffel Tower, rounded to the nearest tenth of a meter, at a height of

s(x) = 0.37 ln x + 0.05 Use s to estimate the population of a city for which the average pedestrian walking speed is 2.9 miles per hour. Round to the nearest hundred-thousand.

a. 50 meters 67. Drag Racing The quadratic function

s1(x) = - 2.25x 2 + 56.26x - 0.28,

b. 125 meters

0 … x … 10

models the speed of a dragster from the start of a race until the dragster crosses the finish line 10 seconds later. This is the acceleration phase of the race. The exponential function

69.

Psychology Industrial psychologists study employee training programs to assess the effectiveness of the instruction. In one study, the percent score P on a test for a person who had completed t hours of training was given by

s2(x) = 8320(0.73)x, 10 6 x … 20 models the speed of the dragster during the 10-second period immediately following the time when the dragster crosses the finish line. This is the deceleration period. How long after the start of the race did the dragster attain a speed of 275 miles per hour? Round to the nearest hundredth of a second. 68. Eiffel Tower The functions

h1(x) = 363.4 - 88.4 ln x,

P =

1 + 30e - 0.088t

a. Use a graphing utility to graph the equation for t Ú 0. b. Use the graph to estimate (to the nearest hour) the number of

hours of training necessary to achieve a 70% score on the test. c. From the graph, determine the horizontal asymptote. d.

16.47 6 x … 61.0

100

Write a sentence that explains the meaning of the horizontal asymptote.

4.5

70.

Psychology An industrial psychologist has determined that the average percent score for an employee on a test of the employee’s knowledge of the company’s product is given by

P =

T =

100 1 + 40e - 0.1t

389

Consumption of Natural Resources A model for how long our coal resources will last is given by

73.

ln(300r + 1) ln(r + 1)

where t is the number of weeks on the job and P is the percent score.

where r is the percent increase in consumption from current levels of use and T is the time, in years, before the resources are depleted.

a. Use a graphing utility to graph the equation for t Ú 0.

a. Use a graphing utility to graph this equation.

b. Use the graph to estimate (to the nearest week) the expected

b. If our consumption of coal increases by 3% per year, in how

many years will we deplete our coal resources?

number of weeks of employment that are necessary for an employee to earn a 70% score on the test.

c. What percent increase in consumption of coal will deplete

the resources in 100 years? Round to the nearest tenth of a percent.

c. Determine the horizontal asymptote of the graph. d.

71.



74.

Ecology A herd of bison was placed in a wildlife preserve that can support a maximum of 1000 bison. A population model for the bison is given by

B =

Effects of Air Resistance on Velocity If we assume that air resistance is proportional to the square of the velocity, then the time t, in seconds, required for an object to reach a velocity v in feet per second is given by

t =

1000 1 + 30e - 0.127t

9 24 + v ln , 0 … v 6 24 24 24 - v

a. Determine the velocity, to the nearest hundredth of a foot

per second, of the object after 1.5 seconds.

where B is the number of bison in the preserve and t is time in years, with the year 1999 represented by t = 0.

b. Determine the vertical asymptote for the graph of this


c.

function. Write a sentence that explains the meaning of the vertical asymptote in the context of this application.

b. Use the graph to estimate (to the nearest year) the number

of years before the bison population reaches 500.

75.

c. Determine the horizontal asymptote of the graph. d.


Terminal Velocity with Air Resistance The velocity v, in feet per second, of an object t seconds after it has been dropped from a height above the surface of the Earth is given by the equation v = 32t, assuming no air resistance. If we assume that air resistance is proportional to the square of the velocity, then the velocity after t seconds is given by

Population Growth A yeast culture grows according to

72.

v = 100a

the equation Y =

- 0.305t

function.


c.

b. Use the graph to estimate (to the nearest hour) the number

c. From the graph, estimate the horizontal asymptote. d.


b

b. Determine the horizontal asymptote for the graph of this

where Y is the number of yeast and t is time in hours.

of hours before the yeast population reaches 35,000.

e0.64t + 1

a. In how many seconds will the velocity be 50 feet per second?

50,000 1 + 250e

e0.64t - 1

76.

Write a sentence that explains the meaning of the horizontal asymptote in the context of this application.

Effects of Air Resistance on Distance The distance s, in feet, that the object in Exercise 75 will fall in t seconds is given by

s =

1002 e 0.32t + e - 0.32t ln a b 32 2

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CHAPTER 4


a. Use a graphing utility to graph this equation for t Ú 0.

b. What is the height of the cable 10 feet to the right of the

y-axis? Round to the nearest tenth of a foot. b. How long does it take for the object to fall 100 feet? Round

c. How far to the right of the y-axis is the cable 24 feet in

to the nearest tenth of a second.

height? Round to the nearest tenth of a foot.

77. Retirement Planning The retirement account for a graphic

designer contains $250,000 on January 1, 2006, and earns interest at a rate of 0.5% per month. On February 1, 2006, the designer withdraws $2000 and plans to continue these withdrawals as retirement income each month. The value V of the account after x months is

79. The following argument seems to indicate that 0.125 7 0.25.

Find the first incorrect statement in the argument. 3 3(log 0.5) log 0.5 3 0.5 3 0.125

V = 400,000 - 150,000(1.005)x If the designer wishes to leave $100,000 to a scholarship foundation, what is the maximum number of withdrawals the designer can make from this account and still have $100,000 to donate?

first incorrect statement in the argument. 4 4 4 4 4

cable shown below is given by -15 … x … 15

where ƒ x ƒ is the horizontal distance, in feet, between P and the y-axis.

2 2(log 0.5) log 0.5 2 0.5 2 0.25

80. The following argument seems to indicate that 4 = 6. Find the

78. Hanging Cable The height h, in feet, of any point P on the

h(x) = 10(e x>20 + e -x>20),

7 7 7 7 7

= = = = =

log 2 16 log 2(8 + 8) log 2 8 + log 2 8 3 + 3 6

81. A common mistake that students make is to write log(x + y)

as log x + log y. If log(x + y) = log x + log y, then what is the relationship between x and y? (Hint: Solve for x in terms of y.)

y P

82. Let f (x) = 2 ln x and g(x) = ln x 2. Does f (x) = g(x) for all x? 83. −15

15

x

84. Find k such that f (t) = 2.2t and g(t) = e - kt represent essen-

a. What is the lowest height of the cable?

SECTION 4.6 Exponential Growth and Decay Carbon Dating Compound Interest Formulas Restricted Growth Models

Explain why the functions F(x) = 1.4x and G(x) = e0.336x represent essentially the same function. tially the same function.

Exponential Growth and Decay PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A29.

PS1. Evaluate A = 1000a 1 + PS2. Evaluate A = 600a 1 +

0.1 12t b for t = 2. Round to the nearest hundredth. [4.2] 12

0.04 4t b for t = 8. Round to the nearest hundredth. [4.2] 4

PS3. Solve 0.5 = e14k for k. Round to the nearest ten-thousandth. [4.5] PS4. Solve 0.85 = 0.5t>5730 for t. Round to the nearest ten. [4.5] PS5. Solve 6 =

70 # for k. Round to the nearest thousandth. [4.5] 5 + 9e-k 12

PS6. Solve 2,000,000 =

3n + 1 - 3 for n. Round to the nearest tenth. [4.5] 2

4.6

EXPONENTIAL GROWTH AND DECAY

391

Exponential Growth and Decay In many applications, a quantity changes at a rate proportional to the amount present. In these applications, the amount present at time t is given by a special function called an exponential growth function or an exponential decay function.

Definition of Exponential Growth and Decay Functions If a quantity N increases or decreases at a rate proportional to the amount present at time t, then the quantity can be modeled by N(t) = N0e kt where N0 is the value of N at time t = 0 and k is a constant called the growth rate constant. If k is positive, N increases as t increases and N(t) = N0e kt is called an exponential growth function. See Figure 4.39. If k is negative, N decreases as t increases and N(t) = N0e kt is called an exponential decay function. See Figure 4.40. N

N

N0

N0

N(t) = N0 e kt; k < 0, t ≥ 0

N(t) = N0 e kt; k > 0, t ≥ 0 t

t

Exponential growth function

Exponential decay function

Figure 4.39

Figure 4.40

Question • Is N(t) = 1450e0.05t an exponential growth function or an exponential decay function?

In Example 1, we find an exponential growth function that models the population growth of a city.

EXAMPLE 1

Find the Exponential Growth Function That Models Population Growth

a.

The population of a city is growing exponentially. The population of the city was 16,400 in 1999 and 20,200 in 2009. Find the exponential growth function that models the population growth of the city.

b.

Use the function from a. to predict, to the nearest 100, the population of the city in 2014. (continued)

Answer • Because the growth rate constant k = 0.05 is positive, the function is an exponential

growth function.

392

CHAPTER 4


Solution a. We need to determine N0 and k in N(t) = N0 e kt. If we represent 1999 by t = 0, then our given data are N(0) = 16,400 and N(10) = 20,200. Because N0 is defined to be N(0), we know that N0 = 16,400. To determine k, substitute t = 10 and N0 = 16,400 into N(t) = N0 e kt to produce N(10) = 16,400e k 20,200 20,200 16,400 20,200 ln 16,400 20,200 1 ln 10 16,400

# 10

= 16,400e 10k

• Substitute 20,200 for N (10).

= e 10k

• Solve for e10k.

= 10k

• Write in logarithmic form.

= k

• Solve for k.

0.0208 L k The exponential growth function is N(t) L 16,400e0.0208t. b.

The year 1999 was represented by t = 0, so we will use t = 15 to represent 2014. N(t) L 16,400e0.0208t

#

N(15) L 16,400e0.0208 15 L 22,400

• Round to the nearest 100.

The exponential growth function yields 22,400 as the approximate population of the city in 2014. Try Exercise 6, page 400

Many radioactive materials decrease in mass exponentially over time. This decrease, called radioactive decay, is measured in terms of half-life, which is defined as the time required for the disintegration of half the atoms in a sample of a radioactive substance. Table 4.11 shows the half-lives of selected radioactive isotopes. Table 4.11

Isotope

Half-Life

14

5730 years

Carbon ( C) 226

Radium (

Ra)

210

Polonium (

Po)

32

Phosphorus ( P) Polonium (214 Po)

EXAMPLE 2

1660 years 138 days 14 days 1>10,000 of a second

Find an Exponential Decay Function

Find the exponential decay function for the amount of phosphorus (32P) that remains in a sample after t days.

4.6


393

Solution When t = 0, N(0) = N0e k(0) = N0. Thus N(0) = N0. Also, because the phosphorus has a half-life of 14 days (from Table 4.11), N(14) = 0.5N0. To find k, substitute t = 14 into N(t) = N0e kt and solve for k. N(14) = N0 # e k 14 0.5N0 = N0 e14k

#

0.5 = e

14k

Study tip

• Divide each side by N0.

ln 0.5 = 14k

Because e-0.0495 L (0.5)1>14, the decay function N(t) = N0 e-0.0495t can also be written as N(t) = N0 (0.5)t>14. In this form, it is easy to see that if t is increased by 14 then N will decrease by a factor of 0.5.

• Substitute 0.5N0 for N (14). • Write in logarithmic form.

1 ln 0.5 = k 14

• Solve for k.

-0.0495 L k The exponential decay function is N(t) L N0 e-0.0495t. Try Exercise 8, page 400

Carbon Dating

P(t)

The bone tissue in all living animals contains both carbon-12, which is nonradioactive, and carbon-14, which is radioactive and has a half-life of approximately 5730 years. See Figure 4.41. As long as the animal is alive, the ratio of carbon-14 to carbon-12 remains constant. When the animal dies (t = 0), the carbon-14 begins to decay. Thus a bone that has a smaller ratio of carbon-14 to carbon-12 is older than a bone that has a larger ratio. The percent of carbon-14 present at time t, in years, is

Amount of carbon-14

N0

0.75 N0

0.50 N0

P(t) = 0.5t>5730

0.25 N0

5730

17,190

28,650

t

The process of using the percent of carbon-14 present at a given time to estimate the age of a bone is called carbon dating.

Time (in years)

P(t) = 0.5t/5730 Figure 4.41

EXAMPLE 3

A Carbon Dating Application

Estimate the age of a bone if it now has 85% of the carbon-14 it had at time t = 0.

Math Matters The chemist Willard Frank Libby developed the carbon dating process in 1947. In 1960 he was awarded the Nobel Prize in chemistry for this achievement.

Solution Let t be the time, in years, at which P(t) = 0.85. 0.85 = 0.5t>5730 ln 0.85 = ln 0.5t>5730 ln 0.85 = 5730 a

t ln 0.5 5730

ln 0.85 b = t ln 0.5 1340 L t

The bone is approximately 1340 years old. Try Exercise 12, page 400

• Take the natural logarithm of each side. • Apply the power property. • Solve for t.

394

CHAPTER 4


Compound Interest Formulas Interest is money paid for the use of money. The interest I is called simple interest if it is a fixed percent r, per time period t, of the amount of money invested. The amount of money invested is called the principal P. Simple interest is computed using the formula I = Prt. For example, if $1000 is invested at 12% for 3 years, the simple interest is I = Prt = $1000(0.12)(3) = $360 The balance after t years is A = P + I = P + Prt. In the preceding example, the $1000 invested for 3 years produced $360 interest. Thus the balance after 3 years is $1000 + $360 = $1360. In many financial transactions, interest is added to the principal at regular intervals so that interest is paid on interest, as well as on the principal. Interest earned in this manner is called compound interest. For example, if $1000 is invested at 12% annual interest compounded annually for 3 years, then the total interest after 3 years is First-year interest Second-year interest Third-year interest

Table 4.12

Number of Years

Balance

3

A3 = P(1 + r)3

4 . . .

A4 = P(1 + r)4 . . .

t

At = P(1 + r)t

$1000(0.12) = $120.00 $1120(0.12) = $134.40 $1254.40(0.12) L $150.53 $404.93

• Total interest

This method of computing the balance can be tedious and time-consuming. A compound interest formula can be used to determine the balance due after t years of compounding. Note that if P dollars is invested at an interest rate of r per year, then the balance after 1 year is A1 = P + Pr = P(1 + r), where Pr represents the interest earned for the year. Observe that A1 is the product of the original principal P and (1 + r). If the amount A1 is reinvested for another year, then the balance after the second year is A2 = (A1) (1 + r) = P(1 + r)(1 + r) = P(1 + r)2 Successive reinvestments lead to the results shown in Table 4.12. The equation At = P(1 + r)t is valid if r is the annual interest rate paid during each of the t years. If r is an annual interest rate and n is the number of compounding periods per year, then the interest rate each period is r>n, and the number of compounding periods after t years is nt. Thus the compound interest formula is as follows.

Compound Interest Formula A principal P invested at an annual interest rate r, expressed as a decimal and compounded n times per year for t years, produces the balance A = P a1 +

EXAMPLE 4

r nt b n

Solve a Compound Interest Application

Find the balance if $1000 is invested at an annual interest rate of 10% for 2 years compounded on the following basis. a.

Monthly

b.

Daily

4.6


395

Solution a.

Because there are 12 months in a year, use n = 12. 0.1 12 b 12

A = $1000a1 + b.

#2

L $1000(1.008333333)24 L $1220.39

Because there are 365 days in a year, use n = 365. A = $1000a 1 +

0.1 365 b 365

#2

L $1000(1.000273973)730 L $1221.37


To compound continuously means to increase the number of compounding periods per year, n, without bound. r 1 To derive a continuous compounding interest formula, substitute for in the comm n pound interest formula A = Pa 1 +

r nt b n

(1)

A = Pa 1 +

1 nt b m

(2)

to produce

m r

r n 1 This substitution is motivated by the desire to express a1 + b as B a1 + b R , n m which approaches e r as m gets larger without bound. 1 r = for n yields n = mr, so the exponent nt can be written m n as mrt. Therefore, Equation (2) can be expressed as Solving the equation

m rt

A = Pa 1 +

1 mrt 1 = P Ba 1 + b R b m m

By the definition of e, we know that as m increases without bound, a1 +

1 m b m

approaches

e

Thus, using continuous compounding, Equation (3) simplifies to A = Pe rt.

Continuous Compounding Interest Formula If an account with principal P and annual interest rate r is compounded continuously for t years, then the balance is A = Pert.

(3)

396

CHAPTER 4

EXAMPLE 5


Solve a Continuous Compound Interest Application

Find the balance after 4 years on $800 invested at an annual rate of 6% compounded continuously. Algebraic Solution


Use the continuous compounding formula with P = 800, r = 0.06, and t = 4.

The following graph of A = 800e0.06t shows that the balance is about $1017.00 when t = 4.

A = Pe rt = 800e0.06(4)

• Substitute given values.

= 800e

• Simplify.

0.24

1400

Y1=800e^(.06X)

L 800(1.27124915) L 1017.00

• Round to the nearest cent. 9.4

0

The balance after 4 years will be $1017.00.

X=4

Y=1016.9993

−200


You have probably heard it said that time is money. In fact, many investors ask the question “How long will it take to double my money?” The following example answers this question for two different investments.

EXAMPLE 6

Double Your Money

Find the time required for money invested at an annual rate of 6% to double in value if the investment is compounded on the following basis. a.

Semiannually

b. Continuously

Solution a.

r nt b with r = 0.06, n = 2, and the balance A equal to twice the n principal (A = 2P). Use A = Pa 1 +

2P = P a1 + 2 = a1 +

0.06 2t b 2

0.06 2t b 2

ln 2 = ln a1 +

0.06 2t b 2

ln 2 = 2t ln a1 + 2t =

0.06 b 2

ln 2 0.06 ln a 1 + b 2

• Divide each side by P. • Take the natural logarithm of each side. • Apply the power property. • Solve for t.

4.6

t =

1# 2


397

ln 2 ln a 1 +

0.06 b 2

t L 11.72 If the investment is compounded semiannually, it will double in value in about 11.72 years. b.

Use A = Pe rt with r = 0.06 and A = 2P. 2P = Pe0.06t 2 = e0.06t ln 2 = 0.06t t =

ln 2 0.06

• Divide each side by P. • Write in logarithmic form. • Solve for t.

t L 11.55 If the investment is compounded continuously, it will double in value in about 11.55 years. Try Exercise 22, page 401

Restricted Growth Models The exponential growth function N(t) = N 0 e kt is an unrestricted growth model that does not consider any limited resources that eventually will curb population growth. The logistic model is a restricted growth model that takes into consideration the effects of limited resources. The logistic model was developed by Pierre Verhulust in 1836.

Definition of the Logistic Model (Restricted Growth Model) The magnitude of a population at time t Ú 0 is given by c P(t) = P(t) 1 + ae-bt c where c is the carrying capacity (the maximum population that can be supported by available resources as t : q ) and b is a positive constant called the growth rate constant. The initial population is P0 = P(0). The constant a is related to the initial population P0 and the carrying capacity c by the formula a =

c - P0 P0

P0 t

P(t) =

c , 0 < P0 < c 1 + ae−bt

In the following example, we determine a logistic growth model for a coyote population.

CHAPTER 4


EXAMPLE 7

Find and Use a Logistic Model

At the beginning of 2007, the coyote population in a wilderness area was estimated at 200. By the beginning of 2009, the coyote population had increased to 250. A park ranger estimates that the carrying capacity of the wilderness area is 500 coyotes. Clm138/Dreamstime.com

398

a.

Use the given data to determine the growth rate constant for the logistic model of this coyote population.

b.

Use the logistic model determined in a. to predict the year in which the coyote population will first reach 400.

Solution a. If we represent the beginning of 2007 by t = 0, then the beginning of 2009 will be represented by t = 2. In the logistic model, make the following substitutions: c - P0 500 - 200 = P(2) = 250, c = 500, and a = = 1.5. P0 200 P(t) =

c 1 + ae-bt

P(2) =

500 # 1 + 1.5e-b 2

• Substitute the given values for t, c, and a.

250 =

500 # 1 + 1.5e-b 2

• P (2) = 250

#

250(1 + 1.5e-b 2) = 500 1 + 1.5e-b

#2

1.5e-b

#2

e-b

#2

=

• Solve for the growth rate constant b.

500 250

= 2 - 1 =

1 1.5

-2b = ln a b = -

1 b 1.5


1 1 ln a b 2 1.5

b L 0.20273255 Using a = 1.5, b = 0.20273255, and c = 500 gives us the following logistic model. P(t) = b.

500 1 + 1.5e-0.20273255t

To determine in which year the logistic model predicts that the coyote population will first reach 400, replace P(t) with 400 and solve for t. 400 =

500 1 + 1.5e-0.20273255t

400(1 + 1.5e-0.20273255t ) = 500

4.6

P(t)

1 + 1.5e-0.20273255t =

500

Coyote population

399

500 400

1.5e-0.20273255t = 1.25 - 1 0.25 e-0.20273255t = 1.5

(8.8, 400)

400


300

-0.20273255t = ln a

200

t =

100

0.25 b 1.5


0.25 1 ln a b - 0.20273255 1.5

• Solve for t.

L 8.8 10

20

30

t

Year (t = 0 represents the beginning of 2007)

P(t) =

500 1 + 1.5e−0.20273255t

According to the logistic model, the coyote population will reach 400 about 8.8 years after the beginning of 2007, which is during 2015. The graph of the logistic model is shown in Figure 4.42. Note that P(8.8) L 400 and that as t : q , P(t) : 500. Try Exercise 38, page 401

Figure 4.42

In Example 8, we use a function of the form v = a(1 - e -kt ) to model the velocity of an object that has been dropped from a high elevation.

EXAMPLE 8

Application to Air Resistance

Assuming that air resistance is proportional to the velocity of a falling object, the velocity (in feet per second) of the object t seconds after it has been dropped is given by v = 82(1 - e-0.39t ). a.

Determine when the velocity will be 70 feet per second.

b.

The graph of v has v = 82 as a horizontal asymptote. Explain the meaning of this asymptote in the context of this example.

Algebraic Solution a.

Visualize the Solution -0.39t

) v = 82(1 - e -0.39t 70 = 82(1 - e ) 70 = 1 - e-0.39t 82 e-0.39t = 1 - 0.39t = ln t =

70 82

6 41

ln(6>41) L 4.9277246 -0.39

a. • Substitute 70 for v. • Divide each side by 82.

90

• Solve for e-0.39t. • Write in logarithmic form. • Solve for t.

The velocity will be 70 feet per second after approximately 4.9 seconds. b.

The horizontal asymptote v = 82 means that, as time increases, the velocity of the object will approach, but never reach or exceed, 82 feet per second. Try Exercise 48, page 403

A graph of y = 82(1 - e-0.39x ) and y = 70 shows that the x-coordinate of the point of intersection is about 4.9.

0 X=4.9 0

Y=70

15

y = 82(1 - e-0.39x)

(Note: The x value shown is rounded to the nearest tenth.)

400

CHAPTER 4


EXERCISE SET 4.6 micrograms remaining in that person after t hours is given by the equation A = 4e - 0.046t.

Population Growth In Exercises 1 to 6, solve the given

problem related to population growth.

a. Graph this equation.

1. The number of bacteria N(t) present in a culture at time t hours

is given by

b. What amount of sodium-24 remains after 5 hours?

N(t) = 2200(2)t c. What is the half-life of sodium-24? d. In how many hours will the amount of sodium-24 be

Find the number of bacteria present when a. t = 0 hours

b. t = 3 hours

2. The population of a city grows exponentially according to the

function f (t) = 12,400(1.14)t

1 microgram? In Exercises 8 to 12, use the half-life information from Table 4.11, page 392, to work each exercise. 8.

amount of polonium (210 Po) that remains in a sample after t days.

for 0 … t … 5 years. Find, to the nearest hundred, the population of the city when t is a. 3 years

b. 4.25 years

9.

Geology Geologists have determined that Crater Lake in Oregon was formed by a volcanic eruption. Chemical analysis of a wood chip assumed to be from a tree that died during the eruption has shown that it contains approximately 45% of its original carbon-14. Estimate how long ago the volcanic eruption occurred.

10.

Radioactive Decay Estimate the percentage of polo-

3. A city had a population of 22,600 in 2000 and a population of

24,200 in 2005. a. Find the exponential growth function for the city. Use t = 0

to represent 2000. b. Use the growth function to predict the population of the city

nium (210Po) that remains in a sample after 2 years. Round to the nearest hundredth of a percent.

in 2015. Round to the nearest hundred. 4. A city had a population of 53,700 in 2002 and a population of

58,100 in 2006.

Radioactive Decay Find the decay function for the

11.

Archeology The Rhind papyrus, named after A. Henry Rhind, contains most of what we know today of ancient Egyptian mathematics. A chemical analysis of a sample from the papyrus has shown that it contains approximately 75% of its original carbon-14. Estimate the age of the Rhind papyrus.

12.

Archeology Estimate the age of a bone if it now contains 65% of its original amount of carbon-14. Round to the nearest 100 years.

a. Find the exponential growth function for the city. Use t = 0

to represent 2002. b. Use the growth function to predict the population of the city

in 2014. Round to the nearest hundred. 5.

The population of Charlotte, North Carolina, is growing exponentially. The population of Charlotte was 395,934 in 1990 and 610,949 in 2005. Find the exponential growth function that models the population of Charlotte and use it to predict the population of Charlotte in 2012. Use t = 0 to represent 1990. Round to the nearest thousand.

Compound Interest In Exercises 13 to 20, solve the given

problem related to compound interest. 13. If $8000 is invested at an annual interest rate of 5% and com-

pounded annually, find the balance after Las Vegas, Nevada, is growing exponentially. The population of Las Vegas was 258,295 in 1990 and 545,147 in 2005. Find the exponential growth function that models the population of Las Vegas and use it to predict the population of Las Vegas in 2013. Use t = 0 to represent 1990. Round to the nearest thousand.

14. If $22,000 is invested at an annual interest rate of 4.5% and

7. Medicine Sodium-24 is a radioactive isotope of sodium that is

15. If $38,000 is invested at an annual interest rate of 6.5% for

6.

used to study circulatory dysfunction. Assuming that 4 micrograms of sodium-24 are injected into a person, the amount A in

a. 4 years

b. 7 years

compounded annually, find the balance after a. 2 years

b. 10 years

4 years, find the balance if the interest is compounded a. annually

b. daily

c. hourly

4.6

16. If $12,500 is invested at an annual interest rate of 8% for 10

years, find the balance if the interest is compounded a. annually

b. daily

c. hourly

17. Find the balance if $15,000 is invested at an annual rate of 10%

for 5 years, compounded continuously.


401

32. P0 = 6200, P(8) = 7100, and the carrying capacity is 9500. 33. P0 = 18, P(3) = 30, and the carrying capacity is 100. 34. P0 = 3200, P(22) L 5565, and the growth rate constant is 0.056. 35. Revenue The annual revenue R, in dollars, of a new company

18. Find the balance if $32,000 is invested at an annual rate of 8%

for 3 years, compounded continuously.

can be closely modeled by the logistic function R(t) =

19. How long will it take $4000 to double if it is invested in a

certificate of deposit that pays 7.84% annual interest compounded continuously? Round to the nearest tenth of a year. 20. How long will it take $25,000 to double if it is invested in a

savings account that pays 5.88% annual interest compounded continuously? Round to the nearest tenth of a year.

625,000 1 + 3.1e-0.045t

where the natural number t is the time, in years, since the company was founded. a. According to the model, what will be the company’s annual

revenue for its first year and its second year (t = 1 and t = 2) of operation? Round to the nearest $1000. b. According to the model, what will the company’s annual

Continuous Compounding Interest In Exercises 21 to 24,

solve the given problem related to continuous compounding interest.

revenue approach in the long-term future? 36. New Car Sales The number of cars A sold annually by an auto-

21. Use the continuous compounding interest formula to derive an

expression for the time it will take money to triple when invested at an annual interest rate of r compounded continuously. 22. How long will it take $1000 to triple if it is invested at an

annual interest rate of 5.5% compounded continuously? Round to the nearest year. 23. How long will it take $6000 to triple if it is invested in a sav-

ings account that pays 7.6% annual interest compounded continuously? Round to the nearest year.

mobile dealership can be closely modeled by the logistic function A(t) =

1650 1 + 2.4e-0.055t

where the natural number t is the time, in years, since the dealership was founded. a. According to the model, what number of cars will the deal-

ership sell during its first year and its second year (t = 1 and t = 2) of operation? Round to the nearest unit. b. According to the model, what will the dealership’s annual

car sales approach in the long-term future? 24. How long will it take $10,000 to triple if it is invested in a sav-

ings account that pays 5.5% annual interest compounded continuously? Round to the nearest year. In Exercises 25 to 30, determine the following constants for the given logistic growth model. a. The carrying capacity b. The growth rate constant c. The initial population P0 25. P(t) =

27. P(t) =

29. P(t) =

1900 1 + 8.5e

-0.16t

157,500 1 + 2.5e-0.04t 2400 1 + 7e

-0.12t

26. P(t) =

28. P(t) =

30. P(t) =

32,550 1 + 0.75e

-0.08t

51 1 + 1.04e-0.03t

37. The population of wolves in a preserve satisfies a logistic model

in which P0 = 312 in 2008, c = 1600, and P(6) = 416. a. Determine the logistic model for this population, where t is

the number of years after 2008. b. Use the logistic model from a. to predict the size of the wolf

population in 2018. 38. The population of groundhogs on a ranch satisfies a logistic

model in which P0 = 240 in 2007, c = 3400, and P(1) = 310. a. Determine the logistic model for this population, where t is

the number of years after 2007.

320 1 + 15e

Population Growth In exercises 37 to 40, solve the given problem related to population growth.

-0.12t

In Exercises 31 to 34, use algebraic procedures to find the logistic growth model for the data. 31. P0 = 400, P(2) = 780, and the carrying capacity is 5500.

b. Use the logistic model from a. to predict the size of the

groundhog population in 2014. 39. The population of squirrels in a nature preserve satisfies a

logistic model in which P0 = 1500 in 2007. The carrying

402

CHAPTER 4


capacity of the preserve is estimated at 8500 squirrels, and P(2) = 1900. a. Determine the logistic model for this population, where t is

44. Law A lawyer has determined that the number of people P(t)

in a city of 1.2 million people who have been exposed to a news item after t days is given by the function

the number of years after 2007. b. Use the logistic model from a. to predict the year in which

the squirrel population will first exceed 4000. 40. The population of walruses on an island satisfies a logistic

model in which P0 = 800 in 2006. The carrying capacity of the island is estimated at 5500 walruses, and P(1) = 900.

P(t) = 1,200,000(1 - e - 0.03t ) a. How many days after a major crime has been reported has

40% of the population heard of the crime? b. A defense lawyer knows it will be difficult to pick an unbi-

ased jury after 80% of the population has heard of the crime. After how many days will 80% of the population have heard of the crime?

a. Determine the logistic model for this population, where t

is the number of years after 2006. b. Use the logistic model from a. to predict the year in which

the walrus population will first exceed 2000. 41. Physics Newton’s Law of Cooling states that if an object at tem-

perature T0 is placed into an environment at constant temperature A, then the temperature of the object, T(t) (in degrees Fahrenheit), after t minutes is given by T(t) = A + (T0 - A)e - kt, where k is a constant that depends on the object. a. Determine the constant k (to the nearest thousandth) for a

canned soda drink that takes 5 minutes to cool from 75°F to 65°F after being placed in a refrigerator that maintains a constant temperature of 34°F. b. What will be the temperature (to the nearest degree) of the

45. Depreciation An automobile depreciates according to the

function V(t) = V0 (1 - r)t, where V(t) is the value in dollars after t years, V0 is the original value, and r is the yearly depreciation rate. A car has a yearly depreciation rate of 20%. Determine, to the nearest 0.1 year, in how many years the car will depreciate to half its original value. 46. Physics The current I(t) (measured in amperes) of a circuit is

given by the function I(t) = 6(1 - e - 2.5t ), where t is the number of seconds after the switch is closed.

a. Find the current when t = 0. b. Find the current when t = 0.5. c. Solve the equation for t.

soda drink after 30 minutes? c. When (to the nearest minute) will the temperature of the

I(t)

Current (in amperes)

soda drink be 36°F? 42. Psychology According to a software company, the users of its

typing tutorial can expect to type N(t) words per minute after t hours of practice with the product, according to the function N(t) = 100(1.04 - 0.99t ). a. How many words per minute can a student expect to type

6

2

1

after 2 hours of practice?

3

t

Time (in seconds)

b. How many words per minute can a student expect to type

after 40 hours of practice? c. According to the function N, how many hours (to the near-

est hour) of practice will be required before a student can expect to type 60 words per minute? 43. Psychology In the city of Whispering Palms, which has a

population of 80,000 people, the number of people P(t) exposed to a rumor in t hours is given by the function P(t) = 80,000(1 - e - 0.0005t ). a. Find the number of hours until 10% of the population has

heard the rumor. b. Find the number of hours until 50% of the population has

heard the rumor.

Air Resistance In Exercises 47 to 50, solve the given

problems related to air resistance. 47. Assuming that air resistance is proportional to velocity, the

velocity v, in feet per second, of a falling object after t seconds is given by v = 32(1 - e - t ). a. Graph this equation for t Ú 0. b. Determine algebraically, to the nearest 0.01 second, when

the velocity is 20 feet per second. c. Determine the horizontal asymptote of the graph of v. d.


4.6


403

48. Assuming that air resistance is proportional to velocity, the

52. Learning Theory A company provides training in the assem-

velocity v, in feet per second, of a falling object after t seconds is given by

bly of a computer circuit to new employees. Past experience has shown that the number of correctly assembled circuits per week can be modeled by

v = 64(1 - e - t/2)

N =

a. Graph this equation for t Ú 0. b. Determine algebraically, to the nearest 0.1 second, when the c. Determine the horizontal asymptote of the graph of v.


53. Medication Level A patient is given three doses of aspirin.

Each dose contains 1 gram of aspirin. The second and third doses are each taken 3 hours after the previous dose is administered. The half-life of the aspirin is 2 hours. The amount of aspirin A in the patient’s body t hours after the first dose is administered is

The distance s (in feet) that the object in Exercise 47 will fall in t seconds is given by the function

49.

a. Graph this equation for t Ú 0.

Find, to the nearest hundredth of a gram, the amount of aspirin in the patient’s body when

b. Determine, to the nearest 0.1 second, the time it takes the

object to fall 50 feet.

a. t = 1

c. Calculate the slope of the secant line through (1, s(1)) and d.

s = 64t + 128(e - t/2 - 1) a. Graph this equation for t Ú 0. b. Determine, to the nearest 0.1 second, the time it takes the

object to fall 50 feet. c. Calculate the slope of the secant line through (1, s(1)) and

(2, s(2)). d.

Write a sentence that explains the meaning of the slope of the secant line you calculated in c.

51. Learning Theory The logistic model is also used in learning

theory. Suppose that historical records from employee training at a company show that the percent score on a product information test is given by P =

100 1 + 25e - 0.095t

where t is the number of hours of training. What is the number of hours (to the nearest hour) of training needed before a new employee will answer 75% of the questions correctly?

b. t = 4

c. t = 9

54.

Medication Level Use the dosage formula in Exercise 53 to determine when, to the nearest tenth of an hour, the amount of aspirin in the patient’s body first reaches 0.25 gram.

55.

Annual Growth Rate The exponential growth function for the population of a city is N(t) = 78,245e0.0245t, where t is in years. Because

Write a sentence that explains the meaning of the slope of the secant line you calculated in c. The distance s (in feet) that the object in Exercise 48 will fall in t seconds is given by the function

50.

0 … t 6 3 3 … t 6 6 t Ú 6

0.5t>2 A(t) = c 0.5t>2 + 0.5(t - 3)>2 0.5t>2 + 0.5(t - 3)>2 + 0.5(t - 6)>2

s = 32t + 32(e - t - 1)

(2, s(2)).

1 + 249e - 0.503t

where t is the number of weeks of training. What is the number of weeks (to the nearest week) of training needed before a new employee will correctly make 140 circuits?

velocity is 50 feet per second.

d.

250

e0.0245t = (e0.0245)t L (1.0248)t we can write the growth function as N(t) = 78,245(1.0248)t L 78,245a1 +

0.0248 1 b 1

#t

In this form we can see that the city’s population is growing by 2.48% per year. The population of the city of Lake Tahoe, Nevada, can be modeled by the exponential growth function N(t) = 22,755e0.0287t. Find the annual growth rate, expressed as a percent, of Lake Tahoe. Round to the nearest hundredth of a percent. 56. Oil Spills Crude oil leaks from a tank at a rate that depends on

1 of the oil 8 in the tank leaks out every 2 hours, the volume V(t) of oil in the tank after t hours is given by V(t) = V0 (0.875)t>2, where V0 = 350,000 gallons, the number of gallons in the tank at the time the tank started to leak (t = 0). the amount of oil that remains in the tank. Because

a. How many gallons does the tank hold after 3 hours?

404

CHAPTER 4


b. How many gallons does the tank hold after 5 hours?

58. A Declining Deer Population The deer population in a

reserve is given by the logistic function c. How long, to the nearest hour, will it take until 90% of the

oil has leaked from the tank?

P(t) =

If P0>c (which implies that 1 0, b > 1

y

x

x

b. Exponential decreasing: y = abx, a > 0, 0 < b < 1

x

c. Logarithmic increasing: y = a + b ln x, b > 0

d. Logarithmic decreasing: y = a + b ln x, b < 0

Figure 4.43

Exponential and logarithmic models

The terms concave upward and concave downward are often used to describe a graph. For instance, Figure 4.44a and 4.44b show the graphs of two increasing functions that join the points P and Q. The graphs of f and g differ in that they bend in different directions. We can distinguish between these two types of “bending” by examining the positions of tangent lines to the graphs. In Figure 4.44c and 4.44d, tangent lines (in red) have been drawn to the graphs of f and g. The graph of f lies above its tangent lines, and the graph of g lies below its tangent lines. The function f is said to be concave upward, and g is concave downward. a. y

b. y Q

Q f g P

P x2 x

x1

c. f is concave upward.

x2 x

x1

d. g is concave downward. y

y Q

Q

f g P

P

x1

x2 x

Figure 4.44

x1

x2 x

406

CHAPTER 4


Definition of Concavity

If the graph of f lies above all of its tangents on an interval 3x1, x24, then f is concave upward on 3x1, x24.

If the graph of f lies below all of its tangents on an interval 3x1, x24, then f is concave downward on 3x1, x24.

An examination of the graphs in Figure 4.43 on page 405 shows that the graphs of all exponential functions of the form y = abx, a 7 0, b 7 0, b Z 1 are concave upward. The graphs of increasing logarithmic functions of the form y = a + b ln x, b 7 0 are concave downward, and the graphs of decreasing logarithmic functions of the form y = a + b ln x, b 6 0, are concave upward. In Example 1, we analyze scatter plots to determine whether the shape of the scatter plot can be best approximated by a function that is concave upward or concave downward. Question • Is the graph of y = 5 - 2 ln x concave upward or concave downward?

EXAMPLE 1

Analyze Scatter Plots

For each of the following data sets, determine whether the most suitable model of the data would be an increasing exponential function or an increasing logarithmic function.

See Section 2.7 to review the steps needed to create a scatter plot on a TI-83/TI-83 Plus/TI-84 Plus calculator.

A = 5(1, 0.6), (2, 0.7), (2.8, 0.8), (4, 1.3), (6, 1.5), (6.5, 1.6), (8, 2.1), (11.2, 4.1), (12, 4.6), (15, 8.2)6 B = 5(1.5, 2.8), (2, 3.5), (4.1, 5.1), (5, 5.5), (5.5, 5.7), (7, 6.1), (7.2, 6.4), (8, 6.6), (9, 6.9), (11.6, 7.4), (12.3, 7.5), (14.7, 7.9)6

Solution For each set, construct a scatter plot of the data. See Figure 4.45. 10

10

0

20 0

0

16 0

Scatter plot of A

Scatter plot of B Figure 4.45

The scatter plot of A suggests that A is an increasing function that is concave upward. Thus an increasing exponential function would be the most suitable model for data set A. The scatter plot of B suggests that B is an increasing function that is concave downward. Thus an increasing logarithmic function would be the most suitable model for data set B. Try Exercise 4, page 411 Answer • The graph of y = 5 - 2 ln x is concave upward because the b-value, - 2, is less than

zero. See Figure 4.43d on page 405.

4.7

MODELING DATA WITH EXPONENTIAL AND LOGARITHMIC FUNCTIONS

407

Modeling Data Integrating Technology Most graphing utilities have built-in routines that can be used to determine the exponential or logarithmic regression function that models a set of data. On a TI-83/TI-83 Plus/ TI-84 Plus calculator, the ExpReg instruction is used to find the exponential regression function and the LnReg instruction is used to find the logarithmic regression function. The TI-83/TI-83 Plus/TI-84 Plus calculator does not show the value of the regression coefficient r or the coefficient of determination unless the DiagnosticOn command has been entered. The DiagnosticOn command is in the CATALOG menu.

The methods of modeling data using exponential or logarithmic functions are similar to the methods used in Section 2.7 to model data using linear or quadratic functions. Here is a summary of the modeling process.

Modeling Process Use a graphing utility to perform the following steps. 1. Construct a scatter plot of the data to determine which type of function will effectively model the data. 2. Find the equation of the modeling function and the correlation coefficient or the coefficient of determination for the equation. 3. Examine the correlation coefficient or the coefficient of determination and view a graph that displays both the modeling function and the scatter plot to determine how well your function fits the data. In the following example we use the modeling process to find an exponential function that closely models the value of a diamond as a function of its weight.

EXAMPLE 2

Model an Application with an Exponential Function

A diamond merchant has determined the values of several white diamonds that have different weights (measured in carats) but are similar in quality. See Table 4.13. Table 4.13 0.50 ct

0.75 ct

1.00 ct

1.25 ct

1.50 ct

1.75 ct

2.00 ct

3.00 ct

4.00 ct

$4600

$5000

$5800

$6200

$6700

$7300

$7900

$10,700

$14,500

Note The value of a diamond is generally determined by its color, cut, clarity, and carat weight. These characteristics of a diamond are known as the four c’s. In Example 2, we have assumed that the color, cut, and clarity of all the diamonds are similar. This assumption enables us to model the value of each diamond as a function of just its carat weight.

Find a function that models the values of the diamonds as a function of their weights, and use the function to predict the value of a 3.5-carat diamond of similar quality. Solution 1. Construct a scatter plot of the data 18,000 L1 .5 .75 1 1.25 1.5 1.75 2

L2 4600 5000 5800 6200 6700 7300 7900

L2(6) =7300

L3

2

0

5 0

Figure 4.46

From the scatter plot in Figure 4.46, it appears that the data can be closely modeled by an exponential function of the form y = abx, a 7 0 and b 7 1. (continued)

408

CHAPTER 4


Math Matters The Hope Diamond, shown below, is the world’s largest deep blue diamond. It has a weight of 45.52 carats. We should not expect the function y L 4067.6 * 1.3816x in Example 2 to yield an accurate value of the Hope Diamond because the Hope Diamond is not the same type of diamond as the diamonds in Table 4.13 on page 407, and its weight is much larger than the weights of the diamonds in Table 4.13.

2. Find the equation of the modeling function The calculator display in Figure 4.47 shows that the exponential regression equation is y L 4067.6(1.3816)x, where x is the carat weight of the diamond and y is the value of the diamond.

ExpReg y=a*b^x a=4067.641145 b=1.381644186 r2=.994881215 r=.9974373238

3. Examine the correlation coefficient or the coefficient of determination The correlation ExpReg display (DiagnosticOn) coefficient r L 0.9974 is close to 1. This Figure 4.47 indicates that the exponential regression function y L 4067.6(1.3816)x provides a good fit for the data. The graph in Figure 4.48 also shows that the exponential regression function provides a good model for the data. To estimate the value of a 3.5-carat diamond, substitute 3.5 for x in the exponential regression function or use the VALUE command in the CALCULATE menu to evaluate the exponential regression function at x = 3.5. See Figure 4.49. 18,000

18,000

Y1=4067.641144727*1.381…

0

5 0

0 X=3.5 0

AP/World Wide

Figure 4.48

The Hope Diamond is on display at the Smithsonian Museum of Natural History in Washington, D.C.

Y=12610.415

5

Figure 4.49

According to the exponential regression function, the value of a 3.5-carat diamond of similar quality is about $12,610. Try Exercise 22, page 412

When you are selecting a function to model a given set of data, try to find a function that provides a good fit to the data and is likely to produce realistic predictions. The following guidelines may facilitate the selection process.

Guidelines for Selecting a Modeling Function 1. Use a graphing utility to construct a scatter plot of the data. 2. Compare the graphical features of the scatter plot with the graphical features of the basic modeling functions available on the graphing utility: linear, quadratic, cubic, exponential, logarithmic, or logistic. Pay particular attention to the concave nature of each function. Eliminate those functions that do not display the desired concavity. 3. Use the graphing utility to find the equation of each type of function you identified in Step 2 as a possible model. 4. Determine how well each function fits the given data, and compare the graphs of the functions to determine which function is most likely to produce realistic predictions.

4.7

EXAMPLE 3


409

Select a Modeling Function and Make a Prediction

Table 4.14 shows the winning times in the women’s Olympic 100-meter freestyle event for the years 1964 to 2008. Table 4.14 Women’s Olympic 100-Meter Freestyle, 1964 to 2008

Year

Time (s)

Year

Time (s)

1964

59.5

1988

54.93

1968

60.0

1992

54.64

1972

58.59

1996

54.50

1976

55.65

2000

53.83

1980

54.79

2004

53.84

1984

55.92

2008

53.12

Source: About.com.

Find a function to model the data, and use the function to predict the winning time in the women’s Olympic 100-meter freestyle event for 2016. Caution When you use a graphing utility to find a logarithmic model, remember that the domain of y = a + b ln x is the set of positive numbers. Thus zero must not be used as an x value of a data point. This is the reason we have used x = 1 to represent 1964 in Example 3.

Solution 65 Construct a scatter plot of the data. See Figure 4.50. (Note: This scatter plot was produced using x = 1 to represent 1964, x = 2 to represent 1968, Á , and x = 12 to represent 2008.) The general shape of the scatter plot suggests that we consider functions whose graphs are decreasing and concave upward. Thus we consider a decreasing exponential function 0 45 and a decreasing logarithmic function as possible models. Use a graphing utility to find the exponential regression function and the logarithmic regression function for the data. See Figure 4.51 and Figure 4.52. ExpReg y=a*b^x a=59.59242185 b=.9897513848 r2=.8277078168 r=−.9097844892

Figure 4.51

20

Figure 4.50

LnReg y=a+b1nx a=60.5509422 b=−2.866898055 r2=.8774890664 r=−.9367438638

Figure 4.52

The exponential function is y L 59.59242185(0.9897513848)x and the logarithmic function is y L 60.5509422 - 2.866898055 ln x. The coefficient of determination r 2 for the logarithmic regression is larger than the coefficient of determination for the exponential regression. See Figure 4.51 and Figure 4.52. Thus the logarithmic regression function provides a better fit to the data than does the exponential regression function. The correlation coefficients r can also be used to determine which function provides (continued)

410

CHAPTER 4


the better fit. For decreasing functions, the function with correlation coefficient closest to -1 provides the better fit. Notice that the graph of the logarithmic function has the desired behavior to the right of the scatter plot. That is, it is a gradually decreasing curve, and this is the general behavior we would expect for future winning times in the 100-meter freestyle event. The graph of the exponential function is almost linear and is decreasing at a rapid pace, which is not what we would expect for results in an established Olympic event. See Figure 4.53. Thus we select the logarithmic function as our modeling function. To predict the winning time for this event in 2016 (represented by x = 14), substitute 14 for x in the equation of the logarithmic function or use the VALUE command in the CALCULATE menu to produce the approximate time of 52.99 seconds, as shown in Figure 4.54.

65

LnReg

ExpReg 0

20 45

Figure 4.53 65 Y1=60.550942196183+-2.86_

X

0

X=14 45

Y=52.985034

20

Figure 4.54


Finding a Logistic Growth Model If a scatter plot of a set of data suggests that the data can be effectively modeled by a logistic growth model, then you can use the Logistic feature of a graphing utility to find the logistic growth model. This process is illustrated in Example 4.

EXAMPLE 4

Find a Logistic Growth Model

Table 4.15 shows the population of deer in an animal preserve for 1994 to 2008. Table 4.15 Deer Population at an Animal Preserve

4800

Year

Population

Year

Population

Year

Population

1994

320

1999

1150

2004

2620

1995

410

2000

1410

2005

2940

1996

560

2001

1760

2006

3100

1997

730

2002

2040

2007

3300

1998

940

2003

2310

2008

3460

Find a logistic model that approximates the deer population as a function of the year. Use the model to predict the deer population in 2014.

0

24 0

Figure 4.55

Solution 1. Construct a scatter plot of the data Enter the data into a graphing utility, and then use the utility to display a scatter plot of the data. In this example, we represent 1994 by x = 0, 2008 by x = 14, and the deer population by y. Figure 4.55 shows that the data can be closely approximated by a logistic growth model.

4.7

Integrating Technology On a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, the logistic growth model is given in the form


411

2. Find the equation of the model On a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, select B: Logistic, which is in the STAT CALC menu. The logistic function for the data is y L

3965.3 1 + 11.445e-0.31152x

See Figure 4.56.

c y = 1 + ae-bx

Logistic y=c/(1+ae^(-bx)) a=11.44466821 b=.3115234553 c=3965.337214

Think of the variable x as the time t and the variable y as P(t).

Figure 4.56

3. Examine the fit A TI-83/TI-83 Plus/TI-84 Plus calculator does not compute the coefficient of determination or the correlation coefficient for a logistic model. However, Figure 4.57 shows that the logistic model provides a good fit to the data. To use the model to predict the deer population in 2014, note that the year 2014 is represented by an x value of 20. Find the y value of the logistic function for x = 20. The VALUE command in the CALCULATE menu shows that the logistic model predicts a deer population of about 3878 in 2014. See Figure 4.58. 4800

4800

Y1=3965.3372136266/(1+1…

0

24 0

Figure 4.57

0 X=20 0

Y=3877.9698 24 Figure 4.58


EXERCISES SET 4.7 In Exercises 1 to 6, use a scatter plot of the given data to determine which of the following types of functions might provide a suitable model of the data. An increasing exponential function y = abx, a 7 0, b 7 1 (See Figure 4.43a, page 405) An increasing logarithmic function y = a + b ln x, b 7 0 (See Figure 4.43c.) A decreasing exponential function y = abx, a 7 0, 0 6 b 6 1 (See Figure 4.43b.) A decreasing logarithmic function y = a + b ln x, b 6 0 (See Figure 4.43d.)

(Note: Some data sets can be closely modeled by more than one type of function.) 1. 5(1, 3), (1.5, 4), (2, 6), (3, 13), (3.5, 19), (4, 27)6 2. 5(1.0, 1.12), (2.1, 0.87), (3.2, 0.68), (3.5, 0.63), (4.4, 0.52)6 3. 5(1, 2.4), (2, 1.1), (3, 0.5), (4, 0.2), (5, 0.1)6 4. 5(5, 2.3), (7, 3.9), (9, 4.5), (12, 5.0), (16, 5.4), (21, 5.8), (26, 6.1)6 5. 5(1, 2.5), (1.5, 1.7), (2, 0.7), (3, -0.5), (3.5, - 1.3), (4, -1.5)6 6. 5(1, 3), (1.5, 3.8), (2, 4.4), (3, 5.2), (4, 5.8), (6, 6.6)6

412

CHAPTER 4


following table shows the percent of municipal solid waste (MSW) that was recycled for selected years from 1960 to 2006.

In Exercises 7 to 10, find the exponential regression function for the data. State the correlation coefficient r. Round a, b, and r to the nearest hundred-thousandth. 7. 5(10, 6.8), (12, 6.9), (14, 15.0), (16, 16.1), (18, 50.0), (19, 20.0)6

MSW Recycling Rates, 1960–2006

8. 5(2.6, 16.2), (3.8, 48.8), (5.1, 160.1), (6.5, 590.2), (7, 911.2)6 9. 5(0, 1.83), (1, 0.92), (2, 0.51), (3, 0.25), (4, 0.13), (5, 0.07)6 10. 5(4.5, 1.92), (6.0, 1.48), (7.5, 1.14), (10.2, 0.71), (12.3, 0.49)6

In Exercises 11 to 14, find the logarithmic regression function for the data. State the correlation coefficient r. Round a, b, and r to the nearest hundred-thousandth.

Year

Recycling Rate

Year

Recycling Rate

1960

6.4%

1995

26.0%

1970

6.6%

2000

29.1%

1980

9.6%

2006

32.5%

1990

16.2%

Source: U.S. Environmental Protection Agency.

a. Find an exponential regression model and a logarithmic

11. 5(5, 2.7), (6, 2.5), (7.2, 2.2), (9.3, 1.9), (11.4, 1.6), (14.2, 1.3)6

regression model for the data. Represent 1960 by x = 60, 2000 by x = 100, and 2006 by x = 106. (Hint: Enter 6.4% as 0.064.)

12. 5(11, 15.75), (14, 15.52), (17, 15.34), (20, 15.18), (23, 15.05)6 13. 5(3, 16.0), (4, 16.5), (5, 16.9), (7, 17.5), (8, 17.7), (9.8, 18.1)6

b. Examine the correlation coefficients of the two regression

models to determine which provides a better fit for the data.

14. 5(8, 67.1), (10, 67.8), (12, 68.4), (14, 69.0), (16, 69.4)6

c. Find a logistic model for the data.

In Exercises 15 to 18, find the logistic regression function for the data. Round the constants a, b, and c to the nearest hundred-thousandth.

d. In your opinion, which of the three models will best predict

the percent of MSW that will be recycled in a given year in the long-term future? Explain.

15. 5(0, 81), (2, 87), (6, 98), (10, 110), (15, 125)6

21.

16. 5(0, 175), (5, 195), (10, 217), (20, 264), (35, 341)6 17. 5(0, 955), (10, 1266), (20, 1543), (30, 1752)6 18. 5(0, 1588), (5, 2598), (10, 3638), (25, 5172)6 19.

Lift Tickets Prices The following table shows the price of

an all-day lift ticket at a ski resort for selected years from 1980 to 2009. All-Day Lift Ticket Prices, 1980–2009

Year

Price

Year

Price

1980

$9

2000

$43

1985

$14

2005

$63

1990

$20

2007

$74

1995

$30

2009

$85

Hypothermia The following table shows the time T, in hours, before a scuba diver wearing a 3-millimeterthick wet suit reaches hypothermia (95°F) for various water temperatures F, in degrees Fahrenheit.

Water Temperature F (°F)

Time T (h)

41

1.1

46

1.4

50

1.8

59

3.7

a. Find an exponential regression model for the data. Round

the constants a and b to the nearest hundred-thousandth. b. Use the model from a. to estimate the time it takes for the

diver to reach hypothermia in water that has a temperature of 65°F. Round to the nearest tenth of an hour.

a. Find an exponential regression model for the data. Represent

1980 by x = 0, 1985 by x = 5, and 2009 by x = 29. b. Use the exponential model to predict the price of an all-day

lift ticket for 2014. Round to the nearest dollar. 20.

Recycling Rates U.S. recycling rates have

been increasing over the last few decades. The

22.

Atmospheric Pressure The following table shows the Earth’s atmospheric pressure y (in newtons per square centimeter) at an altitude of x kilometers. Find a suitable function that models the atmospheric pressure as a function of the altitude. Use the function to estimate the atmospheric pressure at an altitude of 24 kilometers. Round to the nearest tenth of a newton per square centimeter.

4.7

Pressure y (N>cm2)

0

10.3

2

8.0

4

6.4

6

5.1

8

4.0

10

3.2

12

2.5

14

2.0

16

1.6

18

1.3

413

a. Determine whether the data can best be modeled by an

exponential function or a logarithmic function. Let x = 48 represent 1948 and x = 50 represent 1950. b. Assume that a new world record time will be established in

2012. Use the function you chose in a. to predict the world record time in the men’s 400-meter race for 2012. Round to the nearest hundredth of a second. Telecommuting The graph below shows the growth in

25.

the number of telecommuters. Number of telecommuters (in millions)

23.

Altitude x (km)


Hypothermia The following table shows the time T, in hours, before a scuba diver wearing a 4-millimeterthick wet suit reaches hypothermia (95°F) for various water temperatures F, in degress Fahrenheit.

Water Temperature F (°F)

Time T (h)

41

1.5

46

1.9

50

2.4

59

5.2

12 9.6

11

10.4

11.2 11.4

9 6 3 0 1998 2000 2002 2004 2006 Year

Growth in telecommuting

Which type of model, an increasing exponential model or an increasing logarithmic model, is more likely to provide a better fit for the data? Explain. 26.

a. Find an exponential regression model for the data. Round

the constants a and b to the nearest hundred-thousandth.

Population of Hawaii The following table shows the population of the state of Hawaii for selected years from 1950 to 2005.

Population of Hawaii, 1950–2005

b. Use the model from a. to estimate the time it takes for the

Year

Population, P

Year

Population, P

diver to reach hypothermia in water that has a temperature of 65°F. Round to the nearest tenth of an hour. How much greater is this result compared with the answer to Exercise 21b.?

1950

499,000

1985

1,039,698

1955

529,000

1990

1,113,491

1960

642,000

1995

1,196,854

1965

704,000

2000

1,212,125

1970

762,920

2003

1,248,200

1975

875,052

2004

1,262,124

1980

967,710

2005

1,275,194

400-Meter Race The following table lists the pro-

24.

gression of world record times in the men’s 400-meter race from 1948 to 2008. (Note: No new world record times were set during the time period from 2000 to 2008.) World Record Times in the Men’s 400-Meter Race, 1948 to 2008

Year

Time (s)

Year

Time (s)

1948

45.9

1964

44.9

1950

45.8

1967

44.5

1955

45.4

1968

44.1

1956

45.2

1968

43.86

1960

44.9

1988

43.29

1963

44.9

1999

43.18

Source: Track and Field Statistics, http://trackfield.brinkster.net/Main.asp.

Source: Economagic.com, http://www.economagic.com/ em-cgi/data.exe/beapi/a15300.

a. Find a logistic growth model that approximates the popula-

tion of Hawaii as a function of the year. Use t = 0 to represent 1950 and t = 5 to represent 1955. b. Use the model from a. to predict the population of Hawaii

for 2012. Round to the nearest thousand. c. What is the carrying capacity of the model? Round to the

nearest thousand.

414 27.

CHAPTER 4


as the dependent variable (range). State the correlation coefficient for each model. Round a and b to five decimal places and r to six decimal places. Which model provides the better fit for the data?

Optometry The near point p of a person is the clos-

est distance at which the person can see an object distinctly. As one grows older, one’s near point increases. The table below shows data for the average near point of various people with normal eyesight.

Age y (years)

Near Point p (cm)

15

11

20

13

25

15

30

17

35

20

40

23

50

26

q

7.9

12.6

31.6

50.1

79.4

pH

7.0

7.2

7.6

7.8

8.0

b. Use the model you chose in a. to find the q-value associated

with a pH of 8.2. Round to the nearest tenth. 30.

World Population The following table lists the years in which the world’s population first reached 3 billion, 4 billion, 5 billion, and 6 billion.

World Population Milestones

a. Find an exponential regression model for these data. Round

each constant to the nearest thousandth. b. What near point does this model predict for a person 60 years

old? Round to the nearest centimeter. 28.

Chemistry The amount of oxygen x, in milliliters per

Year

Population

1960

3 billion

1974

4 billion

1987

5 billion

1999

6 billion

Source: The World Almanac 2008, p. 845.

liter, that can be absorbed by water at a certain temperature T, in degrees Fahrenheit, is given in the following table.

a. Find an exponential model for the data in the table. Let

x = 0 represent 1960.

Temperature (°F)

Oxygen Absorbed (ml>L)

32

10.5

38

8.4

46

7.6

52

7.1

a. Find an exponential model for the data and use the model to

58

6.8

64

6.5

predict the year in which the panda population p will be reduced to 200. Let t = 0 represent 1980.

b. Use the model to predict the year in which the world’s pop-

ulation will first reach 8 billion. Panda Population One estimate gives the world panda population as 3200 in 1980 and 590 in 2000.

31.

b. a. Find a logarithmic regression model for these data. Round

each constant to the nearest thousandth. b. Using your model, how much oxygen, to the nearest tenth of

a milliliter per liter, can be absorbed in water that is 50°F? 29.

The Henderson-Hasselbach Function The scientists Henderson and Hasselbach determined that the pH of blood is a function of the ratio q of the amounts of bicarbonate and carbonic acid in the blood. a. Determine a linear model and a logarithmic model for the

data. Use q as the independent variable (domain) and pH

32.

Because the exponential model in a. fits the data perfectly, does this mean that the model will accurately predict future panda populations? Explain.

Olympic High Jump The following table on page 415 shows the Olympic gold medal heights for the women’s high jump from 1968 to 2008.

4.7


415

c. Use the exponential model and the logarithmic model to

Women’s Olympic High Jump, 1968–2008

Year

Height (m)

Year

Height (m)

1968

1.82

1992

2.02

1972

1.92

1996

2.05

1976

1.93

2000

2.01

1980

1.97

2004

2.06

1984

2.02

2008

2.05

1988

2.03

predict the number of cinema sites for 2014, which is represented by x = 19. d. In your opinion, which of the predictions in c. is more

realistic. 34.

Source: About.com, http://trackandfield.about.com/od/highjump/qt/ olymhijumpwomen.htm.

a. Determine a linear model and a logarithmic model for the

data, with the height measured in meters. State the correlation coefficient r for each model. Represent 1968 by x = 1, 1972 by x = 2, and 2008 by x = 11.

Temperature of Coffee A cup of coffee is placed in a room that maintains a constant temperature of 70°F. The following table shows both the coffee temperature T after t minutes and the difference between the coffee temperature and the room temperature after t minutes.

Time t (minutes) Coffee Temp. T (°F) T 70°

0

5

10

15

20

25

165°

140°

121°

107°

97°

89°

95°

70°

51°

37°

27°

19°

b. Examine the correlation coefficients to determine which

model provides a better fit for the data.

a. Find an exponential model for the difference T - 70° as a

function of t. c. Use the model you selected in b. to predict the women’s

Olympic gold medal high jump height in 2016, which is represented by x = 13. Round to the nearest hundredth of a meter.

b. Use the model in a. to predict how long it will take (to the

nearest minute) for the coffee to cool to 80°F. 35.

Number of Cinema Sites The following table shows

33.

the number of U.S. indoor cinema sites for the years 1996 to 2007.

World Population The following table lists the years in which the world’s population first reached 3 billion, 4 billion, 5 billion, and 6 billion.

World Population Milestones

Number of U.S. Indoor Cinema Sites


Population

Year


Year

Year

1960

3 billion

1996

7215

2002

5712

1974

4 billion

1997

6903

2003

5700

1987

5 billion

1998

6894

2004

5629

1999

6 billion

1999

7031

2005

5713

2000

6550

2006

5543

2001

5813

2007

5545


a. Find an exponential regression model and a logarithmic

regression model for the data. State the correlation coefficient for each model. Represent 1996 by x = 1 and 2007 by x = 12.

Source: The World Almanac 2008, p. 845.

a. Find a logistic growth model P(t) for the data in the table.

Let t represent the number of years after 1960 (t = 0 represents 1960). b. According to the logistic growth model, what will the

world’s population approach as t : q ? Round to the nearest billion.

36. A Correlation Coefficient of 1 A scientist uses a graphing calb. Examine the correlation coefficients to determine which

model provides a better fit for the data.

culator to model the data set 5(2, 5), (4, 6)6 with a logarithmic function. The following display on page 416 shows the results.

416

CHAPTER 4


b. Explain what the scientist could do so that the data in set A

LnReg y=a+blnx a=4 b=1.442695041 r2=1 r=1

37.

could be modeled by a logarithmic function of the form y = a + b ln x. 39.

Power Functions A function that can be written in the

form y = ax b is said to be a power function. Some data sets can best be modeled by a power function. On a TI-83/ TI-83 Plus/TI-84 Plus calculator, the PwrReg instruction is used to produce a power regression function for a set of data.

What is the significance of the fact that the correlation coefficient for the regression equation is r = 1?

a. Find an exponential regression function and a power regres-

Duplicate Data Points An engineer needs to model the

sion function for the following data. State the correlation coefficient r for each model.

data in set A with an exponential function. A = 5(2, 5), (3, 10), (4, 17), (4, 17), (5, 28)6 Because the ordered pair (4, 17) is listed twice, the engineer decides to eliminate one of these ordered pairs and model the data in set B.

38.

Domain Error A scientist needs to model the data

in set A. A = 5(0, 1.2), (1, 2.3), (2, 2.8), (3, 3.1), (4, 3.3), (5, 3.4)6 The scientist views a scatter plot of the data and decides to model the data with a logarithmic function of the form y = a + b ln x.

1

2

3

4

5

6

y

2.1

5.5

9.8

14.6

20.1

25.8

b. Which of the two regression functions provides the better

B = 5(2, 5), (3, 10), (4, 17), (5, 28)6 Determine whether A and B both have the same exponential regression function.

x

fit for the data? 40.

Period of a Pendulum The following table shows the time t (in seconds) of the period of a pendulum of length l (in feet). (Note: The period of a pendulum is the time it takes the pendulum to complete a swing from the right to the left and back.)

Length l ( ft) Time t (s)

1

2

3

4

6

8

1.11

1.57

1.92

2.25

2.72

3.14

a. When the scientist attempts to use a graphing calculator to

determine the logarithmic regression equation, the calculator displays the message “ERR:DOMAIN” Explain why the calculator was unable to determine the logarithmic regression equation for the data.

a. Determine the equation of the best model for the data. Your

model must be a power function or an exponential function. b. According to the model you chose in a., what is the length

of a pendulum, to the nearest tenth of a foot, that has a period of 12 seconds?

Exploring Concepts with Technology Table 4.16

Using a Semilog Graph to Model Exponential Decay

T(°F)

V

90

700

100

500

110

350

120

250

130

190

140

150

150

120

Consider the data in Table 4.16, which shows the viscosity V of SAE 40 motor oil at various temperatures T in degrees Fahrenheit. The graph of these data is shown in Figure 4.59 on page 417, along with a curve that passes through the points. The graph appears to have the shape of an exponential decay model. One way to determine whether the graph in Figure 4.59 is the graph of an exponential function is to plot the data on semilog graph paper. On this graph paper, the horizontal axis remains the same, but the vertical axis uses a logarithmic scale. The data in Table 4.16 are graphed again in Figure 4.60, but this time the vertical axis is a natural logarithm axis. This graph is approximately a straight line.


V 1n 700

Natural logarithm of viscosity

V 700

Viscosity

600 500 400 300 200 100 90

100

110

120

130

140

150

417

1n 500 1n 300

1n 100

T

90

Temperature (in degrees Fahrenheit)

100

110

120

130

140

150

T

Temperature (in degrees Fahrenheit)

Figure 4.59

Figure 4.60

The slope of the line in Figure 4.60, to the nearest ten-thousandth, is m =

ln 500 - ln 120 L - 0.0285 100 - 150

Using this slope and the point–slope formula, with V replaced by ln V, we have ln V - ln 120 = - 0.0285(T - 150) ln V L - 0.0285T + 9.062 This equation is the equation of the line on a semilog coordinate grid. Now solve the preceding equation for V. e ln V = e-0.0285T + 9.062 V = e-0.0285Te 9.062 V L 8621e-0.0285T This equation is a model of the data in the rectangular coordinate system shown in Figure 4.59. Table 4.17

t

A

1

91.77

4

70.92

8

50.30

15

27.57

20

17.95

30

7.60

1. A chemist wishes to determine the decay characteristics of iodine-131. A 100-milligram sample of iodine-131 is observed over a 30-day period. Table 4.17 shows the amount A, in milligrams, of iodine-131 remaining after t days. a. Graph the ordered pairs (t, A) on semilog paper. (Note: Semilog paper comes in different varieties. Our calculations are based on semilog paper that has a natural logarithm scale on the vertical axis.) b. Use the points (4, 4.3) and (15, 3.3) to approximate the slope of the line that passes through the points. c. Using the slope calculated in b. and the point (4, 4.3), determine the equation of the line. d. Solve the equation you derived in c. for A. e. Graph the equation you derived in d. in a rectangular coordinate system. f. What is the half-life of iodine-131? (continued)

418

CHAPTER 4


2. The birth rates B per thousand people in the United States are given in Table 4.18 for 1986 through 1990 (t = 0 corresponds to 1986).

Table 4.18

t

B

0

15.5

1

15.7

2

15.9

3

16.2

4

16.7

a. Graph the ordered pairs (t, ln B). (You will need to adjust the scale so that you can discriminate among plotted points. A suggestion is given in Figure 4.61.) b. Use the points (1, 2.754) and (3, 2.785) to approximate the slope of the line that passes through the points. c. Using the slope calculated in b. and the point (1, 2.754), determine the equation of the line.

Natural logarithm of birth rate

1n B 2.81

d. Solve the equation you derived in c. for B.

2.80 2.79

e. Graph the equation you derived in d. in a rectangular coordinate system.

2.78

f. If the birth rate continues as predicted by your model, in what year will the birth rate be 17.5 per 1000?

2.77 2.76 2.75 2.74 1

2

3

Year (t = 0 is 1986)

4

t

The difference in graphing strategies between Exercise 1 and Exercise 2 is that in Exercise 1 semilog paper was used. When a point is graphed on this coordinate paper, the y-coordinate is ln y. In Exercise 2, graphing a point (x, ln y) in a rectangular coordinate system has the same effect as graphing (x, y) in a semilog coordinate system.

Figure 4.61


4.1 Inverse Functions Graph the Inverse of a Function A function f has an inverse function if and only if it is a one-to-one function. The graph of f and the graph of its inverse f -1 are symmetric with respect to the line given by y = x.


Composition of Inverse Functions Property If f is a one-to-one function, See Example 2, page 337, and then try then f -1 is the inverse function of f if and only if (f ⴰ f -1)(x) = f [f -1(x)] = x Exercises 3 and 6, page 421. for all x in the domain of f -1 and ( f -1 ⴰ f )(x) = f -1[ f(x)] = x for all x in the domain of f. Find the Inverse of a Function If a one-to-one function f is defined by an equation, then you can often use the following procedure to find the equation of f -1. 1. Substitute y for f (x). 2. Interchange x and y. 3. Solve, if possible, for y in terms of x. 4. Substitute f -1(x) for y.


CHAPTER 4 TEST PREP

419

4.2 Exponential Functions and Their Applications Properties of f(x) bx For positive real numbers b, b Z 1, the exponential function defined by f(x) = bx has the following properties.


• The function f is a one-to-one function. It has the set of real numbers as its domain and the set of positive real numbers as its range. • The graph of f is a smooth, continuous curve with a y-intercept of (0, 1), and the graph passes through (1, b). • If b 1, f is an increasing function and its graph is asymptotic to the negative x-axis. • If 0 b 1, f is a decreasing function and its graph of is asymptotic to the positive x-axis. Graphing Techniques The graphs of some functions can be constructed by translating, stretching, compressing, or reflecting another graph or by combining these techniques.


Natural Exponential Function The number e is defined as the number that


a1 +

1 n b n

approaches as n increases without bound. The value of e accurate to 8 decimal places is 2.71828183. The function f (x) = e x, where x is a real number, is called the natural exponential function. Many applications can be modeled by functions that involve ekx, where k is a constant.

4.3 Logarithmic Functions and Their Applications See Examples 1 and 2, pages 359 and 360, and then try Exercises 39 and 43, page 422.

Exponential and Logarithmic Form The exponential form of y = logb x is b y = x. The logarithmic form of b y = x is y = logb x. Basic Logarithmic Properties logb b = 1 logb 1 = 0 logb (b x) = x

blog b x = x

Properties of f(x) logb x For positive real numbers b, b Z 1, the logarithmic function f (x) = logb x has the following properties. • The domain of f is the set of positive real numbers, and its range is the set of all real numbers. • The graph of f is a smooth, continuous curve with an x-intercept of (1, 0), and the graph passes through (b, 1). • If b 1, f is an increasing function and its graph is asymptotic to the negative y-axis. • If 0 b 1, f is a decreasing function and its graph is asymptotic to the positive y-axis.

See Example 3, page 360, and then try Exercise 16, page 422. See Example 4, page 362, and then try Exercises 31 and 32, page 422.

420

CHAPTER 4


4.4 Properties of Logarithms and Logarithmic Scales Properties of Logarithms • Product property

logb(MN) = logb M + logb N M logb = logb M - logb N N

• Quotient property • Power property • Logarithm-of-each-side property • One-to-one property


logb(M p ) = p logb M M = N implies logb M = logb N logb M = logb N implies M = N

Change-of-Base Formula If x, a, and b are positive real numbers with a Z 1 and b Z 1, then logb x =


loga x loga b

Richter Scale Magnitude An earthquake with an intensity of I has a Richter scale magnitude of

See Examples 5 to 7, page 374, and then try Exercises 75 and 77, page 422.

I M = log a b I0 where I0 is the measure of the intensity of a zero-level earthquake. pH The pH of a solution with a hydronium-ion concentration of [H+] mole per liter is given by pH = - log[H+].


4.5 Exponential and Logarithmic Equations Equality of Exponents Theorem Equations that can be written in the form bx = by can generally be solved by using the Equality of Exponents Theorem, which states that if b x = b y, where b is a positive real number (b Z 1), then x = y.


Exponential Equations Many exponential equations can be solved by writing the equation in its logarithmic form or by taking the logarithm of each side of the equation.


Logarithmic Equations Many logarithmic equations can be solved by using the properties of logarithms and the definition of a logarithm.

See Examples 6 and 7, pages 384–385, and then try Exercises 61 and 62, page 422.

4.6 Exponential Growth and Decay Exponential Growth and Decay Functions The function N(t) = N0e kt is an exponential growth function if k is a positive constant, and it is an exponential decay function if k is a negative constant.

See Examples 1 to 3, pages 391 and 393, and then try Exercises 89 and 90, page 423.

Compound Interest Formula A principal P invested at an annual interest See Example 4, page 394, and then try rate r, expressed as a decimal and compounded n times per year for t years, Exercises 81a and 82a, page 423. produces the balance A = P a1 +

r nt b n


Continuous Compounding Interest Formula If an account with principal P and annual interest rate r is compounded continuously for t years, then the balance is A = Pe rt.

See Example 5, page 396, and then try Exercises 81b and 82b, page 423.

Logistic Model In the logistic model, the magnitude of a population at time t is given by

See Example 7, page 398 and then try Exercise 93, page 424.

P(t) =

421

c 1 + ae-bt

where P0 = P(0) is the population at time t = 0, c is the carrying capacity of the population, and b is the growth rate constant. The constant a is given c - P0 by the formula a = . P0

4.7 Modeling Data with Exponential and Logarithmic Functions Modeling Process 1. Construct a scatter plot of the data to determine which type of function will effectively model the data. 2. Use a graphing utility to find the modeling function and the correlation coefficient or the coefficient of determination for the model. 3. Examine the correlation coefficient or the coefficient of determination and view a graph that displays both the function and the scatter plot to determine how well the function fits the data.


CHAPTER 4 REVIEW EXERCISES In Exercises 1 and 2, draw the graph of the inverse of the given function. 1.

2.

y

(−3, 1)

(3, 3)

m(x) =

3 x - 1

6. p(x) =

x - 5 2x

q(x) =

2x x - 5

5 f

(0, 2)

(−6, 0)

x + 3 x

y

f 5

5. l(x) =

5

(−2, 4) (−1, 2)

x

(2, 14 )

(0, 1) −5

−2

In Exercises 7 to 10, find the inverse of the function. Sketch the graph of the function and its inverse on the same set of coordinate axes.

(1, 12) 2

x

7. f (x) = 3x - 4 9. h(x) = -

In Exercises 3 to 6, use composition of functions to determine whether the given functions are inverse functions. 3. F(x) = 2x - 5 4. h(x) = 1x

G(x) = k(x) = x 2,

x + 5 2 x Ú 0

1 x - 2 2

8. g(x) = - 2x + 3 10. k(x) =

1 x

In Exercises 11 and 12, find the inverse of the given function. 11. f (x) =

2x , where the domain of f is 5x ƒ x 7 16 x - 1

12. g(x) = x 2 + 2x, where the domain of g is 5x ƒ x Ú - 16

422

CHAPTER 4


In Exercises 13 to 24, solve each equation. Do not use a calculator. 13. log 5 25 = x

14. log 3 81 = x

15. ln e 3 = x

16. ln ep = x

17. 32x + 7 = 27

18. 5x - 4 = 625

20. 27(3x) = 3 - 1

21. log x 2 = 6

23. 10log 2x = 14

24. eln x = 64

19. 2x =

22.

1 8

1 log ƒ x ƒ = 5 2

2

In Exercises 25 to 36, sketch the graph of each function. 25. f (x) = (2.5)x 27. f (x) =

3ƒxƒ

x

26. f (x) = a b

1 4

28. f (x) = 4

31. f (x) = log5 x

32. f (x) = log1>3 x

1 33. f (x) = log x 3

34. f (x) = 3 log x

1 ln x 2

1>3

36. f (x) = - ln ƒ x ƒ

In Exercises 37 and 38, use a graphing utility to graph each function. 37. f (x) =

4x + 4 - x 2

38. f (x) =

3x - 3 - x 2

In Exercises 39 to 42, change each logarithmic equation to its exponential form. 39. log 4 64 = 3

40. log1>2 8 = - 3

41. log12 4 = 4

42. ln 1 = 0

In Exercises 43 to 46, change each exponential equation to its logarithmic form. 43. 53 = 125

44. 210 = 1024

45. 100 = 1

46. 81>2 = 212

In Exercises 47 to 50, expand the given logarithmic expression. 2 3

47. log b

x y z

1xy z4

In Exercises 51 to 54, write each logarithmic expression as a single logarithm with a coefficient of 1. 51. 2 log x +

53.

1 log(x + 1) 3

1 ln 2xy - 3 ln z 2

52. 5 log x - 2 log(x + 5)

54. ln x - (ln y - ln z)

In Exercises 55 to 58, use the change-of-base formula and a calculator to approximate each logarithm accurate to six significant digits. 55. log 5 101

56. log 3 40

57. log 4 0.85

58. log 8 0.3

(x - 3)

30. f (x) = 2

35. f (x) = -

50. ln

- ƒ xƒ

29. f (x) = 2 - 3 x

49. ln xy 3

48. log b

1x y2 z

In Exercises 59 to 74, solve each equation for x. Give exact answers. Do not use a calculator. 59. 4x = 30

60. 5x + 1 = 41

61. ln 3x - ln(x - 1) = ln 4

62. ln 3x + ln 2 = 1

63. e ln(x + 2) = 6

64. 10log(2x + 1) = 31

65.

4x + 4 - x = 2 4x - 4 - x

66.

5x + 5 - x = 8 2

67. log(log x) = 3

68. ln(ln x) = 2

69. log 1x - 5 = 3

70. log x + log(x - 15) = 1

71. log 4(log 3 x) = 1

72. log 7(log 5 x 2) = 0

73. log 5 x 3 = log 5 16x

74. 25 = 16log4 x

75. Earthquake Magnitude Determine, to the nearest 0.1, the

Richter scale magnitude of an earthquake with an intensity of I = 51,782,000I0. 76. Earthquake Magnitude A seismogram has an amplitude of

18 millimeters, and the difference in time between the s-wave and the p-wave is 21 seconds. Find, to the nearest tenth, the Richter scale magnitude of the earthquake that produced the seismogram. 77. Comparison of Earthquakes An earthquake had a Richter

scale magnitude of 7.2. Its aftershock had a Richter scale magnitude of 3.7. Compare the intensity of the earthquake with the intensity of the aftershock by finding, to the nearest unit, the ratio of the larger intensity to the smaller intensity.


78. Comparison of Earthquakes An earthquake has an intensity

91.

600 times the intensity of a second earthquake. Find, to the nearest tenth, the difference between the Richter scale magnitudes of the earthquakes.

Cellular Telephone Subscribership The following

table shows the number of U.S. cell phone subscriptions, in thousands, for selected years from 1990 to 2006.

Number of Cell Phone Subscriptions (in thousands)

79. Chemistry Find the pH of tomatoes that have a hydronium-

ion concentration of 6.28 * 10 - 5. Round to the nearest tenth.

Year

80. Chemistry Find the hydronium-ion concentration of rainwa-

ter that has a pH of 5.4. 81. Compound Interest Find the balance when $16,000 is invested

at an annual rate of 8% for 3 years if the interest is compounded a. monthly

b. continuously

82. Compound Interest Find the balance when $19,000 is invested

at an annual rate of 6% for 5 years if the interest is compounded a. daily

b. continuously

83. Depreciation The scrap value S of a product with an expected

- 0.12t

by N(t) = N0 e , where N is the number of square centimeters of unhealed skin t days after the injury and N0 is the number of square centimeters covered by the original wound. a. What percentage of the wound will be healed after 10 days? b. How many days, to the nearest day, will it take for 50% of

the wound to heal?

1990

5283

1992

11,033

1994

24,134

1996

44,043

1998

69,209

2000

109,478

2002

140,767

2004

182,140

2006

233,041

Source: The World Almanac 2008.

life span of n years is given by S(n) = P(1 - r)n, where P is the original purchase price of the product and r is the annual rate of depreciation. A taxicab is purchased for $12,400 and is expected to last 3 years. What is its scrap value if it depreciates at a rate of 29% per year? 84. Medicine A skin wound heals according to the function given

Find the equation of the mathematical model that you believe will most accurately predict the number of U.S. cellular telephone subscriptions for 2012. Explain the reasoning you used to select your model. 92.

Mortality Rate The following table shows the infant mortality rate in the United States for selected years from 1960 to 2005.

U.S. Infant Mortality Rate, 1960–2005 (per 1000 live births)

Year

Rate R

1960

26.0

1970

20.0

1980

12.6

1990

9.2

1995

7.6

2000

6.9

2001

6.8

2002

7.0

ulation was 25,200 in 2007 and 26,800 in 2008. Use t = 0 to represent 2007.

2003

6.9

2004

6.8

b. Use the growth function to predict, to the nearest hundred,

2005

6.9

c. How long, to the nearest day, will it take for 90% of the

wound to heal? In Exercises 85 to 88, find the exponential growth or decay function N(t) N0 e kt that satisfies the given conditions. 85. N(0) = 1, N(2) = 5

86. N(0) = 2, N(3) = 11

87. N(1) = 4, N(5) = 5

88. N( -1) = 2, N(0) = 1

89. Population Growth a. Find the exponential growth function for a city whose pop-

the population of the city in 2014.

423

Source: The World Almanac 2008.

90. Carbon Dating Determine, to the nearest 10 years, the age of a

a. Find an exponential model and a logarithmic model for the

bone if it now contains 96% of its original amount of carbon-14. The half-life of carbon-14 is 5730 years.

infant mortality rate R as a function of the year. Represent 1960 by x = 60.

424

CHAPTER 4


b. Examine the correlation coefficients of the regression mod-

b. Use the model to predict, to the nearest 10, the coyote pop-

els to determine which model provides the better fit. c. Use the model you selected in b. to predict, to the nearest

ulation in 2014. 94. Logistic Growth Consider the logistic function

0.1, the infant mortality rate in 2010. P(t) =

93. Logistic Growth The population of coyotes in a national park

128 1 + 5e - 0.27t

satisfies the logistic model with P0 = 210 in 2001, c = 1400, and P(3) = 360 (the population in 2004).

a. Find P0.

a. Determine the logistic model.

b. What does P(t) approach as t : q ?

CHAPTER 4 TEST 1. Find the inverse of f (x) = 2x - 3. Graph f and f - 1 on the

14. Compound Interest Find the time required for money

invested at an annual rate of 4% to double in value if the investment is compounded monthly. Round to the nearest hundredth of a year.

same coordinate axes. x , where the domain of f is 4x - 8 5x ƒ x 7 26. State the domain and the range of f - 1.

2. Find the inverse of f (x) =

15. Earthquake Magnitude a. What, to the nearest tenth, will an earthquake measure on

the Richter scale if it has an intensity of I = 42,304,000I0?

3. a. Write log b(5x - 3) = c in exponential form. b. Write 3x>2 = y in logarithmic form. 4. Expand log b

z2 y 3 1x

b. Compare the intensity of an earthquake that measures 6.3

on the Richter scale with the intensity of an earthquake that measures 4.5 on the Richter scale by finding the ratio of the larger intensity to the smaller intensity. Round to the nearest natural number.

.

5. Write log(2x + 3) - 3 log(x - 2) as a single logarithm with

a coefficient of 1.

16. a. Find the exponential growth function for a city whose pop-

ulation was 34,600 in 2003 and 39,800 in 2006. Use t = 0 to represent 2003.

6. Use the change-of-base formula and a calculator to approxi-

mate log 4 12. Round your result to the nearest ten-thousandth. b. Use the growth function in a. to predict the population of

7. Graph: f (x) = 3-x>2 8. Graph: f (x) = ln(x + 1)

the city in 2013. Round to the nearest thousand. 17. Determine, to the nearest 10 years, the age of a bone if it now

contains 92% of its original amount of carbon-14. The half-life of carbon-14 is 5730 years.

9. Solve 5x = 22. Round your solution to the nearest ten-

thousandth. 10. Find the exact solution of 45 - x = 7x.

18. a.

5(2.5, 16), (3.7, 48), (5.0, 155), (6.5, 571), (6.9, 896)6

11. Solve: log(x + 99) - log(3x - 2) = 2 12. Solve: ln(2 - x) + ln(5 - x) = ln(37 - x)

b. Use the function to predict, to the nearest integer, the y value

associated with x = 7.8.

13. Find the balance on $20,000 invested at an annual interest rate

of 7.8% for 5 years and compounded a. monthly b. continuously

Find the exponential regression function for the following data.

19.

Women’s Javelin Throw The following table shows the progression of the world record distances for the women’s javelin throw from 1999 to 2008. (Note: No new world record distances were set from 2006 to 2008.)

CUMULATIVE REVIEW EXERCISES

World Record Progression in the Women’s Javelin Throw

Year

Distance d (m)

1999

67.09

2000

68.22

2000

69.48

2001

71.54

2005

71.70

Source: http://www.athletix.org/Statistics/ wrjtwomen.html.

a. Find a logarithmic model and a logistic model for the data.

Use t = 1 to represent 1999 and t = 7 to represent 2005.

425

b. Assume that a new world record distance will be established

in 2012. Use each of the models from a. to predict the women’s world record javelin throw distance for 2012. Round to the nearest hundredth of a meter. 20. Population Growth The population of raccoons in a state

park satisfies a logistic growth model with P0 = 160 in 2007 and P(1) = 190 in 2008. A park ranger has estimated the carrying capacity of the park to be 1100 raccoons. a. Determine the logistic growth model for the raccoon popu-

lation where t is the number of years after 2007. b. Use the logistic model from a. to predict the raccoon popu-

lation in 2014.

CUMULATIVE REVIEW EXERCISES 1. Solve ƒ x - 4 ƒ … 2. Write the solution set using interval notation.

x Ú 1. Write the solution set using set-builder 2. Solve 2x - 6 notation. 3. Find, to the nearest tenth, the distance between the points (5, 2)

11. Find the equations of the vertical and horizontal asymptotes of

the graph of r(x) =

12. Determine the domain and the range of the rational function

R(x) =

and (11, 7). 4. Height of a Ball The height, in feet, of a ball released with an

initial upward velocity of 44 feet per second at an initial height of 8 feet is given by h(t) = - 16t 2 + 44t + 8, where t is the time in seconds after the ball is released. Find the maximum height the ball will reach. 5. Given f (x) = 2x + 1 and g(x) = x 2 - 5, find ( g ⴰ f )(x).

3x - 5 . x - 4

4 x2 + 1

.

13. State whether f (x) = 0.4x is an increasing function or a

decreasing function. 14. Write log4 x = y in exponential form. 15. Write 53 = 125 in logarithmic form. 16. Find, to the nearest tenth, the Richter scale magnitude of an

earthquake with an intensity of I = 11,650,600I0 . 6. Find the inverse of f (x) = 3x - 5. 7. Safe Load The load that a horizontal beam can safely support

varies jointly as the width and the square of the depth of the beam. It has been determined that a beam with a width of 4 inches and a depth of 8 inches can safely support a load of 1500 pounds. How many pounds can a beam of the same material and the same length safely support if it has a width of 6 inches and a depth of 10 inches? Round to the nearest hundred pounds. 8. Use Descartes’ Rule of Signs to determine the number of pos-

sible positive and the number of possible negative real zeros of P(x) = x4 - 3x 3 + x 2 - x - 6. 9. Find the zeros of P(x) = x4 - 5x 3 + x 2 + 15x - 12. 10. Find a polynomial function of lowest degree that has 2, 1 - i,

and 1 + i as zeros.

17. Solve 2e x = 15. Round to the nearest ten-thousandth. 18. Find the age of a bone if it now has 94% of the carbon-14 it had

at time t = 0. The half-life of carbon-14 is 5730 years. Round to the nearest 10 years. 19. Solve

e x - e-x = 12 for x. Round to the nearest ten-thousandth. 2

20. Population Growth The wolf population in a national park sat-

isfies a logistic growth model with P0 = 160 in 2004 and P(3) = 205 (the population in 2007). It has been determined that the maximum population the park can support is 450 wolves. a. Determine the logistic growth model for the data. b. Use the logistic growth model to predict, to the nearest 10,

the wolf population in 2014.


CHAPTER

TRIGONOMETRIC FUNCTIONS

5

mypokcik, 2009/Used under license from Shutterstock.com

5.1 Angles and Arcs 5.2 Right Triangle Trigonometry 5.3 Trigonometric Functions of any Angle 5.4 Trigonometric Functions of Real Numbers 5.5 Graphs of the Sine and Cosine Functions 5.6 Graphs of the Other Trigonometric Functions 5.7 Graphing Techniques 5.8 Harmonic Motion— An Application of the Sine and Cosine Functions

Andrew Brookes/CORBIS

Applications of Trigonometric Functions In this chapter we introduce an important group of functions called trigonometric functions. These functions are often used in applications involving relationships between the sides and the angles of triangles. In Exercise 71 on page 453, right triangles and trigonometric functions are used to find the height and length of the skybridge between the Petronas Towers. In the seventeenth century, a unit circle approach was used to create trigonometric functions of real numbers, which are also referred to as circular functions. These functions allow us to solve a wider variety of applications. For instance, in Exercise 63 page 480, a circular function is used to model a sound wave and find the frequency of the sound wave.

427

428

CHAPTER 5


SECTION 5.1

Angles and Arcs

Degree Measure Radian Measure Arcs and Arc Length Linear and Angular Speed

A point P on a line separates the line into two parts, each of which is called a half-line. The union of point P and the half-line formed by P that includes point A is called a ray, : : and it is represented as PA. The point P is the endpoint of ray PA. Figure 5.1 shows the ray : : PA. and a second ray represented as QR. In geometry, an angle is defined simply as the union of two rays that have a common endpoint. In trigonometry and many advanced mathematics courses, it is beneficial to define an angle in terms of a rotation.

Definition of an Angle

R

An angle is formed by rotating a given ray about its endpoint to some terminal position. The original ray is the initial side of the angle, and the second ray is the terminal side of the angle. The common endpoint is the vertex of the angle.

A

Q

There are several methods used to name an angle. One way is to employ Greek letters. For example, the angle shown in Figure 5.2 can be designated as a or as ∠ a. It also can be named ∠ O, ∠AOB, or ∠ BOA. If you name an angle by using three points, such as ∠AOB, it is traditional to list the vertex point between the other two points. Angles formed by a counterclockwise rotation are considered positive angles, and angles formed by a clockwise rotation are considered negative angles. See Figure 5.3.

P

Figure 5.1

B α

Terminal side

O Vertex

A Initial side

A

α

Figure 5.2

A

O

β

O B

Positive angle

1°

B

Negative angle

Figure 5.3

Degree Measure

Figure 5.4 1 1ⴰ = of a revolution 360

The measure of an angle is determined by the amount of rotation of the initial side. An angle formed by rotating the initial side counterclockwise exactly once until it coincides with itself (one complete revolution) is defined to have a measure of 360 degrees, which can be written as 360°.

Definition of Degree 1 One degree is the measure of an angle formed by rotating a ray of a complete 360 revolution. The symbol for degree is °.

ββ = 30°

Figure 5.5

The angle shown in Figure 5.4 has a measure of 1°. The angle b shown in Figure 5.5 has a measure of 30°. We will use the notation b = 30° to denote that the measure of angle b

5.1

ANGLES AND ARCS

429

60 20 50 0 1 3 1

90

80 70 100 0 11

100 110 80 12 70 0 60 13 50 0

0 14 0 4

4 14 0 0

is 30°. The protractor shown in Figure 5.6 can be used to measure an angle in degrees or to draw an angle with a given degree measure.

30 15 0

0 15 30

170 10

0 10 20 180 170 160

160 20

180 0

Figure 5.6

Protractor for measuring angles in degrees

Angles are often classified according to their measure. 180° angles are straight angles. See Figure 5.7a. 90° angles are right angles. See Figure 5.7b. Angles that have a measure greater than 0° but less than 90° are acute angles. See Figure 5.7c. Angles that have a measure greater than 90° but less than 180° are obtuse angles. See Figure 5.7d. B B

α

B

O

β

A

a. Straight angle (α = 180°)

O

B

θ

θ

O

A

b. Right angle ( β = 90°)

A

O

c. Acute angle (0° < θ < 90°)

A

d. Obtuse angle (90° < θ < 180°)

Figure 5.7

An angle superimposed in a Cartesian coordinate system is in standard position if its vertex is at the origin and its initial side is on the positive x-axis. See Figure 5.8. Two positive angles are complementary angles (Figure 5.9a) if the sum of the measures of the angles is 90°. Each angle is the complement of the other angle. Two positive angles are supplementary angles (Figure 5.9b) if the sum of the measures of the angles is 180°. Each angle is the supplement of the other angle.

y

Terminal side

x Initial side

β β

α

Figure 5.8

An angle in standard position

a. Complementary angles α + β = 90°

α

b. Supplementary angles α + β = 180° Figure 5.9

430

CHAPTER 5


EXAMPLE 1

Find the Measure of the Complement and the Supplement of an Angle

For each angle, find the measure (if possible) of its complement and of its supplement.

90° y

a. 50°

140°

θ = 40°

180°

x

0°

b.

u = 125°

Solution a. Figure 5.10 shows ∠u = 40° in standard position. The measure of its complement is 90° - 40° = 50°. The measure of its supplement is 180° - 40° = 140°. b.

Figure 5.10

u = 40°

Figure 5.11 shows ∠u = 125° in standard position. Angle u does not have a complement because there is no positive number x such that x° + 125° = 90°

90° y

The measure of its supplement is 180° - 125° = 55°. Try Exercise 2, page 438 θ = 125°

Question • Are the two acute angles of any right triangle complementary angles? Explain.

55° 180°

x

Figure 5.11

0°

Some angles have a measure greater than 360°. See Figure 5.12a and Figure 5.12b. The angle shown in Figure 5.12c has a measure less than - 360°, because it is formed by a clockwise rotation of more than one revolution of the initial side.

y

180°

a. 720°

90° x

270°

Figure 5.13

90° y

β

180°

x

270°

Figure 5.14

0°

b. 450°

c . −990°

Figure 5.12

If the terminal side of an angle in standard position lies on a coordinate axis, then the angle is classified as a quadrantal angle. For example, the 90° angle, the 180° angle, and the 270° angle shown in Figure 5.13 are all quadrantal angles. If the terminal side of an angle in standard position does not lie on a coordinate axis, then the angle is classified according to the quadrant that contains the terminal side. For example, ∠ b in Figure 5.14 is a Quadrant III angle. Angles in standard position that have the same sides are coterminal angles. Every angle has an unlimited number of coterminal angles. Figure 5.15 shows ∠u and two of its coterminal angles, labeled ∠1 and ∠2.

90° y

∠1 = 70° 180°

θ = 430° ∠2 = − 290° 270°

Figure 5.15

Answer • Yes. The sum of the measures of the angles of any triangle is 180°. The right angle

has a measure of 90°. Thus the sum of the measures of the two acute angles must be 180° - 90° = 90°.

x

0°

5.1

ANGLES AND ARCS

431

Measures of Coterminal Angles Given ∠ u in standard position with measure x°, then the measures of the angles that are coterminal with ∠u are given by x° + k

#

360°

where k is an integer.

This theorem states that the measures of any two coterminal angles differ by an integer multiple of 360°. For instance, in Figure 5.15, u = 430°, ∠1 = 430° + 1 -12

∠2 = 430° + 1 - 22

# #

360° = 70°, and 360° = - 290°

If we add positive multiples of 360° to 430°, we find that the angles with measures 790°, 1150°, 1510°, and so on, are also coterminal with ∠u.

EXAMPLE 2

Assume the following angles are in standard position. Determine the measure of the positive angle with measure less than 360° that is coterminal with the given angle and then classify the angle by quadrant.

y

a.

Classify by Quadrant and Find a Coterminal Angle

a.

a = 550°

b.

b = - 225°

c. g = 1105°

190°

x

α = 550°

Solution a. Because 550° = 190° + 360°, ∠a is coterminal with an angle that has a measure of 190°. ∠a is a Quadrant III angle. See Figure 5.16a. b.

y

b.

c. 135° x

β = −225°

5 . Thus ∠g is an angle formed by three complete counter 72 5 5 clockwise rotations, plus of a rotation. To convert of a rotation to 72 72 5 degrees, multiply times 360°. 72 5 # 360° = 25° 72 1105° , 360° = 3

Thus 1105° = 25° + 3 # 360°. Hence, ∠g is coterminal with an angle that has a measure of 25°. ∠g is a Quadrant I angle. See Figure 5.16c.

y

c.

Because -225° = 135° + 1 - 12 # 360°, ∠ b is coterminal with an angle that has a measure of 135°. ∠ b is a Quadrant II angle. See Figure 5.16b.

Try Exercise 14, page 438 25° x

γ = 1105°

There are two popular methods for representing a fractional part of a degree. One is the decimal degree method. For example, the measure 29.76° is a decimal degree. It means 29° plus 76 hundredths of 1°

Figure 5.16

A second method of measurement is known as the DMS (degree, minute, second) method. In the DMS method, a degree is subdivided into 60 equal parts, each of which

432

CHAPTER 5


is called a minute, denoted by ¿ . Thus 1° = 60¿. Furthermore, a minute is subdivided into 60 equal parts, each of which is called a second, denoted by –. Thus 1¿ = 60– and 1° = 3600–. The fractions 1° = 1, 60¿

1¿ = 1, 60–

and

1° = 1 3600–

are another way of expressing the relationships among degrees, minutes, and seconds. Each of the fractions is known as a unit fraction or a conversion factor. Because all conversion factors are equal to 1, you can multiply a numerical value by a conversion factor and not change the numerical value, even though you change the units used to express the numerical value. The following illustrates the process of multiplying by conversion factors to write 126°12¿27 – as a decimal degree. 126°12¿27– = 126° + 12¿ + 27 – 1° 1° = 126° + 12¿ a b + 27– a b 60¿ 3600– = 126° + 0.2° + 0.0075° = 126.2075°

Integrating Technology Many graphing calculators can be used to convert a decimal degree measure to its equivalent DMS measure, and vice versa. For instance, Figure 5.17 shows that 31.57° is equivalent to 31°34¿12–. On a TI-83/TI-83 Plus/TI-84 Plus graphing calculator, the degree symbol, °, and the DMS function are in the ANGLE menu. ANGLE 1: ° 2: ' 3: r 4: DMS 31.57° DMS

31°34'12" 31.57

31°34'12"

Figure 5.17

Figure 5.18

To convert a DMS measure to its equivalent decimal degree measure, enter the DMS measure and press ENTER . The calculator screen in Figure 5.18 shows that 31°34¿12– is equivalent to 31.57°. A TI-83/TI-83 Plus/TI-84 Plus calculator needs to be in degree mode to produce the results displayed in Figures 5.17 and 5.18. On a TI-83/TI-83 Plus/TI-84 Plus calculator, the degree symbol, °, and the minute symbol, ¿, are both in the ANGLE menu; however, the second symbol, –, is entered by pressing ALPHA .

Radian Measure Another commonly used angle measurement is the radian. To define a radian, first consider a circle of radius r and two radii OA and OB. The angle u formed by the two radii is

5.1

ANGLES AND ARCS

433

a central angle. The portion of the circle between A and B is an arc of the circle and is μ. We say that AB μ subtends the angle u. The length of AB μ is s (see Figure 5.19). written AB

Definition of a Radian One radian is the measure of the central angle subtended by an arc of length r on a circle of radius r. See Figure 5.20.

s

B θ

r

s=r

r θ

O r

O

B

r

A

A

Figure 5.20

Central angle u has a measure of 1 radian.

Figure 5.19

Figure 5.21 shows a protractor that can be used to measure angles in radians or to construct angles given in radian measure.

1.8

1.6

1.4

2

1.2 1

2.2

2.

4

.8

2.6

.6

2.8

.4

3

.2

Figure 5.21

Protractor for measuring angles in radians

Definition of Radian Measure Given an arc of length s on a circle of radius r, the measure of the central angle s subtended by the arc is u = radians. r 5 5

5

θ

B 5

O

5

A

As an example, consider that an arc with a length of 15 centimeters on a circle with a radius of 5 centimeters subtends an angle of 3 radians, as shown in Figure 5.22. The same result can be found by dividing 15 centimeters by 5 centimeters. To find the measure in radians of any central angle u, divide the length s of the arc that subtends u by the length of the radius of the circle. Using the formula for radian measure, we find that an arc with a length of 12 centimeters on a circle with a radius of 8 centimeters subtends a central angle u whose measure is

Figure 5.22

Central angle u has a measure of 3 radians.

u =

s 12 centimeters 3 radians = radians = radians r 8 centimeters 2

434

CHAPTER 5


Note that the centimeter units are not part of the final result. The radian measure of a central angle formed by an arc with a length of 12 miles on a circle with a radius of 8 miles 3 would be the same, radians. If an angle has a measure of t radians, where t is a real 2 number, then the measure of the angle is often stated as t instead of t radians. For instance, if an angle u has a measure of 2 radians, we can simply write u = 2 instead of u = 2 radians. There will be no confusion concerning whether an angle measure is in degrees or radians, because the degree symbol is always used for angle measurements that are in degrees. Recall that the circumference of a circle is given by the equation C = 2pr. The radian measure of the central angle u subtended by the circumference is 2pr u = = 2p. In degree measure, the central angle u subtended by the circumference is r 360°. Thus we have the relationship 360° = 2p radians. Dividing each side of the equation by 2 gives 180° = p radians. From this last equation, we can establish the following conversion factors.


Radian–Degree Conversion

A calculator shows that 1 radian L 57.29577951ⴰ

To convert from radians to degrees, multiply by a

180° b. p radians

and

To convert from degrees to radians, multiply by a

p radians b. 180°

1ⴰ L 0.017453293 radian

EXAMPLE 3

Convert from Degrees to Radians

Convert each angle in degrees to radians. a.

60°

b. 315°

c. - 150°

Solution

p radians b and simplify. In each case, the degree 180° units in the numerator cancel with the degree units in the denominator. Multiply each degree measure by a

a.

p radians 60 p p 60ⴰ = 60ⴰ/ a b = radians = radians 180ⴰ/ 180 3

b.

315° = 315°/ a

c.

-150° = - 150°/ a

p radians 315 p 7p b = radians = radians / 180 4 180° p radians 5p 150 p b radians = radians b = -a / 180° 180 6


EXAMPLE 4

Convert from Radians to Degrees

Convert each angle in radians to degrees. 3p a. b. 1 radian radians 4

c.

-

5p radians 2

5.1

ANGLES AND ARCS

435

Solution Multiply each radian measure by a

180° b and simplify. In each case, the radian p radians units in the numerator cancel with the radian units in the denominator. a. b.

Table 5.1

Degrees

Radians

0

0

30

p>6

45

p>4

60

p>3

90

p>2

120

2p>3

135

3p>4

150

5p>6

180

p

210

7p>6

225

5p>4

240

4p>3

270

3p>2

300

5p>3

315

7p>4

330

11p>6

360

2p

c.

3 # 180ⴰ 3p radians 180ⴰ 3p b = radians = a ba = 135ⴰ 4 4 p radians 4 180ⴰ 180ⴰ L 57.3ⴰ b = 1 radian = a (1 radian ) a p p radians 5p 5 # 180ⴰ 5p radians 180ⴰ b = radians = a ba = - 450ⴰ 2 2 p radians 2


Table 5.1 lists the degree and radian measures of selected angles. Figure 5.23 illustrates each angle listed in the table as measured from the positive x-axis. 2π 120°, 3 135°, 3π 4 150°, 5π 6

π 90°, 2

60°,

π 3

45°,

π 4

30°,

180°, π

0°, 0

210°, 7π 6 225°, 5π 4 240°, 4π 3

π 6

330°, 315°, 270°, 3π 2

5π 300°, 3

11π 6

7π 4

Figure 5.23

Degree and radian measures of selected angles

100° 1.745329252


Figure 5.24

2.2r

A graphing calculator can convert degree measure to radian measure, and vice versa. For example, the calculator display in Figure 5.24 shows that 100° is approximately 1.74533 radians. The calculator must be in radian mode to convert from degrees to radians. The display in Figure 5.25 shows that 2.2 radians is approximately 126.051°. The calculator must be in degree mode to convert from radians to degrees. On a TI-83/TI-83 Plus/TI-84 Plus calculator, the symbol for radian measure is r, and it is in the ANGLE menu.

126.0507149

Arcs and Arc Length Figure 5.25

s Consider a circle of radius r. By solving the formula u = for s, we have an equation for r arc length.

436

CHAPTER 5


Arc Length Formula Let r be the length of the radius of a circle and u be the nonnegative radian measure of a central angle of the circle. Then the length of the arc s that subtends the central angle is s = ru. See Figure 5.26.

s = arc length B θ

O

r

A

Figure 5.26

s = ru

EXAMPLE 5

Find the Length of an Arc

Find the length of an arc that subtends a central angle of 120° in a circle with a radius of 10 centimeters. Caution The formula s = r u is valid only when u is expressed in radians.

Solution The formula s = ru requires that u be expressed in radians. We first convert 120° to radian measure and then use the formula s = ru. 2p 2p p radians b = radians = u = 120ⴰ = 120ⴰ a ⴰ 180 3 3 s = ru = 110 centimeters2a

2p 20p b = centimeters 3 3


EXAMPLE 6


A pulley with a radius of 10 inches uses a belt to drive a pulley with a radius of 6 inches. Find the angle through which the smaller pulley turns as the 10-inch pulley makes one revolution. State your answer in radians and in degrees. s1

θ1

10 in.

10u1 = 6u2 1012p2 = 6u2

s2

θ2

Solution Use the formula s = ru. As the 10-inch pulley turns through an angle u1, a point on the rim of that pulley moves s1 inches, where s1 = 10u1. See Figure 5.27. At the same time, the 6-inch pulley turns through an angle of u2 and a point on the rim of that pulley moves s2 inches, where s2 = 6u2. Assuming that the belt does not slip on the pulleys, we have s1 = s2. Thus

6 in.

• Solve for u2 when u1 = 2p radians.

10 p = u2 3 The 6-inch pulley turns through an angle of

Figure 5.27


10 p radians, or 600°. 3

5.1

ANGLES AND ARCS

437

Linear and Angular Speed

Peter Gridley/Getty Images

A ride at an amusement park has an inner ring of swings and an outer ring of swings. During each complete revolution, the swings in the outer ring travel a greater distance than the swings in the inner ring. We can say that the swings in the outer ring have a greater linear speed than the swings in the inner ring. Interestingly, all of the swings complete the same number of revolutions during any given ride. We say that all of the swings have the same angular speed. In the following definitions, v denotes linear speed and v (omega) denotes angular speed.

Definition of Linear and Angular Speed of a Point Moving on a Circular Path A point moves on a circular path with radius r at a constant rate of u radians per unit of time t. Its linear speed is v =

s t

where s is the distance the point travels, given by s = ru. The point’s angular speed is v =

u t

Some common units of angular speed are revolutions per second, revolutions per minute, radians per second, and radians per minute.

EXAMPLE 7

Convert an Angular Speed

A hard disk in a computer rotates at 7200 revolutions per minute. Find the angular speed of the disk in radians per second. Solution As a point on the disk rotates 1 revolution, the angle through which the point moves is 2p radians 2p radians. Thus a b will be the conversion factor we will use to convert 1 revolution from revolutions to radians. To convert from minutes to seconds, use the conversion 1 minute factor a b. 60 seconds 7200 revolutions>minute =

1 minute 7200 revolutions 2p radians a ba b 1 minute 1 revolution 60 seconds

= 240p radians>second

• Exact answer

L 754 radians>second

• Approximate answer


We can establish an important relationship between linear speed and angular speed. We start with the linear speed formula and then substitute ru for s, as shown here. v =

s ru u = = r = rv t t t

438

CHAPTER 5


Thus the linear speed of a point moving on a circular path is the product of the radius of the circle and the angular speed of the point.

The Linear Speed–Angular Speed Relationship The linear speed v and the angular speed v, in radians per unit of time, of a point moving on a circular path with radius r are related by v = rv The equation v = rv gives the linear speed of a point moving on a circular path in terms of distance r from the center of the circle and the angular speed v, provided v is in radians per unit of time.

EXAMPLE 8

Find a Linear Speed

A wind machine is used to generate electricity. The wind machine has propeller blades that are 12 feet in length (see Figure 5.28). If the propeller is rotating at 3 revolutions per second, what is the linear speed in feet per second of the tips of the blades? Solution Convert the angular speed v = 3 revolutions per second into radians per second, and then use the formula v = rv.

12 ft

v =

3 revolutions 2p radians 6p radians 3 revolutions = a ba b = 1 second 1 second 1 revolution 1 second

Thus 6p radians b 1 second = 72p feet per second L 226 feet per second

v = rv = (12 feet)a

Figure 5.28


EXERCISE SET 5.1 In Exercises 1 to 12, find the measure (if possible) of the complement and the supplement of each angle. 1. 15°

2. 87°

3. 70°15¿

4. 22°43¿

5. 56°33¿15–

6. 19°42¿05–

7. 1

8. 0.5

9.

p 10. 3

2p 11. 5

p 4

p 12. 6

In Exercises 13 to 18, determine the measure of the positive angle with measure less than 360° that is coterminal with the given angle and then classify the angle by quadrant. Assume the angles are in standard position. 13. a = 610°

14. a = 765°

15. a = - 975°

16. a = - 872°

17. a = 2456°

18. a = - 3789°

5.1

In Exercises 19 to 24, use a calculator to convert each decimal degree measure to its equivalent DMS measure. 19. 24.56°

20. 110.24°

21. 64.158°

22. 18.96°

23. 3.402°

24. 224.282°

In Exercises 25 to 30, use a calculator to convert each DMS measure to its equivalent decimal degree measure. 25. 25°25¿12–

26. 63°29¿42–

27. 183°33¿36–

28. 141°6¿9–

29. 211°46¿48–

30. 19°12¿18–

In Exercises 31 to 42, convert the degree measure to exact radian measure.

64. r = 35.8 meters, s = 84.3 meters

In Exercises 65 to 68, find the length of an arc that subtends a central angle with the given measure in a circle with the given radius. Round answers to the nearest hundredth. 65. r = 8 inches, u =

66. r = 3 feet, u =

p 4

7p 2

32. -45°

33. 90°

67. r = 25 centimeters, u = 42°

34. 15°

35. 165°

36. 315°

68. r = 5 meters, u = 144°

37. 420°

38. 630°

39. 585°

40. 135°

41. -9°

42. - 110°

43.

7p 3

46. -

2p 3

3p 49. 8 52.

6p 5

439

63. r = 5.2 centimeters, s = 12.4 centimeters

31. 30°

In Exercises 43 to 54, convert the radian measure to exact degree measure.

ANGLES AND ARCS

In Exercises 69 and 70, find the number of radians in the revolutions indicated. 69. 1

1 revolutions 2

3 revolution 8

44.

p 4

45.

p 5

70.

47.

p 6

48.

p 9

71. Angular Rotation of Two Pulleys A pulley with a radius of

11p 50. 18 53. -

5p 12

11p 51. 3 54. -

4p 5

In Exercises 55 to 60, convert radians to degrees or degrees to radians. Round answers to the nearest hundredth. 55. 1.5

56. -2.3

57. 133°

58. 427°

59. 8.25

60. - 90°

14 inches uses a belt to drive a pulley with a radius of 28 inches. The 14-inch pulley turns through an angle of 150°. Find the angle through which the 28-inch pulley turns. 72. Angular Rotation of Two Pulleys A pulley with a diameter

of 1.2 meters uses a belt to drive a pulley with a diameter of 0.8 meter. The 1.2-meter pulley turns through an angle of 240°. Find the angle through which the 0.8-meter pulley turns. 73. Angular Speed Find the angular speed, in radians per second,

of the second hand on a clock. 74. Angular Speed Find the angular speed, in radians per second,

In Exercises 61 to 64, find the measure in radians and degrees of the central angle of a circle subtended by the given arc. Round approximate answers to the nearest hundredth.

of a point on the equator of the earth. 75. Angular Speed A wheel is rotating at 50 revolutions per

minute. Find the angular speed in radians per second.

61. r = 2 inches, s = 8 inches 62. r = 7 feet, s = 4 feet

76. Angular Speed A wheel is rotating at 200 revolutions per

minute. Find the angular speed in radians per second.

440

CHAPTER 5


77. Angular Speed The turntable of a record player turns

1 at 33 revolutions per minute. Find the angular speed in radi3 ans per second.

a. Find the distance the container is lifted as the winch is rotated

through an angle of

5p radians. 6

b. Determine the angle, in radians, through which the winch

must be rotated to lift the container a distance of 2 feet. 78. Angular Speed A car with a wheel of radius 14 inches is

moving with a speed of 55 mph. Find the angular speed of the wheel in radians per second. 79. Linear Speed of a Car Each tire on a car has a radius of

15 inches. The tires are rotating at 450 revolutions per minute. Find the speed of the automobile to the nearest mile per hour.

83. Amusement Park Ride A ride at an amusement park

consists of two circular rings of swings. At full speed the swings in the inner ring travel on a circular path with a radius of 32 feet and the swings in the outer ring travel on a circular path with a radius of 38 feet. Each swing makes one complete revolution every 3.75 seconds. How much greater, in miles per hour, is the linear speed of the swings in the outer ring than the linear speed of the swings in the inner ring? Round to the nearest tenth.

80. Linear Speed of a Truck Each tire on a truck has a radius

of 18 inches. The tires are rotating at 500 revolutions per minute. Find the speed of the truck to the nearest mile per hour. 81. Bicycle Gears The chain wheel of Emma’s bicycle has a radius

of 3.5 inches. The rear gear has a radius of 1.75 inches, and the back tire has a radius of 12 inches. If Emma pedals for 150 revolutions of the chain wheel, how far will she travel? Round to the nearest foot.

84. Horse Racing The semicircular turns of a horse race track

each have a radius of 200 feet. During the first turn of a race, the lead horse is running near the inside rail on a path with a 202.0-foot radius, at a constant rate of 24.4 feet per second. A second horse is rounding the same turn on a path with a 206.5foot radius. At what constant rate does the second horse need to run to keep pace with the lead horse during this turn? Round to the nearest tenth of a foot per second. 85. Astronomy At a time when Earth was 93 million miles from

the sun, you observed through a tinted glass that the diameter of the sun occupied an arc of 31¿. Determine, to the nearest ten thousand miles, the diameter of the sun.

Radius 1.75 in.

Radius 12 in.

Radius 3.5 in.

93,000,000 mi

A

31'

82. Rotation versus Lift Distance A winch with a 6-inch radius

Sun

Earth

is used to lift a container. The winch is designed so that, as it is rotated, the cable stays in contact with the surface of the winch. That is, the cable does not wrap on top of itself.

B

(Hint: Because the radius of arc AB is large and its central angle is small, the length of the diameter of the sun is approximately the length of the arc AB.) Radius 6 in.

86. Angle of Rotation and Distance The minute hand on the clock

atop city hall measures 6 feet 3 inches from its tip to its axle. FRA

GILE

LE ND HA ITH W RE CA

a. Through what angle (in radians) does the minute hand pass

between 9:12 A.M. and 9:48 A.M.? b. What distance, to the nearest tenth of a foot, does the tip of

the minute hand travel during this period?

5.1

87. Velocity of the Hubble Space Telescope On April 25, 1990,

the Hubble Space Telescope (HST) was deployed into a circular orbit 625 kilometers above the surface of the earth. The HST completes an Earth orbit every 1.61 hours.

89.

ANGLES AND ARCS

441

Velocity Comparisons Assume that the bicycle in the figure is moving forward at a constant rate. Point A is on the edge of the 30-inch rear tire, and point B is on the edge of the 20-inch front tire.

A

B

a. Which point (A or B) has the greater angular velocity?

NASA and JPL

b. Which point (A or B) has the greater linear velocity?

a. Find the angular velocity, with respect to the center of

Earth, of the HST. Round your answer to the nearest 0.1 radian per hour.

90. Given that s, r, u, t, v, and v are as defined in Section 5.1,

determine which of the following formulas are valid. s = ru v = rv

s u s v = t r =

ru t u v = t

v =

b. Find the linear velocity of the HST. (Hint: The radius

91. Nautical Miles and Statute Miles A nautical mile is the

of Earth is about 6370 kilometers.) Round your answer to the nearest 100 kilometers per hour.

length of an arc, on Earth’s equator, that subtends a 1¿ central angle. The equatorial radius of Earth is about 3960 statute miles. a. Convert 1 nautical mile to statute miles. Round to the near-

88. Estimating the Radius of Earth Eratosthenes, the fifth

librarian of Alexandria (230 B.C.), was able to estimate the radius of Earth from the following data: The distance between the Egyptian cities of Alexandria and Syrene was 5000 stadia (520 miles). Syrene was located directly south of Alexandria. One summer, at noon, the sun was directly overhead at Syrene, whereas at the same time in Alexandria, the sun was at a 7.5° angle from the zenith.

7.5° Alexandria

Syrene A

S 7.5°

est hundredth of a statute mile. b. Determine what percent (to the nearest 1%) of Earth’s cir-

cumference is covered by a trip from Los Angeles, California, to Honolulu, Hawaii (a distance of 2217 nautical miles). 92. Photography The field of view for a camera with a

200-millimeter lens is 12°. A photographer takes a photograph of a large building that is 485 feet in front of the camera. What is the approximate width, to the nearest foot, of the building that will appear in the photograph? (Hint: If the radius of an arc AB is large and its central angle is small, then the length of the line segment AB is approximately the length of the arc AB.) A sector of a circle is the region bounded by radii OA and OB and the intercepted arc AB. See the following figure. The area of the sector is given by

O

B r

1 A r 2U 2 Eratosthenes reasoned that because the sun is far away, the rays of sunlight that reach Earth must be nearly parallel. From this assumption he concluded that the measure of ∠AOS in the figure above must be 7.5°. Use this information to estimate the radius (to the nearest 10 miles) of Earth.

θ

O

r

A

where r is the radius of the circle and U is the measure of the central angle in radians.

CHAPTER 5


In Exercises 93 to 96, find the area, to the nearest square unit, of the sector of a circle with the given radius and central angle.

nearest 10 miles, the given city is from the equator. Use 3960 miles as the radius of Earth. EARTH radius 3960 mi

p 93. r = 5 inches, u = radians 3 94. r = 2.8 feet, u =

N

5p radians 2

New York City 40° 45′ N

95. r = 120 centimeters, u = 0.65 radian

Miami 25° 47′ N

r

Greenwich Meridian

442

Equator

96. r = 30 feet, u = 62°

Latitude describes the position of a point on Earth’s surface in relation to the equator. A point on the equator has a latitude of 0°. The north pole has a latitude of 90°. In Exercises 97 and 98, determine how far north, to the

SECTION 5.2 The Six Trigonometric Functions Trigonometric Functions of Special Angles Applications Involving Right Triangles

Longitude east

Latitude north

97. The city of Miami has a latitude of 25°47¿N. 98. New York City has a latitude of 40°45¿N.

Right Triangle Trigonometry PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A33.

PS1. Rationalize the denominator of

1 . [P.2] 13

PS2. Rationalize the denominator of

2 . [P.2] 12

PS3. Simplify: a , a b [P.5]

a 2

PS4. Simplify: a b , a

a 2

13 ab [P.5] 2

PS5. Solve

x 12 = for x. Round your answer to the nearest hundredth. [1.1] 2 5

PS6. Solve

x 13 for x. Round your answer to the nearest hundredth. [1.1] = 3 18

The Six Trigonometric Functions The study of trigonometry, which means “triangle measurement,” began more than 2000 years ago, partially as a means of solving surveying problems. Early trigonometry used the length of a line segment between two points of a circle as the value of a trigonometric function. In the sixteenth century, right triangles were used to define a trigonometric function. We will use a modification of this approach.

5.2

RIGHT TRIANGLE TRIGONOMETRY

443

When working with right triangles, it is convenient to refer to the side opposite an angle or the side adjacent to (next to) an angle. Figure 5.29 shows the sides opposite and adjacent to the angle u. Figure 5.30 shows the sides opposite and adjacent to the angle b. In both cases, the hypotenuse remains the same.

Hypotenuse

Opposite side

β

Hypotenuse

Adjacent side

θ

Adjacent side

Opposite side

Figure 5.29

Figure 5.30

Adjacent and opposite sides of ∠u

Adjacent and opposite sides of ∠ b

Six ratios can be formed by using two lengths of the three sides of a right triangle. Each ratio defines a value of a trigonometric function of a given acute angle u. The functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

Definitions of Trigonometric Functions of an Acute Angle Let u be an acute angle of a right triangle. See Figure 5.29. The values of the six trigonometric functions of u are length of opposite side length of adjacent side sin u = cos u = length of hypotenuse length of hypotenuse length of opposite side length of adjacent side tan u = cot u = length of adjacent side length of opposite side length of hypotenuse length of hypotenuse csc u = sec u = length of adjacent side length of opposite side We will write opp, adj, and hyp as abbreviations for the length of the opposite side, adjacent side, and hypotenuse, respectively.

EXAMPLE 1

Evaluate Trigonometric Functions

Find the values of the six trigonometric functions of u for the triangle given in Figure 5.31. Solution Use the Pythagorean Theorem to find the length of the hypotenuse. hyp

hyp = 332 + 42 = 125 = 5

3

From the definitions of the trigonometric functions, θ

4

Figure 5.31

opp 3 = hyp 5 adj 4 = cot u = opp 3 sin u =


adj 4 = hyp 5 hyp 5 sec u = = adj 4

cos u =

opp 3 = adj 4 hyp 5 = csc u = opp 3 tan u =

444

CHAPTER 5


Given the value of one trigonometric function of the acute angle u, it is possible to find the value of any of the remaining trigonometric functions of u.

EXAMPLE 2

Find the Value of a Trigonometric Function

Given that u is an acute angle and cos u =

5 , find tan u. 8

Solution cos u =

adj 5 = 8 hyp

Sketch a right triangle with one leg of length 5 units and a hypotenuse of length 8 units. Label as u the acute angle that has the leg of length 5 units as its adjacent side (see Figure 5.32). Use the Pythagorean Theorem to find the length of the opposite side. 8

1opp22 + 52 = 82 1opp22 + 25 = 64

opp

1opp22 = 39

θ

opp = 139 5

Therefore, tan u =

Figure 5.32

opp 139 = . adj 5


Trigonometric Functions of Special Angles

B 45° r= 2a

a

45° A

a

Figure 5.33

C

In Example 1, the lengths of the legs of the triangle were given, and you were asked to find the values of the six trigonometric functions of the angle u. Often we will want to find the value of a trigonometric function when we are given the measure of an angle rather than the measure of the sides of a triangle. For most angles, advanced mathematical methods are required to evaluate a trigonometric function. For some special angles, however, the value of a trigonometric function can be found by geometric methods. These special acute angles are 30°, 45°, and 60°. First, we will find the values of the six trigonometric functions of 45°. (This discussion is based on angles measured in degrees. Radian measure could have been used without changing the results.) Figure 5.33 shows a right triangle with angles 45°, 45°, and 90°. Because ∠A = ∠ B, the lengths of the sides opposite these angles are equal. Let the length of each equal side be denoted by a. From the Pythagorean Theorem, r 2 = a2 + a2 = 2a2 r = 22a2 = 12 a The values of the six trigonometric functions of 45° are 1 a 12 = = 2 12 a 12 a tan 45° = = 1 a

sin 45° =

sec 45° =

12 a = 12 a

1 12 a = = 2 12 a 12 a cot 45° = = 1 a cos 45° =

csc 45° =

12 a = 12 a

5.2

30° a 3a h= 2 O

60° a 2

445

The values of the trigonometric functions of the special angles 30° and 60° can be found by drawing an equilateral triangle and bisecting one of the angles, as Figure 5.34 shows. The angle bisector also bisects one of the sides. Thus the length of the side opposite the 30° angle is one-half the length of the hypotenuse of triangle OAB. Let a denote the length of the hypotenuse. Then the length of the side opposite the a 30° angle is . The length of the side adjacent to the 30° angle, h, is found by using the 2 Pythagorean Theorem.

A

C


B

a 2 a2 = a b + h2 2

Figure 5.34

a2 =

a2 + h2 4

3a2 = h2 4

• Subtract

13 a 2

h =

a2 from each side. 4

• Solve for h.

The values of the six trigonometric functions of 30° are

sin 30° = tan 30° = sec 30° =

a>2 a

=

1 2

a>2

=

13 a>2

13a>2

cos 30° = 1 13 = 3 13

cot 30° =

a 2 2 13 = = 13 a>2 3 13

csc 30° =

=

a 13a>2 a>2

13 2

= 13

a = 2 a>2

The values of the trigonometric functions of 60° can be found by again using Figure 5.34. The length of the side opposite the 60° angle is

13a , and the length of the side adjacent 2

a to the 60° angle is . The values of the trigonometric functions of 60° are 2 sin 60° = tan 60° =

sec 60° =

13a>2 a 13a>2 a>2

13 2

cos 60° =

= 13

cot 60° =

=

a = 2 a>2

csc 60° =

a>2 a

=

a>2 13 a>2

1 2 =

1 13 = 3 13

a 213 2 = = 13 a>2 3 13

Table 5.2 on page 446 summarizes the values of the trigonometric functions of the special angles 30° 1p>62, 45° 1p>42, and 60° 1p>32.

446

CHAPTER 5


Table 5.2

Study tip Memorizing the values given in Table 5.2 will prove to be extremely useful in the remaining trigonometry sections.

Trigonometric Functions of Special Angles

U

sin U

cos U

tan U

csc U

sec U

cot U

p 30°; 6

1 2

13 2

13 3

2

213 3

13

45°;

p 4

12 2

12 2

1

12

12

1

60°;

p 3

13 2

1 2

13

213 3

2

13 3

Question • What is the measure, in degrees, of the acute angle u for which sin u = cos u,

tan u = cot u, and sec u = csc u?

EXAMPLE 3

Find the exact value of sin2 45° + cos2 60°. Note: sin2 u = 1sin u21sin u2 = 1sin u22 and cos2 u = 1cos u21cos u2 = 1cos u22.

Study tip The patterns in the following chart can be used to memorize the sine and cosine of 30°, 45°, and 60°. 11 2 12 sin 45° = 2 13 sin 60° = 2 sin 30° =

Evaluate a Trigonometric Expression

13 2 12 cos 45° = 2 11 cos 60° = 2

Solution Substitute the values of sin 45° and cos 60° and simplify. sin2 45° + cos2 60° = a

cos 30° =

12 2 1 2 2 1 3 b + a b = + = 2 2 4 4 4


From the definition of the sine and cosecant functions, 1sin u21csc u2 = =

opp hyp

#

hyp = 1 opp

or

1sin u21csc u2 = 1

By rewriting the last equation, we find sin u =

1 csc u

and

csc u =

1 , provided sin u Z 0 sin u

The sine and cosecant functions are called reciprocal functions. The cosine and secant are also reciprocal functions, as are the tangent and cotangent functions. Table 5.3 shows each trigonometric function and its reciprocal. These relationships hold for all values of u for which both of the functions are defined. Table 5.3 Trigonometric Functions and Their Reciprocals

Answer • 45°.

sin u =

1 csc u

cos u =

1 sec u

tan u =

1 cot u

csc u =

1 sin u

sec u =

1 cos u

cot u =

1 tan u

5.2


447

Integrating Technology The values of the trigonometric functions of the special angles 30°, 45°, and 60° shown in Table 5.2 are exact values. If an angle is not one of these special angles, then a graphing calculator often is used to approximate the value of a trigonometric function. For instance, to find sin 52.4° on a TI-83/TI-83 Plus/TI-84 Plus calculator, first check that the calculator is in degree mode. Then use the sine function key

冷 SIN 冷 to key in sin(52.4) and press 冷 ENTER 冷. See Figure 5.35. To find sec 1.25, first check that the calculator is in radian mode. A TI-83/TI-83 Plus/TI-84 Plus calculator does not have a secant function key, but because the secant function is the reciprocal of the cosine function, we can evaluate sec 1.25 by evaluating 1/(cos 1.25). See Figure 5.36. Select Degree in the Mode menu.

Select Radian in the Mode menu.

Normal Sci Eng Float 0123456789 Radian Degree Func sin(52.4) Cong .7922896434 Sequ Real Full

Normal Sci Eng Float 0123456789 Radian Degree Func 1/(cos(1.25)) Cong 3.171357694 Sequ Real Full

Figure 5.35

Figure 5.36

Line of sight

Angle of elevation

Horizontal line

Angle of depression Line of sight

Figure 5.37

When you evaluate a trigonometric function with a calculator, be sure the calculator is in the correct mode. Many errors are made because the correct mode has not been selected.

Applications Involving Right Triangles Some applications concern an observer looking at an object. In these applications, angles of elevation or angles of depression are formed by a line of sight and a horizontal line. If the object being observed is above the observer, the acute angle formed by the line of sight and the horizontal line is an angle of elevation. If the object being observed is below the observer, the acute angle formed by the line of sight and the horizontal line is an angle of depression. See Figure 5.37.

EXAMPLE 4 h

64.3° 115 ft

Figure 5.38

Solve an Angle-of-Elevation Application

From a point 115 feet from the base of a redwood tree, the angle of elevation to the top of the tree is 64.3°. Find the height of the tree to the nearest foot. Solution From Figure 5.38, the length of the adjacent side of the angle is known (115 feet). Because we need to determine the height of the tree (length of the opposite side), (continued)

448

CHAPTER 5


we use the tangent function. Let h represent the length of the opposite side. opp h = tan 64.3° = adj 115 h = 115 tan 64.3° L 238.952 The height of the tree is approximately 239 feet.

• Use a calculator to evaluate tan 64.3°.


Note The significant digits of an approximate number are every nonzero digit the digit 0, provided it is between two nonzero digits or it is to the right of a nonzero digit in a number that includes a decimal point For example, the approximate number 502 has 3 significant digits. 3700 has 2 significant digits. 47.0 has 3 significant digits. 0.0023 has 2 significant digits. 0.00840 has 3 significant digits.

Because the cotangent function involves the sides adjacent to and opposite an angle, we could have solved Example 4 by using the cotangent function. The solution would have been adj 115 = opp h 115 h = L 238.952 feet cot 64.3°

cot 64.3° =

The accuracy of a calculator is sometimes beyond the limits of measurement. In Example 4 the distance from the base of the tree was given as 115 feet (three significant digits), whereas the height of the tree was shown to be 238.952 feet (six significant digits). When using approximate numbers, we will use the conventions given below for calculating with trigonometric functions.

A Rounding Convention: Significant Digits for Trigonometric Calculations Angle Measure to the Nearest

Significant Digits of the Lengths

Degree

Two

Tenth of a degree

Three

Hundredth of a degree

Four

EXAMPLE 5

Solve an Angle-of-Depression Application

Distance measuring equipment (DME) is standard avionic equipment on a commercial airplane. This equipment measures the distance from a plane to a radar station. If the distance from a plane to a radar station is 160 miles and the angle of depression is 33°, find the number of ground miles from a point directly below the plane to the radar station.

33° 57° 160 mi

x

Figure 5.39

Solution From Figure 5.39, the length of the hypotenuse is known (160 miles). The length of the side opposite the angle of 57° is unknown. The sine function involves the hypotenuse and the opposite side, x, of the 57° angle. x 160 x = 160 sin 57° L 134.1873 Rounded to two significant digits, the plane is 130 ground miles from the radar station. sin 57° =


5.2

EXAMPLE 6


449

Solve an Angle-of-Elevation Application

An observer notes that the angle of elevation from point A to the top of a space shuttle is 27.2°. From a point 17.5 meters further from the space shuttle, the angle of elevation is 23.9°. Find the height of the space shuttle. y

Solution From Figure 5.40, let x denote the distance from point A to the base of the space shuttle, and let y denote the height of the space shuttle. Then

23.9° 27.2°

tan 27.2° =

(1)

x A

y x

and

(2)

tan 23.9° =

y x + 17.5

y Solving Equation (1) for x, x = = y cot 27.2ⴰ , and substituting into tan 27.2 ⴰ Equation (2), we have

17.5 m

Figure 5.40

tan 23.9° =

y y cot 27.2° + 17.5

y = 1tan 23.9°21y cot 27.2° + 17.52 y - y tan 23.9° cot 27.2° = 1tan 23.9°2117.52 y11 - tan 23.9° cot 27.2°2 = 1tan 23.9°2117.52 1tan 23.9°2117.52 y = L 56.2993 1 - tan 23.9° cot 27.2°

Note The intermediate calculations in Example 6 were not rounded off. This ensures better accuracy for the final result. Using the rounding convention stated on page 448, we round off only the last result.

• Solve for y.

To three significant digits, the height of the space shuttle is 56.3 meters. Try Exercise 68, page 452

EXERCISE SET 5.2 In Exercises 1 to 12, find the values of the six trigonometric functions of U for the right triangle with the given sides. 1.

5.

6. 8

5

5

2.

θ θ

7

12

2

7.

θ

θ

3

5

8. 3

θ

10

5

2

3.

θ

4. 7 θ

9 4

θ

3

9. 6

θ

10. θ

3 0.8

1

450

CHAPTER 5


11.

12. θ

2

5

36. cos

θ

p p p tan + 2 tan 4 6 3

1

37. 2 csc

p p p - sec cos 4 3 6

38. 3 tan

p p p + sec sin 4 6 3

6

In Exercises 13 to 15, let U be an acute angle of a right 3 triangle for which sin U . Find 5 13. tan u

14. sec u

15. cos u

In Exercises 16 to 18, let U be an acute angle of a right 4 triangle for which tan U . Find 3 16. sin u

17. cot u

18. sec u

In Exercises 19 to 21, let B be an acute angle of a right 13 triangle for which sec B . Find 12 19. cos b

20. cot b

21. csc b

In Exercises 22 to 24, let U be an acute angle of a right 2 triangle for which cos U . Find 3 22. sin u

23. sec u

In Exercises 39 to 50, use a calculator to find the value of the trigonometric function to four decimal places. 39. tan 32°

40. sec 88°

41. cos 63°20¿

42. cot 55°50¿

43. cos 34.7°

44. tan 81.3°

45. sec 5.9°

46. sin

48. sec

3p 8

p 5

47. tan

49. csc 1.2

p 7

50. sin 0.45

51. Vertical Height from Slant Height A 12-foot ladder is rest-

ing against a wall and makes an angle of 52° with the ground. Find the height to which the ladder will reach on the wall. 52. Distance Across a Marsh Find the distance AB across the

marsh shown in the accompanying figure.

24. tan u A

In Exercises 25 to 38, find the exact value of each expression. 25. sin 45° + cos 45°

26. csc 45° - sec 45°

27. sin 30° cos 60° - tan 45°

28. csc 60° sec 30° + cot 45°

52° B

C 31 m

29. sin 30° cos 60° + tan 45° 30. sec 30° cos 30° - tan 60° cot 60°

31. sin

p p + cos 3 6

33. sin

p p + tan 4 6

34. sin

p p p cos - tan 3 4 4

35. sec

p p p cos - tan 3 3 6

32. csc

53. Width of a Ramp A skateboarder wishes to build a jump

p p - sec 6 3

ramp that is inclined at a 19.0 angle and that has a maximum height of 32.0 inches. Find the horizontal width x of the ramp.

32.0 in. 19.0° x

5.2

54. Time of Closest Approach At 3:00 P.M., a boat is 12.5 miles

due west of a radar station and traveling at 11 mph in a direction that is 57.3° south of an east–west line. At what time will the boat be closest to the radar station?


451

from the building. Find the height of the building to the nearest tenth of a foot.

Transit 27.8°

131 ft

12.5 mi

5.5 ft

57.3°

60. Width of a Lake The angle of depression to one side of a lake,

55. Placement of a Light For best illumination of a piece of

measured from a balloon 2500 feet above the lake as shown in the accompanying figure, is 43°. The angle of depression to the opposite side of the lake is 27°. Find the width of the lake.

art, a lighting specialist for an art gallery recommends that a ceiling-mounted light be 6 feet from the piece of art and that the angle of depression of the light be 38°. How far from a wall should the light be placed so that the recommendations of the specialist are met? Notice that the art extends outward 4 inches from the wall.

4 in.

38° 6 ft

43°

27°

2500 ft A

B

61. Astronomy The moon Europa rotates in a nearly circular 56. Height of the Eiffel Tower The angle of elevation from a

point 116 meters from the base of the Eiffel Tower to the top of the tower is 68.9°. Find the approximate height of the tower. 57. Distance of a Descent An airplane traveling at 240 mph is

orbit around Jupiter. The orbital radius of Europa is approximately 670,900 kilometers. During a revolution of Europa around Jupiter, an astronomer found that the maximum value of the angle u formed by Europa, Earth, and Jupiter was 0.056°. Find the distance d between Earth and Jupiter at the time the astronomer found the maximum value of u. Round to the nearest million kilometers.

descending at an angle of depression of 6°. How many miles will the plane descend in 4 minutes? Europa

58. Time of a Descent A submarine traveling at 9.0 mph is

descending at an angle of depression of 5°. How many minutes, to the nearest tenth, does it take the submarine to reach a depth of 80 feet?

r = 670,900 km

Earth

= 0.056 d Jupiter

59. Height of a Building A surveyor determines that the angle of

elevation from a transit to the top of a building is 27.8. The transit is positioned 5.5 feet above ground level and 131 feet

Not drawn to scale.

452

CHAPTER 5


62. Astronomy Venus rotates in a nearly circular orbit around

the sun. The largest angle formed by Venus, Earth, and the sun is 46.5°. The distance from Earth to the sun is approximately 149 million kilometers. See the following figure. What is the orbital radius r of Venus? Round to the nearest million kilometers.

51.9° 36.4° 350 ft

66. Height of a Building Two buildings are 240 feet apart. The

angle of elevation from the top of the shorter building to the top of the other building is 22°. If the shorter building is 80 feet high, how high is the taller building?

Venus

r

67. Height of the Washington Monument From a point A on a

46.5 Earth

Sun

149,000,000 km

line from the base of the Washington Monument, the angle of elevation to the top of the monument is 42.00°. From a point 100 feet away from A and on the same line, the angle to the top is 37.77°. Find the height of the Washington Monument.

63. Area of an Isosceles Triangle Consider the following isosce-

les triangle. The length of each of the two equal sides of the triangle is a, and each of the base angles has a measure of u. Verify that the area of the triangle is A = a2 sin u cos u.

37.77°

42.00° A

100.0 ft a

a

θ

θ

68. Height of a Tower The angle of elevation from a point A to the

b

top of a tower is 32.1. From point B, which is on the same line but 55.5 feet closer to the tower, the angle of elevation is 36.5. Find the height of the tower.

64. Area of a Hexagon Find the area of

the hexagon. (Hint: The area consists of six isosceles triangles. Use the formula from Exercise 63 to compute the area of one of the triangles and multiply by 6.)

4 in.

4 in. 60° 60° 4 in.

65. Height of a Pyramid The angle of elevation to the top of the

Egyptian pyramid of Cheops is 36.4°, measured from a point 350 feet from the base of the pyramid. The angle of elevation from the base of a face of the pyramid is 51.9°. Find the height of the Cheops pyramid.

36.5°

32.1° B

A

55.5 ft

69. Length of a Golf Drive The helipad of the Burj al Arab

hotel is 211 meters above the surrounding beaches. A golfer drives a golf ball off the edge of the helipad as shown in the

5.2


453

following figure. Find the length (horizontal distance AB) of the drive.

Helipad A

Horizontal line B

C

32.0

D

211 meters

AB = 412 ft CAB = 53.6° AB is at ground level CAD = 15.5°

C

B A

70. Size of a Sign From point A, at street level and 205 feet from

the base of a building, the angle of elevation to the top of the building is 23.1°. Also, from point A the angle of elevation to the top of a neon sign, which is atop the building, is 25.9°. a. Determine the height of the building.

72. An Eiffel Tower Replica Use the information in the

accompanying figure to estimate the height of the Eiffel Tower replica that stands in front of the Paris Las Vegas Hotel in Las Vegas, Nevada.

Tony Craddock/Getty Images

b. How tall are the letters in the sign?

Street level 25.9 23.1

46.3° 65.5°

235 ft

205 ft

A

73. Radius of a Circle A circle is inscribed in a regular hexagon

with each side 6.0 meters long. Find the radius of the circle.

71. The Petronas Towers The Petronas Towers in Kuala Lumpur,

Malaysia, are the world’s tallest twin towers. Each tower is 1483 feet in height. The towers are connected by a skybridge at the forty-first floor. Note the information given in the following figure.

74. Area of a Triangle Show that the area A of the triangle given

in the figure is A =

1 ab sin u. 2

a. Determine the height of the skybridge.

a

b. Determine the length of the skybridge.

θ

c

b

454

CHAPTER 5


75. Find a Maximum Length Find the length of the longest piece

76. Find a Maximum Length In Exercise 75, suppose that the hall

of wood that can be slid around the corner of the hallway in the figure following. Round to the nearest tenth of a foot.

is 8 feet high. Find the length of the longest piece of wood that can be taken around the corner. Round to the nearest tenth of a foot.

3 ft

θ

3 ft

SECTION 5.3

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Trigonometric Functions of Quadrantal Angles Signs of Trigonometric Functions The Reference Angle


3 . [P.1] 4

PS1. Find the reciprocal of PS3. Evaluate: 120 - 180

3 2

PS5. Simplify: p -

[P.1]

p [P.1] 2

PS2. Find the reciprocal of PS4. Simplify: 2p -

21 5 . [P.1] 5

9p [P.1] 5

PS6. Simplify: 31 - 322 + 1 - 522 [P.1]

Trigonometric Functions of Any Angle The applications of trigonometry would be quite limited if all angles had to be acute angles. Fortunately, this is not the case. In this section we extend the definition of a trigonometric function to include any angle. Consider angle u in Figure 5.41 in standard position and a point P1x, y2 on the terminal side of the angle. We define the trigonometric functions of any angle according to the following definitions.

y

P(x, y) r y

θ

x

O

Figure 5.41

Definitions of the Trigonometric Functions of Any Angle x

Let P1x, y2 be any point, except the origin, on the terminal side of an angle u in standard position. Let r = d1O, P2, the distance from the origin to P. The six trigonometric functions of u are y r r csc u = , y

sin u =

x r r sec u = , x cos u =

y Z 0

where r = 2x2 + y2.

x Z 0

y tan u = , x x cot u = , y

x Z 0 y Z 0

5.3

y

P(a′, b ′) r′ b′

P(a, b)

r

b

θ

x

a a′

Figure 5.42

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

455

The value of a trigonometric function is independent of the point chosen on the terminal side of the angle. Consider any two points on the terminal side of an angle u in standard position, as shown in Figure 5.42. The right triangles formed are similar triangles, so b¿ b the ratios of the corresponding sides are equal. Thus, for example, = . Because a a¿ b¿ b b¿ tan u = = , we have tan u = . Therefore, the value of the tangent function is a a¿ a¿ independent of the point chosen on the terminal side of the angle. By a similar argument, we can show that the value of any trigonometric function is independent of the point chosen on the terminal side of the angle. Any point in a rectangular coordinate system (except the origin) can determine an angle in standard position. For example, P1- 4, 32 in Figure 5.43 is a point in the second quadrant and determines an angle u in standard position with r = 21 - 422 + 32 = 5. y

P(− 4, 3) 5 θ

x

Figure 5.43

The values of the trigonometric functions of u as shown in Figure 5.43 are sin u =

3 5

cos u =

-4 4 = 5 5

tan u =

3 3 = -4 4

csc u =

5 3

sec u =

5 5 = -4 4

cot u =

-4 4 = 3 3

EXAMPLE 1


Find the exact value of each of the six trigonometric functions of an angle u in standard position whose terminal side contains the point P1- 3, - 22. Solution The angle is sketched in Figure 5.44. Find r by using the equation r = 2x2 + y2, where x = - 3 and y = - 2. r = 21 -322 + 1 -222 = 19 + 4 = 113

y

4

Now use the definitions of the trigonometric functions.

2 θ

−4 P(−3, −2)

−2

2 r

−2

Figure 5.44

4

x

2 113 -2 = 13 113 113 113 csc u = = -2 2 sin u =


3113 -3 = 13 113 113 113 sec u = = -3 3 cos u =

-2 2 = -3 3 -3 3 cot u = = -2 2

tan u =

456

CHAPTER 5


Trigonometric Functions of Quadrantal Angles Recall that a quadrantal angle is an angle whose terminal side coincides with the x- or y-axis. The value of a trigonometric function of a quadrantal angle can be found by choosing any point on the terminal side of the angle and then applying the definition of that trigonometric function. The terminal side of 0° coincides with the positive x-axis. Let P1x, 02, x 7 0, be any point on the x-axis, as shown in Figure 5.45. Then y = 0 and r = x. The values of the six trigonometric functions of 0° are

y

P(x, 0) x

Figure 5.45

sin 0° =

0 = 0 r

csc 0° is undefined.

0 = 0 x

cos 0° =

x x = = 1 r x

tan 0° =

sec 0° =

x r = = 1 x x

cot 0° is undefined.

Question • Why are csc 0° and cot 0° undefined?

In like manner, the values of the trigonometric functions of the other quadrantal angles can be found. The results are shown in Table 5.4. Table 5.4

U

Values of Trigonometric Functions of Quadrantal Angles

sin U

0° y

θ

x

tan U

csc U

sec U

cot U

1

0

Undefined

1

Undefined

Undefined

1

Undefined

0

0

Undefined

-1

Undefined

Undefined

-1

Undefined

0

90°

1

0

180°

0

-1

270°

-1

0

Signs of Trigonometric Functions The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies. For example, if u is an angle whose terminal side lies in Quadrant III and y x P1x, y2 is on the terminal side of u, then both x and y are negative, and therefore and x y y x are positive. See Figure 5.46. Because tan u = and cot u = , the values of the tangent x y and cotangent functions are positive for any Quadrant III angle. The values of the other four trigonometric functions of any Quadrant III angle are all negative. Table 5.5 lists the signs of the six trigonometric functions in each quadrant. Figure 5.47 is a graphical display of the contents of Table 5.5.

P(x, y)

Figure 5.46

y

Sine and cosecant positive

0

cos U

All functions positive

Table 5.5

Signs of the Trigonometric Functions

x

Tangent and cotangent positive

Cosine and secant positive

Figure 5.47

Sign of

Terminal Side of U in Quadrant I

II

III

IV

sin u and csc u

Positive

Positive

Negative

Negative

cos u and sec u

Positive

Negative

Negative

Positive

tan u and cot u

Positive

Negative

Positive

Negative

Answer • P1x, 02 is a point on the terminal side of a 0° angle in standard position. Thus

csc 0° =

x r , which is undefined. Similarly, cot 0° = , which is undefined. 0 0

5.3


457

In the next example we are asked to evaluate two trigonometric functions of the angle u. A key step is to use our knowledge about trigonometric functions and their signs to determine that u is a Quadrant IV angle.

EXAMPLE 2 Given tan u = -

2

−2

7 and sin u 6 0, find cos u and csc u. 5

Solution The terminal side of angle u must lie in Quadrant IV; that is the only quadrant in which sin u and tan u are both negative. Because

y

−2


6

θ

8

x

tan u = -

y 7 = x 5

−4 −6

74 (5, −7)

−8

and the terminal side of u is in Quadrant IV, we know that y must be negative and x must be positive. Thus the preceding equation is true for y = - 7 and x = 5. Now r = 252 + 1 -722 = 274. See Figure 5.48. Hence,

Figure 5.48

cos u =

5 174 5 x = = r 74 174

csc u =

r 174 174 = = y -7 7

and Study tip In this section, r is a distance and hence nonnegative.


The Reference Angle We will often find it convenient to evaluate trigonometric functions by making use of the concept of a reference angle.

Study tip

Definition of a Reference Angle

The reference angle is an important concept that will be used repeatedly in the remaining trigonometry sections.

Given a nonquadrantal ∠u in standard position, its reference angle u¿ is the acute angle formed by the terminal side of ∠ u and the x-axis.

Figure 5.49, on page 458, shows ∠u and its reference angle u¿ for four cases. In every case the reference angle u¿ is formed by the terminal side of ∠u and the x-axis (never the y-axis). The process of determining the measure of ∠u¿ varies according to which quadrant contains the terminal side of ∠u.

458

CHAPTER 5


y

y

θ' = θ

y

θ

θ'

y

θ

x

θ

x

θ'

x θ'

If 0° < θ < 90°, then θ' = θ.

If 90° < θ < 180°, then θ' = 180° − θ.

If 180° < θ < 270°, then θ' = θ − 180°.

x

If 270° < θ < 360°, then θ' = 360° − θ.

Figure 5.49

EXAMPLE 3

Find the Measure of a Reference Angle

Find the measure of the reference angle u¿ for each angle. 13p 7p a. u = 120° b. u = 345° c. u = d. u = 4 6 Solution For any angle in standard position, the measure of its reference angle is the measure of the acute angle formed by its terminal side and the x-axis. a.

b.

y

y

θ = 345° θ = 120°

θ ' = 60°

θ ' = 15°

x

x

u¿ = 180° - 120° = 60°

u¿ = 360° - 345° = 15°

Integrating Technology c. A TI-83ⲐTI-83 PlusⲐTI-84 Plus graphing calculator program is available to compute the measure of the reference angle for a given angle. This program, REFANG, can be found on our website at http://www.cengage. com/math/aufmann/algtrig7e.

d.

y

y

θ = 7π 4

θ' = π 6 θ' = π 4

u¿ = 2p -

x

7p p = 4 4

x θ = 13π 6

u¿ =

p 13 p - 2p = 6 6


It can be shown that the absolute value of a trigonometric function of u is equal to the trigonometric function of u¿ . This relationship allows us to establish the following procedure.

5.3


459

Reference Angle Evaluation Procedure Let u be a nonquadrantal angle in standard position with reference angle u¿. To evaluate a trigonometric function of u, use the following procedure. 1. Determine the reference angle u¿. 2. Determine the sign of the trigonometric function of u. 3. The value of the trigonometric function of u equals the value of the trigonometric function of u¿, prefixed with the correct sign. 1 2

EXAMPLE 4

Use the Reference Angle Evaluation Procedure

Evaluate each function. a.

sin 210°

b. cos 405°

c. tan

Solution a. The reference angle for u = 210° is u¿ = 30°. See Figure 5.50. The terminal side of u is in Quadrant III; thus sin 210° is negative. The value of sin 210° is the value of sin 30° with a negative sign prefix.

y

θ = 210°

sin 210° = - sin 30° = -

x

θ ′ = 30°

c

u

c

c

1 2

Correct u¿ sign prefix

1 2 b. The reference angle for u = 405° is u¿ = 45°. See Figure 5.51. The terminal side of u is in Quadrant I; thus cos 405° is positive. The value of cos 405° is the value of cos 45°. sin 210ⴰ = -

Figure 5.50 y

θ = 405°

5p 3

θ ′ = 45°

cos 405° = + cos 45° =

c

x

cos 405 ⴰ =

Figure 5.51

22 2

c

c

u Correct sign prefix

u¿

5p p is u = . See Figure 5.52. The terminal side of u 3 3 5p 5p p is in Quadrant IV; thus tan is negative. The value of tan is the value of tan 3 3 3 with a negative sign prefix. p 5p = -tan = - 23 tan 3 c 3

c. The reference angle for u =

y

θ=5 3 x

θ′=

3

22 2

c

e

c

u Correct u¿ sign prefix

Figure 5.52

tan

5p = - 23 3


460

CHAPTER 5


EXERCISE SET 5.3 In Exercises 1 to 8, find the value of each of the six trigonometric functions for the angle, in standard position, whose terminal side passes through the given point. 1. P12, 32

2. P13, 72

3. P1- 2, 32

4. P1 -3, 52

5. P1-8, -52

6. P1- 6, -92

7. P1 -5, 02

8. P10, 22

In Exercises 9 to 20, evaluate the trigonometric function of the quadrantal angle, or state that the function is undefined. 9. sin 180°

10. cos 270°

11. tan 180°

12. sec 90°

13. csc 90°

14. cot 90°

p 15. cos 2

3p 16. sin 2

p 17. tan 2

18. cot p

p 19. sin 2

20. cos p

In Exercises 21 to 26, let U be an angle in standard position. State the quadrant in which the terminal side of U lies. 21. sin u 7 0,

cos u 7 0

22. tan u 6 0,

sin u 6 0

23. cos u 7 0,

tan u 6 0

24. sin u 6 0,

cos u 7 0

25. sin u 6 0,

cos u 6 0

26. tan u 6 0,

cos u 6 0

In Exercises 27 to 36, find the exact value of each expression. 27. sin u = -

1 , 180° 6 u 6 270°; find tan u. 2

28. cot u = - 1, 90° 6 u 6 180°; find cos u.

p 6 u 6 p ; find cot u. 2 213 3p 30. sec u = , 6 u 6 2p; find sin u. 3 2 1 31. sin u = - and cos u 7 0; find tan u. 2 29. csc u = 1 2,

32. tan u = 1 and sin u 6 0; find cos u.

1 and tan u = 13; find csc u. 2 12 ; find sec u. 34. tan u = 1 and sin u = 2 33. cos u =

35. cos u = -

13 1 and sin u = ; find cot u. 2 2

1 2 13 and sin u = - ; find cot u. 3 2

36. sec u =

In Exercises 37 to 48, find the measure of the reference angle U œ for the given angle U. 37. u = 160°

38. u = 255°

40. u = 48°

41. u =

11 p 5

42. u = - 6

44. u =

18 p 7

45. u = 1406°

43. u =

8 3

46. u = 840°

47. u = - 475°

39. u = 351°

48. u = - 650°

In Exercises 49 to 60, use the Reference Angle Evaluation Procedure to find the exact value of each trigonometric function. 49. sin 225°

50. cos 300°

51. tan 405°

52. sec 150°

4 53. csc a p b 3

7 54. cot a p b 6

55. cos

17p 4

58. csc 1 - 510°2

56. tan a -

p b 3

59. cot 540°

57. sec 765° 60. cos 570°

In Exercises 61 to 72, use a calculator to approximate the given trigonometric function to six significant digits. 61. sin 127°

62. sin 1 - 257°2

63. cos 1 - 116°2

64. cot 398°

65. sec 578°

66. sec 740°

67. sin a -

p b 5

70. tan 1 - 4.122

68. cos

3p 7

71. sec 1 - 4.452

69. csc

9p 5

72. csc 0.34

In Exercises 73 to 80, find (without using a calculator) the exact value of each expression. 73. sin 210° - cos 330° tan 330° 74. tan 225° + sin 240° cos 60°

5.4

13 3 13 91. sin u = 2

75. sin2 30° + cos2 30° 76. cos p sin 77. sin a

11p 7p - tan 4 6

5p 5p b + cos2 a b 4 4

80. tan2 a

7p 7p b - sec2 a b 4 4

y 2 y2 x2 + y2 x 2 x2 . cos2 U sin2 U a b a b 2 2 r r r r r2

In Exercises 81 to 86, find two values of U, 0° ◊ U3 2 2 a. r = = and a1 = 4. Thus an = 4a b . 4 3 3 b.

r =

- 10 = - 2 and a1 = 5. Thus an = 5(- 2)n - 1. 5


EXAMPLE 2

Determine Whether a Sequence Is a Geometric Sequence

Determine whether the given sequence is a geometric sequence. n-1

a.

1 4, -2, 1, Á , 4 a- b 2

,Á

b.

1, 4, 9, Á , n2, Á

Solution To determine whether the sequence is a geometric sequence, calculate the ratio of successive terms.

a.

ai + 1 = ai

1 i 4a- b 2 1 i-1 4a- b 2

1 i - (i - 1) 1 1 1 = a- b = a- b = 2 2 2

Because the ratio of successive terms is a constant, the sequence is a geometric sequence. b.

(i + 1)2 ai + 1 1 2 i +1 2 i 12 b + b b = = a = a = a1 + ai i i i i i2 Because the ratio of successive terms is not a constant, the sequence is not a geometric sequence. Try Exercise 36, page 869

Finite Geometric Series The sum of the first n terms of a geometric sequence is a finite geometric series. Adding the terms of a geometric sequence, we can define the nth partial sum of a geometric sequence in a manner similar to that of an arithmetic sequence. Consider sequence of partial sums for the geometric sequence 1, 2, 4, 8, Á , 2n - 1, Á .

11.3

S1 S2 S3 S4 ## # Sn

= = = =

1 1 + 2 = 1 + 2 + 1 + 2 + ## # = 1 + 2 +

GEOMETRIC SEQUENCES AND SERIES

863

3 4 = 7 4 + 8 = 15 4 + 8 + Á + 2n - 1

The first four terms of the sequence of partial sums are 1, 3, 7, and 15. To find a general formula for Sn, the nth term of the sequence of partial sums of a geometric sequence, let Sn = a1 + a1r + a1r 2 + Á + a1r n - 1 Multiply each side of this equation by r. Sn = a1 + a1r + a1r 2 + Á + a1r n - 2 + a1r n - 1 rSn = a1r + a1r 2 + Á + a1r n - 2 + a1r n - 1 + a1r n Subtract the two equations. Sn - rSn = a1 - a1r n Sn(1 - r) = a1(1 - r n) a1(1 - r n) Sn = 1 - r

• Factor out the common factors. •r 1

This proves the following theorem.

Formula for the nth Partial Sum of a Geometric Sequence The nth partial sum of a geometric sequence with first term a1 and common ratio r is Sn =

a1(1 - r n) , r Z 1 1 - r

Question • If r = 1, what is the nth partial sum of a geometric sequence?

EXAMPLE 3 a.

Find a Partial Sum of a Geometric Sequence

Find the sum of the first four terms of the geometric sequence 5, 15, 45, Á , 5(3)n - 1, Á . n-1

b.

17 3 Evaluate the finite geometric series a 3a b 4

.

n=1

Solution a. We have a1 = 5, r = 3, and n = 4. Thus S4 =

5(1 - 34) 5( - 80) = = 200 1 - 3 -2 (continued)

Answer • When r = 1, the sequence is the constant sequence a1. The nth partial sum of a constant

sequence is na1.

864

CHAPTER 11

SEQUENCES, SERIES, AND PROBABILITY

b.

9 When n = 1, a1 = 3. The first term is 3. The second term is . 4 3 Therefore, the common ratio is r = . Thus 4 S17 =

331 - (3>4)174 1 - (3>4)

L 11.909797


Infinite Geometric Series Following are two examples of geometric sequences for which ƒ r ƒ 6 1. 3 3 3 3 3 3, , , , , ,Á 4 16 64 256 1024

•r =

1 1 1 1 1 2, -1, , - , , - , , Á 2 4 8 16 32

•r = -

1 4 1 2

Note that when the absolute value of the common ratio of a geometric sequence is less than 1, the terms of the geometric sequence approach zero as n increases. We write, for ƒ r ƒ 6 1, n ƒ r ƒ : 0 as n : q. Consider again the geometric sequence 3 3 3 3 3 , ,Á 3, , , , 4 16 64 256 1024 The nth partial sums for n = 3, 6, 9, and 12 are given in Table 11.1, along with the values of r n. As n increases, Sn is closer to 4 and r n is closer to zero. By finding more values of Sn for larger values of n, we would find that Sn : 4 as n : q. As n becomes larger Sn is the nth partial sum of ever more terms of the sequence. The sum of all the terms of a sequence is called an infinite series. If the sequence is a geometric sequence, we have an infinite geometric series. Table 11.1

n

Sn

rn

3

3.93750000

0.01562500

6

3.99902344

0.00024414

9

3.99998474

0.00000381

12

3.99999976

0.00000006

Sum of an Infinite Geometric Series If an is a geometric sequence with ƒ r ƒ 6 1 and first term a1, then the sum of the infinite geometric series is S =

a1 1 - r

11.3

Caution The sum of an infinite geometric series is not defined when ƒ r ƒ Ú 1. For instance, the infinite geometric series 2 + 4 + 8 + Á + 2n + Á with r = 2 increases without bound. However, applying the a1 formula S = with r = 2 and 1 - r a1 = 2 gives S = - 2, which is not correct.


865

A formal proof of this formula requires topics that typically are studied in calculus. We can, however, give an intuitive argument. Start with the formula for the nth partial sum of a geometric sequence. a1(1 - r n) 1 - r

Sn =

When ƒ r ƒ 6 1, ƒ r ƒ n L 0 when n is large. Thus a1(1 - r n) a1(1 - 0) a1 L = 1 - r 1 - r 1 - r

Sn =

q

An infinite series is represented by a an. n=1

EXAMPLE 4

Find the Sum of an Infinite Geometric Series q

2 n-1 Evaluate the infinite geometric series a a- b . 3 n=1 Solution

n-1

2 The general term is an = a- b . To find the first term, let n = 1. Then 3 2 1-1 2 0 2 = a- b = 1. The common ratio is r = - . a1 = a- b 3 3 3 a1 1 - r

S =

1

S =

2 1 - a- b 3

1

=

1 +

2 3

=

1 3 = 5 5 3


Consider the repeating decimal 0.6 =

6 6 6 6 + + + + Á 10 100 1000 10,000

The right-hand side is a geometric series with a1 =

6 1 . and common ratio r = 10 10

Thus S =

6>10 1 - (1>10)

=

6>10 9>10

=

2 3

2 . We can write any repeating decimal as a ratio of two inte3 gers by using the formula for the sum of an infinite geometric series. The repeating decimal 0.6 =

866

CHAPTER 11


EXAMPLE 5

Write a Repeating Decimal as the Ratio of Two Integers in Simplest Form

Write 0.345 as the ratio of two integers in simplest form. Solution 0.345 =

3 45 45 45 + B + + + ÁR 10 1000 100,000 10,000,000

The terms in the brackets form an infinite geometric series. Evaluate that series with 45 1 3 and r = a1 = , and then add the term . 1000 100 10 45>1000 45 45 1 45 + + + Á = = 1000 100,000 10,000,000 22 1 - (1>100) Thus 0.345 =

1 19 3 . + = 10 22 55


Applications of Geometric Sequences and Series Ordinary Annuities

In an earlier chapter, we discussed compound interest by using exponential functions. As an extension of this idea, suppose that, for each of the next 5 years, P dollars are deposited on December 31 into an account earning i % annual interest compounded annually. Using the compound interest formula, we can find the total value of all the deposits. Table 11.2 shows the growth of the investment. Table 11.2

Deposit Number

Value of Each Deposit P(1 + i) P(1 + i)3 P(1 + i)2 P(1 + i) P 4

1 2 3 4 5

Value of first deposit after 4 years Value of second deposit after 3 years Value of third deposit after 2 years Value of fourth deposit after 1 year Value of fifth deposit

The total value of the investment after the last deposit, called the future value of the investment, is the sum of the values of all the deposits. A = P + P(1 + i) + P(1 + i)2 + P(1 + i)3 + P(1 + i)4 This is a geometric series with first term P and common ratio 1 + i. Thus, using the formula for the nth partial sum of a geometric sequence, S = we have A =

a1(1 - rn) 1 - r

P31 - (1 + i)54 1 - (1 + i)

=

P3(1 + i)5 - 14 i

Deposits of equal amounts at equal intervals of time are called annuities. When the amounts are deposited at the end of a compounding period (as in our example), we have an ordinary annuity.

11.3


867

Future Value of an Ordinary Annuity i and m = nt, where i is the annual interest rate, n is the number of n compounding periods per year, and t is the number of years. Then the future value A of an ordinary annuity after m compounding periods is given by

Let r =

A =

P3(1 + r)m - 14 r

where P is the amount of each deposit.

EXAMPLE 6

Find the Future Value of an Ordinary Annuity

An employee savings plan allows any employee to deposit $25 at the end of each month into a savings account earning 6% annual interest compounded monthly. Find the future value of this savings plan if an employee makes the deposits for 10 years. Solution We are given P = 25, i = 0.06, n = 12, and t = 10. Thus 0.06 i = = 0.005 and m = nt = 12(10) = 120 n 12 253(1 + 0.005)120 - 14 L 4096.9837 A = 0.005 r =

The future value after 10 years is $4096.98. Try Exercise 70, page 870

Note Remember that the sum of an infinite geometric series for which a1 . For the infinite ƒ r ƒ 6 1 is 1 - r D(1 + g) series at the right, a 1 = 1 + i 1 + g . Therefore, and r = 1 + i a1 = 1 - r

D(1 + g) 1 + i 1 + g b 1- a 1 + i

=

D(1 + g) i - g

There are many applications of infinite geometric series to the area of finance. Two such applications are stock valuation and the multiplier effect. Stock Valuation

The Gordon model of stock valuation, named after Myron Gordon, is used to determine the value of a stock whose dividend is expected to increase by the same percentage each year. The value of the stock is given by q 1 + g n D(1 + g) b = ,g 6 i Stock value = a D a 1 + i i - g n=1

where D is the dividend of the stock when it is purchased, g is the expected percent growth rate of the dividend, and i is the growth rate the investor requires. An example of stock valuation is given in Example 7.

EXAMPLE 7

Find the Value of a Stock

Suppose a stock is paying a dividend of $1.50 and it is estimated that the dividend will increase 10% per year. The investor requires a 15% return on an investment. Using the Gordon model of stock valuation, determine the price per share the investor should pay for the stock. (continued)

868

CHAPTER 11


Solution Substitute the given values into the Gordon model. D(1 + g) i - g 1.50(1 + 0.10) = 0.15 - 0.10 = 33

Stock value =

• D 1.50, g 0.10, i 0.15

The investor should pay $33 per share for the stock. Try Exercise 76, page 870

If the dividend of a stock does not grow but remains constant, then the formula for the stock value is q

D D Stock value (no dividend growth) = a n = (1 + i) i n=1 where D is the dividend and i is the investor’s required rate of return. For instance, to find the value of a stock whose dividend is $2.33 and from which an investor requires a 20% rate of return, use the formula above with D = 2.33 and i = 0.20. Stock value (no dividend growth) = =

D i 2.33 = 11.65 0.20

The investor should pay $11.65 for each share of stock. The Multiplier Effect

Note The amount that consumers will spend is referred to by economists as the marginal propensity to consume. For the example at the right, the marginal propensity to consume is 75%.

A phenomenon called the multiplier effect can occur in certain economic situations. We will examine this effect when applied to a reduction in income taxes. Suppose the federal government enacts a tax reduction of $5 billion. Suppose also that an economist estimates that each person receiving a share of this reduction will spend 75% and save 25%. This means that 75% of $5 billion, or $3.75 billion (0.75 # 5 = 3.75), will be spent. The amount that is spent becomes income for other people, who in turn spend 75% of that amount, or $2.8125 billion (0.75 # 3.75 = 2.8125). This money becomes income for other people, and so on. This process is illustrated in the table below. Amount Available to Spend

Amount Spent

New Amount Available to Spend

$5 billion

0.75(5) = 3.75

$3.75 billion

$3.75 billion

0.75(3.75) = 0.7530.75(5)4 = (0.75)2 (5) = 2.8125

$2.8125 billion

$2.8125 billion

0.75(2.8125) = 0.753(0.75)2 54 = (0.75)3(5) = 2.109375

$2.109375 billion

$2.109375 billion

0.75(2.109375) = 0.753(0.75)3 54 = (0.75)4(5) = 1.58203125

$1.58203125 billion

11.3


869

Note that the values in the middle column form a geometric sequence. The net effect of all the spending is found by summing an infinite geometric series. q

5 + 0.75(5) + (0.75)2(5) + Á + (0.75)n - 1(5) + Á = a 5(0.75)n - 1 n=1

=

5 = 20 1 - 0.75

This means that the original tax cut of $5 billion results in actual spending of $20 billion. Some economists believe this is good for economic growth; others see it as contributing to inflation and therefore bad for the economy.

EXERCISE SET 11.3 In Exercises 1 to 20, find the nth term of the geometric sequence. 1. 2, 8, 32, Á

2. 1, 5, 25, Á

3. -4, 12, -36, Á

4. -3, 6, -12, Á

8 3

5. 6, 4, , Á

7. - 6, 5, -

25 ,Á 6

9 2

6. 8, 6, , Á

4 3

8 9

8. -2, , - , Á

4 2 3 9

10. 8, - , , Á

11. 1, -x, x 2, Á

12. 2, 2a, 2a 2, Á 14. -x 2, x 4, -x 6, Á

3 3 3 15. , , ,Á 100 10,000 1,000,000 16.

and whose eighth term is

1 32

23. The second term of a geometric sequence whose third term is

4 32 and whose sixth term is 3 81 24. The fifth term of a geometric sequence whose fourth term is

9. 9, -3, 1, Á

13. c 2, c 5, c 8, Á

22. The fourth term of a geometric sequence whose third term is 1

7 7 7 , , ,Á 10 10,000 10,000,000

and whose seventh term is

64 243

In Exercises 25 to 36, determine whether the sequence is arithmetic, geometric, or neither. 25. an =

1 n

26. an =

2

27. an = 2n - 7 29. an = a-

31. an =

n

6 b 5

n 2n

19. 0.45, 0.0045, 0.000045, Á

33. an = - 3n

35. an =

20. 0.234, 0.000234, 0.000000234, Á

In Exercises 21 to 24, find the requested term of the geometric sequence. 21. The third term of a geometric sequence whose first term is 2

and whose fifth term is 162

3n 2

n 3n

28. an = 5 - 3n 30. an = 0.23n - 1

32. an = en

17. 0.5, 0.05, 0.005, Á 18. 0.4, 0.004, 0.00004, Á

8 9

34. an =

(- 1)n n

36. an =

n! nn

In Exercises 37 to 46, find the sum of the finite geometric series. 5

37. a 3n n=1

7

38. a 2n n=1

870

CHAPTER 11

6


n

14

39. a a b n=1 3

2

8

n

2

4

7

41. a a- b 5 n=0 10

n

name. Each recipient then added his or her name to the bottom of the list and sent the letter to five friends. Assuming no one broke the chain, how much money would each recipient of the letter receive?

40. a a b n=1 3

n

42. a a- b 3 n=0

1

72. Prosperity Club The population of the United States in 1935

was approximately 127 million people. Assuming no one broke the chain in the Prosperity Club chain letter (see Exercise 71) and no one received more than one letter, how many levels would it take before the entire population received a letter?

7

n-1

43. a ( -2)

n

44. a 2(5)

n=1

n=0

9

10

45. a 5(3)n

46. a 2(- 4)n

n=0

73. Medicine The concentration (in milligrams per liter) of an

n=0

antibiotic in the blood is given by the geometric series

In Exercises 47 to 56, find the sum of the infinite geometric series. q

q

n

1 47. a a b 3 n=1 q

q

n

49. a a- b 3 n=1 q

2

n

50. a a- b 5 n=1 q

n

51. a a b n = 1 100

9

n

3 48. a a b 4 n=1

3

n

52. a a b n = 1 10

q

7

tain concentration of a medication in the blood for an extended period. In such a case, the amount of medication in the blood can be approximated by the infinite geometric series

54. a (0.5)n

n=1

n=1

q

A + Aekt + Ae 2kt + Á + Ae(n - 1)kt + Á

q

55. a (- 0.4)n

56. a (-0.8)n

n=0

where A is the number of milligrams in one dose of the antibiotic, n is the number of doses, t is the time between doses, and k is a constant that depends on how quickly the body metabolizes the antibiotic. Suppose one dose of an antibiotic increases the blood level of the antibiotic by 0.5 milligram per liter. If the antibiotic is given every 4 hours and k = - 0.867, find the concentration, to the nearest hundredth, of the antibiotic just before the fifth dose. 74. Medicine To treat some diseases, a patient must have a cer-

q

53. a (0.1)n

A + Aekt + Ae 2kt + Á + Ae(n - 1)kt

n=0

In Exercises 57 to 68, write each rational number as the ratio of two integers in simplest form. 57. 0.3

58. 0.5

59. 0.45

60. 0.63

61. 0.123

62. 0.395

63. 0.422

64. 0.355

65. 0.254

66. 0.372

67. 1.2084

68. 2.2590

69. Future Value of an Annuity Find the future value of an ordi-

nary annuity that calls for depositing $100 at the end of every 6 months for 8 years into an account that earns 9% interest compounded semiannually.

where A is the number of milligrams in one dose of the medication, t is the time between doses, and k is a constant that depends on how quickly the body metabolizes the medication. If the medication is given every 12 hours, k = - 0.25, and the required concentration of medication is 2 milligrams per liter, find the amount of the dosage. Round your result to the nearest hundredth of a milligram. (Suggestion: Solve the equation A + Ae kt + Ae 2kt + Á + Ae(n - 1)kt + Á = 2 for A.) 75. Gordon Model of Stock Valuation Suppose Myna Alton

purchases a stock for $67 per share that pays a divided of $1.32. If Myna requires a 20% return on her investment, use the Gordon model of stock valuation to determine the dividend growth rate Myna expects. 76. Gordon Model of Stock Valuation Use the Gordon model

70. Future Value of an Annuity To save for the replacement of a

computer, a business deposits $250 at the end of each month into an account that earns 8% annual interest compounded monthly. Find the future value of the ordinary annuity in 4 years.

of stock valuation to determine the price per share the manager of a mutual fund should pay for a stock whose dividend is $1.87 and whose dividend growth rate is 15% if the manager requires a 20% rate of return on the investment.

71. Prosperity Club In 1935, the “Prosperity Club” chain letter

77. Gordon Model of Stock Valuation Suppose that a stock

was started. A letter listing the names and addresses of six people was sent to five other people, who were asked to send $0.10 to the name at the top of the list and then remove that person’s

is paying a constant dividend of $2.94 and that an investor wants to receive a 15% return on his investment. What price per share should the investor pay for the stock?

11.4

871

the counterfeit money is used, 40% of it is detected and removed from circulation. How much counterfeit money, to the nearest thousand dollars, is used in transactions before all of it is removed from circulation? This problem represents another application of the multiplier effect. (See pages 868– 869.)

78. Gordon Model of Stock Valuation Suppose that a stock is

paying a constant dividend of $3.24 and that an investor pays $16.00 for one share of the stock. What rate of return does the investor expect? 79.

MATHEMATICAL INDUCTION

Stock Valuation Explain why g must be less than i in

the Gordon model of stock valuation. (See page 867.)

83. Genealogy Some people can trace their ancestry back 10 gen-

erations, which means two parents, four grandparents, eight great-grandparents, and so on. How many grandparents does such a family tree include?

80. Multiplier Effect Sometimes a city will argue that having a

professional sports franchise in the city will contribute to economic growth. The rationale for this statement is based on the multiplier effect. Suppose a city estimates that a professional sports franchise will create $50 million of additional income and that a person receiving a portion of this money will spend 90% and save 10%. Assuming the multiplier effect model is accurate, what is the net effect of the $50 million? (See pages 868– 869.)

Parents

Grandparents

81. Multiplier Effect Suppose a city estimates that a new conven-

tion facility will bring $25 million of additional income. If each person receiving a portion of this money spends 75% and saves 25%, what is the net effect of the $25 million? (See pages 868– 869.)

GreatGrandparents

82. Counterfeit Money Circulation Suppose that $500,000 of

counterfeit money is currently in circulation and that each time

MID-CHAPTER 11 QUIZ n . 2n

5. Find the sum of the first 25 terms of the arithmetic sequence

2. Find the third and fifth terms of the sequence given by a1 = - 3,

6. Find the nth term of the geometric sequence whose first three

1. Find the fourth and eighth terms of the sequence given by an =

whose nth term is an = 5 - n.

an = - 2an - 1. 5

3. Evaluate: a

terms are - 2, (- 1)k - 1

k=1

k2

8 4 ,- . 3 9

7. Find the sum of the first eight terms of the geometric sequence

whose nth term is an = ( -3)n.

4. Find the 20th term of the arithmetic sequence whose first 3

terms are 1, 4, 7.

SECTION 11.4 Principle of Mathematical Induction Extended Principle of Mathematical Induction

8. Write 0.43 as the ratio of two integers in simplest form.

Mathematical Induction PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A75. n

1

n

PS1. Show that a when n = 4. [11.1] = n + 1 i = 1 i(i + 1) PS2. Write k(k + 1)(2k + 1) + 6(k + 1)2 as a product of linear factors. [P.4] PS3. Simplify:

k 1 [P.5] + k + 1 (k + 1)(k + 2)

872

CHAPTER 11


PS4. What is the smallest natural number for which n2 7 2n + 1? [P.1] PS5. Let an = n and Sn =

n(n + 1) . Write Sn + an + 1 in simplest form. [P.3/11.1] 2

PS6. Let an = 2n and Sn = 2n + 1 - 2. Show that Sn + an + 1 = 2(2n + 1 - 1). [P.2/11.1]

Consider the sequence 1 1 1 1 , , ,Á, ,Á 1#2 2#3 3#4 n(n + 1) and its sequence of partial sums, as shown below. 1

1 2

S1 =

1#2

S2 =

1 1 2 + # = 1#2 2 3 3 1

=

1

S4 =

1 1 1 1 4 + # + # + # = 1#2 2 3 3 4 4 5 5

+

3#4

3 4

1#2

+

2#3

1

S3 =

=

Question • What does the pattern above suggest for the value of S5?

This pattern suggests the conjecture that Sn =

Table 11.3

1

n2 n 41 41 prime

2

43 prime

3

47 prime

4

53 prime

5

61 prime

n

1 1 1 1 n + # + # + Á + = 1#2 2 3 3 4 n(n + 1) n + 1

How can we be sure that the pattern does not break down when n = 50, n = 2000, or some other large number? As we will show, this conjecture is true for all values of n. As a second example, consider the conjecture that the expression n 2 - n + 41 is a prime number for all positive integers n. To test this conjecture, we will try various values of n. See Table 11.3. The results suggest that the conjecture is true. But again, how can we be sure? In fact, this conjecture is false when n = 41. In that case we have n 2 - n + 41 = (41)2 - 41 + 41 = (41)2 and (41)2 is not a prime. The last example illustrates that just verifying a conjecture for a few values of n does not constitute a proof of the conjecture. To prove theorems about statements involving positive integers, a process called mathematical induction is used. This process is based on an axiom called the induction axiom.

Answer • S5 =

5 5 = 5 + 1 6

11.4


873

Induction Axiom Suppose S is a set of positive integers with the following two properties: 1. 1 is an element of S. 2. If the positive integer k is in S, then k + 1 is in S. Then S contains all positive integers.

Part 2 of this axiom states that if some positive integer—say, 8—is in S, then 8 + 1, or 9, is in S. But because 9 is in S, Part 2 says that 9 + 1, or 10, is in S, and so on. Part 1 states that 1 is in S. Thus 2 is in S; thus 3 is in S; thus 4 is in S; and so on. Therefore, all positive integers are in S.

Principle of Mathematical Induction The induction axiom is used to prove the Principle of Mathematical Induction.

Principle of Mathematical Induction Let Pn be a statement about a positive integer n. If 1. P1 is true and 2. the truth of Pk implies the truth of Pk + 1 then Pn is true for all positive integers.

In a proof that uses the Principle of Mathematical Induction, the first part of Step 2, the truth of Pk , is referred to as the induction hypothesis. When applying this step, we assume that the statement Pk is true (the induction hypothesis), and then we try to prove that Pk + 1 is also true. As an example, we will prove that the first conjecture we made in this section is true for all positive integers. Every induction proof has the two distinct parts stated in the Principle of Mathematical Induction. First, we show that the result is true for n = 1. Second, we assume that the statement is true for some positive integer k and, using that assumption, we prove that the statement is true for n = k + 1. Prove that Sn =

1

1#2

+

1

2#3

+

1

3#4

+ Á +

1 n = n(n + 1) n + 1

for all positive integers n. Proof 1. For n = 1, S1 =

1 1 and = 1(1 + 1) 2

The statement is true for n = 1.

n 1 1 = = n + 1 1 + 1 2

874

CHAPTER 11


2. Assume the statement is true for some positive integer k. Sk =

1

1#2

+

1

2#3

+

1

3#4

+ Á +

1 k = k(k + 1) k + 1

• Induction hypothesis

Now verify that the formula is true when n = k + 1. That is, verify that Sk + 1 =

k + 1 k + 1 = (k + 1) + 1 k + 2

• This is the goal of the induction proof.

It is helpful, when proving a theorem about sums, to note that Sk + 1 = Sk + ak + 1 Begin by noting that because an = Sk + 1 =

Sk

1 1 ,a . = n(n + 1) k + 1 (k + 1)(k + 2)

+ ak+1

=

1 k + k + 1 (k + 1)(k + 2)

=

k(k + 2) 1 + (k + 1)(k + 2) (k + 1)(k + 2)

• By the induction hypothesis and substituting for ak +1

(k + 1)2 k(k + 2) + 1 k2 + 2k + 1 = = (k + 1)(k + 2) (k + 1)(k + 2) (k + 1)(k + 2) k + 1 = k + 2

= Sk + 1

Because we have verified the two parts of the Principle of Mathematical Induction, we can conclude that the statement is true for all positive integers. N

EXAMPLE 1

Prove a Statement by Mathematical Induction

n(n + 1)(2n + 1) Prove that 12 + 22 + 32 + Á + n2 = for all positive integers n. 6 Solution Verify the two parts of the Principle of Mathematical Induction. 1.

Let n = 1. S1 = 12 = 1 and

2.

n(n + 1)(2n + 1) 1(1 + 1)(2 # 1 + 1) 6 = = = 1 6 6 6

Assume the statement is true for some positive integer k. k(k + 1)(2k + 1) Sk = 12 + 22 + 32 + Á + k 2 = 6 Verify that the statement is true when n = k + 1. Show that Sk + 1 =

(k + 1)(k + 2)(2k + 3) 6

Because an = n 2, ak + 1 = (k + 1)2.


11.4

Sk + 1 =


+ ak + 1

Sk

=

k(k + 1)(2k + 1) + (k + 1)2 6

=

6(k + 1)2 k(k + 1)(2k + 1) + 6(k + 1)2 k(k + 1)(2k + 1) + = 6 6 6

= Sk + 1 =

875

(k + 1)3k(2k + 1) + 6(k + 1)4 6

=

(k + 1)(2k 2 + 7k + 6) 6

(k + 1)(k + 2)(2k + 3) 6

By the Principle of Mathematical Induction, the statement is true for all positive integers. Try Exercise 8, page 877

Mathematical induction can also be used to prove statements about sequences, products, and inequalities.

EXAMPLE 2 Prove that a1 +

Prove a Product Formula by Mathematical Induction 1 1 1 1 b a1 + b a1 + b Á a1 + b = n + 1 for all positive n 1 2 3

integers n. Solution 1. Verify for n = 1. a1 + 2.

1 b = 2 1

and 1 + 1 = 2

Assume the statement is true for some positive integer k. Pk = a1 +

1 1 1 1 b a1 + b a1 + b Á a1 + b = k + 1 1 2 3 k


Verify that the statement is true when n = k + 1. That is, prove that Pk + 1 = k + 2. Pk + 1 = a1 +

1 1 1 1 1 b a1 + b a1 + b Á a1 + b a1 + b 1 2 3 k k + 1

= Pk a 1 +

1 1 b = (k + 1) a1 + b = k + 1 + 1 k + 1 k + 1

Pk + 1 = k + 2 By the Principle of Mathematical Induction, the statement is true for all positive integers. Try Exercise 12, page 877

876

CHAPTER 11


EXAMPLE 3

Prove an Inequality by Mathematical Induction

Prove that 1 + 2n … 3n for all positive integers n. Solution 1. Let n = 1. Then 1 + 2(1) = 3 … 31. The statement is true when n is 1. 2.

Assume the statement is true for some positive integer k. 1 + 2k … 3k


Now prove that the statement is true for n = k + 1. That is, prove that 1 + 2(k + 1) … 3k + 1. 3k + 1 = 3k(3) Ú (1 + 2k)(3) = 6k + 3 7 2k + 2 + 1 = 2(k + 1) + 1

• Because, by the induction hypothesis, 1 + 2k … 2k • 6k 7 2k and 3 = 2 + 1

Thus 1 + 2(k + 1) … 3k + 1. By the Principle of Mathematical Induction, 1 + 2n … 3n for all positive integers. Try Exercise 16, page 877

Extended Principle of Mathematical Induction The Principle of Mathematical Induction can be extended to cases in which the beginning index is greater than 1.

Extended Principle of Mathematical Induction Let Pn be a statement about a positive integer n. If 1. Pj is true for some positive integer j and 2. for k Ú j the truth of Pk implies the truth of Pk + 1 then Pn is true for all positive integers n Ú j.

EXAMPLE 4

Prove an Inequality by Mathematical Induction

For all integers n Ú 3, prove that n2 7 2n + 1. Solution 1. Let n = 3. Then 32 = 9 and 2(3) + 1 = 7. Thus n2 7 2n + 1 for n = 3. 2.

Assume the statement is true for some positive integer k Ú 3. k 2 7 2k + 1


Verify that the statement is true when n = k + 1. That is, show that (k + 1)2 7 2(k + 1) + 1 = 2k + 3

11.4


877

(k + 1)2 = k2 + 2k + 1 7 (2k + 1) + 2k + 1


7 2k + 1 + 1 + 1

• 2k 7 1

= 2k + 3 Thus (k + 1) 7 2k + 3. 2

By the Extended Principle of Mathematical Induction, n2 7 2n + 1 for all n Ú 3. Try Exercise 20, page 877

EXERCISE SET 11.4 In Exercises 1 to 12, use mathematical induction to prove each statement for all positive integers n. 1. 1 + 4 + 7 + Á + 3n - 2 =

n(3n - 1) 2

16. If a 7 1, show that a n + 1 7 a n for all positive integers n. 17. 1 # 2 # 3 # Á # n 7 2n, n Ú 4, n is an integer

n2(n + 1)2 4

4. 2 + 4 + 8 + Á + 2n = 2(2n - 1)

18.

5. 3 + 7 + 11 + Á + 4n - 1 = n(2n + 1) 6. 3 + 9 + 27 + Á + 3n =

10.

1

1#3

+

1

3#5

+

1

5#7

+ Á +

12. a1 -

n(n + 1)(n + 2) 3

n 1 = (2n - 1)(2n + 1) 2n + 1

n(n + 1)(2n + 1)(3n + 3n - 1) 30

1 1 1 1 1 b a 1 - b a 1 - b Á a1 b = 2 3 4 n + 1 n + 1

13. a b 7 n + 1, n Ú 4, n is an integer

3 2

+

1 13

1 + Á + Ú 1n, n is a positive integer 1n

increasing function, log(n + 1) … log(n + n).)

In Exercises 21 to 30, use mathematical induction to prove each statement. 21. 2 is a factor of n2 + n for all positive integers n. 22. 3 is a factor of n3 - n for all positive integers n. 23. 4 is a factor of 5n - 1 for all positive integers n.

Hint: 5k + 1 - 1 = 5 # 5k - 5 + 4

2

In Exercises 13 to 20, use mathematical induction to prove each inequality. n

1 12

20. log n 6 n for all positive integers n (Hint: Because log x is an

n 1 1 1 1 = + + + Á + 2#4 4#6 6#8 2n(2n + 2) 4(n + 1)

11. 1 + 16 + 81 + Á + n4 =

+

integers n.

7. 1 + 27 + 125 + Á + (2n - 1)3 = n2(2n2 - 1)

9.

1 11

19. For a 7 0, show that (1 + a)n Ú 1 + na for all positive

3(3n - 1) 2

8. 2 + 6 + 12 + Á + n(n + 1) =

4 3

15. If 0 6 a 6 1, show that a n + 1 6 a n for all positive integers n.

2. 2 + 4 + 6 + Á + 2n = n(n + 1) 3. 1 + 8 + 27 + Á + n3 =

n

14. a b 7 n, n Ú 7, n is an integer

24. 5 is a factor of 6n - 1 for all positive integers n. 25. (xy)n = x n y n for all positive integers n. n

26. a b =

x y

xn for all positive integers n. yn

27. For a Z b, show that (a - b) is a factor of a n - b n, where n

is a positive integer. Hint: a k + 1 - b k + 1 = (a # a k - ab k ) + (ab k - b # b k )

878

CHAPTER 11


28. For a Z - b, show that (a + b) is a factor of a2n + 1 + b 2n + 1,

where n is a positive integer. Hint: a2k + 3 + b2k + 3 = (a2k + 2 + b2k + 2)(a + b) - ab(a2k + 1 + b 2k + 1) n

29. a ar k - 1 k=1

a(1 - r n) = for r Z 1. 1 - r

n

n3(n + 1)a + 2b4

k=1

2

30. a (ak + b) =

n

35. Prove that a

2 + 4 + 8 + Á + 2n = 2n + 1 + 1 a. Show that if we assume the formula is true for some posi-

tive integer k then the formula is true for k + 1. b. Show that the formula is not true for n = 1.

32. Let an be a sequence for which there is a number r and an

c. Show that the formula is not valid for any value of n by

an + 1 6 r for n Ú N. Show that an 6 aN r k for each positive integer k.

showing that the left side is always an even number and the right side is always an odd number.

integer N for which

d. 33. For constant positive integers m and n, show that (x m)n = x mn.

SECTION 11.5 Binomial Theorem ith Term of a Binomial Expansion Pascal’s Triangle

n + 1 n b 6 3 for all integers n Ú 3. n

by mathematical induction, it is important that both parts of the Principle of Mathematical Induction be verified. For instance, consider the formula

Using a calculator, find the smallest integer N for which log N ! 7 N. Now prove that log n! 7 n for all n 7 N.

aN + k

1

36. Steps in a Proof by Mathematical Induction In every proof

In Exercises 31 to 35, use mathematical induction to prove each statement. 31.

1

34. Prove that a … 3 - for all positive integers n. n i = 0 i!

Explain how this shows that both steps of the Principle of Mathematical Induction must be verified.

Binomial Theorem PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A75.

PS1. Expand: (a + b)3 [P.3] PS2. Evaluate: 5! [11.1] PS3. Evaluate: 0! [11.1] PS4. Evaluate

n! when n = 6 and k = 2. [11.1] k! (n - k)!

PS5. Evaluate

n! when n = 7 and k = 3. [11.1] k! (n - k)!

PS6. Evaluate

n! when n = 10 and k = 10. [11.1] k! (n - k)!

Binomial Theorem In certain situations in mathematics, it is necessary to write (a + b)n as the sum of its terms. Because (a + b) is a binomial, this process is called expanding the binomial. For small values of n, it is relatively easy to write the expansion by using multiplication. Earlier in the text we found that (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3

11.5

BINOMIAL THEOREM

879

Building on these expansions, we can write a few more. (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 We could continue to build on previous expansions and eventually have quite a comprehensive list of binomial expansions. Instead, however, we will look for a theorem that will enable us to expand (a + b)n directly, without multiplying. Look at the variable parts of each expansion above. Note that for each n = 1, 2, 3, 4, 5, the following statements are true. The first term is an. The exponent on a decreases by 1 for each successive term. The exponent on b increases by 1 for each successive term. The last term is b n. The degree of each term is n. Question • What is the degree of the fourth term in the expansion of (a + b)11 ?

To find a pattern for the coefficients in each expansion of (a + b)n, first note that there are n + 1 terms and that the coefficient of the first and last term is 1. To find the remaining coefficients, consider the expansion of (a + b)5. (a + b)5 = a5 + 5a 4b + 10a3b 2 + 10a2b3 + 5ab 4 + b 5 5 5#4 5#4#3 5#4#3#2 = 5 = 10 = 10 = 5 # # # 1 2 1 3 2 1 4#3#2#1 Observe from these patterns that there is a strong relationship to factorials. In fact, we can express each coefficient by using factorial notation. 5! = 5 1! 4!

5! = 10 2! 3!

5! = 10 3! 2!

5! = 5 4! 1!

In each denominator, the first factorial is the exponent on b, and the second factorial is the exponent on a. In general, we will conjecture that the coefficient of the term an-kb k in the expansion n! . Each coefficient of a term of a binomial expansion is called a of (a + b)n is k! (n - k)! n binomial coefficient and is denoted by a b. k

Formula for a Binomial Coefficient In the expansion of (a + b)n, n a positive integer, the coefficient of the term whose variable part is an - k b k is n n! a b = k k! (n - k)! The first term of the expansion of (a + b)n can be thought of as anb0. In this case, we can calculate the coefficient of this term as n n! n! a b = = # = 1 0 0! (n - 0)! 1 n! Answer • 11

880

CHAPTER 11


EXAMPLE 1

Evaluate a Binomial Coefficient

Evaluate each binomial coefficient.

a.

9 a b 6

b.

a

10 b 10

Solution 9! 9 # 8 # 7 # 6! 9 9! a. a b = = = = 84 6! (9 - 6)! 6! 3! 6! # 3 # 2 # 1 6 b.

a

10 10! 10! b = = = 1 10 10! (10 - 10)! 10! 0!

• Remember that 0! = 1.


We are now ready to state the Binomial Theorem for positive integers.

Binomial Theorem for Positive Integers If n is a positive integer, then n

n (a + b)n = a a b a n - i b i i=0 i n n n n = a b a n + a b a n - 1b + a b a n - 2 b 2 + Á + a b bn 0 1 2 n EXAMPLES

5 5 5 5 5 5 (x + 4)5 = a b x 5 + a b x4(4) + a b x 3(4)2 + a b x 2(4)3 + a b x(4)4 + a b(4)5 0 1 2 3 4 5 = x 5 + 20x 4 + 160x 3 + 640x 2 + 1280x + 1024 4 4 4 4 4 (x - 2)4 = a b x 4 + a b x 3( -2) + a b x 2( - 2)2 + a b x( - 2)3 + a b ( - 2)4 0 1 2 3 4 = x 4 - 8x 3 + 24x 2 - 32x + 16

EXAMPLE 2

Expand a Sum of Two Terms

Expand: (2x 2 + 3)4 Solution Use the Binomial Theorem with a = 2x 2, b = 3, and n = 4. 4 4 4 4 4 (2x 2 + 3)4 = a b (2x 2)4 + a b (2x 2)3(3) + a b(2x 2)2(3)2 + a b (2x 2)(3)3 + a b(3)4 0 1 2 3 4 = 16x8 + 96x6 + 216x4 + 216x2 + 81 Try Exercise 22, page 882

11.5

EXAMPLE 3

BINOMIAL THEOREM

881

Expand a Difference of Two Terms 5

Expand: ( 1x - 2y)

Solution Use the Binomial Theorem with a = 1x, b = - 2y, and n = 5.

( 1x

5 5 5 5 5 4 3 - 2y) = a b ( 1x ) + a b ( 1x ) ( - 2y) + a b ( 1x ) ( - 2y)2 0 1 2

Note If exactly one of the terms a or b in (a + b)n is negative, then the terms of the expansion alternate in sign.

5 5 5 2 + a b (1x ) ( - 2y) 3 + a b (1x )( -2y)4 + a b ( - 2y)5 3 4 5 = x 5/2 - 10x 2 y + 40x 3/2y 2 - 80xy 3 + 80x 1/2y 4 - 32y 5 Try Exercise 28, page 882

ith Term of a Binomial Expansion The Binomial Theorem also can be used to find a specific term in the expansion of (a + b)n.

Formula for the ith Term of a Binomial Expansion Note

The ith term of the expansion of (a + b)n is given by a

The exponent on b is 1 less than the term number.

EXAMPLE 4

n b a n - i + 1b i - 1 i - 1

Find the i th Term of a Binomial Expansion

Find the fourth term in the expansion of (2x 3 - 3y 2)5. Solution Use the preceding theorem with a = 2x 3, b = - 3y 2, i = 4, and n = 5. 5 a b (2x 3)2( - 3y 2)3 = - 1080x 6y 6 3 The fourth term is -1080x6y6. Try Exercise 38, page 882

Pascal’s Triangle A pattern for the coefficients of the terms of an expanded binomial can be found by writing the coefficients in a triangular array known as Pascal’s Triangle. See Figure 11.1 on page 882.

882

CHAPTER 11


Each row begins and ends with the number 1. Any other number in a row is the sum of the two closest numbers above it. For example, 4 + 6 = 10. Thus each succeeding row can be found from the preceding row.

Note If Pascal’s Triangle is written as shown below, each sum of a diagonal is a term of the Fibonacci sequence. 1 1

1

1 1 2 1 1 1 3 3 1 2 1 4 6 4 1 3 1 5 10 10 5 1 5 8

a

b 1:

a

b 2:

a

b 3:

a

b 4:

a

b 5:

a

b 6:

1 1

2

1

1

4

3

1

6

5 6

1

3

1 1

1

10 15

4

1

10 20

5 15

1 6

1

Figure 11.1

Pascal’s Triangle can be used to expand a binomial for small values of n. For instance, the seventh row of Pascal’s Triangle is 1

7

21

35

35

21

7

1

Therefore, (a + b)7 = a 7 + 7a6b + 21a5b 2 + 35a4b 3 + 35a3b4 + 21a2b 5 + 7ab6 + b 7

EXERCISE SET 11.5 In Exercises 1 to 8, evaluate the binomial coefficient. 1. a b

2. a b

3. a b

4. a

5. a

6. a b

7. a

8. a

7 4

12 b 9

8 6

10 b 5

9 2

11 b 0

6 5

14 b 14

In Exercises 33 to 40, find the indicated term without expanding. 33. (3x - y)10; eighth term

34. (x + 2y)12; fourth term

35. (x + 4y)12; third term

36. (2x - 1)14; thirteenth term

9

37. (1x - 1y ) ; fifth term

38. (x - 1>2 + x 1>2)10; sixth term

In Exercises 9 to 32, expand the binomial. 39. a

9. (x + y)5

10. (x + y)7

11. (a - b)4

12. (a - b)6

13. (x + 5)4

14. (x + 2)6

15. (a - 3)5

16. (a - 2)7

17. (2x - 1)7

18. (2x + y)6

19. (x + 3y)6

20. (x - 4y)5

21. (2x - 5y)4

22. (3x + 2y)4

23. (x2 - 4)7

24. (x - y3)6

25. (2x2 + y3)5

26. (2x - y3)6

27. (x + 1y )5

28. (2x - 1y )7

29. a

a b 3 30. a + b b a

31. (s

-2

+ s )

2 6

32. (2r

40. a

3 x 13 - b ; seventh term x 3

In Exercises 41 to 48, find the requested term. 41. Find the term that contains b 8 in the binomial expansion of

(2a - b)10. 42. Find the term that contains s7 in the binomial expansion of

(3r + 2s)9. 43. Find the term that contains y 8 in the binomial expansion of

x 4 2 - b x 2 -1

a b 11 + b ; ninth term b a

(2x + y 2)6. -1 5

+ s )

44. Find the term that contains b 9 in the binomial expansion of

(a - b 3)8.

11.6

45. Find the middle term of the binomial expansion of (3a - b)10. 46. Find the middle term of the binomial expansion of (a + b 2)8.

PERMUTATIONS AND COMBINATIONS

In Exercises 49 to 54, use the Binomial Theorem to simplify the power of the complex number. 49. (2 - i )4

50. (3 + 2i)3

47. Find the two middle terms of the binomial expansion of

51. (1 + 2i)5

52. (1 - 3i)5

48. Find the two middle terms of the binomial expansion of

53. a

(s- 1 + s)9.

(x1/2 - y1/2)7.

SECTION 11.6

883

12 8 12 + i b 2 2

54. a

13 6 1 + i b 2 2

Permutations and Combinations

Fundamental Counting Principle Permutations Combinations


PS1. Evaluate: 7! [11.1] PS2. Evaluate: (7 - 3)! [11.1] PS3. Evaluate a b when n = 7 and k = 1. [11.5]

n k

PS4. Evaluate a b when n = 8 and k = 5. [11.5]

n k

PS5. Evaluate

n! when n = 10 and k = 2. [11.1] (n - k)!

PS6. Evaluate

n! when n = 6 and k = 6. [11.1] (n - k)!

Fundamental Counting Principle T1 A1

T2 T3

T1 A2

T2 T3

Figure 11.2

S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2

Suppose that an electronics store offers a three-component stereo system for $250. A buyer must choose one amplifier, one tuner, and one pair of speakers. If the store has two models of amplifiers, three models of tuners, and two speaker models, how many different stereo systems could a consumer purchase? This problem belongs to a class of problems called counting problems. The problem is to determine the number of ways in which the conditions of the problem can be satisfied. One way to do this is to make a tree diagram using A1 and A2 for the amplifiers; T1, T2 , and T3 for the tuners; and S1 and S2 for the speakers. See Figure 11.2. By counting the possible systems that can be purchased, we find there are 12 different systems. Another way to arrive at this result is to find the product of the numbers of options available. Number of number of number of number of * * = amplifiers tuners speakers systems 2 * 3 * 2 = 12 In some states, a standard car license plate consists of a nonzero digit, followed by three letters, followed by three more digits. What is the maximum number of car license plates of

884

CHAPTER 11


this type that could be issued? If we begin a list of the possible license plates, it soon becomes apparent that listing them all would be time-consuming and impractical. 1AAA000,

1AAA001,

1AAA002,

1AAA003, Á

Instead, the following counting principle is used. This principle forms the basis for all counting problems.

Fundamental Counting Principle Let T1 , T2 , T3 , . . . , Tn be a sequence of n conditions. Suppose that T1 can occur in m1 ways, T2 can occur in m2 ways, T3 can occur in m3 ways, and so on, until finally Tn can occur in mn ways. Then the number of ways of satisfying the conditions T1 , T2 , T3 , . . . , Tn in succession is given by the product m1m2 m3 Á mn Table 11.4

Condition

Number of Ways

T1: a nonzero digit

m1 = 9

T2: a letter

m2 = 26

T3: a letter

m3 = 26

T4: a letter

m4 = 26

T5: a digit

m5 = 10

T6: a digit

m6 = 10

T7: a digit

m7 = 10

To apply the Fundamental Counting Principle to the license plate problem, first find the number of ways each condition can be satisfied, as shown in Table 11.4. Thus we have Number of car license plates = 9 # 26 # 26 # 26 # 10 # 10 # 10 = 158,184,000

Question • Suppose that a license plate begins with two letters. How many different ways could

the license plate begin?

EXAMPLE 1

Apply the Fundamental Counting Principle

An automobile dealer offers three midsize cars. A customer selecting one of these cars must choose one of three different engines, one of five different colors, and one of four different interior packages. How many different selections can the customer make? Solution T1 : midsize car T2 : engine

m1 = 3 m2 = 3

T3 : color T4 : interior

The number of different selections is 3 # 3 # 5 # 4 = 180.

m3 = 5 m4 = 4


Permutations One application of the Fundamental Counting Principle is to determine the number of arrangements of distinct elements in a definite order.

Definition of a Permutation A permutation is an arrangement of distinct objects in a definite order. EXAMPLE

abc and bca are two of the possible permutations of the three letters a, b, and c.

Answer • 26 * 26 = 676

11.6


885

Consider a race with 10 runners. In how many different orders can the runners finish first, second, and third (assuming no ties)? Any one of the 10 runners could finish first:

m1 = 10

Any one of the remaining 9 runners could be second: m2 = 9 Any one of the remaining 8 runners could be third: m3 = 8

Integrating Technology Some graphing calculators use the notation nPr to represent the number of permutations of n objects taken r at a time. The calculation of the number of permutations of 15 objects taken 4 at a time is shown below.

By the Fundamental Counting Principle, there are 10 # 9 # 8 = 720 possible first-, second-, and third-place finishes for the 10 runners. Using the language of permutations, we would say, “There are 720 permutations of 10 objects (the runners) taken 3 (the number of possible finishes) at a time.” Permutations occur so frequently in counting problems that a formula, rather than the Fundamental Counting Principle, is often used.

Formula for the Number of Permutations of n Distinct Objects Taken r at a Time

15 nPr 4

The number of permutations of n distinct objects taken r at a time is

32760

P(n, r) =

EXAMPLE 2

n! (n - r)!

Find the Number of Permutations

In how many ways can a president, a vice president, a secretary, and a treasurer be selected from a committee of 15 people? Solution There are 15 distinct people to place in four positions. Thus n = 15 and r = 4. P(15, 4) =

15! 15 # 14 # 13 # 12 # 11! 15! = = = 32,760 (15 - 4)! 11! 11!

There are 32,760 ways to select the officers. Try Exercise 16, page 888

EXAMPLE 3

Find the Number of Seating Permutations

Six people attend a movie and sit in the same row containing six seats. a.

Find the number of ways the group can sit together.

b.

Find the number of ways the group can sit together if two particular people in the group must sit side by side.

c.

Find the number of ways the group can sit together if two particular people in the group refuse to sit side by side. (continued)

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Solution a. There are six distinct people to place in six distinct positions. Thus n = 6 and r = 6. P(6, 6) =

6! 6! 6! = = = 720 (6 - 6)! 0! 1

There are 720 arrangements of the six people. b.

Think of the two people who must sit together as a single object and count the number of arrangements of the five objects (AB), C, D, E, and F. Thus n = 5 and r = 5. P(5, 5) =

5! 5! 5! = = = 120 (5 - 5)! 0! 1

There are also 120 arrangements with A and B reversed [(BA), C, D, E, F]. Thus the total number of arrangements is 120 + 120 = 240. c.

From a., there are 720 possible seating arrangements. From b., there are 240 arrangements with two specific people next to each other. Thus there are 720 - 240 = 480 arrangements in which two specific people are not seated together. Try Exercise 22, page 888

Combinations Up to this point, we have been counting the number of distinct arrangements of objects. In some cases, we may be interested in determining the number of ways of selecting objects without regard to the order of the selection. For example, suppose we want to select a committee of three people from five candidates denoted by A, B, C, D, and E. One possible committee is A, C, D. If we select D, C, A, we still have the same committee because the order of the selection is not important. An arrangement of objects in which the order of the selection is not important is a combination.

Note Recall that a binomial coefficient n! n is given by a b = , r !(n - r)! r which is the same as C(n, r).

Formula for the Number of Combinations of n Objects Taken r at a Time The number of combinations of n objects taken r at a time is C(n, r) =

EXAMPLE 4

n! r! (n - r)!

Find the Number of Combinations

A standard deck of playing cards consists of 52 cards. How many five-card hands can be chosen from this deck? Solution We have n = 52 and r = 5. Thus

Integrating Technology Some calculators use the notation nCr to represent a combination of n objects taken r at a time.

C(52, 5) =

52! 52! 52 # 51 # 50 # 49 # 48 # 47! = = = 2,598,960 5! (52 - 5)! 5! 47! 5 # 4 # 3 # 2 # 1 # 47!

There are 2,598,960 five-card hands. Try Exercise 20, page 888

11.6

EXAMPLE 5


887

Find the Number of Combinations

A chemist has nine solution samples, of which four are type A and five are type B. If the chemist chooses three of the solutions at random, determine the number of ways in which the chemist can choose exactly one type A solution. Solution The chemist has chosen three solutions, one of which is type A. If one is type A, then two are type B. The number of ways of choosing one type A solution from four type A solutions is C(4, 1). C(4, 1) =

4! 4! = = 4 1! (4 - 1)! 1! 3!

The number of ways of choosing two type B solutions from five type B solutions is C(5, 2). C(5, 2) =

5! 5! = = 10 2! (5 - 2)! 2! 3!

By the Fundamental Counting Principle, there are

C(4, 1) # C(5, 2) = 4 # 10 = 40

ways to choose one type A and two type B solutions. Try Exercise 30, page 888

The difficult part of solving a counting problem is determining whether to use the counting principle, the permutation formula, or the combination formula. Following is a summary of guidelines.

Guidelines for Solving Counting Problems 1. The Fundamental Counting Principle will always work but is not always the easiest method to apply. 2. When reading a problem, ask yourself, “Is the order of the selection process important?” If the answer is yes, the arrangements are permutations. If the answer is no, the arrangements are combinations.

EXERCISE SET 11.6 In Exercises 1 to 10, evaluate each quantity.

12. Color Monitors A computer monitor produces color by

1. P(6, 2)

2. P(8, 7)

3. C(8, 4)

4. C(9, 2)

5. P(8, 0)

6. P(9, 9)

7. C(7, 7)

8. C(6, 0)

9. C(10, 4)

10. P(10, 4)

11. Computer Systems A computer manufacturer offers a com-

puter system with three different disk drives, two different monitors, and two different keyboards. How many different computer systems could a consumer purchase from this manufacturer?

blending colors on palettes. If a computer monitor has four palettes and each palette has four colors, how many blended colors can be formed? Assume each palette must be used each time. 13. Computer Science A four-digit binary number is a sequence

of four digits, each consisting of 0 or 1. For instance, 0011 and 1011 are binary numbers. How many four-digit binary numbers are possible?

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14. Computer Memory An integer is stored in a computer’s

26. Test Questions A 20 question, four-option multiple-choice

memory as a series of 0s and 1s. Each memory unit contains eight spaces for a 0 or a 1. The first space is used for the sign of the number, and the remaining seven spaces are used for the integer. How many positive integers can be stored in one memory unit of this computer?

examination is given as a preemployment test. In how many ways could a prospective employee answer the questions on this test just by guessing? Assume that all questions are answered. 27. State Lottery A state lottery game requires a person to select

15. Scheduling In how many different ways can six employees be

6 different numbers from 40 numbers. The order of the selection is not important. In how many ways can this be done?

assigned to six different jobs? 28. Test Questions A student must answer 8 of 10 questions on 16. Contest Winners First-, second-, and third-place prizes are to

an exam. How many different choices can the student make?

be awarded in a dance contest in which 12 contestants are entered. In how many ways can the prizes be awarded?

29. Acceptance Sampling A warehouse receives a shipment of

17. Mailboxes There are five mailboxes outside a post office. In

10 computers, of which 3 are defective. Five computers are then randomly selected from the 10 and delivered to a store.

how many ways can three letters be deposited into the five boxes? 18. Committee Membership How many different committees

of three people can be selected from nine people? 19. Test Questions A professor provides to a class 25 possible

essay questions for an upcoming test. Of the 25 questions, the professor will ask 5 of the questions on the exam. How many different tests can the professor prepare? 20. Tennis Matches Twenty-six people enter a tennis tourna-

ment. How many different first-round matches are possible if each player can be matched with any other player?

a. In how many ways can the store receive no defective

computers? b. In how many ways can the store receive one defective

computer? c. In how many ways can the store receive all three defective

computers? 30. Contest Fifteen students, of whom seven are seniors, are

selected as semifinalists for a literary award. Of the 15 students, 10 finalists will be selected. a. In how many ways can 10 finalists be selected from the

21.

Employee Initials A company has more than 676 em-

15 students?

ployees. Explain why there must be at least 2 employees who have the same first and last initials.

b. In how many ways can the 10 finalists contain three seniors?

22. Seating Arrangements A car holds six passengers, three in

c. In how many ways can the 10 finalists contain at least five

the front seats and three in the back seats. How many different seating arrangements of six people are possible if one person refuses to sit in front and one person refuses to sit in back?

seniors? 31. Serial Numbers A television manufacturer uses a code for

24. Arranging Numbers The numbers 1, 2, 3, 4, 5, and 6 are to

the serial number of a television set. The first symbol is the letter A, B, or C and represents the location of the manufacturing plant. The next two symbols (01, 02, . . . , 12) represent the month in which the set was manufactured. The next symbol is a 5, a 6, a 7, an 8, or a 9 and represents the year the set was manufactured. The last seven symbols are digits. How many serial numbers are possible?

be arranged. How many different arrangements are possible under each of the following conditions?

32. Playing Cards Five cards are chosen at random from a stan-

23. Committee Membership A committee of six people is cho-

sen from six senators and eight representatives. How many committees are possible if there are to be three senators and three representatives on the committee?

a. All the even numbers come first.

dard deck of playing cards. In how many ways can the cards be chosen under each of the following conditions?

b. The arrangements are such that the numbers alternate

a. All are hearts.

b. All are the same suit.

c. Exactly three are kings.

d. Two or more are aces.

between even and odd. 25. Test Questions A true–false examination contains 10 ques-

tions. In how many ways can a person answer the questions on this test just by guessing? Assume that all questions are answered.

33. Acceptance Sampling A quality control inspector receives a

shipment of 10 computer disk drives and randomly selects 3 of the drives for testing. If 2 of the disk drives in the shipment are

11.6


889

defective, find the number of ways in which the inspector could select at most 1 defective drive. 34. Basketball Teams A basketball team has 12 members. In how

many ways can 5 players be chosen under each of the following conditions? a. The selection is random. b. The two tallest players are always among the 5 selected. 35. Arranging Numbers The numbers 1, 2, 3, 4, 5, and 6 are

arranged randomly. In how many ways can the numbers 1 and 2 appear next to one another in the order 1, 2? 36. Occupancy Problem Seven identical balls are randomly put

in seven containers so that two balls are in one container and each of the remaining six containers receives at most one ball. Find the number of ways in which this can be accomplished. 37. Lines in a Plane Seven points lie in a plane in such a way that

no three points lie on the same line. How many lines are determined by the seven points? 38. Chess Matches A chess tournament has 12 participants. How

45. Ice Cream Cones An ice cream store offers 31 flavors of ice

cream. How many different triple-decker cones are possible? (Note: Assume that different orders of the same flavors are not different cones. Thus a scoop of rocky road followed by two scoops of mint chocolate is the same as one scoop of mint chocolate followed by one scoop of rocky road followed by a second scoop of mint chocolate.) 46. Computer Screens A typical computer monitor consists of

pixels, each of which can, in some cases, be assigned any one of 216 different colors. If a computer screen has a resolution of 1024 pixels by 768 pixels, how many different images can be displayed on the screen? (Suggestion: Write your answer as a power of 2.) 47. Dartboards How many different arrangements of the integers

1 through 20 are possible on a typical dartboard, assuming that 20 is always at the top?

many games must be scheduled if every player must play every other player exactly once?

5

20

1

12 39. Contest Winners Eight couples attend a benefit dinner at

which two prizes are given. In how many ways can two names be randomly drawn so that the prizes are not awarded to the same couple? 40. Geometry Suppose there are 12 distinct points on a circle.

How many different triangles can be formed with vertices at the given points? 41. Test Questions In how many ways can a student answer a 20-

question true–false test if the student randomly marks 10 of the questions true and 10 of the questions false? 42. Committee Membership From a group of 15 people, a com-

mittee of 8 is formed. From the committee, a president, a secretary, and a treasurer are selected. Find the number of ways in which the two consecutive operations can be carried out. 43. Committee Membership From a group of 20 people, a com-

mittee of 12 is formed. From the committee of 12, a subcommittee of 4 people is chosen. Find the number of ways in which the two consecutive operations can be carried out.

18

9

4

14

13

11

6

8

10 16

15 7

19

3

17

2

48. Lines in a Plane Generalize Exercise 37. That is, given n

points in a plane, no three of which lie on the same line, how many lines are determined by the n points? 49. Birthdays Seven people are asked the month of their birth. In

how many ways can each of the following conditions exist? a. No two people have a birthday in the same month. b. At least two people have a birthday in the same month.

44. Checkerboards A checkerboard consists of eight rows and

50. Sums of Coins From a penny, a nickel, a dime, and a quarter,

eight columns of squares as shown in the following figure. Starting at the top left square of a checkerboard, how many possible paths will end at the bottom right square if the only way a player can legally move is right one square or down one square from the current position?

how many different sums of money can be formed using one or more of the coins? 51. Biology Five sticks of equal length are broken into a short

piece and a long piece. The 10 pieces are randomly arranged in

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five pairs. In how many ways will each pair consist of a long stick and a short stick? (This exercise actually has a practical side. When cells are exposed to harmful radiation, some chromosomes break. If two long sides unite or two short sides unite, the cell dies.)

53. Random Walk An aimless tourist, standing on a street corner,

52. Arranging Numbers Four random digits are drawn (repeti-

tions are allowed). Among the four digits, in how many ways can two or more repetitions occur?

SECTION 11.7 Sample Spaces and Events Probability of an Event Independent Events Binomial Probabilities

tosses a coin. If the result is heads, the tourist walks one block north. If the result is tails, the tourist walks one block south. At the new corner, the coin is tossed again and the same rule applied. If the coin is tossed 10 times, in how many ways will the tourist be back at the original corner? This problem is an elementary example of what is called a random walk. Random walk problems have many applications in physics, chemistry, and economics.

Introduction to Probability PREPARE FOR THIS SECTION Prepare for this section by completing the following exercises. The answers can be found on page A75.

PS1. What is the fundamental counting principle? [11.6] PS2. How many ways can a two-digit number be formed from the digits 1, 2, 3, and 4

if no digit can be repeated in the number? [11.6] PS3. Evaluate: P(7, 2) [11.6] PS4. Evaluate: C(7, 2) [11.6]

Math Matters The beginning of probability theory is frequently associated with letters sent between Pascal (of Pascal’s Triangle) and Fermat (of Fermat’s Last Theorem) in which they discuss the solution of a problem posed to them by Antoine Gombaud, Chevalier de Mere, a French aristocrat who liked to gamble. The basic question was “How many tosses of two dice are necessary to have a better than 50–50 chance of rolling two sixes?” Although the correct analysis of this problem by Fermat and Pascal presaged probability theory, historical records indicate that commerce and the need to insure ships, cargo, and lives was another motivating factor to calculating probabilities. The first merchant insurance companies were established in the fourteenth century to insure ships. Life insurance companies were established at the end of the seventeenth century. Lloyd’s of London was established sometime before 1690.

PS5. Evaluate a b p kq n-k when n = 8, k = 5, p =

n k

1 3 , and q = . [11.5] 4 4

PS6. A light switch panel has four switches that control the lights in four different

areas of a room. In how many on–off configurations can the four switches be placed? [11.6]

Many events in the world around us have random character, such as the chances of an accident occurring on a certain freeway, the chances of winning a state lottery, and the chances that the nucleus of an atom will undergo fission. By repeatedly observing such events, it is often possible to recognize certain patterns. Probability is the mathematical study of random patterns. When a weather reporter predicts a 30% chance of rain, the forecaster is saying that similar weather conditions have led to rain 30 times out of 100. When a fair coin is tossed, 1 1 we expect heads to occur , or 50%, of the time. The numbers 30% (or 0.3) and are the 2 2 probabilities of the events.

Sample Spaces and Events An activity with an observable outcome is called an experiment. Examples of experiments include 1. Flipping a coin and observing the side facing upward 2. Observing the incidence of a disease in a certain population 3. Observing the length of time a person waits in a checkout line in a grocery store The sample space of an experiment is the set of all possible outcomes of that experiment.

11.7

INTRODUCTION TO PROBABILITY

891

Consider the experiment of tossing one coin three times and recording the outcome of the upward side of the coin for each toss. The sample space is S = 5HHH, HHT, HTH, THH, HTT, THT, TTH, TTT6

EXAMPLE 1

List the Elements of a Sample Space

Suppose that among five batteries, two are defective. Two batteries are randomly drawn from the five and tested for defects. List the elements in the sample space. Solution Label the nondefective batteries N1, N2, and N3 and the defective batteries D1 and D2. The sample space is S 5N1D1, N2D1, N3D1, N1D2, N2D2, N3D2, N1N2, N1N3, N2N3, D1D26


An event E is any subset of a sample space. For the sample space defined in Example 1, several of the events we could define are E1: There are no defective batteries. E2: At least one battery is defective. E3: Both batteries are defective. Because an event is a subset of the sample space, each of these events can be expressed as a set. E 1 = 5N1N2, N1N3, N2N36 E 2 = 5N1D1, N2D1, N3D1, N1D2, N2D2, N3D2, D1D26 E 3 = 5D1D26

There are two methods by which elements can be drawn from a set: with replacement and without replacement. With replacement means that after the element is drawn, it is returned to the set. The same element could be selected on the next drawing. When elements are drawn without replacement, an element drawn is not returned to the set and therefore is not available for any subsequent drawing.

EXAMPLE 2

List the Elements of an Event

A two-digit number is formed by choosing from the digits 1, 2, 3, and 4, both with replacement and without replacement. Express each event as a set. a.

E1:

The second digit is greater than or equal to the first digit.

b.

E2 :

Both digits are less than zero.

Solution a. With replacement:

E1 = 511, 12, 13, 14, 22, 23, 24, 33, 34, 446

Without replacement: E1 = 512, 13, 14, 23, 24, 346

(continued)

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b.

E2 = Choosing from the digits 1, 2, 3, and 4, this event is impossible. The impossible event is denoted by the empty set or null set. Try Exercise 14, page 897

Probability of an Event The probability of an event is defined in terms of the concepts of sample space and event.

Definition of the Probability of an Event Let n(S) and n(E) represent the number of elements in the sample space S and the number of elements in the event E, respectively. The probability of event E is P(E) =

n(E) n(S)

Because E is a subset of S, n(E) … n(S). Thus P(E) … 1. If E is an impossible event, then E = and n(E) = 0. Thus P(E) = 0. If E is an event that always occurs, then E = S and n(E) = n(S). Thus P(E) = 1. Thus we have, for any event E, 0 … P(E) … 1 Question • Is it possible for the probability of an event to equal 1.25?

EXAMPLE 3

Calculate the Probability of an Event

A coin is tossed three times. What is the probability of each outcome? a.

E1 :

Two or more heads will occur.

b.

E2 :

At least one tail will occur.

Solution First determine the number of elements in the sample space. The sample space for this experiment is S = 5HHH, HHT, HTH, THH, HTT, THT, TTH, TTT6

Therefore, n(S) = 8. Now determine the number of elements in each event. Then calculate the probability of the event by using P(E) = n(E)>n(S). a.

E1 = 5HHH, HHT, HTH, THH6

P(E1 ) = b.

n(E1 ) 4 1 = = n(S) 8 2

E2 = 5HHT, HTH, THH, HTT, THT, TTH, TTT6 P(E2 ) =

n(E2 ) 7 = n(S) 8

Try Exercise 22, page 897 Answer • No. All probabilities must be between 0 and 1, inclusive.

11.7


893

The expression “one or the other of two events occurs” is written as the union of the two sets. For example, suppose an experiment leads to the sample space S = 51, 2, 3, 4, 5, 66 and the events are Draw a number less than four, E1 = 51, 2, 36 Draw an even number, E2 = 52, 4, 66

Then the event E1 ´ E2 is described by drawing a number less than four or an even number. Thus E1 ´ E2 = 51, 2, 36 ´ 52, 4, 66 = 51, 2, 3, 4, 66 Two events E1 and E2 that cannot occur at the same time are mutually exclusive events. Using set notation, if E1 ¨ E2 = , then E1 and E2 are mutually exclusive. For example, using the same sample space 51, 2, 3, 4, 5, 66, a third event is Draw an odd number, E3 = 51, 3, 56 Then E2 ¨ E3 = and the events E2 and E3 are mutually exclusive. On the other hand, E1 ¨ E2 = 526 so the events E1 and E2 are not mutually exclusive. One of the axioms of probability involves the union of mutually exclusive events.

Probability Axiom If E1 and E2 are mutually exclusive events, then P(E1 ´ E2 ) = P(E1 ) + P(E2 )

If the events are not mutually exclusive, the addition rule for probabilities can be used.

Addition Rule for Probabilities If E1 and E2 are two events, then P(E1 ´ E2 ) = P(E1 ) + P(E2 ) - P(E1 ¨ E2 )

EXAMPLE 4

Use Addition Rules for Probability

Let S = 5Natural numbers less than or equal to 1006, and consider the experiment of drawing one number from S. a.

If E1 is the event of drawing a number less than 11 and E2 is the event of drawing a number greater than 90, what is the probability of E1 or E2?

b.

If E3 is the event of drawing a number less than 11 and E4 is the event of drawing an even number, what is the probability of E3 or E4? (continued)

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CHAPTER 11


Solution a. E1 ¨ E2 = . Therefore, the events are mutually exclusive. n(E1) = 10 and n(E2) = 10. P(E1 ´ E2) = P(E1) + P(E2)

b.

=

n(E2) n(E1) + n(S) n(S)

=

10 1 10 + = 100 100 5

• Probability axiom

E3 ¨ E4 = 52, 4, 6, 8, 106. Therefore, the events are not mutually exclusive. n(E3) = 10, n(E4 ) = 50, and n(E3 ¨ E4) = 5. P(E3 ´ E4) = P(E3) + P(E4) - P(E3 ¨ E4) =

n(E4) n(E3 ¨ E4) n(E3) + n(S) n(S) n(S)

=

50 5 11 10 + = 100 100 100 20

• Addition rule


EXAMPLE 5

Use Counting Principles to Calculate a Probability

A state lottery game allows a person to choose five numbers from the integers 1 to 40. Repetitions of numbers are not allowed. If three or more numbers match the numbers chosen by the lottery, the player wins a prize. Find the probability that a player will match the following. a.

Exactly three numbers

b.

Exactly four numbers

Solution The sample space S is the set of ways in which 5 numbers can be chosen from 40 numbers. This is a combination because the order of the drawing is not important. n(S) = C(40, 5) =

40! = 658,008 5! 35!

We will call the 5 numbers chosen by the state lottery “lucky” and the remaining 35 numbers “unlucky.” a.

Let E1 be the event that a player has 3 lucky and therefore 2 unlucky numbers. The 3 lucky numbers are chosen from the 5 lucky numbers. There are C(5, 3) ways to do this. The 2 unlucky numbers are chosen from the 35 unlucky numbers. There are C(35, 2) ways to do this. By the fundamental counting principle, the number of ways the event E1 can occur is n(E1 ) = C(5, 3) # C(35, 2) = 10 # 595 = 5950 n(E1 ) C(5, 3) # C(35, 2) 5950 P(E1 ) = = = L 0.009042 n(S) C(40, 5) 658,008

11.7

b.


895

Let E2 be the event that a player has four lucky numbers and one unlucky number. The number of ways a person can select four lucky numbers and one unlucky number is C(5, 4) # C(35, 1). P(E2 ) =

n(E2 ) C(5, 4) # C(35, 1) 175 = = L 0.000266 n(S) C(40, 5) 658,008


Independent Events Two events are independent if the outcome of the first event does not influence the outcome of the second event. As an example, consider tossing a fair coin twice. The outcome of the first toss has no bearing on the outcome of the second toss. The two events are independent. Now consider drawing two cards in succession, without replacement, from a standard deck of playing cards. The probability that the second card drawn will be an ace depends on the card drawn first.

Probability Rule for Independent Events If E1 and E2 are two independent events, then the probability that both E1 and E2 will occur is P(E1 ) # P(E2 )

EXAMPLE 6

Calculate a Probability for Independent Events

A coin is tossed and then a die is rolled. What is the probability that the coin will show a head and the die will show a six? Solution The events are independent because the outcome of one does not influence the outcome 1 1 of the other. P(head) = and P(six) = . Thus the probability of tossing a head and 2 6 rolling a six is P(head) # P(six) =

1 1#1 = 2 6 12


Binomial Probabilities Some probabilities can be calculated from formulas. One of the most important of these formulas is the binomial probability formula. This formula is used to calculate probabilities for independent events.

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Math Matters Airlines “overbook” flights. That is, they sell more tickets than there are seats on the plane. An airline company can determine the probability that some number of passengers will be “bumped” on a certain flight by using the binomial probability formula. For instance, suppose a plane has 200 seats and the airline sells 240 tickets. If the probability that a person will show up for this flight is 0.8, then the probability that one or more people will have to be bumped is given by 40 240 200 + k 0.240 - k a a 200 + k b 0.8 k=1

A computer calculation reveals that this sum is approximately 0.08. That is, there is only an 8% chance that someone will have to be bumped, even though the number of tickets sold exceeds the plane’s capacity by 20%.

Binomial Probability Formula Let an experiment consist of n independent trials for which the probability of success on a single trial is p and the probability of failure is q = 1 - p. Then the probability of k successes in n trials is given by n a b pkqn - k k

EXAMPLE 7

Use the Binomial Formula

A multiple-choice exam consists of 10 questions. For each question there are four possible choices, of which only one is correct. If someone randomly guesses at the answers, what is the probability of guessing exactly six answers correctly? Solution Selecting an answer is one trial of the experiment. Because there are 10 questions, n = 10. There are four possible choices for each question, of which only one is correct. Therefore, p =

1 4

and

q = 1 - p = 1 -

1 3 = 4 4

A success for this experiment occurs each time a correct answer is guessed. Thus k = 6. By the binomial probability formula, 10 1 6 3 4 b a b a b L 0.016222 4 4 6

P = a

The probability of guessing exactly six answers correctly is approximately 0.0162. Try Exercise 46, page 898

Following are five guidelines for calculating probabilities.

Guidelines for Calculating a Probability 1. The word or usually means to add the probabilities of each event. 2. The word and usually means to multiply the probabilities of each event. 3. The phrase at least n means n or more. At least 5 is 5 or more. 4. The phrase at most n means n or less. At most 5 is 5 or less. 5. Exactly n means just that. Exactly five heads in seven tosses of a coin means five heads and therefore two tails.

EXERCISE SET 11.7 In Exercises 1 to 10, list the elements in the sample space defined by the given experiment. 1. Two people are selected from two senators and three represen-

tatives.

2. A letter is chosen at random from the word Tennessee. 3. A fair coin is tossed, and then a random integer between 1 and

4, inclusive, is selected.

11.7


897

22. Tossing Coins A coin is tossed four times. Find the probabil-

4. A fair coin is tossed four times.

ity of each event. 5. Two identical tennis balls are randomly placed in three tennis

ball cans. Let A, B, and C represent the three cans. Use an ordered pair to represent each outcome. For example, (A, B) represents the event that ball 1 is in can A and ball 2 is in can B. 6. Two people are selected from among one Republican, one

Democrat, and one Independent. 7. Three cards are randomly chosen from the ace of hearts, ace of

spades, ace of clubs, and ace of diamonds. 8. Three letters addressed to A, B, and C, respectively, are ran-

domly put into three envelopes addressed to A, B, and C, respectively.

a. Two of the coins land heads and two land tails. b. There are at least three heads. 23. Tossing Coins A coin is tossed four times. Find the proba-

bility of each event. a. All the coins land tails. b. There is at least one head. 24. Playing Cards One card is drawn from a standard deck of

playing cards. Let E1 be the event that the card is a spade, and let E2 be the event the card is a heart. a. Are the events mutually exclusive? b. What is the probability of E1 or E2?

9. Two vowels are randomly chosen from a, e, i, o, and u. 10. Three computer disks are randomly chosen from one defective

disk and three nondefective disks. In Exercises 11 to 15, use the sample space defined by the experiment of tossing a fair coin four times. Express each event as a subset of the sample space.

than 1006, and consider the experiment of drawing one number from S. Let E1 be the event that the first digit of the number is 5, and let E2 be the event that the number is a perfect square.

a. Are the events mutually exclusive? b. What is the probability of E1 or E2? 26. Playing Cards One card is drawn from a standard deck of

11. There are no tails.

playing cards. Let E1 be the event that the card is a spade, and let E2 be the event the card is a jack, a queen, or a king.

12. There are exactly two heads.

a. Are the events mutually exclusive?

13. There are at most two heads.

b. What is the probability of E1 or E2? 27. Number Theory Let S = 5Natural numbers less than or equal

14. There are more than two heads.

to 1006, and consider the experiment of drawing one number from S. Let E1 be the event the number is divisible by 5, and let E2 be the event the number is a divisible by 7.

15. There are 12 tails.

In Exercises 16 to 20, use the sample space defined by the experiment of choosing two random numbers, in succession, from the integers 1, 2, 3, 4, 5, and 6. The numbers are chosen with replacement. Express each event as a subset of the sample space. 16. The sum of the numbers is 7. 17. The two numbers are the same.

19. The second number is a 4. 20. The sum of the two numbers is greater than 1. 21. Playing Cards From a standard deck of playing cards, one

card is chosen at random. Find the probability of each event. b.

a. Are the events mutually exclusive? b. What is the probability of E1 or E2? 28. Number Theory A single number is chosen from the digits 1,

2, 3, 4, 5, and 6. Find the probability that the number is an even number or a number divisible by 3. 29. Economics An economist predicts that the probability of an

18. The first number is greater than the second number.

a. The card is a king.

25. Number Theory Let S = 5Two-digit natural numbers less

The card is a spade.

increase in gross domestic product (GDP) is 0.64 and that the probability of an increase in inflation is 0.55. The economist also predicts that the probability of an increase in GDP and inflation is 0.22. Find the probability of an increase in GDP or an increase in inflation. 30. Number Theory Four digits are selected from the digits

1, 2, 3, and 4, and a number is formed. Find the probability that the number is greater than 3000, assuming digits can be repeated.

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CHAPTER 11


31. Building Industry An owner of a construction company has

42. National Defense A missile radar detection system consists

bid for the contracts on two buildings. If the contractor estimates 1 that the probability of getting the first contract is , that of get2 1 ting the second contract is , and that of getting both contracts 5 1 is , find the probability that the contractor will get at least 10 one of the two building contracts.

of two radar screens. The probability that either one of the radar screens will detect an incoming missile is 0.95. If radar detections are assumed to be independent events, what is the probability that a missile that enters the detection space of the radar will be detected?

32. Acceptance Sampling A shipment of 10 calculators contains

2 defective calculators. Two calculators are chosen from the shipment. Find the probability of each event. a. Both are defective.

b. At least one is defective.

33. Number Theory Five random digits are selected from 0 to 9

with replacement. What is the probability (to the nearest hundredth) that 0 does not occur?

43. Oil Industry An oil drilling venture involves drilling four wells

in different parts of the country. For each well, the probability that it will be profitable is 0.10, and the probability that it will be unprofitable is 0.90. If these events are independent, what is the probability of drilling at least one unprofitable well? 44. Manufacturing A manufacturer of CD-ROMs claims that

only 1 of every 1000 CD-ROMs manufactured is defective. If this claim is correct and if defective CD-ROMs are independent events, what is the probability (to the nearest tenthousandth) that, of the next three CD-ROMs produced, all are not defective?

34. Queuing Theory Six people are arranged in a line. What is

the probability that two specific people, say A and B, are standing next to each other? 35. Lottery A box contains 500 envelopes, of which 50 have $100

in cash, 75 have $50 in cash, and 125 have $25 in cash. If an envelope is selected at random from this box, what is the probability that it will contain at least $50?

45. Preference Testing A software firm is considering marketing

two newly designed spreadsheet programs, A and B. To test the appeal of the programs, the firm installs them in four corporations. After 2 weeks, the firm asks each corporation to evaluate each program. If the corporations have no preference, what is the probability that all four will choose product A? 46. Agriculture A fruit grower claims that one-fourth of the

15 women and 15 men. What is the probability (to the nearest ten-thousandth) that the jury will have 6 men and 6 women?

orange trees in a grove crop have suffered frost damage. Find the probability (to the nearest ten-thousandth) that among eight orange trees, exactly three have frost damage.

37. Queuing Theory Three girls and three boys are randomly

47. Quality Control A quality control inspector receives a ship-

placed in six adjacent seats. What is the probability that the boys and girls will be in alternating seats?

ment of 20 computer monitors. From the 20 monitors, the inspector randomly chooses 5 for inspection. If the probability of a monitor being defective is 0.05, what is the probability (to the nearest ten-thousandth) that at least one of the monitors chosen by the inspector is defective?

36. Jury Selection A jury of 12 people is selected from 30 people:

38. Committee Membership A committee of four is chosen

from three accountants and five actuaries. Find the probability that the committee has exactly two accountants.

48. Lottery Consider a lottery that sells 1000 tickets and awards 39. Extrasensory Perception A magician claims to be able to

read minds. To test this claim, five cards numbered 1 to 5 are used. A subject selects two cards from the five and concentrates on the numbers. What is the probability that the magician can correctly identify the two cards by just guessing? 40. Playing Cards One card is randomly drawn from a standard

deck of playing cards. The card is replaced and another card is drawn. Are the events independent? What is the probability that both cards drawn are aces?

two prizes. If you purchase 10 tickets, what is the probability that you will win a prize? 49. Airline Scheduling An airline estimates that 75% of the peo-

ple who make a reservation for a certain flight will actually show up for the flight. Suppose the airline sells 25 tickets on this flight and the plane has room for 20 passengers. What is the probability (to the nearest ten-thousandth) that 21 or more people with tickets will show up for the flight? 50. Airline Scheduling Suppose that an airplane’s engines oper-

41. Scheduling A meeting is scheduled by randomly choosing a

weekday and then randomly choosing an hour between 8:00 A.M. and 4:00 P.M. What is the probability that Monday at 8:00 A.M. is chosen?

ate independently and that the probability that any one engine will fail is 0.03. A plane can make a safe landing if at least onehalf of its engines operate. Is a safe flight more likely to occur in a two-engine or a four-engine plane?


51. Spread of a Rumor A club has nine members. One member

starts a rumor by telling it to a second club member, who repeats the rumor to a third member, and so on. At each stage, the recipient of the rumor is chosen at random from the nine club members. What is the probability that the rumor will be told three times without returning to the originator? 52. Extrasensory Perception As a test for extrasensory percep-

tion (ESP), 10 cards, five black and five white, are shuffled, and then a person looks at each card. In another room, the ESP subject attempts to guess whether the card is black or white. The ESP subject must guess black five times and white five times. If the ESP subject has no extrasensory perception, what is the probability that the subject will correctly name 8 of the 10 cards?

that the probability of Player A winning two consecutive points after a game is tied is given by the infinite geometric series p2(1 + 32p(1 - p)4 + 32p(1 - p)42 + Á) where p is the probability that Player A will win any particular point. Suppose the probability that you will win a particular point is 0.55 (that is, p = 0.55). What is the probability that you will go on to win a game that is presently tied? Round to the nearest thousandth. 54. Gambling One way that a player can win in a game of craps

is to bet that, when two dice are rolled, a sum of 6 will occur before a sum of 7 occurs. The probability of this outcome is given by the infinite geometric series 11 2 5 11 11 2 a1 + + a b + a b + Áb 36 36 36 36

53. Sports In some games, such as tennis, the winning player must

win by at least two points. If a game is tied, play is continued until one player wins two consecutive points. It can be shown

899

What is the probability of winning this bet?


Mathematical Expectation Expectation E is a number used to determine the fairness of a gambling game. It is defined as the probability P of winning a bet multiplied by the amount A available to win. E = P#A A game is called fair if the expectation of the game equals the amount bet. For example, if you and a friend each bet $1 on who can guess the side facing up on the 1 flip of a coin, then the expectation is E = # $2 = $1. Because the amount of your 2 bet equals the expectation, the game is fair. When a game is unfair, it benefits one of the players. If you bet $1 and your friend 1 bets $2 on who can guess the flip of a coin, your expectation is E = # $3 = $1.50. 2 Because your expectation is greater than the amount you bet, the game is advantageous to you. Your friend’s expectation is also $1.50, which is less than the amount your friend bet. This is a disadvantage to your friend. Keno is a game of chance played in many casinos. In this game, a large basket contains 80 balls numbered from 1 to 80. From these balls, the dealer randomly chooses 20 balls. The number of ways in which the dealer can choose 20 balls from 80 is the number of combinations of 80 objects chosen 20 at a time, or C(80, 20). In one particular game, a gambler can bet $1 and mark five numbers. The gambler will win a prize if three of the five numbers marked are included in the 20 numbered balls chosen by the dealer. By the Fundamental Counting Principle, there are C(20, 3) # C(60, 2) = 2,017,800 ways the gambler can do this. The probability of this C(20, 3) # C(60, 2) event is L 0.0839. The amount the gambler wins for this event is $2 C(80, 5) (the $1 bet plus $1 from the casino), so the expectation of the gambler is approximately 0.0839 # $2 = $0.17. (continued)

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CHAPTER 11


Each casino has different rules and different methods of awarding prizes. The tables below give the prizes for a $2 bet for some of the possible choices a gambler can make at four casinos. In each case, the Mark column indicates how many numbers the gambler marked, and the Catch column shows how many of the numbers marked by the gambler were also chosen by the dealer. Complete the Expectation columns. Adding the expectations in each column gives you the total expectation for marking six numbers. Find the total expectation for each casino. Which casino offers the gambler the greatest expectation? Casino 1

Casino 2

Mark

Catch

6

4

6 6

Win

Expectation

Mark

Catch

$8

6

4

$6

5

$176

6

5

$160

6

$2960

6

6

$3900

Mark

Catch

Win

Casino 3

Win

Expectation

Casino 4

Mark

Catch

Win

Expectation

6

4

$8

6

4

$6

6

5

$180

6

5

$176

6

6

$3000

6

6

$3000

Expectation


11.1 Infinite Sequences and Summation Notation Infinite Sequence An infinite sequence is a function whose domain is the set of natural numbers and whose range is a set of real numbers. The terms of a sequence are frequently designated as a1, a2, a3, Á , an , Á , where an is the value of the function at n.


n Factorial n factorial, written n!, is the product of the first n natural numbers. That is, n! = 1 # 2 # 3 # Á # (n - 1) # n. This is also written in the reverse order as n! = n # (n - 1) # Á # 3 # 2 # 1.


nth Partial Sum The nth partial sum of a sequence is the sum of the first n terms of the sequence. The nth partial sum of the sequence


n

a1, a2, a3, Á , an , Á is given in summation notation as a ai. i=1

11.2 Arithmetic Sequences and Series Arithmetic Sequence Let d be a real number. A sequence an is an arithmetic sequence if and only if ai + 1 - ai = d for all positive integers i. The nth term of an arithmetic sequence is given by an = a1 + (n - 1)d.


CHAPTER 11 TEST PREP

901

Sum of an Arithmetic Series The nth partial sum of an arithmetic sequence See Example 2, page 857, and then try n n Exercises 32 and 46, page 903. an with common difference d is Sn = (a1 + an), or Sn = 32a1 + (n - 1)d4. 2 2 Arithmetic Mean The arithmetic mean of two numbers a and b is

a + b . 2


11.3 Geometric Sequences and Series Geometric Sequence Let r be a nonzero real number. The sequence an is ai + 1 a geometric sequence if and only if = r for all i. The nth term of a ai geometric sequence is given by an = a1r n - 1.

See Examples 1 and 2, page 862, and then try Exercises 37, 51, and 53, page 903.

Sum of a Finite Geometric Series The nth partial sum of a geometric See Example 3, page 863, and then try a1(1 - r n) Exercises 41 and 45, page 903. , r Z 1. sequence an with first term a1 and common ratio r is Sn = 1-r Sum of an Infinite Geometric Sequence The sum of an infinite geometric a1 , ƒ r ƒ 6 1. sequence an with first term a1 and common ratio r is S = 1 - r


11.4 Mathematical Induction Principle of Mathematical Induction Let Pn be a statement about a positive integer n. If (1) P1 is true and (2) the truth of Pk implies the truth of Pk + 1, then Pn is true for all positive integers.


Extended Principle of Mathematical Induction Let Pn be a statement about a positive integer n. If (1) Pj is true for some positive integer j and (2) for k Ú j the truth of Pk implies the truth of Pk + 1, then Pn is true for all positive integers n Ú j.


11.5 Binomial Theorem Binomial Coefficient In the expansion of (a + b)n, with n a positive integer, the coefficient of the term whose variable part is a n - k bk is n n! n a b = . The value of a b is called a binomial coefficient. k k k!(n - k)!


Binomial Theorem for Positive Integers If n is a positive integer, the n n expansion of (a + b)n is given by (a + b)n = a a b a n - ibi. i=0 i


ith Term of a Binomial Expansion The ith term of a binomial expansion n b a n - i + 1b i - 1. of (a + b)n is given by a i - 1


11.6 Permutations and Combinations Fundamental Counting Principle The Fundamental Counting Principle lets us determine the total number of ways in which a sequence of events can occur. See page 884.


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CHAPTER 11


Permutation A permutation is an arrangement of distinct objects in a definite order. The formula for the permutation of n distinct objects taken n! r at a time is P(n, r) = . (n - r)!

See Examples 2 and 3, page 885, and then try Exercises 77 and 78, page 904.

Combination A combination is an arrangement of distinct objects for which the order is not important. The formula for the combination of n n! distinct objects taken r at a time is C(n, r) = . r! (n - r)!


11.7 Introduction to Probability Sample Space An activity with an observable outcome is called an experiment. The sample space of an experiment is the set of all possible outcomes of the experiment.


Event An event is a subset of a sample space.


Probability of an Event Let n(S) and n(E) represent the number of elements, respectively, in the sample space S and the event E. Then the n(E) probability of E is P(E) = . n(S)


Addition Rules for Probabilities Two events E1 and E2 are called mutually exclusive if E1 ¨ E2 = . If two events are mutually exclusive, then P(E1 ´ E2) = P(E1) + P(E2). If two events are not mutually exclusive, then P(E1 ´ E2) = P(E1) + P(E2) - P(E1 ¨ E2).


Independent Events Two events are independent when the outcome of the first event has no influence on the outcome of the second event.


Binomial Probability Formula Let an experiment consist of n independent trials for which the probability of success on a single trial is p and the probability of failure is q = 1 - p. Then the probability of k successes in n n trials is given by a b pkq n - k. k


CHAPTER 11 REVIEW EXERCISES In Exercises 1 to 18, find the third and seventh terms of the sequence defined by an. 1. an = 3n + 1

2. an = 2n - 1

3. an = n2

4. an = n3 - 2n

1 5. an = n

n 6. an = n + 1

7. an = 2n 9. an = a b

2 3

8. an = 3n n

10. an = a b

1 2

n

11. a1 = 2, an = 3an - 1

12. a1 = - 1, an = 2an - 1

13. a1 = 1, an = - nan - 1

14. a1 = 2, an = 2nan - 1


15. a1 = 1, a2 = 2, an = an - 1an - 2 16. a1 = 1, a2 = 2, an =

903

In Exercises 37 to 44, find the requested term or sum for the geometric sequence.

an - 1 an - 2


terms are 4, 2, 1.

17. a1 = 1, a2 = 2, an = 2an - 2 - an - 1 18. a1 = 2, a2 = 4, an = 2an - 2 + an - 1


terms are 3, 6, 12. 39. Find the nth term of the geometric sequence whose first three

In Exercises 19 to 26, evaluate the expression. 19. 5! + 3!

terms are 5,

20. 6! - 5!

10! 21. 6!

22. 3! # 4!

12 23. a b 3

15 24. a b 8

15 45 , . 4 16


terms are 9, -6, 4. 41. Find the sum of the first eight terms of the geometric sequence

whose nth term is an = 2n - 1.

42. Find the sum of the first 12 terms of the geometric sequence 5

25. a k 2 k=1

6

1 26. a j! j=1

1 n-1 whose nth term is an = 5a b . 3

43. Find the sum of the infinite geometric series for the sequence

In Exercises 27 to 34, find the requested term or partial sum for the given arithmetic sequence. 27. Find the 25th term of the arithmetic sequence whose first

3 terms are 3, 7, 11. 28. Find the 19th term of the arithmetic sequence whose first

1 n-1 whose nth term is an = a- b . 2

44. Find the sum of the infinite geometric series for the sequence

2 n whose nth term is an = a b . 5

3 terms are 2, 5, 8. In Exercises 45 to 48, evaluate the given series. 29. The 10th term of an arithmetic sequence is 25, and the 1st term

is 2. Find the 15th term. 30. The 8th term of an arithmetic sequence is 33, and the 1st term

is 5. Find the 12th term.

5

45. a 2a b 5 k=1 q

1

k-1

47. a a- b 6 n=1

5

25

46. a (3 - 4k) k=1

n-1

q

48. a 2a b 4 n=1

3

n-1

31. Find the sum of the first 20 terms of the arithmetic sequence

given by an = 3n - 4. 32. Find the sum of the first 50 terms of the arithmetic sequence

given by an = 1 - 4n. 33. Find the sum of the first 100 terms of the arithmetic sequence

whose first 3 terms are 6, 8, 10. 34. Find the sum of the first 75 terms of the arithmetic sequence

whose first 3 terms are 1, 3, 5. In Exercises 35 and 36, insert the arithmetic means.

In Exercises 49 and 50, write each number as the ratio of two integers in simplest form. 49. 0.23

50. 0.145

In Exercises 51 to 58, determine whether the sequence is arithmetic, geometric, or neither. 51. an = n2

52. an = n!

53. an = ( -2)n

54. an = a b

n + 1 2

35. Insert 4 arithmetic means between 13 and 28.

55. an =

36. Insert 5 arithmetic means between 19 and 43.

57. an = n2n

2 3

n

56. an = 1 - 3n 58. an = 2(0.1)n

904

CHAPTER 11


In Exercises 59 to 66, use mathematical induction to prove each statement. n

59. a (5i + 1) = i=1

n(5n + 7) 2

60. a (3 - 4i) = n(1 - 2n) i=1 n

1

i

n

62. a ( -1)i = i=0

displayed on a shelf. How many different arrangements are possible? 79. Scheduling The emergency staff at a hospital consists of

n

61. a a- b = 2 i=0

78. Arranging Books Three of five different books are to be

231 - (-1>2)n + 14 3

4 supervisors and 12 regular employees. How many shifts of 4 people can be formed if each shift must contain exactly 1 supervisor? 80. Committee Membership From 12 people, a committee of

5 people is formed. In how many ways can this be accomplished if there are 2 people among the 12 who refuse to serve together on the committee?

1 - ( -1)n + 1 2

81. Playing Cards How many different four-card hands can be 63. nn Ú n!, n is a positive integer

drawn without replacement from a standard deck of playing cards?

64. n! 7 4n, n Ú 9, n is an integer 82. Playing Cards Two cards are drawn, without replacement, 65. 3 is a factor of n3 + 2n for all positive integers n.

from the four aces of a standard deck of playing cards. List the elements of the sample space.

66. Let a1 = 12 and an = (12 )an - 1. Prove that an 6 2 for all

positive integers n.

83. Number Theory Three numbers are drawn from the digits 1

through 5, inclusive. In Exercises 67 to 72, use the Binomial Theorem to expand each binomial or find the requested term. 67. (4a - b)5

68. (x + 3y)6

69. (a - b)7

70. (3a - 2b)4

71. Find the fifth term in the binomial expansion of (3x - 4y)7. 72. Find the eighth term in the binomial expansion of (1 - 3x)9. 73. Car Options The buyer of a new car is offered 12 exterior col-

ors and 8 interior colors. How many different color selections are possible? 74. Dinner Options A restaurant offers a prix fixe dinner that

includes one of five appetizers, one of six entrees, and one of four desserts. How many different dining options are available for the prix fixe dinner? 75. Computer Passwords A computer password consists of eight

letters. How many different passwords are possible? Assume there is no difference between lowercase and uppercase letters.

a. If the numbers are drawn with replacement, is 431 one of

the elements of the sample space? b. If the numbers are drawn with replacement, is 313 one of

the elements of the sample space? 84. Tossing Coins A coin is tossed five times. List the elements

in the event that there is exactly one tail. 85. Dice Two dice are tossed. List the elements in the event that

the sum of the values on the upward faces is 10. 86. Number Theory Two numbers are drawn, without replacement,

from the digits 1 to 3. What is the probability that the second number drawn is greater than the first number drawn? 87. Number Theory Let S = 5Natural numbers less than or equal

to 1006 and consider the experiment of drawing one number from S. Let E1 be the event that the number is prime, and let E2 be the event that the number is greater than 50. a. Are the events mutually exclusive? b. What is the probability of E1 or E2? 88. Playing Cards A deck of 10 cards contains 5 red and 5 black

76. Serial Numbers The serial number on an airplane consists of

the letter N, followed by six numerals, followed by one letter. How many serial numbers are possible?

cards. If 4 cards are drawn from the deck, what is the probability that 2 are red and 2 are black? 89. Playing Cards Which of the following has the greater

77. Committee Membership From a committee of 15 members,

a president, a vice president, and a treasurer are elected. In how many ways can this be accomplished?

probability: drawing an ace and a ten-card (10, jack, queen, or king) from one standard deck of playing cards, or drawing an ace and a ten-card from two standard decks of playing cards?

CHAPTER 11 TEST

905

90. Sums of Coins A nickel, a dime, and a quarter are tossed.

93. Employee Badges A room contains 12 employees who are

What is the probability that the nickel and dime will show heads and the quarter will show tails? What is the probability that only one of the coins will show tails?

wearing badges numbered 1 to 12. If 3 employees are randomly selected, what is the probability that the person wearing badge 6 will be included?

91. Medicine A company claims that its cold remedy is success-

94. Gordon Model of Stock Valuation Suppose a stock pays a

ful in reducing the symptoms of a cold in 90% of the people who use it. Assuming the company’s claim is true, what is the probability that 8 of 10 people with a cold who take the cold remedy will report it reduced their cold symptoms? Round to the nearest hundredth.

dividend of $1.27 and has a dividend growth rate of 3%. If an investor requires a 12% return on an investment, use the Gordon model of stock valuation to determine the price per share the investor should pay for the stock. Round to the nearest cent.

92. Community Government A survey of members in a city

95. Multiplier Effect Suppose a city estimates that a new sports

council indicates that 75% are in favor of creating a new park. If six members of the city council are interviewed, what is the probability that exactly four of them will be in favor of the new park? Round to the nearest hundredth.

facility will bring in $15 million of additional income. If each person receiving a portion of this money will spend 80% and save 20%, what is the net effect of the $15 million in additional income?

CHAPTER 11 TEST In Exercises 1 and 2, find the third and fifth terms of the sequence defined by an. 1. an =

n

2 n!

In Exercises 14 to 16, use the Binomial Theorem. 14. Write the binomial expansion of (x - 2y)5.

2. a1 = 3, an = 2an - 1 15. Write the binomial expansion of ax +

In Exercises 3 to 5, classify each sequence as arithmetic, geometric, or neither. n-1

3. an = - 2n + 3

4. an = 2n2

5. an =

(- 1) 3n

1 6 b . x

16. Find the sixth term in the binomial expansion of (3x + 2y)8. 17. Playing Cards Three cards are randomly chosen from a stan-

dard deck of playing cards. In how many ways can the cards be chosen? In Exercises 6 to 8, evaluate the given series. 6

10

1

6. a i=1 i

20

1

8. a (3k - 2)

7. a j j=1 2

k=1

9. The third term of an arithmetic sequence is 7 and the eighth term

is 22. Find the twentieth term. q

The first three characters are uppercase letters of the alphabet. The next two characters are selected from the digits 1 through 9. The last two characters are uppercase letters of the alphabet. How many serial numbers are possible if no letter or number can be used twice in the same serial number?

k

10. Evaluate the infinite geometric series a a b . k=1 8

3

18. Serial Numbers A serial number consists of seven characters.

11. Write 0.15 as the ratio of two integers in simplest form.

19. Playing Cards Five cards are randomly selected from a deck of

cards containing 8 black cards and 10 red cards. What is the probability that 3 black cards and 2 red cards are selected? 20. Botany A company that sells roses claims that 95% of all its

In Exercises 12 and 13, use mathematical induction to prove the statement. n

12. a (2 - 3i) = i=1

13. n! 7 3n,

n(1 - 3n) 2

n Ú 7

rose plants will survive at least 1 year. If a gardener purchases eight rose plants from this company, what is the probability that seven will survive at least 1 year? Round to the nearest hundredth.

906

CHAPTER 11


CUMULATIVE REVIEW EXERCISES 1 Find the linear regression equation for the set 5(1, 5), (3, 8),

(4, 11), (6, 15), (8, 16)6. Round constants to the nearest tenth. 2. Find the value of x in the domain of F(x) = 5 +

x for which 3

F(x) = - 3. 3. Solve: 2x 2 - 3x = 4 4. Write logb a

xy2 3

z

12. Solve: e

x 2 + y 2 + xy = 10 x - y = 1

3 13. Find the product of C - 2 1

b in terms of the logarithms of x, y, and z.

1 R. -3

175 v ln a1 b . Determine the velocity of the sky 32 175 diver after 5 seconds. Round to the nearest foot per second. t = -

15. Given sin u = -

2x - 3y = 8 x + 4y = - 7 -1

7. Given A = C 5

0

2 7 3 S and B = C 6 3 1

-3 5 S , find 3A - 2B. -2

8. Let g(x) = x 2 - x + 4 and h(x) = x - 2. Find a b(- 3).

h g

9. Find the horizontal asymptote of the graph of

F(x)

x3 - 8 x5

.

10. Evaluate: log 12 64 11. Solve 4

1 4

sky diver to reach a velocity of v feet per second is given by

16x 2 + 25y 2 - 96x + 100y - 156 = 0

2x + 1

3 0

14. Velocity of a Sky Diver The time t in seconds required for a

5. Find the eccentricity of the graph of

6. Solve: e

2 2 1S B -2 -4

x-2

= 3

16. Express

1 223 , find cot u. and sec u = 2 3

1 + cos x sin x + in terms of csc x. 1 + cos x sin x

the triangle ABC if A = 40°, B = 65°, and c = 20 centimeters.

17. Solve

18. Find the angle of rotation a that eliminates the xy term

of 9x2 + 4xy + 6y2 + 12x + 36y + 44 = 0. Round to the nearest degree. 19. Evaluate: sin a

4 1 cos-1 b 2 5

20. Given v = 2i + 5j and w = 3i - 6j, find 2v - 3w.

. Round to the nearest tenth.

SOLUTIONS TO THE TRY EXERCISES Exercise Set P.1, page 14 2. a. Integers: 31, 51

Exercise Set P.2, page 29 10.

b. Rational numbers:

5 1 , 31, -2 , 4.235653907493, 51, 7 2

0.888... c. Irrational number:

36.

5 17

d. Prime number: 31 e. Real numbers: all the numbers are real numbers.

52.

4-2 2-3

=

23

=

42

( -3a2b3)2

8 1 = 16 2

#

=

#

#

(- 3)1 2a2 2b 3 2

# # # ( -2ab4)3 ( -2)1 3a1 3b 4 3 9a4b6 9a = = - 6 3 12 - 8a b 8b

(6.9 * 1027)(8.2 * 10-13) 4.1 * 10

15

6. In absolute value, the four smallest integers are 0, 1, 2, and 3.

=

Replacing x in x2 - 1 by these values, we obtain - 1, 0, 3, and 8.

16. A ¨ B = 5-2, 0, 26 and A ¨ C = 50, 1, 2, 36. Therefore,

(A ¨ B) ´ (A ¨ C) = 5-2, 0, 1, 2, 36.

40. The interval ( - q, 34 includes all real numbers less than or

equal to 3. The interval (2, 6) includes all real numbers between 2 and 6, not including 2 and not including 6. Therefore, (- q, 34 ¨ (2, 6) = (2, 34. The graph is −5 −4 −3 −2 −1 0 1 2 3 4 5

50. 5x ƒ -3 … x 6 06 ´ 5x ƒ x Ú 26 is the set of all real numbers

62. a

68. ( -5x1>3)( -4x1>2) = ( -5)( -4)x1>3 + 1>2

= 20x 2>6 + 3>6 = 20x 5>6 84. 218x2y5 = 29x2y4 12y = 3 ƒ x ƒ y2 12y

92. - 3x 254x4 + 2216x7 = - 3x233 # 2x4 + 2 224x7 3

3

3

= = = =

= 4(- 16) = - 64

90. (z - 2y) - 3z 2

3

= 3( -1) - 2( -2)42 - 3( -1)3

= 3 -1 + 442 - 3( -1) = 32 + 3 = 9 + 3 = 12

102. Commutative property of addition 106. Substitution property of equality 120. 6 + 332x - 4(3x - 2)4 = 6 + 332x - 12x + 84

= 6 + 33-10x + 84 = 6 - 30x + 24 = - 30x + 30

3

3 3 3

3 3 - 3x 23 x 12x + 2 223x6 12x 3 2 3 - 3x(3x 12x) + 2(2x 12x) 3 3 - 9x 2 1 2x + 4x2 1 2x 2 3 - 5x 12x

= 9 # 5y - 121 5y - 121 5y + 16 = 45y - 241 5y + 16

70. d(z, 5) = ƒ z - 5 ƒ ; therefore, ƒ z - 5 ƒ 7 7.

80. (3 - 5)2(32 - 5 2) = (- 2)2(9 - 25)

3

102. (3 15y - 4)2 = (3 15y - 4)(3 15y - 4)

ƒ x + 6 ƒ = x + 6 and ƒ x - 2 ƒ = - (x - 2). Thus, ƒ x + 6 ƒ + ƒ x - 2 ƒ = x + 6 - (x - 2) = 8.

- (6 # 6 # 6) -216 8 = = 76. = 4 (3)(3)( 3)(3) 81 3 ( -3)

4.1 * 1015 56.58 * 1014

16 3>2 16 1>2 3 4 3 64 b = ca b d = a b = 25 25 5 125

−5 −4 −3 −2 −1 0 1 2 3 4 5

-6 3

(6.9)(8.2) * 1027 - 13

4.1 * 1015 = 13.8 * 10-1 = 1.38 * 100

between - 3 and 0, including -3 but excluding 0, together with (union) all real numbers greater than or equal to 2. The graph is

60. When 0 6 x 6 2, x + 6 7 0 and x - 2 6 0. Therefore,

=

112.

116.

2 4

14y

=

2 4

4

# 24 4y

3

14y 24y 3

=

4 22 4y 3

2y

=

4 2 4y 3

y

-7 -7 # 312 + 5 = 312 - 5 312 - 5 312 + 5 - 211 2 - 35 = 18 - 25 -211 2 - 35 = = 31 2 + 5 -7

Exercise Set P.3, page 37 12. a. -12x4 - 3x2 - 11 c. - 12, - 3, - 11

b. 4 d. - 12

e. - 12x , - 3x , - 11 4

2

S1

S2

SOLUTIONS TO THE TRY EXERCISES

24. (5y2 - 7y + 3) + (2y2 + 8y + 1) = 7y2 + y + 4

3x + 4x 3

38.

x + 7

2

2.

3x - 2 - 6x3 - 8x2 + 2x - 14

16.

9x + 12x - 3x + 21x 4

3

Exercise Set P.5, page 57

2

9x + 6x - 11x + 23x - 14 4

3

2

50. (3a - 5b)(4a - 7b) = 12a2 - 21ab - 20ab + 35b2

= 12a2 - 41ab + 35b2 56. (4x 2 - 3y)(4x 2 + 3y) = (4x 2)2 - (3y)2 = 16x 4 - 9y 2

22.

72. - x 2 - 5x + 4

- (-5)2 - 5(- 5) + 4 = - 25 + 25 + 4 = 4 82.

• Replace x with -5. • Simplify.

2x 2 - 5x - 12 2x + 5x + 3 2

x 2 - 16

#x

- 4x - 21

,

3x2 - xy - 2y2

2y - 5

(3y - 1)( y - 3)

3y2 2y2 - y) 2x + 3y = - y) 2x - 3y (2y - 5)(3y + 1)

(3y 2 - 10y + 3) - (6y 2 - 13y - 5) (3y + 1)( y - 3) - 3y 2 + 3y + 8 = (3y + 1)( y - 3) =

-4

b. 4.3 * 10 (5000) - 2.1 * 10 (5000)

= 106.45 seconds c. 4.3 * 10-6(10,000)2 - 2.1 * 10-4(10,000)

= 427.9 seconds


34.

2 3 # y2 - 1 2 3 # (y + 1)(y - 1) = y y + 1 y + 4 y y + 1 y + 4 3(y + 1)(y - 1) 2 = y (y + 1)(y + 4) 3(y - 1) 2 = y y + 4 3(y - 1) y 2#y + 4 # = y y + 4 y + 4 y (2y + 8) - (3y2 - 3y) y(y + 4) 2 - 3y + 5y + 8 = y(y + 4)

6. 6a3b 2 - 12a2b + 72ab 3 = 6ab(a2b - 2a + 12b 2)

=

12. b 2 + 12b - 28 = (b + 14)(b - 2) 18. 57y 2 + y - 6 = (19y - 6)(3y + 1) 24. b 2 - 4ac = 82 - 4(16)(- 35) = 2304 = 482

The trinomial is factorable over the integers. 40. y4 - 1 = (y2)2 - 12

= (y2 - 1)(y2 + 1) = (y - 1)(y + 1)(y2 + 1) 46. b 2 - 24b + 144 = (b - 12)2 52. b 3 + 64 = (b + 4)(b 2 - 4b + 16) 64. z4 + 3z2 - 4 = (z2 + 4)(z2 - 1)

= (z2 + 4)(z + 1)(z - 1) 70. a2y 2 - ay 3 + ac - cy = ay 2(a - y) + c(a - y)

= (a - y)(ay + c) 2

76. 81y - 16 = (9y - 4)(9y + 4) 2

x - 4 x + 1

30. 3y + 1 - y - 3 = (3y + 1)( y - 3) - ( y - 3)(3y + 1)

= 4.09 seconds

4

=

4x2 - 9y2 2x2 + xy - 3y2 2 6x + 13xy + 6y2 2x2 + xy # 2 = 4x2 - 9y2 3x - xy (3x + 2y)(2x + 3y)(2x + 3y)(x = (2x - 3y)(2x + 3y)(3x + 2y)(x

84. a. 4.3 * 10-6(1000)2 - 2.1 * 10-4(1000) 2

2

6x2 + 13xy + 6y2

It is possible to form 1330 different committees.

-6

(2x + 3)(x + 1)

x + 7x + 12 x 2 - 4x (x - 4)(x + 4)(x + 3)(x - 7) x - 7 = = (x + 3)(x + 4)x(x - 4) x 2

3y - 1

1 1 1 3 n - n2 + n 6 2 3 1 1 1 (21)3 - (21)2 + (21) = 1330 6 2 3

(2x + 3)(x - 4)

=

3y - 2 3y - 2 2 2 y y - 1 y y - 1 y(y - 1) # = 54. y y y(y - 1) y - 1 y - 1 3y - 2 2# # y(y - 1) y(y - 1) y y - 1 = y # y(y - 1) y - 1 2(y - 1) - y(3y - 2) = y2 =

2

= (3y - 2)(3y + 2)(9y 2 + 4)

2y - 2 - 3y 2 + 2y y2 - 3y + 4y - 2 2

=

y2


8. 6 - 2(4x + 1) = 3x - 2(2x + 5)

1 1 f ef e-2 - f -1 f - e2 e2 , = = 60. 2 ef 1 ef e f = 64. a.

b.

6 - 8x - 8x - 8x + x -7x - 7x + 4

f - e2 1 f - e2 # = ef e 2f e 3f 2

v1 + v2 1.2 * 108 + 2.4 * 108 = L 3.4 * 108 mph 8 8 v1v2 )(2.4 * 10 ) (1.2 * 10 1+ 2 1+ c (6.7 * 108)2 c 2(v1 + v2) c 2(v1 + v2) v1 + v2 = = 2 v1v2 c + v1v2 v1v2 1 + 2 c 2 a1 + 2 b c c

14.

18. (5 - 3i) - (2 + 9i) = 5 - 3i - 2 - 9i = 3 - 12i

(5 + 8i)(1 5 - 25i + 5 - 25i + 5 - 25i + 45 - 17i

8 - i 8 - i # 2 - 3i = 48. 2 + 3i 2 + 3i 2 - 3i 16 - 24i - 2i + 3i 2 = 22 + 32 16 - 24i - 2i + 3( -1) = 4 + 9 16 - 26i - 3 = 13 13(1 - 2i) 13 - 26i = = 13 13 = 1 - 2i 60.

1 i

83

=

1 i

3

=

1 #i 1 i i = = = i = 2 -i -i i 1 -i

= = = = = =

3x - 4x - 10 - x - 10 - x + x - 10 - 10 - 10 - 4 - 14 -14 = -7 = 2

2x + 28 - x = 38 x = 38 - 28 x = 10

8. 6 - 1 -1 = 6 - i 34. (5 + 2 1 -16) (1 - 1 -25) = 35 + 2(4i)4(1 - 5i)

- 2 + 4 + 4 + 4 - 4 - 7x -7x -7 x

1 19 1 x + 7 - x = 2 4 2 1 1 19 4a x + 7 - xb = 4 a b 2 4 2


= = = = =

S3

1 6x + 1 = 3 3 6x + 1 6x + 1 = 3 3

• Multiply each side by 4.

• Collect like terms.

24. 2x +

- 5i) 8i - 40i 2 8i - 40(-1) 8i + 40

• Rewrite the left side.

The left side of this equation is now identical to the right side. Thus the original equation is an identity. ƒ 2x - 3 ƒ = 21

38.

2x - 3 = 21 2x = 24 x = 12

or

2x - 3 = - 21 2x = - 18 x = -9

The solutions of ƒ 2x - 3 ƒ = 21 are - 9 and 12. 50. Replace LBM by 55 and H by 175 in the equation

LBM = 0.3281W + 0.3393H - 29.5336, and then solve for W. LBM 55 55 25.1561 77

= = = = L

0.3281W + 0.3393H - 29.5336 0.3281W + 0.3393(175) - 29.5336 0.3281W + 29.8439 0.3281W W

The person should weigh approximately 77 kilograms. 52. Substitute 22 for m in the given equation and solve for s.

Exercise Set 1.1, page 81 4.

4x 4x - 7x -3x - 3x - 11

- 11 - 11 - 11 + 11 -3x - 3x -3

= = = = =

7x + 20 7x - 7x + 20 20 20 + 11 31 31 = -3 31 x = 3

1 ƒ s - 55 ƒ + 25 2 -44 = ƒ s - 55 ƒ - 50 22 = -

6 = ƒ s - 55 ƒ s - 55 = 6 or s = 61

• Substitute 22 for m.

• Multiply each side by -2 to clear the equation of fractions. • Add 50 to each side. s - 55 = - 6 s = 49

Kate should drive her car at either 61 miles per hour or 49 miles per hour to obtain a gas mileage of 22 miles per gallon.

S4


Exercise Set 1.2, page 92 4. A = P + Prt

A = P(1 + rt) A P = (1 + rt)

• Factor.

32. Let x be the amount of money invested at 5%. Then 7500 - x

is the amount of money invested at 7%. 0.05x + 0.07(7500 - x) = 405 0.05 x 7500 - x

0.07

• Solve for P.

14. Substitute 105 for w.

SMOG reading grade level = 1105 + 3 L 10.2 + 3 = 13.2 According to the SMOG formula, the estimated reading grade level required to fully understand A Tale of Two Cities is 13.2. (Note: A different sample of 30 sentences likely would produce a different result. It is for this reason that reading grade levels are often estimated by using several different samples and then computing an average of the results.) 18.

P = 2l + 2w,

1 w = l + 1 2

1 110 = 2l + 2a l + 1b • Substitute for P and w. 2 110 = 2l + l + 2 • Simplify. 108 = 3l 36 = l l = 36 meters 1 1 w = l + 1 = (36) + 1 = 19 meters 2 2 24. Let x be the length of the person’s shadow. Using similar

triangles,

x x + 10 . = 6 25

Solve the equation. x x + 10 = 6 25 x + 10 x b 150 a b = 150a 6 25 25x = 6x + 60 19x = 60 x L 3.2 The person’s shadow is approximately 3.2 feet long. 26. Let x be the number of glasses of orange juice.

Profit = revenue - cost 2337 = 0.75x - 0.18x 2337 = 0.57x 2337 = 4100 x = 0.57 The owner must sell 4100 glasses of orange juice.

0.05x + 525 - 0.07x - 0.02x x 7500 - x

= = = =

405 - 120 6000 1500

$6000 was invested at 5%. $1500 was invested at 7%. 36. Let t1 be the time it takes to travel to the island.

Let t2 be the time it takes to make the return trip. t1 + t2 = 7.5 t2 = 7.5 - t1 The distance traveled to the island is the same as the distance traveled on the return trip. 15t1 15t1 15t1 25t1 t1 d

= = = = = =

10t2 • Substitute for t2. 10(7.5 - t1 ) 75 - 10t1 75 3 hours 15t1 = 15(3) = 45 nautical miles

44. Let x be the number of liters of the 40% solution to be mixed

with the 24% solution. 0.40x + 0.24(4) = 0.30(4 + x) 0.40 x 0.24 0.30

4 4+x

0.40x + 0.96 = 1.2 + 0.30x 0.10x = 0.24 x = 2.4

Thus 2.4 liters of 40% sulfuric acid solution should be mixed with 4 liters of 24% sulfuric acid solution to produce the 30% solution. 54. Let x be the amount of the alloy that costs $6.50 per ounce.

Then 20 - x is the amount of the alloy that costs $8.00 per ounce. The alloys are mixed to form 20 ounces of a new alloy that costs $7.40 per ounce. a

value of Value of b + a b $6.50 alloy $8.00 alloy 6.50x + 8.00(20 - x) 6.50x + 160 - 8x -1.5x + 160 - 1.5x x

= a = = = = =

value of b new alloy 7.40(20) 148 148 - 12 8

The jeweler should use 8 ounces of the alloy that costs $6.50 per ounce and 12 ounces of the alloy that costs $8.00 per ounce.


56. Let x be the number of hours needed to print the report if

both the printers are used. 1 Printer A prints of the report every hour. 3 1 Printer B prints of the report every hour. 4 Thus 1 1 x + x = 1 3 4 4x + 3x = 12 # 1 7x = 12 12 x = 7

46. x 2 - 6x + 10 = 0

x 2 - 6x = - 10 x 2 - 6x + 9 = - 10 + 9 (x - 3)2 = - 1 x - 3 = 1- 1

The solutions are 3 - i and 3 + i. 58.

2x 2 x = =

Exercise Set 1.3, page 106

=

6. 12x 2 - 41x + 24 = 0

(4x - 3)(3x - 8) = 0

x =

3x - 8 = 0

3 4

x =

The solutions are

• Factor. • Apply the zero product principle.

8 3

8 3 and . 4 3

= = = = =

42. 2x 2 + 10x - 3 = 0

The solutions are

- 2 - 16 -2 + 16 and . 2 2

72. x 2 + 3x - 11 = 0

b2 - 4ac = 32 - 4(1)(- 11) = 9 + 44 = 53 7 0 Thus the equation has two distinct real solutions.

x2 x2 x x

The solutions are -2 - 2i 17 and -2 + 2i 17.

x = -

The solutions are

The legs of this right triangle each measure 90 feet.The distance from home plate to second base is the length of the hypotenuse x of this right triangle.

- 28 1- 28 i 128 2i 17 - 2 2i 17

2x 2 + 10x = 3 2(x 2 + 5x) = 3 3 x 2 + 5x = 2 3 25 25 = + x2 + 5x + 4 2 4 2 5 31 ax + b = 2 4 31 5 = x + 2 A4

2x2 + 4x = 1 + 4x - 1 = 0 -4 242 - 4(2)(- 1) 2(2) -4 124 - 4 116 + 8 = 4 4 -4 216 - 2 16 = 4 2

82. Home plate, first base, and second base form a right triangle.

28. (x + 2)2 + 28 = 0

(x + 2)2 x + 2 x + 2 x + 2 x

2 1 • Add c ( -6) d to each side. 2 • Factor the left side. • Use the square root procedure.

x - 3 = i x = 3i

5 It would take 1 hours to print the report. 7

4x - 3 = 0

S5

= = = L

902 + 902 16,200 116,200 127.3

To the nearest tenth of a foot, the distance from home plate to second base is 127.3 feet. (Note: We have not considered - 116,200 as a solution because we know that the distance must be positive.) 94. Let w be the width, in inches, of the new candy bar. Then the

1 • Multiply each side by . 2 • Complete the square.

5 131 2 2

- 5 - 131 -5 + 131 and . 2 2

length, in inches, of the new candy bar is 2.5w, and the height is 0.5 inch. The volume of the original candy bar is 5 # 2 # 0.5 = 5 cubic inches. Thus the volume of the new candy bar is 0.80(5) = 4 cubic inches. Substitute in the formula for the volume of a rectangular solid to produce lwh (2.5w)(w)(0.5) 1.25w 2 w2 w

= = = = = L

V 4 4 3.2 13.2 1.8 (continued)

S6


The width of the new candy bar should be about 1.8 inches and the length should be about 2.5(1.8) = 4.5 inches. Disregard the negative solution because the width must be positive. 96. When the ball hits the ground, h = 0. Thus we need to solve

0 = - 16t 2 + 52t + 4.5 for t.

0 = - 16t 2 + 52t + 4.5 - (52) 2(52)2 - 4( -16)(4.5) t = 2(-16) -52 12992 = -32 L 3.3

- 41x + 7 = - 20 (1x + 7)2 = (5)2 x + 7 = 25 x = 18 ⱨ Check: 118 + 7 - 2 118 - 9 125 - 2 ⱨ 19 5 - 2ⱨ3 3 = 3 The solution is 18. 50. 4x 4>5 - 27 = 37

The ball will hit the ground in about 3.3 seconds. Disregard the negative solution because the time must be positive.

4x3 + 5x2 - 16x - 20 x (4x + 5) - 4(4x + 5) (4x + 5)(x2 - 4) (4x + 5)(x + 2)(x - 2) 4x + 5 = 0 x + 5 x = 4 2

= = = = 2

0 0 0 0 = 0

x = -2

x = 32 The solutions are 32 and - 32. x - 2 = 0 x = 2

5 The solutions are -2, - , and 2. 4 24.

x - 4 10x + 13 2x + 1 = x - 3 x + 5 x + 5 x - 4 2x + 1 b(x - 3)(x + 5) a x - 3 x + 5 10x + 13 = ab(x - 3)(x + 5) x + 5 (2x + 1)(x + 5) - (x - 4)(x - 3) = -(10x + 13)(x - 3) 2x2 + 11x + 5 - (x2 - 7x + 12) = -(10x2 - 17x - 39) x2 + 18x - 7 = - 10x2 + 17x + 39 11x2 + x - 46 = 0 (11x + 23)(x - 2) = 0 11x + 23 = 0 x - 2 = 0 23 x = x = 2 11 The solutions are -

28.

110 - x 10 - x -x x

= = = =

23 and 2. 11

4 Check: 110 - (-6) ⱨ 4 116 ⱨ 4 16 6 4 = 4 -6

The solution is -6. 30.

1x + 7 - 2 = 1x - 9 (1x + 7 - 2)2 = (1x - 9)2 x + 7 - 41x + 7 + 4 = x - 9

• Divide each side by 4. • Raise each side of the equation to the 5兾4 (the reciprocal of 4兾5) power. • Because the numerator of the exponent in x 4兾5 is an even number, use absolute value.

ƒ x ƒ = 32


• Add 27 to each side.

4x4>5 = 64 x4>5 = 16 4>5 5>4 (x ) = 165>4

54. x 4 - 10x 2 + 9 = 0

• Let u = x 2 .

u - 10u + 9 = 0 (u - 9)(u - 1) = 0 u - 9 = 0 u - 1 = 0 u = 9 u = 1 2 x2 = 1 x = 9 x = 3 x = 1 2

The solutions are 3, - 3, 1, and - 1. 62. 6x 2>3 - 7x1>3 - 20 = 0

• Let u = x1>3.

6u - 7u - 20 = 0 (3u + 4)(2u - 5) = 0 2

3u + 4 = 0

2u - 5 = 0 5 u = 2 5 x1>3 = 2

4 u = 3 4 x1>3 = 3

5 3 (x1>3)3 = a b 2

4 3 b 3 64 x = 27

(x1>3)3 = a-

The solutions are -

x =

125 8

64 125 . and 27 8

70. Let r be Maureen’s running rate. Then Hector’s running rate

is r - 2. 1 = time for Hector 2 1 12 12 + = r 2 r - 2 12 1 12 2r(r - 2)a + b = 2r(r - 2)a b r 2 r - 2

Time for Maureen +

(continued)


2(r - 2)(12) + r(r - 2) = 2r(12) 24r - 48 + r 2 - 2r = 24r r 2 - 2r - 48 = 0 (r - 8)(r + 6) = 0 r - 8 = 0 r + 6 = 0 r = 8 r = -6

0.04(1 + (6 - x)2) = 1 - 0.25x + 0.015625x 2 1.48 - 0.48x + 0.04x2 = 1 - 0.25x + 0.015625x 2 0.024375x 2 - 0.23x + 0.48 = 0 Using the quadratic formula, x L 3.1 and x L 6.3. The value 6.3 would make 6 - x a negative number and therefore is not a possible answer. The answer is to run the line above ground for 3.1 miles from the power station.

A running rate of -6 miles per hour is not possible. Maureen’s running rate is 8 miles per hour. Hector’s running rate is 6 miles per hour. 72. First, find the time t it would take the roofer working alone to

repair the roof. 6 6 + t 14 6 6 14ta + b t 14 84 + 6t 84 21 2

S7

Exercise Set 1.5, page 133 6. -4(x - 5) Ú 2x + 15

- 4x + 20 Ú 2x + 15 - 6x Ú -5 5 x … 6

= 1 = 14t # 1 = 14t = 8t

The solution set is e x ƒ x …

= t

5 6

21 hours to repair the roof. 2 Now let x be the additional time it would take the assistant to repair the roof after the roofer and the assistant work together for 2 hours.

− 5 − 4 −3 −2 −1 0 1 2 3 4 5

It takes the roofer, working alone,

portion done by Portion done by b + a b + a roofer in 2 hr assistant in 2 hr portion done by assistant a b after roofer leaves 2 x 2 + + 21 14 14 2 4 2 x + + 21 14 14 4 2 x 42a + + b 21 14 14 8 + 6 + 3x 3x

2x 6 4

21 2

and

x 6 2

x 7 -

21 21 f ¨ 5x ƒ x 6 26 = e x ` 6 x 6 2f 2 2

The solution set is e x ` -

21 6 x 6 2 f. 2

− 21 2 −12−10 −8−6 −4 −2 0 2 4 6 8

= 1 = 42 # 1

84. Cost over land = $0.125x million; cost under water =

0.125x + 0.221 + (6 - x)2 = 1 0.221 + (6 - x)2 = 1 - 0.125x (0.221 + (6 - x)2 )2 = (1 - 0.125x)2

2x + 5 6 9

and

= 1

repair.

0.125x + 0.2 21 + (6 - x)2 = 1

and

2x 7 - 21

=1

28 1 , or 9 , hours to complete the 3 3

$0.221 + (6 - x)2 million; total cost = $1 million.

10. 2x + 5 7 - 16

ex ƒ x 7 -

= 42 = 28 28 x = 3

It would take the assistant

5 f. 6

18. ƒ 2x - 9 ƒ 6 7

- 7 6 2x - 9 6 7 2 6 2x 6 16 1 6 x 6 8 In interval notation, the solution set is (1, 8). 34.

x2 + 5x + 6 6 0 (x + 2)(x + 3) 6 0 x = - 2 and

x = -3

• Critical values

Use a test number from each of the intervals (- q , -3), (- 3, - 2), and (- 2, q ) to determine where x 2 + 5x + 6 is negative. +++++++ − +++++ −3 −2

0

In interval notation, the solution set is ( -3, -2).

S8

50.


3x + 1 x - 2 3x + 1 - 4 x - 2 3x + 1 - 4(x - 2) x - 2 -x + 9 x - 2 x = 2 and

26.

Ú 4 Ú 0 Ú 0 Ú 0

x = 9

• Critical values

30. The general variation is

f =

Company A: 29 + 0.12m

Company B: 22 + 0.21m

29 + 0.12m 6 22 + 0.21m 77.7 6 m Company A is less expensive if you drive at least 78 miles per day. 58. Substitute 1.25 for A. Then solve the following inequality.

1.25 6 - 0.05x + 1.73 - 0.48 6 - 0.05x 9.6 7 x

32.

Adding 9.6 to 2000 yields 2009.68. According to the given mathematical model, we should first expect annual theater attendance to decline to less than 1.25 billion people in 2009. 163 - m ` 6 2.33 62. ` 1.79 163 - m 6 2.33 1.79 -4.1707 6 163 - m 6 4.1707 - 167.1707 6 - m 6 - 158.8293 - 2.33 6

34.

d2 k

L = kwd 2 200 = k # 2 # 6 2 25 200 = k = 2 # 9 2 6 Thus L =

38.

3 # 100 = 7.5 inches. 40

k

102 5000 = k 5000 5000 5000 L 22.2 footcandles. Thus I = = = 2 2 225 d 15

167.1707 7 m 7 158.8293

d = kw 6 = k # 80 6 = k 80 3 k = 40

I = 50 =

The last inequality can be written as 158.8293 6 m 6 167.1707. The range of mean weights of women is between 158.8 and 167.2 pounds.


k l

where f is the frequency, in vibrations per second, of the vibrating string and l is the length of the string in inches. We are given that f = 144 when l = 20 inches. k Solving 144 = for k yields k = 2880. Thus the specific 20 2880 variation is f = . When l = 18 inches, we find l 2880 f = = 160. The frequency of a guitar string with a 18 length of 18 inches is 160 vibrations per second. (Note: We have assumed that the tension is the same for both strings.)

56. Let m = the number of miles driven.

Thus d =

7 # 2 65 L 164.3 feet. 180

Thus r =

Use a test number from each of the intervals (- q , 2), -x + 9 (2, 9), and (9, q ) to determine where is positive. x - 2 The solution set is (2, 94.

22.

r = kv2 140 = k # 602 140 = k 602 7 = k 180

25 # # 2 1600 L 178 pounds. 4 4 = 9 9

L = k#

wd 2 l 4 # 82 800 = k # 12 12 # 800 = k 4 # 82 37.5 = k Thus L = 37.5 #

3.5 # 62 = 295.3125 L 295 pounds. 16

S9



f. g(c + 5) = 2(c + 5)2 + 3 = 2c 2 + 20c + 50 + 3

= 2c 2 + 20c + 53

6. d = 2(x 2 - x1)2 + (y2 - y1)2

d = 2[ - 10 - ( -5)] 2 + (14 - 8)2 = 2( -5)2 + 6 2 = 125 + 36 = 161 y

26.

y

30.

3

2

domain of f is the set of all real numbers.

−2

44. Because 15 - x is not a real number when x 7 5, the

domain of h is 5x ƒ x … 56.

−3

32.

Because 0 … 0 … 5, Q(0) = 4. Because 6 6 e 6 7, Q(e) = - e + 9. Because 1 6 n 6 2, Q(n) = 4. Because 1 6 m … 2, 8 6 m2 + 7 … 11. Thus Q(m2 + 7) = 2(m2 + 7) - 7 = 2m2 = m.

28. - 2x + 1 is a real number for all values of x. Therefore, the

x

−6

3 x

−3

26. a. b. c. d.

y

40. y-intercept: a 0, -

5

x-intercept: (5, 0)

15 b 4

x

-4

1

4

5

3

2

1

0

y = f (x) = 15 - x

y

y

4 −3

3 x

−4

8 x

4

6 4

64. r = 2(1 - (-2))2 + (7 - 5)2

2

= 19 + 4 = 113 Using the standard form

− 10

(x - h) + ( y - k) = r 2

2

−8

−6

−4

−2

2

with h = - 2, k = 5, and r = 113 yields

2

4

6

8

x

−2 −4

(x + 2) + (y - 5) = ( 113) 2

66.

2

x 2 + y 2 - 6x - 4y + 12 x 2 - 6x + y 2 - 4y 2 x - 6x + 9 + y 2 - 4y + 4 (x - 3)2 + ( y - 2)2

= = = =

2

0 - 12 - 12 + 9 + 4 12

54.

The center is (3, 2), and the radius is 1.

• Solve for y.

- 2y = - x + 2 1 y = x2 - 1 2 2

y is a function of x because each x value will yield only one y value. 18. Given g(x) = 2x 2 + 3 a. g(3) = 2(3)2 + 3 = 18 + 3 = 21 b. g( -1) = 2( -1)2 + 3 = 2 + 3 = 5 c. g(0) = 2(0)2 + 3 = 0 + 3 = 3 2

d. ga b = 2 a b + 3 =

1 7 1 1 + 3 = 2 2 2 2 e. g(c) = 2(c)2 + 3 = 2c 2 + 3

-2 -2 0 0 a - 7 a

• Replace f (a) with a 2 - 5a - 16. • Solve for a. = 0 = 7

There are two values of a, -2 and 7, for which f (a) = - 2.

Exercise Set 2.2, page 180 4. x 2 - 2y = 2

f (a) = a2 - 5a - 16 = a2 - 5a - 14 = (a + 2)(a - 7) = a + 2 = 0 or a = - 2 or

66.

f (x) = 0 2x 2 + 3x - 5 = 0 (2x + 5)(x - 1) = 0 2x + 5 = 0 or x 5 x = or 2 5 The zeros are - and 1. 2

• To find the zeros, let f (x) = 0. • Replace f (x) with 2x 2 + 3x - 5. • Solve for x. 1 = 0 x = 1

74. a. [0, q ) b. 31,250 is in the interval 8025 … x 6 32,550.

Use T(x) = 0.15(x - 8025) + 802.50. Then

T(23,225) = 0.15(23,225) + 802.50 = 4286.25 The tax is $4286.25. (continued)

S10


c. 78,900 is in the interval 78,850 … x 6 164,550.

Use T(x) = 0.28(x - 78,850) + 16,056.25. Then T(50) = 0.28(50) + 16,056.25 = 16,070.25 The tax is $16,070.25.

76. a. This is the graph of a function. Every vertical line

intersects the graph in at most one point. b. This is not the graph of a function. Some vertical lines

intersect the graph at two points. x = - 2 intersects the graph at more than one point. d. This is the graph of a function. Every vertical line intersects the graph at exactly one point. 92. v(t) = 44,000 - 4200t, 0 … t … 8

b. Domain: 5x ƒ 0 6 x 6 156

98. AB = 21 + x 2. The time required to swim from A to

21 + x 2 hours. 2 BC = 3 - x. The time required to run from B to C at 3 - x hours. 8 miles per hour is 8 Thus the total time to reach point C is 3 - x 21 + x 2 + t = hours. 2 8 B at 2 miles per hour is

Exercise Set 2.3, page 195 1 - 4 3 = 5 - ( -2) 7 y

3 −3

y2 - y1 x2 - x1 - 10 -4 - 6 = = -5 = - 3 - ( -5) 2

m =

y - y1 y - 6 y - 6 y

= = = =

m(x - x1) - 5(x - ( - 5)) - 5x - 25 - 5x - 19

52. Solve x + 3y = 4 for y.

= (900 - 120x + 4x )x = 900x - 120x 2 + 4x 3 2

The slope is -1 and the y-intercept is (0, 1).

46. Find the slope of the line.

The equation of the line is y = - 5x - 19.

94. a. V(x) = (30 - 2x)2x

16. m = - 1 and b = 1

y = - 2x + 5

Use the point–slope formula.

c. This is not the graph of a function. The vertical line at

2. m =

36. y - 5 = - 2(x - 0)

3

x

−3

32. Solve 3x - 4y = 8 for y.

x + 3y = 4 3y = - x + 4 1 4 y = - x + 3 3 1 The slope of the line parallel to the graph of the given line is - . 3 Use the point–slope formula with the given point (-3, -1). y - y1 = m(x - x1) 1 y - ( -1) = - (x - ( -3)) 3 1 y + 1 = - x - 1 3 1 y = - x - 2 3 The equation of the line that is parallel to the graph of x + 3y = 4 and passes through the point (-3, -1) is 1 y = - x - 2. 3 70. a. Using the data for 2003 and 2008, two ordered pairs on

the line are (2003, 791.9) and (2008, 975.3).

3x - 4y = 8 - 4y = - 3x + 8 3 y = x - 2 4 3 The slope is ; the y-intercept is (0, -2). 4 y 6

4 2

(4, 1) −6 −4 −2 2 4 x −2 (0, − 2) −4 −6

m =

975.3 - 791.9 = 36.68 2008 - 2003

Use the point–slope formula to find the equation of the line between the given points. C - C1 C - 791.9 C - 791.9 C

= = = =

m(t - t1) 36.68(t - 2003) 36.68t - 73,470.04 36.68t - 72,678.14

Using function notation, the linear function is C(t) = 36.68t - 72,678.14.


70. b. To find the year when consumer debt first exceeds $1.1

k = f a-

trillion ($1100 billion), let C(t) = 1100 and solve for t. C(t) 1100 73,778.14 2011.4

= = = L

36.68t - 72,678.14 36.68t - 72,678.14 36.68t t

= 2a-

P(x) P(x) P(x) 45.5x - 5005 45.5x x

= = = = = =

R(x) - C(x) 124x - (78.5x + 5005) 45.5x - 5005 0 5005 110 • The break-even point

• Replace f (x) with 2x 2 + 6x - 5. • Solve for x. • Factor. • Use the zero product x - 2 = 0 or x + 5 = 0 principle to solve for x. x = 2 or x = -5

. 115 The slope of the linear path of the rock is - 115. The path of the rock is given by y - 1 = - 115 (x - 115 ) y - 1 = - 115 x + 15 y = - 115 x + 16

The values of x for which f (x) = 15 are 2 and -5. 38. f (x) = - x 2 - 6x

= - (x 2 + 6x) = - (x 2 + 6x + 9) + 9 = - (x + 3)2 + 9

Every point on the wall has a y value of 14. Thus 14 = - 115 x + 16 - 2 = - 115 x 2 x = L 0.52 115

The maximum value of f is 9 when x = - 3. 48. The soccer ball hits the ground when h(t) = 0.

h(t) = - 4.9t 2 + 12.8t

The x-coordinate of the point where the rock hits the wall is 0.52.

x = −3 y

10. f (x) = x 2 + 6x - 1

= x + 6x + 9 + (-1 - 9) = (x + 3)2 - 10

x

−3

50. a. l + w = 240, so w = 240 - l.

A = 240l - l 2 is (−3, − 10)

−10

-

240 b = = 120 2a 2(- 1)

Thus l = 120 meters and w = 240 - 120 = 120 meters are the dimensions that produce the greatest area. 68. Let x be the number of parcels.

32. Determine the y-coordinate of the vertex of the graph of

6 3 b = = 2a 2(2) 2

- 12.8 L 2.6 - 4.9

c. The l value of the vertex point of the graph of

Vertex: (3, -9) f (x) = (x - 3)2 - 9

h = -

• Solve for t.

b. A = lw = l(240 - l) = 240l - l 2

Vertex: ( - 3, -10) Axis of symmetry: x = - 3

f (x) = 2x 2 + 6x - 5. f (x) = 2x 2 + 6x - 5

• Replace h(t) with 0.

0 = t(- 4.9t + 12.8)

The soccer ball hits the ground in approximately 2.6 seconds.

2

-6 b = = 3 2a 2(1) k = f (3) = 32 - 6(3) = - 9

0 = - 4.9t 2 + 12.8t

t = 0 or t =


20. h = -

3 2 19 3 b + 6a- b - 5 = 2 2 2

2x 2 + 6x - 5 = 15 2x 2 + 6x - 20 = 0 2(x - 2)(x + 5) = 0

1

88. The slope of the radius from (0, 0) to ( 115, 1) is

• Find the y-coordinate of the vertex.

19 3 The vertex is a- , - b. Because the parabola opens 2 2 19 up, is the minimum value of f. Therefore, the range 2 19 f. of f is e y ƒ y Ú 2 To determine the values of x for which f (x) = 15, replace f (x) with 2x 2 + 6x - 5 and solve for x. f (x) = 15

According to the model, revolving consumer debt will first exceed $1.1 trillion in 2011. 80.

3 b 2

S11

• a = 2, b = 6, c = - 5. • Find the x-coordinate of the vertex.

a. R(x) = xp = x(22 - 0.01x) = - 0.01x 2 + 22x b. P(x) = R(x) - C(x)

= ( -0.01x 2 + 22x) - (2025 + 7x) = - 0.01x 2 + 15x - 2025 (continued)

S12


c. -

62. a. The graph of y = f (x + 3) + 2 is the graph of f shifted to

15 b = = 750 2a 2(- 0.01)

the left 3 units and up 2 units. y 6

The maximum profit is P(750) = - 0.01(750)2 + 15(750) - 2025 = $3600

4 2

d. The price per parcel that yields the maximum profit is

p(750) = 22 - 0.01(750) = $14.50

−6 −4 −2 −2

e. The break-even points occur when R(x) = C(x).

−6

b. The graph of y = f (x - 2) - 1 is the graph of f shifted to

-(-15) 2(-15)2 - 4(0.01)(2025) x= 2(0.01)

the right 2 units and down 1 unit. y

x = 150 and x = 1350 are the break-even points.

4

Thus the minimum number of parcels the air freight company must ship to break even is 150.

2 −6 −4 −2 −2

70. h(t) = - 16t 2 + 64t + 80

70. a., b.

which the projectile achieves this maximum height is at time t = 2 seconds.

E(−x)

2 − 4 −2 −2

b. The vertex of the graph of h is (2, 144), so the time at

4 x

2 −E(x)

72.

y 8

• Solve for t with h = 0

- 5) = 0 - 5) = 0 or t - 5 = 0 or t = 5

6 4 2

The time cannot be negative. The projectile will have a height of 0 feet at time t = 5 seconds.

−2 −2

2

4 x y

74. a.


−8

−6

−4

−2

replacing y with -y leaves the equation unaltered. b. The graph is not symmetric with respect to the y-axis because replacing x with -x alters the equation. 24. The graph is symmetric with respect to the origin because

(-y) = (-x) - (- x) simplifies to - y = - x + x, which is equivalent to the original equation y = x 3 - x. 3

44. Even, because h(- x) = (-x)2 + 1 = x 2 + 1 = h(x). g(x − 3) g(x) − 2

y

−4

2

4 x

4

6

x

8

y

b.

1 −8

−6

−4

−2

2

4

6

8

x

−2

Exercise Set 2.6, page 234 Domain: 5x ƒ x f (x) - g(x) = 1x - 4 + x Domain: 5x ƒ x Domain: 5x ƒ x f (x) # g(x) = - x1x - 4 f (x) 1x - 4 Domain: 5x ƒ x = g(x) x

10. f (x) + g(x) = 1x - 4 - x

2

− 4 −2 −2

2 −2

14. a. The graph is symmetric with respect to the x-axis because

58. a., b.

x

4

−6 y 4

a. The vertex (2, 144) gives the maximum height of 144 feet.

3

2

−4

64 b = = 2 2a 2(- 16) h(2) = - 16(2)2 + 64(2) + 80 = - 64 + 128 + 80 = 144 t = -

-16(t 2 - 4t - 16(t + 1)(t t + 1 = 0 t = -1

x

4

−4

-0.01x 2 + 22x = 2025 + 7x 0 = 0.01x 2 - 15x + 2025

c. - 16t 2 + 64t + 80 = 0

2

Ú 46 Ú 46 Ú 46 Ú 46


14. ( f + g)(x) = (x 2 - 3x + 2) + (2x - 4) = x 2 - x - 2

( f + g)( -7) = ( -7) - (-7) - 2 = 49 + 7 - 2 = 54 2

30.

f (x + h) - f (x) = h

34(x + h) - 54 - (4x - 5)

h 4x + 4(h) - 5 - 4x + 5 = h 4(h) = 4 = h

b. On 32, 2.54,

a = 2 ¢t = 2.5 - 2 = 0.5 s(a + ¢t) = s(2.5) = 6(2.5)2 = 37.5

50. ( f ⴰ g)(4) = f 3g(4)4 = f 342 - 5(4)4 = f 3- 44 = 2( -4) + 3 = - 5 74. a. l = 3 - 0.5t for 0 … t … 6. l = - 3 + 0.5t for t 7 6.

Area (in square inches)

In either case, l = ƒ 3 - 0.5t ƒ . w = ƒ 2 - 0.2t ƒ as in Example 7. b. A(t) = ƒ 3 - 0.5t ƒ ƒ 2 - 0.2t ƒ c. A is decreasing on 30, 64 and on 38, 104. A is increasing on 36, 84 and on 310, 144.

s(2.5) - s(2) 0.5 37.5 - 24 = 0.5 13.5 = 27 feet per second = 0.5

Average velocity =

38. ( g ⴰ f )(x) = g3 f (x)4 = g32x - 74

= 332x - 74 + 2 = 6x - 19 ( f ⴰ g)(x) = f 3g(x)4 = f 33x + 24 = 233x + 24 - 7 = 6x - 3

S13

c. On 32, 2.14,

a = 2 ¢t = 2.1 - 2 = 0.1 s(a + ¢t) = s(2.1) = 6(2.1)2 = 26.46 s(2.1) - s(2) 0.1 26.46 - 24 = 0.1 2.46 = 24.6 feet per second = 0.1

Average velocity =

d. On 32, 2.014,

a = 2 ¢t = 2.01 - 2 = 0.01 s(a + ¢t) = s(2.01) = 6(2.01)2 = 24.2406

A(t)

6 5 4 3 2 1

s(2.01) - s(2) 0.01 24.2406 - 24 = 0.01 0.2406 = 24.06 feet per second = 0.01

Average velocity = 2 4 6 8 10 12 t (in seconds)

t

d. The highest point on the graph of A for 0 … t … 14

occurs when t = 0 seconds.

80. a. On 32, 34,

a = 2 ¢t = 3 - 2 = 1 s(a + ¢t) = s(3) = 6 # 32 = 54 s(a) = s(2) = 6 # 22 = 24 s(a + ¢t) - s(a) Average velocity = ¢t s(3) - s(2) = 1 = 54 - 24 = 30 feet per second This is identical to the slope of the line through (2, s(2)) and (3, s(3)) because m =

s(3) - s(2) = s(3) - s(2) = 54 - 24 = 30 3 - 2

e. On 32, 2.0014,

a = 2 ¢t = 2.001 - 2 = 0.001 s(a + ¢t) = s(2.001) = 6(2.001)2 = 24.024006 s(2.001) - s(2) 0.001 24.024006 - 24 = 0.001 0.024006 = = 24.006 feet per second 0.001

Average velocity =

f. On 32, 2 + ¢t4,

s(2 + ¢t) - s(2) 6(2 + ¢t)2 - 24 = ¢t ¢t 6(4 + 4(¢t) + (¢t)2) - 24 = ¢t (continued)

S14


24 + 24(¢t) + 6(¢t)2 - 24 ¢t 24¢t + 6(¢t)2 = 24 + 6(¢t) = ¢t

=

As ¢t approaches zero, the average velocity approaches 24 feet per second.

兩

26. 3 2 -1

2

P(c) = P(3) = 53 36.

兩

-6 1

4 -6 1 -2

18. Enter the data in the table. Then use your calculator to find a. The linear regression equation is

y = 3.1410344828x + 65.09359606 b. Evaluate the linear regression equation when x = 58.

y = 3.1410344828(58) + 65.09359606 L 263 The ball will travel approximately 263 feet. 32. Enter the data in the table. Then use your calculator to find

the quadratic regression model. a. y = 0.05208x 2 - 3.56026x + 82.32999

兩

5 -1

3 -4

-5 1

-4 4

1

4

-1

-4

0

The reduced polynomial is x 3 + 4x 2 - x - 4. x 4 + 5x 3 + 3x 2 - 5x - 4 = (x + 1)(x 3 + 4x 2 - x - 4)

Exercise Set 3.2, page 282 2. Because an = - 2 is negative and n = 3 is odd, the graph of

P goes up to the far left and down to the far right. 18. P(x) = x3 - 6x2 + 8x

= x(x2 - 6x + 8) = x(x - 2)(x - 4)

b. The speed at which the bird has minimum oxygen con-

sumption is the x-coordinate of the vertex of the graph of the regression equation. Recall that the x-coordinate of the b vertex is given by x = . 2a

The speed that minimizes oxygen consumption is approximately 34 kilometers per hour.


3x + x - 5x + 2 3

2

x 2 - 2x + 2 1 12. 5 5 6 - 8 25 155 735 5 31 147 736

兩

+ -

3x 5x 6x 11x 14x 3x

+ 7 + 2 + 2 + 14 - 12

= 3x + 7 +

The factor x can be written as (x - 0). Apply the Factor Theorem to determine that the real zeros of P are 0, 2, and 4.

兩

-6 0 -6

1 0 1

• P(0) = 1

兩

-6 3 -3

1 -3 -2

• P(1) = -2

24. 0 4 -1

0 4 -1

-3.56026 b = L 34 2a 2(0.05208)

x 2 - 2x + 2 冄 3x3 + x 2 3x 3 - 6x 2 7x 2 7x 2

- 90 90 0

54. - 1 1

the linear regression equation.

8.

- 27 12 - 15

A remainder of 0 indicates that x + 6 is a factor of P(x).


x = -

-1 54 53

3 15 18

6 5

3x - 12 x 2 - 2x + 2

5x 3 + 6x 2 - 8x + 1 736 = 5x 2 + 31x + 147 + x - 5 x - 5

1 4 -1 4 4 3

P is a polynomial function. Also, P(0) and P(1) have opposite signs. Thus, by the Intermediate Value Theorem, we know that P must have a real zero between 0 and 1. 34. The exponent of (x + 2) is 1, which is odd. Thus the graph of

P crosses the x-axis at the x-intercept ( -2, 0). The exponent of (x - 6)2 is even. Thus, the graph of P intersects but does not cross the x-axis at (6, 0).

42. Far-left and far-right behavior: The leading term of

P(x) = x3 + 2x2 - 3x is 1x3. The leading coefficient, 1, is positive, and the degree of the polynomial, 3, is odd. Thus the graph of P goes down to the far left and up to the far right. The y-intercept: P(0) = 03 + 2(0)2 - 3(0) = 0. The y-intercept is (0, 0). The x-intercept or intercepts: Try to factor x3 + 2x2 - 3x. x3 + 2x2 - 3x = x(x2 + 2x - 3) = x(x + 3)(x - 1) (continued)


Use the Factor Theorem to determine that (0, 0), ( - 3, 0), and (1, 0) are the x-intercepts. Apply the Even and Odd Powers of (x - c) Theorem to determine that the graph of P will cross the x-axis at each of its x-intercepts. Additional points: ( -2, 6), (-1, 4), (0.5, - 0.875), (1.5, 3.375) Symmetry: The function P is neither an even nor an odd function. Thus the graph of P is not symmetric with respect y to either the y-axis or the origin. 8 Sketch the graph.

S15

Thus the cubic regression function is f (x) = - 0.0019920568x 3 + 0.0640002412x 2 - 0.3245133779x + 1.341589267 Graph the data and the cubic regression function on the same screen. Use the value command, in the CALCULATE menu, to predict the vehicle sales in the year 2011 (represented by x = 24). Press 2ND [CALC] Enter 24 Enter to produce the screen below. 6

Y1=-.00199205679081X^3+._

4

−4

−2

2

4 x 0 X=24 0

−4

Y=2.8792141

25

The cubic model predicts about 2.9 million sport/cross utility vehicles will be sold in 2011.

−8 P(x) = x3 + 2x2 − 3x

66. The volume of the box is V = lwh, with h = x, l = 18 - 2x,

and w =

42 - 3x . Therefore, the volume is 2

Exercise Set 3.3, page 295 12. p = 1, 2, 4, 8

q = 1, 3 p 1 2 4 8 = 1, 2, 4, 8, , , , q 3 3 3 3

42 - 3x bx 2 = 3x3 - 69x2 + 378x

V(x) = (18 - 2x)a

20.

Use a graphing utility to graph V. The graph is shown below. The value of x that produces the maximum volume is 3.571 inches (to the nearest 0.001 inch). (Note: Your x-value may differ slightly from 3.5705971 depending on the values you use for Xmin and Xmax. The maximum volume is approximately 606.6 cubic inches.) Volume (in cubic inches)

800

兩

11 0 1 1 1 2 1 0 2 1 2 3 1 0 3 1 3 4 1 0 4 1 4 5 1 0 5

兩兩兩

0

Maximum X=3.570598

Y=606.55979

0

6

Height (in inches)

68. Enter the x-values (years) and the sales data into a graphing

utility. To find the cubic regression function on a TI-83/TI-83 Plus/TI-84 Plus calculator, select 6: CubicReg in the STAT CALC menu. The following screen shows the results. CubicReg y=ax3+bx2+cx+d a= -.0019920568 b=.0640002412 c= -.3245133779 d=1.341589267 R2=.9969553409

兩

1

5

兩

-19 1 -18 -19 4 -15 -19 9 -10 -19 16 -3 -19 25

-28 -18 -46 -28 -30 -58 -28 -30 -58 -28 -12 -40 -28 30

-1 1

6

2

1

None of these numbers are negative, so 5 is an upper bound.

1 -2 1

兩

1 -3 1

兩

1 -4 1

兩

1 -5 1

兩

0 -1 -1 0 -2 -2 0 -3 -3 0 -4 -4 0 -5

-19 1 -18 -19 4 -15 -19 9 -10 -19 16 -3 -19 25

-28 18 -10 -28 30 2 -28 30 2 -28 12 -16 - 28 - 30

-5

6

-58

These numbers alternate in sign, so - 5 is a lower bound.

30. P has one positive real zero because P(x) has one variation

in sign. P(- x) = ( -x)3 - 19(- x) - 30 = - x 3 + 19x - 30 P has two or no negative real zeros because P( -x) = - x 3 + 19x - 30 has two variations in sign.

S16


44. P has one positive and two or no negative real zeros (see

Exercise 30).

兩

5 1 0 - 19 - 30 5 25 30 1 5 6 0 The reduced polynomial is x 2 + 5x + 6 = (x + 3)(x + 2) which has -3 and -2 as zeros. Thus the zeros of P(x) = x3 - 19x - 30 are -3, -2, and 5. 76. We need to find the natural number solution of

n3 - 3n2 + 2n = 504, which can be written as n3 - 3n2 + 2n - 504 = 0 The constant term has many natural number divisors, but the following synthetic division shows that 10 is an upper bound for the zeros of P(n) = n3 - 3n2 + 2n - 504.

兩

10 1 - 3 10 1 7

2 70 72

- 504 720 216

The following synthetic division shows that 9 is a zero of P.

兩

9 1 -3 9 1 6

2 54 56

- 504 504 0

Thus the given group of cards consists of exactly nine cards. There is no need to seek additional solutions because any increase or decrease in the number of cards will increase or decrease the number of ways we can select three cards from the group of cards. 80. The volume of the tank is equal to the volume of the two

hemispheres plus the volume of the cylinder. Thus 4 3 px + 6px 2 = 9p 3 Dividing each term by p and multiplying by 3 produces 4x 3 + 18x 2 = 27 Intersection Method Use a graphing utility to graph y = 4x 3 + 18x 2 and y = 27 on the same screen, with x 7 0. The x-coordinate of the point of intersection of the two graphs is the desired solution. The graphs intersect at x L 1.098 (rounded to the nearest thousandth of a foot). The length of the radius is approximately 1.098 feet.

Exercise Set 3.4, page 305 2. Use the Rational Zero Theorem to determine the possible

rational zeros. p = 1, 5 q The following synthetic division shows that 1 is a zero of P.

兩

1 1 -3 1 1 -2

7 -2 5

-5 5 0

Use the quadratic formula to find the zeros of the reduced polynomial x2 - 2x + 5. x =

- (- 2) 2( -2)2 - 4(1)(5) = 1 2i 2(1)

The zeros of P(x) = x3 - 3x2 + 7x - 5 are 1, 1 - 2i, and 1 + 2i. The linear factored form of P is

P(x) = 1(x - 1)(x - 31 - 2i4)(x - 31 + 2i4) or

P(x) = (x - 1)(x - 1 + 2i)(x - 1 - 2i)

兩

-29 15 + 9i 3 -14 + 9i

92 -97 + 3i - 5 + 3i

5 - 3i 3 -14 + 9i 15 - 9i 3 1

兩

- 5 + 3i 5 - 3i 0

18. 5 + 3i 3

34 - 34 0

1 as a zero. The zeros 3 1 of P(x) = 3x 3 - 29x 2 + 92x + 34 are 5 + 3i, 5 - 3i, and - . 3 The reduced polynomial 3x + 1 has -

兩

22. 3i 1 - 6 + 0i

0 + 3i

22 + 0i -9 - 18i

- 64 + 0i 54 + 39i

117 + 0i -117 - 30i

-90 90

- 3i 1 - 6 + 3i 0 - 3i 1 -6

13 - 18i 0 + 18i 13

-10 + 39i 0 - 39i - 10

0 - 30i 30i 0

0

兩

p = 1, 2, 5, 10 q

兩

2 1 -6 2 1 -4

13 -8 5

- 10 10 0

Use the quadratic formula to solve x 2 - 4x + 5 = 0. x = =

- (- 4) 2(- 4)2 - 4(1)(5) 4 1-4 = 2(1) 2 4 2i = 2i 2

The zeros of P(x) = x5 - 6x4 + 22x3 - 64x2 + 117x - 90 are 3i, -3i, 2, 2 + i, and 2 - i. 30. The graph of P(x) = 4x3 + 3x2 + 16x + 12 is shown on

page S17. Applying Descartes’ Rule of Signs, we find that the real zeros are all negative numbers. By the Upper- and LowerBound Theorem, there is no real zero less than -1, and by the Rational Zero Theorem, the possible rational zeros (that are (continued)


p 1 1 3 = - , - , and - . q 2 4 4 3 From the graph, it appears that - is a zero. 4 negative and greater than -1) are

x 2 - 6x + 9 = 0 (x - 3)(x - 3) = 0 x = 3

38. Vertical asymptote:

1 = 1 (the Theorem on 1 Horizontal Asymptotes) because the numerator and denominator both have degree 2. The graph crosses the 3 horizontal asymptote at a , 1b. The graph intersects, 2 but does not cross, the x-axis at (0, 0). See the following graph. The horizontal asymptote is y =

30

−3

S17

3

−15

y

3 Use synthetic division with c = - . 4 3 4 3 16 12 4 -3 0 -12

10

兩

4

0

16

0

3 is a zero, and by the Factor Theorem, 4 3 4x3 + 3x2 + 16x + 12 = a x + b(4x2 + 16) = 0 4

Solve 4x2 + 16 = 0 to find that x = - 2i and x = 2i. The 3 solutions of the original equation are - , -2i, and 2i. 4 48. Because P has real coefficients, use the Conjugate Pair

Theorem.

(x - 33 + 2i4)(x - 33 - 2i4)(x - 7) (x - 3 - 2i)(x - 3 + 2i)(x - 7) (x 2 - 6x + 13)(x - 7) x 3 - 13x 2 + 55x - 91

x + 1

44. x 2 - 3x + 5 冄 x 3 - 2x 2 + 3x + 4

x 3 - 3x 2 + 5x + x 2 - 2x + x 2 - 3x + x x2 - 3x + 5 Slant asymptote: y = x + 1 x 2 - x - 12 2

x2 - 4 = 0 (x - 2)(x + 2) = 0

(x - 4)(x + 3) x + 3 = ,x Z 4 (x - 4)(x + 2) x + 2

x = −2 y=1

x = -2

degree of the denominator is larger than the degree of the numerator. y

2

(0, − ) 1 2

(4, 76 )

x 4 x-axis

x=2

2

4

6

x

−3

16. The horizontal asymptote is y = 0 (x-axis) because the

22. Vertical asymptote: x - 2 = 0

y

3

−4

The vertical asymptotes are x = 2 and x = - 2.

x = 2 Horizontal asymptote: y = 0 No x-intercepts 1 y-intercept: a0, - b 2

=

x - 2x - 8 The function F is undefined at x = 4. Thus the graph of F x + 3 7 is the graph of y = with an open circle at a 4, b . x + 2 6 The height of the open circle was found by evaluating x + 3 at x = 4. y = x + 2

10. Set the denominator equal to zero.

or

4 4 5 1

x - 1

F(x) = x + 1 +

62. F(x) =


x = 2

x

5 x=3

(0, 0)

Thus -

P = = = =

( 32 , 1)

y=1

68. a. Set r = 60 and solve for x.

30 + x = 60 3 1 + x 4 70 3 1 + xb 4 70 6 30 + x = 45 + x 7 30 + x = 60a

• Simplify. (continued)

S18


x -

6 x = 45 - 30 7 1 x = 15 7 x = 105

2 −2

You need to drive 105 miles at 70 miles per hour to bring your average speed up to 60 miles per hour. b. The numerator and denominator of r are both of degree 1.

f 3 g(x)4 = f 32x + 34 1 3 = (2x + 3) 2 2 3 3 = x + 2 2 = x

1 = 70 1 a b 70

0.0006(1000) + 9(1000) + 401,000 1000 $410.60 0.0006(10,000)2 + 9(10,000) + 401,000 10,000 $55.10 0.0006(100,000)2 + 9(100,000) + 401,000 100,000 $73.01

= C (10,000) = = C (100,000) = =

The following shows that g3 f (x)4 = x for all real numbers x. 1 3 g3 f (x)4 = gc x - d 2 2 1 3 = 2a x - b + 3 2 2 = x - 3 + 3 = x Thus f and g are inverses. 32.

b. Graph C using the minimum feature of a graphing utility. 100

0

Minimum X=25852.147 Y=40.022572

−35

Exercise Set 4.1, page 342 10. Because the graph of the given function is a line that passes

through (0, 6), (2, 3), and (6, -3), the graph of the inverse will be a line that passes through (6, 0), (3, 2), and (- 3, 6). See the following figure. Notice that the line shown in the figure is a reflection of the line given in Exercise 10 across the line given by y = x. Yes, the inverse relation is a function.

f (x) y x x + 8

= = = =

4x - 8 4x - 8 4y - 8 4y

1 (x + 8) = y 4 1 y = x + 2 4 1 f -1(x) = x + 2 4

50,000

The minimum average cost per cell phone is $40.02. The minimum is achieved by producing approximately 25,852 cell phones.

(6, 0) x

and g3 f (x)4 = x for all x in the domain of f. The following shows that f 3 g(x)4 = x for all real numbers x.

2

C (1000) =

4

20. Check to see whether f 3 g(x)4 = x for all x in the domain of g

The further you drive at 70 miles per hour, the closer your average speed for the entire trip will be to 70 miles per hour. 72. a.

(3, 2)

−4

The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 1/70. Thus the graph of r has a horizontal asymptote of y =

y

(− 3, 6)

• Solve for x.

38.

f (x) = y = x = x( y - 2) xy - 2x xy - y y(x - 1)

= = = =

y = f -1(x) =

x ,x Z 2 x - 2 x x - 2 y y - 2 y y 2x 2x 2x x - 1 2x ,x Z 1 x - 1

• Replace f (x) with y. • Interchange x and y. • Solve for y.

• Replace y with f -1(x).

• Replace f (x) with y. • Interchange x and y.

• Solve for y.

• Replace y with f -1(x) and indicate any restrictions.


44.

f (x) y x x2 x2 - 4 -x 2 + 4 f -1(x)

= = = = = = =

14 14 14 4 -y y - x2

- x, x … 4 - x - y y

• Replace f(x) with y. • Interchange x and y. • Solve for y.

K(x) y x x + 4.7 x + 4.7 1.3

= = = =

+ 4, x Ú 0

• Replace y with f -1(x) and indicate any restrictions.

1.3x - 4.7 1.3x - 4.7 1.3y - 4.7 1.3y

• Replace K(x) with y. • Interchange x and y. • Solve for y.

= y

K -1(x) =

48. a. A(45) = 200e-0.014(45)

After 45 minutes the patient’s bloodstream will have about 107 milligrams of medication. b. Use a graphing calculator to graph y = 200e-0.014x and

y = 50 in the same viewing window, as shown below. 250

Plot1 Plot2 Plot3 \Y 1 = 2 0 0 * e^ ( - .0 1 4 * X ) \Y 2 = 5 0 \Y 3 = \Y 4 = \Y 5 = \Y 6 = \Y 7 =

0

150

Intersection X=99.021026 Y=50 − 100

x + 4.7 1.3

• Replace y with K -1(x).

x + 4.7 can be used to convert a 1.3 U.K. men’s shoe size to its equivalent U.S. shoe size. The function K -1(x) =

2. f (3) = 53 = 5 # 5 # 5 = 125

1 52

=

1 1 = 5#5 25 x

22. The graph of f (x) = a b has a y-intercept of (0, 1), and the

5 2

The x-coordinate (which represents time in minutes) of the point of intersection is about 99.02. Thus it will take about 99 minutes before the patient’s medication level is reduced to 50 milligrams. 54. a. P(0) =

3600

1 + 7e-0.05(0) 3600 = 1 + 7 3600 = 8 = 450

Exercise Set 4.2, page 354 f ( -2) = 5-2 =

5 x 2 produced by reflecting the graph of f across the x-axis.

30. Because F(x) = - c a b d = - f (x), the graph of F(x) can be

L 106.52

The range of f is 5 y ƒ y Ú 06. Therefore, the domain of f -1 is 5x ƒ x Ú 06, as indicated above. 52.

S19

5 graph passes through a1, b. Plot a few additional points, 2 5 2 25 such as a- 1, b and a2, b. Because the base is greater 5 4 2 than 1, we know that the graph must have all the properties of an increasing exponential function. Draw a smooth increasing curve through the points. The graph should be asymptotic to the negative portion of the x-axis, as shown in the following figure. y 6

Immediately after the lake was stocked, the lake contained 450 bass. b. P(12) =

3600

1 + 7e-0.05(12) L 743.54

After 1 year (12 months), there were about 744 bass in the lake. c. As t : q , 7e-0.05t =

7 e0.05t

approaches 0. Thus, as t : q ,

3600 3600 will approach = 3600. As time 1+0 1 + 7e-0.05t goes by, the bass population will increase, approaching 3600. P(t) =

4 2 −4 −2 x+5

28. Because F(x) = 6

Exercise Set 4.3, page 366 2

4x

= f (x + 5), the graph of F(x) can be produced by shifting the graph of f horizontally to the left 5 units.

4. The exponential form of y = logb x is b y = x. Thus the

exponential form of 3 = log4 64 is 43 = 64.

14. The logarithmic form of b y = x is y = logb x. Thus the

logarithmic form of 53 = 125 is 3 = log5 125.

32. By property 3, log10(106) = 6.

S20


44. To graph y = log6 x, use the equivalent exponential equation

x = 6 y. Choose some y values, such as -1, 0, and 1, and then calculate the corresponding x values. This yields the 1 ordered pairs a , -1 b, (1, 0), and (6, 1). Plot these ordered 6 pairs and draw a smooth curve through the points to produce the following graph.

Exercise Set 4.4, page 377 2. ln

z3 1xy

y 6

3

3

6

= ln z 3 - ln 1xy = ln z 3 - ln(xy)1>2 1 = 3 ln z - ln(xy) 2 1 = 3 ln z - (ln x + ln y) 2 1 1 = 3 ln z - ln x - ln y 2 2 1 log2 u + 4 log2 v = log2 t 3 - log2 u1>3 + log2 v 4 3 t3 = log 2 + log2 v 4 u1>3 t 3v 4 = log 2 u1>3 t 3v 4 = log 2 3 1u

18. 3 log 2 t -

9 x

52. log4(5 - x) is defined only for 5 - x 7 0, which is equivalent

to x 6 5. Using interval notation, the domain of k(x) = log4(5 - x) is (- q , 5).

66. The graph of f (x) = log6(x + 3) can be produced by shifting

the graph of f (x) = log6 x (from Exercise 44) 3 units to the left. y 6

log 37 L 2.2436 log 5

34. log5 37 =

ln(5 - x) ln(5 - x) into Y1 on a , so enter ln 8 ln 8 graphing calculator.

46. log8(5 - x) =

3

5

3

6 x 7

−3

86. a. S(0) = 5 + 29 ln(0 + 1) = 5 + 0 = 5. When starting,

the student had an average typing speed of 5 words per minute. S(3) = 5 + 29 ln(3 + 1) L 45.2. After 3 months the student’s average typing speed was about 45 words per minute. b. Use the intersection feature of a graphing utility to find

the x-coordinate of the point of intersection of the graphs of y = 5 + 29 ln(x + 1) and y = 65. 75

0

Intersection X=6.9166293 Y=65 0

−5

68. M = log a

398,107,000I 0 I b = log a b = log 398,107,000 I0 I0

L 8.6 70. log a

I b I0 I I0 I I

= 9.5 = 109.5 = 109.5I0 L 3,162,277,660I0

72. In Example 7, we noticed that if an earthquake has a Richter 16

The graphs intersect near (6.9, 65). The student will achieve a typing speed of 65 words per minute in about 6.9 months.

scale magnitude of M1 and a smaller earthquake has a Richter scale magnitude of M2 then the larger earthquake is 10M1 - M2 times as intense as the smaller earthquake. In this exercise, M1 = 9.5 and M2 = 8.3. Thus 10M1 - M2 = 109.5 - 8.3 = 101.2 L 15.8. The 1960 earthquake in Chile was about 15.8 times as intense as the San Francisco earthquake of 1906.


76. M = log A + 3 log 8t - 2.92

= log 26 + 3log38 # 174 - 2.92 L 1.4150 + 6.4006 - 2.92 L 4.9

• Substitute 26 for A and 17 for t.

36. ln x =

ln x =

78. pH = - log3H+4 = - log(1.26 * 10-3) L 2.9

ln x =

Thus vinegar is an acid. 80.

pH 5.6 - 5.6 10-5.6

= = = =

- log3H+4 - log3H+4 log3H+4 3H+4

ln x = ln x = x = x2 = 0 = 0 = x = 5

The hydronium-ion concentration is 10-5.6 L 2.51 * 10-6 mole per liter.


Check:

2. 3x = 243

3x = 35 x = 5 10.

18.

42.

3x-2 log 3x-2 (x - 2)log 3 x log 3 - 2 log 3 x log 3 - 2 log 3 - 2x log 4 x log 3 - 2x log 4 x(log 3 - 2 log 4)

= = = = = = =

42x+1 log 42x + 1 (2x + 1)log 4 2x log 4 + log 4 log 4 log 4 + 2 log 3 log 4 + 2 log 3 log 4 + 2 log 3 x = log 3 - 2 log 4 x L - 2.141

22. log(x 2 + 19) = 2

= = = =

102 100 81 9

A check shows that 9 and - 9 are both solutions of the original equation. 26. log3 x + log3 (x + 6) = 3

log3 3x (x + 6)4 33 27 2 x + 6x - 27 (x + 9)(x - 3) x = - 9 or x = 3

= = = = =

1 5 1 ln a2x + b + ln 2 2 2 2 1 5 c ln a2x + b + ln 2 d 2 2 1 5 ln c 2 a2x + b d 2 2 1 ln(4x + 5) 2 ln(4x + 5)1>2 14x + 5 4x + 5 x 2 - 4x - 5 (x - 5)(x + 1) or x = - 1 5 1 1 ln a10 + b + ln 2 2 2 2 1.6094 L 1.2629 + 0.3466 ln 5 =

Because ln( - 1) is not defined, - 1 is not a solution. Thus the only solution is x = 5.

6x = 50 log(6x) = log 50 x log 6 = log 50 log 50 L 2.18 x = log 6

x 2 + 19 x 2 + 19 x2 x

S21

3 x(x + 6) x 2 + 6x 0 0

Because log3 x is defined only for x 7 0, the only solution is x = 3.

10x + 10-x 2 10x + 10-x 10x(10x + 10-x) 102x + 1 2x 10 - 16(10x) + 1 u2 - 16u + 1

= 8 = = = = =

u = 10x = log 10x = x =

16 (16)10x • Multiply each side by 10x. 16(10x) 0 • Let u = 10x. 0 2 16 216 - 4(1)(1) = 8 317 2 8 317 • Replace u with 10x. log(8 317) log(8 317) L 1.20241

9 24 + v ln 24 24 - v 9 24 + v 1.5 = ln 24 24 - v 24 + v 4 = ln 24 - v 24 + v e4 = • N = ln M means e N = M. 24 - v (24 - v)e4 = 24 + v - v - ve4 = 24 - 24e4 v( -1 - e4) = 24 - 24e4 24 - 24e4 L 23.14 v = - 1 - e4 The velocity is about 23.14 feet per second.

74. a.

t =

b. The vertical asymptote is v = 24. c. Due to the air resistance, the object can never reach or exceed

a velocity of 24 feet per second.

S22


18. P = 32,000, r = 0.08, t = 3

Exercise Set 4.6, page 400 6. Let t = 0 represent 1990 and let t = 15 represent 2005. Start

by substituting 15 for t in N(t) = N0e kt. N(15) 545,147 545,147 258,295 545,147 ln a b 258,295 545,147 1 ln a b 15 258,295 0.049796876

#

= N0 e k 15 = 258,295e15k

• Substitute.

= e15k

• Solve for k.

A = Pert = 32,000e3(0.08) L $40,679.97 ln 3 , r = 0.055 r ln 3 t = 0.055 t L 20 years (to the nearest year)

22. t =

38. a. Represent 2007 by t = 0; then 2008 will be represented

= 15k

by t = 1. Use the following substitutions: P0 = 240, P(1) = 310, c = 3400, and

= k

a =

L k

The exponential growth function is N(t) = 258,295e0.049796876t The year 2013 is represented by t = 23.

#

N(23) = 258,295e0.049796876 23 L 812,000 The exponential growth function yields 812,000 as the approximate population of Las Vegas in 2013. = = = = =

N0 e kt N0 e138k N0 e138k e138k 138k ln 0.5 k = L - 0.005023 138 N(t) = N0(0.5)t>138 L N0 e-0.005023t

8.

N(t) N(138) 0.5N0 0.5 ln 0.5

12.

N(t) 0.65N0 0.65 ln 0.65

= = = =

N0(0.5)t>5730 N0(0.5)t>5730 (0.5)t>5730 ln(0.5)t>5730 ln 0.65 L 3600 t = 5730 ln 0.5

Using a = 13.16667, b = 0.27833, and c = 3400 gives the following logistic model. P(t) L

3400 L 1182 1 + 13.16667e-0.27833(7) According to the model, there will be about 1182 groundhogs in 2014. P(7) L

16. a. P = 12,500, r = 0.08, t = 10, n = 1

0.08 10 b L $26,986.56 1

b. n = 365

48 a.

0.08 3650 A = 12,500a1 + b L $27,816.82 365

3400

1 + 13.16667e-0.27833t b. Because 2014 is 7 years past 2007, 2014 is represented by t = 7.

The bone is approximately 3600 years old.

A = 12,500a1 +

c - P0 3400 - 240 = L 13.16667. P0 240 c P(t) = 1 + ae-bt 3400 P(1) = 1 + 13.16667e-b(1) 3400 310 = 1 + 13.16667e-b 310(1 + 13.16667e-b) = 3400 3400 1 + 13.16667e-b = 310 3400 - 1 13.16667e-b = 310 13.16667e-b L 9.96774 9.96774 e-b L 13.16667 9.96774 - b L ln 13.16667 b L 0.27833

v 50

c. n = 8760

A = 12,500a1 +

0.08 87600 L $27,819.16 b 8760 5

t

(continued)


b. Here is an algebraic solution. An approximate solution can

be obtained from the graph. v 50 50 64 50 1 64 50 b ln a 1 64

= 64(1 - e-t>2) = 64(1 - e-t>2)

0

25

0 Xscl = 5

= e-t>2 t 2

t = - 2 ln a1 -

50 b L 3.0 64

The velocity is 50 feet per second in approximately 3.0 seconds. c. As t : q , e

12

ExpReg y=a*b^x a=10.14681746 b=.8910371309 r2=.9997309204 r=-.9998654511

= (1 - e-t>2)

= -

-t>2

: 0. Therefore, 64(1 - e horizontal asymptote is v = 64.

-t>2

) : 64. The

24. a. Enter the data into a graphing utility. Because a world record

time was set twice in 1968, we have entered the domain value 68 (which represents 1968) twice in L1. L1


30

0 Xscl = 5

Yscl = 1

L3

1

L1(9) =68

4. The following scatter plot suggests that the data can be mod-

0

L2 44.9 44.9 44.5 44.1 43.86 43.29 43.18

63 64 67 68 68 88 99

approach, but never reach or exceed, 64 feet per second.

8

Yscl = 1

The correlation coefficient r L - 0.99987 is close to -1. This indicates that the function y L 10.1468(0.89104)x provides a good fit for the data. The graph of y also indicates that the regression function provides a good model for the data. When x = 24 kilometers, the atmospheric pressure is about 10.1468(0.89104)24 L 0.6 newton per square centimeter.

d. Because of air resistance, the velocity of the object will

eled by an increasing function that is concave down. Thus the most suitable model for the data is an increasing logarithmic function.

S23

Perform an exponential regression and a logarithmic regression. See the calculator displays below. The logarithmic regression function provides a slightly better fit than the exponential regression function, as determined by comparing the correlation coefficients. ExpReg y=a*b^x a=48.57147035 b=.9987164563 r2=.8921939256 r=-.9445601758

LnReg y=a+b1nx a=61.75560609 b= -4.108626059 r2=.9299128774 r=-.9643199041

22. From the scatter plot in the following figure, it appears that

the data can be closely modeled by a decreasing exponential function of the form y = ab x, with b 6 1.

b. To predict the world record time in 2012, evaluate the

logarithmic function at x = 112. The graph above shows that the predicted world record time in the 400-meter race for 2012 is about 42.37 seconds.

12

50 Y2=61.755606089805+-4.1_ 0

25

0 Xscl = 5

Yscl = 1

The calculator display in the following figure shows that the exponential regression equation is y L 10.1468(0.89104)x, where x is the altitude in kilometers and y is the pressure in newtons per square centimeter.

48

X=112 40

Y=42.369059 Xscl=10 Yscl=1

120

S24


26. a. Use a graphing utility to perform a logistic regression on

74. The Earth makes 1 revolution (u = 2p) in 1 day.

t = 24 # 3600 = 86,400 seconds u 2p v = = L 7.27 * 10-5 radian/second t 86,400

the data. The following figure shows the results obtained by using a TI-83/TI-83 Plus/TI-84 Plus graphing calculator. Logistic y=c/(1+ae^(-bx)) a=2.229711876 b=.0438206742 c=1531680.577

80. C = 2pr = 2p(18 inches) = 36p inches

Thus one conversion factor is (36p inches/1 revolution). 500 revolutions 36p inches 500 revolutions = a b 1 minute 1 minute 1 revolution 18,000p inches = 1 minute Now convert inches to miles and minutes to hours.

The logistic regression function for the data is P(t) =

1,531,680.577 1 + 2.229711876e-0.0438206742t

18,000p inches 1 minute 18,000p inches 1 foot 1 mile 60 minutes a ba ba b = 1 minute 12 inches 5280 feet 1 hour L 54 miles per hour

b. The year 2012 is represented by t = 62.

P(62) =

1,531,680.577 1 + 2.229711876e-0.0438206742(62)

L 1,335,000

The logistic model predicts that Hawaii’s population will be about 1,335,000 in 2012. c. The carrying capacity c, to the nearest thousand, of the

logistic model is 1,532,000.

Exercise Set 5.1, page 438 2. The measure of the complement of an angle of 87° is

(90° - 87°) = 3° The measure of the supplement of an angle of 87° is (180° - 87°) = 93°

14. Because 765° = 2 # 360° + 45°, ⬔a is coterminal with an

Exercise Set 5.2, page 449 6. adj = 282 - 52

adj = 164 - 25 = 139 hyp opp 8 5 csc u = = sin u = = opp hyp 8 5 hyp adj 8139 139 8 sec u = = cos u = = = hyp 8 adj 39 139 adj opp 5 5139 139 cot u = tan u = = = = opp adj 39 5 139 18. Because tan u =

angle that has a measure of 45°. ⬔a is a Quadrant I angle.

32. -45° = - 45°a 44.

hyp = 232 + 42 = 5 hyp 5 sec u = = adj 3

p radians p b = - radian 180° 4

p p 180° radian = radian a b = 45° 4 4 p radians

68. s = r u = 5a144° #

34. sin

p b = 4p L 12.57 meters 180°

72. Let u2 be the angle through which the pulley with a diameter

of 0.8 meter turns. Let u1 be the angle through which the pulley with a diameter of 1.2 meters turns. Let r2 = 0.4 meter be the radius of the smaller pulley, and let r1 = 0.6 meter be the radius of the larger pulley. u1 = 240° =

56.

4 p radians 3

u2 =

0.6 4 a pb = 2p radians or 360° 0.4 3

p p 13 # 12 p cos - tan = - 1 3 4 4 2 2 16 - 4 16 - 1 = = 4 4 h tan 68.9° = 116 h = 116 tan 68.9° h L 301 meters h (three significant digits) 68.9°

116 m

Thus r2 u2 = r1u1 4 0.4 u2 = 0.6a pb 3

opp 4 = , let opp = 4 and adj = 3. adj 3

58.

5° 80 ft d

(continued)


80 d 80 d = sin 5° d L 917.9 feet Change 9 miles per hour to feet per minute. 9 miles # 5280 feet # 1 hour miles = r = 9 hour 1 hour 1 mile 60 minutes 9(5280) feet feet = 792 = 60 minute minute d t = r 917.9 feet t L L 1.2 minutes (to the nearest tenth 792 feet per minute of a minute) 68. Here are two methods that can be used to solve this exercise. sin 5° =

S25

Solve Equation (1) for x. y tan 36.5° x = y cot 36.5° x =

Substitute y cot 36.5° for x into Equation (2) and solve for y. tan 32.1° =

y y cot 36.5° + 55.5

(y cot 36.5° + 55.5)(tan 32.1°) = y

• Solve for y.

y cot 36.5° tan 32.1° + 55.5 tan 32.1° = y y cot 36.5° tan 32.1° - y = - 55.5 tan 32.1° y(cot 36.5° tan 32.1° - 1) = - 55.5 tan 32.1° y =

-55.5 tan 32.1° (cot 36.5° tan 32.1° - 1)

y L 228.6640425 To three significant digits, the height of the tower is 229 feet.

Exercise Set 5.3, page 460 6. x = - 6, y = - 9, r = 2( - 6)2 + ( -9)2 = 1117 = 3113 36.5°

32.1° B

A

55.5 ft

Method 1: Let x denote the distance from point B to the center of the base of the tower. Let y denote the height of the tower. Then y y and (2) tan 32.1° = (1) tan 36.5° = x x + 55.5

sin u =

y 3 3213 -9 = = = r 13 3213 213

x -6 2 2213 = = = r 13 3213 213 y -9 3 tan u = = = x -6 2

cos u =

Solve each equation for y and equate the results. y = x tan 36.5°

y = (x + 55.5) tan 32.1°

30. sec u =

213 r = 3 x

y = 2(2 13)2 - 32 = 13

Now solve for x.

y = - 13 because y 6 0 in Quadrant IV

Substitute this x-value into Equation (1). y L 309.0217178 tan 36.5° y L 228.6640425 To three significant digits, the height of the tower is 229 feet. Method 2: Let x denote the distance from point B to the center of the base of the tower. Let y denote the height of the tower. Then y y (1) tan 36.5° = and (2) tan 32.1° = x x + 55.5

sin u =

213 3

sec u = -

213 2

cot u =

2 3

• Let r = 213 and x = 3.

Thus x tan 36.5° = (x + 55.5) tan 32.1° x tan 36.5° = x tan 32.1° + 55.5 tan 32.1° x tan 36.5° - x tan 32.1° = 55.5 tan 32.1° x (tan 36.5° - tan 32.1°) = 55.5 tan 32.1° 55.5 tan 32.1° x = (tan 36.5° - tan 32.1°) x L 309.0217178

csc u = -

- 13 1 = 2 213

38. u¿ = 255° - 180° = 75° 50. The reference angle for u = 300° is

u¿ = 360° - 300° = 60° The terminal side of u is in Quadrant IV. Thus cos 300° is positive. cos 300° = + cos 60° =

c u

c

c

Correct u¿ sign prefix

cos 300° =

1 2

1 2

S26



tan2 t = sec2 t - 1

7p ; W(t) = P(x, y) where 10. t = 4 x = cos t y = sin t = cos a= cos = W a-

7p b 4

tan t = 2sec2t - 1 7p b 4

= sin a-

p 4

= sin

12 2

=

78. 1 + tan2 t = sec2 t

p 4

3p 6 t 6 2p, tan t is negative. Thus 2 tan t = - 2sec2t - 1.

Because

82. March 5 is represented by t = 2.

12 2

T(2) = - 41 cosa

p = - 41 cosa b + 36 3

7p 12 12 b = a , b 4 2 2

16. The reference angle for -

sec a -

5p p is . 6 6

5p p b = - sec 6 6 = -

• sec t 6 0 for t in Quadrant III

= - 41(0.5) + 36 = 15.5°F July 20 is represented by t = 6.5. T(6.5) = - 41 cosa

223 3

p # 6.5b + 36 6

L - 41(-0.9659258263) + 36 L 75.6°F

44. F(-x) = tan(- x) + sin(- x)

= - tan x - sin x

• tan x and sin x are odd functions.

= - (tan x + sin x)


= - F(x) Because F(- x) = - F(x), the function defined by F(x) = tan x + sin x is an odd function. 56.

p # 2b + 36 6

y P1(x, y)

22. y = -

3 sin x 2

a = `-

3 3 ` = 2 2

Period = 2p

t t−π

x A(1, 0)

32. y = sin

P2(− x, − y)

tan t =

a = 1

y x

tan(t - p) =

3p x 4

Period = -y y = -x x

• From the unit circle

Therefore, tan t = tan(t - p). 1 1 + sin t + 1 - sin t 1 + = 72. 1 - sin t 1 + sin t (1 - sin t)(1 + sin t) = =

y 2 1

−2

y 8 3

1

2p 2p 8 = = b 3 3p>4

3 x

1

−1

34. y = cos 3px

y

a = 1

1

2p 2p 2 Period = = = b 3p 3

2π x

π

−1

1 3 2 1 3

−1

x

2 1 - sin2t 2

cos2t = 2 sec2t

40. y =

a =

1 px sin 2 3

1

1 2

Period =

y

3

2p 2p = = 6 b p>3

6 x

S27


48. y = -

3 cos 5x 4

π 5

Period =

x

2p 2p = b 5

34. y =

2 54. y = - ` 3 sin x ` 3

y 3π 2

y

4

2π 5

1

3 3 a = `- ` = 4 4 Period =

32. y = - 3 tan 3x

y

p p = b 3

x

1 cot 2x 2

y

1

p p Period = = b 2

3π

π 6

π − 6

π 4

x

π 2

x

−3

60.

40. y = 3 csc

px 2

y 4

y

Period =

1

−1

π 2

2π 3

π

4π x 3

44. y = sec

Because the graph passes through the origin, we start with an equation of the form y = a sin bx. The graph completes one 4p 4p cycle in units. Thus the period is . Use the equation 3 3 4p 2p to solve for b. = b 3 4p 2p = b 3 6p = 4bp 3 = b 2

Period =

2p 2p = = 4p b 1>2

a = 1


Period = p

y =

= 22. y = tan(x - p)

p = p Period = b

40. y = 2 sin a

1

a = 2 x

−π 2

π 2

π 6

x

7π 6

y

1

c b -p = = p 1

y

2π 3

-p>3 p = 2 6

Phase shift = -

1 tan x 3

y

c b

Period = p

3 3 sin x. 2 2

4π x

−1

− 4π

1

Exercise Set 5.6, page 489 24. y =

p b 3

Phase shift = -

3 3 for a and for b in y = a sin bx to produce 2 2

x


3 3 minimum height of - . Thus its amplitude is a = . 2 2 Substitute

4

y

• Multiply each side by 3b.

3 and a 2

2

x 2

20. y = cos a2x -

The graph has a maximum height of

2

2p 2p = 4 = b p>2

px + 1b - 2 2

−1

π 2

π

3π x 2

π

2π

y 4

2

Period = 4

x

c Phase shift = b = -

1 2 = p 2>p

−4

S28


42. y = - 3 cos(2px - 3) + 1

78. y = x cos x

y 4

a = 3

2

Period = 1

−4π

x

−2

c 3 Phase shift = - = b 2p

−4

2

4

6

y


8

1 , period = p p 2p Because = p, we have b = 2. Thus y = 3 cos 2t. b

20. Amplitude = 3, frequency = 4

x + cos x 2

28. Amplitude = ƒ - 1.5 ƒ = 1.5

6π x

3π

f =

y

y= π

y=

x 2

π

y = cos x y = − sin x π

−2

c 64. a. Phase shift = - = b

4π

x

y = − sin x + cos x

a-

7 pb 12

p a b 6

= 3.5 months,

p tb. Because the phase shift 6 is 3.5 months, shift the graph of y1 3.5 units to the right to produce the graph of y2. Now shift the graph of y2 upward 4 units to produce the graph of S.

b. First graph y1 = 2.7 cosa

Shoe sales in hundreds of pairs

y

−2

7.05

a. f has pseudoperiod

10 , (2p) L 1.59

.012 y = .01

f

8

15.05 y = −.01

(10.5, −.01)

−.012

Exercise Set 6.1, page 520 2. We will try to verify the identity by rewriting the left side so

that it involves only sines and cosines.

4 3.5

=

10

y1

y2

2p = 2p. 1

Thus f completes only one full oscillation on 0 … t … 10.

tan x sec x sin x =

S

6

1 1 bt d = - 1.5 cos t 6p 3

b. The following graph of f shows that ƒ f (t) ƒ 6 0.01 for t 7 10.5.

2p 2p = = 12 months b p>6

6

1 , period = 6p 6p

−12.5

3π 2π

=

12.5

0

2

3

x

2π

y

#1

y = a cos 2p ft = - 1.5 cosc 2p a 34.

y = cos x

−1

56. y = cos x - sin x

x 2

1 1 k 3 1 = = 2p A m 2p A 27 2p

+ cos x

1

period =

4π

−13

x 48. y = csc + 4 3 Period = 6p

52. y =

13

t

Time in months

c. 3.5 months after January 1 is the middle of April.

sin x # 1 # sin x cos x cos x sin2 x

cos2 x sin x 2 b = a cos x = (tan x)2 = tan2 x


12. sin4 x - cos4 x = (sin2 x + cos2 x)(sin2 x - cos2 x)

= 1(sin x - cos x) = sin x - cos x 2

2

2

2

2 sin x cot x + sin x - 4 cot x - 2 2 cot x + 1 (sin x)(2 cot x + 1) - 2(2 cot x + 1) = 2 cot x + 1 (2 cot x + 1)(sin x - 2) = sin x - 2 = 2 cot x + 1 1 1 1 1 + + sin x cos x sin x cos x # sin x cos x 34. = 1 1 sin x cos x 1 1 sin x cos x sin x cos x 24.

cos x cos x cos x = cos x =

+ + -

=

sin x sin x sin x # cos x - sin x sin x cos x - sin x cos2 x - sin2 x

cos2 x - 2 sin x cos x + sin2 x cos2 x - sin2 x = 1 - 2 sin x cos x 44. Rewrite the left side so that it involves only sines and/or cosines. cos x 2 # sin x 2 cot x = cot x + tan x cos x sin x + sin x cos x cos x 2# sin x = cos x cos x sin x sin x + sin x cos x cos x sin x cos x 2# sin x = 2 cos x + sin2 x sin x cos x cos x 2# sin x = 1 sin x cos x cos x # sin x cos x = 2# sin x 1 = 2 cos2 x

Exercise Set 6.2, page 529 4. Use the identity cos(a - b) = cos a cos b + sin a sin b

with a = 120° and b = 45°.

S29

cos(120° - 45°) = cos 120° cos 45° + sin 120° sin 45° 12 13 12 1 b + a ba b = a- b a 2 2 2 2 16 12 + = 4 4 16 - 12 = 4 20. The value of a given trigonometric function of u, measured in

degrees, is equal to its cofunction of 90° - u. Thus cos 80° = sin(90° - 80°) = sin 10° 26. sin x cos 3x + cos x sin 3x = sin(x + 3x) = sin 4x

24 24 7 , with 0° 6 a 6 90°; sin a = , cos a = 7 25 25 8 sin b = - , with 180° 6 b 6 270° 17 8 15 cos b = , tan b = 17 15

38. tan a =

a. sin(a + b) = sin a cos b + cos a sin b

15 7 8 24 bab + a ba- b 25 17 25 17 56 416 360 = = 425 425 425

= a

b. cos(a + b) = cos a cos b - sin a sin b

15 24 8 7 bab - a ba- b 25 17 25 17 105 192 87 + = 425 425 425 tan a - tan b 1 + tan a tan b 24 24 8 8 7 15 7 15 105 # = 192 105 24 8 1 + 1 + a ba b 105 7 15 360 - 56 304 = 105 + 192 297

= a = c. tan(a - b) =

=

=

50. cos(u + p) = cos u cos p - sin u sin p

= (cos u)(- 1) - (sin u)(0) = - cos u 62. cos 5x cos 3x + sin 5x sin 3x = cos(5x - 3x) = cos 2x

= cos(x + x) = cos x cos x - sin x sin x = cos2 x - sin2 x 76. sin(u + 2p) = sin u cos 2p + cos u sin 2p

= (sin u)(1) + (cos u)(0) = sin u

S30


Exercise Set 6.3, page 538 2. 2 sin 3u cos 3u = sin32(3u)4 = sin 6u 10. cos a =

24 with 270° 6 a 6 360° 25

7 25

-7>25 24>25 7 = 24

sin 2a = 2 sin a cos a

cos 2a = cos2 a - sin2 a

sin a = = -

C

1 - a

= 2a= tan 2a =

24 2 b 25

tan a =

24 7 ba b 25 25

= a

336 625

1 (330°), we can find cos 165° by using the 2 a half-angle identity for cos with a = 330°. The angle 2 a = 165° lies in Quadrant II, and the cosine function is 2 negative in Quadrant II. Thus cos 165° 6 0, and we must select the minus sign that precedes the radical in a 1 + cos a cos = to produce the correct result. 2 A 2 1 + cos 330° cos 165° = 2 A

28. Because 165° =

=

7 2 24 2 b - a- b 25 25

= -

527 625

2 tan a 7 b 24

7 12 = = 49 7 2 1 1 - a- b 576 24 -

#

= 576 336 = 576 527

1 - cos 2x 1 + 2 cos 2x + cos2 2x ba b = a 2 4

= = = = = = =

1 2 + 4 cos 2x + 1 + cos 4x (1 - cos 2x)a b 8 2 1 (1 - cos 2x)(3 + 4 cos 2x + cos 4x) 16 1 (3 + 4 cos 2x + cos 4x - 3 cos 2x - 4 cos2 2x 16 - cos 2x cos 4x) 1 2 (3 + cos 2x + cos 4x - 4 cos 2x - cos 2x cos 4x) 16 1 1 + cos 4x a3 + cos 2x + cos 4x - 4a b - cos 2x cos 4xb 16 2 1 (3 + cos 2x + cos 4x - 2(1 + cos 4x) - cos 2x cos 4x) 16 1 (3 + cos 2x + cos 4x - 2 - 2 cos 4x - cos 2x cos 4x) 16 1 (1 + cos 2x - cos 4x - cos 2x cos 4x) 16

C

a

2 + 13 b 2

#

1 2

C

38. Because a is in Quadrant III, cos a 6 0. We now solve for

cos a.

cos a = - 21 - sin2 a = -

C

1 - a-

7 2 b 25

49 1 A 625 24 = 25

= -

1 + cos 4x 1 b = (1 - cos 2x)a1 + 2 cos 2x + 8 2 =

2

2 + 13 4 22 + 13 = 2

= -

22. sin2 x cos4 x

= sin2 x (cos2 x)2 1 - cos 2x 1 + cos 2x 2 ba b = a 2 2

S

2 13 + 2 2 = S 2

1 - tan2 a 2a -

13 2

1 +

Because 180° 6 a 6 270°, we know that 90° 6 Thus

a 6 135°. 2

a a a is in Quadrant II, sin 7 0, cos 6 0, and 2 2 2

a 6 0. 2 Use the half-angle formulas. tan

1 - cos a a = sin = S 2 A 2 =

A

1 - a-

24 b 25

2

25 + 24 712 49 = = A 50 50 10 1 + a-

cos

24 b 25

a 1 + cos a = = S 2 A 2 2 25 - 24 1 12 = = = A 50 10 A 50

(continued)


7 25 24 1 + a- b 25 7 25 = -7 = 1 25

a sin a tan = = 2 1 + cos a

-

1 1 50. = 1 - cos 2x 1 - 1 + 2 sin2 x 1 = csc2 x = 2 2 2 sin x 1

68. cos2

x 1 + cos x 2 d = c 2 A 2 1 + cos x = 2 1 + cos x # sec x = 2 sec x sec x + 1 = 2 sec x

Exercise Set 6.4, page 546 3u - 5u 3u + 5u cos 2 2 = 2 cos 4u cos(-u) = 2 cos 4u cos u 1 3sin(5x + 3x) + sin(5x - 3x)4 2 1 (sin 8x + sin 2x) 2 1 (2 sin 4x cos 4x + 2 sin x cos x) 2 sin 4x cos 4x + sin x cos x

22. cos 3u + cos 5u = 2 cos

36. sin 5x cos 3x =

= = =

5x - 3x 5x + 3x sin 2 2 cos 5x - cos 3x = 44. sin 5x + sin 3x 5x - 3x 5x + 3x cos 2 sin 2 2 sin 4x sin x = = - tan x sin 4x cos x -2 sin

62. a = 1, b = 13 , k = 2(13)2 + (1)2 = 2. Thus a is a

first-quadrant angle. 13 1 and cos a = 2 2 p Thus a = . 3 sin a =

y = k sin(x + a) p y = 2 sin ax + b 3

S31

70. From Exercise 62, we know that

y = sin x + 13 cos x = 2 sin ax +

p b 3

The graph of y = sin x + 13 cos x has an amplitude p of 2 and a phase shift of - . It is the graph of 3 p y = 2 sin x shifted units to the left. 3 y

2 −π

2π 3

3

x

5π 3

Exercise Set 6.5, page 557 2. y = sin-1

12 implies 2

p p 12 for … y … 2 2 2 p Thus y = . 4 28. Because tan(tan-1 x) = x for all real numbers x, we have 1 1 tan c tan-1 a b d = . 2 2 sin y =

50. Let x = cos-1

cos x =

3 5

3 . Thus 5 3 2 4 1 - a b = C 5 5 4>5 3 4 sin x = b = tan x = = 5 cos x 3 3>5

and sin x =

y = tan acos-1 56. y = cos asin-1

Let a = sin-1 b = cos-1

3 5 + cos-1 b 4 13 3 2 17 3 3 1 - a b = , sin a = , cos a = . 4 4 C 4 4

5 5 5 2 12 , cos b = , sin b = . 1 - a b = 13 13 C 13 13

y = cos(a + b) = cos a cos b - sin a sin b =

17 4

#

5 3 13 4

#

12 517 36 517 - 36 = = 13 52 52 52

4 p = 5 6 4 p -1 - cos-1 sin x = 6 5 4 p sin(sin-1 x) = sin a - cos-1 b 6 5

66. sin-1 x + cos-1

(continued)

S32


4 4 p p cosacos-1 b - cos sin acos-1 b 6 5 6 5 1 # 4 13 # 3 4 - 3 13 = = 2 5 2 5 10

x = sin

72. Let a = cos-1 x and b = cos-1(- x). Thus cos a = x

and cos b = - x. We know that sin a = 21 - x2 and sin b = 21 - x2 because a is in Quadrant I and b is in Quadrant II. cos-1 x + cos-1( -x) = a + b

= cos-13cos(a + b)4

= cos-1(cos a cos b - sin a sin b)

= cos-1 C x( -x) - 21 - x2 # 21 - x2 D

= cos-1( -x2 - 1 + x2)


2 cos2 x + 1 = - 3 cos x 2 cos2 x + 3 cos x + 1 = 0 (2 cos x + 1)(cos x + 1) = 0 2 cos x + 1 = 0 or cos x + 1 = 0 1 cos x = cos x = - 1 2 2p 4p , x = x = p 3 3 2p , p, and The solutions in the interval 0 … x 6 2p are 3 4p . 3

52. sin x + 2 cos x = 1

sin x = 1 - 2 cos x

= cos-1( -1) = p

(sin x)2 = (1 - 2 cos x)2

76. The graph of y = f (x - a) is a horizontal shift of the graph

of y = f (x). Therefore, the graph of y = cos-1(x - 1) is the graph of y = cos-1 x shifted 1 unit to the right. π

y = cos−1 (x − 1)

2 x

−2

84. a.

The solutions in the interval 0 … x 6 360° are 90° and 323.1°. (Note: x = 270° and x = 36.9° are extraneous solutions. Neither of these values satisfies the original equation.)

1100

V

56. 2 cos2 x - 5 cos x - 5 = 0

10

0 − 150

b. Although the water rises 0.1 foot in each case, there is a

greater volume of water at the 4.9- to 5.0-foot level near the diameter of the cylinder than at the 0.1- to 0.2-foot level near the bottom. c.

V(4) = 12c25 cos-1 a

5 - (4) b - 35 - (4)4210(4) - (4)2 d 5

1 = 12c25 cos-1 a b - 124 d 5 L 352.04 cubic feet d.

1 - cos2 x = 1 - 4 cos x + 4 cos2 x 0 = cos x(5 cos x - 4) or cos x = 0 5 cos x - 4 = 0 4 x = 90°, 270° cos x = 5 x L 36.9°, 323.1°

y

y = cos−1 x

sin2 x = 1 - 4 cos x + 4 cos2 x

5 2( - 5)2 - 4(2)(- 5) 5 165 = 2(2) 4 cos x L 3.27 or cos x L - 0.7656 cos x =

The solutions in the interval 0° … x 6 360° are 140.0° and 220.0°. 66. cos 2x = -

13 2

7p 5p + 2kp or 2x = + 2kp, k an integer 6 6 5p 7p x = + kp or x = + kp, k an integer 12 12

2x =

84. 2 sin x cos x - 212 sin x - 13 cos x + 16 = 0

2 sin x(cos x - 12 ) - 13 (cos x - 12 ) = 0

1100

(cos x - 12 )(2 sin x - 13 ) = 0

V

cos x = 12 or

V = 288 0 Intersection

x L 140.0°, 220.0°

No solution

X=3.4476991 Y=288

10

−150

When V = 288 cubic feet, x L 3.45 feet.

No solution

13 2 p 2p x = , 3 3

sin x =

The solutions in the interval 0 … x 6 2p are

p 2p . and 3 3


86. The following graph shows that the solutions in the interval

30, 2p) are x = 0 and x = 1.8955.


3

S33

C

78° a 2π

0

28° 44

B X=1.8954943 Y=1.8954943

92. When u = 45°, d attains its maximum of 4394.5 feet. 94. a. Enter the data into a graphing utility and perform a

sine regression. Each time, given in the hour:minute format, needs to be converted to hours. For instance, 16:55 is 55 L 16 + 0.917 = 16.917 hours 60

The following screens show the results obtained on a TI-83/TI-83-Plus/TI-84 Plus graphing calculator using 16 as the number of iterations to be performed. L1 1 61 122 183 245 306 L1(1)=1

b c = sin B sin C

a c = sin A sin C

b 44 = sin 28° sin 78°

a 44 = sin 74° sin 78°

b =

a =

44 sin 74° L 43 sin 78°

and c 16.92. Use the Law of Sines to find the measure of angle C. a c = sin A sin C 24.42 16.92 = sin 54.32° sin C 24.42 sin C = 16.92 sin 54.32° sin C =

16.92 sin 54.32° 24.42

sin C L 0.562813 There are two angles C, where C is between 0° and 180°, such that sin C = 0.562813

y L 1.527518 sin(0.016579x - 1.205750) + 18.289850 b. Graph the regression function and use the value

command in the CALC menu to determine that the sunset time for March 22, 2012, occurs at 18.523764 hours.

One of the angles is an acute angle, and the other is an obtuse angle. The inverse sine function can be used to find the acute angle. C L sin-1 0.562813 C L 34.25°

21

Y1=1.527518378158*sin(.0–

X=82

44 sin 28° L 21 sin 78°

14. We need to solve triangle ABC, given A 54.32°, a 24.42,

L2 L3 1 16.917 SinReg 16,L1,L2, 18.1 365.25,Y1 19.217 19.867 SinReg 18.783 17.15 y=a*sin(bx+c)+d a=1.527518378 b=0.0165791284 c=–1.205749695 d=18.28984953

The sine regression function is

0

A

A = 180° - 78° - 28° = 74°

−3

16 +

b

Y=18.523764

365

The obtuse angle is the supplement of 34.25°, which is 180° 34.25° 145.75°. We can determine that the 145.75° value for C is not a valid result, because A 54.32° and the sum of 54.32° and 145.75° is greater than 180°. (The sum of the measures of the angles in any triangle is 180°.) Thus, the only possibility for C is C L 34.25°. We can now estimate the measure of angle B.

16

B = 180° - A - C L 180° - 54.32° - 34.25°

Convert the decimal portion of this result (0.523764) to minutes by multiplying by 60. 0.523764 * 60 = 31.42584 Thus the sunset time is 18:31 in the hour:minute format, rounded to the nearest minute.

L 91.43° The Law of Sines can now be used to find b. b =

a 24.42 (sin B) L (sin 91.43°) L 30.05 sin A sin 54.32°

(continued)

S34


There is only one triangle with the given dimensions, and the solution is

24. We need to solve triangle ABC, given B 52.7°, b 12.3,

and c 16.3. Use the Law of Sines to find the measure of angle C.

B L 91.43°, C L 34.25°, b L 30.05 20. We need to solve triangle ABC, given B 22.6°, b 5.55,

b c = sin B sin C

and a 13.8.

Use the Law of Sines to find the measure of angle A. b a = sin A sin B 5.55 13.8 = sin A sin 22.6° 13.8 sin 22.6° = 5.55 sin A

12.3 16.3 = sin 52.7°° sin C

• The Law of Sines

12.3 sin C = 16.3 sin 52.7° • a 13.8, B 22.6°, b 5.55

sin C =

• Solve for sin A.

sin C L 1.054164

13.8 sin 22.6°°° 5.55 sin A L 0.955545

sin A =

There are two angles A, where A is between 0° and 180°, such that sin A 0.955545. One of the angles is an acute angle, and the other is an obtuse angle. The inverse sine function can be used to find the acute angle. A L sin-1 0.955545 A L 72.9°

Because 1.054164 is not in the range of the sine function, we know that there is no C value such that sin C = 1.054164 Thus no triangle can be formed using the given values. 30. The angle with its vertex at the position of the helicopter

measures 180° - (59.0° + 77.2°) = 43.8°. Let the distance from the helicopter to the carrier be x. Using the Law of Sines, we have

(to the nearest tenth of a degree)

The obtuse angle is the supplement of 72.9°, which is

x 7620 = sin 77.2° sin 43.8°

180° - 72.9° = 107.1° The 107.1° value is a valid result because B 22.6° and the sum of 22.6° and 107.1° is less than 180°. Thus, there are two triangles with the given dimensions. In one triangle A L 72.9°, and in the other triangle A L 107.1°. Case 1: If A L 72.9°, then

x =

7620 sin 77.2° sin 43.8°

L 10,700 feet 40.

N

C L 180° - A - B L 180° - 72.9° - 22.6° = 84.5° and

16.3 sin 52.7° 12.3

(to three significant digits)

C Fire b

8°

50° A

65°

5.55 b (sin C) L (sin 84.5°) L 14.4 c = sin B sin 22.6° Case 2: If A L 107.1°, then C L 180° - A - B L 180° - 107.1° - 22.6° = 50.3° and c =

b 5.55 (sin C) L (sin 50.3°) L 11.1 sin B sin 22.6°

The two solutions are

20 α = 65°

B

a = 65° B = 65° + 8° = 73° A = 180° - 50° - 65° = 65° C = 180° - 65° - 73° = 42° b c = sin B sin C b 20 = sin 73° sin 42° 20 sin 73° sin 42°

Case 1: A L 72.9°, C L 84.5°, c L 14.4

b =

Case 2: A L 107.1°, C L 50.3°, c L 11.1

b L 29 miles


Exercise Set 7.2, page 597 12. c2 = a2 + b2 - 2ab cos C

c = 14.2 + 9.30 - 2(14.2)(9.30) cos 9.20° c = 214.22 + 9.302 - 2(14.2)(9.30) cos 9.20° c L 5.24 2

2

Exercise Set 7.3, page 613 10. a = 3 - 3 = 0

2

b2 + c2 - a2 2bc 1322 + 1602 - 1082 cos A = L 0.7424 2(132)(160)

b = 0 - ( -2) = 2

A vector equivalent to P 1 P 2 is v = 80, 29.

12. 7 v 7 = 262 + 102

= 236 + 100 = 2136 = 2234

18. cos A =

u = tan-1

1 ac sin B 2 1 K = (32)(25) sin 127° L 320 square units 2

30. K =

A unit vector in the direction of v is u = h 24.

a2 sin B sin C 2 sin A 8.52 sin 102° sin 27° L 21 square units K = 2 sin 51°

K =

1 (a + b + c) 2 1 = (10.2 + 13.3 + 15.4) = 19.45 2

40. s =

C

A 66 mi

B

a A b c a2 a2 a a

270° - 254° = 16° 16° + 90° + 32° = 138° 4 # 16 = 64 miles 3 # 22 = 66 miles b2 + c2 - 2bc cos A 642 + 662 - 2(64)(66) cos 138° 2642 + 662 - 2(64)(66) cos 138° 120 miles 1 60. S = (324 + 412 + 516) = 626 2 K = 2626(626 - 324)(626 - 412)(626 - 516) = 24,450,284,080

Cost = 4.15 A 24,450,284,080 B L $276,848, or $277,000 to the nearest $1000

3 , 3 i - 8 - 6, -49 2

30. 3u + 2v = 3(3i - 2j) + 2( -2i + 3j)

= (9i - 6j) + ( - 4i + 6j) = (9 - 4)i + ( -6 + 6)j = 5i + 0j = 5i 8p L - 1.8 7 8p a2 = 2 sin L - 0.9 7

40. a1 = 2 cos

v = a1 i + a2 j L - 1.8i - 0.9j

254°

= = = = = = = L

3 3 u - 2v = 8 - 2, 49 - 28 -3, - 29 4 4

9 = h , 7i 2

32° 64 mi α

6 10 3134 5134 , i = h , i 34 34 2134 2134

= h-

K = 2s(s - a)(s - b)(s - c) = 219.45(19.45 - 10.2)(19.45 - 13.3)(19.45 - 15.4) L 66.9 square units 52.

10 5 = tan-1 L 59.0° 6 3

Thus v has a direction of about 59° as measured from the positive x-axis.

A L cos-1(0.7424) L 42.1°

32. A = 180° - 102° - 27° = 51°

S35

44.

y

C B

60° α

D x

θ A

327°

AB = 18 cos 123°i + 18 sin 123°j L - 9.8i + 15.1j AD = 4 cos 30°i + 4 sin 30°j L 3.5i + 2j AC = AB + AD L (- 9.8i + 15.1j) + (3.5i + 2j) = - 6.3i + 17.1j (continued)

S36


7 AC 7 = 2(- 6.3)2 + (17.1)2 L L 18 a = tan-1 `

17.1 17.1 ` = tan-1 L 70° -6.3 6.3

u L 270° + 70° = 340° The speed of the boat is about 18 miles per hour at an approximate heading of 340°.

48. In the following figure, we need to find 7 -F 1 7 and 7 F2 7 .

cos u =

3(6) + ( -4)( -12)

225 2180 66 cos u = L 0.9839 52180 u L 10.3° 72. projw v =

projw v =

w 1

From the right triangle formed by w, F1, and F2, we see that 7 F1 7 7 F2 7 and cos 31.8° = sin 31.8° = . 7w7 7w7

sin 31.8° =

7 F1 7

811

• 7 w 7 = 811

.

23 ` = tan-1 23 = 60° 1

a = tan-1 `

u = a = 60°, z = 2 cis 60° 5p 5p 1 23 + i sin b = 4a i b = 2 - 2i 23 3 3 2 2

56. 23 - i = 2 cis(- 30°); 1 + i 23 = 2 cis 60°

A 23 - i B A 1 + i23 B = 2 cis(- 30°) # 2 cis 60° = 2 # 2 cis(- 30° + 60°)

L 427

= 4 cis 30°

Now 7 -F 1 7 = 7 F 1 7 . Thus the magnitude of the force needed to keep the motorcycle from rolling down the ramp is approximately 427 pounds.

cos 31.8° =

• 7 w 7 = 811

811

60. z1 = 1 + i

r1 = 212 + 12 = 22

z1 = 22 (cos 45° + i sin 45°) = 22 cis 45° + 3j) # (4i

- 2j)

= 5(4) + 3( -2) = 20 - 6 = 14

cos u =

23 1 + i b 2 2

1 a1 = tan-1 ` ` = 45°; u1 = 45° 1

L 689

70. cos u =

= 4a

= 223 + 2i

7w7 7 F2 7 7

The magnitude of the force the motorcycle exerts against the ramp is approximately 689 pounds.

60.

= 4(cos 30° + i sin 30°)

7 F2 7

7 F2 7 = 811 cos 31.8°

v # w = (5i

33 33117 = L 8.0 17 117

12. r = 212 + ( 132)2 = 2

30. z = 4 acos

7 F1 7 = 811 sin 31.8°

b. cos 31.8° =

=


F

7w7 7 F1 7

2( -4)2 + 12

W = 100 # 25 # cos 42° W L 1858 foot-pounds

F2

31.8°

a. sin 31.8° =

7w7

8 - 7, 59 # 8 - 4, 199

80. W = 7 F 7 7 s 7 cos a

31.8° 58.2°

v#w

v#w

7v7 7w7

(3i - 4j) # (6i - 12j) 232 + (- 4)2

262 + (- 12)2

z2 = 1 - i r2 = 212 + ( -1)2 = 22 a2 = tan-1 `

-1 ` = 45°; u2 = 315° 1

z2 = 22 (cos 315° + i sin 315°) = 22 cis 315° z1 12 cis 45° = z2 12 cis 315° = cis(-270°) = cos 270° - i sin 270° = 0 - ( -i) = i



Vertex: (0, 0)

= 16 cis 1320° = 16 cis 240° = 16(cos 240° + i sin 240°)

x2 + 5x 25 x2 + 5x + 4 5 2 ax + b 2 4p p

= - 8 - 8i23 12 12 12 + i b = [1(cos 135° + i sin 135°)]12 2 2 = 1

[cos(12 # 135°) + i sin (12 # 135°)]

12

= - 1 + 0i, or -1

120° + 360°k 120° + 360°k + i sin b , w k = 4 acos 3 3 1/3

k = 0, 1, 2 120° 120° w 0 = 41/3 acos + i sin b L 1.216 + 1.020i 3 3 120° + 360° 120° + 360° w 1 = 41/3 acos + i sin b 3 3 L - 1.492 + 0.543i

L 0.276 - 1.563i

3

+ i sin

120° + 360° # 2 3

= 4ay +

25 4

• Complete the square.

29 b 16

29 5 •h = - ,k = 2 16 • Compare to

= 4 = 1

(x - h)2 = 4p( y - k)2.

y

2 2

y2 = 4(5)x y2 = 20x 32. Vertex: (2, - 3); focus: (0, - 3)

(h, k) = (2, - 3), so h = 2 and k = - 3. Focus is (h + p, k) = (2 + p, - 3) = (0, - 3). Therefore, 2 + p = 0 and p = - 2. (y - k)2 = 4p(x - h) (y + 3)2 = 4(- 2)(x - 2) (y + 3)2 = - 8(x - 2) 40.

k = 0, 1 22 w 0 = 21/2(cos 60° + i sin 60°) = + 2 120° 120° + 360° + i sin w 1 = 21/2 acos 2

26 i 2 + 360° b 2

= 21/2(cos 240° + i sin 240°)

x2 = 4py 40.52 = 4p(16) 40.52 64 p L 25.6 feet p =

Exercise Set 8.2, page 654 22. 25x2 + 12y2 = 300

26 22 i = 2 2

y2 x2 + = 1 12 25

• a2 = 25, b2 = 12, c2 = 25 - 12 a = 5, b = 213, c = 113

Center: (0, 0)


1 . 16

= 4y + 1 +

y2 = 4px

b

120° + 360°k 120° + 360°k w k = 21/2 acos + i sin b, 2 2

or p = -

= 4y + 1

y

Vertices: (0, 5) and (0, - 5) 1 1 y, we have 4p = - , 4 4

x

30. Vertex: (0, 0); focus: (5, 0); p = 5 because focus is (p, 0).

30. - 1 + i 23 = 2(cos 120° + i sin 120°)

6. Comparing x2 = 4py with x2 = -

−4

13 5 Focus: (h, k + p) = a- , - b 2 16 45 Directrix: y = k - p = 16

28. - 2 + 2i23 = 4(cos 120° + i sin 120°)

w 2 = 41/3 acos

x

29 5 Vertex: a- , - b 2 16

= cos 1620° + i sin 1620°

120° + 360° # 2

1

22. x2 + 5x - 4y - 1 = 0

23 1 bd = 16 c - + ia 2 2

16. a-

y

1 Focus: a0, - b 16 1 Directrix: y = 16

6. (2 cis 330°)4 = 24 cis(4 # 330°)

S37

Foci: (0, 113) and (0, - 113)

2 2

x

S38

28.


9x2 + 16y2 + 36x - 16y - 104 9x2 + 36x + 16y2 - 16y - 104 9(x2 + 4x) + 16(y2 - y) 1 9(x2 + 4x + 4) + 16 ay2 - y + b 4

58. The mean distance is a = 67.08 million miles.

= 0 = 0 = 104

Aphelion = a + c = 67.58 million miles. Thus c = 67.58 - a = 0.50 million miles. b = 2a2 - c2 = 267.082 - 0.502 L 67.078 An equation of the orbit of Venus is

= 104 + 36 + 4

1 2 b = 144 2 1 2 ay - b 2 (x + 2) 2 + = 1 16 9 1 Center: a- 2, b 2 9(x + 2)2 + 16 ay -

x2

= 1. 67.082 67.0782 60. The length of the semimajor axis is 50 feet. Thus c2 = a2 - b2 322 = 502 - b2 b2 = 502 - 322

y

a = 4, b = 3, c = 242 - 32 = 17

4

2

b = 2502 - 322 b L 38.4 feet

x

1 1 Vertices: a2, b and a- 6, b 2 2 1 1 Foci: a- 2 + 17, b and a-2 - 17, b 2 2 44. The center ( - 4, 1) = (h, k). Therefore, h = - 4 and k = 1.


The length of minor axis is 8, so 2b = 8, or b = 4. The equation of the ellipse is of the form (x - h)2 a2 (x + 4)2 a2 (0 + 4)2 a2

+

(4 - 1) = 1 16

Foci: (0, 161) and (0, - 161) Asymptotes: y =

• The point (0, 4) is on the satgraph. Thus x 0 and y 4 isfy the equation.

is (0, 0), the midpoint of the line segment between (0, - 3) and (0, 3). c e = a 1 3 1 = •e = 4 4 a a = 12 32 = 122 - b2 • c2 a2 b2 2 b = 144 - 9 = 135 • Solve for b 2. The equation of the ellipse is

y2 x2 + = 1. 135 144

5 x and 6

y

4 8 x

5 y = - x 6

16

50. Because the foci are (0, - 3) and (0, 3), c = 3 and the center

c2 = a2 + b2 = 25 + 36 = 61

Center: (0, 0)

• h 4, k 1, b 4

9 + = 1 • Solve for a 2. 2 16 a 7 16 = 16 a2 256 a2 = 7 (x + 4)2 (y - 1)2 The equation of the ellipse is + = 1. 16 256>7

b2 = 36

a = 5 b = 6 c = 161 The transverse axis is on the y-axis because the y2 term is positive.

2

+

y2 x2 = 1 25 36 a2 = 25

(y - k)2

= 1 b2 (y - 1)2 + = 1 16

y2

+

Vertices: (0, 5) and (0, - 5) 28.

16x2 - 9y2 - 32x - 54y + 79 = 0 16(x - 2x + 1) - 9(y2 + 6y + 9) = - 79 + 16 - 81 2

= - 144 (y + 3)2 (x - 1)2 = 1 16 9 The transverse axis is parallel to the y-axis because the y2 term is positive. The center is at (1, - 3); a2 = 16, so a = 4. Vertices: (h, k + a) = (1, 1) (h, k - a) = (1, - 7) c2 = a2 + b2 = 16 + 9 = 25 c = 125 = 5 Foci: (h, k + c) = (1, 2) (h, k - c) = (1, - 8)

(continued)


Because b2 = 9 and b = 3, the asymptotes are 4 y + 3 = (x - 1) and 3 4 y + 3 = - (x - 1). 3

A¿ = A cos2 a + B cos a sin a + C sin2 a

y

= 0a

2 1

x

50. Because the vertices are (2, 3) and (- 2, 3), a = 2 and the

12 2 12 12 2 12 1 b + 1a ba b + 0a b = 2 2 2 2 2

C¿ = A sin2 a - B cos a sin a + C cos2 a = 0a

12 2 12 1 12 12 2 b - 1a ba b + 0a b = 2 2 2 2 2

center is (0, 3).

F¿ = F = 10

c e = c2 = a2 + b2 a 5 c 52 = 22 + b2 = 2 2 b2 = 25 - 4 = 21 c = 5 Substituting into the standard equation yields

(x¿)2 (y¿)2 1 1 2 (x¿) - (y¿)2 + 10 = 0 or = 1 2 2 20 20

4

8

8

x'

y'

y

(y - 3) x = 1. 4 21 2

45°

4

x

−

8

56. a. Because the transmitters are 300 miles apart, 2c = 300 and

c = 150.

2a = rate * time 2a = 0.186 * 800 = 148.8 miles Thus a = 74.4 miles. b = 2c2 - a2 = 21502 - 74.42 L 130.25 miles The ship is located on the hyperbola given by y2 x2 = 1. 74.42 130.252 b. The ship will reach the coastline when x 6 0 and y = 0. Thus x2 74.42

-

02 130.252 x2

= 1

= 1 74.42 x2 = 74.42 x = - 74.4

The ship reaches the coastline 74.4 miles to the left of the origin, at the point (- 74.4, 0).


xy = - 10 xy + 10 = 0 A = 0, B = 1, C = 0, D = 0, E = 0, F = 10 cot 2a =

A - C 0 - 0 = = 0 B 1

Thus 2a = 90° and a = 45°.

−8

−

2

S39

20. x2 + 4xy + 4y2 - 215x + 15y = 0

A = 1, B = 4, C = 4, D = - 215, E = 15, F = 0 A - C 1 - 4 3 cot 2a = = = B 4 4 The following figure shows the second quadrant angle 3 2a for which cot 2a = - . 4 y (−3, 4)

5 4

−3

2␣

x

3 From the figure we see that cos 2a = - . 5 Use the half-angle formulas to find sin a and cos a. 3 1 - a- b 5 215 sin a = = S 2 5 3 1 + a- b 5 15 = cos a = 2 5 S a L 63.4° A¿ = A cos2 a + B cos a sin a + C sin2 a 15 2 15 215 215 2 = 1a b + 4a ba b + 4a b = 5 5 5 5 5 (continued)

S40


y'

y 8

20. Because ƒ a ƒ = ƒ b ƒ = 2, the graph of r = 2 - 2 cos u,

- p … u … p, is a cardioid.

4

24. The equation r 2 = 25 sin 2u is of the form r 2 = a2 sin 2u

with a = 5. Thus the graph is a lemniscate that is symmetric with respect to the line u = p>4 and to the pole. Start by plotting a few points in the interval 0 … u … p>4.

x'

C¿ = A sin2 a - B cos a sin a + C cos2 a 15 2 15 2 15 215 2 b - 4a ba b + 4a b = 0 = 1a 5 5 5 5 D¿ = D cos a + E sin a 15 215 b + 15a b = 0 = - 215a 5 5 E¿ = - D sin a + E cos a 215 15 b + 15a b = 5 = 215a 5 5 5(x¿)2 + 5y¿ = 0 or y¿ = - (x¿)2

0

P 12

P 6

P 4

r 2 = 25 sin 2u

0

25 2

2513 2

25

r = 51sin 2u

0

L 3.54

L 4.65

5

U

63.4° 8 x

−8 −8

26. Use y1 and y2 as in Equation (4) and Equation (5) that

precede Example 4. Store the following constants. A = 2, B = - 8, C = 8, D = 20, E = - 24, F = - 3

The points from the above table are plotted in the following figure.

Graph y1 and y2 on the same screen to produce the following parabola.

π 5π 2 12 π 3 π 4 π 6

3π 4

5

π

−5

3

5π 4

4 −1

32. Because

B2 - 4AC = ( - 1013)2 - 4(11)(1) = 300 - 44 7 0 the graph is a hyperbola.

The complete graph can now be produced by using symmetry with respect to the line u = p>4. See the following figure. π 5π 2 12 π 3 π 4 π 6

3π 4

π

4

7π 4

3π 2


π 12 5 0

3

5π 4

3π 2

5

7π 4

28. r = 4 - 4 sin u

16. r = 5 cos 3u

Because 3 is odd, this is a rose with three petals.

u min = 0 u max = 2p u step = 0.1

π 12 0

3.25 −9.4

9.4

8 −9.75


34. r = - 4 sec u

58.

u min = 0 u max = 2p u step = 0.1 6.25

−9.4

y = r sin u

r =

p = (2)csin a - b d 3

p = (2)ccos a- b d 3

56. Since x 12 and y 5, we see that

r2 = x2 + y2 = (12)2 + (- 5)2 = 169 Thus r = 13 or - 13. Now find u. y -5 5 tan u = = = x 12 12

cos u sin u

r(r sin u) = r cos u

A 2x2 + y2 B y = x

• Subtract 2y from each side. • Complete the square.

y 12

≈ 157.4° 12 ≈ –22.6°

• cot u =

2

2

• Multiply each side by r.

2

• Square each side.

y4 + x2y2 - x2 = 0 78.

2x - 3y = 6 2r cos u - 3r sin u = 6 r(2 cos u - 3 sin u) = 6

• x = r cos u; y = r sin u

6 2 cos u - 3 sin u

Exercise Set 8.6, page 695 8 2 - 4 cos u 4 r = 1 - 2 cos u

2. r =

• Divide numerator and denominator by 2.

e = 2, so the graph is a hyperbola. The transverse axis is on the polar axis because the equation involves cos u. Let u = 0. 8 8 r = = = -4 2 - 4 cos 0 2 - 4 Let u = p.

x

(12, –5) (13, –22.6°) (–13, 157.4°)

The directions for this exercise require 0° … u 360°. The angle in (13, - 22.6°) is less than 0°, so it is not one of our answers. The above drawing shows that the 337.4° angle is coterminal with the 22.6° angle. Therefore, the rectangular coordinates (12, 5) can also be written in polar coordinates as (13, 337.4°). Hence, the answer for this exercise is given by (13, 157.4°) and (13, 337.4°).

cos u sin u

• y = r sin u; x = r cos u

(x + y )y = x 2

r =

5 Use a calculator to approximate u = tan-1 a - b to the 12 nearest tenth of a degree. u L - 22.6° The point (12, 5) lies in Quadrant IV. Thus we can write it in polar coordinates as (13, - 22.6°) or ( -13, 157.4°).

13

• r2 x 2 y 2; y r sin u

r sin u = cos u

13 1 = (2) a = (2)a b = 1 b = - 13 2 2 The rectangular coordinates of the point are (1, - 13 ).

≈ 337.4° –6

• Multiply each side by r.

r = cot u

66.

−6.25

–12

2 sin u 2r sin u 2y 0 1 12

A rectangular form of r = 2 sin u is x2 + (y - 1)2 = 12. The graph of each of these equations is a circle with center (0, 1) and radius 1.

9.4

44. x = r cos u

= = = = = =

r r2 x2 + y2 x2 + (y2 - 2y) 2 x + (y2 - 2y + 1) x2 + (y - 1)2

S41

r =

8 8 4 = = 2 - 4 cos p 2 + 4 3

4 The vertices are ( - 4, 0) and a , pb. 3

8

S42


4. r =

r =


6 3 + 2 cos u 2

6. Plotting points for several values of t yields the following • Divide numerator and denominator by 3.

2 1 + a b cos u 3

graph. y 4

2 , so the graph is an ellipse. The major axis is on the 3 polar axis because the equation involves cos u. Let u = 0. e =

r =

6 6 6 = = 3 + 2 cos 0 3 + 2 5

−4 −4

12. x = 3 + 2 cos t, y = - 1 - 3 sin t, 0 … t 6 2p

Let u = p.

y + 1 x - 3 , sin t = 2 3 2 2 cos t + sin t = 1

6 The vertices on the major axis are a , 0b and (6, p). 5 p Let u = . 2 6 6 r = = = 2 3 + 0 p 3 + 2 cosa b 2

r =

−4

y + 1 2 x - 3 2 b + ab = 1 2 3

4

x

−4

( y + 1)2 (x - 3)2 + = 1 4 9 14. x = 1 + t 2, y = 2 - t 2, t R

y 5

x = 1 + t

(1, 2)

t2 = x - 1

3 + 2 cosa

3p b 2

=

y = -x + 3 p b and 2

3p b. 2

5x

−5

y = 2 - (x - 1)

6 = 2 3 + 0

Two additional points on the ellipse are a2, a 2,

a

2

3p . 2 6

y 4

cos t =

6 6 r = = = 6 3 + 2 cos p 3 - 2

Let u =

4 x (− 1, 1)

−5

Because x = 1 + t 2 and t 2 Ú 0 for all real numbers t, x Ú 1 for all t. Similarly, y … 2 for all t. 24. Eliminate the parameter to determine an equation in x and y

that is equivalent to the two parametric equations x = t + 1, y = 1t. First, solve x = t + 1 for t.

x = t + 1 t = x - 1

8

Substitute x - 1 for t in the equation y = 1t.

y = 1x - 1 24. e = 1, r cos u = - 2

Thus the directrix is x = - 2. The distance from the focus (pole) to the directrix is d = 2. r =

ed 1 - e cos u

r =

(1)(2) 1 - (1) cos u

r =

2 1 - cos u

• Use Eq (2) because the vertical directrix is to the left of the pole. • e = 1, d = 2

Squaring both sides produces y2 = x - 1, with y Ú 0. (Note: y is nonnegative because of the information provided by the parametric equation y = 1t.) Thus the graph of the parametric equations x = t + 1, y = 1t for t Ú 0 is the upper half of a parabola. When t = 0, P is at (1, 0); when t = 4, P is at (5, 2). Thus P starts at (1, 0) and moves along the upper portion of the parabola y2 = x - 1 until it reaches (5, 2) at time t = 4. y (5, 2) 2

(1, 0)

4

x


32. The maximum height of the cycloid is 2a = 2(3) = 6.

The cycloid intersects the x-axis every 2pa = 2p(3) = 6p units. 7

−1

24. e

The maximum height is about 195 feet when t L 3.50 seconds. The range is 1117 feet when t L 6.99 seconds.


3x - 8y = - 6 -5x + 4y = 10

(1) (2)

- 10x + 8y = -7x = x = 3(- 2) - 8y = -8y = y =

20 14 -2 -6 0 0

• 2 times Eq. (2)

• Substitute -2 for x in Eq. (1). • Solve for y.

The solution is ( -2, 0).

(1) 8x + 3y = - 7 x = 3y + 15 (2)

8(3y + 15) + 3y = - 7 24y + 120 + 3y = - 7 27y = - 127 127 y = 27

• Replace x in Eq. (1). • Simplify.

127 8 x = 3 ab + 15 = 27 9

127 • Substitute for y in 27 Eq. (2).

8 127 b. The solution is a , 9 27

28. e

4x + y = 2 (1) 8x + 2y = 4 (2)

- 8x - 2y = - 4 8x + 2y = 4 0 = 0

• -2 times Eq. (1)

Because the equation 0 = 0 is an identity, an ordered pair that is a solution of Equation (1) is also a solution of 0 = 0. Thus the solutions of the original system are the solutions of Equation (1). Solving Equation (1) for y yields y = - 4x + 2 Because x can be replaced with any real number c, the solutions of the original system are all the ordered pairs of the form

3x - 4y = 8 (1) 6x - 8y = 9 (2)

8y = 6x - 9 9 3 y = x 4 8 9 3 3x - 4a x - b = 8 4 8 9 3x - 3x + = 8 2 9 = 8 2

• Simplify.

3x - 8y = - 6

1200 0

18. e

• Replace y in Eq. (1).

This is a true statement; therefore, the system of equations 5 is dependent. Let x = c. Then y = - c + 1. Thus the 2 5 solutions are ordered triples of the form ac, - c + 1b . 2

200

6. e

(2)

5 x + 1b = 2 2 5x - 5x + 2 = 2 2 = 2

36␲

0

(1)

5 y = - x + 1 2

5x + 2 a-

−1

36.

5x + 2y = 2 20. c

S43

• Solve Eq. (2) for y.

• Replace y in Eq. (1). • Simplify.

This is a false equation. Therefore, the system of equations is inconsistent and has no solution.

(c, -4c + 2) x = 25p - 500 x = - 7p + 1100 by using the substitution method.

42. Solve the system of equations e

25p - 500 = - 7p + 1100 32p = 1600 p = 50 The equilibrium price is $50. 46. Let r be the rate of the canoeist.

Let w be the rate of the current. Rate of canoeist with the current: r + w Rate of canoeist against the current: r - w

(continued)

S44


r#t (r + w) # 2 (r - w) # 4 r + w r - w 2r r 4.5 + w w

= = = = = = = = =

d 12 12 6 3 9 4.5 6 1.5

(1) (2) • Divide Eq. (1) by 2. • Divide Eq. (2) by 4.

x + 3y - 2z = 2 (1) 2y - 3z = 4 (4) 0 = 1 (6) This system of equations contains a false equation. The system is inconsistent and has no solution. c

2x + 3y - 6z = • Substitute r = 4.5 in the equation r + w = 6.

Rate of canoeist = 4.5 miles per hour Rate of current = 1.5 miles per hour

Exercise Set 9.2, page 738 3x + 2y - 5z = 6 (1) 12. c 5x - 4y + 3z = - 12 (2) 4x + 5y - 2z = 15 (3)

4

18. c 3x - 2y - 9z = - 7

2x + 5y - 6z = 6x + 9y - 18z = - 6x + 4y + 18z = 13y = y =

(1) (2) (3)

8 12 14 26 2 (4)

2x + 3y - 6z = 4 (1) -2x - 5y + 6z = -8 -2y = -4 y = 2 (5) y = 2 (4) - y = -2 0 = 0 (6)

• 3 times Eq. (1) • -2 times Eq. (2) • Divide by 13. • -1 times Eq. (3) • Divide by 2.

15x + 10y - 15x + 12y 22y 11y

-

25z 9z 34z 17z

= = = =

30 36 66 33 (4)

• 5 times Eq. (1) • -3 times Eq. (2) • Divide by 2.

The equations are dependent. Let z = c.

12x + 8y - 12x - 15y - 7y y

+ +

20z 6z 14z 2z

= 24 = -45 = -21 = 3 (5)

• 4 times Eq. (1) • -3 times Eq. (3)

2x + 3(2) - 6c = 4 x = 3c - 1

11y - 17z - 11y - 22z -39z z

= 33 (4) = -33 = 0 = 0 (6)

11y - 17(0) = 33 y = 3 3x + 2(3) - 5(0) = 6 x = 0

• Divide by -7. • -11 times Eq. (5)

• Substitute z = 0 in Eq. (4). • Substitute z = 0 in Eq. (1).

The solution is (0, 3, 0). x + 3y - 2z = 2 (1) z = 0 (2) -3x - 7y + 3z = - 1 (3) Eliminate x from Equation (2) by multiplying Equation (1) by 2 and adding it to Equation (2). Eliminate x from Equation (3) by multiplying Equation (1) by 3 and adding it to Equation (3).

16. c -2x - 4y +

x + 3y - 2z = 2 (1) 2y - 3z = 4 (4) 2y - 3z = 5 (5) Eliminate y from Equation (5) by multiplying Equation (4) by -1 and adding it to Equation (5). c

• -1 times Eq. (5)

• Substitute y = 2 and z = c in Eq. (1).

The solutions are ordered triples of the form (3c - 1, 2, c). 20. e

x 3x - 3x 3x

+ -

3y 8y 9y 8y y

+ -

4z = 9 (1) 2z = 4 (2) 12z = - 27 (2) 2z = 4 (3) 14z = - 23 y = 14z - 23

x - 3(14z - 23) + 4z = 9 x = 38z - 60

• -3 times Eq. (1)

• Solve Eq. (3) for y. • Substitute 14z - 23 for y in Eq. (1). • Solve for x.

Let z = c. The solutions are ordered triples of the form (38c - 60, 14c - 23, c). 5x + 2y + 3z = 0 32. c 3x + y - 2z = 0 4x - 7y + 5z = 0 15x + 6y + 9z = -15x - 5y + 10z = y + 19z =

(1) (2) (3) 0 0 0 (4)

20x + 8y + 12z = 0 -20x + 35y - 25z = 0 43y - 13z = 0 (5)

• 3 times Eq. (1) • -5 times Eq. (2) • 4 times Eq. (1) • -5 times Eq. (3)

(continued)


-43y - 817z 43y - 13z -830z z

= = = =

0 0 (5) 0 0 (6)

• -43 times Eq. (4)

36. x 2 + y 2 + ax + by + c = 0

c

0 + 36 + a(0) + b(6) + c = 0 1 + 25 + a(1) + b(5) + c = 0 49 + 1 + a(- 7) + b(-1) + c = 0

c

6b + c = - 36 (1) a + 5b + c = - 26 (2) - 7a - b + c = - 50 (3)

7a + 35b + 7c = - 182 - 7a - b + c = -50 (3) 34b + 8c = -232 17b + 4c = -116 (4) -24b - 4c = 144 17b + 4c = -116 (4) -7b = 28 b = -4 17(-4) + 4c = - 116 c = - 12 -7a - (- 4) - 12 = - 50 a = 6

• Let x = 0, y = 6. • Let x = 1, y = 5. • Let x = -7, y = -1.

• 7 times Eq. (2)

• Divide by 2. • -4 times Eq. (1)

• Divide by -7. • Substitute -4 for b in Eq. (4). • Substitute -4 for b and -12 for c in Eq. (3).

42. Let x1 , x2 , x3 , and x4 represent the numbers of cars per hour

that travel AB, BC, CD, and DA, respectively. Using the principle that the number of cars entering an intersection must equal the number of cars leaving the intersection, we can write the following equations. = = = =

x1 x2 x3 x4

+ + + +

60 100 50 40

The equations for the traffic intersections result in the following system of equations. x1 x1 d x2 x3

-

x4 x2 x3 x4

= 15 (1) = 50 (2) = 5 (3) = - 40 (4)

Subtracting Equation (2) from Equation (1) gives x1 - x4 = 15 x1 - x2 = 50 x2 - x4 = - 35

(5)

(6)

Because Equation (5) and Equation (6) are the same, the system of equations is dependent. Because we want to know the number of cars per hour between B and C, solve the system in terms of x2 . x1 = x2 + 50 x3 = x2 - 5 x4 = x2 + 35 Because there cannot be a negative number of cars per hour between two intersections, to ensure that x3 Ú 0, we must have x2 Ú 5. The minimum number of cars traveling between B and C is 5 cars per hour.

Exercise Set 9.3, page 745 x - 2y = 3 (1) xy = - 1 (2) x = 2y + 3 (2y + 3)y = - 1

8. e

An equation of the circle whose graph passes through the three given points is x 2 + y 2 + 6x - 4y - 12 = 0.

A: 75 + x4 B: x1 + 50 C: x2 + 45 D: x3 + 80

Adding Equation (3) and Equation (4) gives x2 - x3 = 5 x3 - x4 = - 40 x2 - x4 = - 35

Solving by back substitution, the only solution is (0, 0, 0).

S45

• Solve Eq. (1) for x. • Replace x with 2y + 3 in Eq. (2).

2y 2 + 3y + 1 = 0 (2y + 1)( y + 1) = 0 y = -

1 2

x - 2a-

• Solve for y.

y = -1

or

1 b = 3 2 x = 2

x - 2(- 1) = 3

The solutions are a2, 16. e

x = 1

• Substitute for y in Eq. (1).

1 b and (1, -1). 2

3x 2 - 2y 2 = 19 (1) x 2 - y 2 = 5 (2)

3x 2 - 2y 2 -3x 2 + 3y 2 y2 y x - ( -2) x2 - 4 x2 x 2

x 2 - 22 x2 - 4 x2 x

2

= = = =

= 19 (1) = - 15 = 4 = 2 = = = =

5 5 9 3

5 5 9 3

• Multiply Eq. (2) by -3. • Solve for y. • Substitute -2 for y in Eq. (2).

• Substitute 2 for y in Eq. (2).

The solutions are (3, - 2), ( -3, - 2), (3, 2), and ( -3, 2).

S46


20. e

Substitute 1.6x for y in Equation (1) and solve for x.

2x 2 + 3y 2 = 11 (1) 3x 2 + 2y 2 = 19 (2)

x 2 + (1.6x)2 = 252 3.56x 2 = 252 252 x2 = 3.56 25 x = 13.56 x L 13.24997

2

Use the elimination method to eliminate y . 4x 2 + 6y 2 = 22 - 9x 2 - 6y 2 = -57 = -35 - 5x 2 2 7 x = 2(7) + 3y 2 = 11 3y 2 = -3 y 2 = -1

• 2 times Eq. (1) • -3 times Eq. (2)

• Substitute for x 2 in Eq. (1).

To find the height, multiply by 1.6. y = 1.6x L 1.6(13.24997) = 21.199952

y 2 = - 1 has no real number solutions. The graphs of the equations do not intersect. The system is inconsistent and has no solution. (x + 2)2 + ( y - 3)2 = 10 (x - 3)2 + ( y + 1)2 = 13 2 x + 4x + 4 + y 2 - 6y + 9 = 10 (1)

28. e


x 2 - 6x + 9 + y 2 + 2y + 1 = 13 (2) 10x - 5

- 8y + 8 = -3

• Subtract.

10x - 8y = -6 5x + 3 (3) • Solve for y. 4 5x - 9 2 (x + 2)2 + a b = 10 • Substitute 4 for y. 25x 2 - 90x + 81 2 = 10 • Solve for x. x + 4x + 4 + 16 16x 2 + 64x + 64 + 25x 2 - 90x + 81 = 160 41x 2 - 26x - 15 = 0 (41x + 15)(x - 1) = 0 y =

x = y =

y =

15 41

or

y =

12 41

y = 2

The solutions are a-

x = 1

5(1) + 3 4

5 15 3 a- b + 4 41 4

• Substitute for x into Eq. (3).

15 12 , b and (1, 2). 41 41

36. Let x be the width of the sign.

Let y be the height of the sign. Applying the Pythagorean Theorem yields x 2 + y 2 = 252. The height is 1.6 times the width, so y = 1.6x. We need to solve the following system of equations. e

x 2 + y 2 = 252 (1) y = 1.6x (2)

The width of the sign is approximately 13.2 feet, and the height is approximately 21.2 feet.

7x + 44 = (x x2 + 10x + 24 7x + 44 = A(x + 6) 7x + 44 = (A + B)x 7 = A + B e 44 = 6A + 4B

7x + 44 A B = + + 4)(x + 6) x + 4 x + 6 + B(x + 4) + (6A + 4B)

The solution is A = 8, B = - 1. 7x + 44

=

8 -1 + x + 4 x + 6

x + 10x + 24 B C A x - 18 + + 22. = 2 x x 3 x(x - 3) (x - 3)2 2 x - 18 = A(x - 3) + Bx(x - 3) + Cx x - 18 = Ax2 - 6Ax + 9A + Bx2 - 3Bx + Cx x - 18 = (A + B)x2 + ( -6A - 3B + C)x + 9A 0 = A + B c 1 = - 6A - 3B + C -18 = 9A 2

The solution is A = - 2, B = 2, C = - 5. x - 18

2 -5 -2 + + x x 3 x(x - 3) (x - 3)2 3 2 24. x - x + 10x - 10 = (x - 1)(x 2 + 10) A Bx + C 9x 2 - 3x + 49 = + 2 2 x 1 (x - 1)(x + 10) x + 10 9x 2 - 3x + 49 = A(x 2 + 10) + (Bx + C)(x - 1) 9x 2 - 3x + 49 = (A + B)x 2 + ( -B + C)x + (10A - C) 2

=

9 = A + B c -3 = - B + C 49 = 10A - C The solution is A = 5, B = 4, C = 1. 9x 2 - 3x + 49 x - x + 10x - 10 3

2

=

5 4x + 1 + 2 x - 1 x + 10


30.

2x3 + 9x + 1

Ax + B

=

36.

Cx + D

+

2 0 μ 9 1

2

2

2

2

= A = B = 7A + C = 7B + D

x + 14x + 49

46.

=

2x

2

x + 7

+

-5x + 1

(x 2 + 7)2 x + 1 34. 2x 2 + 3x - 2 ƒ 2x 3 + 5x 2 + 3x - 8 2x 3 + 3x 2 - 2x 4

2

2x 2 + 5x - 8 2x 2 + 3x - 2

e

The solution is A = - 2, B = 2. 2x + 3x - 2 2

-2 2 = x + 1 + + 2x - 1 x + 2


12.

y

2

x

26.

4 x

2

y

2 2

x

y … 0.85(208 - 18.2) y … 0.85(189.8)

y Ú 151.84

y … 161.33

Exercise Set 9.6, page 768 14. C = 4x + 3y

(x, y)

C

(0, 8)

24

(2, 4)

20

(5, 2)

26

(11, 0)

44

8

(20, 0)

80

4

(20, 20)

140

(0, 20)

60

y

• Minimum

20 16 12

4

8 12 16 20

x

y = hours of machine 2 use

−4 −2 −2

y

y Ú 0.80(208 - 18.2) y Ú 0.80(189.8)

26. x = hours of machine 1 use

−4

20.

y … 0.85(208 - 0.7x) y … 0.85(208 - 0.7(26))

The minimum is 20 at (2, 4). y 4

2

2

y Ú 0.80(208 - 0.7x) y Ú 0.80(208 - 0.7(26))

2

2 = A + 2B - 6 = 2A - B

2x 3 + 5x 2 + 3x - 8

Substitute the sprinter’s age, 26, in the first inequality to find the minimum value of the targeted exercise heart rate range. Substitute 26 in the second inequality to find the maximum value.

2x - 6

= x + 1 +

2x + 3x - 2 2x + 3x - 2 A B 2x - 6 = + (2x - 1)(x + 2) 2x - 1 x + 2 2x - 6 = A(x + 2) + B(2x - 1) 2x - 6 = Ax + 2A + 2Bx - B 2x - 6 = (A + 2B)x + (2A - B) 2

x

Rounding to the nearest beat per minute yields a targeted exercise heart rate range of 152 to 161 beats per minute.

2x - 6 2x 3 + 5x 2 + 3x - 8

2 1 1 2 3

The solution is A = 2, B = 0, C = - 5, D = 1. 2x 3 + 9x + 1

38. Because the solution sets of the

y

inequalities do not intersect, the system has no solution.

(x + 7) x + 7 (x + 7) 2x 3 + 9x + 1 = (Ax + B)(x 2 + 7) + Cx + D 2x 3 + 9x + 1 = Ax 3 + Bx 2 + (7A + C)x + (7B + D) 2

S47

4

4

x

Cost = 28x + 25y 4x + 13y Ú 160 Constraints: c 5x + 10y Ú 100 x Ú 0, y Ú 0

(x, y)

Cost

(0, 20)

500

(12, 4)

436

(20, 0)

560

• Minimum

y

5

(12, 4) 5 10

To achieve the minimum cost, use machine 1 for 12 hours and machine 2 for 4 hours.

x

S48


28. Let x be number of standard models.

x - 3y + 5z = - 18 d x - 3y - 5z = - 14 13 x - 3y - 5z = 5

Let y be number of deluxe models. Profit = 25x + 35y 24 10 16 0

By back substitution, the solution is a

16

(0, 8)

280

12

(6, 4)

290

8

(3, 7)

320

(8, 0)

200

5

-3R1 + R2 -5R1 + R3

m

0

4

• Maximum 8 10


y - z = - 2 or y = z - 2 x - 3(z - 2) + 2z = 4 x - 3z + 6 + 2z = 4 x = z - 2

3 -1 S 4

2

m

4 1 2

-4

-3 -5 4

1 C0 0

m

1 R3 35

1 C0 0

m

-

1 1 - 12 S - 4 R3 -4

4 1 2

-4

1 2 4 C 0 1 12 0 0 1

1 8 -2R + R -3 † 2 S -1R1 + R2 1 3 1 1

-7R2 + R3

4 -1 -4

1 2 C0 1 0 0

1 1 1 S - 2 R2 -4

1 C0 0

m

1 18. C 2 1

2 -2 0

m

4 -1 -4

1 2 C0 1 0 0

5

m

2 -2 0

-3 1 0 -3 1 0

5 2 12

24. C 1

3 Constant matrix: C -1 S 4 1 C0 0

8 1 -5 † - 14 S 91 35 8 1 - 5 † - 14 S 13 1 5

• Solve for x.

Let z be any real number c. The solutions of the system are the ordered triples (c - 2, c - 2, c).

2 0S 3

1 -2R + R 3 S -3R1 + R2 1 3 -1

• Solve Eq. (2) for z. • Substitute z 2 for y in Eq. (1).

1 1S -4 1 - 12 S -4

1 - 12 S 1

-3 1 7

8 1 - 5 † - 14 S -7 0

-1 2 1 R 4 R1 -3 † 5 S 2 C2 10 1 5 5 1 2 -3 C0 1 8 † -11 S -15 0 2 16

m

-3 -1 -2

x - 3y y

-2R1 + R2 -5R1 + R3

2 5 12

5 -3 2 † -1 S 10 1

m

0 Coefficient matrix: C 2 3

-3 2 -1 0 -2 3

c

1 C0 0 x + 2y - 3z c x - 3y + 8z x - 3y - 50 -2R2 + R3

m

3

1 C0 0

-4R2 + R3

5 2 -3 1 8 † -11 S 7 0 0 = - 15 = - 11 = - 17

Because 0 = 7 is a false equation, the system of equations has no solution. -1 -3 1

3 4 -2

- 5 10 1 † 7S 6 -2

-2R1 + R2 -3R1 + R3

1 C0 0

-1 -1 4

1 40. C 2 3

m

0 2. Augmented matrix: C 2

4 7 8

x

4 -3 2 -5 2 † 4 S -11 6 12 1 -3 2 4 C0 1 - 1 † -2 S 0 4 - 4 -8

m

2

To maximize profits, produce three standard models and seven deluxe models.

2 2 6

4 2 1 R 4 R1 2 † 4S 2 C3 6 12 5 1 -3 2 4 1 C0 4 - 4 † - 8 S 4 R2 0 4 -4 -8 4 1 -3 2 C0 1 -1 † -2 S 0 0 0 0 + 2z = - 4 (1) - z = - 2 (2) 0 = - 0 (3)

12 13 , - 1, b . 5 5

m

22. C 1

y

(0, 0)

1 6. C 2 3

-5 -3 -11

3

Profit

-1R2

1 C0 0

m

(x, y)

… … … Ú

m

2x + 3y 2x + 3y Constraints: μ 2x + 3y x Ú 0, y

-1 1 4

3 -2 -11

3 2 - 11

-5 10 11 † - 13 S 13 - 24

10 -5 - 11 † 13 S - 24 13

(continued)

m


-1 1 0

1 C0 0

-1 3 1 2 0 1

1 C0 0

-4R2 + R3

1 R3 19

m

-

3 2 -19

Solving the system of equations by back substitution yields a0 = 0, a1 = 2, a2 = - 2, and a3 = 1. The interpolating polynomial is p(x) = x 3 - 2x 2 + 2x.

10 -5 -11 † 13 S -76 57

-5 10 -11 13 S -3 4

Exercise Set 10.2, page 807 2

t - u + 3v - 15w = 10 (1) c t - u + 2v - 11w = 13 (2) t - u + 2v - 13w = 14 (3)

6. a. A + B = C 3

1 2

v = 3w + 4 • Substitute 3w + 4 u + 2(3w + 4) - 11w = 13 u = 5w + 5 for v in Eq. 2. • Substitute t - (5w + 5) + 3(3w + 4) - 5w = 10 t = w + 3 5w + 5 for u and 3w + 4 for v in Eq. (1). Let w be any real number c. The solutions of the system of equations are the ordered triples (c + 3, 5c + 5, 3c + 4, c). 48. Because there are four given points, the degree of the inter-

polating polynomial will be at most 3. The form of the polynomial will be p(x) = a3 x 3 + a2 x 2 + a1 x + a0 . Use this polynomial to create a system of equations. p(x) = a3 x 3 + a2 x 2 + a1 x + a0 p( -1) = a3(- 1)3 + a2(- 1)2 + a1(- 1) + a0 = - a3 + a2 - a1 + a0 = - 5 p(0) = a3(0)3 + a2(0)2 + a1(0) + a0 = a0 = 0 p(1) = a3(1)3 + a2(1)2 + a1(1) + a0 = a3 + a2 + a1 + a0 = 1 p(2) = a3(2)3 + a2(2)2 + a1(2) + a0 = 8a3 + 4a2 + 2a1 + a0 = 4

+ a2 - a1 + a2 - a1 + a2 + a1 4a2 + 2a1

+ + + +

a0 a0 a0 a0

= = = =

-5 -0 -1 -4

-1 0 D 1 8

1 0 1 4

-1 0 1 2

0.5 1 0 0

0.25 -0.5 1 0

0.125 0.5 0.75 -3 ∞ T 0.5 2 1 0

a3 + 0.5a2 + 0.25a1 + 0.125a0 = 0.5 a2 - 0.5a1 + 0.75a0 = - 3 μ a1 + 0.5a0 = 2 a0 = -0

1 -1 c. 2B = 2C 2

-4

8 1 -2 S = C 5 3 -3

-2 -1 4S - C 2 0 -4

8 3 -2 S = C 1 3 5

8 -2 -2 S = C 4 3 -8 2

d. 2A - 3B = 2C 3

1 -1 16. AB = C 2 -2

2 -1 2

-1 5 -1

-10 6S -3

16 -4 S 6

-2 -1 4 S - 3C 2 0 -4

0 2 1S C1 -1 0

6 2S 3

8 7 -28 -2 S = C 0 14 S 3 14 - 9

0 -1 S 3

( -1)(2) + (2)(1) + (0)(0) = C (2)(2) + ( -1)(1) + (1)(0) (- 2)(2) + (2)(1) + (- 1)(0)

( - 1)(0) + (2)(-1) + (0)(3) (2)(0) + ( - 1)( - 1) + (1)(3) S ( -2)(0) + (2)(- 1) + ( -1)(3)

1 -5 1 0 T ∞ 1 1 1 4

The augmented matrix in row echelon form and the resulting system of equations are 1 0 D 0 0

b. A - B = C 3

-2 -1 4S + C 2 0 -4

(- 1)( - 1) + (2)(5) + (0)(- 1) (2)(- 1) + (- 1)(5) + (1)(- 1) ( -2)( -1) + (2)(5) + (- 1)(- 1)

The system of equations and the associated augmented matrix are - a3 -a3 μ a3 8a3 +

S49

0 = C 3 -2 2 BA = C 1 0

11 -8 13 -1 5 -1

-2 4S -5 -1 0 -1 S C 2 3 -2

2 -1 2

0 1S -1

(2)(- 1) + ( - 1)(2) + (0)(-2) = C (1)(- 1) + (5)(2) + ( - 1)( -2) (0)(- 1) + ( - 1)(2) + (3)(- 2) (2)(2) + ( -1)( -1) + (0)(2) (1)(2) + (5)(-1) + (- 1)(2) (0)(2) + ( -1)( -1) + (3)(2) (2)(0) + ( -1)(1) + (0)(- 1) (1)(0) + (5)(1) + ( - 1)( - 1) S (0)(0) + ( -1)(1) + (3)(- 1) (continued)


-4 = C 11 -8 2 36. C 3

4 2x C 3x 4x 2x c 3x 4x

-1 6S -4

5 -5 7

0 5 x 9 -5 1 S C y S = C 7 S -7 6 z 14 9 + 5z - 5y + z S = C 7 S 14 - 7y + 6z - 5y + 5z = 19 - 5y + 5z = 17 - 7y + 6z = 14

2 1 A3 = E 4 6 5

• Multiply the matrices.

• Use equality of matrices to write the system of equations.

-4 -2 2 for the matrix of the vertices is C -1 1 - 5 S , where we have 1 1 1 written the vertices in the order of points A, B, and C. Any order that goes around the figure is a valid matrix of the vertices. Reflect the figure across the y-axis. 0 1 0

0 -4 0 S C -1 1 1

4 = C -1 1

2 1 1

-2 -5 S 1

-2 1 1

2 -5 S 1

A 30.25

1

1

-2 -1 -5 S = C 0 1 0

0 -1 0

0 4 0 S C -1 1 1

-4 = C 1 1

-2 -1 1

2 5S 1

2

2 1 1

-2 -5 S 1

62. The adjacency matrix is the 5 * 5 matrix whose aij is a 1 if

there is one edge from vertex i to vertex j and 0 if there is no edge from vertex i to vertex j. 0 0 0 1 1

1 0 0 1 0

1 1 1 0 0

1 C0 0

R1 + R2 -2R1 + R3

1 C0 0

1 - R2 2

1 1 0U 0 0

A3 is the matrix that gives the walks of length 3 between any two vertices.

-3R2 + R1

1 C0 0

0 1 0

B 0.6094

0 0S 1

-2 1 4 † 1 1 -2

3 -2 0

0 1 0

0 -3 0 2 0 † - 29 - 12 2 S 1 -2 0 1

3 1 0

3

21

0 1 0

-4

0 2 2 0 † - 29 - 12 1 -2 0 21 2

3 2 1 2

2S 1 -4 2 S. 1

The inverse matrix is C - 2 -2 0 9

1 10. C 2

3

-2 -3 -6

-2R1 + R2 -3R1 + R3

2 1 1 † 0 6 0 1 C0 0

0 0S 1

-2 1 0 0 -2 † - 12 - 12 0 S 1 -2 0 1

3 1 0

1 C0 0

2R1 + R2 3R3 + R1

The vertices of the transformed triangle are A¿(- 4, 1), B¿( -2, -1), and C¿(2, 5).

0 0 A = E1 1 1

-2 1 6 † 0 -3 0

3 -5 6

6. C -1

m

2 1 1

A 0.02 5 R L 30.391 0.95


m

4

5 4 2U 1 0

After 5 months, store A has approximately 39.1% of the town’s customers.

Rotate the resulting figure 180° about the origin. R180 # C -1

6 5 4 2 1

B 0.98 0.754 B 0.05

m

2 -1 -5 S = C 0 1 0

70.

m

-2 1 1

4 2 2 4 2

a34 = 4. There are 4 walks of length 3 between vertex 3 and vertex 4.

54. Create a matrix using the vertices of the triangle. One possibility

-4 Ry # C - 1 1

1 0 2 5 4

m

S50

0 1 0

-2 1 0

0 0S 1 2 1 -3 † - 2 0 -3

0 1 0

0 0S 1

Because there are zeros in a row of the original matrix, the matrix does not have an inverse.


1 20. C 2

3

2 3 6

-1 x 5 -1 S C y S = C 8 S -2 z 14

3

42. † 1

2

0 The inverse of the coefficient matrix is C - 1 -3

2 -1 0

-2 2 -2

-1 1 4† = - †3 3 2

2 -2 -2

4 -1 † 3

1 = - †0 0

2 -8 -6

4 -13 † -5

1 = 8†0 0

2 1 -6

1 = 8†0 0

2 1

-1 1 S. 1

Multiply each side of the equation by the inverse: x 0 C y S = C -1 z -3

2 -1 0

-1 5 2 1S C 8S = C 1S 1 14 -1

The solution is (2, 1, -1). 26. Write a system of equations.

40 + 40 + 25 + x2 105 + x2 = 4 4 d 25 + 60 + 40 + x1 125 + x1 x2 = = 4 4

= 8(1)(1)a

x1 =

Rewrite the system of equations. e

4. x1 =

Solve the system of equations using an inverse matrix.

B 4 15 C 1 15

4 -1

1 4 15 SB 4 -1 15

-1 x1 105 RB R = B R 4 x2 125 4

-1 x1 15 RB R = C 1 4 x2 15

1 105 15 SB R 4 125 15

109

x 3 B 1 R = C 121 S x2

x2 =

3

24. x3 =

Exercise Set 10.4, page 831 2 -6

9 ` = 2 # 2 - (- 6)(9) = 4 + 54 = 58 2

14. M13 = `

1 6

3 ` = 1( -2) - 6(3) = - 2 - 18 = - 20 -2

C13 = ( -1)1 + 3 # M13 = 1 # M13 = 1( -20) = - 20 20. Expanding by cofactors of row 1 yields

3 †2 8

-2 -3 -2

0 2 † = 3C11 + (- 2)C12 + 0 # C13 5 -3 2 2 2 ` + 2` ` + 0 = 3` -2 5 8 5 = 3(- 15 + 4) + 2(10 - 16) + 0 = 3( -11) + 2( -6) = - 33 + ( - 12) = - 45

9 8

5 ` 7

4 13 8 19 4

†

• -3R 1 + R 2 • -2R 1 + R 3 1 • - R2 8 • 6R 2 + R 3

19 b = 38 4

2 5 2 ` 5

5 ` 7 9 ` 8

`

5 ` 7

`

2 5

=

23 23 63 - 40 = = 14 - 25 - 11 11

=

16 - 45 -29 29 = = 14 - 25 - 11 11

The solution is a-

The temperatures, to the nearest tenth of a degree, are x1 = 36.3°F and x2 = 40.3°F.

2. `

-5

• R1 4 R 2

Exercise Set 10.5, page 836 `

4x1 - x2 = 105 - x1 + 4x2 = 125

0

4

13 8 †

S51

23 29 , b. 11 11

2 5 1 7 ∞ 4 0 3 2

-3 4 3 0

-3 -1 ∞ 1 0

2 5 1 7 ∞ 4 0 3 2

-5 8 1 -1

-3 -1 ∞ 1 0

=

157 168

Exercise Set 11.1, page 852 (- 1)n + 1 (- 1)1 + 1 1 , a1 = = , n(n + 1) 1(1 + 1) 2 (- 1)2 + 1 (- 1)3 + 1 1 1 = - , a3 = = , a2 = 2(2 + 1) 6 3(3 + 1) 12 (- 1)8 + 1 1 a8 = = 8(8 + 1) 72

6. an =

28. a1 = 1, a2 = 22 # a1 = 4 # 1 = 4, a3 = 32 # a2 = 9 # 4 = 36

S52

44.


12! 12 # 11 # 10 # 9 # 8! 12 # 11 # 10 # 9 = = = 495 4! 8! 4! 8! 4#3#2#1 6

6

54. a (2i + 1)(2i - 1) = a (4i 2 - 1) i=1

i=1

= (4 # 12 - 1) + (4 # 22 - 1) + (4 # 32 - 1) + (4 # 42 - 1) + (4 # 52 - 1) + (4 # 62 - 1) = 3 + 15 + 35 + 63 + 99 + 143 = 358

Exercise Set 11.2, page 859 16. a6 = - 14, a8 = - 20

a8 a8 - a6 2 - 20 - (- 14) 2 -3 an a6 - 14 a1

= = = =

= a6 + 2d = d

• Solve for d.

= d

1 1 - 1 = 2 2 1 5 2 a3 - a2 = - = 9 2 18 a2 - a1 =

The difference between successive terms is not constant. The sequence is not arithmetic. 1 a2 2 1 = = a1 1 2 2 a3 9 4 = = a2 1 9 2 The ratio of successive terms is not constant. The sequence is not geometric.

= d

a1 + (n - 1)d a1 + (6 - 1)( -3) a1 + ( -15) 1

a15 = 1 + (15 - 1)( -3) = 1 + (14)(-3) = - 41 22. S20 =

n! . Find the first three terms of the sequence. nn 1 2 1! 2! 2 3! 6 a1 = 1 = 1, a2 = 2 = = , a3 = 3 = = 4 2 27 9 1 2 3

36. an =

20 (a + a20) 2 1

a1 = 1 - 2(1) = - 1 a20 = 1 - 2(20) = - 39 S20 = 103-1 + (- 39)4 = 10( -40) = - 400

The sequence an = 40. r =

c1 c2 c3 c4 c5

= = = = =

a1 a1 a1 a1 a1

+ + + + +

• There are seven terms, so n = 7.

d = 7 + 2 = 9 2d = 7 + 4 = 11 3d = 7 + 6 = 13 4d = 7 + 8 = 15 5d = 7 + 10 = 17


a2 3 6 = = = r a1 8 4 an = a1 r n - 1 3 n-1 an = 8a b 4

4 4 , a1 = , n = 14 3 3

Sn =

S14 =

a1(1 - rn) 1 - r 4 14 4 c1 - a b d 3 3

4 - 263,652,487 c d 3 4,782,969

= 4 1 1 3 3 1,054,609,948 = L 220.49 4,782,969

34. a = 7, c1 , c2 , c3 , c4 , c5 , b = 19

an = a1 + (n - 1)d 19 = 7 + (7 - 1)d 19 = 7 + 6d d = 2

n! is neither arithmetic nor geometric. nn

3 n 4 3 1 3 To find the first term, let n = 1. Then a1 = a b = . 4 4 3 The common ratio is r = . 4

48. The general term is an = a b .

S =

a1 1 - r 3 4

3 4

3 4 = S = = = 3 1 3 3 1 1 - a b 4 4 4 95>1000 95 95 3 3 + + + Á = + 10 1000 100,000 10 1 - 1>100 3 95 392 196 = + = = 10 990 990 495

62. 0.395 =


70. A =

A =

P3(1 + r)m - 14 r

;

P = 250, r =

0.08 48 250c a 1 + b - 1d 12

0.08 12 The future value is $14,087.48. D(1 + g) i - g 1.87(1 + 0.15) = 0.20 - 0.15 = 43.01

S53

Prove the statement is true for n = k + 1. That is, prove

0.08 , m = 12(4) = 48 12

P k + 1 = a1 -

L 14,087.48

=

1 1 1 1 b a1 - b Á a1 b a1 b 2 3 k + 1 k + 2

1 k + 2

Because an = a 1 -

1 1 b , ak + 1 = a1 b. n + 1 k + 2

P k + 1 = P k # ak + 1 =

1 # 1 a1 b k + 1 k + 2 1 #k + 1 1 = = k + 1 k + 2 k + 2

76. Stock value =

• D = 1.87, g = 0.15, i = 0.20

By the Principle of Mathematical Induction, the statement is true for all positive integers n.

The manager should pay $43.01 for each share of stock.

16. If a 7 1, show that an + 1 7 an for all positive integers n.

Exercise Set 11.4, page 877 8. Sn = 2 + 6 + 12 + Á + n(n + 1) =

n(n + 1)(n + 2) 3

1. Because a 7 1, a # a 7 a # 1 or a2 7 a. Thus the statement is true when n = 1. 2. Assume the statement is true for n = k. ak + 1 7 ak

1(1 + 1)(1 + 2) = 2. 1. When n = 1, S1 = 1(1 + 1) = 2; 3 Therefore, the statement is true for n = 1.

Because a k + 1 7 a k and a 7 0,

2. Assume the statement is true for n = k. Sk = 2 + 6 + 12 + Á + k(k + 1) k(k + 1)(k + 2) = • Induction hypothesis 3 Prove the statement is true for n = k + 1. That is, prove Sk + 1

(k + 1)(k + 2)(k + 3) = . 3

Because an = n(n + 1), ak + 1 = (k + 1)(k + 2). k(k + 1)(k + 2) + (k + 1)(k + 2) 3 k(k + 1)(k + 2) + 3(k + 1)(k + 2) = 3 (k + 1)(k + 2)(k + 3) = • Factor out (k + 1) 3 and (k + 2) from each term.

Sk + 1 = Sk + ak + 1 =

By the Principle of Mathematical Induction, the statement is true for all positive integers n. 12. Pn = a1 -

1 1 1 1 b a1 - b Á a1 b = 2 3 n + 1 n + 1

1. Let n = 1; then P1 = a1 -

1 1 1 1 b = ; = . 2 2 1 + 1 2

The statement is true for n = 1. 2. Assume the statement is true for n = k. P k = a1 -

1 1 1 1 b a1 - b Á a1 b = 2 3 k + 1 k + 1


Prove the statement is true for n = k + 1. That is, prove a k + 2 7 a k + 1. a(a k + 1) 7 a(a k ) ak + 2 7 ak + 1 By the Principle of Mathematical Induction, the statement is true for all positive integers n. 20. 1. Let n = 1. Because log10 1 = 0, log10 1 6 1.

The inequality is true for n = 1. 2. Assume log10 k 6 k is true for some positive integer k (induction hypothesis). Prove the inequality is true for n = k + 1. That is, prove log10 (k + 1) 6 k + 1. Because log10 x is an increasing function, log10(k + 1) … log10(k + k) = log10 2k = log10 2 + log10 k 6 1 + k Thus log10(k + 1) 6 k + 1. By the Principle of Mathematical Induction, log10 n 6 n for all positive integers n.

Exercise Set 11.5, page 882 4. a

10 # 9 # 8 # 7 # 6 # 5! 10 # 9 # 8 # 7 # 6 10 10! = = # # # # b = 5! 5! 5! 5! 5 4 3 2 1 5 = 252

22. (3x + 2y)4

= (3x)4 + 4(3x)3(2y) + 6(3x)2(2y)2 + 4(3x)(2y)3 + (2y)4 = 81x4 + 216x3y + 216x2y2 + 96xy3 + 16y4

S54


c. “At least 5 seniors” means 5, 6, or 7 seniors are finalists

28. (2x - 1y ) 7 = a b(2x)7 + a b(2x)6 ( - 1y )

7 0

7 1

7 7 + a b (2x)5 ( - 1y )2 + a b(2x)4 ( - 1y )3 2 3 7 7 + a b (2x)3 ( - 1y )4 + a b(2x)2 ( - 1y )5 4 5 7 7 + a b (2x) ( - 1y ) 6 + a b ( - 1y )7 6 7 7 6 = 128x - 448x 1y + 672x 5y - 560x 4y1y + 280x 3y 2 - 84x 2y 2 1y + 14xy 3 - y 3 1y 38. a

10 10 b (x-1>2 )10 - 6 + 1(x1>2 )6 - 1 = a b (x-1>2)5(x1>2)5 = 252 6 - 1 5


(there are only 7 seniors). Because the events are related by or, sum the number of ways each event can occur.

C(7, 5) # C(8, 5) + C(7, 6) # C(8, 4) + C(7, 7) # C(8, 3) = 21 # 56 + 7 # 70 + 1 # 56 = 1176 + 490 + 56 = 1722

There are 1722 ways in which the finalists can include at least 5 seniors.

Exercise Set 11.7, page 896 6. Let R represent the Republican, D the Democrat, and I the

Independent. The sample space is 5RD, RI, DI6.

14. 5HHHH, HHHT, HHTH, HTHH, THHH6

22. S = 5HHHH, HHHT, HHTH, HTHH, THHH,

HHTT, HTTH, TTHH, HTHT, THTH, TTHH, TTTH, TTHT, THTT, HTTT, TTTT6

12. Because there are four palettes and each palette contains

four colors, by the Fundamental Counting Principle there are 4 # 4 # 4 # 4 = 256 possible colors.

Therefore, n(S) = 16.

a. E1 = 5HHTT, HTTH, TTHH, HTHT, THTH, TTHH6

16. There are three possible finishes (first, second, and third) for

the 12 contestants. Because the order of finish is important, this is a permutation of 12 contestants selected 3 at a time. P(12, 3) =

P(E1) =

b. E2 = 5HHHH, HHHT, HHTH, HTHH, THHH6

12! 12! = = 12 # 11 # 10 = 1320 (12 - 3)! 9!

P(E2) =

There are 1320 possible finishes. 20. Player A matched against Player B is the same tennis match

Let E1 = 52, 4, 66 (the number is even). Let E2 = 53, 66 (the number is divisible by 3). Then E1 ¨ E2 = 566, so the events are not mutually exclusive. The probability of E1 ´ E2 (an even number or a number divisible by 3) is

26! 26 # 25 # 24! 26! = = = 325 2!(26 - 2)! 2! 24! 2 # 1 # 24!

There are 325 possible first-round matches.

P(E1 ´ E2) = P(E1) + P(E2) n(E2) n(E1) + = n(S) n(S) 3 2 1 = + - = 6 6 6

22. The person who refuses to sit in the back can be placed in

any one of the three front seats. Similarly, the person who refuses to sit in the front can be placed in any of the three back seats. The remaining four people can sit in any of the remaining seats. The number of seating arrangements is 3 # 3 # 4 # 3 # 2 # 1 = 216. 30. a. The number of ways in which 10 finalists can be selected

from 15 semifinalists is a combination of 15 students selected 10 at a time. C(15, 10) = 3003

n(E2) 5 = n(S) 16

28. S = 51, 2, 3, 4, 5, 66. Therefore, n(S) = 6.

as Player B matched against Player A. Therefore, this is a combination of 26 players selected 2 at a time. C(26, 2) =

n(E1) 6 3 = = n(S) 16 8

38.

C(3, 2) # C(5, 2) 3 # 10 3 = = C(8, 4) 70 7

40. Yes, because the card was replaced. The probability of an ace

on each draw is

There are 3003 ways in which the finalists can be chosen. b. The number of ways in which the 10 finalists can include

3 seniors is the product of a combination of 7 seniors selected 3 at a time and a combination of 8 remaining students selected 7 at a time. C(7, 3) # C(8, 7) = 35 # 8 = 280

There are 280 ways in which the finalists can include 3 seniors.

- P(E1 ¨ E2) n(E1 ¨ E2) n(S) 4 2 = 6 3

P(two aces) =

4 1 = . 52 13 1 # 1 1 = 13 13 169

46. This is a binomial experiment; p =

1 3 , q = , n = 8, and 4 4

k = 3. 8 1 3 3 5 243 1 a b a b a b = 56a b a b L 0.2076 3 4 4 64 1024

ANSWERS TO SELECTED EXERCISES Exercise Set P.1, page 14 1. Integers: 0, - 44, 181, 53; rational numbers: -

1 , 0, - 44, 3.14, 181, 53; irrational numbers: p, 5.05005000500005 . . .; prime number: 53; 5

real numbers: all the numbers are real numbers.

3. 2, 4, 6, 8

11. 50, 1, 2, 36

15. 51, 36

13.

25. 5x ƒ - 2 6 x 6 36, 29. 5x ƒ x Ú 26, 35. 30, 14, 41. 47.

37.

−5 −4 −3 −2 −1 0 1 2 3 4 5

43. 49.

−5 −4 −3 −2 −1 0 1 2 3 4 5

57. p2 + 10

31. (3, 5),

−5 −4 −3 −2 −1 0 1 2 3 4 5

61. 3x - 1

59. 9

73. 0 6 ƒ x - 4 ƒ 6 1

45.

−5 −4 −3 −2 −1 0 1 2 3 4 5

65. ƒ x - 3 ƒ

39.

33. 3 -2, q ),

83. 8

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

53. -5

−5 −4 −3 −2 −1 0 1 2 3 4 5

67. ƒ x + 2 ƒ = 4

81. -45

79. 7

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

51.

−5 −4 −3 −2 −1 0 1 2 3 4 5

77. 29

9. 5- 3, -2, -1, 0, 1, 2, 3, 4, 66

23. B is a subset of A.

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

63. ƒ m - n ƒ

75. -2000

7. 0, 1, 2, 3 21.

19. A

27. 5x ƒ - 5 … x … - 16,

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

5. 3, 5, 7, 9

17. 5- 2, 0, 1, 2, 3, 4, 66

69. ƒ a - 4 ƒ 6 5

87. -72

85. 12

89. 19

55. 12

71. ƒ x + 2 ƒ 7 4 93. -3

91. 13

95. Associative property of multiplication 97. Distributive property 99. Commutative property of multiplication 101. Identity property of multiplication 103. Reflexive property of equality 105. Transitive property of equality 107. Inverse property of multiplication 109. No. (8 , 4) , 2 = 2 , 2 = 1, 8 , (4 , 2) = 8 , 2 = 4

111. All but the multiplicative inverse property

9 115. - 12x + 6y + 5 117. 2a 119. 21a + 6 121. a + 4 4

123. 6 square inches

113. 6x - 13

125. 66 beats per minute

127. 100 feet

Prepare for This Section (P.2), page 17 PS1. 32

PS2.

1 16

PS3. 64

PS4. 314,000

PS5. False

PS6. False

Exercise Set P.2, page 29 1. - 125

3. 1

29. -18m5n6

5. 31.

1 16

7. 32

1

33.

x6

1 4a4b 2

47. 2.7 * 108

49. 1.5 * 10-11

67. -12x11>12

69. 3x 2y 3

87. -1312

4

71.

89. -10 13

11. -2

9. 27

35. 2x

37.

51. 7.2 * 1012

4z 2>5 3

45 + 1915 20

3

91. 17y14y

119. 213 + 5

129. L3.13 * 107 seeds

b10 a10

2 x

15. 5z6

4

75.

3a1>12 b

3

93. -14x2y1y

121.

57. - 16

55. 8 77. 315

3 21x 111. x

123.

133. 8 minutes

21. -

19. 6a3b9

41. 5.62 * 10-10

95. 17 + 715 3

3 14 109. 2

2x + 71x + 6 9 - 4x

131. L 1.38 * 10-2

17. x4 y7

39. 2.011 * 1012

53. 8 * 10-16

73. 6x 5>6y 5>6

110 103. x + 41x - 3 + 1 105. 12 107. 6 117. -

13.

3

79. 213 97. - 7

1 27

x - 20

61. 3

81. - 315

2 3

4a

4

25.

2y 2 3x 2

12

27.

a3b

45. - 0.0000023

63. 16

65. 8ab 2

83. 2 ƒ x ƒ y 16y

99. 12z + 1z - 6

125.

3

23.

43. 31,400,000

313 - 12 113. 13

x2 - 315x + 10 2

59.

28x 2 y

3

85. 2ay2 12y

101. x + 41x + 4

315 - 3 115. 4

315 - 31x 5 - x

127.

1 14 + h + 2

135. L 1.66 * 10 -24 grams

A1

A2

ANSWERS TO SELECTED EXERCISES

139. a. 7.5 * 10-13 b. 4.8 * 10-11 c. 0.750628 d. 19.1987 e. 68.0755 f. The percent error is very small for everyday speeds. g. As the speed of the object approaches the speed of light, the denominator of the kinetic energy equation

137. a. 56%

b. 24%

approaches 0, which implies that the kinetic energy is approaching infinity. Thus it would require an infinite amount of energy to move a particle at the speed of light.

Prepare for This Section (P.3), page 32 PS1. -6a + 12b

PS2. - 4x + 19

PS4. -x 2 - 5x - 1

PS3. 3x 2 - 3x - 6

PS5. False

PS6. False

Exercise Set P.3, page 37 1. D

3. H

5. G

d. 1

e. x , -1

7. B

15. a. 2x + 3x + 4x + 5

3

4

23. 5x + 11x + 3 2

3

2

25. 9w + 8w - 2w + 6 3

33. 9x - 18x + 23x - 6 3

11. a. x 2 + 2x - 7

9. J

2

4

39. 6x4 - 19x 3 + 26x 2 - 29x + 10 59. 16w2 + 8wz + z2 73. -1

75. 33

c. 2, 3, 4, 5

27. -2r + 3r - 12 2

41. 10x 2 + 22x + 4

51. 18x 2 + 55xy + 25y 2

61. x 2 + 10x + 25 - y 2

83. 11,175 matches

4

3

2

e. 2x , 3x , 4x , 5

29. - 3u - 2u + 4

13. a. x 3 - 1 17. 3

19. 5

c. 1, -1

b. 3

21. 2

31. 8x + 18x - 67x + 40

2

3

2

37. 2x - 3x - 2x + 33x - 30 4

3

43. y 2 + 3y + 2

53. 6p2 - 11pq - 35q2

63. 12d 2 + 4d - 8

b. 3.6 pounds

e. x 2, 2x, - 7

d. 1

d. 2

2

3

77. a. 1.6 pounds

b. 0.085 second

b. 4

c. 1, 2, -7

35. 10x + 18x - 36x + 8x + 24

2

49. 10x 2 - 57xy + 77y 2

b. 2

45. 4z2 - 19z + 12 55. 9x 2 - 25

65. r 3 + s 3

79. a. 72p cubic inches

85. 14.8 seconds; 90.4 seconds

2

47. a2 + 3a - 18

57. 9x4 - 6x 2y + y 2

67. 60c 3 - 49c 2 + 4

b. 300p cubic centimeters

69. 29

71. -17

81. a. 0.076 second

87. Yes. The ball is approximately 4.4 feet high when it crosses

home plate.

Mid-Chapter P Quiz, page 39 1. 120 [P.1]

2. 14x - 27 [P.1]

8. 4a2 + 28a + 49 [P.3]

3.

4y7 x5

3

[P.2] 4. -72ab4 [P.2] 5. 2ab 3c 2 22ac 2 [P.2] 6. -

6 + 415 [P.2] 7. 6x2 + 7xy - 20y2 [P.3] 11

9. 8x3 - 2x2 - 29x + 21 [P.3]

Prepare for This Section (P.4), page 40 PS1. 3x 2

PS2. -36x6

PS3. a. (x 2)3

b. (x3)2

PS4. 3b3

PS5. 7

PS6. 1

Exercise Set P.4, page 48 1. 5(x + 4)

3. -3x(5x + 4)

13. (x + 5)(x + 1)

5. 2xy(5x + 3 - 7y)

15. (6x + 1)(x + 4)

23. Factorable over the integers 31. (2a + 7)(2a - 7)

25. Not factorable over the integers

33. (1 + 10x)(1 - 10x)

41. x(x + 3)(x - 3)(x2 + 9)

43. (x + 5)2

53. (2x - 3y)(4x + 6xy + 9y ) 2

61. (xy + 5)(xy - 1) 71. (3w + 2)(2w - 5) 2

7. (x - 3)(2a + 3b)

17. (17x + 4)(3x - 1)

2

63. 4x(x2 - 2)(x2 + 1) 73. 2(3x - 1)(3x + 1)

4

49. (z2 + 2w2)2

57. (x - 3)(x - 3x + 3) 2

65. (z + 2)(z - 2)(z2 + 5)

51. (x - 2)(x2 + 2x + 4)

59. (x 2 - 3)(x 2 + 2)

67. (3x + 1)(x2 + 2)

75. (2x - 1)(2x + 1)(4x + 1) 2

29. (x + 3)(x - 3)

39. (x + 5)(x - 5)(x2 + 25)

37. 6(x + 6)(x - 6)

47. (2x + 3)2

55. (2 - x )(4 + 2x + x ) 2

11. (a - 12)(a + 2)

21. (6x + 5)(x + 3)

27. Not factorable over the integers

35. (x + 3)(x - 1)

45. (a - 7)2 2

9. (x + 3)(x + 4)

19. (3x + 8y)(2x - 5y)

69. (x - 1)(ax + b)

77. a(3x - 2y)(4x - 5y)

79. b(3x + 4)(x - 1)(x + 1) 81. 2b(6x + y) 83. (w - 3)(w - 12w + 39) 85. (x + 3y - 1)(x + 3y + 1) 87. Not factorable over the integers 89. (2x - 5)2(3x + 5) 91. (2x - y)(2x + y + 1) 93. 8 95. 64 2

2


A3

Prepare for This Section (P.5), page 49 8 5

PS1.

PS2.

xz wy

PS3. x + 3

PS4. 2x(2x - 3)

PS5. (x - 6)(x + 1)

PS6. (x - 4)(x 2 + 4x + 16)

Exercise Set P.5, page 57 1.

x + 4 3

x - 3 x - 2

3.

5.

(2y + 3)(3y - 4) 19. (2y - 3)( y + 1)

a2 - 2a + 4 a - 2

1 21. a - 8

7. -

x + 8 x + 2

3p - 2 23. r

9. -

4y 2 + 7 y + 7

3y - 4 25. y + 4

8

11. -

3

ab

13.

8x(x - 4) 27. (x - 5)(x + 3)

10 27q 29.

2

15.

x(3x + 7) 2x + 3

-2x2 + 14x - 3 (x - 3)(x + 3)(x + 4)

33.

(2x - 1)(x + 5) x(x - 5)

35.

- q 2 + 12q + 5 (q - 3)(q + 5)

37.

3x2 - 7x - 13 (x + 3)(x + 4)(x - 3)(x - 4)

41.

4x + 1 x - 1

x - 2y y( y - x)

45.

(5x + 9)(x + 3) (x + 2)(4x + 3)

47.

(b + 3)(b - 1) (b - 2)(b + 2)

49.

x - 1 x

55.

-x - 7 x + 6x - 3 2

57.

2x - 3 x + 3

59.

a + b ab(a - b)

61.

(b - a)(b + a) ab(a + b ) 2

2

x + 3 2x + 3

7z(2z - 5) (2z - 3)(z - 5)

31.

43.

17.

51. 2 - m2

(x + 2)(3x - 1) x2

-x2 + 5x + 1

53.

63. a. L136.55 miles per hour

39.

x2 2v1v2 v1 + v2

b.

65.

2x + 1 x(x + 1)

3x2 - 4 67. x(x - 2)(x + 2)

Prepare for This Section (P.6), page 59 PS1. 15x2 - 22x + 8

PS2. 25x2 - 20x + 4

PS3. 416

PS4. -54 + 15

PS5.

17 + 8 12 7

PS6. b

Exercise Set P.6, page 65 1. 9i

3. 7i 12

21. - 40

5. 4 + 9i

23. - 10

25. 19i

7. 5 + 7i

9. 8 - 3i 12

27. 20 - 10i

7 2 15 29 i 43. 1 + i 45. i 41. 53 53 41 41 59. -i

61. -1

63.

11. 11 - 5i

29. 22 - 29i

31. 41

13. -7 + 4i

15. 8 - 5i

17. - 10

33. 12 - 5i

35. -114 + 42i 12

19. -2 + 16i

37. -6i

39. 3 - 6i

5 12 + i 49. 2 + 5i 51. -16 - 30i 53. - 11 - 2i 55. - i 57. -1 47. 13 13

1 13 3 13 + i 65. - + i 2 2 2 2

67.

1 1 + i 2 2

Chapter P Review Exercises, page 70 1. Integer, rational, real, prime [P.1] 6. 3, 5, 7, 9 [P.1] 10. {x| x 7 -1},

7. {1, 2, 3, 5, 7, 11} [P.1] −5 −4 −3 −2 −1 0 1 2 3 4 5

12. ( - q , -1] ´ (3, q ), 18. -x - 7 [P.1]

2. Irrational, real [P.1] 8. {5} [P.1]

27. Distributive property [P.1]

20. 5 [P.1]

9. {x | -3 … x 6 2},

[P.1] 11. (-4, 2],

−5 −4 −3 −2 −1 0 1 2 3 4 5

19. 10 [P.1]

3. Rational, real [P.1]

14. p - 2 [P.1]

22. -144 [P.1]

28. Commutative property of addition [P.1]

30. Closure property of addition [P.1]

31. Identity property of addition [P.1]

33. Symmetric property of equality [P.1]

5. 0, 1, 4, 9 [P.1]

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

[P.1] 13. 7 [P.1]

21. -256 [P.1]

4. Rational, real [P.1]

23. -9 [P.1]

[P.1]

[P.1] 15. 4 - p [P.1] 24. 0 [P.1]

16. 11 [P.1]

25. 23 [P.1]

26. 104 [P.1]

29. Associative property of multiplication [P.1] 32. Identity property of multiplication [P.1]

34. Transitive property of equality [P.1]

35. - 6x + 23 [P.1]

17. 3 [P.1]

36. 50x - 63 [P.1]

A4


37. -

1 [P.2] 32

38. -1 [P.2]

44. 0.000000431 [P.2] 52.

9b 20 a10

2y 23 x 18

101x + 15 [P.2] 4x - 9

81. 4x + 20x + 25 [P.3]

68. 4 - 121x + 9x [P.2]

82. 16x - 25y [P.3]

90. (3x + 5)(x - 3) [P.4]

2

86. (3a - 4)(3x + 8) [P.4] 91. 6xy (x - 6)(x + 4) [P.4]

95. (x 2 - 6)(x 2 + 1) [P.4]

74. -45 [P.3]

x(3x + 10) [P.5] (x + 3)(x - 3)(x + 4)

108.

110.

x + 4 [P.5] 5x + 8

111. 5 + 8i [P.6]

116. 29 + 22i [P.6]

112. 2 - 3i22 [P.6]

117. 8 + 6i [P.6]

118. i [P.6]

6. 1.37 * 10-3 [P.2] 12. -

6 + 917 [P.2] 59

7.

x 5>6 y 9>4

13. -

2. [-3, 4),

−5 −4 −3 −2 −1 0 1 2 3 4 5

76. 2b2 + 8b - 8 [P.3]

80. 10x 2 - 33x - 7 [P.3]

2

89. (2x + 3)(x + 4) [P.4]

93. (3x - 10)(3x + 10) [P.4]

97. (x - 3)(x 2 + 3x + 9) [P.4]

3x - 2 [P.5] x + 4

2x - 5

104.

4x2 - 10x + 25

x(5x - 7) [P.5] (x + 3)(x + 4)(2x - 1) 114. - 2 + 10i [P.6]

120. -

109.

3

8. 7xy 13xy [P.2]

9 + 4 15 [P.2] 2

9. 22 - 1613 [P.2]

2x - 9 [P.5] 3x - 17

115. 8 + 6i [P.6]

14 23 i [P.6] 25 25

3. 2x - 4 [P.1]

4.

4 4 2

9x y

[P.2]

20. (x - 4)(x + 4)(x2 + 1) [P.4]

21. -

x + 3 [P.5] x + 5

26. 7 + 2i15 [P.6]

10. x - 41x + 4 + 8 [P.2]

14. 3x 3 - 4x 2 - 4x + 2 [P.3]

17. (7x - 1)(x + 5) [P.4]

x(2x - 1) [P.5] 2x + 1

[P.1]

[P.5]

5.

96bc 2 a5

[P.2]

4

[P.2]

16. 6x 3 + 3x 2 - 34x - 10 [P.3]

25.

23 + 9 17 [P.2] 6

100. (a - b)(a + b)(2a + b) [P.4]

113. 6 - i [P.6]

119. - 3 - 2i [P.6]

71.

Chapter P Test, page 73 1. Distributive property [P.1]

[P.2]

84. 6a b (4a + 2ab - 3b 2 ) [P.4] 2 3

92. -2a b (a - 2)(a + 3) [P.4]

103.

107.

323x [P.2] x

88. (x - 5)(x + 3) [P.4]

96. (x - 1)(x + 1)(x 2 + 3) [P.4]

102. 3xy 2(x - 2)(x + 2)(x 2 + 1) [P.4]

2x + 1 [P.5] x + 3

70.

2 3

101. 2b 2(3a + 5b)(4a - 9b) [P.4] 106.

27y 9

65. - 9 + 515 [P.2]

2

79. 3x 2 + 2x - 8 [P.3]

2 2

99. (x + y)(x - y)(2x + 1)(2x - 1) [P.4]

2x + 3 [P.5] 2x - 5

3

75. - a 2 - 2a - 1 [P.3]

98. 3(x + 4)(x 2 - 4x + 16) [P.4]

105.

3b

x6

51. -

[P.2]

5

3

3 12 [P.2] 2

69.

87. (x - 2)(x + 9) [P.4]

2

2a2

64. 2x 2y 3 22x2y [P.2]

83. 2xy (6x y + 5xy - 17) [P.4] 2

50.

43. 35,000 [P.2]

[P.2] 57. 4x1>2 [P.2] 58. x3>4y2 [P.2] 59. 4ab3 13b [P.2]

a b

78. 12y 4 - 23y 3 + 10y 2 - 32y + 40 [P.3] 2

85. (3x - y)(-x + 5) [P.4]

1 1>6 11>4

63. 8a2b2 12b [P.2]

73. -x 3 - 7x 2 + 4x + 5; 3, -1, 5 [P.3]

2

49. - 24x7y 5 [P.2]

48. 9 [P.2]

62. -5y 2 12x [P.2]

67. 44 - 16 17 [P.2]

77. 6x 3 + 8x 2 - 35x + 18 [P.3]

94. (5x - 3y)2 [P.4]

1 [P.2] 6

47.

42. 1.7 * 10-6 [P.2]

41. 6.2 * 105 [P.2]

[P.2]

x4

3

61. -3y 2 25x 2y [P.2]

66. - 12 + 912 [P.2]

y3

[P.2] 55. x1>4 [P.2] 56.

3

60. 2a 13ab [P.2]

72.

[P.2] 54. -

y10

40.

46. -9 [P.2]

45. 5 [P.2]

4x 2

[P.2] 53.

39. 2z4 [P.2]

22.

28x [P.2] 2

15. 6a 2 - 13ab - 63b 2 [P.3]

18. (a - 4b)(3x - 2) [P.4]

(x - 6)(x + 1) [P.5] (x + 3)(x - 2)(x - 3)

27. 2 + 2i [P.6]

11.

28. 22 - 3i [P.6]

29.

19. 2x(2x - y)(4x 2 + 2xy + y 2) [P.4]

x + 1 [P.5] x + 4

23.

11 23 + i [P.6] 26 26

24.

x(x + 2) [P.5] x - 3

30. i [P.6]

Exercise Set 1.1, page 81 1. 15

3. - 4

5.

27. Contradiction

9 2

7. 1

9.

29. Identity

108 23

11.

2 9

13. 12

15. 16

31. Conditional equation

17. 9

33. - 4, 4

19.

22 13

35. 7, 3

21.

95 18

23. Identity

37. 8, -3

39. 2, -8

25. Conditional equation 41. 20, -12


43. No solution 45. 12, -18 53. 72 square yards

a + b a - b , 2 2

47.

55. 4.6 minutes

49. 882 cubic centimeters

A5

51. After 3 hours and after 5 hours 24 minutes

57. Maximum 166 beats per minute, minimum 127 beats per minute

Prepare for This Section (1.2), page 83 PS1. 23

1 2

PS2.

4 15

PS3. Distributive property

PS4. Associative property of multiplication

PS5.

ab 11 x PS6. 15 a + b

Exercise Set 1.2, page 92 S - a1 an - a1 I Fd 2 5. m1 = 7. d = 9. r = 11. 96.25 13. 9.5 15. 12.0 17. Width, 30 feet; Pr Gm 2 n - 1 S pr length, 57 feet 19. 12 centimeters, 36 centimeters, 36 centimeters 21. 15 feet 23. 17.6 feet 25. 850 pairs 27. $10.05 for book, $0.05 for bookmark 29. $937.50 31. $7600 invested at 8%, $6400 invested at 6.5% 33. $3750 35. 240 meters 37. 2 hours 2 39. 1384 feet 41. 140 kilometers per hour 43. 18 grams 45. 64 liters 47. 10 grams 49. 12 pounds of the $6.50 grade, 11 8 pounds of the $4.25 grade 51. Cranberries, 7 pounds; granola, 18 pounds 53. $8 coffee, 18.75 pounds; $4 coffee, 31.25 pounds 1. h =

3V

3. t =

2

7 8

55. 7 hours

57. 6 hours

59. 10

2 hours 3

Prepare for This Section (1.3), page 96 PS1. (x + 6)(x - 7)

PS2. (2x + 3)(3x - 5)

PS3. 3 + 4i

PS4. 1

PS5. 1

PS6. 0

Exercise Set 1.3, page 106 1. -3, 5 3. 23. 0, -6 39.

1 3 , 1 5. -24, 2 8

25. 6 3 12

-3 113 2

55. 1 13

57.

-1 2i13 69. 2

41.

7. 0,

7 3

9. 2, 8

27. -6 2i13

-2 16 2

-1 15 2

43.

59.

4 113 3

-2 315 2

71. 81; two real solutions

79. -111; no real solutions

29.

95. 1.8 seconds and 11.9 seconds

13. 2 16

- 8 312 2

-3 141 4

63. - 3 2i

65.

83. Width, 43.2 inches; height, 32.4 inches

99. 9 people

101. 2031

1 5i 2

75. 0; one real solution

89. 12 feet by 48 feet or 32 feet by 18 feet

97. No

17. - 1, 11

19. - 2 3 13

67. -1 2i 17 77. 2116; two real solutions

85. 48.6 yards

91. 0.3 mile and 3.9 miles

103. a. 9700 objects

93. 1.7 seconds

b. 2030

Mid-Chapter 1 Quiz, page 109 1.

7 13 [1.1] 2. [1.1] 3. -1, 6 [1.3] 4. 2 2

-2 - 16, - 2 + 16 [1.3] 5. 3 - i 13, 3 + i 13 [1.3] 6. 2 hours [1.2]

7. 200 milliliters of 9% solution; 300 milliliters of 4% solution [1.2]

21. 4 5i

3 4 i 115 33. -3, 5 35. - , 4 37. -3 2 12 2 2 3 5 1 i - 1 i 12 47. 49. 51. -3, 5 53. - , 2 3 4 3

45. - 2 i 61.

15. 4i

31.

73. -116; no real solutions

81. 26.8 centimeters

87. 5800 racquets or 11,000 racquets

11. -9, 9

8. 6 hours [1.2]

A6


Prepare for This Section (1.4), page 110 PS1. x(x + 4)(x - 4)

PS2. x 2 (x + 6)(x - 6)

PS3. 4

PS4. 64

PS6. x - 41x + 3 + 7

PS5. x + 21x - 5 - 4

Exercise Set 1.4, page 120 5. - 15, 15, 3

1. -5, 0, 5 3. -1, 1, 2 15. No solution

17. 31

39. -2, -1

37. 1, 5 3

3

57. 12 , - 13 73. 10 hours

19. 4

21.

4 3

41. - , 3

75. 27.5 seconds

23. 2, 7

43. 8

9. 2, - 1 + i 13, - 1 - i13

25. 1 - 121, 1 + 121

45. -243, 243

47. 9

27. 40

49. - 125, 125

77. 3 inches

79. 10.5 millimeters

81. 131 feet

11. -3, 2, 29. 7

51. 81

1 256 , 64 63. -1, 1, -3i, 3i 65. -1, 32 67. , 16 27 81

61. -

59. 1, 16

7 2

2 3

7. -3, - , 3

3 + 3i 13 3 - 3i13 , 2 2

31. 7

33. No solution

53. 17 , 12

69. 9 miles per hour

13. 35. 4, 5

16 2

55. 2, 71. 13

7 3

1 hours 3

85. 12.1 kilometers

Prepare for This Section (1.5), page 123 PS1. 5x ƒ x 7 56 PS2. 38

PS3. 2

PS4. (2x + 3)(5x - 3) PS5.

5 7 PS6. , 3 2 2

Exercise Set 1.5, page 133 1. 5x ƒ x 6 46,

13 f, 5. e x ƒ x Ú 8 9. e x ` -

3. 5x ƒ x 6 - 66,

−3 −2 −1 0 1 2 3 4 5 6 7

−11−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

7. 5x ƒ x 6 26,

− 13 8 −5 −4 −3 −2 −1 0 1 2 3 4 5

3 6 x … 4 f, 4

11. e x `

− 34 − 5 −4 −3 −2 −1 0 1 2 3 4 5

13. 5x ƒ x 6 - 3 or x Ú - 16,

− 5 − 4 −3 −2 −1 0 1 2 3 4 5

4 19. (- q , -84 ´ 32, q ) 21. c - , 8 d 3

− 5 − 4 −3 −2 −1 0 1 2 3 4 5

1 11 … x … f, 3 3

15. 5x ƒ x 6 16,

28 23. (- q , - 44 ´ c , q b 5

1 3

− 5 − 4 −3 −2 −1 0 1 2 3 4 5

− 5 − 4 −3 −2 −1 0 1 2 3 4 5

29 , - 8b 2

45. c-4, -

7 b 2

63. 183.7 pounds 6 m 6 196.3 pounds

57. 0 inches 6 h … 26 inches

65. 130.0 to 137.5 centimeters

5 2

3 2

39. ( - q , - 44 ´ 3- 2, 24 ´ 34, q 4

47. (- q , - 1) ´ (2, 4) 49. ( - q , 5) ´ 312, q ) 51. a-

55. If you write more than 57 checks a month

17. a- q , - b ´ a , qb

25. (- q , q ) 27. 546 29. ( - q , -7) ´ (0, q )

31. 3- 4, 44 33. (- 5, - 2) 35. ( - q , - 44 ´ 37, q ) 37. (- q , -4) ´ (1, 4) 43. c-

11 3

2 5 , 0b ´ a , q b 3 2

59. 54,000 miles

67. ($0, $210)

41. ( -4, 1)

53. (- q , 5)

61. 20° … C … 40°

69. At least 9791 books

71. More than

1 second but less than 3 seconds

Prepare for This Section (1.6), page 136 PS1. 65 PS2. 45 PS3. 27 PS4. 28.125 PS5. The area becomes four times as large. PS6. No. The volume becomes nine times as large.

Exercise Set 1.6, page 141 1. d = kt

3. y =

k x

5. m = knp

7. V = klwh

9. A = ks 2

11. F =

km1 m2 d

2

13. y = kx, k =

4 3

15. r = kt 2, k =

1 81


A7

7 19. V = klwh, k = 1 21. 1.02 liters 23. 62 semester hours 25. 11.7 fluid ounces 27. a. 3.3 seconds 25 b. 3.7 feet 29. 40 revolutions per minute 31. 0.245 watts per square meter 33. a. V is nine times as large. b. V is three times as large. c. V is 27 times as large. 35. V is six times as large. 37. 2.97 39. 3950 pounds 41. 142 million miles 17. T = krs 2, k =

Chapter 1 Review Exercises, page 148 V 1 5 1 1 11 [1.1] 2. [1.1] 3. [1.1] 4. [1.1] 5. 1, 5 [1.1] 6. -9, -1 [1.1] 7. -3, 2 [1.1] 8. - , 5 [1.1] 9. h = [1.2] 4 4 2 4 3 pr 2 2A - hb2 3 4 A - P P - 2l 10. t = [1.2] 11. b1 = [1.2] 12. w = [1.2] 13. 2, 3 [1.3] 14. - , [1.3] 15. 2 - 5 12, 2 + 512 [1.3] Pr h 2 2 3 1. -

16. -4 - 3i, - 4 + 3i [1.3] 20.

17. 3 - 110, 3 + 110 [1.3]

18.

1 13 1 13 i, + i [1.3] 21. Real number solutions [1.3] 2 2 2 2

24. -2, 0, 2 [1.4] 31. 5 [1.4] 39. 45. c

3 2

25. - 2, - , 2 [1.4]

32. No solution [1.4]

26. -1,

33. -4, -

1 - 113 1 + 113 1 - 12 1 + 12 , , [1.3] 19. [1.3] 2 2 6 6 22. Nonreal complex number solutions [1.3]

23. 0,

5 [1.4] 3

1 2 38 1 [1.4] 28. - [1.4] 29. , 3 [1.4] 30. -1 [1.4] , 1 [1.4] 27. 3 15 2 3

3 7 1 [1.4] 34. - , - [1.4] 35. 4 [1.4] 36. 7 [1.4] 37. 16 [1.4] 38. -27, 27 [1.4] 2 4 4

6 213 110 4 [1.4] 40. [1.4] 41. (- q , 24 [1.5] 42. c , q b [1.5] 43. ( - q , -3) ´ (2, q ) [1.5] 44. (-2, 2] [1.5] , 3 2 9 7

145 , 35 d [1.5] 46. (86, 149) [1.5] 9

50. (a - b, a) ´ (a, a + b) [1.5] 54. (- q , -7) ´ 30, 54 [1.5]

47. a , 2b [1.5]

2 3

48. ( - q , 14 ´ 32, q ) [1.5]

51. (- q , -3) ´ 32, q ) [1.5]

55. a - q ,

5 5 d ´ (3, q ) [1.5] 56. c , 5b [1.5] 57. Width, 12 feet; length, 15 feet [1.2] 2 2 63. $864 [1.2]

66. Inez, 6 miles per hour; Olivia, 4 miles per hour [1.4] 69. 15 ounces [1.2]

75. 2.0 seconds [1.3]

59. 22.5 feet [1.2]

64. $1750 at 4%, $3750 at 6% [1.2]

60. 4 feet [1.2]

71. 18 hours [1.4]

61. 13 feet [1.2]

65. 24 nautical miles [1.2]

67. 5%, 400 milliliters, 11%, 200 milliliters [1.2]

70. Raisins, 12.5 pounds; nuts, 7.5 pounds [1.2]

74. 116 by 116 centimeters [1.3]

53. ( - q , -3) ´ (4, q ) [1.5]

52. (- q , -4) ´ (- 2, 4) [1.5]

58. Length, 12 inches; width, 8 inches or length, 8 inches; width, 12 inches [1.2] 62. Calculator, $20.50; battery, $0.50 [1.2]

49. (1, 2) ´ (2, 3) [1.5]

68. 60 milliliters [1.2]

72. 10 minutes [1.4]

73. 40 yards [1.3]

77. 63.1 inches 6 m 6 64.5 inches [1.5]

76. 41 6 x 6 59, where x is an integer [1.5]

78. 38.0 inches 6 m 6 40.0 inches [1.5] 79. 9.39 to 9.55 inches [1.5] 80. More than 0.6 miles but less than 3.6 miles from the city center [1.5] 81. 3 feet per second squared [1.6] 82. 23.85 feet [1.6] 83. 6000 players [1.6] 84. 10 pounds [1.6] 85. 1.64 meters per second squared [1.6]

Chapter 1 Test, page 151 1. -

1 8 15 c - cd 4 114 , a Z c [1.2] 4. - , [1.3] 5. [1.1] 2. -9, 4 [1.1] 3. x = [1.3] 6. 2 - 3i, 2 + 3i [1.3] 2 3 2 2 a - c

7. Discriminant: 1; two real solutions [1.3] 13. 4, -2 - 2i13, -2 + 2i13 [1.4]

8. 3 [1.4]

9. 1, 5 [1.4]

14. a. 5x ƒ x … 86 [1.5]

10. 32 [1.4]

b. 3 -2, 5) [1.5]

11. -

1 14 [1.4] 12. - 2, - , 2 [1.4] 3 2

15. a - q , - b ´ (3, q ) [1.5]

1 3

16. (-1, 6) [1.5]

17. 3 -4, -1) ´ 33, q ) [1.5] 18. 2.25 liters [1.2] 19. 15 hours [1.2] 20. 4.3 feet [1.2] 21. Ground beef, 30 pounds; ground sausage, 20 pounds [1.2] 22. 0.6 second, 9.4 seconds [1.3] 23. 10 miles per hour [1.4] 24. 30.0 6 p 6 77.5 [1.5] 25. 4.4 miles per second [1.6]

A8


Cumulative Review Exercises, page 152 2. 1.7 * 10-4 [P.2]

1. - 11 [P.1] 8.

3. 8x 2 - 30x + 41 [P.3]

4. (8x - 5)(x + 3) [P.4]

5.

2x + 17 [P.5] x - 4

6. a11>12 [P.2]

7. 29 [P.6]

10 2 110 110 [1.1] 9. [1.3] 10. 1, 5 [1.1] 11. 5 [1.4] 12. -6, 0, 6 [1.4] 13. 13 , [1.4] 3 2 2

14. 5x ƒ x … - 1 or x 7 16 [1.5]

15. (- q , 44 ´ 38, q ) [1.5]

18. 9475 to 24,275 printers [1.5]

19. 68 to 100 [1.5]

16. e x `

10 3 … x 6 f [1.5] 17. Length, 58 feet; width, 42 feet [1.2] 7 2

20. Between 14.3 and 23.1 [1.5]


3. a. 2001 and 2002

y

(−2, 1)

15. - x110

(2, 4)

b. $23,100

17. (12, 0), (-4, 0)

7. 11261

5. 715 19. (3, 2)

21. (6, 4)

9. 189

11. 338 - 1216

23. ( -0.875, 3.91)

25.

13. 22a 2 + b 2 y

x

(−5,−3)

27.

4

(0, −3)

−4

y 3

29.

y

y

31.

33.

y

−3 4

x

5

−3

y

35.

3

3

4

−4

x

3 x

−2 3

x

x

37.

y

2

x

−4

x

−4

−6

39. (6, 0), a0,

y 6

12 b 5

41. (5, 0); (0, 15 ), (0, - 15 )

−6

y 4

x

6

−4

5

x

−4

43. (- 4, 0); (0, 4), (0, -4)

45. (2, 0), (0, 2)

y

y

4

4

4 x

−4

−4

−4

47. ( 4, 0), (0, 4)

49. Center (0, 0), radius 6

y

55. Center (8, 0), radius

4 −4

x

−4

4

−4

4

x

59. ax -

1 2


53. Center ( - 2, -5), radius 5

57. (x - 4)2 + ( y - 1)2 = 22

1 2 1 2 2 b + ay - b = ( 15 ) 2 4

63. (x - 1)2 + ( y - 3)2 = 52

61. (x - 0)2 + ( y - 0)2 = 52


67. Center (7, - 4), radius 213


69. Center a , - b, radius

1 2

77. (13, 5)

3 2

79. (7, - 6)

71. Center a-

5 2

137 3 , 3b , radius 2 2

73. (x + 1)2 + ( y - 7)2 = 25

A9

75. (x - 7)2 + ( y - 11)2 = 121

81. x 2 - 6x + y 2 - 8y = 0

Prepare for This Section (2.2), page 166 PS2. D = 5- 3, -2, - 1, 0, 26; R = 51, 2, 4, 56

PS1. -4

PS3. 158

PS4. x Ú 3

PS5. - 2, 3

PS6. 13

Exercise Set 2.2, page 180 1. Yes

3. No

19. a. 15 23. a. 1

5. No

b. 3

d. - 1

−2

2

f. - 1

e. 1

−2 −4

25. a. -11

−6

−4

−2

43.

−4

2

4

6 x

d. - k 2 - 2k + 10

2

−2

−4

−2

2

−2

y

51. 4

53. -3 and 1

2

59. 2

61. -

x

b.

2 5

101. a. A(x) = a

63. -2 and 2

−2

2

4

x

57. No real number values for a

65. -3 and 8

79. Increasing on (- q , q )

0.60

81. Decreasing on (- q , -34; increasing on

0.40

83. Constant on ( - q , 04; increasing on 30, q )

3-3, 04; decreasing on 30, 34; increasing on 33, q )

1

2

3

4 w

85. Decreasing on ( - q , 04; constant on 30, 14; increasing on 31, q ) 87. g and F

93. a. C(x) = 2000 + 22.80x

b. R(x) = 37.00x

99. d = 2(45 - 8t)2 + (6t)2

b. 74.72, 67.68, 64.34, 64.79, 70

109. 1 is not in the range of f.

55. -4 and 4

−2

0.80

1 5 1 2 + bx - x + 25 b. 25, 17.27, 14.09, 15.46, 21.37, 31.83 4p 16 2

107. c = - 2 or c = 3

−4

77. Decreasing on ( - q , 04; increasing on 30, q )

97. d = 29t 2 + 2500

103. a. L(x) = 2900 + x 2 + 2400 + (40 - x)2

−6

75. a, b, and d.

91. v(t) = 80,000 - 6500t, 0 … t … 10

95. h = 15 - 5r

y

1.00

Weight (ounces)

b. A = 25l - l

c + 4

27. All real numbers

int [102(2.3458) + 0.5]

C(w)

0

c. P(x) = 14.20x - 2000

x

1 2 + hƒ ƒ

= 2.35 102 int [103(34.05622) + 0.5] = 34.056 69. 103 67.

0.20

89. a. w = 25 - l

4

f. 3k + 5

f.

2

45.

y

−6

x

4

2

= 0.0895 73. a. $.78

2

37. 5x ƒ x 7 - 46

1

e.

2

Cost (dollars)

104

d. 1

2

−2

int [104(0.08951) + 0.5]

5 3

4

−4

71.

c.

2

−2

−2

1 2

e. 3k - 1

d. 1

4

2 −6

b.

c. - 1

4

49.

y 4

1 2

21. a.

c. 3c + 1

b. 6

b. -4

17. a. 5

35. 5x ƒ -2 … x … 26

y

−6

47.

15. Yes

f. 2c 2 + 5

33. 5x ƒ x Ú - 76

x

4

13. No

e. 2r 2 + 2r + 6

41.

2 −4

11. Yes

31. 5x ƒ x Z - 26

29. All real numbers

−6

9. No

d. 121

c. 3

c. - 1

b. 1

39.

7. Yes

c. 30, 204 c. 30, 404

105. 275, 375, 385, 390, 394

A10


111.

113.

2

115.

1 −4.7

−4.7

117. 1, -3

2

119. c. Thursday

4.7

4.7

−4.7

4.7

−5

−5

−4


PS2. -1

PS3. -

8 5

3 x - 3 PS6. 2 5

PS5. y =

PS4. y = - 2x + 9

Exercise Set 2.3, page 195 1. -

3 2

3. -

1 2

5. Undefined

7. 6

9.

9 19

11.

f (3 + h) - f (3) h

13.

f (h) - f (0) h

15. m = 2

y-intercept: (0, - 4) y

3 x −3

3 4 y-intercept: (0, 1)

17. m =

19. m = - 2

y-intercept: (0, 3)

y

−4

−2

2

2 2

−2

27. m = - 2

4

x

−6 −4 −2 −2

4 3 y-intercept (0, 4)

29. m = -

y-intercept (0, 5) y

2


2

x

35. y = x + 3 47. y = 2x - 8

x

2

37. y =

3 1 x + 4 2

49. y = -

51. y =

2

2

y

y

6

−6 −4 −2 −2

2 x 5

53. y = -

x

2

−2 −2

4 x

2


4

39. y = (0)x + 4, or y = 4

3 x - 1 4

y 4

33. m = -

2

2

y-intercept (0, 0)

y

−2

x

2

31. m =

y

2

4

−2

6 x

25. m = 1

y-intercept (0, 0)

y

4

2

23. m = 2

y-intercept (0, 3)

y 6

4

−6

21. m = 0

4 2

4

6

2

x

−6 −4 −2 −2

41. y = - 4x - 10

1 5 x 2 2

55. y =

43. y = -

4 x + 8 3

2

4

6

3 13 x + 4 4

x

45. y =

57. y = 4x - 18

12 29 x 5 5

59. - 2

61 -

1 2

63. 4 65. -20 67. m = 2.875. The value of the slope indicates that the speed of sound in water increases 2.875 meters per second for a 1°C increase in temperature. 69. a. H(c) = c + 8 b. 26 miles per gallon 71. a. N(t) = 2500t - 4,962,000 b. 2008 73. a. B(d) = 30d - 300 b. The value of the slope means that a 1-inch increase in the diameter of a log 32 feet long results in an increase of 30 board-feet of lumber that can be obtained from the log. c. 270 board-feet 75. Line A, Michelle; line B, Amanda; line C, distance between Michelle and Amanda 77. a. y = - 0.93x + 79.96 b. 57 years 79. P(x) = 40.50x - 1782, x = 44, the break-even point


81. P(x) = 79x - 10,270; x = 130, the break-even point 85. a. C(t) = 19,500.00 + 6.75t 89. a. Q = (3, 10), m = 5

b. R(t) = 55.00t

83. a. $275

b. $283

c. $355

c. P(t) = 48.25t - 19,500.00

b. Q = (2.1, 5.41), m = 4.1

d. $8 per unit

d. Approximately 405 days

c. Q = (2.01, 5.0401), m = 4.01

d. 4

87. - 5

93. a ,

9 81 b 2 4

Mid-Chapter 2 Quiz, page 200 2. Center (3, -2), radius 115 [2.1]

1. Midpoint (-1, 1), length 2113 [2.1]

3. 28 [2.2]

4. (- q , 2] [2.2]

5. -3 and 4 [2.2]

1 2 2 [2.3] 7. y = - x + 1 [2.3] 8. m = 2 3 3 y-intercept: (0, 1)

6. -

[2.3]

4 2 −4 −2 −2

2

4

−4

Prepare for This Section (2.4), page 200 PS1. (3x - 2)(x + 4)

PS2. x 2 - 8x + 16 = (x - 4)2

PS3. 26

PS4. -

1 ,1 2

PS5.

- 3 117 2

PS6. 1, 3

Exercise Set 2.4, page 209 1. d

3. b

5. g

7. c

11. f (x) = (x - 4)2 - 11

9. f (x) = (x + 2)2 - 3

Vertex: (- 2, - 3)

Vertex: (4, - 11)

Axis of symmetry: x = - 2

Axis of symmetry: x = 4 y

y

5 3 2 b 2 4 5 3 Vertex: a- , - b 2 4

13. f (x) = ax +

Axis of symmetry: x = y

3 x

−2

−5

4

x

1 x

−3

15. f (x) = - (x - 2)2 + 6

17. f (x) = - 3ax -

31 1 2 b + 2 4

1 31 , b 2 4

Vertex: (2, 6)

Vertex: a

Axis of symmetry: x = 2

1 Axis of symmetry: x = 2 y

y

21. Vertex: (0, - 10), f (x) = x 2 - 10 23. Vertex: (3, 10), f (x) = - (x - 3)2 + 10

3 47 3 2 47 b, f (x) = 2a x - b + 4 8 4 8

25. Vertex: a ,

1 17 1 2 17 b, f (x) = - 4 ax - b + 8 16 8 16

27. Vertex: a ,

4

29. 5 y ƒ y Ú - 26, - 1 and 3

3 x

2

19. Vertex: (5, -25), f (x) = (x - 5)2 - 25

x

1

31. e y ƒ y …

17 3 f , 1 and 8 2

3 2

A11

A12


35. -4 and - ; (-4, 0), a-

3 2

33. - 6 and 4; (- 6, 0), (4, 0) 43. -11, minimum

45. 35, maximum

600 - 2l 3

51. a. w =

47. 18.4 seconds

2 2 l 3

b. A = 200l -

59. a. 41 miles per hour

b. 34 miles per gallon

1 8

41. - , minimum

39. 11, maximum

5 feet c. L 20.1 feet from the center 16

53. a. 12:43 P.M.

b. 91°F

55. 6.1 joules

57. Yes

61. 740 units yield a maximum revenue of $109,520. 65. P(x) = - 0.1x 2 + 50x - 1840, break-even points: x = 40 and x = 460

c. t = 8 seconds

b. 256 feet

b. 22

49. a. 27 feet

b. P(x) = - 0.25x 2 + 27.50x - 180

67. a. R(x) = - 0.25x + 30.00x 2

73. r =

37. -16, minimum

c. w = 100 feet, l = 150 feet

63. 85 units yield a maximum profit of $24.25. 69. a. t = 4 seconds

3 , 0b 2

c. $576.25

d. 55 tickets

71. 30 feet

48 L 6.72 feet, h = r L 6.72 feet 4 + p

Prepare for This Section (2.5), page 213 PS1. x = - 2

PS4. 3, -1, -3, -3, - 1

PS5. (0, b)

PS6. (0, 0)


3.

y

5.

y

7.

y

B(5, 3)

C(−5, 3)

R(−2, 3)

A(2, 3)

x

A(−5, −3)

B(−2, −3)

P(5, −3)

C(2, −3)

27. Yes

b. Yes

15. a. No

29. Yes

31.

b. No

3

−3

x

17. a. Yes

y

−3 x

x

2

−2

x

−3

x

b. Yes

23. No

A(4, −5)

b. Yes

33.

y

11.

y

C (4, 5)

B(−4, 5)

T (−4, −5)

13. a. No

9.

y

19. a. Yes 35.

y

b. Yes

21. a. Yes 37.

y

25. Yes

y (0, 8)

3 (1, 0)

(−1, 0)

(1, 0) x (0, 0)

x (0, − 1)

(−1, 0)

3

x

3

(4, 0)

39.

41.

y

5

43. Even

y

45. Odd

47. Even

49. Even

x

−2 −3

(0, 0)

x (4, 0)

57. a., b.

Intercept: (a, 0), a 0

59. a.

y f(x) + 3

b.

y

f(x − 3)

y

2

2

x

−4 −2

2

4x

−4 −2

2

4x

51. Even

53. Even

x

55. Neither


61. a.

y

63. a. ( - 5, 5), (- 3, - 2), (- 2, 0)

4

4

65. a.

2

2

b.

y

−6 −4 −2 −2

2

4

x

−6 −4 −2 −2

−4

−4

−6

−6

67. a. (1, 3), (-2, -4)

b. (- 1, - 3), (2, 4)

2

b.

y

y 4

2

2

6 x

4

b. ( -2, 6), (0, - 1), (1, 1)

−4 −2

4x

−2

−4

4x

2

−4

69. a., b.

71.

y 4

y

F(−x)

6

2 − 4 −2

−2

−F(x)

4

4 x

2

−4

−6 −4

6 x

−2 −4 −6

73. a.

b.

y

y

2 1 −6

−2 −1

75. a.

1 2

x

6

−6

−2 −1

2

b.

y

y

2

2

−2π

−π

π

x

2π

−2π

−π

π

6

3

y = √x + 3

79.

J(x) for

y = √x 3

y = √x − 1 5

81.

7

3

−5

x

2π

−2

−2

77.

x

6

−4

7

y = 1x 2 2

c=2

y = x2

c=0 c = −1

y = 2x 2

6 −4

−4

83.

4 −1

−3

85. a.

5

b.

y

y

6 y = 1 (|x −1| − |x|) −5

5

2

4

y = |x −1| − |x|

4

y = 4 (|x −1| − |x|) −5

2

2

2

4

6

x

4

6

x

A13

A14 c.


y

12 10 8 6 4 2

2

4

6

x

Prepare for This Section (2.6), page 227 PS1. x 2 + 1

PS2. 6x 3 - 11x 2 + 7x - 6

PS3. 18a2 - 15a + 2

PS4.

2h2 + 3h PS5. All real numbers except x = 1 PS6. [4, q )

Exercise Set 2.6, page 234 1. f(x) + g(x) = x 2 - x - 12,

Domain is the set of all real numbers. 3. f (x) + g(x) = 3x + 12, Domain is the set of all real numbers. f (x) - g(x) = x 2 - 3x - 18, Domain is the set of all real numbers. f (x) - g(x) = x + 4, Domain is the set of all real numbers. f (x) # g(x) = x 3 + x 2 - 21x - 45, Domain is the set of all real f (x) # g(x) = 2x 2 + 16x + 32, Domain is the set of all real numbers. numbers. f (x) f (x) = x - 5, Domain 5x ƒ x Z - 36 = 2, Domain 5x ƒ x Z - 46 g(x) g(x)

5. f(x) + g(x) = x 3 - 2x 2 + 8x,

7. f (x) + g(x) = 2x 2 + 7x - 12,

Domain is the set of all real numbers. f (x) - g(x) = x 3 - 2x 2 + 6x, Domain is the set of all real numbers. Domain is the set of all real f (x) # g(x) = x 4 - 2x 3 + 7x 2, numbers. f (x) = x 2 - 2x + 7, Domain 5x ƒ x Z 06 g(x)

9. f(x) + g(x) = 1x - 3 + x,

f (x) - g(x) = 1x - 3 - x,

f (x) # g(x) = x 1x - 3, f (x) 1x - 3 = , g(x) x 13. 18

15. -

9 4

35. -8x - 4h

Domain 5x ƒ x Ú 36

11. f (x) + g(x) = 24 - x 2 + 2 + x,


f (x) - g(x) = 24 - x - 2 - x, 2

f (x) # g(x) = ( 24 - x 2)(2 + x),


17. 30

19. 12

21. 300

37. ( g ⴰ f )(x) = 6x + 3

1 - 5x x + 1 2 ( g ⴰ f )(x) = 3x - 4

23. -

384 125

25. -

5 2

27. -

2 39. ( g ⴰ f )(x) = x + 4x + 1

( f ⴰ g)(x) = x + 8x + 11

21 - x2 ƒxƒ 1 ( f ⴰ g)(x) = x - 1

45. ( g ⴰ f )(x) =

Domain 5x ƒ -2 … x … 26

Domain 5x ƒ -2 … x … 26

Domain 5x ƒ -2 … x … 26

f (x) 24 - x 2 = , Domain 5x ƒ -2 6 x … 26 g(x) 2 + x


( f ⴰ g)(x) = 6x - 16 43. ( g ⴰ f )(x) =

Domain is the set of all real numbers. f (x) - g(x) = - 2x2 + x - 2, Domain is the set of all real numbers. f (x) # g(x) = 8x3 - 2x2 - 41x + 35, Domain is the set of all real numbers. f (x) 5 4x - 7 , Domain e x ƒ x Z 1, x Z - f = 2 g(x) 2 2x + 3x - 5

1 4

29. 2

31. 2x + h

33. 4x + 2h + 4

3 41. ( g ⴰ f )(x) = - 5x - 10x

( f ⴰ g)(x) = - 125x 3 - 10x 2ƒ5 - xƒ 47. ( g ⴰ f )(x) = 3 3ƒxƒ ( f ⴰ g)(x) = ƒ 5x + 2 ƒ 2

A15


49. 66

53. -4

51. 51

55. 41

57. -

3848 625

59. 6 + 2 13

73. a. A(t) = p(1.5t)2, A(2) = 9p square feet L 28.27 square feet 75. a. d(t) = 2(48 - t) - 4

61. 16c 2 + 4c - 6

63. 9k 4 + 36k 3 + 45k 2 + 18k - 4

b. V(t) = 2.25pt 3, V(3) = 60.75p cubic feet L 190.85 cubic feet

b. s(35) = 13 feet, d(35) L 12.37 feet 77. (Y ⴰ F)(x) converts x inches to yards. 79. a. 99.8 (mg兾L)兾h; this is identical to the slope of the line through 30, C(0)4 and 31, C(1)4. b. 156.2 (mg兾L)兾h c. -49.7 (mg兾L)兾h d. -30.8 (mg兾L)兾h e. - 16.4 (mg兾L)兾h f. 0 (mg兾L)兾h 2

2

Prepare for This Section (2.7), page 237 1 3 2 ; y-intercept: (0, 4) PS2. Slope: ; y-intercept: (0, - 3) PS3. y = - 0.45x + 2.3 PS4. y = - x - 2 3 4 3 PS6. 3

PS1. Slope: PS5. 19

Exercise Set 2.7, page 244 1. No relationship

3. Linear

7. y = 2.00862069x + 0.5603448276

5. Figure A

11. y = 2.222641509x - 7.364150943

9. y = - 0.7231182796x + 9.233870968

13. y = 1.095779221x 2 - 2.69642857x + 1.136363636

15. y = - 0.2987274717x 2 - 3.20998141x + 3.416463667

17. a. y = 23.55706665x - 24.4271215

b. 1248 centimeters

19. a. y = 0.1094224924x + 0.7978723404 b. 4.3 meters per second 21. a. y = 0.1628623408x - 0.6875682232 b. 25 23. No, because the linear correlation coefficient is close to 0. 25. a. Yes, there is a strong linear correlation. b. y = - 0.9082843137x + 79.23480392 c. 57 years 27. a. positively b. 1098 calories 29. y = - 0.6328671329x 2 + 33.61608392x - 379.4405594 31. a. y = - 0.0165034965x 2 + 1.366713287x + 5.685314685

b. 32.8 miles per gallon

5-pound: s = 0.6130952381t - 0.0714285714t + 0.1071428571 10-pound: s = 0.6091269841t 2 - 0.0011904762t - 0.3 15-pound: s = 0.5922619048t 2 + 0.3571428571t - 1.520833333 b. All the regression equations are approximately the same. Therefore, the equations of motion of the three masses are the same. 2

33. a.

Chapter 2 Review Exercises, page 253 1. 1181 [2.1] 5.

2. 415 [2.1]

[2.1] 6.

y

4

1 , 10b [2.1] 4. (2, -2) [2.1] 2 [2.1] 7.

y 6

4

x

−6 −4 −2 −2

−4

2

4

−6 −4 −2 −2

6 x

10. Intercepts:

(-1, 0), (0, 1), (0, -1).

4

−6 −4 −2 −2

[2.1]

y 6

2

4

−6

12. Intercept:

(4, 0), (0, 3).

(2, 0) [2.1]

y 6

4

4

4

2

2

2

2

2

4

x

−6 −4 −2 −2

2

4

x

−6 −4 −2 −2

2

4

x

[2.1]

y 6

4

−6 −4 −2 −2

6 x

−4

11. Intercepts:

(-4, 0), (4, 0), (0, 4), (0, -4). [2.1]

y 6

2

−6

−6

9. Intercepts:

2 6 x

−4

−4

−6

[2.1]

y 6

4

2

2 2

[2.1] 8.

y

4

4

2 −6 −4 −2 −2

3. a-

−6 −4 −2 −2

−4

−4

−4

−4

−6

−6

−6

−6

2

4

x

A16


13. Center (3, - 4); radius 9 [2.1]

14. Center (-5, -2), radius 3 [2.1]

16. (x + 5)2 + (y - 1)2 = 82, radius = 8 [2.1] 21. a. 2

22. a. 155 24. a. 1

c. 3t + 4t - 5

b. 139

c. -2 [2.2]

b. 10

2

4

2

2

−6 −4 −2 −2

2

x

4

−4

−6

−6

41.

39. 0 [2.3] 2

[2.3] 42.

y

2 2

4

2 2 4

40. -

[2.3] 43.

c. -3 [2.2]

30. 1 [2.2]

37. -1 [2.3]

38. Undefined [2.3]

1 [2.3] 5

[2.3] 44.

y 6

4

4

2

2

2

2

4

x

−6 −4 −2 −2

2

4

x

[2.3]

y 6

4

−6 −4 −2 −2

x

4

c. 4 [2.2]

x

4

y 6

4

6

29. -3 and 1 [2.2]

36. a. -1 b. 0

−6 −4 −2 −2

−4

b. - 4

[2.2] 33. -3 [2.2] 34. -2 and 6 [2.2] 35. a. 6 b. -2 c. - 4 [2.2]

y 6

4

23. a. 6

f. 27t 2 + 12t - 5 [2.2]

26. Domain: 5x ƒ x … 66 [2.2]

28. Domain: 5x ƒ x Z - 3, x Z 56 [2.2]

[2.2] 32.

y 6

20. Yes [2.2]

2

f. 2216 - t 2 [2.2]

25. Domain: 5x ƒ x is a real number6 [2.2]

27. Domain: 5x ƒ -5 … x … 56 [2.2] 31.

e. 2264 - t 2

19. No [2.2]

e. 9t + 12t - 15

2

d. 264 - x 2

c. 0

18. No [2.2]

d. 3x + 6xh + 3h + 4x + 4h - 5

2

b. 10

17. Yes [2.2]

15. (x - 2)2 + (y + 3)2 = 5 2 [2.1]

−6 −4 −2 −2

−4

−4

−4

−6

−6

−6

2

4

x

6

2 7 x + 1 [2.3] 49. y = x - 7 [2.3] 4 3 2 23 2 5 65 5 50. y = x [2.3] 51. y = x - 3 [2.3] 52. y = - x + 11 [2.3] 53. f (x) = x [2.3] 54. f (t) = 28.5t + 65 [2.3] 5 5 3 2 3 3 2 3

45. y = - x [2.3]

46. y = - 2x - 2 [2.3]

55. f (x) = (x + 3)2 + 1 [2.4]

47. y = x + 5 [2.3]

56. f (x) = 2(x + 1)2 + 3 [2.4]

48. y =

57. f (x) = - (x + 4)2 + 19 [2.4]

59. f (x) = - 3 ax -

2 2 11 [2.4] 60. f (x) = (x - 3)2 + 0 [2.4] 61. (1, 8) [2.4] b 3 3

64. (- 4, 30) [2.4]

65. 6 [2.4]

69. 61,250 square feet [2.4]

66. -5.125 [2.4]

70.

67. 43.0625 feet [2.4]

[2.5] 71.

y

c.

72. Symmetric to the y-axis [2.5] 74. Symmetric to the origin [2.5]

68. a. R = 13x

−2 −2

x

4 −4

b.

2

a.

−4

2

x

c.

a.

73. Symmetric to the x-axis [2.5] 75. Symmetric to the x-axis, the y-axis, and the origin [2.5]

76. Symmetric to the x-axis, the y-axis, and the origin [2.5] 77. Symmetric to the origin [2.5] 79. Symmetric to the origin [2.5]

78. Symmetric to the x-axis, the y-axis, and the origin [2.5]

3 2 5 b - [2.4] 4 4

63. (5, 161) [2.4]

b. P = 12.5x - 1050

[2.5]

4 4

−4

62. (0, -10) [2.4]

y

b.

58. f (x) = 4ax -

c. x = 84 [2.3]


80.

81.

y

y

82.

y

2 1 −2

1

6 x

12

x

a. Domain is the set of all

−3

real numbers. Range: 5y ƒ y … 46 b. Even [2.5]

83.

a. Domain is the set of all

numbers. Range is the set of all real numbers. b. g is neither even nor odd. [2.5]

real numbers. Range: 5y ƒ y Ú 46 b. Even [2.5]

y

−2

[2.5] 88.

y

2

2

x

4

−6 −4 −2 −2

2

x

4

−6 −4 −2 −2

−4

−4

−4

−6

−6

−6

[2.5] 90.

[2.5] 91.

y

4

4

2

2

2

x

4

−6 −4 −2 −2

2

4

x

−4

−4

−6

−6

−6

[2.5] 93.

[2.5] 94.

y

x

4

[2.5]

−6 −4 −2 −2

−4

y

2

y

4

2

[2.5]

y

2

−6 −4 −2 −2

2

x

4

[2.5] 95.

y

4

4

4

2

2

2

2

2

x

4

−6 −4 −2 −2

2

4

x

−6 −4 −2 −2

−4

−4

−4

−6

−6

−6

b. 1

c. x - 2x - 3 2

99. a. 18 feet per second

d. 3x + 4x - 5x - 2 [2.6] 2

b. 15 feet per second

b. -11 c. x - 12x + 32 2

3

4

101. a. 79

x

2

4

x

−6

98. 3x + 3xh + h - 1 [2.6] 2

d. 12.03 feet per second b. 56

−6 −4 −2 −2 −4

97. 8x + 4h - 3 [2.6]

c. 13.5 feet per second

d. x + 4x - 8 [2.6] 2

2

[2.5]

y

4

−6 −4 −2 −2

100. a. 5

Range: 5y ƒ y is an even integer6

b. g is neither even nor odd. [2.5]

4

2

x

4

a. Domain: 5x ƒ x is a real number6

4

y

96. a. 11

numbers. Range is the set of all real numbers. b. Odd [2.5]

4

−6 −4 −2 −2

2

a. Domain is the set of all real

[2.5] 87.

y

92.

2

2 x

Range: 5y ƒ 0 … y … 46 b. Even [2.5]

89.

y

4

x

a. Domain: 5x ƒ -4 … x … 46

86.

85.

2

2 1 12

x

3

a. Domain is the set of all real

84.

y

A17

2

e. 12 feet per second [2.6]

c. 2x - 4x + 9 2

d. 2x2 + 6 [2.6]

102. a. y = 1.171428571x + 5.19047619 b. 19 meters per second [2.7] 103. a. h = 0.0047952048t 2 - 1.756843157t + 180.4065934 b. No c. The regression equation is a model of the data and is not based on physical principles.

A18


Chapter 2 Test, page 257 1. Midpoint (1,1), length 2 113 [2.1]

2. (-4, 0) (0, - 12), (0, 12)

y

[2.1] 3.

2

2

5. 5x ƒ x Ú 46 ´ 5x ƒ x … - 46 [2.2]

x

2

−2

4. Center (2, -1), radius = 3 [2.1]

−2 −2

7. -

6. -8 and 2 [2.2]

2 2 [2.3] 10. f (x) = (x + 3)2 - 11, (-3, -11), x = -3 [2.4] 11. -12, minimum [2.4] 3 3 c. Neither [2.5] 13. a. x-axis b. Origin c. y-axis [2.5] [2.5] 15.

y 8

6

4

4

2

2

−10 −8 −6 −4 −2 −2

16.

2

4

6

x

8

−4

−4

−6

−6

−8

−8

−10

−10

[2.5] 17.

y 8

6

4

4

2

2 2

4

6

x

8

−10 −8 −6 −4 −2 −2

−4

−4

−6

−6

−8

−8

−10

−10

[2.5] 19. a. x 2 - 3x + 3

y 8

2

4

6

8

[2.5]

2

4

6

b. - 40

8

x

c. 22

d. 2x 2 - 2x + 3 [2.6]

21. x = 20 feet, y = 40 feet

6

20. 2x + h [2.6]

4

22. a. 25 feet per second

2

23. a. y = - 7.98245614x + 767.122807

−10 −8 −6 −4 −2 −2

2

4

6

8

b. Odd

x

y 8

6

−10 −8 −6 −4 −2 −2

18.

−10 −8 −6 −4 −2 −2

12. a. Even

[2.5]

y 8

6

x

2

5 [2.3] 8. y = - 2x + 7 [2.3] 6

9. y = - x +

14.

[2.1]

y

b. 22.5 feet per second

c. 20.05 feet per second [2.6]

b. 57 calories [2.7]

x

−4 −6 −8 −10

Cumulative Review Exercises, page 258 1. Commutative property of addition [P.1] 7.

x + 9 [P.5] x + 3

8.

-2 [P.5] (2x - 1)(x - 1)

2.

6 , 12 [P.1] p

9. 0 [1.1]

10.

3. 8x - 33 [P.1]

4. 128x5 y10 [P.2]

5.

4 3b2

[P.2]

6. 6x 2 - 5x - 21 [P.3]

1 15 7 2 [1.3] 11. - , 1 [1.3] 12. x = - y + 5 [1.1] 2 2 3


13. 12, i [1.4] 19. Yes [2.4]

15. 117 [2.1]

14. x 7 - 4 [1.5]

16. -15 [2.2]

17. y = -

A19

1 x - 2 [2.3] 18. 100 ounces [1.1] 2

20. 0.04°F per minute [2.3]

Exercise Set 3.1, page 268 10 x + 3

1. 5x 2 - 9x + 10 -

9. x 3 - x 2 + 2x - 1 +

3. x 3 + 2x 2 - x + 1 +

-x + 3 2x + 2x - 3

403 x - 4

29. -2230

27. 45

31. - 80

55. (x - 4)(x + 3x + 3x + 1) 3

2

61. a. 11,880 ways

33. - 187

57. a. 336

7

6

b. 255,024 ways 5

4

3

344 x - 2

35. Yes

67. 12

7. x 2 + 3x - 2 +

-x + 5 2x2 - x + 1

1 x + 1

19. 8x2 + 6

716 x + 3

23. x 5 - 3x 4 + 9x 3 - 27x 2 + 81x - 242 +

37. No

63. a. 304 cubic inches 2

25 x - 3

13. 4x 2 - 4x + 2 +

b. 336; they are the same.

65. x + x + x + x + x + x + x + x + 1 8

17 x - 2

17. x 4 + x 3 + x 2 + x + 1

21. x 7 + 2x 6 + 5x 5 + 10x 4 + 21x 3 + 42x 2 + 85x + 170 + 25. 25

5. x 2 + 4x + 10 +

11. 4x 2 + 3x + 12 +

2

15. x 4 + 4x 3 + 6x 2 + 24x + 101 +

1 x - 2

39. Yes

41. Yes

59. a. 100 cards

53. (x - 2)(x 2 + 3x + 7)

43. No

b. 610 cards

b. 892 cubic inches

69. -12

71. 13

73. Yes


PS2.

PS3. 3- 1, q )

9 8

PS4. 31, q )

PS5. (x + 1)(x - 1)(x + 2)(x - 2)

PS6. a , 0b, a-

2 3

1 , 0b 2

Exercise Set 3.2, page 282 1. Up to the far left, up to the far right

3. Down to the far left, up to the far right 9. a 6 0

7. Down to the far left, up to the far right 11. Relative maximum y L 5.0 at

x L - 2.1, relative minimum y L - 16.9 at x L 1.4

5. Down to the far left, down to the far right

13. Relative maximum y L 31.0 at

x L - 2.0, relative minimum y L - 77.0 at x L 4.0

15. Relative maximum y L 2.0 at x L 1.0,

relative minima y L - 14.0 at x L - 1.0 and y L - 14.0 at x L 3.0

40

6

20

4

−4

8

−4

6

−4

− 80

− 20

17. -3, 0, 5

19. -3, -2, 2, 3

− 20

21. -2, -1, 0, 1, 2

33. Crosses the x-axis at ( -1, 0), (1, 0), and (3, 0)

35. Crosses the x-axis at (7, 0); intersects but does not cross at (3, 0)

37. Crosses the x-axis at (1, 0); intersects but does not cross at a , 0b

3 2

39. Crosses the x-axis at (0, 0); intersects but does not cross at (3, 0) y 4

41.

y 8

43.

y 4

45.

y 4

47. −8

4x

−4 −4

8x

−8 −8

−4

y 2

49. 8 x

8 x

−8

4x −14

A20


y

51.

y 8

53.

y

55.

2

8 4 x

−4

8 x

−8

8 x

−8

−4

57. Shift the graph of P vertically upward 2 units. 59. Shift the graph of P horizontally 1 unit to the right. 61. Shift the graph of P horizontally 2 units to the right and reflect this graph about the x-axis. Then shift the resulting graph vertically upward 3 units. 63. a. 20.69 milligrams 67. a. 2530 watts

b. 118 minutes

65. a. V(x) = x(15 - 2x)(10 - 2x) = 4x 3 - 50x 2 + 150x

b. 12.6 meters per second

c. The power increases by a factor of 8.

b. 1.96 inches

d. The power increases by a factor of 27.

69. a. f (x) = 3.325304325x 3 - 51.92834943x 2 + 13.21021571x + 7228.323232,

f (x) = - 0.9423441142x 4 + 27.82625129x 3 - 262.6095692x 2 + 681.7360544x + 6640.300505 b. Answers will vary. 71. a. f (x) = 0.00015385409x 3 - 0.0297742717x 2 + 1.674968246x + 2.136506708, f (x) = - 0.000005619394x 4 + 0.00099676312x 3 - 0.0696117578x 2 + 2.292098069x + 0.4928341073 b. 24.4 miles per gallon; 18.5 miles per gallon c. Answers will vary; however, the downward trend for speeds greater than 50 mph suggests that 18.5 miles per gallon is the more realistic value. 73. (5, 0) 75. Shift the graph of y = x3 horizontally 2 units to the right and vertically upward 1 unit.

Prepare for This Section (3.3), page 287 PS1.

2 7 , 3 2

PS2. 2x 2 - x + 6 -

19 x + 2

PS3. 3x 3 + 9x 2 + 6x + 15 +

40 x - 3

PS4. 1, 2, 3, 4, 6, 12

PS5. 1, 3, 9, 27

PS6. P(- x) = - 4x 3 - 3x 2 + 2x + 5

Exercise Set 3.3, page 295 1. 3 (multiplicity 2), - 5 (multiplicity 1)

-3 (multiplicity 2) 7. 13. 1, 2, 4,

5 (multiplicity 2) 5. 2 (multiplicity 1), -2 (multiplicity 1), 3 3 1 9. 1, 2, 4, 8 11. 1, 2, 3, 4, 6, 12, , 2 2

3. 0 (multiplicity 2), -

5 7 (multiplicity 2), (multiplicity 1) 3 2

1 1 2 4 1 , , , , 2 3 3 3 6

15. 1, 7,

7 1 7 1 , , , 2 2 4 4

17. 1, 2, 4, 8, 16, 32

19. Upper bound 2, lower bound -5 21. Upper bound 4, lower bound - 4 23. Upper bound 1, lower bound -4 25. Upper bound 4, lower bound -2 27. Upper bound 2, lower bound -1 29. One positive zero, two or no negative zeros 31. Two or no positive zeros, one negative zero 33. One positive zero, three or one negative zeros 35. Three or one positive zeros, one negative zero 37. One positive zero, no negative zeros 39. Two or no positive zeros; four, two, or no negative zeros 41. Seven, five, three, or one positive zeros; no negative zeros 43. 2, -1, - 4 53. 5,

45. 3, -4,

1 2

47.

1 1 1 , - , -2 (multiplicity 2) 49. , 4, 13, - 13 51. 6, 1 + 15, 1 - 15 2 3 2

1 2 , 2 + 13, 2 - 13 55. 1, -1, -2, - , 3 + 13, 3 - 13 57. 2, - 1 (multiplicity 2) 59. 0, - 2, 1 + 12, 1 - 12 2 3

3 , 1 (multiplicity 2), 8 65. 0, 4, - 4 67. n = 9 inches 69. x = 4 inches 2 71. a. 26 pieces b. 7 cuts 73. 7 rows 75. x = 0.084 inch 77. 25 or 29.05 inches 79. 16.9 feet 81. B = 15. The absolute value of each of the given zeros is less than B. 83. B = 11. The absolute value of each of the zeros is less than B. 61. -1 (multiplicity 3), 2

63. -


A21

Mid-Chapter 3 Quiz, page 299 2 3 5. P(3) = -52, which is less than 0, and P(4) = 100, which is greater than 0. Therefore, by the Intermediate Value Theorem, 2 4 8 1 the continuous polynomial function P has a zero between 3 and 4. [3.2] 6. , 1, , 2, , 4, , 8 [3.3] 3 3 3 3 1 7. Relative minimum y L 2.94 at x L 1.88 [3.2] 8. , 2 (multiplicity 2), 5 [3.3] 2 1. 535 [3.1]

2. Yes [3.1]

3. Down to the far left, up to the far right. [3.2]

4. -3, , 3 [3.3]

Prepare for This Section (3.4), page 299 PS1. 3 + 2i

PS2. 2 - i15

PS3. x3 - 8x2 + 19x - 12

PS4. x2 - 4x + 5

PS5. - 3i, 3i

PS6.

1 1 1 1 - i 119, + i 119 2 2 2 2

Exercise Set 3.4, page 305 1. 2, -3, 2i, - 2i; P(x) = (x - 2)(x + 3)(x - 2i)(x + 2i)

3.

1 1 , -2, 1 + i, 1 - i; P(x) = 2ax - b(x + 2)(x - 1 - i)(x - 1 + i) 2 2

5. 1 (multiplicity 3), 3 + 2i, 3 - 2i; P(x) = (x - 1)3(x - 3 - 2i)(x - 3 + 2i) 7. - 3, 9. 4, 2, 11.

1 1 , 2 + i, 2 - i; P(x) = 2(x + 3)a x + b(x - 2 - i)(x - 2 + i) 2 2

1 3 1 3 1 3 1 3 + i, - i; P(x) = 2(x - 4)(x - 2) ax - - i b a x - + i b 2 2 2 2 2 2 2 2

5 5 , 1 + i 13, 1 - i 13; P(x) = 2ax - b(x - 1 - i 13 )(x - 1 + i 13 ) 2 2

13. -2,

1 1 , 1 + i 12, 1 - i 12 ; P(x) = 2(x + 2) ax - b(x - 1 - i 12 )(x - 1 + i 12 ) 2 2

15. -1 (multiplicity 2),

1 4 4 , i 15 , - i 15 ; P(x) = 3(x + 1)2 ax - b(x - i 15 )(x + i 15 ) 17. 1 - i, 3 3 2

23. 1 - 3i, 1 + 2i, 1 - 2i 31. -

2 3 5 , , 3 4 2

25. 2i, 1 (multiplicity 3)

33. -i, i, 2 (multiplicity 2)

39. P(x) = x 3 - 3x 2 + 4x - 12

27. 5 - 2i,

4

21. 2 + 3i, i, - i

3 7 13 7 13 1 17 1 17 + i, i 29. , - + i, - i 2 2 2 2 2 2 2 2 2

35. -3 (multiplicity 2), 1 (multiplicity 2)

37. P(x) = x3 - 3x2 - 10x + 24

41. P(x) = x 4 - 10x 3 + 63x 2 - 214x + 290

43. P(x) = x - 22x + 212x - 1012x 2 + 2251x - 1830 5

19. i, -3

3

49. P(x) = x 4 - 18x 3 + 131x 2 - 458x + 650 53. P(x) = 3x 3 - 12x 2 + 3x + 18

45. P(x) = 4x 3 - 19x 2 + 224x - 159

47. P(x) = x 3 + 13x + 116

51. P(x) = x 5 - 4x 4 + 16x 3 - 18x 2 - 97x + 102

55. P(x) = - 2x 4 + 4x 3 + 36x 2 - 140x + 150

57. The Conjugate Pair Theorem does not apply

because some of the coefficients of the polynomial are not real numbers.


x - 3 3 , x Z - 3 PS2. x - 5 2

PS6. x + 4 +

7x - 11 x2 - 2x

PS3.

1 3

PS4. x = 0, - 3,

5 2

PS5. The degree of the numerator is 3. The degree of the denominator is 2.

A22


Exercise Set 3.5, page 320 1. All real numbers except 0

y

7. All real numbers except 0, -2, and 6

y

x=3

( )

(0, )

0,

1 4

4 3

y 4

y-axis

x

2

x-axis

x-axis −3

−4

x

x=2

(−4, 0) 4

x (0, 0)

−6

y 5

(0, 2) y=1

−2

x x-axis

−2

3 and 6 2

11. x = -

21. x = - 4, y = 0 y 4

5. All real numbers except

1 1 4 ,x = 13. x = 0, x = , x = 6 15. y = 4 17. y = 30 19. y = 12 2 3 4 23. x = 3, y = 0 25. x = 0, y = 0 27. x = - 4, y = 1 29. x = 2, y = - 1

9. x = 0, x = - 3

x = −4

3. All real numbers

x

y = −1

x = −4

−5

31. x = 3, x = - 3, y = 0

33. x = - 3, x = 1, y = 0

y

y

y 4

1

x

x-axis

−4

x=3

x = −3

1

2

(0, 0)

x

4

−3

x-axis

(−1, 0)

43. y = 3x - 7

45. y = x

47. y = - 4x - 5 49. x = 0, y = x y

5

(0, −4) −3

x=3

x = −1 +

53. x = 4, y = 2x + 13

y (−1, 0)

2

x = −2

y

(0, 0)

y = 2x + 13

(0, − ) 4 3

(

−

3 , 2

0

6

)

57. x = 2, x = - 2, y = x

y

10

(−1, 0)

(4, 0)

4

y=x−3

x

5

(1, 0)

(−1, 0)

x

x=4

y

61.

4

y=x

(− 2, 4)

4

6 x

y 10

x=3

y

65.

5

(0, −1)

y=1

6 x

(−1, −1)

4

x

−4

4

x

1 4

y=x

63. y = x2

(0, − )

−20

x = −2 y

59.

(2, 0)

y=x

55. x = - 2, y = x - 3

y 70 x

y-axis

(−2, 0)

x

3

x = −1 − 2

51. x = - 3, y = x - 6

x

y=1

6 x (1, 0)

10

x = −2

(0, 5)

y

3

x = −3

x

41. x = - 1 + 12, x = - 1 - 12, y = 1

0, 9

y=2

−15

y

4

x=1

2

y=x−6

horizontal asymptote: y = 0

y=1

0, − 3

x = −3

39. x = 3, x = - 3, y = 2 y 6

37. No vertical asymptote;

x-axis

2

1 0, − 9

x = −3 −6

35. x = - 2, y = 1

y = −x − 3

(5, 0) x

−6

(0, − ) 5 2

x = −2

x=2

x


67. a. 1.2 amperes

c. y = 0. The current approaches 0 amperes as the resistance of the variable resistor increases without bound.

b. 33 ohms

b. y = 0.43. As the number of golf balls produced increases, the average cost per golf ball approaches $0.43.

69. a. $76.43, $8.03, $1.19 71. a. $1333.33

A23

b. $8000

C

c.

73. a. 3500 CDs, 5800 CDs, 6500 CDs c. The sales will approach 0 CDs.

10,000

b. The seventh month

x = 100

5,000

75. a. 3.8 centimeters

Surface area (in square centimeters)

40 A

p

80

b. No

c. As the radius r increases without bound, the surface area approaches twice the area of a circle with radius r. 77. ( -2, 2)

Minimum is near (3.8, 277)

79. Answers will vary; however, one possible function is f (x) =

100 r

2 6 Radius (in centimeters)

2x 2 x2 - 9

.

Chapter 3 Review Exercises, page 328 22 [3.1] 2. x 3 + 2x2 - 8x - 9 [3.1] 3. 77 [3.1] 4. 22 [3.1] 5. 33 [3.1] 6. 558 [3.1] x - 3

1. 4x 2 + x + 8 +

7–10. The verifications in Exercises 7–10 use the concepts from Section 3.1. 11. Yes [3.1]

12. No [3.1]

13. Up to the far left, down to the far right [3.2]

14. Down to the far left, down to the far right [3.2]

15. Relative maximum y L 2.015 at x L - 0.560, relative minimum y L -1.052 at x L 0.893 [3.2] 16. Relative minimum y L -1.056 at x L -1.107, relative maximum y L 1.130 at x L 0.270, relative minimum y L 0.927 at x L 0.838 [3.2] 17. P(2) 6 0 and P(3) 7 0. Thus we can conclude by the Intermediate Value Theorem that P has a zero between 2 and 3. [3.2] 18. P(-2) 7 0 and P(-1) 6 0. Thus we can conclude by the Intermediate Value Theorem that P has a zero between -2 and -1. [3.2] 19. Crosses the x-axis at (-3, 0), intersects but does not cross the x-axis at (5, 0) [3.2] 20. Intersects but does not cross the x-axis at (4, 0), crosses the x-axis at (-1, 0) [3.2] 21.

y

y

[3.2] 22.

1 −2

1

y

x

y

[3.2] 24.

[3.2] 25.

2

x

y

[3.2]

10

2

2

4 2 x

26.

[3.2] 23.

2 x

2

x

3 5 15 1 [3.2] 27. 1, 2, 3, 6 [3.3] 28. 1, 2, 3, 5, 6, 10, 15, 30, , , , [3.3] 2 2 2 2

y 4 1

x

29. 1, 2, 3, 4, 6, 12,

1 2 4 1 2 3 4 6 12 1 2 4 , , , , , , , , , , , [3.3] 3 3 3 5 5 5 5 5 5 15 15 15

30. 1, 2, 4, 8, 16, 32, 64 [3.3]

31. 1 [3.3]

33. No positive real zeros, three or one negative real zeros [3.3] 35. One positive real zero, one negative real zero [3.3] 37. 1, -2, - 5 [3.3]

38. 2, 5, 3 [3.3]

32. 1, 2,

1 1 2 1 , , , [3.3] 6 3 2 3

34. Three or one positive real zeros, one negative real zero [3.3]

36. Five, three, or one positive real zeros; no negative real zeros [3.3]

39. - 2 (multiplicity 2), -

4 1 , - [3.3] 2 3

A24

40. 43.


1 1 , -3, i, - i [3.4] 41. 1 (multiplicity 4) [3.3] 42. - , 2 + 3i, 2 - 3i [3.4] 2 2

1 , 2, 1 + 2i, 1 - 2i; 2

44. 1, 3, 1 + 3i, 1 - 3i; 45. -1, 3, 1 + 2i [3.4]

P(x) = 2ax -

1 b(x - 2)(x - 1 - 2i)(x - 1 + 2i) [3.4] 2

P(x) = 1(x - 1)(x - 3)(x - 1 - 3i)(x - 1 + 3i) [3.4] 46. - 5, 2, 2 - i [3.4]

47. P(x) = 2x3 - 3x2 - 23x + 12 [3.4]

50. P(x) = x4 + 2x3 + 6x2 + 32x + 40 [3.4]

49. P(x) = x4 - 3x3 + 27x2 - 75x + 50 [3.4] 52. All real numbers except 55. y = 6 [3.5] 59.

1 and 4 [3.4] 6

56. y = 0 [3.5]

6

x=2

y = 2x

–5

y

[3.5] 64.

−2

y

[3.5] 65.

[3.5]

3

x

5

y=4

x

2

y

[3.5] 66.

[3.5]

5 1

6 3

−3

y

(2, 3)

x

4

x

y 12

[3.5] 62.

y

y=1 −2

63.

54. x = - 3, x = 3 [3.5]

x = −2

6

y=3

51. All real numbers [3.4]

58. y = 3x + 6 [3.5]

[3.5] 61.

y

6

2

53. x = 0, x = - 3, x = 4 [3.5]

57. y = 2x - 1 [3.5]

[3.5] 60.

y

48. P(x) = x4 + x3 - 5x2 + x - 6 [3.4]

x

−6

y=3 3 x

x-axis −1

−12

y = −x

−5

5

x = −3

x=1

2

1

x

x

x=3

67. a. $12.59, $6.43 b. y = 5.75. As the number of skateboards produced increases, the average cost per skateboard approaches $5.75. [3.5] 68. a. 15°F b. 0°F [3.5] 69. a. f (x) = - 0.3030450372x 3 + 10.61192716x 2 - 131.8295333x + 1790.343407 b. 895,000 thefts c. Answers will vary; however, cubic and quartic regression functions are often unreliable at predicting future results. [3.2] 70. a. As the radius of the blood vessel approaches 0, the resistance increases.

b. As the radius of the blood vessel gets larger, the

resistance approaches 0. [3.5]

Chapter 3 Test, page 331 1. 3x2 - x + 6 -

13 [3.1] 2. 43 [3.1] 3. The verification for Exercise 3 uses concepts from Section 3.1. x + 2

4. Up to the far left, down to the far right [3.2]

5. 0,

2 , -3 [3.2] 3

6. P(1) 6 0, P(2) 7 0. Therefore, by the Intermediate Value Theorem, the polynomial function P has a zero between 1 and 2. [3.2] 7. 2 (multiplicity 2), -2 (multiplicity 2),

1 3 3 1 1 (multiplicity 1), -1 (multiplicity 3) [3.3] 8. 1, 3, , , , [3.3] 2 2 2 3 6

9. Four, two, or no positive zeros; no negative zeros [3.3] 12. 0, 1 (multiplicity 2), 2 + i, 2 - i [3.4]

asymptote: y = 3 [3.5]

10.

2 1 5 , 3, - 2 [3.3] 11. 2 - 3i, - , - [3.4] 2 3 2

13. P(x) = x4 - 5x3 + 8x2 - 6x [3.4]

14. Vertical asymptotes: x = 3, x = 2; horizontal


15.

[3.2] 16.

y

y

1 2

4 6 x

− 4 −2

x

2 4 6

−4 −2

−4

5 x

−5

[3.5]

4 2

2

y=1

5

y

[3.5] 17.

(−1, ) 4

A25

y = 2x x=3

x = −1

18. a. f (x) = - 0.1022006728x 4 + 3.494215719x 3 - 31.90479504x 2 - 16.68786302x + 2996.908654 19. a. 5 words per minute, 16 words per minute, 25 words per minute

b. 228 hours

b. 1,810,000 burglaries [3.2]

c. 70 words per minute [3.5]

20. 2.42 inches, 487.9 cubic inches [3.3]

Cumulative Review Exercises, page 332 1. -1 + 2i [P.6]

2.

1 15 [1.3] 3. 2, 10 [1.4] 4. 5x ƒ - 8 … x … 146 [1.5] 5. 1281 [2.1] 2

6. Shift the graph of y = x 2 to the right 2 units and up 4 units. [2.5] 9. x 3 - x 2 + x + 11 [2.6]

10. 4x 3 - 8x 2 + 14x - 32 +

14. 1, 2, 4,

13. 0.3997 [3.2]

7. 2x + h - 2 [2.6]

8. 32x2 - 92x + 60 [2.6]

59 [3.1] 11. 141 [3.1] 12. The graph goes down. [3.2] x + 2

1 2 4 , , [3.3] 15. No positive real zeros, three or one negative real zeros [3.3] 3 3 3

16. -2, 1 + 2i, 1 - 2i [3.4]

17. P(x) = x3 - 4x2 - 2x + 20 [3.4] 18. (x - 2)(x + 3i)(x - 3i) [3.4] x = - 3, x = 2; horizontal asymptote: y = 4 [3.5] 20. y = x + 4 [3.5]

19. Vertical asymptotes:


3. -3

5. 3

7. Range

y

9. Yes

(−5, 6) (−3, 0)

(3, 2) −8

y

11. Yes (6, 8)

8

(0, −4)

8

x

−8

15. No

y

17. Yes

29. 5(1, 0), (2, 1), (4, 2), (8, 3), (16, 4)6

−8

8

49. f

x

35. f −8

41. f

−8

8

-1

19. Yes

21. No

1 5 (x) = - x + 2 2

23. Yes

37. f

-1

(x) =

25. Yes 31. f

-1

4

(0, −4) 8

(−2, − 8)

y

13. Yes

8

x

−4

4

x

−4

(4, −6)

27. 5(1, -3), (2, - 2), (5, 1), (- 7, 4)6

(x) =

7 1 1 x - 2 33. f -1(x) = x + 2 3 3

x x + 1 , x Z 2 39. f -1(x) = ,x Z 1 x - 2 1 - x

-1

(x) = 1x - 1, x Ú 1 43. f -1(x) = x 2 + 2, x Ú 0 45. f -1(x) = 1x + 4 - 2, x Ú - 4 47. f -1(x) = - 1x + 5 - 2, x Ú -5

-1

(x) =

9 x + 32; f -1(x) is used to convert x degrees Celsius to an equivalent Fahrenheit temperature. 5

53. a. c(30) = $22. The company charges $22 per person to cater a dinner for 30 people.

b. c-1(x) =

51. s-1(x) =

300 x - 12

1 x - 12 2

c. 100

55. E -1(s) = 20s - 50,000. The executive can determine the value of the software that must be sold to achieve a given monthly income.

A26


57. a. p(10) L 0.12 = 12%; p(30) L 0.71 = 71% b. The graph of p, for 1 … n … 60, is an increasing function. Thus p has an inverse that is a function. c. p-1 (0.223) = 14, which is the number of people required to be in the group to achieve a 22.3% probability that at least two people share a birthday. 59. a. 25 47 71 67 47 59 53 71 33 47 43 27 63 47 53 39 b. PHONE HOME c. Answers will vary. 61. Because the function is increasing and 4 is between 2 and 5, c must be between 7 and 12. 69. slope:

b 1 ; y-intercept: a 0, - b m m

63. Between 2 and 5

65. No

67. No

71. The reflection of f across the line given by y = x yields f. Thus f is its own inverse.


PS2.

1 81

PS3.

17 8

PS4.

40 9

1 1 1 , 1, 10, and 100 PS6. 2, 1, , and 10 2 4

PS5.

Exercise Set 4.2, page 354 1. f (0) = 1; f (4) = 81

3. g(- 2) =

13. 9.74

15. a. k(x)

b. g(x)

17.

y

1 9 8 ; g(3) = 1000 5. h(2) = ; h( - 3) = 100 4 27

c. h(x)

5

y

21.

23.

x 2

x

2

−2

9. 9.19

11. 9.03

y

2

2

10 1

1 16

d. f (x)

y 20

19.

7. j(- 2) = 4; j(4) =

x

x

2

25. Shift the graph of f vertically upward 2 units. 27. Shift the graph of f horizontally to the right 2 units. 29. Reflect the graph of f across the y-axis. 31. Stretch the graph of f vertically away from the x-axis by a factor of 2. 33. Reflect the graph of f across the y-axis and then shift this graph vertically upward 2 units. 35. Shift the graph of f horizontally to the right 4 units and then reflect this graph across the x-axis. 37. Reflect the graph of f across the y-axis and then shift this graph vertically upward 3 units. 12

39. No horizontal asymptote

6

41. No horizontal asymptote

3

−3

4.7

−4.7

−6

−1

43. Horizontal asymptote: y = 0

1 8

−1

45. Horizontal asymptote: y = 10

12

0

−14

0

12

47. a. 6400 bacteria; 409,600 bacteria b. 11.6 hours 49. a. 145 items per month; 34 items per month b. The demand will approach 0 items per month. 51. a. 69.2% b. 7.6 53. a. 363 beneficiaries; 88,572 beneficiaries b. 13 rounds 55. a. 141°F b. After 28.3 minutes 57. a. 261.63 vibrations per second b. No. The function f (n) is not a linear function. Therefore, the graph of f (n) x does not increase at a constant rate. 61. 63. ( - q , q ) 65. 30, q ) y f(x) = e 2 y=x

2

x


A27


PS2. 3

PS4. f

PS3. 5

-1

PS5. 5x ƒ x Ú 26

3x 2 - x

(x) =

PS6. The set of all positive real numbers

Exercise Set 4.3, page 366 1. 101 = 10

3. 82 = 64

5. 70 = x

19. ln y = x

21. log 100 = 2

9. e0 = 1

23. 2 = ln(x + 5)

y

43.

7. e4 = x

25. 2

11. 102 = 3x + 1 27. - 5

y

45.

2

29. 3 47.

2

13. log3 9 = 2

31. - 2

33. - 4

15. log4 35. 12

1 = - 2 17. logb y = x 16

37. 8

39.

2 5

41.

10 3

y 4

2 4

8

12

x

16

48

−2

x

144

51. (3, q )

8

−2

−2

y

49.

96

x

16

−4

53. ( - q , 11)

55. ( - q , - 2) ´ (2, q )

57. (4, q )

59. ( -1, 0) ´ (1, q )

2 3

−2

61. a

6

11 , qb 2

x

63. a , q b

7 3

65.

y

67. y

2

4

4

8

12

16

20

2

x

−2

69.

48

71. y

y 6

73. a. k(x)

b. f (x)

96

c. g(x)

144

x

d. h(x)

5

75.

Y1=-21n(X)

2

4 2

12 4

−2

16

20

5

9.4

0

10 x

x

8

−5 2.5

77.

0.5

79.

−12

12

−0 5

b. 45 months

91. a. Answers will vary.

c. 3385 digits

b. 2750 units

d. 12,978,189 digits

95. Range of f: 5y ƒ - 1 6 y 6 16; range of g: all real numbers

4.4

−14.4

−2

− 0.5

87. a. 3298 units; 3418 units; 3490 units

b. 96 digits

Y1=31og(abs(2X+10))

10

9.4

−0 5

85. a. 2.0%

5

83.

Y1=1og(X+10)

0 0

1.5

81.

Y1=1og(Xˆ(1/3))

Y1=abs(1n(X))

89. 2.05 square meters

93. f and g are inverse functions.

A28


Prepare for This Section (4.4), page 369 PS1. L0.77815 for each expression

PS2. L 0.98083 for each expression

PS3. L1.80618 for each expression

PS4. L3.21888 for each expression



Exercise Set 4.4, page 377 1. log b x + logb y + logb z

3. ln x - 4 ln z

5.

1 1 1 log2 x - 3 log2 y 7. log7 x + log7 z - 2 log7 y 9. 2 + ln z 2 2 2

1 1 1 1 1 log 4 z - 2 - 3 log4 z 13. log x + log z 15. ln z + 17. log3x2(x + 5)4 19. ln(x + y) 3 2 4 3 6 xy 2 (2x + 5)1w x 2y4 (x + 4)2 3 d 29. ln c b 25. log6 a b 27. ln c d 21. log3x 3 # 1y (x + 1)4 23. log a z x + 2 x(x 2 - 3) yz 2 11.

33. 1.5395

35. 0.8672

37. -0.6131

3

45.

Y1=(1og(X))/(1og(4))

39. 0.6447

41. 8.1749

4

47.

4

2

51.

Y1=(1og((X–3)2))/(1og(3))

8.4

−1

(x + 3)y 3 d x - 3

43. 0.8735 49.

Y1=(1og(X–3))/(1og(8))

31. ln c

Y1=-1og(abs(X–2))/log(5)

6.7

−2.7 8.4

−1

−3

−2

−2

53. False; log 10 + log 10 = 2 but log(10 + 10) = log 20 Z 2.

63. 2

65. 500501

−2

55. True

57. False; log 100 - log 10 = 1 but log(100 - 10) = log 90 Z 1. 61. False; (log 10)2 = 1 but 2 log 10 = 2.

8.4

−1

59. False;

67. 5.0

log 100 2 = = 2, but log 100 - log 10 = 1. log 10 1

69. 106.5I0, or about 3,162,277.7I0

71. 100 to 1

73. 101.8 to 1 or about 63 to 1 75. 5.5 77. 10.4; base 79. 3.16 * 10-10 mole per liter 81. a. 82.0 decibels b. 40.3 decibels c. 115.0 decibels d. 152.0 decibels 83. 10 times as great 85. 1:870,551; 1:757,858; 1:659,754; 1:574,349; 1:500,000

Mid-Chapter 4 Quiz, page 380 2. f

-1

(x) =

4x + 5 , x Z 24 [4.1] 3. 0.0907 [4.2] 4. e6 = x [4.3] 5. x - 24

[4.3] 6. ln x + 3 ln y - 2 [4.4]

y

x = −3

5

5

7. log a

x 5y 2 z2

x

b [4.4] 8. 2.8943 [4.4] 9. 5.9 [4.4] 10. 10 2.8 L 631 times as great [4.4]

Prepare for This Section (4.5), page 380 PS1. log3 729 = 6

PS2. 54 = 625

PS3. log a b = x + 2

PS4. x =

4a 7b + 2c

PS5. x =

3 44

PS6. x =

17.

ln 2 - ln 3 ln 6

100(A - 1) A + 1

Exercise Set 4.5, page 386 1. 6 21. 7

3. -

3 2

23. 4

5. -

6 5

7. 3

25. 2 + 212

9.

log 70 log 5

11. -

log 120 log 3

27.

199 95

29. -1

31. 3

13.

log 315 - 3 2

33. 1010

35. 2

15. ln 10

37. No solution

39. 5

19.

3 log 2 - log 5 2 log 2 + log 5

41. log (20 + 1401)


43.

1 3 log a b 2 2

45. ln (15 4 114 )

59. a. 8500; 10,285

b. In 6 years

61. a. 60°F

120

69. a.

47. ln (1 + 165 ) - ln 8

49. 1.61

b. 27 minutes

51. 0.96

63. 3.7 years

55. - 1.93

53. 2.20

65. 6.9 months

A29

57. - 1.34

67. 6.67 seconds and 10.83 seconds

b. 48 hours

Percent score

c. P = 100 d. As the number of hours of training increases, the test scores approach 100%.

0

120

0 Hours of training

1200

71. a.

c. B = 1000

b. In 27 years, or the year 2026

d. As the number of years increases, the bison

Number of bison

population approaches, but never reaches or exceeds, 1000.

0

100

0 Years

250

b. 78 years

c. 1.9%

Years

73. a.

0

1

0 Percent (written as a decimal) increase in consumption

75. a. 1.72 seconds

b. v = 100

c. The object cannot fall faster than 100 feet per second.

79. The second step. Because log 0.5 6 0, the inequality sign must be reversed.

81. x =

77. 138 withdrawals

y y - 1

83. e0.336 L 1.4

Prepare for This Section (4.6), page 390 PS1. 1220.39

PS2. 824.96

PS3. -0.0495

PS4. 1340

PS5. 0.025

PS6. 12.8

Exercise Set 4.6, page 400 1. a. 2200 bacteria Micrograms of Na

7. a.

b. 17,600 bacteria

3. a. N(t) L 22,600e0.01368t

A

b. 3.18 micrograms

4

11. L2378 years old

3

15. a. $48,885.72

2 1 30 60 90 Time (in hours)

33. P(t) L

100 1 + 4.55556e-0.22302t

39. a. P(t) L

t

c. L15.07 hours b. $49,282.20

ln 3 21. t = r

23. 14 years

29. a. 2400

b. 0.12

1 + 4.66667e-0.14761t

b. 2016

41. a. 0.056

c. $49,283.30 b. 0.16

31. P(t) L

b. $625,000 b. 42°F

9. L 6601 years ago

b. $11,256.80

25. a. 1900

c. 300

5. N(t) L 395,934e0.0289177304t, 748,000

d. L30.14 hours

13. a. $9724.05

35. a. $158,000; $163,000

8500

b. 27,700

17. $24,730.82 c. 200

19. 8.8 years

27. a. 157,500

b. 0.04

c. 45,000

5500 1 + 12.75e-0.37263t

37. a. P(t) L

c. After 54 minutes

1600 1 + 4.12821e-0.06198t 43. a. 211 hours

b. 497 wolves b. 1386 hours

A30


45. 3.1 years

47. a.

v 30 25 20 15 10 5

b. 0.98 second

d. As time increases, the velocity approaches, but

never reaches or exceeds, 32 feet per second.

1

49. a.

c. v = 32

2

v

3

4

t

b. 2.5 seconds

c. L24.56 feet per second

d. The average speed of the object was approximately

24.56 feet per second during the period from t = 1 to t = 2 seconds.

80 60 40 20 1

51. 45 hours

2

3

4

53. a. 0.71 gram

t

b. 0.96 gram

c. 0.52 gram

59. a. 3 minutes 39.41 seconds; 3 minutes 34.75 seconds

55. 2.91%

57. a. 1500

b. 1000

b. 3 minutes 19.13 seconds

Prepare for This Section (4.7), page 404 PS1. Decreasing

PS2. Decreasing

PS3. 36

PS4. 840

PS5. 15.8

PS6. P = 55

Exercise Set 4.7, page 411 1. Increasing exponential function

3. Decreasing exponential function;

5. Decreasing logarithmic function

decreasing logarithmic function 32

4

3.2

7

−2 7

0 4.7

0 −1

−1

7. y L 0.99628(1.20052)x; r L 0.85705

−2.8

9. y L 1.81505(0.51979)x; r L - 0.99978

11. y L 4.89060 - 1.35073 ln x; r L - 0.99921

235.58598 2098.68307 17. y L 1 + 1.90188e-0.05101x 1 + 1.19794e-0.06004x 19. a. f (x) = 9.283629464(1.079805006)x b. $126 21. a. T L 0.06273(1.07078)F b. 5.3 hours 23. a. T L 0.07881(1.07259)F b. 7.5 hours; 2.2 hours 25. An increasing logarithmic model provides a better fit because of the concave-downward nature of the graph. 27. a. p L 7.862(1.026)y b. 36 centimeters 29. a. LinReg: pH L 0.01353q + 7.02852, r L 0.956627; LnReg: pH L 6.10251 + 0.43369 ln q, r L 0.999998. The logarithmic model provides a better fit. b. 126.0 13. y L 14.05858 + 1.76393 ln x; r L 0.99983

31. a. p L 3200(0.91894)t; 2012

15. y L

b. No. The model fits the data perfectly because there are only two data points.

33. a. ExpReg: y L 7337.932474(0.9733252907)x, r L - 0.9219739145

LnReg: y L 7524.106468 - 802.5769482 ln x, r L - 0.9108469173 c. Exponential model: 4390; logarithmic model: 5161

b. The exponential model provides a better fit.

d. Answers will vary.

11.26828 b. 11 billion people 37. A and B have different exponential regression functions. 1 + 2.74965e-0.02924t 39. a. ExpReg: y L 1.81120(1.61740)x, r L 0.96793; PwrReg: y L 2.09385(x)1.40246, r L 0.99999 b. The power regression function provides the better fit. 35. a. P(t) L


A31

Chapter 4 Review Exercises, page 421 y

1.

y

[4.1] 2. f −1

5

4

(3, 3)

( 14 , 2) ( 1 , 1) 2

x

(2, 0)5 (1, −3) ( 0, −6)

−5

[4.1] 3. Yes [4.1] 4. Yes [4.1] 5. Yes [4.1] 6. No [4.1]

(1,0)

−4

7. f

-1

y

−1

−2 4

-1

(x) =

16. p [4.3] 25. [4.2]

y

y

h −1

f

11. f

f −1

x + 4 1 1 3 [4.1] 8. g-1 (x) = - x + [4.1] 9. h-1(x) = - 2x - 4 [4.1] 10. k -1(x) = k(x) = [4.1] 3 2 2 x

(x) = y 4

f

x

4 (2, −1) (4, −2)

2

x

x

−2

g

both k and k −1

2

x

x

4

g−1

h

x , 5x ƒ x 7 26 [4.1] 12. g-1(x) = 1x + 1 - 1, 5x ƒ x Ú - 16 [4.1] 13. 2 [4.3] 14. 4 [4.3] 15. 3 [4.3] x - 2 17. - 2 [4.5]

19. - 3 [4.5] 20. -4 [4.5] 21. 1000 [4.5] 22. 1010 [4.5] 23. 7 [4.5] y y y 27. [4.2] 28. [4.2]

18. 8 [4.5] 26. [4.2]

y

24. 8 [4.5]

1 2

5

12

y

30. [4.2]

4

y

31. [4.3]

32. [4.3]

y

3

4 2

x

1

x

1

1 x

x

y

29. [4.2]

5

4

x

1

4

f(x) = log5 x

x

5

5 −3

x

x

f(x) = log1/3 x

−4

33. [4.3]

y

y

34. [4.3]

35. [4.3]

1

1 3

x

y 2

2

x

2

y 5

36. [4.3] x

−5

5 x −5

12

37. [4.2]

4.7

−4.7 4.7

− 4.7 −2 Xscl = 1 Yscl = 1

1 -3 [4.2] 39. 43 = 64 [4.3] 40. a b = 8 [4.3] 2

7

38.

−7 Xscl = 1 Yscl = 1

A32


( 12 ) 4

41.

= 4 [4.3] 42. e0 = 1 [4.3] 43. log5 125 = 3 [4.3] 44. log2 1024 = 10 [4.3] 45. log10 1 = 0 [4.3]

46. log8 212 =

1 1 [4.3] 47. 2 logb x + 3 logb y - logb z [4.4] 48. logb x - 2 logb y - logb z [4.4] 49. ln x + 3 ln y [4.4] 2 2

12xy xz 1 1 x5 3 ln x + ln y - 4 ln z [4.4] 51. log (x 2 1x + 1 ) [4.4] 52. log [4.4] 53. ln 3 [4.4] 54. ln [4.4] 55. 2.86754 [4.4] 2 2 2 y (x + 5) z

50.

56. 3.35776 [4.4] 63. 4 [4.5]

57. -0.117233 [4.4]

64. 15 [4.5]

65.

58. -0.578989 [4.4]

59.

log 41 1 ln 30 [4.5] 60. - 1 [4.5] 61. 4 [4.5] 62. e [4.5] ln 4 log 5 6

ln(8 317 ) ln 3 2 [4.5] 66. [4.5] 67. 101000 [4.5] 68. e (e ) [4.5] 69. 1,000,005 [4.5] 2 ln 4 ln 5

15 + 1265 [4.5] 71. 81 [4.5] 72. 15 [4.5] 73. 4 [4.5] 74. 5 [4.5] 75. 7.7 [4.4] 76. 5.0 [4.4] 77. 3162 to 1 [4.4] 2

70.

78. 2.8 [4.4] 79. 4.2 [4.4] 80. L 3.98 * 10-6 mole per liter [4.4] 81. a. $20,323.79 b. $20,339.99 [4.6] 82. a. $25,646.69 b. $25,647.32 [4.6] 83. $4,438.10 [4.6] 84. a. 69.9% b. 6 days c. 19 days [4.6] 85. N(t) L e0.8047t [4.6] 86. N(t) L 2e0.5682t [4.6] 88. N(t) L e-0.6931t [4.6]

87. N(t) L 3.783e0.0558t [4.6] 91. Answers will vary. [4.7]

89. a. P(t) L 25,200e0.06155789t

b. 38,800 [4.6]

90. 340 years [4.6]

92. a. ExpReg: R L 163.0844341(0.963625525) , r L - 0.9890222722; x

LnReg: R L 165.1522017 - 34.30348409 ln x, r L - 0.982011144 c. 5.3 per 1000 live births [4.7]

b. The exponential model provides a better fit for the data.

1400 17 -0.22458t 1 + e 3

93. a. P(t) L

b. 1070 coyotes [4.6]

94. a. 21

1 3

b. P(t) : 128 [4.6]

Chapter 4 Test, page 424 1. f

-1

(x) =

3 1 8x x + [4.1] 2. f -1(x) = 2 2 4x - 1 1 f 4 Range f -1: 5y ƒ y 7 26 [4.1] Domain f -1: e x ƒ x 7

y

2 f −1

−2

2 f

x

−4

x 1 2x + 3 [4.3] 4. 2 log b z - 3 log b y - log b x [4.4] 5. log [4.4] 6. 1.7925 [4.4] 2 2 (x - 2)3 5 ln 4 y 8. [4.3] 9. 1.9206 [4.5] 10. [4.5] 11. 1 [4.5] 12. - 3 [4.5] ln 28

3. a. b c = 5x - 3 [4.3] 7. [4.2]

b. log3 y =

y

x = −1

5

5

x

5

13. a. $29,502.36

5

x

19. a. LnReg: d L 67.35500994 + 2.540152486 ln t; logistic: d L

b. $29,539.62 [4.6]

14. 17.36 years [4.6]

15. a. 7.6 b. 63 to 1 [4.4] 16. a. P(t) L 34,600e0.04667108t b. 55,000 [4.6] 17. 690 years [4.6] 18. a. y L 1.67199(2.47188)x b. 1945 [4.7]

72.03782781

1 + 0.1527878996e-0.6775213733t 1100 b. Logarithmic: 74.06 meters; logistic: 72.04 meters [4.7] 20. a. P(t) L b. About 457 raccoons [4.6] 1 + 5.875e-0.20429t


A33

Cumulative Review Exercises, page 425 1. 32, 64 [1.5]

2. 5x ƒ 3 6 x … 66 [1.5]

7. 3500 pounds [1.6]

3. 7.8 [2.1]

4. 38.25 feet [2.4]

5. 4x 2 + 4x - 4 [2.6]

6. f

-1

(x) =

1 5 x + [4.1] 3 3

9. 1, 4, - 13, 13 [3.3]

8. Three or one positive real zeros; one negative real zero [3.3]

10. P(x) = x - 4x + 6x - 4 [3.4] 11. Vertical asymptote: x = 4; horizontal asymptote: y = 3 [3.5] 12. Domain: all real numbers; range: 5y ƒ 0 6 y … 46 [3.5] 13. Decreasing function [4.2] 14. 4 y = x [4.3] 15. log5 125 = 3 [4.3] 16. 7.1 [4.4] 17. 2.0149 [4.5] 3

2

18. 510 years old [4.6]

20. a. P(t) L

19. 3.1798 [4.5]

450 1 + 1.8125e-0.13882t

b. 310 wolves [4.6]

Exercise Set 5.1, page 438 1. 75°, 165°

3. 19°45¿, 109°45¿

15. 105°, Quadrant II

5. 33°26¿45–, 123°26¿45–

17. 296°, Quadrant IV

7.

19. 24°33¿36–

p - 1, p - 1 2

21. 64°9¿28.8–

9.

p 3p , 4 4

11.

23. 3°24¿7.2–

p 3p , 10 5

13. 250°, Quadrant III

25. 25.42°

27. 183.56°

p p 11p 7p 13p p 33. 35. 37. 39. 41. 43. 420° 45. 36° 47. 30° 49. 67.5° 51. 660° 6 2 12 3 4 20 53. - 75° 55. 85.94° 57. 2.32 59. 472.69° 61. 4, 229.18° 63. 2.38, 136.63° 65. 6.28 inches 67. 18.33 centimeters 5p 5p p 69. 3p 71. radians or 75° 73. radian per second L 0.105 radian per second 75. radians per second L 5.24 radians 12 30 3 10p radians per second L 3.49 radians per second 79. 40 mph 81. 1885 feet 83. 6.9 mph 85. 840,000 miles per second 77. 9 87. a. 3.9 radians per hour b. 27,300 kilometers per hour 89. a. B b. Both points have the same linear velocity. 29. 211.78°

31.

91. a. 1.15 statute miles

b. 10%

93. 13 square inches

95. 4680 square centimeters

97. 1780 miles


13 3

PS2. 12

PS3. 2

13 3

PS4.

PS5. 3.54

PS6. 10.39

Exercise Set 5.2, page 449 13 12

3. sin u =

cos u =

5 13 sec u = 13 5

tan u =

12 5

1. sin u =

7. sin u =

12 13

121 7

csc u =

cot u =

5 12

csc u =

121 3

csc u =

7 4

cos u =

133 7

sec u =

7133 33

cos u =

129 2 129 sec u = 29 2

tan u =

4133 33

cot u =

133 4

tan u =

5 2

9. sin u =

13 2

csc u =

1 2

sec u = 2

cos u =

217 17 sec u = 7 2

cos u =

tan u =

13 2

tan u = 13 cot u =

cot u =

2 13 3

3 4 3 12 13 15. 17. 19. 21. 4 5 4 13 5 37. 2 12 - 13 39. 0.6249 41. 0.4488 13.

5. sin u =

213 3

13 3

5129 29

129 5

4 7

11. sin u =

csc u =

cot u = 6 161 61

2 5 csc u =

161 6

cos u =

161 5161 sec u = 61 5

tan u =

6 5

cot u =

5 6

3 5 3 - 13 3 3 12 + 213 25. 12 27. 29. 31. 13 33. 35. 2 4 4 6 3 43. 0.8221 45. 1.0053 47. 0.4816 49. 1.0729 51. 9.5 feet 53. 92.9 inches

23.

A34


55. 5.1 feet

57. 1.7 miles

71. a. 559 feet

59. 74.6 feet

61. 686 million kilometers

73. 127 meters L 5.2 meters

b. 193 feet

65. 612 feet

67. 555.6 feet

69. 338 meters

75. L 8.5 feet

Prepare for This Section (5.3), page 454 PS1. -

4 3

PS2.

15 2

PS3. 60

p 5

PS4.

PS6. 134

PS5. p

Exercise Set 5.3, page 460 3113 113 csc u = 13 3 2113 113 cos u = sec u = 13 2 3 2 tan u = cot u = 2 3

3113 13 2 113 cos u = 13 3 tan u = 2

1. sin u =

7. sin u = 0

3. sin u =

csc u is undefined.

cos u = - 1

sec u = - 1

tan u = 0

cot u is undefined.

25. Quadrant III 47. 65°

49. -

65. -1.26902 83. 150°, 210°

27.

12 2

13 3

9. 0

29. -1 53. -

51. 1

11. 0

213 3

13. 1

13 3

31. -

55.

67. - 0.587785

69. - 1.70130

85. 225°, 315°

3p 7p , 87. 4 4

113 5189 189 5. sin u = csc u = 3 89 5 113 8 189 189 sec u = cos u = sec u = 2 89 8 8 2 5 cot u = cot u = tan u = 3 8 5 csc u =

2 13 3

33.

12 2

15. 0

17. Undefined

35. -

57. 12

13 3

37. 20°

21. Quadrant I

39. 9°

59. cot 540° is undefined.

71. -3.85522

73. 0

77. -

75. 1

p 2p , 91. 3 3

5p 11p , 89. 6 6

19. 1

41.

p 5

23. Quadrant IV

43. p -

79. 1

45. 34°

63. - 0.438371

61. 0.798636

3 2

8 3

81. 30°, 150°

Prepare for This Section (5.4), page 461 PS1. Yes

PS2. Yes

PS3. No

PS4. 2p

PS5. Even function

PS6. Neither


13 1 , b 2 2

17. -

2 13 3

19. - 1

35. a. -0.8 63. sec t 83. -

3. a -

b. 0.6

65. -tan t

sin 2 t cos t

2

13 1 ,- b 2 2 21. -

213 3

37. 0.4, 2.7 67. - cot t

85. csc t sec t

97. (tan t + 2)(tan t - 3)

5. a , -

1 2

13 b 2

23. 0.9391 39. 3.4, 6.0 2

69. cos t

87. 1 - 2 sin t + sin2 t

7. a

13 1 ,- b 2 2

25. -1.1528 41. Odd

9. ( -1, 0) 27. - 0.2679

43. Neither 2

71. 2 csc t

2

99. (2 sin t + 1)( sin t - 1)

101.

12 2

1 2

13 b 2

29. 0.8090

45. Even

47. Odd

49. 2p

77. 21 - cos t

91. cos 2 t

93. 2 csc t

103. -

13 3

13. -

31. 48.0889

75. 1

73. csc t

89. 1 - 2 sin t cos t

11. a- , -

2

13 3

15. -

33. a. 0.9

51. p

53. 2p

79. 21 + cot t 2

1 2 b. -0.4 61. sin t

81. 750 miles

95. (cos t - sin t) (cos t + sin t)


A35

Mid-Chapter 5 Quiz, page 472 1. a.

7p 12

7. a. -

b. 108°

1 2

13 3

b.

10p 4141 feet 3. 13 mph 4. sin u = 3 41 141 csc u = 4

2.

13 1 ,- b 2 2

8. 137 feet 9. a

5141 41 141 sec u = 5

4 7 5. 5 12 5 cot u = 4

cos u =

tan u =

6. 60°

10. sec t = 21 + tan2 t

Prepare for This Section (5.5), page 473 PS3. Reflect the graph of y = f (x) across the x-axis.

PS2. -0.7

PS1. 0.7

1 y = f (x) toward the y-axis by a factor of . 2

PS5. 6p

PS4. Compress the position of each point on the graph of

PS6. 5p

Exercise Set 5.5, page 479 3. 1, p

1. 2, 2p 19.

5.

y

1 ,1 2

7. 2, 4p

1 , 2p 2

9.

y

21.

1

11. 1, 8p

13. 2, 6

y 4

23.

3

15. 3, 3p

17. 4.7, 2.5

y 4

25.

y

27.

1

2 2π x

π

x

π 2π

−2

π

2π x

π

x

2π

−4

x

2π 3

−1

−4

29.

y

31. 2π 3

1

4π 3

y

1 x

−1

39.

−1

y 2

33.

1

2

3

43.

2 x

2

4 x

−2

49.

51.

y

2π

4π x −2

59. y = 2 sin

2 x 3

2π 3

3π

1

4π 3

x

57. y = cos 2x

y

2 x

6π x

−2

2π 3

−2

55.

y

67.

2 6π x

x

63. a. V = 4 sin pt, 0 … t … 8 milliseconds

2 −3π

y

y

2 π

−2

3π

−2

−3

61. y = - 2 cos px

y

65.

π 3

x

47. π 2

2 x

2

4π x

2π

y

3

53.

2

−4

45.

2

2 −2

y

y

37.

x

1

−2

−2

y

x

2

2 1

y 4

35.

−1

y

41.

4

y 1

−2π

−2

2π

4π x

1 cycle per millisecond 2

11

69.

y2 = 2 cos x

b.

−1

20

y1 = 2 cos x

2 −11

6π x

A36


20

71.

75. y = 2 sin

73. y 3

2 x 3

77. y = 2.5 sin

5p x 8

2

−1

20

1 π

2π

3π

4π

x

1 Maximum = e, minimum = L 0.3679, period = 2p e

−20

79. y = 3 cos

4p x 5


PS3. Stretch the position of each point on the graph of y = f (x) away from the x-axis by a factor of 2.

PS2. 0.6

PS4. Shift the graph of y = f (x) 2 units to the right and 3 units up.

PS5. 2p

4 p 3

PS6.


p + kp, k an integer 2 y

23.

3.

p + kp, k an integer 2 y

25.

3 −π 2

33.

π

−3

y

x

2p 3

29.

y

37.

15. 8p

17. 1

2π

−3

39.

y

−π

41.

y

43.

y

−π

51. y = cot

3 x 2

2

−1 4

y

49.

2 2

2 x 3

55. y = sec

3 x 4

π

4

57.

−1.5π

x

2

−π

4 x

−3

1 4

y

x

59.

2 x

−2

2π x

π

53. y = csc

x

−2

47.

−2 −2π

π

x

3

2

π

3

y

45.

−2

2

−1

6π x

3π

−2

y

2π x

π

−1

y

21. 8.5

2

2 x

19. 5

31.

x

2π

1

3

p 3

13.

1

−2

y

35.

11.

2

x

π 2

9. 2p

y

27.

3

−3

7. p

5. 2p

6

1.5π −π −4

2π

−2

x


61. a. h = 1.4 tan x

b. d = 1.4 sec x

20

c.

A37

d. The graph of d is above the graph of h, but the distance

between the graphs approaches 0 as x approaches 8 x 3

65. y = sec

63. y = tan 3x

d

h π /2

0 0

67. y = cot

Xscl = π/8, Yscl = 1

p x 2

p . 2

69. y = csc

4p x 3

Prepare for This Section (5.7), page 491 2 3

PS1. Amplitude 2, period p

PS2. Amplitude , period 6p

PS3. Amplitude 4, period 1

PS5. -3

PS4. 2

PS6. y-axis

Exercise Set 5.7, page 497 p , 2p 2

1. 2,

3. 1,

p ,p 8

17. y 1

5 2p 2p , , 4 3 3

p p , 8 2

27.

−1

− 3π 4

y

29.

p ,p 16

13.

23.

1 10π x 3

−1

− 2π 3

11. - 3p, 6p y

4π 3

1

y 2

9. 21.

5π 2

x

25.

7.

y

19. π 2

p , 3p 4

5. 4, -

15. - 12p, 4p

y

1 x

π 4

−1

π 4

x

9π 4

y

31.

y

2

2

1 −π 4

π 4

5π 4

x

7π 4

−1

3π 2

35. y

33. y

15π x 2

9π 2

37.

y

y 6

39.

1

1

3π 12

−2

41.

x

2 1

3

5

7

π

y

45. π 2

5π 2

47.

1 x

−π

y 1

−2 −4

49. y

π

51.

y y = x − sin x

1 x

π x

2π

4π x

π

y=x y = −sin x

π

53.

y y=x

π

y = x + sin 2x y = sin 2x x π

55.

y

1 −1

57. y = sin a2x -

y = sin x + cos x y = sin x

y = cos x

2π

x

2π x

x

−4

1

x

y

2π x

43. y

7π 12

4 π

2π x

1

8π x

2π

−2

p b 3

59. y = csc a

x - pb 2

2π

x

61. y = sec ax -

p b 2

A38


63. a. 7.5 months, 12 months

y

b.

c. August

65. L 20 parts per million

Sales in hundreds of suits

10 7

S

6 7.5

y1

−2

6

−4

t

10

y2

Time in months

67. s = 7 cos 10 pt + 5

69. s = 400 tan

p t, t in seconds 5 73.

s

s 12

71. y = 3 cos

400 −2

t 0.2 min

p t + 9, 12 feet at 6:00 P.M. 6

0

0

4π

−3

−2

6

77.

79.

0

4π

8π

−0.2

p x - b 2 4

4

− 1.5

− 25

85. y = tan a

83. y = 2 sin a 2x -

1.5

81.

25

0

−6

87. cos 2 x + 2

1.5

89.

91.

6

−2π −4π

The graph above does not show that the function is undefined at x = 0.

Prepare for This Section (5.8), page 500 3 2p

PS2.

5 2

PS3. 4

PS4. 3

PS5. 4

Exercise Set 5.8, page 504 1. 2, p,

1 p

3. 3, 3p,

1 3p

5. 4, 2,

1 2

7.

3 1 , 4, 4 4

2π

4π

−0.5

PS1.

4π

10 t

5

−400

3

75.

2

PS6. y = 4 cos px

−6

2p b 3

A39


11. y =

9. y = 4 cos 3pt

3 4p cos t 2 3

13. y = 2 sin 2t

y

y

15. y = sin pt y

y

4

2

19. y =

21. y = 2.5 cos pt

35. a. 10

π

t

3

29. a. 196 cycles per second,

2 4

2π

t

1 cos 4t 2

y

1

1 2

17. y = 2 sin 2pt

1 2p cos t 2 3

23. y =

t

t

25. y = 4 cos 4t

1

t

t 1 , 2 feet; y = - 2 cos 4p 2

27. 4p,

1 p second b. The amplitude needs to increase. 31. h = - 37cos a tb + 41 33. a. 3 b. 59.8 seconds 196 22.5

b. 71.0 seconds 37. a. 10 43. Yes 45. Yes

b. 9.1 seconds

39. a. 10

b. 6.1 seconds

41. The new period is three times the

original period.

Chapter 5 Review Exercises, page 510 1. Complement: 28°, Supplement: 118° [5.1]

7p 5. [5.1] 4

6. 114.59° [5.1]

2. No complement, Supplement: 5° [5.1]

7. 3.93 meters [5.1] 12. sin u =

11. 55 radians per second [5.1]

cos u =

5134 34

tan u =

3 [5.2] 5

134 3

sec u =

134 5

cot u =

5 3

10. 37.7 feet per second [5.1]

15 [5.2] 2

13.

4. 340°, Quadrant IV [5.1]

14.

2 16 + 4 13 [5.2] 15. [5.2] 3 4

413 + 16 1 213 + 1 3 110 [5.2] 17. 2 [5.2] 18. [5.2] 19. sin u = [5.3] 3 4 4 10

20. sin u =

csc u =

4 5

cos u = -

3 5

tan u = -

4 p p [5.3] 21. 70° [5.3] 22. 80° [5.3] 23. [5.3] 24. [5.3] 3 4 6

5 5 3 sec u = cot u = 4 3 4

25. a. -

213 3

28. a. -

213 3

31. even [5.4]

c. -1

b. 1

b. 2 [5.3]

d. -

p , 0 [5.6] 3 [5.5]

y 2

2

1 [5.3] 26. a. - 0.5446 b. 0.5365 2

c. - 3.2361

12 2

b. a , -

29. a.

34. sec2f [5.4]

41. No amplitude, 46.

9. 8 p radians per second [5.1]

8. 0.3 [5.1]

3 134 34

csc u = 16.

3. 145°, Quadrant II [5.1]

b. -1 [5.3]

35. tan f [5.4]

42. 2, 47.

30. a. ( - 1, 0)

36. sin f [5.4]

2p p , - [5.5] 3 9

43. 1, p,

x

37. tan2f [5.4]

p [5.5] 3

[5.5]

y 1

48.

1

51.

c. a-

y

[5.5]

3π

x

4π 3

2π 3

[5.7]

y 3

52.

−π 8

8

−3

b.

40. 3, p,

2π 3

π

x

π

p [5.6] 4

x

3π

−1

[5.6]

y

−π

p [5.5] 2

[5.7]

y 1

53.

[5.6]

y

3 π 3

13 [5.3] 3

45. No amplitude, 2p,

x

1 x 7π

1 2

d. (1, 0) [5.4]

39. 0 [5.4]

p 3p , [5.6] 2 8 49.

27. a. -

12 12 ,b 2 2

38. csc 2f [5.4]

2 π

[5.7]

y

13 b 2

44. No amplitude,

−1

50.

1 2

d. 3.0777 [5.3]

π

x

1 π 2

π

x

A40


54.

[5.6]

y

55.

[5.6]

y

56.

[5.7]

y

6 4

1

4 2 π 2

π 4

−2

π 2

2

3π 4 π x

π

2π

3π

4π

π x

x

−2 −4

−4

−6

57.

[5.7]

y

58.

[5.7]

y

59.

3

[5.7]

y 6

60.

y

[5.7]

π 2

2

−π 8 −3

61.

3π 8

π 6

x

7π 8

[5.7]

y 5

7π 6

62.

x

π −3 4 −6

[5.7]

y

9π 4

5π 4

63.

2π x

π 2

[5.7]

y

2

x

π

x

−6 π 3

[5.7]

65. y

−4

[5.7] 64. π 6

π

x

y 1

4

−1

x

π

1

66.

2π 3

π

x

[5.7]

y

67.

[5.7]

y

2

−2

π 2

4

4

2

2

2π x

π

−2

68. 0.089 mile [5.2]

69. 12.3 feet [5.2]

74. y = 3 cos 4pt [5.8]

π

π 2

3π 2

2π

3π 2

x

π

2π

x

−2

70. 1.7 feet per second [5.1]

75. Amplitude = 0.5, f =

π 2

71. 46 feet [5.2]

1 , p = p, y = - 0.5 cos 2t [5.8] p

72. 2.5,

p 25 , [5.8] 25 p

73. y = 5 sin 2pt [5.8]

76. 7.2 seconds [5.8]

Chapter 5 Test, page 512 1. a. 6.

5p 6

b. 72° [5.1]

2.

158 215 [5.2] 7. sin u = 7 5 csc u = -

11.

p [5.1] 12

p [5.6] 3

15 2

3. 13.1 centimeters [5.1]

cos u = -

15 5

sec u = - 15

4. 12p radians>second [5.1]

tan u = 2 [5.3] cot u =

12. Amplitude 3, period p, phase shift -

1 2

p [5.7] 4

8.

13 - 6 [5.3] 6

9. a

5. 80 centimeters>second [5.1]

13 1 , - b [5.4] 2 2

10. sin2 t [5.4]


13. Period 3, phase shift -

1 [5.7] 2

14.

[5.5]

y 4

15.

2

A41

[5.6]

y 4

2 π

−2

2π

3π

4π x

π

−2

−4

2π

3π

4π

x

−4

p 16. Shift the graph of y = 2 sin 2x, units to the right and 1 unit down. [5.7] 4

17.

[5.7]

y 4

2 −2

18.

[5.7]

y 4

2

π

2π

3π

19. 25.5 meters [5.2]

20. y = 13 sin

π

2π

3π

4π x

2p t [5.8] 5

4π x

−2

Cumulative Review Exercises, page 513 1. (x + y)(x - y) [P4]

2. 13 [P5]

6. (- q , 4) ´ (4, q ) [2.2/3.5]

3. 12 square inches [1.2]

4. Odd function [2.5]

5. f -1(x) =

3x [4.1] 2x - 1

7. 30, 24 [2.2]

8. Shift the graph of y = f (x) horizontally 3 units to the right. [2.5] 9. Reflect the 5 5p 13 + 1 [5.1] 11. 225° [5.1] 12. 1 [5.3] 13. [5.2] 14. [5.2] graph of y = f (x) across the y-axis. [2.5] 10. 3 2 4 p 3 15. Negative [5.3] 16. 30° [5.4] 17. [5.4] 18. ( - q , q ) [5.4] 19. 3 -1, 14 [5.4] 20. [5.2] 3 5

Exercise Set 6.1, page 520 57. Identity

59. Identity

61. Identity

13 2 + 13 left side is and the right side is . 2 2

p , the left side is 2 and the right side is 1. 4 69. If x = 0, the left side is - 1 and the right side is 1.

63. Not an identity

65. If x =

67. If x = 0°, the

Prepare for This Section (6.2), page 522 PS1. Both function values equal

equal.

1 . 2

PS2. Both function values equal

1 . 2

PS3. For each of the given values of u, the function values are

PS4. For each of the given values of u, the function values are equal.

PS5. Both function values equal

23 . 3

PS6. 0

Exercise Set 6.2, page 529 16 + 12 16 + 12 1 16 + 12 - 16 + 12 3. 5. 2 - 13 7. 9. 11. 2 + 13 13. 0 15. 17. 13 4 4 4 4 2 19. cos 48° 21. cot 75° 23. csc 65° 25. sin 5x 27. cos x 29. sin 4x 31. cos 2x 33. sin x 35. tan 7x 13 77 84 77 63 56 63 63 56 33 77 84 37. a. b. c. 39. a. b. c. 41. a. b. c. 43. a. b. c. 85 85 36 65 65 16 65 65 56 85 85 84 33 16 63 56 63 16 45. a. b. c. 47. a. b. c. 75. - cos u 77. tan u 79. sin u 81. Identity 83. Identity 65 65 16 65 65 63 1.

A42


Prepare for This Section (6.3), page 532 2 tan a PS4. For each of the given values of a, the function values are equal. 1 - tan2 a PS5. Let a = 45°; then the left side of the equation is 1, and the right side of the equation is 12. PS6. Let a = 60°; then the left side of 13 1 the equation is , and the right side of the equation is . 2 4 PS1. 2 sin a cos a

PS2. cos2 a - sin2 a

PS3.

Exercise Set 6.3, page 538 24 7 24 240 161 11. sin 2 = , cos 2 = , tan 2 = , cos 2 = , 25 25 7 289 289 240 527 336 240 161 240 336 tan 2 = , cos 2a = , tan 2a = , cos 2a = , tan 2a = 13. sin 2 = 15. sin 2a = 161 625 625 527 289 289 161 720 1519 720 1 , cos 2a = , tan 2a = (3 + 4 cos 2x + cos 4x) 17. sin 2a = 19. 3(1 + cos 2x) 21. 1681 1681 1519 8 1. sin 4a

23.

3. cos 10b

5. cos 6a

9. sin 2 = -

7. tan 6a

1 (1 - cos 2x - cos 4x + cos 2x cos 4x) 16

25.

22 + 13 2

27. 22 + 1

22 + 12 2

29. -

31.

22 - 12 2

33.

22 - 12 2

a 22 - 13 a 5126 a 126 a a 5134 a 3134 5 , cos = , tan = 5 39. sin = , cos = , tan = 37. sin = 2 2 26 2 26 2 2 34 2 34 2 3 2 a 15 a 215 a 1 a 12 a 7 12 1 a , cos = , tan = , cos = , tan = L 2.61 41. sin = 43. sin = 91. a. 12 - 12 2 5 2 5 2 2 2 10 2 10 2 7 1 b. a = 2 sin- 1 a b c. a decreases. 93. Identity 95. Identity M 35.

Mid-Chapter 6 Quiz, page 540 3.

16 - 12 [6.2] 4

4. a. tan 3a

b. cos 5a [6.2]

5.

215 [6.2] 25

6. -

3 [6.3] 5

7.

22- 13 [6.3] 2

Prepare for This Section (6.4), page 541 PS1. sin a cos b

PS2. cos a cos b

PS3. Both function values equal -

1 . 2

PS4. sin x + cos x

PS5. Answers will vary.

PS6. 2

Exercise Set 6.4, page 546 13 - 2 4 3 1 2 sin 3u cos u 19. 2 cos 2u cos u 21. - 2 sin 4u sin 2u 23. 2 cos 4u cos 3u 25. 2 sin 7u cos 2u 27. - 2 sin u sin u 2 2 12 3 u 1 5 2 sin u sin u sin u 49. y = 12 sin(x - 135°) 51. y = sin(x - 60°) 53. y = sin(x - 45°) 31. 2 cos 4 4 12 12 2 p 3p b 61. y = sin ax + b y = 3 12 sin(x + 135°) 57. y = p12 sin(x - 45°) 59. y = 12 sin ax + 4 6 3p 2p y = 20 sin ax + b 65. y = 5 12 sin ax + b 3 4

1. sin 3x - sin x 17. 29. 55. 63. 67.

3.

1 (sin 8x - sin 4x) 2

y

69. 8π 3

2 2π 3

y

2 x

5. sin 8x + sin 2x

π − 4

71. 7π 4

7.

y

1 4

11. -

x

12 4

75. 10

17π 6 5π 6

9.

y

73.

2 x

1 (cos 4x - cos 6x) 2

13. -

1 4

15.

y 12

−2π 3 4π 3

−10

x

−6

π 6

13π 6

x


77. a. s(t) = sin(2p # 1336t) + sin(2p # 770t) 81. Identity 83. Identity

b. s(t) = 2 sin(2106pt) cos(566pt)

c. 1053 cycles per second

A43

79. Identity

Prepare for This Section (6.5), page 548 PS1. PS2. PS3. PS6.

A one-to-one function is a function for which each range value ( y value) is paired with one and only one domain value (x value). If every horizontal line intersects the graph of a function at most once, then the function is a one-to-one function. f 3g(x)4 = x PS4. f 3 f - 1(x)4 = x PS5. The graph of f - 1 is the reflection of the graph of f across the line given by y = x. No

Exercise Set 6.5, page 557 p 2p 17. 19. a. 1.0014 b. 0.2341 21. a. 1.1102 3 6 p 1 3 1 p p x b. 0.2818 23. u = cos- 1 a b 25. 27. 2 29. 31. 1 33. 35. 37. 39. Undefined 41. 0.4636 43. 7 2 5 2 6 4 6 2 - 12 2 + 115 13 4115 24 24 1 12 55. 57. (3 17 - 413) 59. 61. 2 63. 45. 47. 49. 51. 0 53. 3 15 25 25 6 5 13 2 2 21 - x 7 12 5p L 0.2588 69. 65. 67. cos 10 12 x 1.

p 2

5p 6

3.

5. -

y

75.

p 4

7.

9.

y

77.

π

p 3

p 3

11. -

p 4

13. -

y

79.

p 3

81.

15.

−4

y

4

x

83. a. s = 3960 cos- 1 a

x

−2

b. 17,930 miles

π

1

−π

x

1

85. f (x) Z g (x)

−π

x

π 2

87.

3960 b a + 3960

π

89.

2π

91.

y

3

f g −1

f and g have the same graph in Quadrant I x 1

− 12.5

− 2.35

−9.5

−2

97. y =

1 tan 5x 3

12.5

2.35

9.5

0

− 2π

π − 2

99. y = 3 + cos a x -

p b 3

Prepare for This Section (6.6), page 560 PS1. x =

5 173 6

PS2. 1 - cos2x

PS3.

5 9 13 p, p, and p 2 2 2

PS4. (x + 1)ax -

13 b 2

PS5.

100

PS6. 0, 1

40

0 0

A44


Exercise Set 6.6, page 568 p 7p p 4p p p 3p 3p p 3p 3. 5. 7. , , , , , , 4 4 3 3 4 2 4 2 2 2 p 3p 5p 7p p 5p 4p 5p , p, , , , , 17. 0, , 19. 21. 0, 4 4 4 4 6 6 3 3 29. No solution 31. 12.8°, 167.2° 33. 15.5°, 164.5° 1.

41. 0°, 120°, 240° 51. 53.1°, 180°

43. 70.5°, 289.5° 53. 72.4°, 220.2°

p 5p 7p 11p p p 3p 11p p 3p p p 5p 11. 13. 15. , , , , , , , , , 6 4 4 6 4 4 6 2 6 6 6 6 6 3p p , p, 23. 41.4°, 318.6° 25. No solution 27. 68.0°, 292.0° 2 2 35. 0°, 33.7°, 180°, 213.7° 37. No solution 39. No solution 9.

45. 68.2°, 116.6°, 248.2°, 296.6° 55. 50.1°, 129.9°, 205.7°, 334.3°

p 2kp + , where k is an integer 10 5

47. 19.5°, 90°, 160.5°, 270° 57. No solution

49. 60°, 90°, 300°

59. 22.5°, 157.5°

61.

p kp + , where k 8 2

5p p + 2kp, p + 2kp, + 2kp, where k is an integer 3 3 p 5p 7p 3p 11p p p 5p + kp, + kp, where k is an integer 69. 0 + 2kp, where k is an integer 71. 0, p 73. 0, , , , p, , , 67. 2 6 6 2 6 6 2 6 p 3p p 2p 4p 5p 4p 5p p 3p 5p 7p p 5p 83. , p, , , , p, , , , p 85. 0.7391 75. 0, , 77. 0, , 79. 81. 0, , 2 2 3 3 3 3 3 3 4 4 4 4 6 6 87. - 3.2957, 3.2957 89. 1.16 91. 14.99° and 75.01° is an integer

63.

65. 0 + 2kp,

The sine regression functions in Exercises 93, 95, and 97 were obtained on a TI-83/TI-83 Plus/TI-84 Plus calculator by using an iteration factor of 16. The use of a different iteration factor may produce a sine regression function that varies from the regression functions listed below. 93. a. y L 1.358096 sin(0.015952x + 1.877512) + 6.143353 b. 4:55 95. a. y L 49.486134 sin(0.212889x + 2.920261) + 46.653448 b. 73% 97. a. y L 32.226740 sin(0.399289x + 2.917528) + 26.974425 b. 40.3°

p p , 103. 6 2

5p ,0 105. 3

99. b. 42° and 79°

p p 3p 5p 3p 7p , p, , , 107. 0, , , 4 2 4 4 2 4

c. 60°

101. 0.93 foot, 1.39 feet

Chapter 6 Review Exercises, page 576 1.

16 - 12 [6.2] 4

2. 13 - 2 [6.2]

7.

22 - 12 [6.3] 2

8. -

22. 32. 57. 60. 63.

22 - 13 [6.3] 2

16 - 12 [6.2] 4

4. 12 - 16 [6.2]

9. 12 + 1 [6.3]

10.

5. -

22 + 12 [6.3] 2

16 + 12 [6.2] 4 11. a. 0

6.

b. 13

12 + 16 [6.2] 4 c.

1 [6.2/6.3] 2

13 16 - 12 22 - 13 b. - 13 c. [6.2/6.3] 14. a. b. - 23 c. 1 [6.2/6.3] 2 2 4 3p 5p cos 32º [6.2] 16. cot 18º [6.2] 17. sin [6.2] 18. csc [6.2] 19. sin 6x [6.3] 20. tan 3x [6.2] 21. sin 3x [6.2] 10 14 cos 4u [6.3] 23. tan 2u [6.1] 24. tan u [6.3] 29. 2 sin 3u sin u [6.4] 30. - 2 cos 4u sin u [6.4] 31. 2 sin 4u cos 2u [6.4] 4 13 4 56 7 3 2 cos 3u sin 2u [6.4] 51. [6.5] 52. [6.5] 53. [6.5] 54. [6.5] 55. [6.6] 56. [6.6] 5 5 65 25 2 5 p 30°, 150°, 240°, 300° [6.6] 58. 0°, 45°, 135° [6.6] 59. + 2kp, 3.8713 + 2kp, 5.5535 + 2kp, where k is an integer [6.6] 2 7p 19p 3p 7p p p 5p 13p 17p - + kp, 1.2490 + kp, where k is an integer [6.6] 61. , , , , , , [6.6] 62. [6.6] 4 12 12 12 12 12 12 4 4 5p p p 4p y = 2 sin ax + b [6.4] 64. y = 212 sin ax + b [6.4] 65. y = 2 sin a x + b [6.4] 66. y = sin ax - b [6.4] 6 4 3 6

12. a. 0 15.

3.

b. - 2

c.

1 [6.2/6.3] 2

13. a.

y

2 −

π 6

y

y 3 11π x 6

−5π 4

y 1

2

3π 4

−3

x

−

4π 3

2π 3

x

π 6

−1

13π 6

x


67. 0.4 [6.5]

68.

p [6.5] 6

69.

[6.5]

y

70.

[6.5]

y

71.

π 2

π

72.

π 2 x

1

1

[6.5]

y

[6.5]

y

π 2

x

1

1

x

73. a. y L 0.888804(0.015538x + 1.880197) + 6.504205

A45

x

b. 6 : 57

Chapter 6 Test, page 578 5.

- 16 + 12 [6.2] 4

14. 0.701 [6.5]

15.

6. -

12 [6.2] 10

5 [6.5] 13

16.

[6.5]

π 2

12.

2 - 13 [6.4] 4

−1

1

2

3

13. y = sin ax +

17. 41.8°, 138.2° [6.6]

y = sin−1 x

−

p 2p 4p , , [6.6] 2 3 3

7 [6.3] 25

y

y = sin− 1 (x + 2) −3

19.

9. -

8. sin 9x [6.2]

5p b [6.4] 6 p 11p 18. 0, , p, [6.6] 6 6

x

π 2

20. a. y L 2.781047 sin(0.016734x - 1.308844) + 12.121309

b. 14.902 hours

Cumulative Review Exercises, page 579 1. (x - y)(x2 + xy + y2) [P.4]

2. 2, 8 [1.1]

3. Shift the graph of y = f (x) to the left 1 unit and up 2 units. [2.5]

4. Reflect the graph of x [4.1] 8. log2 x = 5 [4.3] 9. 3 [4.3] y = f (x) across the x-axis. [2.5] 5. x = 2 [3.5] 6. Odd function [2.5/5.5] 7. f - 1(x) = x - 5 4p 2 15 p 1 13 10. [5.1] 11. 300° [5.1] 12. [5.2] 13. Positive [5.3] 14. 50° [5.3] 15. [5.3] 16. x = , y = [5.4] 3 5 3 2 2 p p p p 17. 0.43, p, [5.7] 18. [6.5] 19. 3- 1, 14 [6.5] 20. a - , b [6.5] 12 6 2 2

Exercise Set 7.1, page 589 1. C = 77°, b L 16, c L 17 3. B = 38°, a L 18, c L 10 5. C L 15°, B L 33°, c L 7.7 7. C = 45.1°, b L 39.4, c L 30.2 9. C = 32.6°, c L 21.6, a L 39.8 11. B = 47.7°, a L 57.4, b L 76.3 13. A L 58.5°, B L 7.3°, a L 81.5 15. A L 71.7°, B L 44.1°, b L 55.3 or A L 108.3°, B L 7.5°, b L 10.4 17. A L 61.7°, C L 35.7°, a L 8.21 19. No triangle is formed. 21. C = 19.8°, B = 145.4°, b L 10.7 or C = 160.2°, B = 5.0°, b L 1.64 23. No triangle is formed. 25. C = 51.21°, A = 11.47°, c L 59.00 27. B L 130.9°, C L 28.6°, b L 22.2 or B L 8.1°, C L 151.4°, b L 4.17 29. L 68.8 miles 31. 231 yards 33. L 110 feet 35. 4840 feet 37. L 96 feet 39. L 33 feet 41. L 8.1 miles 43. L 1200 miles 45. L 260 meters 49. 48 Minimum value of L L 11.19 meters

90

0

Minimum X=49.854625 Y=11.1941 − 15

A46



PS3. C = cos- 1 a

PS2. 25.5 square inches

a2 + b2 - c2 b 2ab

PS4. 12.5 meters

PS5. 6

PS6. c2 = a2 + b2

Exercise Set 7.2, page 597 1. L 13 3. L 150 5. L 29 7. L 9.5 9. L 10 11. L 40.1 13. L 90.7 15. L 39° 17. L 90° 19. L 47.9° 21. L 116.67° 23. L 80.3° 25. a L 11.1, B L 62.0°, C L 78.6° 27. A L 34.2°, B L 104.6°, C L 41.3° 29. L 140 square units 31. L 53 square units 33. L 81 square units 35. L 299 square units 37. L 36 square units 39. L 7.3 square units 41. L 710 miles 43. L 74 feet 45. L 60.9° 47. L 350 miles 49. 40 centimeters 51. L 2800 feet 53. 402 miles, S62.6°E 55. L 47,500 square meters 57. 162 square inches 59. L $41,000 61. L 6.23 acres 63. Triangle DEF has an incorrect dimension. 65. L 12.5° 69. L 140 cubic inches


PS2. - 6.691

PS3. 30°

PS4. 157.6°

PS5.

15 5

PS6.

14117 17

Exercise Set 7.3, page 613 5. a = 7, b = - 1; 87, - 19 7. a = - 7, b = - 5; 8 - 7, - 59 9. a = 0, 15 - 2 15 3 4 15 - 215 b = 8; 80, 89 11. 5, L 126.9°, h - , i 13. L 44.7, L 296.6°, h , i 15. L 4.5, L 296.6°, h , i 5 5 5 5 5 5 11 7 7 158 - 3158 , i 19. 8- 6, 129 21. 8 - 1, 109 23. h , i 25. 215 27. 21109 17. L 45.7, L 336.8°, h 58 58 6 3 11 1 29. - 8i + 12j 31. 14i - 6j 33. i + j 35. 1113 37. a1 L 4.5, a2 L 2.3, 4.5i + 2.3j 39. a1 L 2.8, a2 L 2.8, 12 2 2.8i + 2.8j 41. L 380 miles per hour 43. L 250 miles per hour at a heading of 86° 45. 293 pounds 47. a. 131 pounds b. 319 pounds 49. The forces are in equilibrium. 51. The forces are not in equilibrium. F4 = 0i + 10j 53. The forces are in 46 equilibrium. 55. - 3 57. 0 59. 1 61. 0 63. L 79.7° 65. 45° 67. 90°, orthogonal 69. 180° 71. 5 14129 11 15 L 2.6 75. 15 L 2.2 77. L - 4.9 79. L 954 foot-pounds 81. L 779 foot-pounds 73. 29 5 1. a = 4, b = 2; 84, 29

83.

3. a = - 5, b = 4; 8 - 5, 49

86, 99

y

85. The vector from P1(3, -1) to P2(5, - 4) is equivalent to 2i - 3j.

(6, 9)

9

v

u+v

u+

v+

w

w

(1, 4)

(−1, 1) u 6

x

87. Because v # w = 0, the vectors are perpendicular.

89. 87, 29 is one example.

91. No

95. The same amount of work is done.

Mid-Chapter 7 Quiz, page 615 1. 22.8 inches [7.1] 2. 109 miles [7.2] 3. 28.2º [7.2] 4. a. 85, 299 6. 81.2º [7.3] 7. 120 pounds [7.3] 8. 2396 foot-pounds [7.3]

b. 134

c. 117

d. 29 [7.3]

5. 72 square meters [7.2]


A47

Prepare for This Section (7.4), page 616 PS1. 1 + 3i

PS2.

1 1 + i 2 2

PS3. 2 - 3i

PS4. 3 + 5i

PS5. -

1 13 i 2 2

PS6. - 3i, 3i

Exercise Set 7.4, page 621 9. 12 cis 315°

Im

1–7.

11. 2 cis 330°

13. 3 cis 90°

15. 5 cis 180°

17. 16 cis 120°

19. 4 cis 240°

Re

−2 − 2i

−2i

3−i 3 − 5i

ƒ - 2 - 2i ƒ = 2 12 ƒ 13 - i ƒ = 2 ƒ - 2i ƒ = 2 ƒ 3 - 5i ƒ = 134 21. 12 + i12 35.

23.

9 13 9 - i 2 2

49. 3i

51. -

12 12 i 25. - 3 12 + 3i12 2 2

37. L - 0.832 + 1.819i

313 3i + 2 2

61. 0 - 12 i = - 12 i

39. 6 cis 255°

53. L - 2.081 + 4.546i 63. 16 - 16i

65. -

27. 8

29. - 13 + i

41. 12 cis 335°

43. 10 cis

55. L 2.7321 - 0.7321i

13 3 + i 8 8

31. - 3i

67. 59.0 + 43.0i

16p 15

33. - 412 + 4i12 45. 24 cis 6.5

57. 6 + 0i = 6

59. -

47. - 4 - 4i 13

13 1 + i 2 2

69. r2 or a2 + b2

Prepare for This Section (7.5), page 622 PS1. i

PS2. 2

PS3. 3

PS4. 2 12 cis

p 4

PS5. - 13 + i

PS6. 1

Exercise Set 7.5, page 625 1. - 128 - 128i 13 3. - 16 + 16i13 5. 1612 + 16i 12 7. 64 + 0i = 64 9. 0 - 32i = - 32i 11. - 4 + 0i = - 4 13. 1024 - 1024i 15. 0 - 1i = - i 17. 3 + 0i = 3 19. 2 + 0i = 2 21. 0.809 + 0.588i 23. 1 + 0i = 1 25. 1.070 + 0.213i 27. - 0.276 + 1.563i

i 13 1 + i13 - 0.309 + 0.951i - 0.213 + 1.070i - + + i13 - 1 + 0i = - 1 - 1.070 - 0.213i 2 2 + 0i = - 2 - 0.309 - 0.951i 0.213 - 1.070i 1 i 13 - - i13 0.809 - 0.588i 2 2 - i13 4 29. 2 12 + 2i 16 31. 2 cis 60° 33. cis 67.5° 35. 3 cis 0° 37. 3 cis 45° 39. 1 2 cis 75° 4 - 2 12 - 2i16 2 cis 180° cis 157.5° 3 cis 120° 3 cis 135° 1 2 cis 165° 4 2 cis 300° cis 247.5° 3 cis 240° 3 cis 225° 1 2 cis 255° 4 cis 337.5° 3 cis 315° 12 cis 345° 47. For n Ú 2, the sum of the nth roots of 1 is 0. - 3 + 0i = - 3

1 -1 -2 -1 1

- 1.216 - 1.020i 1.492 - 0.543i

3 41. 1 2 cis 80°

3 2 cis 200° 1 3 2 cis 320° 1

Chapter 7 Review Exercises, page 630 1. C = 41.4°, a L 227, b L 131 [7.1] 2. C = 97.9°, b L 51.1, c L 82.8 [7.1] 3. B L 48°, C L 95°, A L 37° [7.2] 4. A L 47°, B L 75°, C L 58° [7.2] 5. c L 13, A L 55°, B L 90° [7.2] 6. a L 169, B L 37°, C L 61° [7.2]

A48


7. No triangle is formed. [7.1] 8. No triangle is formed. [7.1] 9. A L 27.9°, B L 30.6°, a L 29.9 [7.1] 10. B L 55.5°, C L 79.3°, c L 85.2 or B L 124.5°, C L 10.3°, c L 15.5 [7.1] 11. L 360 square units [7.2] 12. L 31 square units [7.2] 13. L 920 square units [7.2] 14. L 46 square units [7.2] 15. L 790 square units [7.2] 16. L 210 square units [7.2] 17. L 170 square units [7.2] 18. L 140 square units [7.2] 19. a1 = 5, a2 = 3, 85, 39 [7.3] 20. a1 = 1, a2 = 6, 81, 69 [7.3] 21. L 4.5, 153.4° [7.3] 22. L 6.7, 333.4° [7.3] 23. L 3.6, 123.7° [7.3] 24. L 8.1, 240.3° [7.3]

3134 7 1193 12 1193 5126 126 5134 ,i [7.3] 27. i + j [7.3] 28. i j [7.3] 193 193 26 26 34 34 17 47 13 29. 8- 7, - 39 [7.3] 30. 818, 79 [7.3] 31. - 6i j [7.3] 32. - i j [7.3] 33. 386 miles [7.1] 34. 464 feet [7.1] 2 6 6 35. 420 miles per hour [7.3] 36. L 7° [7.3] 37. 18 [7.3] 38. - 21 [7.3] 39. - 9 [7.3] 40. 20 [7.3] 41. 86° [7.3] 25. h -

8 189 5189 , i [7.3] 89 89

42. 138° [7.3] 48.

26. h

43. 125° [7.3]

[7.4]

Im 4i

44. 157° [7.3]

49.

45.

Im

27129 10 141 [7.3] 46. [7.3] 47. L 662 foot-pounds [7.3] 41 29 [7.4] 50. 212 cis 315° [7.4] 51. 213 cis 120° [7.4]

−5 + i 3

4 Re

2

−2

Re

z = 2 − 3i

113, L 304° L 5.29, L 161° 5 12 512 i [7.4] 53. - 3 - 3i13 [7.4] 54. - 512 - 512 i [7.4] 55. L - 8.918 + 8.030i [7.4] 52. 2 2 56. L - 27.046 + 7.247i [7.4] 57. L - 6.012 - 13.742i [7.4] 58. 3 cis(- 100°) or 3 cis 260° [7.4] 59. 3 cis 110° [7.4] 1 i 13 [7.5] 64. 64 - 64i 13 [7.5] + 2 2 4 4 4 4 65. 32,768i [7.5] 66. 3 cis 30°, 3 cis 150°, 3 cis 270° [7.5] 67. 1 8 cis 22.5°, 1 8 cis 112.5°, 1 8 cis 202.5°, 1 8 cis 292.5° [7.5] 68. 4 cis 30°, 4 cis 120°, 4 cis 210°, 4 cis 300° [7.5] 69. cis 12°, cis 84°, cis 156°, cis 228°, cis 300° [7.5] 60. 5 cis(- 59°) or 5 cis 301° [7.4]

61. 12 cis 285° [7.4]

62. 0 - 729i [7.5]

63. -

Chapter 7 Test, page 631 1. B = 103.3°, a L 75.9, b L 81.2 [7.1] 3. b L 34.7, A L 55.4°, C L 73.1° [7.2] 7. - 9.2i - 7.7j [7.3] 13. - 1 + 0i [7.5]

2. B = 72.9°, C L 48.3°, c L 37.1 or B = 107.1°, C L 14.1°, c L 12.1 [7.1] 4. B L 96.77° [7.2] 5. K L 39 square units [7.2] 6. K L 260 square units [7.2]

8. - 19i - 29j [7.3]

9. - 1 [7.3]

10. 103° [7.3]

11. 313 cis 145° [7.4]

15. L - 15.556 - 1.000i [7.5]

14. 2.5 cis(- 95°) or 2.5 cis 265° [7.4]

12.

5 12 512 i [7.4] 2 2

313 3 3 313 + i, + i, 0 - 3i [7.5] 17. L 27 miles [7.3] 18. L 169 miles per hour at a heading of 82.2° [7.3] 2 2 2 2 19. cis 9°, cis 81°, cis 153°, cis 225°, cis 297° [7.5] 20. $66,000 [7.2] 16.

Cumulative Review Exercises, page 632 1. ( f ⴰ g)(x) = cos(x2 + 1) [2.6] 5. L 16.7 centimeters [5.2]

6.

2. f

-1

(x) =

1 x - 4 [4.1] 2 [5.5]

y 4

2 −4 −2 −2 −4

7. 8.

2

4 x

9. 13.

3 3 4 , cos u = , tan u = [5.2] 5 5 4 p Amplitude: 4; period: p; phase shift: [5.7] 4 7p p 12 sin ax + b or 12 sin ax - b [6.4] 4 4 12 Odd [5.4] 11. [6.5] 12. sin( - x) or - sin x [6.2] 5 11p p 7p 11p p [6.6] 14. 0, , p, [6.6] , , 2 6 6 6 6 3. 270° [5.1]

4. sin u =


15. Magnitude: 5; angle: 126.9° [7.3] 18. 26° [7.1]

19. 16 [7.5]

16. 176.8° [7.3]

17. About 439 miles per hour at a heading of 54.6° [7.2]

12 12 12 12 + i and i [7.5] 2 2 2 2

20.

Exercise Set 8.1, page 641 1. a. iii

b. i

c. iv

3. Vertex: (0, 0)

d. ii 7. Vertex: (2, -3)

5. Vertex: (0, 0)

Focus: (0, -1)

1 Focus: a , 0b 12

Directrix: y = 1

Directrix: x = -

1 12

2

Focus: (1, - 4)

7 Focus: a- , 1b 2

Directrix: y = - 5

Directrix: x = 3

Directrix: x = -

2

x

4

y 2

y

2

−4

x

−2

11. Vertex: ( - 4, 1)

Focus: (2, - 1)

y

y

9. Vertex: (2, - 4)

y

2

x

2

−4

x

2

−4

−4

17. Vertex: a- ,

15. Vertex: (-4, -10)

13. Vertex: (2, 2)

5 Focus: a 2, b 2 Directrix: y =

Focus: a - 4, -

39 b 4 41 Directrix: y = 4

3 2

y

7 3 b 4 2 3 Focus: a - 2, b 2 Directrix: x = -

y 8 4

23. Vertex: a2, -

5 b 4 3 Focus: a 2, - b 4 Directrix: y = -

y

y

3

2

2

x

3 13 b 2 12 3 1 Focus: a- , b 2 3 11 Directrix: y = 6

21. Vertex: a- ,

y 2

2

x

4

x

x

−4

x

2

9 Focus: a- , -3b 2 11 Directrix: x = 2

x

−4

2

19. Vertex: (- 5, -3)

3 2

9 2

25. Vertex: a , - 1b

7 4

9 1 27. Vertex: a1, b 29. x2 = - 16y 31. (x + 1)2 = 4(y - 2) 2 9 35 31 Focus: a , -1 b Focus: a1, b 8 36 23 37 Directrix: x = Directrix: y = 8 36 y

y 4

y 8

4 x

−2 4

2

x −2

33. (x - 3)2 = 4( y + 4) 35. (x + 4)2 = 4( y - 1) 41. 6.0 inches 49. 4

51. 4 ƒ p ƒ

43. a. 5900 square feet 53.

55.

−4

(

4

2

(

3 4

,

47. Vertex: (250, 20); focus: a

57. x2 + y2 - 8x - 8y - 2xy = 0

y

2

(

39. On axis of symmetry 4 feet above vertex

45. a = 1.5 inches

3 4

F

x

1 feet = 4 feet 2 inches 6

b. 56,800 square feet

y 2

37. 4

2

2

x

3240 , 20b 13

A49

A50


Prepare for This Section (8.2), page 645 PS2. - 8, 2

PS1. Midpoint: (2, 3); length: 2113 PS6.

PS3. 1 13

PS4. x2 - 8x + 16 = (x - 4)2

PS5. y = 24 - (x - 22

y

−2

6x

4

2

−2 −4 −6

Exercise Set 8.2, page 654 1. a. iv

b. i

c. ii

d. iii

3. Vertices: (0, 5), (0, - 5)

5. Vertices: (3, 0), (-3, 0)

Center: (0, 0)

Center: (0, 0)

Foci: (0, 3), (0, -3)

Foci: (15, 0), (- 15, 0)

y 6

4

Center: (0, 0)

Center: (0, 0) Foci: a0,

Foci: (12, 0), (- 12, - 0)

155 155 b , a0, b 2 2

y

y

y

3

4

4

x

x −4

−3

11. Vertices: (8, -2), (-2, -2)

9. Vertices: (0, 4), (0, - 4)

7. Vertices: (3, 0), (- 3, 0)

4

x

−4

−4

4 −4

15. Vertices: (1 + 121, 3), (1 - 121, 3)

13. Vertices: (- 2, 5), (- 2, -5)

Center: (3, -2)

Center: (-2, 0)

Center: (1, 3)

Foci: (6, -2), (0, -2)

Foci: ( -2, 4), (-2, - 4)

Foci: (1 + 117, 3), (1 - 117, 3)

y

y

y

2

6 4 2

2 2 4 6

x

−6

−2 −2

2 4

x −2

−8

17. Vertices: (1, 2), (1, -4)

Center: (1, -1) Foci: a1, - 1 +

165 165 b , a1, -1 b 3 3

y

19. Vertices: (2, 0), (-2, 0)

x

Foci: (1, 0), (-1, 0)

Foci: (0, 3), (0, -3) y 6

1 1

−4

21. Vertices: (0, 5), (0, - 5)

Center: (0, 0)

y

4

x

Center: (0, 0)

2 −2

2 4

x

4x

x


23. Vertices: (0, 4), (0, -4)

27. Vertices: ( - 1, - 3), (5, -3)

25. Vertices: (3, 6), (3, 2)

Center: (0, 0) 139 139 b, (0, b Foci: a0, 2 2 y

Center: (2, -3)

Center: (3, 4) Foci: (3, 4 + 13), (3, 4 - 13)

Foci: (0, - 3), (4, - 3)

y

2

y 2

4 x

1

2 2

x

6

−2

4 x

Center: (2, 0)

Center: (-1, 1)

1 11 , -1b, a , -1b 2 2 Center: (3, -1)

Foci: (2, 17), (2, - 17)

Foci: ( -1, 4), ( - 1, - 2)

Foci: a3 +

29. Vertices: (2, 4), (2, -4)

31. Vertices: ( -1, 6), (- 1, -4)

33. Vertices: a

y

y

117 117 , - 1b, a3 , - 1b 2 2

y

6

4

6

x

6

x

4 x

−2

35.

(x - 2)2 y2 y2 y2 (x + 2)2 (y - 4)2 (y - 4)2 x2 x2 x2 + = 1 37. + = 1 39. + + = 1 43. = 1 = 1 41. + 25 9 36 16 36 16 7 25 81>8 25>24

45.

(x - 5)2 (y - 1)2 y2 y2 (x - 1)2 ( y - 3)2 y2 x2 x2 x2 + = 1 47. + = 1 49. + = 1 51. + = 1 53. + = 1 16 25 25 21 20 36 25 21 80 144

55. On the major axis of the ellipse, 41 centimeters from the emitter

ax 61.

67. y =

9115 2 b y2 2 = 1 + 324 81>4

63. 24 feet

- 36 21296 - 36(16x2 - 108) 18

69. y =

884.74

2

9.4

2

+

y2 883.352

= 1 59. 40 feet

- 18 2324 - 36(4x2 + 24x + 44) 18

−4

− 6.2

73.

x2

65. a. 17 feet to the right and left of O. b. 8 feet

6.2

−9.4

57.

y2 (x - 1)2 (y - 2)2 x2 9 + = 1 75. + = 1 77. 36 27 16 12 2

0.7

−2

71. p115 square units

A51

A52


Prepare for This Section (8.3), page 658 PS1. Midpoint: (1, - 1); length: 2113 PS6.

PS2. -4, 2

PS3. 12

PS4. 4(x2 + 6x + 9) = 4(x + 3)2

PS5. y =

3 2x2 - 4 2

y

−2

2

−2

4

6x

−4 −6

Exercise Set 8.3, page 666 1. a. iii

b. ii

c. i

d. iv

3. Center: (0, 0)

5. Center: (0, 0)

7. Center: (0, 0)

9. Center: (0, 0)

Vertices: (4, 0)

Vertices: (0, 2)

Vertices: ( 17, 0)

3 Vertices: a , 0b 2

Foci: A 241, 0 B

Foci: A 0, 229 B

Foci: (4, 0)

Foci: a

5 Asymptotes: y = x 4


Asymptotes: y =

y

317 x 7


y 4

y

4

173 , 0b 2

y 4

2 x

2

6

x

4 x

−4

−4

−4

11. Center: (3, -4)

4 x

−4

13. Center: (1, -2)

15. Center: ( - 2, 0)

Vertices: (7, - 4), ( - 1, - 4)

Vertices: (1, 0), (1, -4)

Vertices: (1, 0), (- 5, 0)

Foci: (8, -4), ( - 2, -4)

Foci: (1, -2 2 15)

3 Asymptotes: y + 4 = (x - 3) 4

1 Asymptotes: y + 2 = (x - 1) 2

Foci: (- 2 134, 0) 5 Asymptotes: y = (x + 2) 3

y

y

4

y

2 x

4

6

2

x

2 x

−4 −4

17. Center: (1, - 1)

19. Center: (0, 0)

7 1 Vertices: a , -1b , a- , -1 b 3 3 197 Foci: a1 , - 1b 3 9 Asymptotes: y + 1 = (x - 1) 4 4

−4

Vertices: (3, 0)

Vertices: (0, 3)

Foci: A322, 0 B

Foci: (0, 5)

Asymptotes: y = x


y 6

y

2 −4

4

x

21. Center: (0, 0)

y 6

6 x

23. Center: (0, 0)

2 Vertices: a0, b 3 25 Foci: a0, b 3 Asymptotes: y = 2x y

6 x

1

x


27. Center: (- 2, - 1)

25. Center: (3, 4)

29. y =

Vertices: (3, 6), (3, 2)

Vertices: ( -2, 2), (-2, - 4)

Asymptotes: y - 4 = (x - 3)

3 Asymptotes: y + 1 = (x + 2) 2

Foci: A 3, 4 222 B

-6 236 + 4(4x2 + 32x + 39) -2 14

Foci: A -2, - 1 213 B

−8.75

3

y 6

y

4

A53

−5

4 x

8 x

64 24096 + 64(9x2 - 36x + 116) -32

31. y =

33. y =

18 2324 + 36(4x2 + 8x - 6) - 18

6

35.

y2 y2 x2 x2 = 1 37. = 1 9 7 20 5

6

−3.75

−6.75

8

5

−8

−8

39.

(y - 2)2 y y (y - 3) (x - 4)2 (x - 7)2 (x - 4) (y + 2)2 x x = 1 43. = 1 45. = 1 = 1 41. = 1 47. 9 16 64 4 5 3 12 36>7 144>41 225>41

49.

(y - 7)2 (x - 1)2 =1 1 3

2

2

2

2

2

51.

y2 x2 =1 4 12

53.

2

(x - 4)2 36>7

-

(y - 1)2 (y - 1)2 (x - 4)2 = 1 and =1 4 4 36>7

2

55. a.

y x2 = 1 2162.25 13,462.75

61. Ellipse

b. 221 miles

63. Parabola

y

57. y2 - x2 = 10,0002, hyperbola

65. Parabola

67. Ellipse

y 10

y

10 10

2

73.

x

−6

2 −2 −6

b. D(4, 0.25), P(1, 12.0444) 2

6

x

6

x

−8

75. a. Parabolic: x2 = 64y; hyperbolic:

y

−2

x

x

(y - 7)2 5

2

-

x2

A2214 B 2

c. 12.6694 inches

x2 22

69.

y

4

4

59. a.

=1

-

y2 0.52

= 1 b. 6.25 inches

y2 x2 = 1 1 3

71.

y2 x2 = 1 9 7

A54


Prepare for This Section (8.4), page 670 PS1. cos a cos b - sin a sin b

PS2. sin a cos b + cos a sin b

p 6

PS3.

PS4. 150°

PS5. Hyperbola

y 4

PS6.

2 −2

2

4

x

−4

Exercise Set 8.4, page 676 1. 45°

3. 36.9°

5. 73.5°

7. 22.5°

9.

(x¿)2 (y¿)2 = 1 8 8

11.

(x¿)2 (y¿)2 + = 1 9 3

y

8

x'

y' 4

8

4 x

−4

17. (y¿)2 = 2(x¿ - 2)

y

2

21.

4 4

4

−4

4

x' 36.9°

6

x

−4

25.

6.5

−9.4

6.5

−9.4

9.4

9.4

4.7 0

27. y =

x

4 −4

23.

− 4.7

35. Parabola

−6.5

−6.5

2 - 16 2 + 16 x and y = x 2 2 37. Hyperbola

−8

−4

−4

x' 36.9°

4 4

−4

6

4

y'

4 4 x' 30° x 4

8

x' ≈18.4° x

y

y'

29. a

115 3115 3 115 115 , b and a,b 5 5 5 5

39. Ellipse or circle

31. Hyperbola

33. Parabola

43. 9x2 - 4xy + 6y2 = 100

Mid-Chapter 8 Quiz, page 678 (x - 5)2

2.

3. Vertex: ( -3, - 1) ; Focus: a -

17 19 , -1 b ; Directrix: x = [8.1] 6 6

12

+

(y + 3)2

1. (y - 2)2 = - 8(x - 6) [8.1]

(15)2

= 1

[8.2] 4. Center: (4, 3); Vertices: (1, 3), (9, 3); [8.2]

Foci: ( -4 + 121, 3), (-4 - 121, 3)

y

3 (−3, −1)

y (−4, 5) 7

(3, 1) 5

−5

(3, −3)

(1, 3)

(−9, 3) x

−4 (−4, 1)

= 1

1>3 y

19. 15(x¿)2 - 10(y¿)2 + 6x¿ + 28y¿ + 11 = 0

y

y'

(y¿)2

8

4

−8

-

1>2

4

4

−4

(x¿)2

y'

45° 4

15. y¿ = (x¿)2 + 4

y

y'

4

13.

x

4 −4 −8

x' ≈26.6° x


6. (x¿)2 + 9(y¿)2 = 9 or [8.4]

5. Center: (3, 1); vertices: (6, 1), (0, 1); [8.3]

(x¿)2

Foci: (3 - 113, 1), (3 + 113, 1) ; Asymptotes: y - 1 =

32

+

(y¿)2

= 1

12

2 2 (x - 3), y - 1 = - (x - 3) 3 3 y x′

y

y′

3

3 α = 60°

3

5 (6,1)

(0,1)

x

3

x

3 –5

Prepare for This Section (8.5), page 678 PS1. Odd

PS2. Even

PS3.

2p 5p , 3 3

PS4. 240°

PS5. r2

PS6. (4.2, 2.6)

Exercise Set 8.5, page 690 1–7.

9.

5π

−3, 3

11.

13.

15.

(2, 60°) π

−2, 4

4

6

4

6

(1, 315°)

0 … u … 2p 17.

19.

0 … u … p 21.

5

2

0 … u … p

7π π 2π 12 2 3π 3 4

23.

6

0 … u … 2p

0 … u … 2p

π

0 … u … 2p

25.

5π 12 π 3 π 4

5

4

0

0 … u … 2p

3π 2

p 3p … u … 4 4 4

27.

−3

9

−4

0 … u … 2p

4

29.

−6

4

31.

6

−6

6

33.

6

−9

9

−4

−4

−6

0 … u … p

0 … u … p

0 … u … p

A55

A56


4

35.

−6

41.

48

12

−72

10

−8

72

8

−4

−8

−48

−10

0 … u … 4p

0 … u … 6p

0 … u … 2p

0 … u … 2p

13 x 3

77. r = 2

85.

51. a 3 12,

69. y = 2x + 6

67. y2 + 4x - 4 = 0

79. r cos2u = 8 sin u

3

2p 5p b , a 2, b 3 3

5p p b , a- 312, b 4 4

57. x2 + y2 - 3x = 0 59. x = 3

55. (5, 53.1°), (-5, 233.1°)

65. x4 - y2 + x2y2 = 0

75. r = 3 sec u

49. a-2,

47. (- 4, -4 13)

45. ( -4, - 4)

53. A 7 12, 135° B , A - 712, 315° B

83.

39.

−12

6

3 3 13 b 43. a , 2 2

63. y =

8

37.

71. r = 2 csc u

61. x2 + y2 = 16 73. u =

p 3

81. r2(cos 2u) = 25 5

87.

2

2

89. −4

−4

4

−2

−4

−3

10

95.

−3

97.

−15

3

4

−2

−2

2

93.

−4

−3

0 … u … 2p

−5

5

-4p … u … 4p

4

b. −5

5

−4

−4

0 … u … 5p

0 … u … 20p


3 5

PS2. x = - 1

PS3. y =

2 1 - 2x

PS4.

3p 2

PS5. e 7 1

PS6.

30

3 −1

0 … u … p

20

−30

−10

4

99.

3

15

−2

101. a.

4

2

4 2 - cos x

−20

- 30 … u … 30


A57

Exercise Set 8.6, page 695 1. Hyperbola

3. Ellipse

5. Parabola

8

7. Hyperbola

9. Ellipse with holes at

a2,

4

4

p 3p b and a2, b 2 2

15 5

11. Ellipse with holes at (2, 0) and (2, p)

15. 3x2 - y2 + 16x + 16 = 0

13. Parabola

5

19. x2 - 6y - 9 = 0 31. r =

3 1 - 2 sin u

5

21. r =

2 1 - 2 cos u

23. r =

2 1 + sin u

6.2

33.

25. r =

8 3 - 2 sin u

4.4

9.4

−6.2

p radian. 6

counterclockwise p radians.

−3

Rotate the graph in Exercise 5 clockwise

4

43.

p radian. 6

45.

4

4

−5 −5

−6.2

6.2

−3.2

Rotate the graph in Exercise 3

4

41.

4 1 - cos u

9.4

− 6.2

6.2

29. r =

−9.4

Rotate the graph in Exercise 1 counterclockwise

6 2 + 3 cos u

37.

9.4 −9.4

− 14.4

27. r =

9.2

35.

−9.4

39.

17. 16x2 + 7y2 + 48y - 64 = 0

−4

5

5

−2

−4

0 … u … 12p

Rotate the graph in Exercise 7 clockwise p radians.

Prepare for This Section (8.7), page 696 PS1. y2 + 3y +

3 2 9 = ay + b 4 2

PS2. y = 4t2 + 4t + 1

PS3. Ellipse

PS4. 1

PS5. t = e y

PS6. Domain: ( - q , q);

range: 3 -3, 34

A58


Exercise Set 8.7, page 702 y

1.

y

3.

4

y

5.

4

4 x

4 x

11. x2 - y2 - 1 = 0

1

x Ú 1 yR

17. y = x2 - 1

19. y = ln x

y

x Ú 0

-1 … y … 1

x

yR

23. The point traces a line segment,

y

( y - 3)

(x - 2)

2

2

as shown in the figure. The point starts at (1, 1) and moves along the line segment until it reaches the point (5, 4) at time t 3.

4

+ = 1 , as shown in 32 22 (−1, 3) 2 the figure. The point starts at (5, 3) and moves counterclockwise along the ellipse −2 until it reaches the point (1, 3) at time t . 25. The point traces a portion of the top

y

y

2

2

(−1, −1)

(5, 3)

2

6 x

4

(−1, 2)

2

4

6 x

(0, 1) −1

1

x 2 π < t ≤ 3π/2

x

33. a. x = 6, y = 60t for t Ú 0

5

31.

−1

4.5π

700

37.

1500

(−1, 1) 2

C2 is a line. C1 is a ray.

(1, 2)

39.

5

−7

0

(5, 4) 4

−2

−1 500

y

27. C1 : y = - 2x + 5, x Ú 2; C2 : y = - 2x + 5, x R.

y

branch of the hyperbola y2 x2 1, as shown in the figure. The point starts at (1, 12) and moves along the hyperbola until it reaches p the point (1, 12) at time t = . 2

x 2 0 < t ≤ π/2

x

2

21. The point traces the top half of the ellipse

35.

x

1

2

4 x

29.

y

1

y

x 7 0

y Ú -1

4 x

-1 … x … 1

(2, 3)

2

4

4

15. x2/3 + y2/3 = 1

2

y Ú 3

x

1

y

4 x

y

x … 2

9.

4

4 x

13. y = - 2x + 7

y

y

7.

4

0

0

Max height (nearest foot) of 462 feet is attained when t L 5.38 seconds. Range (nearest foot) of 1295 feet is attained when t L 10.75 seconds.

7

3500 0

−5

Max height (nearest foot) of 694 feet is attained when t L 6.59 seconds. Range (nearest foot) of 3084 feet is attained when t L 13.17 seconds.

b. The Hummer


47. x = (b - a) cos u + a cosa

45. x = a cos u + au sin u

5

41.

−5

b - a ub a

b - a ub a

y = (b - a) sin u - a sin a

y = a sin u - au cos u

5

A59

−5

Chapter 8 Review Exercises, page 711 1. Center: (0, 0)

Vertices: (- 1, - 1), (7, -1) Foci: (3 2 13, - 1)

Focus: (4, 0) Directrix: x = - 4

[8.3]

y

[8.1]

y

Vertices: (0, -3), ( -4, -3) Foci: (- 2 17, - 3) 13 Asymptotes: y + 3 = (x + 2) 2 y [8.3]

[8.2]

y

8

4

4. Center: (2, 3)

3. Center: (3, -1)

2. Vertex: (0, 0)

Vertices: (2, 0) Foci: (2 12, 0) Asymptotes: y = x

−4

2

x

4

x

4

−8 −4 −4

4

4

8 x

−3

x

−8

5. Vertex: ( -2, 1)

7. Center: ( -2, 1)

6. Vertex: (3, 1)

29 Focus: a - , 1 b 16 35 Directrix: x = 16

21 Focus: a , 1b 8 27 Directrix: x = 8

8. Center: (2, -1)

Vertices: (- 2, - 2), ( - 2, 4)

Vertices: (7, - 1), ( -3, -1)

Foci: (- 2, 1 15)

Foci: (8, -1), ( -4, - 1) Asymptotes: y + 1 =

[8.1]

y

[8.1]

y

4

[8.2]

y

8

111 (x - 2) 5

[8.3]

y 8

4

4

6 −8

x

−4

4

8 x

4

x

−8

2 b 3

10. Center: a - 2,

2 2 Vertices: a - 5, b, a7, b 3 3 2 Foci: a1 2 113, b 3 2 2 y - = (x - 1) 3 3 2 2 Asymptotes: y - = 6 (x - 1) 3 3 y [8.3]

x

−8

−8

9. Center: a1,

4

1 b 2 1 1 Vertices: a2, b, a - 6, b 2 2 1 Foci: a - 2 17, b 2

11. Vertex: a- , -1 b

7 2 7 Focus: a- , -3b 2

Focus: a 3,

Directrix: y = 1

[8.2]

y

12. Vertex: (3, 2)

17 b 4

Directrix: y = -

[8.1]

y

1 4

[8.1]

y

8

8

−4

2 4

x

−8

4 −8

x

−2

x −8

−4 −8

4

8 x

A60

13.

17. 20.


3 15 [8.2] 7 (x + 2)2 4 (x + 2)2 9

+

14.

113 [8.3] 2

(y - 2)2 5 (y + 1)2 5

(x - 2)2

15.

25

16

= 1 [8.2]

18. (y + 3)2 = - 8(x - 4) [8.1]

= 1 [8.2]

21.

y2 x2 = 1 [8.3] 36 4>9

31. Parabola [8.4] π 2

26. 6(x¿)2 +

16.

(x - 1)2

-

9

19. x2 =

(y - 1)2

3(y + 2) 2

22. y = (x - 1)2 [8.1]

= 1 [8.3]

7

or (y + 2)2 = 12x [8.1]

23. 40 inches [8.1]

24. 119.54 inches [8.2]

9 12 12 x¿ + y¿ - 12 = 0, parabola [8.4] 2 2

1 1 2 (x¿) - (y¿)2 - 12x¿ - 1 = 0, hyperbola [8.4] 29. Hyperbola [8.4] 30. Ellipse or 2 2 32. Ellipse or circle [8.4] 33. a. No b. No c. Yes [8.5] 34. a. No b. Yes c. No [8.5]

27. (x¿)2 - 4y¿ + 8 = 0, parabola [8.4]

35.

(y - 3)2

= 1 [8.3]

25. (x¿)2 + 2(y¿)2 - 4 = 0, ellipse [8.4]

circle [8.4]

+

[8.5]

28.

π 2

36.

[8.5]

[8.5]

37.

[8.5]

38.

r sin = 3

4 π

2

4

π

0

2

3π 2

3π 2

[8.5]

39.

[8.5]

40.

6

6

51. y2 = 8x + 16 [8.5]

[8.6]

6

52. x2 - 3x + y2 + 4y = 0 [8.5]

[8.6]

y 4

8

49. 3r cos u - 2r sin u = 6 [8.5]

−4

50. r2 sin 2u = 8 [8.5]

53. x4 + y4 + 2x2y2 - x2 + y2 = 0 [8.5]

[8.6]

57.

4 x

−4

[8.5]

46.

6

48. r + 4 cos u + 3 sin u = 0 [8.5]

56.

[8.5]

42.

4

[8.5]

45.

4

47. r sin2 u = 16 cos u [8.5]

55.

[8.5]

41.

6

[8.5]

44.

0

4

r cos = 2

4

[8.5]

43.

2

58.

54. y = (tan 1)x [8.5]

8 x

−8 1

2

3

[8.6]

y 8

−8


59. y =

3 5 x + [8.7] 4 2

60. y = 2x + 1, x … 1 [8.7]

61.

y2 x2 + = 1 [8.7] 16 9

y 4

y

4

y2 x2 = 1 [8.7] 1 16

62.

y 4

y

2 x

4

4 x

−4

2

x

4 x

−4

−4

63. y = - 2x, x 7 0 [8.7]

64. (x - 1)2 + (y - 2)2 = 1 [8.7]

65. y = 2-x , x Ú 0 [8.7] 2

y

y 4

y

2 4

x

4 x

−4

−4

2

−4

66. The point traces the right half of the ellipse given by

(x - 1)2 3

2

+

x

( y - 5)2 42

67. The point traces a portion of the parabola given by

= 1 x Ú 1

y = 2 + 1x 0 … x … 4

as shown in the following figure.

as shown in the following figure.

y

y

t=0 (1,9)

t=3 (4, 4) 4

t=π 2 (4, 5)

5

t = –1 (0, 2)

t=π (1, 1)

4 4

x

The point starts at (1, 9) and moves clockwise along the right half of the ellipse until it reaches the point (1, 1) at time t = p. [8.7] 68.

[8.7]

7

69.

The point starts at (0, 2) and moves along the parabola, as shown by the arrowheads, until it reaches the point (4, 4) at time t = 3. [8.7]

−1

[8.4]

15

−15

0

x

15

70.

−4.7

4.7

200 −15

[8.5]

3.1

− 3.1

A61

A62 71.


[8.7]

300

0

2000 0

Max. height (nearest foot) of 278 feet is attained when t L 4.17 seconds.

Chapter 8 Test, page 713 1. Vertex: (0, 0)

2.

Focus: (0, 2) Directrix: y = - 2 [8.1]

3. Vertices: (3, 4), (3, 6)

6. a. 15 feet

4.

Foci: (3, 3), (3, 5) [8.2]

3 −4

5. 1 inch [8.1]

[8.2]

y

( y + 3)2 x2 + = 1 [8.2] 45 9

4 x

−2

b. 6 feet [8.2]

7.

[8.3]

y

8. 30° [8.4]

9. Ellipse [8.4]

10. P(2, 300°) [8.5]

4 −8

x

4 −6

[8.5]

11.

2

4

15. y2 - 8x - 16 = 0 [8.5]

[8.5]

12.

6

2

16. x2 + 8y - 16 = 0 [8.6]

4

6

2

1 y [8.7] 2

17. (x + 3)2 =

5 5 13 b [8.5] 14. a , 2 2

[8.5]

13.

18.

3

(y - 2)2 x2 + = 1 [8.7] 16 1

y

y

6

6

3 −6

19.

[8.7]

5

−1

24π

−1

Xscl = 2p

20. 443 feet [8.7]

−3

3

6 x

−6

−3

3

−3

−3

−6

−6

6 x


A63

Cumulative Review Exercises, page 714 1. 2, i 12 [1.4]

-x + 7 [P.5] (x - 1)(x + 2)

2.

7. x = - 3, y = 2 [3.5]

8. 129 [2.1]

3. - 4 - h [2.6]

9.

4. - 19 [2.6]

[4.2]

y

10. 1 [4.5]

5 3 6. y = - x - [2.3] 2 2

5. 6 [3.4] 11.

4

2 − 8 −4 −2

12. f -1(x) = 17. Period:

1 x + 4 [4.1] 2

2

13. - 4, -2i [3.4]

p ; amplitude: 3 [5.5] 2

19.

[2.5]

y 8

x

4

x

−8

14. Odd [2.5]

p p 5p 3p , , , [6.6] 6 2 6 2

8

4

−4

15.

2p [5.1] 3

16. 13 centimeters [5.2]

20. 8 [7.3]

Exercise Set 9.1, page 726 1. (2, -4)

3. a-

19. ac, -

4 c + 2b 3

35. (213, 3)

6 27 , b 5 5

5. (3, 4)

21. (2, -4)

37. a

38 3 , b 17p 17

7. (1, - 1)

23. (0, 3) 39.

9. (3, - 4)

25. a c, c b

3 5

( 12, 13 )

41. $125

45. Boat: 25 miles per hour; current: 5 miles per hour 49. 40% gold: 8 grams; 60% gold: 12 grams 57. 90 people

51.

11. (2, 5) 27. a-

1 2 , b 2 3

13. ( -1, - 1)

15. a

29. No solution

62 34 , b 25 25

31. ( - 6, 3)

17. No solution 33. a 2, -

3 b 2

43. Plane: 120 miles per hour; wind: 30 miles per hour

47. $12 per kilogram for iron; $16 per kilogram for lead

9 square units 5

53. 8

55. 42, 56, 70; 42, 40, 58; 42, 144, 150; 42, 440, 442

59. Supply pump: 20,000 gallons per hour; each outlet pump: 2500 gallons per hour

Prepare for This Section (9.2), page 728 PS1. y =

2 x - 3 5

PS2. z = - c + 13

PS3. a

18 , 4b 5

PS4. (2, - 5)

PS5. ( -2, - 10)

PS6. (c, -4c + 9)

Exercise Set 9.2, page 738 1. (2, -1, 3) 15. (3, - 1, 0) 25. (0, 0, 0)

3. (2, 0, -3)

17. No solution 27. a

5 4 c, c, cb 14 7

37. Center ( -7, -2), radius 13

BD: 262 to 312 cars per hour 47. A = -

13 2

5. (2, -3, 1)

7. (- 5, 1, - 1)

9. (3, - 5, 0)

1 1 19. a (50 - 11c), (11c - 18), cb 11 11 29. (- 11c, -6c, c)

31. (0, 0, 0)

11. (0, 2, 3)

21. No solution

33. y = 2x 2 - x - 3

13. (5c - 25, 48 - 9c, c) 23. a

1 1 (25 + 4c), (55 - 26c), c b 29 29

35. x 2 + y 2 - 4x + 2y - 20 = 0

39. 90 to 190 cars per hour 41. CA: 258 to 308 cars per hour; DC: 209 to 259 cars per hour; 43. 7 inches from the 9-ounce chime and 6 inches from the 2-ounce chime 45. 3x - 5y - 2z = - 2

49. A Z - 3, A Z 1

A64


Prepare for This Section (9.3), page 740 PS1. - 1 13

PS2. (1, -3)

PS3. Parabola

PS4. Hyperbola

PS5. Two

PS6. Four


1. (1, 0), (2, 2) 11. a

2 + 12 - 6 + 12 2 - 12 - 6 - 12 , b, a , b 2 2 2 2

19 11 ,b, (1, 1) 29 29

13. ( -2, 9), (1, -3), ( -1, 1)

12 1 26 3 , b, (2, 1) 23. a , - b, (1, - 2) 5 5 5 5 -3 - 13 1 - 13 19 22 a , b 29. a , b, (1, 4) 31. No solution 2 2 13 13

large carpet: 28 by 28 feet 45. (0, 1), (1, 2)

25. a

49. (1.7549, 1.3247)

9. a-

3 , - 4b, (2, 3) 2

17. (4, 2), (-4, 2), (4, - 2), ( - 4, -2)

39 7 -3 + 13 1 - 13 , - b, (3, 2) 27. a , b, 10 10 2 2

33. Width: 5 inches; height: 7.5 inches

37. Large radius: 16.0 inches; small radius: 8.0 inches

47. (0.7035, 0.4949)

7. (4, 6), (6, 4)

15. (- 2, 1), (- 2, - 1), (2, 1), (2, -1)

21. a

19. No solution

5. (5, 18)

41. r Ú

39. 82 units

53. (- 1, 1), (1, - 1)

51. (1, 5)

35. Small carpet: 9 by 9 feet;

1

A5

or

15 5

43. $45

55. (1, - 2), ( - 1, 2)

Mid-Chapter 9 Quiz, page 747 1. (3, 7) [9.1] 2. Ordered pairs of the form (c, 2c + 3) [9.1] 5. (3, 1), (3, -1), (- 3, 1), (-3, -1) [9.3]

3. Answers will vary. [9.1]

4. y = 2x 2 - 3x + 1 [9.2]

Prepare for This Section (9.4), page 748 PS1. (x 2 + 7)2

PS2.

6x + 9 (x - 1)(x + 2)

PS3.

x 2 + 2x + 7

PS4. ( -1, 2)

x(x - 1)

2

PS6. x + 3 +

PS5. (2, - 2, -1)

2x - 35 x 2 - 7x

Exercise Set 9.4, page 754 1. A = - 3, B = 4

3. A = -

7 -4 + x - 9 x + 2

11.

3 5 + x x + 4

21.

1 2 -28 + + x x + 7 (x + 7)2

13.

2 1 ,B = 5 5

23.

5. A = 1, B = - 1, C = 4 15.

5 3 + 2x + 3 2x + 5

2 3x - 1 + 2 x x - 3x + 1

25.

17.

7. A = 1, B = 3, C = 2

20 -3 + 11(3x + 5) 11(x - 2)

2 -1 4 + + 2 x + 3 (x + 3)2 x + 1

27.

9. A = 1, B = 0, C = 1, D = 0 19. x + 3 +

1 -1 + x - 2 x + 2

3 5 + x - 4 (x - 4)2

3 4x 1 3x - 1 1 1 -2 + 2 31. 33. x + 35. 2x - 2 + 2 + + 2k(k - x) 2k(k + x) x x - 1 x 2 + 10 (x + 10)2 x - x - 1 1 4 1 4 2 2x + 7 3 -2 37. 39. 41. + + 2 + 4 + + 5(x + 2) 5(x - 3) x x - 2 3(x - 1) x x 3(x 2 + x + 1) 29.


PS2.

y 8

4 −8 −4 −4 −8

PS3.

y 8

4 4

8 x

−8 −4 −4 −8

PS4.

y 8

4

4 4

8 x

−8 − 4 −4 −8

PS5.

y 8

4

8 x

−8 −4 −4 −8

y 8

4 4

8 x

−8 −4 −4 −8

4

8 x


PS6.

y 8

4 −8 −4 −4

8 x

4

−8


3.

y

5.

y

7.

y

9.

y

y

4

3

4

2

3

−2

x

2

x

2

−2

x

3

x x

2

−4

11.

13.

y

15.

y

17.

y

4 4 x −4

21.

2

6 x

−2

19.

y

4

4

x

4

y

x

23.

y

x

4

−4

25.

y

27. No solution

y

29.

y

2 2

33.

y 4

35.

y

4

4 x

x

4

x

2

31.

2

2

37.

y

39.

4

y

8

4 x

2

4 x

x

4

x

−2

4 x

−10

41.

43.

y

4

45. 101 to 138 beats per minute

y

47.

10 −3 x

2

51.

10

53.

y

2

2

x

49.

y

y

3

4

3

x

x

y

y

4

4

4 xy > 1

x

If x is a negative number, then the inequality is reversed when both sides of the inequality are divided by a negative number. x

4 1

y> x

4 x

A65

A66



PS2.

y 8

PS3. 20, 18, 22, 27

y 8

PS4. 95, 111, 115, 105

PS5. (1, 3)

PS6. (1, 6)

4

4 −8 −4 −4

8 x

4

−8 −4 −4

8 x

4

−8

−8

Exercise Set 9.6, page 768 1. Minimum at (2, 3): 18 3. Maximum at (5, 19): 74.5 5. Minimum at (0, 8): 16 7. Maximum at (6, 5): 71 9. Maximum at (0, 12): 72 11. Minimum at (0, 32): 32 13. Maximum at (0, 8): 56 15. Minimum at (2, 6): 18 17. Maximum at (3, 4): 25 19. Minimum at (2, 3): 12 21. Maximum at (100, 400): 3400 23. 32 cups of Oat Flakes and 6 cups of Crunchy O’s; the minimum cost is $14.08. 25. 20 acres of wheat and 40 acres of barley 27. 60 economy boards and 48 superior boards produce the maximum weekly profit of $3576. 29. 24 ounces of food group B and 0 ounces of food group A yield a minimum cost of $2.40. 31. Two 4-cylinder

engines and seven 6-cylinder engines yield a maximum profit of $2050.

Chapter 9 Review Exercises, page 775 1. a -

18 15 ,b [9.1] 7 28

2. a , - 3b [9.1]

3 2

7. a (5 - 3c), c b [9.1]

1 2

1 1 (7c - 3), (16c - 43), cb [9.2] 11 11

15. a

14 2 c, c, cb [9.2] 11 11 20. a-

24. (0, -3), (2, 1) [9.3]

a

12. a

1 , 0b , (1, 3) [9.3] 2

31.

3 4 [9.4] + x - 2 x + 1

35.

2 4 5 [9.4] + + x x - 1 x + 1

32.

17. a (c + 1),

1 2

1 2 [9.4] + x - 1 (x - 1)2 x + 2

14. a 1, -

[9.4]

1 18 b, (1, 2) [9.3] 5 5

3 2

32 26 , b [9.3] 27. (2, 1), (-2, -1) [9.3] 28. (1, 3), (- 1, - 3), 25 25 115 115 115 115 30. a , b, a,b, (1, -1), ( -1, 1) [9.3] 15 15 15 15

26. (0, 2), a-

33.

6x - 2 5(x + 1) 2

+

7 -6 8 3 [9.4] 34. + [9.4] + 2 5(x + 2) 3(x - 2) 3(x + 1) (x - 2)

y

37.

[9.5] 38.

[9.5]

y

4 x x

−7

−4

−4

39.

[9.5]

y

40.

[9.5]

y

4

−2

2

x

−6

2 1 , b [9.2] 3 4

23. a ,

22. a , 3b , (1, - 1) [9.3]


x2 + 1

16 10 29 , , - b [9.2] 3 27 45

65 - 11c 19 - c 1 (3c - 1), c b [9.2] 18. a , , c b [9.2] 4 16 8

29. (2, -3), ( -2, 3) [9.3]

36. 1 +

6. ( -1, 1) [9.1]

5. (3, 1) [9.1]

74 1 3 1 ,, b [9.2] 13. a2, (3c + 2), c b [9.2] 31 31 31 2

18 64 , - b [9.3] 17 17

16 16 16 16 ,b , a, b [9.3] 6 3 6 3

10. a

1 2

16. (0, 0, 0) [9.2]

25. (2, 0), a

4. ( -4, 7) [9.1]

9. a , 3, - 1b [9.2]


11. a

19. (2, - 3) [9.3]

3. (- 3, - 1) [9.1]

41.

[9.5] 42.

y

[9.5]

y 4

x

2

2 x

2

x


43.

[9.5]

y

44.

[9.5]

y

[9.5] 46.

y

2

4

−4

[9.5]

y

48.

[9.5]

y

x

2

x

4

[9.5]

y

2

3

10 x

47.

45.

A67

49.

x

3

[9.5] 50.

y

[9.5]

y

4

4

4 x 4 x

51.

[9.5] 52.

y 12

−4

[9.5] 53.

y

8 x

55. y

−4 −4

[9.5]

56. y

[9.5]

4

8 12 x

y 4

57.

[9.5] 58.

3

[9.5]

y

60.

−2

[9.5] 61. The maximum is 18 at (4, 5). [9.6] 63. The minimum is 8 at (0, 8). [9.6]

x

4

x

62. The maximum is 44 at (6, 4). [9.6] 64. The minimum is 20 at (10, 0). [9.6]

65. The minimum is 27 at (2, 5). [9.6]

x

2

[9.5]

x

6 y 4

59.

y

4 x

x

6

x

8

6

2

[9.5]

y

5

4

5

4

[9.5] 54.

8

x

4

2

y 12

4

8

−8 −4 −4

x

4 x

66. 0 starter sets and 18 professional sets [9.6]

67. y =

11 2 2 21 10 5 47 x - x + [9.2] 68. x 2 + y 2 x y + = 0 [9.2] 6 2 3 11 11 11

69. z = - 2x + 3y + 3 [9.2]

70. 15 liters [9.1] 71. Wind: 28 miles per hour; plane: 143 miles per hour [9.1] 3 quarters; 1 nickel, 7 dimes, 2 quarters [9.2] 73. (0, 0, 0), (1, 1, 1), (1, - 1, - 1), (- 1, - 1, 1), (- 1, 1, -1) [9.2]

72. 4 nickels, 3 dimes,

Chapter 9 Test, page 777 1. ( -3, 2) [9.1] 5. a

2. a (6 + c), c b [9.1]

1 2

3. a

173 29 4 , , - b [9.2] 39 39 3

1 1 9 1 (c + 10), (5c + 11), c b [9.2] 6. a c, c, cb [9.2] 13 13 14 14

9.

[9.5] 10.

y 4

[9.5] 11.

y 4

4. a (c + 3) ,

1 4

7. (2, 5), (-2, 1) [9.3]

−4

4 x

−4 −4

−4

x

−2

8. ( -2, 3), (-1, - 1) [9.3]

[9.5] 12. No graph; the solution set is the empty set. [9.5]

y 4

4 4 x

−4

1 (7c + 1), c b [9.2] 8

A68


13. y

[9.5]

14. y

[9.5] 15.

7 8 + [9.4] 5(x - 4) 5(x + 1)

16.

-x + 2 1 + 2 [9.4] x x + 1

4 2 x

4

x

2

17. Length: 120 meters; width: 100 meters [9.1]

$1.75 [9.1]

18. Fee for first hour: $4.00; fee for each additional half-hour or portion of the half-hour: 400 680 19. acres of oats and acres of barley [9.6] 20. x2 + y2 - 2y - 24 = 0 [9.2] 7 7

Cumulative Review Exercises, page 777 1. {y ƒ y … - 3} [2.4]

2. log6[8x3(x - 5)] [4.4]

3. (x - 4)2 = - 25(y - 2) [8.1]

4. even [2.5] 5.

300 [4.5] 199

(x - 6)2 (y - 2)2 = 1 [8.3] 7. 5 [2.2] 8. 30 [2.6] 9. x 2 + 0.4x - 0.8 [2.7] 10. x 3 + 2x 2 + 9x + 18 [3.4] 16 128 r - 2 13 y 11. Q-1(r) = [4.1] 12. y = 2x + 1 [3.5] 13. 81 [4.2] 14. [4.2] 15. [5.4] 16. B = 44° [7.2] r 2 8 6.

4 −8 −4 −4

8 x

4

−8

17.

16 - 12 [6.2] 18. 109.7° [7.3] 19. 0 [6.5] 20. ( - 512, - 512) [8.5] 4

Exercise Set 10.1, page 788 2 1. C 3 1

-3 1 1 2 - 2 3 † 0 S, C 3 0 5 4 1

-3 1 1 -2 3 S , C 0 S 0 5 4

-3 2 -1 -3

2 0 3. D 1 3

-4 1 2 -2

1 2 0 2 ∞ T, 0 4 0 1

2 -3 0 2 D 1 -1 3 -3

-4 1 2 -2

1 2 0 2 T, D T 0 4 0 1

Answers to Exercises 5–12 are not unique.

-1 1 0

2 -1 1

2 -6 S - 27 2

m

m

m

m

R1 4 R2 1 - 2R1 + R2 The following answers list the elementary row operations used to produce the row echelon forms shown. 5. - 3R1 + R3 C 0 - 5R2 + R3 0 - 12 R3 R1 4 R3 1 -2 -1 3 R1 4 R3 - 3R1 + R2 - 2R1 + R2 1 -1 1 2 1 -3 4 2 1 - 4R1 + R2 - 4R1 + R3 11 3 12 R1 + R3 7. 1 D0 1 2 - 2 T 9. - 2R1 + R3 C 0 1 - 5 - 5 S 11. 1 C0 1 -1 -2 -1 S 2 R2 3 R2 1 0 0 0 11 0 0 0 1 3 13 R - 3R2 + R3 5 2 R2 + R3 0 0 1 - 6 1 5R + R 2 3 - 3 R3

13. (1, 3)

15. (2, - 3)

17. (2, -1, 1)

27. (7c + 6, -11c - 8, c) 39. a

19. (1, -2, -1)

29. (c + 2, c, c)

27 5 c + 39, c + 10, -4c - 10, cb 2 2

47. p(x) = x 3 - 2x 2 - x + 2

21. a 2 - 2c, 2c +

31. No solution

41. ac1 -

49. p(x) = 2x + 5

1 , cb 2

33. (2, -2, 3, 4)

23. No solution

35. a

12 6 9 1 c2 + , c1 - c2 + , c1 , c2b 7 7 7 7 51. z = 2x + 3y - 2

77c + 151 -25c - 50 14c + 34 , , , -3c - 7, cb 57. a 3 3 3

25. (16c, 6c, c)

21 8 2 5 ,- , ,- b 10 5 5 2 43. p(x) = 2x - 3

37. a 3, -

45. p(x) = x 2 - 2x + 3

53. x 2 + y 2 + 2x - 4y - 20 = 0

59. All real values of a except a = 1 and a = - 6

3 , 1, -1b 2

55. (1, 0, - 2, 1, 2)

61. a = - 6


A69

Prepare for This Section (10.2), page 791 PS2. - c

PS1. 0

PS3. 1

PS4.

-3 7 -7

1

1 c

PS5. 3 * 1

PS6. C 0

0

4 -5 S 13

Exercise Set 10.2, page 807 1. a. B d. B

1 5

2 R 4

b. B

2 -4 0 6 -2 S 6 -2 4

c. C4

-2 -1 -2

9 15. C 0

4 25. D-

1 3 1 3 1 3

5. a. C 3

2 -7 -17

- 53 -

-2 -3 -4 -1

27. C 3

7

2x1 - x2 4x1 + x2 + 2x3 37. μ 6x1 + x3 5x1 + 2x2 - x3

-2 4

5 -5 S -4

12 - 11 9

d. C -6

-6 4 2 S, C 2 -4 4 4 3T 4 3

c. B

1

-5 0 R - 15 5

9 -4

-4 R 2

3 1

6 4S 3 1 2S -2

6 R 2

d. B

-7 b. C 1

-4

-2 7S 0

9. B

17. 30, 84

29. B

1 1

-11 R 3

7 0

3 -1 S 4

-10 6

3. a. B

8 c. C 2

6

17 0 R, B -8 1

2 -4 S -8 22 R - 18

-3 3

0 5

d. C

5 R -5

- 18 1 - 11 10 14

7 31. C 1

5

-1 1 2 0S -1 4

33. e

5 0S 12

21. B

3 -1

-2 -5

1 R 1

-1

1 2 2

7. a. C 2

-1

6 14 R, B -7 0

11. B

19. The product is not possible.

-3 R -2

b. B

0 0

-1 R -11 0 R 0

c. B

-1 1S 5

-6 4

2 10 -3

b. C -2

-7

4 R -6 5 -4 4

-1 3S 1

0 -4 5 5 - 13 0 3 S, B R 5 -4 -3 -2 1

13. C 6

23. The product is not possible.

x - 3y - 2z = 6 3x - 8y = 11 35. c 3x + 3y - 2z = 2 4x + 3y = 11 2x - 4y + 5z = 1

+ -

2x4 = 5 3x4 = 6 39. a. 3 * 4. Three different fish were caught in four different samples. b. Fish A was caught in 2x4 = 10 4x4 = 8 1.96 1.37 2.94 1.37 26 8 sample 4. c. Fish B 41. C 0.78 1.08 1.96 0.88 S 43. a. C 21 13 S ; the matrix represents the total number of wins and losses for 3.53 1.18 4.41 1.47 18 16 2 -2 each team for the season. b. C 7 -7 S ; the matrix represents the difference between performance at home and performance away. 2 -2 50 150 140 65 88.5 15 170 370 72 S; T2 45. A - B = D T ; A - B represents the number of each item sold during the week. 47. C 54 85 250 130 68.5 94.5 80 115 25 49. P¿(2, - 5), Q¿(-3, -6)

51. P¿(1, - 3), Q¿(3, - 5)

0 1 57. a. D 1 0

1 0 1 1

1 1 0 1

2 0 1 1 T b. D 1 1 0 2

0 1 61. a. E 1 0 0

1 0 1 1 1

1 1 0 1 1

0 1 1 0 0

53. A¿(4, 2), B¿( -3, 4), C¿(3, 6)

1 3 2 1

1 2 3 1

2 0 1 1 T ; 2 walks 59. a. D 1 1 2 0

0 2 1 7 1 U b. E 7 0 2 0 2

7 6 7 7 7

7 7 6 7 7

2 7 7 2 2

1 0 1 1

1 1 0 0

2 0 7 0 7 U ; 2 walks 63. a. E 0 2 1 2 1

0 2 1 4 T b. D 0 3 1 0 0 0 2 0 0

0 2 0 1 1

1 0 1 0 0

55. A¿( -1, 10), B¿(- 1, 6), C¿( -6, 6), D¿( -6, 10)

4 2 4 3

3 4 2 1

1 3 T ; 3 walks 1 0

1 8 0 0 1 U b. E 16 0 0 0 0

0 24 0 16 16

16 0 40 0 0

0 16 0 12 12

0 16 0 U ; 12 walks 12 12

A70


0 1 65. a. D 0 1

1 0 2 1

24 -7 73. E 32 19 29

21 -8 10 -15 9

0 2 0 0

1 2 7 1 T b. D 0 2 0 3 -12 32 3 21 -32 1 -17 30 -28 13

79. a. y = - x - 2

7 2 12 7

0 20 5U 20 -6

2 12 0 2

3 7 T 2 2

67. a. 45.6%

46 82 75. E 73 212 68

b. y = - 3x - 1

c. y =

- 100 -93 -10 -189 -22

b. 46.7%

69. a. 22.9%

36 273 93 19 27 97 - 23 109 83 U 52 37 156 54 221 58

2x - 3 x - 2

-8 16 0 -4 -20

76 14 77. E 39 0 56

d. y = x 2 - 4x + 3

b. 27.9%

e. x = y 2

71. 11 months

-25 -10 -45 23 - 22

30 14 22 83 7

6 2 27 U -16 5

f. x = - y 2 + 2y - 3

Prepare for This Section (10.3), page 813 PS1. -

3 2

1 PS2. C 0

0

0 1 0

0 0S 1

1 PS3. See Section 7.1.

PS4. C 0

0

-2 3 -4

3 -2 S 11

Exercise Set 10.3, page 821 -5 1. B -2 19 2 7 4 11. E 7 -2 1 4

-3 R -1 -

-

1 2 1 4 1 2 1 4

5 3. B -1 -

3 2 1 4 1 2 1 4

-

3 2 3 4 1U 2 1 4

-2

1R 2

-16 5. C 7 -3

-2 1 0

7 -3 S 1

3 5 - 75 14 5 - 85

7 5 2 5 1 5 2 5

4 5 4 5 3U 5 1 5

2 13. E

4 -6 3

-

-

-1

15 11 7. C - 2 3

1 2

0

15. (2, 1)

PS6. e

PS5. X = A-1B

7 2 5 2

-2 1

2T

-1

0

1

-4 3 2S

9. D -

-1

17. a , -

7 4

25 b 8

2x + 3y = 9 4x - 5y = 7

-2

19. (1, -1, 2)

21. (23, - 12, 3)

23. (0, 4, -6, - 2) 25. x1 : 49.7°F, x2 : 53.7°F 27. x1 : 55°F, x2 : 57.5°F, x3 : 52.5°F, x4 : 55°F 29. On Saturday, 80 adults, 20 children; on Sunday, 95 adults, 25 children 31. Sample 1: 500 grams of additive 1, 200 grams of additive 2, 300 grams of additive 3; Sample 2:

400 grams of additive 1, 400 grams of additive 2, 200 grams of additive 3 -0.150

- 5.667 - 27.667 33. D - 19.333 -15

- 3.667 - 18.667 - 13.333 - 10

- 25 - 24 -17 -13

- 0.333 - 2.333 T - 1.667 -1

-0.217 0.302 -0.024 0.013 S 37. $194.67 million worth of manufacturing, $157.03 million worth of transportation, $121.82 million 0.217 -0.200 -0.195 worth of service 39. $39.69 million worth of coal, $14.30 million worth of iron, $32.30 million worth of steel

35. C 0.248

Mid-Chapter 10 Quiz, page 823 6 1. (-6, 40 ,14) [10.1]

2. p(x) = x 2 - x - 4 [10.1]

3. C 2

6 -4 6. C - 5

-5

-11 12 - 13 14 S [10.3] - 14 15

-2 3 4

-2 - 8 S [10.2] -5

6 4. C 6

4

-8 4 S [10.2] -4

10 5. C - 8

14

-10 26 S [10.2] 10


A71


PS2. 1

PS3. 4

PS4. 1

PS5. B

-6 -9

1

3 R - 15

-2 -3 S 9

3 5 -12

PS6. C 0

0

Exercise Set 10.4, page 831 3. -15

1. 13

5. 0

7. 0

9. 19, 19

11. 1, -1

15. -9, - 9

13. - 9, -9

17. 10

19. 53

21. 20

23. 46

25. 0

27. Row 2 consists of zeros, so the determinant is zero. 29. 2 was factored from row 2. 31. Row 1 was multiplied by -2 and added to row 2. 33. 2 was factored from column 1. 35. The matrix is in triangular form. The value of the determinant is the product of the elements on the main diagonal. 37. Row 1 and row 3 were interchanged, so the sign of the determinant was changed. 39. Each row of the determinant was multiplied by a. 41. 0 43. 0 45. 6 47. - 90 49. 21 51. 3 53. -38.933 55.

9 square units 2

57. 46

1 square units 2

61. 7x + 5y = - 1

Prepare for This Section (10.5), page 833 PS1. -11

PS3. B

PS2. 9

2 3

-7 R 5

PS5. -

PS4. 10

7 13

PS6. No


44 29 , b 31 31

15. a-

3. a , -

1 3

2 b 3

29 25 19 ,,- b 64 64 32

5. (2, - 7)

17. a

7. (0, 0)

50 62 4 , , b 53 53 53

11. a

9. (1.28125, 1.875) 21. x2 = -

19. (0, 0, 0)

35 19

13. a

21 3 29 ,, b 17 17 17

23. x1 = -

121 131

32 13 6 , , b 49 49 7

25. x4 =

4 3

27. The determinant of the coefficient matrix is zero, so Cramer’s Rule cannot be used. The system of equations has infinitely many solutions. 29. all real values of k except k = 0

31. all real values of k except k = 2

Chapter 10 Review Exercises, page 840 1. B

3 2

5 -7 3

3. C 1

-2

1 5 6 R , B R [10.1] 2. C 2 -7 -1 5

6 3 R, B -1 2 -3 -4 0

2 0 1

1 0 -2

4 3 -2 S, C 1 0 -2

2 0 1

-3 -4 0

-6 1 2

0 -4 -6

1 4 0 S , C - 2 S [10.1] -2 0

2 1 0 S, C 2 3 5

0 -4 -6

-6 2 1 S , C 0 S [10.1] 2 3

0

-4 2 0

2 0 -7

4. C 3

1

5 0 2 S, C 3 -3 1

-4 2 0

2 5 0 S , C 2 S [10.1] -7 -3

Answers to Exercises 5–12 are not unique. The following answers list the elementary row operations used to produce the row echelon forms shown. - 13 R1 6. -1 6R1 + R2 18 R2

C

8. 1-R1 + R2 4 R2

m

C

-1

0

1

7 12

1 2 2 C 0 1 23 0 0 0

m

R1 4 R2 - 3R1 + R2 4R1 + R3 11. - 13R2 1 2 R3 - 6R2 + R3

1

2 3

S [10.1]

R1 4 R3 10R1 + R2 5R1 + R3 9. - 1R 3 2 8R2 + R3 3R3

-2

2 3

0

1

- 23

-1 1 0

1 C0 0

1 - 43 1

S [10.1]

7. - 13R1 + R2

-1 7 3S

-1

1 3 R1

[10.1]

- 4 R2

B

1 0

1 1

- 43 R [10.1] 0

R1 4 R2 - 2R1 + R2 10. - R2 - 4R2 + R3

1 C0 0

2 1 0

-4 -2 1

9 4 S [10.1] -1

m

1 6 R1

1

m

8 R [10.1] -21

m

2 1

m

m

1 B 0

1 2 R1

4 10 3 S

-11

[10.1]

4R1 + R2 12. - 5R1 + R3 3 2 R2 + R3

1 12 C0 1 0 0

m

- R1 5. - 3R1 + R2

- 12 -5 0

1 - 4 S [10.1] 0

13. (2, -1) [10.1]

14. (1, -3) [10.1]

A72


15. (3, 0) [10.1]

16. a

40 42 , - b [10.1] 17. (3, 1, 0) [10.1] 18. (2, 1, 3) [10.1] 19. (1, 0, -2) [10.1] 20. (0, 2, 3) [10.1] 29 29

21. (3, -4, 1) [10.1] 22. (4, -2, -2) [10.1]




26. ( -2, 0, 0) [10.1]

27. a c +

5 6 4 9 1 8 7 , c + , cb [10.1] 28. a- , + c, cb [10.1] 29. (- c - 2, - c - 3, c) [10.1] 30. (5, -2, 0) [10.1] 31. (1, -2, 2, 3) [10.1] 7 7 7 7 5 5 2 32. (2, 3, -1, 4) [10.1] 33. (- 37c + 2, 16c, - 7c + 1, c) [10.1] 34. (63c + 2, - 14c + 1, 5c, c) [10.1] 35. y = x 2 + 3x - 2 [10.1] 36. y = x 2 - 2x + 3 [10.1] 37. B

41. B

-1 7

- 15 18 R [10.2] 42. B 1 -3

42 26 51. B

7 - 10

56. e

-x + 3y = 5 [10.2] 4x - y = 6

24 9 7 R [10.2] 52. B -22 1 - 10

0 1 59. A = E 0 1 0 61. B

27 -7

70. E

74. E

1 0 1 1 1

0 1 0 0 0

1 1 0 0 0

-1 1 R [10.3] - 32 1

66. C 4

- 12 -2 3

2x - y + 3z = 6

0 2 1 1 0 U ; A2 = E 1 0 1 0 1 62. B

1

-1

- 47

2 7

- 17

9 7

3 -2

5 1 S [10.3] -1

-1

2

20 7

-

14

- 33 2

-6

14

- 31 2

-7

-9

21 2 - 52

4 -1

19 7

-

58. c

2x + 3y + 7z = 6

- 51 7

- 50 7

0 4 9 -5 5 -1 13 -14 0 R [10.2] 38. C - 8 - 4 S [10.2] 39. B R [10.2] 40. B R [10.2] -3 1 -4 6 -6 10 - 17 -2 6 -6 -4 2 -8 4 - 10 12 28 -5 0 10 S [10.2] 44. C - 4 12 18 S [10.2] 45. C 2 6 0 S [10.2] [10.2] 43. C 14 -7 - 7 6 -15 10 - 13 6 16 -1 -36 - 4 124 4 S [10.2] 48. Not possible [10.2] 49. Not possible [10.2] 50. Not possible [10.2] -32 -6 -1 -5 4 24 9 2x - 3y = 5 2 - 32 S [10.3] 54. -2 [10.4] 55. e [10.2] R [10.2] 53. C 12 -22 1 4x + 5y = - 1 0 -2 1

57. c x - 5y + 4z = 10 [10.2]

46 7 17 7

116 7 - 45 7

2

8 R - 27

-11 -12 - 4 S [10.2] 47. C 48 -9 -9

108 24 64

46. C 10

-3 6

6 9

5 2 5 2 3U 2 1 2

1 4 0 1 0

1 0 1 1 1

1 1 1 2 1

-4 R [10.3] 3 - 10

67. C - 5

3

U [10.3]

20 9 -6

2x - 6y + 5z = 1 8x + 4y + z = 13 [10.2] - 2x + 3y + 4z = 5

1 0 0 0 1 U ; 1 walk [10.2] 60. A = E 0 1 1 1 0 - 27

63. C

1 7

3 14 S 1 7

[10.3]

64. C

-3 27 - 1 S [10.3] 68. C - 7 1 4

71. singular matrix [10.3]

-

1 11 3 22

-39 10 -6

0 0 1 1 0

0 1 0 0 0 2 11 S 5 22

1 1 0 0 1

0 0 0 0 0 U ; A3 = E 1 1 3 0 0 2

[10.3]

5 - 1 S [10.3] 1


0 0 2 4 0

1 2 0 0 1

3 4 0 0 3

0 0 1 U ; 4 walks [10.2] 3 0

-2

1 - 1 S [10.3] -1 - 1 1 - 1 -7 4 2 - 6 -3 2 3 69. D T [10.3] 1 2 -1 -1 -2 0 0 1 65. C 0

3 2


[10.3] 75. a. (18, -13) b. (-22, 16) [10.3] 76. a. (41, 17) b. (-39, -16) [10.3] 77. a. a-

6 18 23 , ,- b 7 7 7


A73

b. a-

31 20 3 4 , , b [10.3] 78. a. a- , -1, 2b b. ( - 9, -11, 6) [10.3] 79. 13 [10.4] 80. 0 [10.4] 81. -6, -6 [10.4] 82. -2, 2 [10.4] 14 7 7 3 83. 11, -11 [10.4] 84. 3, 3 [10.4] 85. 51 [10.4] 86. 40 [10.4] 87. -115 [10.4] 88. -69 [10.4] 89. 204 [10.4] 90. 42 [10.4] 2 16 91. -2 [10.4] 92. 1 [10.4] 93. -1 [10.4] 94. -1 [10.4] 95. 0 [10.4] 96. 3 [10.4] 97. 0 [10.4] 98. 0 [10.4] 99. a , - b [10.5] 19 19 1 21 13 1 17 17 19 13 18 26 38 21 12 40 100. a , - b [10.5] 101. a , , - b [10.5] 102. a- , , - b [10.5] 103. a , - , b [10.5] 104. a , , - b [10.5] 13 26 44 4 44 22 22 22 23 69 69 83 83 83 115 289 105. x3 = [10.5] 106. x2 = [10.5] 107. A¿(2, - 2), B¿( - 3, 7) [10.2] 108. A¿(1, - 2), B¿( - 6, 2), C¿(0, 1) [10.2] 126 230 109. City: 357,000; suburbs: 293,000 [10.2] 110. QuikPro: 2.4 million; PhotoPro: 1.1 million [10.2] 111. x1 = 52°F, x2 = 58°F [10.3] 112. x1 = 52.5°F, x2 = 60°F, x3 = 50°F, x4 = 57.5°F [10.3] 113. Computer division: $34.47 million; monitor division: $14.20 million; disk drive division: $23.64 million [10.3] 114. Lumber division: $30.82 million; paper division: $20.86 million; prefabrication division: $11.79 million [10.3]

Chapter 10 Test, page 844 2

3 0 -4

1. C 3

4

-3 2 4 2 † -1 S, C 3 2 4 3


4 9. C 8

11

1 0 0 -6

13. C -15

1

6. (3c - 5, -7c + 14, 4 - 3c, c) [10.1]

3 - 19 S [10.2] 10 -1 - 25 3

10. B

-19 -27 S [10.2] 31

17. M21 = - 8, C21

-3 4 3x - 2y + 5z - 4w = - 9 2 S , C -1 S [10.1] 2. c 2x + 3y - 5z + 4w = - 8 [10.1] 3. (2, -1, 2) [10.1] 4. (3, -1, -1) [10.1] 2 3 2x + 3y + 3z + 2w = - 1

3 0 -4

16 15

-1 -11

-2 R [10.2] -3

7. B

3 -3

17 14

-4 - 15

11. B

-9 -12

-4 R [10.2] -2

9

12. CA is not defined. [10.2]

6 13 -2 12 S [10.2] - 3 11

15. C - 3

14. A2 is not defined. [10.2]

-6 R [10.2] 8. A + B is not defined. [10.2] 3

18 16. C 4

-5 -1 1

7 2 S [10.3] -1

20 -3 140 = 8 [10.4] 18. 49 [10.4] 19. - 1 [10.4] 20. [10.5] 21. (10, 3, 21) [10.3] 22. p(x) = x 3 - 2x + 1 [10.1] 41

0 0 23. A¿(2, - 3), B¿( -3, 1), C ¿(-1, 4) [10.2] 24. E 0 1 1

0 0 1 1 1

0 1 0 1 0

1 1 1 0 0

1 1 0 U [10.2] 0 0

1 25. £ C 0 0

0 1 0

0 0.15 0 S - C 0.08 1 0.16

0.23 0.10 0.11

0.11 0.05 S ≥ 0.07

-1

50 C 32 S [10.3] 8

Cumulative Review Exercises, page 845 1 2

1. y = - x + 3 [2.3]

2. 2x 2 + x - 10 [3.1]

7. x = - 5, x = 1 [3.5] 8.

4 2

17. 20 m2 [7.2]

18. sec u [5.4]

4. (1, 1) [8.1]

5. (8, 5) [9.1]

[2.5] 9. ( - q , -2) ´ ( - 1, q ) [1.5] 10.

y

−4 − 2 −2 −4

3. 0.3679 [4.2]

11. 2x - 3 + h [2.6] 2 4

12. -

x

14. 0.3828 [4.5]

15.

19. z = 4(cos 300° + i sin 300°) [7.4]

6. {x ƒ -3 … x … 3} [2.2]

(x - 3)2 (y + 4)2 + = 1 [8.2] 36 20

4 2 3 + [4.5] 13. [9.4] 3 7(x - 6) 7(x + 1)

1 13 + 416 [5.2] 16. - 32 + 12 [6.3] 4 2 20. 0,

p 5p , [6.6] 3 3

A74


Exercise Set 11.1, page 852 1 256 1 1 1 2 4 8 9. , , , - , - , a8 = , a8 = 3 6 9 24 3 9 27 6561 12 13 12 0, 2, 0, a8 = 2 13. 1.1, 1.21, 1.331, a8 = 2.14358881 15. 1, , , a8 = 17. 1, 2, 6, a8 = 40,320 2 3 4 0, 0.3010, 0.4771, a8 L 0.9031 21. 1, 4, 2, a8 = 4 23. 3, 3, 3, a8 = 3 25. 5, 10, 20 27. 2, 4, 12 29. 2, 4, 16 31. 2, 8, 48 5 9 25 6 53. 72 55. 22 3, 13, 13 35. 2, , 37. 4, 7, 11 39. 4320 41. 72 43. 56 45. 100 47. 15 49. 40 51. 2 4 12 6 7 4 4 1 1 3 log 2 59. 256 61. a 2 63. a 2i(- 1)i + 1 65. a (7 + 3i) 67. a i 69. L 2.6457520 i=1 i i=1 i=0 i=1 2 The sum of the first n terms of the Fibonacci sequence equals the (n + 2) term minus 1; 143 73. F10 = 55, F15 = 610

1. 0, 2, 6, a8 = 56 11. 19. 33. 57. 71.

3. 0,

7 1 2 , , a8 = 2 3 8

5. 1, -

1 1 1 , , a8 = 4 9 64

7. -

Prepare for This Section (11.2), page 854 PS1. - 2

PS2.

5 2

PS3.

6525 4

PS4. 21

PS5. -5

PS6. Yes

Exercise Set 11.2, page 859 1. a9 = 38, a24 = 98, an = 4n + 2 3. a9 = - 10, a24 = - 40, an = 8 - 2n 5. a9 = 16, a24 = 61, an = 3n - 11 7. a9 = 25, a24 = 70, an = 3n - 2 9. a9 = a + 16, a24 = a + 46, an = a + 2n - 2 11. a9 = log 7 + 8 log 2, a24 = log 7 + 23 log 2, an = log 7 + (n - 1)log 2 13. a9 = 9 log a, a24 = 24 log a, an = n log a 15. 45 17. -79 19. 185 21. - 555 23. 468 25. 525 27. - 465

3 5 31. 210x 33. 3, 7, 11, 15, 19 35. , 2, , 1 39. 20 in the sixth row; 135 in the six rows 2 2 49. an = 7 - 3n 51. a50 = 197

41. $1500; $48,750

29. 78 + 12x

43. 784 feet

45. 32


PS2.

15 8

PS3. 33

PS4. S = a(1 + r)

PS5. -

3 3 3 , ,2 4 8

PS6. 2, 6, 14

Exercise Set 11.3, page 869 5 n-1 1 n-3 1 n 9. a- b 11. ( -x)n - 1 13. c3n - 1 15. 3a b b 17. 5(0.1)n 6 3 100 19. 45(0.01)n 21. 18 23. -2 25. Neither 27. Arithmetic 29. Geometric 31. Neither 33. Arithmetic 35. Geometric 279,091 1330 1 2 9 1 5 1 5 37. 363 39. 41. 3. - 341 45. 147,620 47. 49. 51. 53. 55. 57. 59. 729 390,625 2 5 91 9 7 3 11 41 422 229 997 61. 63. 65. 67. 69. $2271.93 71. $1562.50 73. 0.52 milligrams per liter 75. 17.68% 77. $19.60 333 999 900 825 79. If g Ú 1, the common ratio of the geometric series is greater than or equal to 1 and the sum of the infinite geometric series is not defined. 81. $100 million 83. 2044 1. 22n - 1

3. -4( -3)n - 1

5. 6a b

2 3

n-1

7. -6 a-

Mid-Chapter 11 Quiz, page 871 1 1 3019 ,a = [11.1] 2. a3 = -12, a5 = -48 [11.1] 3. [11.1] 4 8 32 3600 13 7. 4920 [11.3] 8. [11.3] 30 1. a4 =

4. 58 [11.2]

5. -200 [11.2]

6. an = - 2a-

2 n-1 b [11.3] 3


A75

Prepare for This Section (11.4), page 871 PS2. (k + 1)(k + 2)(2k + 3)

PS3.

k + 1 k + 2

PS4. 3

PS5.

(n + 1)(n + 2) 2

Exercise Set 11.4, page 877 No answers are provided because each exercise is a verification.

Prepare for This Section (11.5), page 878 PS1. a 3 + 3a 2b + 3ab 2 + b 3

PS2. 120

PS3. 1

PS4. 15

PS5. 35

PS6. 1


3. 36

5. 220

7. 1

9. x 5 + 5x 4y + 10x 3y 2 + 10x 2y 3 + 5xy 4 + y5

13. x + 20x + 150x + 500x + 625 4

3

2

11. a 4 - 4a 3b + 6a 2b 2 - 4ab 3 + b 4

15. a - 15a + 90a - 270a + 405a - 243 5

4

3

2

17. 128x 7 - 448x 6 + 672x 5 - 560x 4 + 280x 3 - 84x 2 + 14x - 1 19. x 6 + 18x 5y + 135x 4y 2 + 540x 3y 3 + 1215x 2y 4 + 1458xy 5 + 729y 6

- 28x

12

+ 336x

21. 16x 4 - 160x 3y + 600x 2y 2 - 1000xy 3 + 625y 4

- 2240x + 8960x - 21504x + 28672x - 16384 25. 32x 10 + 80x 8y 3 + 80x 6y 6 + 40x 4y 9 + 10x 2y 12 + y 15 16 16 15 6 1 x4 27. x 5 + 5x 4y 1>2 + 10x3y + 10x 2y 3>2 + 5xy 2 + y 5>2 29. - x 2 + 6 - 2 + 4 31. s 12 + 6s 8 + 15s 4 + 20 + 4 + 8 + 12 16 x x s s s 165b5 3 7 10 2 2 2 2 8 2 8 5 5 -1 33. -3240x y 35. 1056x y 37. 126x y 1x 39. 41. 180a b 43. 60x y 45. -61,236a b 47. 126s , 126s a5 49. -7 - 24i 51. 41 - 38i 53. 1 23. x

14

10

8

6

4

2


PS2. 24

PS3. 7

PS4. 56

PS5. 90

PS6. 720

Exercise Set 11.6, page 887 1. 30 3. 70 5. 1 7. 1 9. 210 11. 12 13. 16 15. 720 17. 125 19. 53,130 21. There are 676 ways to arrange 26 letters taken 2 at a time. If there are more than 676 employees, then at least 2 employees will have the same first and last initials. 23. 1120 25. 1024 27. 3,838,380 29. a. 21 b. 105 c. 21 31. 1.8 * 109 33. 112 35. 120 37. 21 39. 112 41. 184,756 43. 62,355,150 45. 5456 47. 19! 49. a. 3,991,680 b. 31,840,128 51. 120 53. 252

Prepare for This Section (11.7), page 890 PS1. See page 638.

PS2. 12

PS3. 42

PS4. 21

PS5.

189 8192

PS6. 16

Exercise Set 11.7, page 896 1. 5S1 R1 , S1 R2 , S1 R3 , S2 R1 , S2 R2 , S2 R3 , R1 R2 , R1 R3 , R2 R3 , S1S 2 6 3. 5H1, H2, H3, H4, T1, T2, T3, T46 5. 5(A, A), (A, B) (A, C), (B, B ), (B, C ), (B, A), (C, C ), (C, A), (C, B)6 7. 5HSC, HSD, HCD, SCD6 9. 5ae, ai, ao, au, ei, eo, eu, io, iu, ou6 11. 5HHHH6 13. 5TTTT, HTTT, THTT, TTHT, TTTH, TTHH, THTH, HTHT, THHT, HTTH, HHTT6 15. 17. 5(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)6

19. 5(1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4)6

21. a.

1 1 b. 13 4

23. a.

1 16

b.

15 16

A76


25. a. Yes 45.

1 16

b.

8 45

47. 0.2262

27. a. No

b.

49. 0.2137

8 25

51.

29. 0.97

49 64

3 5

31.

33. 0.59

35. 0.25

37. 0.1

39. 0.1

41. 0.025

43. 0.9999

53. 0.599

Chapter 11 Review Exercises, page 902 1. a3 = 10, a7 = 22 [11.1]

2. a3 = 5, a7 = 13 [11.1] 3. a3 = 9, a7 = 49 [11.1] 4. a3 = 21, a7 = 329 [11.1] 1 1 3 7 5. a3 = , a7 = [11.1] 6. a3 = , a7 = [11.1] 7. a3 = 8, a7 = 128 [11.1] 8. a3 = 27, a7 = 2187 [11.1] 3 7 4 8 8 128 1 1 , a7 = 9. a3 = [11.1] 10. a3 = , a7 = [11.1] 11. a3 = 18, a7 = 1458 [11.1] 12. a3 = - 4, a7 = - 64 [11.1] 27 2187 8 128 13. a3 = 6, a7 = 5040 [11.1]

14. a3 = 48, a7 = 645,120 [11.1]

17. a3 = 0, a7 = - 20 [11.1]

18. a3 = 8, a7 = 128 [11.1]

23. 220 [11.5]

24. 6435 [11.5]

31. 550 [11.2]

32. -5050 [11.2]

37. an = 4a-

1 b 2

n-1

[11.3]

42.

1,328,600 [11.3] 177,147

50.

8 [11.3] 55

43.

25. 55 [11.1]

1237 [11.1] 720

26.

39. an = 5a b

3 4

38. an = 3(2)n - 1 [11.3]

51. Neither [11.3]

45.

52. Neither [11.3]

1562 [11.3] 625

20. 600 [11.1]

16. a3 = 2, a7 = 1 [11.1] 21. 5040 [11.1]

28. -56 [11.2]

27. 99 [11.2]

34. -5625 [11.2]

33. 10,500 [11.2]

2 2 [11.3] 44. [11.3] 3 3

15. a3 = 2, a7 = 256 [11.1]

19. 126 [11.1]

29. 40 [11.2]

35. 16, 19, 22, 25 [11.2]

n-1

[11.3]

40. an = 9a-

46. -1225 [11.2]

53. Geometric [11.3]

47.

2 b 3

22. 144 [11.1] 30. 49 [11.2]

36. 23, 27, 31, 35, 39 [11.2] n-1

[11.3]

6 [11.3] 11

54. Geometric [11.3]

41. 255 [11.3]

48. 8 [11.3]

49.

7 [11.3] 30

55. Arithmetic [11.3]

56. Arithmetic [11.3] 57. Neither [11.3] 58. Geometric [11.3] 59– 66. No answers are provided because each exercise is a verification. [11.4] 67. 1024a5 - 1280a4b + 640a3b2 - 160a2b3 + 20ab4 - b5 [11.5] 68. x6 + 18x5y + 135x4y 2 + 540x3y3 + 1215x2y4 + 1458xy5 + 729y6 [11.5] 69. a7 - 7a6b + 21a5b2 - 35a4b3 + 35a3b4 - 21a2b5 + 7ab6 - b7 [11.5] 70. 81a4 - 216a3b + 216a2b2 - 96ab3 + 16b4 [11.5] 8

75. 26 [11.6]

6

76. 10 26 [11.6]

77. 2730 [11.6]

82. 5SH, SD, SC, HD, HC, DC6 [11.7] 85. 5(4, 6), (5, 5), (6, 4)6 [11.7] 86. 90.

1 3 , [11.7] 8 8

91. 0.19 [11.7]

71. 241,920x3y4 [11.5]

78. 60 [11.6]

83. a. Yes

b. Yes [11.7]

72. - 78,732x7 [11.5]

79. 880 [11.6]

80. 672 [11.6]

73. 96 [11.6]

74. 120 [11.6]

81. 6,497,400 [11.6]

84. 5HHHHT, HHHTH, HHTHH, HTHHH, THHHH6 [11.7]

10 1 13 [11.7] 87. a. No b. [11.7] 88. [11.7] 89. Drawing an ace and a ten-card from one deck [11.7] 2 20 21 1 92. 0.30 [11.7] 93. [11.7] 94. $14.53 [11.3] 95. $75 million [11.3] 4

Chapter 11 Test, page 905 1. a3 = 7.

4 4 ,a = [11.1] 3 5 15

1023 [11.3] 1024

2. a3 = 12, a5 = 48 [11.1]

8. 590 [11.2]

15. x 6 + 6x 4 + 15x 2 + 20 + 20. 0.28 [11.7]

9. 58 [11.2]

15 x2

+

6 x4

+

1 x6

10.

3. Arithmetic [11.3]

3 [11.3] 5

11.

5 [11.3] 33

4. Neither [11.3]

5. Geometric [11.3]

6.

49 [11.1] 20

14. x 5 - 10x 4y + 40x 3y 2 - 80x 2y 3 + 80xy 4 - 32y 5 [11.5]

[11.5] 16. 48,384x 3y 5 [11.5] 17. 132,600 [11.6] 18. 568,339,200 [11.6] 19.

5 [11.7] 17


A77

Cumulative Review Exercises, page 906 1. y = 1.7x + 3.6 [2.7]

2. -24 [2.3]

-17 3 -2

3.

3 3 141 [1.3] 4. logb x + 2 logb y - 3 logb z [4.4] 5. [8.2] 6. x = 1, y = - 2 [9.1] 4 5

12 5 -1 S [10.2] 8. [2.6] 9. y = 0 [3.5] 10. -6 [4.3] 11. - 2.1 [4.5] 16 13 2 9 11 -3 1 + 113 -1 + 113 1 - 113 -1 - 113 , b, a , b [9.3] 13. C - 6 -6 12. a 2 -5 S [10.3] 14. 105 ft/sec [4.5] 2 2 2 2 10 3 -15 13 7. C

15. - 13 [5.3]

16. 2 csc x [6.1]

17. C = 75°, a = 13 cm, b = 19 cm [7.1]

18. 27° [8.4]

19.

110 [6.5] 20. -5i + 28j [7.3] 10


INDEX Abscissa, 154 Absolute maximum and minimum, 273 Absolute value of complex number, 616 of cosine function, 478 description of, 7–8 equations, 79–80 inequalities with, 126–127, 756–757 of trigonometric functions, 458 Acid, 377 Acidity, 376–377 Actual velocity, 607 Acute angle definition of, 429 illustration of, 583 trigonometric functions of, 443 Addition of complex numbers, 61 of functions, 227 of matrices, 791–794 of ordinates, 495–496 of polynomials, 33 of rational expressions, 51, 53 of real numbers, 11–12, 123 Addition Rule for Probabilities, 893 Additive identity of matrix, 794 of real number, 12 Additive inverse of a vector, 604 of matrix, 793–794 of polynomial, 33–34 of real number, 11 Adjacency matrices, 805–806 Agnesi, Maria, 158 Air resistance, 384–385, 399 Airspeed, 607 Algorithms, 37, 783 Alignment charts. See Nomograms Alkalinity, 376–377 a b identities, 523–524 Alternating sequences, 849 Altitude, 570 Ambiguous case, 583–587 Amperage, 499 Amplitude, 474, 477 of the motion, 501 Amplitude–Time–Difference Formula, 375 Analytic geometry, 154–155 See also Conic sections Angle(s) acute. See Acute angle central, 433–434 complementary, 429 coterminal, 430–431 definition of, 428 degree measure of, 428–432 initial side of, 428 measure of, 428, 432

negative, 428 obtuse, 429 positive, 428 quadrantal, 430, 456–457 radian measure of, 432–435 reference, 457–459 right, 429 60°, 445 in standard position, 429 straight, 429 supplementary, 429 terminal side of, 428 30°, 445 trigonometric functions of, 454–460 vertex of, 428 Angle measurements, 463 Angle of depression, 447–448 Angle of elevation, 447–449 Angle of rotation, 440, 670 Angular rotation, 439 Angular speed, 437–440 Annuities, ordinary, 866–867 Aphelion, 652 Apollonius, 634 Apparent magnitude, 368 Application problems, strategies for solving, 86 Arc(s), 433, 435–436 Arc length, 435–436 Archimedes, 2, 300 Area of triangle, 453, 595–596 Argument, 617 Arithmetic means, 858–859 Arithmetic sequences, 854–859 Arithmetic series, 855–859 Associative property, 12–13 Astronomy, 451–452 Asymptotes horizontal, 307–312, 314 hyperbola, 660, 662–663 parabolic, 323 slant, 316–318 vertical, 307–311, 314 Augmented matrices, 780 Average velocity, 229–230 Axis conjugate, 659 coordinate. See Coordinate axis imaginary, 616 major, 646, 649 minor, 646, 649 polar, 678 semimajor, 646 semiminor, 646 of symmetry, 634–635, 637–638 transverse, 659, 662 x-, 647, 649, 659, 662 y-, 647, 649, 659, 662 Axis of symmetry, 213–214, 217 of parabola, 201

Back substitution, 730 Base of exponential expression, 9 of exponential function, 346–347 of logarithmic function, 359, 372–373 Base (pH), 377 Bearing, 588 Bernoulli, Johann, 700–701 Bertalanffy’s equation, 387–388 Binet form, 853 Binomial(s) definition of, 33 expanding, 878–882 factoring, 43–46 product of, 34 Binomial coefficients, 879–880 Binomial Probability Formula, 895–896 Binomial Theorem, 878–882 Birthday problem, 344 Blood pressure, 494–495 Boyle’s Law, 138–139 Brachistochrone problem, 700–701 Break-even point, 194 Calculators. See Graphing calculators Carbon dating, 393 Cardioid, 682 Carrying capacity, 397 Cartesian coordinates, 154–155 Cauchy, Augustin Louis, 298 Cauchy’s Bound Theorem, 298 Center, of circle, 161 Central angle, 433–434 Change-of-base formula, 372–373 Charles’s Law, 142 Circle, 84, 161–163 description of, 441 involute of, 704 polar equations of, 681–682 radius of, 453 unit, 462, 467, 469 Circular functions, 427, 463 Closed intervals, 6 Closure property, 12 Coefficient, 11, 32 of determination, 242–243, 281, 407–409 leading, 33, 272 Coefficient matrices, 780 Cofactors of a matrix, 825–827 Cofunctions, 524–525 Column matrices, 796 Column operations, 828 Combinations, 886–887 Combined variation, 141 Common denominator, 52–54 Common difference, 854 Common logarithms, 364–365 Common ratio, 861, 863

I1

I2

INDEX

Commutative property, 12–13 Complementary angles, 429 Completing the square, 99–100, 163 Complex conjugates, 63 Complex fractions, 55–56 Complex numbers, 59–64 argument of, 617 fifth roots of, 624–625 modulus of, 617 polar form of, 617 power of, 623 product property of, 618–620 quotient property of, 618–620 in standard form, 618 trigonometric form of, 616–622 Complex plane, 616 Complex solutions, of quadratic equation, 103 Complex zeros, of polynomial, 299–305 Composite numbers, 3 Composition of functions, 230–234 with inverse function, 336–338 logarithmic and exponential, 359 Compound inequalities, 125–126 Compound interest, 394–397, 866–867 Compressing graphs, 221–223 Computational complexity, 37 Computer algebra systems, 323–324 Concavity, 405–406 Conditional equations, 78 Conic(s) focus–directrix definitions of, 692 general equation of a nondegenerate, 670 identification theorem, 674 polar equations of, 691–695 rotation theorem for, 670–673 Conic sections degenerate, 634 description of, 633 ellipses. See Ellipse hyperbolas. See Hyperbola parabolas. See Parabola Conjugate of complex number, 63 of radical expression, 28 Conjugate axis, 659 Conjugate Pair Theorem, 302–304 Consistent systems of equations, 718, 729, 732 Constant functions, 175 Constant matrices, 780 Constant of proportionality, 137 Constant polynomials, 33 Constant sequences, 852 Constant terms, 11, 33 Constraints, 762–764 Continuous curves, 175, 271 Contradictions, 78 Conversion factor, 432 Coordinate(s) description of, 7, 154–155 polar. See Polar coordinate system rectangular, 154–155, 686–688 Coordinate axis real number line, 7

symmetry with respect to, 213–214, 217 in the plane, 154 in three dimensions, 728 Correlation coefficient, 240–242, 407–409 cos1 x, 550 Cosecant, 443 Cosecant function graph of, 486–487, 495 inverse, 552 period of, 467 Cosine definition of, 443, 446 double-angle identity for, 532 identity verification using, 517–519 See also Law of Cosines Cosine function absolute value of, 478 as cofunctions, 525 graph of, 476–479 harmonic motion. See Harmonic motion period of, 467 Cost, 88, 132–133, 194, 199 Cotangent, 443 Cotangent function graph of, 484–486 inverse, 552 period of, 468 Coterminal angles, 430–431 Counting Principle, 883–884, 887 Counting problems, 887 Cramer’s Rule, 833–836 Critical value method, 127–130 Cube roots, 45, 624 Cubes, sum or difference of, 45 Cubic equations, 110, 294–295 Cubic regression model, 280–282 Curve, 697–698 Curve fitting, 736–737 See also Interpolating polynomials Cycloid, 701 Damped harmonic motion, 503–504 Damping factor, 496 De Moivre’s Theorem, 622–625, 705 Decibels (dB), 379 Decimal(s) definition of, 2 repeating, 2, 866 in scientific notation, 20 Decimal degree method, 431 Decreasing functions, 175, 335 Degenerate conic section, 634 Degenerate form, 658 Degree of angles, 428–432 definition of, 428 fractional part of, 431–432 of a monomial, 32 of a polynomial, 32 radians conversions, 434–435 Demand–supply problems, 724–725 Denominator, 12 rationalizing, 27–28

Dependent systems of equations in three variables, 729, 732–734, 786 in two variables, 718, 720, 724 Dependent variable, 167 Depreciation, straight-line, 178 Depressed polynomials. See Reduced polynomials Descartes, René, 60, 154 Descartes’ Rule of Signs, 290–292 Determinants, 824–830 of 2 2 matrix, 824–825 conditions for zero determinant, 829 of matrix in triangular form, 828 product property of, 830 solving linear systems with, 833–836 Diagonal of a matrix, 780 Difference of cubes, 45 of real numbers, 11 of two functions, 496 See also Subtraction Difference identities, 523–524, 526–528 Difference quotient, 229–230 Direct variation, 137–138 Direction angle of vector, 603 Directrix of parabola, 636, 638–639 Discontinuities, 175 Discriminant, 102–103 Disjoint sets, 5 Displacement, 501 Distance between points in plane, 155–156 between points on real number line, 7–8 of falling object, 138, 856 in uniform motion, 89 Distributive property, 12–13 Dividend, 261 Division of complex numbers, 63–64 of exponential expressions, 18 of functions, 227 of inequality by real number, 123–124 of polynomials, 260–264 of radical expressions, 25, 27–28 of rational expressions, 51–52 of real numbers, 12 Divisor, 261 DMS method, 431–432 Domain, 167–168, 170 composition of functions and, 231, 233 of inverse function, 334, 340 of logarithmic function, 363–364 operations on functions and, 227–228 of rational expression, 50 of rational function, 307 of trigonometric functions of real numbers, 465 Dominant term, of polynomial, 272 Dot product, 609–610, 612 Double root, 97 Double solution, 97, 103 Double-angle identities, 532–534

INDEX

e (base of natural exponential function), 351 Earth orbit of, 652–653 radius of, 441 Earthquakes, 373–376, 379 Eccentricity of ellipse, 651–652 of hyperbola, 663–664 Edges, 805 Eiffel Tower, 451, 453 Einstein, Albert, 1, 23 Elementary row operations, 781–783 determinant and, 828–830 Elements of a matrix, 780 of a set, 3 Eliminating the parameter of a pair of parametric equations, 698–699 Elimination methods for linear systems, 721–724, 730–734, 784–787 for nonlinear systems, 743 Ellipse applications of, 652–653 with center at (0, 0), 646–648 with center at (h, k), 648–650 definition of, 645, 692 eccentricity of, 651–652 equation of, 650 foci of, 645, 647–649 graph of, 646, 650, 677 parametric equations for, 704 in polar form, 694 reflective property of, 653 vertices of, 646–649 Empty set, 4 Endpoint, 428 Epicycloid, 704 Equality of matrices, 792 of ordered pairs, 155 of polynomials, 750 properties of, 14 of rational expressions, 50 Equality of Exponents Theorem, 381 Equations absolute value, 79–80 classifying, 79 conditional, 78 contradiction as, 78 cubic, 110, 294–295 definition of, 14, 76–77 of ellipse, 650 equivalent, 76 exponential, 380–383 formulas, 84–85 general equation of a nondegenerate conic, 670 general second-degree equation in two variables, 670 of a graph, 479 identity as, 78 inverse trigonometric, 555 of a line, 190–191

linear in one variable, 77–78, 80–81, 84–85 logarithmic, 384–386 of parabolas, 636, 639 parametric. See Parametric equations polar. See Polar equations polynomial, 110, 294–295, 304 quadratic. See Quadratic equations quadratic in form, 116–117 radical, 112–114 rational, 110–112 with rational exponents, 114–115 rectangular, 688–689 transformation, 637 trigonometric. See Trigonometric equations in two variables, 157–163 variations, 136–141 See also Linear systems of equations; Nonlinear systems, of equations; Solutions Equilibrium, 501 Equilibrium price, 724–725 Equivalent equations, 76 Equivalent expression, 525 Equivalent inequalities, 123 Equivalent rational expressions, 50 Equivalent systems of equations, 721–722, 730 Equivalent vectors, 602 Eratosthenes, 441 Euler, Leonhard, 2, 167, 351, 849 Even and Odd Powers of (x c) Theorem, 278–280 Even functions, 216–217, 466–467 Events, 891–895 Existence theorems, 300 Expanding the logarithmic expression, 370–371 Expectation, 899–900 Experiment, 890 Exponent(s) equality of, 381 integer, 17–20 natural number, 9 negative, 17 properties of, 18–19 rational, 21–23, 114–115 restriction agreement for, 18 in scientific notation, 20 simplest form of expressions with, 19–20 zero and, 17–18 Exponential decay, 391–393, 416–418 Exponential equations, 380–383 Exponential functions, 346–354 definition of, 346–347 evaluating, 347 graphs of, 347–351 logarithmic functions and, 358–361 models based on, 333, 353–354 natural, 354 properties of, 349 Exponential growth, 391–393 Exponential notation, 9 Exponential regression, 408–409

I3

Exponential time algorithm, 37 Extended Principle of Mathematical Induction, 876–877 Extraneous solutions, 113 Factor Theorem, 266–267 Factorable over the integers, 40, 42 Factorials, 849–850 Factoring polynomials, 40–48 difference of squares, 44 general strategy for, 47–48 greatest common factor in, 40–41 by grouping, 46–47 solving equations with, 96–97, 110 sum or difference of cubes, 45 trinomials, 41–45 zeros and, 266–267, 276, 278, 280, 287–288, 300–301 Factoring trigonometric equations, 561 Factorization Theorem, 43 Factors of a real number, 3 Fair game, 899 Falling objects air resistance, 384–385, 399 distance fallen, 138, 856 height of, 105–106 Family of curves, 218 Feasible solutions, 763 Feedback, 66 Fermat, Pierre de, 154, 890 Fibonacci, Leonardo, 849 Fibonacci sequence, 849, 853 Final demand, 819 Finite sets, 4 Floor function, 175–178 Focus/foci of ellipses, 645, 647–649 of hyperbolas, 658, 661–663 of parabola, 636, 638–639 of paraboloid, 640 FOIL method, 34–35, 41 Force, 614 Formulas, 84–85, 178–179 Fraction(s) complex, 55–56 equations containing, 78, 110–112 rationalizing the denominator, 27–28 See also Rational expressions Fractional form, of synthetic division, 263 Frustum of a cone, 84 Functions algebraic operations on, 227–228 applications of, 178–179 composition of, 230–234 constant, 175 decreasing, 175 definition of, 166–167 difference quotient of, 229–230 domain of, 167–168, 170 evaluating, 168–169, 228 even, 216–217 families of, 218 of the form f(x) = a sin x + b cos x, 544–545 graphs of, 170–172

I4

INDEX

Functions (Continued ) greatest integer, 175–178 identifying, 168 increasing, 175 inverse, 334–342 maximum or minimum of, 274–275 notation for, 168–170 odd, 216–217 one-to-one, 175 piecewise-defined, 169, 248–249 range of, 167–168 trigonometric. See Trigonometric functions vertical line test for, 174–175 Fundamental Counting Principle, 883–884, 887 Fundamental Linear Programming Theorem, 764 Fundamental Theorem of Algebra, 299 Future value of annuity, 867 Fuzzy sets, 4 Galileo, 248, 856 Galois, Evariste, 101 Gauss, Carl Friedrich, 299–300 Gaussian elimination, 784–787 GCF (greatest common factor), 40–41 General equation of a nondegenerate conic, 670 General second-degree equation in two variables, 670 Geometric formulas, 84–85 Geometric sequences, 847, 861–862 Geometric series, 862–869 Germain, Sophie, 3 Germain prime number, 3 Golden mean, 75 Googol, 9 Gordon model of stock valuation, 867 Graph(s) amplitude of, 474 of circles, 161–163 compressing, 221–223 of conic in polar form, 693–694 of cosecant function, 486–487, 495 of cosine function, 476–479 of cotangent function, 484–486 of curve, 698–699 cycle of, 474 of cycloid, 701 definition of, 805 difference of two functions, 496 of ellipses, 646, 650, 677 equation of, 479 of equations in three variables, 728–730 of equations in two variables, 157–163 of even functions, 217 of exponential functions, 347–351 function of the form f (x) = a sin x + b cos x, 545 of functions, 170–172 of hyperbola, 663, 677 of inequalities, 755–760 of inverse functions, 335–336, 338, 556–557 of lemniscate, 686 of limaçon, 682–683 of linear functions, 186–195

of linear systems, 718 of logarithmic functions, 361–363 of nonlinear systems, 740–741 of odd functions, 217 of parabolas, 201–203, 637, 677 of piecewise-defined functions, 248–249 of polar equation, 679, 681 of polynomial functions, 271–282 product of two functions, 496 of r a, 680 of rational functions, 307–320 reflections of, 220–221 of rose curve, 684 scatter plots, 154 of secant function, 488–489 of second-degree equations in two variables, 674–676 semilog, 416–418 of sine function, 473–476 stretching, 221–223 sum of two functions, 495–496 symmetries of, 201, 213–217 of tangent function, 481–484 translations of, 217–220, 492–493 vectors added using, 602 walks through, 806 Graphing calculators absolute value functions, graphing, 159 adjusting settings of, 178 combinations, 886 complex numbers, 63, 303 connected vs. dot mode, 176–177, 248 CUTOUT program, 276 decimal degree measure conversion to DMS measure, 432 degree measure conversion to radian measure, 435 determinants, evaluating, 827 equations in two variables, graphing, 158–159, 161 exponential equations, solving, 381–382 exponential expressions, evaluating, 19, 352 exponential functions, graphing, 352, 354 families of curves, graphing, 218 functions, graphing, 172, 174, 178 greatest integer function, 176 inverse functions, graphing, 336–338 inverse of a matrix, 816 iterations, 66 linear systems, solving, 723, 772 LIST feature, 218 logarithmic functions, graphing, 364–365, 373 logarithms, evaluating, 364–365 logistic models, 411 matrix operations, 798 maximum and minimum, 274–276 modeling guidelines, 408 nonlinear systems, solving, 742 nth roots of z, 705 parabolas, 159, 669 permutations, 885 piecewise functions, graphing, 248–249 polar equation, 683 polynomial equations, solving, 294–295, 304

polynomial functions, graphing, 274–275, 290, 293 projectile path, 702 quadratic equations, solving, 105 reference angles, 458 regression analysis, 239–240, 243, 407–408 regression models, 281–282 rose curve, 684–685 row echelon form, 783, 790 scientific notation, 20 second-degree equations in two variables, 674–676 sinusoidal data, 566–567 sinusoidal families, 505 SOLVE feature, 144–145 special angles, 447 SQUARE viewing window, 338 synthetic division, 264 TABLE feature, 158 translations of figures, 812–813 ZERO feature, 161 zeros of polynomial functions, 290, 293 Greatest common factor (GCF), 40–41 Greatest integer function, 175–178 Ground speed, 607 Growth models exponential, 391–393 logistic, 397–399, 410–411 Gunning-Fog Index, 93 Hale Telescope, 643 Half-angle identities, 535–537 Half-life, 392 Half-line, 428 Half-open intervals, 6 Halley’s comet, 656, 664 Harmonic motion damped, 503–504 definition of, 501 simple, 501–504 Heading, 588 Heron’s formula, 596–597 Herschel, Caroline, 664 Holes in graphs, 271 Homogeneous systems of equations, 735–736 Hooke’s Law, 142 Hooper, Grace Murray, 849 Horizontal asymptotes, 307–312, 314–315 Horizontal axis of symmetry, 638 Horizontal compressing and stretching, 222–223 Horizontal line test, 175 Horizontal lines, 188, 191 Horizontal translations, 218–220, 491 Hubble Space Telescope, 441 Hydronium-ion concentration, 377 Hyperbola applications of, 664–665 asymptotes of, 660, 662–663 with center at (0, 0), 659–661 with center at (h, k), 661–663 definition of, 658, 692 degenerate form of, 658

INDEX

eccentricity of, 663–664 foci of, 658, 661–663 graphing of, 663, 677 navigational uses of, 664–665 in polar form, 693–694 reflective property of, 665 transverse axis of, 659, 662 vertices of, 659, 661 Hypocycloid, 704 Hypotenuse, 103–104 i (imaginary unit), 60, 64 Ideal Gas Law, 143 Identities, 78 a b, 523–524 cofunction, 525 definition of, 515 description of, 468–470 difference, 523–524, 526–528 double-angle, 532–534 fundamental, 516 half-angle, 535–537 inverses, 555–556 odd–even, 516 power-reducing, 534–535 product-to-sum, 541–542 Pythagorean, 468–469, 516–517 ratio, 468–469, 516 reciprocal, 469, 516 sines used to verify, 517–519 sum-to-product, 542–543 verification of, 516–520, 528, 534, 537 Identity matrix, 800 Identity properties of matrices, 794, 800 of real numbers, 12–13 Ill-conditioned systems of equations, 772 Imaginary axis, 616 Imaginary numbers, 60 Imaginary part, 60 Imaginary unit, 60, 64 Inconsistent systems of equations linear, 718–720, 729, 734, 786–787 nonlinear, 743 Increasing functions, 175, 335 Independent events, 895 Independent systems of equations, 718, 720, 731–732 Independent variable, 167 Index of radical, 23 of summation, 851 Index property of radicals, 25 Induction, mathematical, 871–877 Induction axiom, 873 Induction hypothesis, 873 Inequalities, 123–133 with absolute values, 126–127, 756–757 applications of, 131–133 compound, 125–126 critical values of, 127 equivalent, 123 graphs of, 755–760 linear, 124–125, 756–759, 762–768

nonlinear, 756, 759–760 polynomial, 127–129 proof of, by mathematical induction, 876–877 properties of, 123–124 rational, 130–131 in two variables, 755–760 Infinite geometric series, 864–866 Infinite sequences, 848–849 See also Sequences Infinite sets, 4 Infinity symbol, 6, 308–309 Initial point, 602 Initial side, 428 Inner product, 609 Input-output analysis, 819–821 Input-output matrix, 819 Integers, 2–4 Intercepted arc, 441 Intercepts, 160 See also x-intercepts; y-intercepts Interest compound, 394–397, 866–867 simple, 88–89, 394 Intermediate Value Theorem, 276–277 Interpolating polynomials, 788 Intersection of events, 893 of lines, 718, 720, 722–723 of sets, 4–5 of solution sets of inequalities, 125 Interval notation, 6–8 Inverse functions, 334–342 logarithmic and exponential, 359 Inverse of a matrix, 813–818 condition for existence of, 830 solving linear systems with, 816–818 Inverse properties of real numbers, 12 Inverse relations, 335 Inverse trigonometric equation, 555 Inverse trigonometric functions composition of, 553–556 cosecant function, 552 description of, 549–552 evaluation of, 551–552 graphs of, 556–557 polynomials used to approximate, 572 secant function, 552 Inverse variation, 138–140 Involute of a circle, 704 Irrational numbers, 2–3, 351 Isosceles triangle, 452 Iteration, 66 Joint variation, 140–141 Kepler’s Laws, 144 Latus rectum of a parabola, 644 of an ellipse, 657 Law of Cosines applications of, 593–594 dot product formula from, 610

I5

law of sines vs., 594 triangles solved using, 592 uses of, 593 Law of Sines ambiguous case of, 583–587 applications of, 587–588 description of, 582–583 law of cosines vs., 594 triangles solved using, 582–587 Leading coefficient of polynomial, 33, 272 Leading term of polynomial, 272 Least common denominator (LCD), 53–54 Least-squares regression line, 238–239 Lemniscate, 682, 685–686 Leontief, Wassily, 819 Libby, Willard Frank, 393 Lick Telescope, 643 Like radicals, 26 Like terms, 13, 32 Limaçon, 682–683 Line(s) equations of, 190–191 parallel, 191–192, 718 perpendicular, 191–192 polar equations of a, 679 slopes of, 186–190 symmetry with respect to, 201, 213–214, 217 Line of best fit. See Linear regression Linear correlation coefficient, 241 Linear equations in one variable, 77–78, 80–81, 84–85 in three variables, 728–730 in two variables, 189–190 Linear Factor Theorem, 300–301 Linear functions, 186, 190–195 Linear inequalities, 124–125, 756 systems of, 757–759, 762–768 Linear motion, 498 Linear programming, 762–768 Linear regression, 237–243 Linear speed, 437–438, 440 Linear systems of equations applications of, 724–726, 736–738 condition for unique solution, 836 Cramer’s Rule used in solving, 833–836 dependent, 718, 720, 724, 729, 732–734, 786 elimination for solving of, 721–724, 730–734, 784–787 graphing calculator for solving of, 723, 772 homogeneous, 735–736 ill-conditioned, 772 inconsistent, 718–720, 729, 734, 786–787 inverses of matrices for solving of, 816–818 matrix representations of, 780–781, 801–802 in n variables, 835 nonsquare, 734–735, 787 solutions of, 718, 720, 735, 787 substitution for solving of, 719–721, 729–730 in three variables, 728–730 triangular form of, 730–734 in two variables, 718–726 Lissajous figures, 704 Lithotripter, 655

I6

INDEX

Local minimum, 274 Logarithm(s) change-of-base formula for, 372–373 changing to exponential form, 359–360 common, 364–365 definition of, 359 natural, 364–365 properties of, 360–361, 369–371 Logarithmic equations, 384–386 Logarithmic functions, 358–366 applications of, 365–366, 373–377, 379 common, 364–365 definition of, 359 domains of, 363–364 exponential functions and, 358–360 graphs of, 361–364, 373 natural, 364–365 properties of, 362–363 Logarithmic scales, 373–377, 379 Logarithm-of-each-side property, 370 Logistic models, 397–399, 410–411 Long division, 262 LORAN, 667 Lovell Telescope, 643 Lucas sequence, 853 Magnitude of the vector, 601 Main diagonal of matrix, 780 Major axis, 646, 649 Marginal cost or revenue, 199 Marginal propensity to consume, 868 Mars, 657 Mathematica, 323–324 Mathematical induction, 871–877 Matrices addition of, 791–794 additive inverse of, 793–794 adjacency, 805–806 applications of, 779, 788, 806–807, 819–821 cofactors of, 825–827 definition of, 779–780 determinants of, 824–830, 833–836 elementary row operations on, 781–783 elements of, 780 equality of, 792 identity, 800 input-output, 819 inverse of, 813–818, 830 linear systems represented with, 780–781, 801–802 linear systems solved with, 784–787, 816–818 main diagonal of, 780 minors of, 825–827 multiplication of, 794–800 notation for, 792 order (dimension) of, 780 real number used in multiplication of, 794–796 row echelon form of, 781 stochastic, 837–838 subtraction of, 793–794 transformation, 779, 802–805 triangular form of, 828 zero matrix, 794

Maximum and minimum in linear programming, 763–768 of polynomial function, 273–276 of quadratic function, 204–206 Means, arithmetic, 858–859 Midpoint formula, 155–157 Millennium Wheel, 464 Minor axis, 646, 649 Minors of a matrix, 825–827 Minute, 432 Mixture problems, 90–91 Modeling data, 237–244, 407–410 Modulus, 617 Mollweide’s formulas, 600 Monomials, 32 Motion harmonic. See Harmonic motion uniform, 89 See also Falling objects; Speed Motion of a point, 700 Multiple zeros of a polynomial function, 287–288 Multiplication of binomials, 35 of complex numbers, 62–63 of exponential expressions, 19 of functions, 227 of inequality by real number, 123–124 of matrices, 796–800 of matrix by real number, 794–796 of polynomials, 34 of radical expressions, 25, 28 of rational expressions, 51–52 of real numbers, 11, 13 Multiplicative identity for matrices, 800 for real numbers, 12 Multiplicative inverse of matrix, 814–816, 830 of real number, 12 Multiplier effect, 868–869 Mutually exclusive events, 893–894 n! (n factorial), 850 Napier, John, 358 Natural exponential function, 354 Natural logarithms, 364–365 Natural numbers, 3–4 Nautical mile, 441 Negative angles, 428 Negative exponents, 17 Negative infinity symbol, 6 Negative integers, 3–4 Newton’s Method, 853 Nomograms, 379 Nonfactorable over the integers, 42–43 Nonlinear inequalities, 756 Nonlinear systems of equations, 740–745 of inequalities, 759–760 Nonsingular matrices, 816 Nonsquare systems of equations, 734–735, 787 nth roots, 623–625, 705 Null set, 4

Numbers complex. See Complex numbers sets of, 2–4 Numerator, 12 Numerical coefficients, 11, 32 leading, 33, 272 Objective function, 762 Oblique triangle, 582, 594 Observation angle, 571 Obtuse angles, 429 Odd functions, 216–217, 466–467, 526 Odd–even identities, 516 One-to-one functions, 175, 335, 338 One-to-one property, 370 Open intervals, 6 Optimization problems, 762–768 Orbits of comets, 664 of planets, 652–653, 656–657 Order of Operations Agreement, 9–11, 54 Ordered pairs as coordinates, 154 equality of, 155 of function, 167, 170 of relation, 166 as solutions of equations, 157 as solutions of inequalities, 755 as solutions of systems, 718, 720–721, 735 Ordered triples, 728, 733–734 Ordinate, 154 Orientation, 700 Origin, 7, 154 symmetry with respect to, 215–217 Orthogonal vectors, 611 Parabolas applications of, 639–641 axis of symmetry of, 634–635, 637–638 definition of, 201, 634, 692 directrix of, 636, 638–639 equation in standard form of, 639 equation of, 636 focus of, 636, 638–639 graphing of, 677 reflective property of, 640 with vertex at (0, 0), 634–637 with vertex at (h, k), 637–639 vertex of, 201–203, 634 See also Quadratic functions Parabolic asymptotes, 323 Paraboloid, 640 Parallel lines, 191–192, 718 Parallel vectors, 611–612 Parallelogram, 84, 602 Parameter of family of functions, 218 time as, 700 Parametric equations brachistochrone problem, 700–701 curve and, 697–698 definition of, 697 eliminating the parameter of a pair of, 698–699

INDEX

for ellipse, 704 projectile motion and, 702 time as parameter, 700 Partial fractions, 748–753 Partial sums, 850–851, 856–857, 863–864 Pascal’s Triangle, 881–882 Percent mixture problems, 90–91 Perfect cubes, 45 Perfect squares, 43 Perfect-square trinomials, 44–45, 99 Perihelion, 652 Periodic function, 467 Permutations, 884–886 Perpendicular lines, 191–192 Perpendicular vectors, 611–612 Petronas Towers, 453 pH of a solution, 376–377 pi (p), 2 Piecewise-defined functions, 169, 248–249 Plane coordinates in, 154–155 as graph of equation, 728–729 Point(s) plotting, in coordinate plane, 154, 158–159, 170–171 of real number line, 7 symmetry with respect to, 215, 217 Point–slope form, 190 Polar axis, 678 Polar coordinate system definition of, 678 graph of equations in, 679–686 polar axis, 678 rectangular coordinates and, 686–688 Polar equations of circle, 681–682 of conics, 691–695 definition of, 679 graph of, 681 of lemniscates, 685–686 rectangular equations and, 688–689 Polar form of complex number, 617 Pole, 678 Polynomial(s), 32–37 addition of, 33 applications of, 36–37, 275–276, 293–295 approximating inverse trigonometric functions, 572 basic terminology about, 32–33 with complex coefficients, 298–299 definition of, 32 division of, 260–264 dominant term of, 272 equality of, 750 evaluating, 36 Even and Odd Powers of (x c) Theorem, 278–280 factoring. See Factoring polynomials far-left and far-right behavior, 272–273, 279 finding, with given zeros, 304–305 graphing procedure, 279–280 graphs of, 271–282 Intermediate Value Theorem for, 276–277 interpolating, 788

maxima and minima of, 273–276 multiplication of, 34 nonfactorable over the integers, 42–43 reduced, 267–268, 292–293, 300 sign property of, 127 standard form of, 33 subtraction of, 34 See also Linear functions; Quadratic functions; Zeros of a polynomial Polynomial equations, 110, 294–295, 304 Polynomial inequalities, 127–129 Polynomial time algorithm, 37 Population growth, 391–392, 397–399 Position equation, 105 Positive angles, 428 Positive integers, 3–4 Power(s) direct variation as, 137 of exponential expressions, 18–19 of i, 64 inverse variation as, 139 of radical expressions, 24 restrictions on zero, 18 See also Exponents Power functions, 416 Power principle, 112–113 Power property, of logarithm, 370 Power-reducing identities, 534–535 Price, equilibrium, 724–725 Prime numbers, 3 Principal, 394 Principal square root, 23 Principle of Mathematical Induction, 873 extended, 876–877 Probability, 890–896 expectation and, 899–900 guidelines for, 896 Product definition of, 11 of two functions, 496 See also Multiplication Product property of complex numbers, 618–620 of logarithm, 370 Product-to-sum identities, 541–542 Profit, 88, 132–133, 194 Projectile, 565–566, 702, 713–714 Proportionality constant, 137 Protractor, 429 p-waves, 375 Pyramid, 452 Pythagorean identities, 468–469, 516–517 Pythagorean Theorem, 103–104, 444, 616 Quadrantal angle description of, 430 trigonometric functions of, 456 Quadrants, 154 Quadratic equations, 96–106 applications of, 103–106 classifying the solutions of, 103 completing the square used in solving of, 99–100 definition of, 96

I7

discriminant of, 102–103 factoring used in solving of, 96–97 quadratic formula used in solving of, 101–102 square roots used in solving of, 97–98 standard form of, 96 Quadratic formula, 101–102 trigonometric equation solved using, 562–563 Quadratic functions applications of, 206–209 definition of, 201 maximum or minimum of, 204–206 range of, 204 standard form of, 202–203 See also Parabolas Quadratic in form, 45–46, 116–117 Quadratic regression, 242–244 Quartic regression model, 280–282 Quotient definition of, 12, 261 difference, 229–230 See also Division Quotient property of complex numbers, 620 of logarithm, 370 Radian angle measurements using, 432–435 definition of, 433 degree conversions, 434–435 Radical equations, 112–114 Radical expressions, 23–28 Radicand, 23 Radius, of circle, 161, 453 Random walk, 890 Range of function, 167–168 of inverse function, 334 of quadratic function, 204 of trigonometric functions of real numbers, 465 Ratio identities, 468–469, 516 Rational equations, 110–112 Rational exponents, 21–23 equations with, 114–115 Rational expressions, 49–57 application of, 56–57 arithmetic operations on, 51–54 complex fractions and, 55–56 critical values of, 130 definition of, 50 domains of, 50 least common denominators of, 53–54 order of operations agreement with, 54 partial fraction decomposition of, 748–753 properties of, 50 simplifying, 51–53 Rational functions applications of, 318–320 asymptotes of graphs of, 307–313, 316–318, 323 with common factor, 318 definition of, 307 domains of, 307 graphing procedure for, 313–316 sign property of, 313

I8

INDEX

Rational inequalities, 130–131 Rational numbers, 2–3 Rational Zero Theorem, 288–289, 292–293 Rationalizing the denominator, 27–28 Ray, 428 Real number line, 7–8 Real numbers, 2–4, 11–13 trigonometric functions of, 461–472 Real part, of complex number, 60 Reciprocal, 12 Reciprocal functions, 446 Reciprocal identities, 469, 516 Rectangle, 84 Rectangular coordinates, 154–155, 686–688 Rectangular equations, 688–689 Rectangular form, 616 Rectangular solid, 84 Recursion. See Iteration Recursively defined sequence, 849 Reduced polynomials, 267–268, 292–293, 300 Reduction formulas, 528–529 Reference angle, 457–459 Reflection matrices, 802–803 Reflections of graphs, 220–221 of exponential functions, 351 Reflective property of ellipse, 653 of hyperbola, 665 of parabola, 640 Reflexive property of equality, 14 Regression analysis, 237–244, 566 Regression models, 280–282 Relations, 166, 168 inverse, 335 Relative maximum and minimum, 274 Relativity theory, 1, 23, 31 Remainder, in polynomial division, 261, 264–267 Remainder Theorem, 264–266 Repeating decimals, 2, 866 Resolving a vector, 607 Restriction Agreement, 18 Resultant vector, 602 Revenue, 88, 132–133, 194 marginal, 199 Richter, Charles F., 373 Richter scale, 373–376 Right angles, 429 Right circular cone, 84 Right circular cylinder, 84 Right triangles, 103–104 applications involving, 447–449 trigonometric functions, 442–444 Roots, 623–624 cube, 45, 624 double, 97 of an equation, 76, 287 fifth, 624–625 nth, 623–625, 705 in radical expressions, 23–28 rational exponents and, 21 See also Solution(s); Square roots; Zeros of a polynomial

Rose curves, 682, 684 Rotation matrices, 779, 804–805 Rotation of axes definition of, 670 formulas for, 671 Rounding numbers, 177, 448 Row echelon form, 781 Row matrices, 796 Row operations, 781–783, 828–830 Rule of Signs, Descartes’, 290–292 Sample spaces, 890, 892 Satellite, 471 Satellite dish, 640–642 Satisfying an equation, 76 Saturn, 656 Scalar, 601 Scalar multiplication description of, 602, 605 of matrix, 794–796 Scalar product, 609 Scalar projection, 611 Scalar quantities, 601 Scatter diagram, 281 Scatter plots, 154, 405–406, 567 See also Regression analysis Scientific notation, 20–21 Secant, 443, 446 Secant function graph of, 488–489 inverse, 552 period of, 467 Second, 432 Second-degree equations in two variables general, 670 graphing of, 674–676 Sector, 441 Seismograms, 375–376, 379 Semilog graphs, 416–418 Semimajor axis, 646 Semiminor axis, 646 Semiperimeter, 592 Sequences, 851 alternating, 849 arithmetic, 854–859 constant, 852 Fibonacci, 849, 853 geometric, 847, 861–862 infinite, 848–849 Lucas, 853 recursively defined, 849 Series, 851 arithmetic, 855–859 geometric, 862–869 Set(s), 2–5 disjoint, 5 elements of, 3 empty (null), 4 fuzzy, 4 infinite, 4 intersection of, 4–5 interval notation for, 6–8 of numbers, 2–4 union of, 4–5

Set-builder notation, 4 Sign diagrams, 128 Sign property of rational functions, 313 Significant digits, 448 Signs of rational expressions, 50 Similar triangles, 87–88 Simple harmonic motion, 501–504 Simple interest, 88–89, 394 Simple zero, 287 Simplifying complex fractions, 55–56 exponential expressions, 19–20, 23 radical expressions, 23–28 rational expressions, 51–53 trigonometric expressions, 469 variable expressions, 11–14 sin1 x, 549 Sine definition of, 443 identity verification using, 517 See also Law of Sines Sine function as cofunctions, 525 graph of, 473–476 harmonic motion. See Harmonic motion inverse, 550 period of, 467 Sine regression, 566–567 Singular matrices, 816 Sinusoidal data, 566–567 Slant asymptotes, 316–318 Slope-intercept form, 188–190 Slopes, 186–190 SMOG readability formula, 92–93 Smooth continuous curves, 271 Solution(s) definition of, 76 double, 97 of equation in two variables, 157 extraneous, 113 feasible, in linear programming, 763 polar equation, 679 of quadratic equation, 97, 103 of system of equations, 718, 720–721, 735, 787 x-intercepts and, 190 See also Zeros of a polynomial Solution set of inequality in one variable, 123, 125–126 of inequality in two variables, 755 of system of inequalities, 757 Sonic boom, 668 Sound wave, 480 Special product formulas, 35 Speed average for a round trip, 56–57 average over a time interval, 230 in uniform motion, 89 Sphere, 84 Spring constant, 502 Square(s) difference of, 44 formula, 84 Square matrices, 780

INDEX

Square roots of negative numbers, 60 of perfect squares, 43–44 of real numbers, 23–24 solving quadratic equations with, 97–98 Sørenson, Søren, 376 Standard form of the equation of a circle, 162–163 Standard position, 429 Statute mile, 441 Step, 805 Step functions, 176 Stochastic matrices, 837–838 Stock valuation, 867–868 Straight angles, 429 Stretching graphs, 221–223 of exponential functions, 351 Subsets, 4 Substitution methods for equations quadratic in form, 116–117 for linear systems, 719–721, 729–730 for nonlinear systems, 741, 744 Substitution property of equality, 14 Subtraction of complex numbers, 61 of functions, 227 of matrices, 793–794 of polynomials, 34 of rational expressions, 51, 53–54 of real number from inequality, 123 of real numbers, 11 Sum. See Addition Sum identities, 523–524, 526–528 Sum of first n positive integers, 858 Sum of two functions, 495–496 Summation notation, 851–852 Sum-to-product identities, 542–543 Supplementary angles, 429 Supply–demand problems, 724–725 s-waves, 375 Sylvester, James, 780 Symmetric property of equality, 14 Symmetries of graphs of function and its inverse, 336 of polynomial functions, 280 of rational functions, 314–315 with respect to a line, 201, 213–214, 217 with respect to a point, 215, 217 See also Axis of symmetry Synthetic division, 262–264 bounds for real zeros and, 289–290 with complex numbers, 303 Systems of linear equations. See Linear systems of equations of linear inequalities, 757–759 of nonlinear equations, 740–745 of nonlinear inequalities, 759–760 Tangent, 443 Tangent function double-angle identity for, 532 graph of, 481–484 period of, 468

Tangent lines, concavity and, 405 Telescope, 712 Terminal point, 602 Terminal side, 428 Terms of polynomial, 32 of sequence, 848 of sum, 11 of variable expression, 11 Test value, 128 Third-degree equations. See Cubic equations Tides, 499 Time as parameter, 700 in uniform motion, 89 Touch-tone phones, 515 Tower, 452 Transformation equations, 637 Transformation matrix, 779, 802–805 Transitive property of equality, 14 of inequalities, 123 Translation matrices, 802 Translations graphing uses of, 492–493 horizontal, 491 of trigonometric functions, 491–495 vertical, 491 Translations of graphs, 217–220 of exponential functions, 350–351 of logarithmic functions, 364–365 Transverse axis, 659, 662 Triangle(s), 84, 87–88, 103–104 area of, 453, 595–596 Isosceles, 452 Law of Cosines used to solve, 592 Law of Sines used to solve, 582–587 oblique, 582, 594 Pascal’s, 881–882 right. See Right triangles similar, 87–88 Triangular form of matrix, 828 of system of equations, 730–734 Trigonometric equations factoring used to solve, 561 inverse, 555 quadratic formula used to solve, 562–563 solving, 560–566 squaring each side of, 561–562 Trigonometric expression difference of, 543 evaluation of, 524, 554 simplifying, 527, 533 Trigonometric form of complex numbers, 616–622 Trigonometric functions absolute value of, 458 acute angle, 443 of angles. See Angle(s) angular speed, 437–438 applications of, 427 arcs, 435–436 complex number written in form of, 617

I9

composition of, 553–556 definition of, 442 evaluation of, 527, 533, 535–537 even, 466–467 inverse. See Inverse trigonometric functions linear speed, 437–438 list of, 442–444 odd, 466–467 of quadrantal angles, 456 of real numbers, 427, 461–472 signs of, 456–457 of special angles, 444–447 translations of, 491–495 Trigonometric identities. See Identities Trinomials definition of, 33 factoring, 41–43, 45 perfect-square, 44–45, 99 quadratic in form, 45–46 Trivial solution of linear system, 735 Turning points, 272–273 Uniform motion, 89 See also Speed Union of events, 893 of sets, 4–5 of solution sets of inequalities, 125 Unit circle, 462, 467, 469 Unit fraction, 432 Unit vectors, 604–606 Upper- and Lower-Bound Theorem, 289–290 Variable, 167–168 Variable expressions, 11–14 Variable part, 11 Variable terms, 11 Variation, 136–141 Vector additive inverse of a, 604 angle between, 610 applications of, 607–608 components of, 603–604, 608 definition of, 601 direction angle of, 603 dot product of, 609–610, 612 equivalent, 602 fundamental operations, 604 horizontal component of, 606 magnitude of, 601, 609–610 orthogonal, 611 parallel, 611–612 perpendicular, 611–612 resolving the, 607 resultant, 602 scalar multiplication of, 602 triangle method for adding, 602 unit, 604–606 vertical component of, 606 zero, 604 Vector quantities, 601 Velocity. See Speed Venus, 656 Verhulst population models, 66, 397

I10

INDEX

Vertex at (0, 0), 634–637 of angle, 428 at (h, k), 637–639 in linear programming, 764 of parabola, 201–203, 634 of paraboloid, 640 Vertical asymptotes, 307–311, 314–316 Vertical axis of symmetry, 637 Vertical line test, 174–175 Vertical lines, 188 Vertical stretching and compressing, 221–222 Vertical translations, 217–220, 491 Vertices description of, 805 of ellipse, 646–649 of hyperbola, 659, 661 Voltage, 499 Walk definition of, 805 in graphs, 806 length of, 805 random, 890 Washington Monument, 452 Whispering gallery, 653, 656

Work, 612 Work problems, 91–92 Wrapping function, 462–463 x-axis, 154, 647, 649, 659, 662 reflection across, 220 symmetry with respect to, 213–214 x-coordinate, 154 x-intercepts, 160 of rational functions, 314–315 real solutions and, 190 zeros of a polynomial and, 278 xy-plane, 154 xyz-coordinate system, 728 y-axis, 154, 647, 649, 659, 662 reflection across, 220 symmetry with respect to, 213–214, 217 y-coordinate, 154 y-intercepts, 160 of lines, 188–189 of polynomial functions, 279 of rational functions, 314–315 z-axis, 728 Zeller’s Congruence, 185

Zero, in exponential expressions, 17–18 Zero determinant, 829 Zero factorial, 850 Zero matrix, 794 Zero product principle, 96–97 Zero vector, 604 Zeros of a polynomial applications of, 293–295 complex, 299–305 complex coefficients and, 298–299 definition of, 127, 260 Descartes’ Rule of Signs and, 290–292 factors and, 266–267, 276, 278, 280, 287, 300–301 finding polynomial, given zeros, 304–305 finding with Mathematica, 323–324 guidelines for finding, 292–293 Intermediate Value Theorem and, 276–277 multiple, 287–288 number of, 288 rational, 288–289 sign of polynomial between, 127 simple, 287 upper and lower bounds for, 289–290, 298 x-intercepts and, 278 Zeros of a rational function, 313


A Library of Functions 2.3 Identity function y 4

2.3 Linear function y

f (x) = x

2.3 Constant function y

f (x) = mx + b

y 6

f(x) = c

(0, c)

2

2.2 Absolute value function

m

(0, b)

4

f(x) = |x|

1 −4

−2

x

4 x

2

2

x

−2

−4

−2 −2

−4

2.4 Squaring function

−4

3.2 Cubing function

y 6

y 4

4

2 f (x) = x 2

2 −2

4 x

2

2

−4

−2

Square root function

f(x) = x 3

2 −2

4 x

4.2 Exponential function

2 −4 2

4

−2

4.3 Logarithmic function y

f(x) = log b x, b > 1 (b, 1)

(1, b)

(1, 0) x

(1, 0)

x

(1, b) f(x) = log b x, 0 < b < 1

x

x

4.6 Logistic function

4.6 Logistic function

P(t)

P(t)

c

3.5 Reciprocal function y 4

c 2 c P(t) = , a>1 1 + ae −bt

P0

c P(t) = ,0 0, b > 0

f (x) =

y=a 2

x

4 x

P0 t

4 x

−4

(b, 1)

(0, 1)

2 −2

6 x

y

f (x) = b x, 0 < b < 1

(0, 1)

3

f(x) = x

f(x) = x

4.3 Logarithmic function

y

f(x) = b x, b > 1

4

−2

4.2 Exponential function

y

y 4

−2

−4

−2

y 6

2

4 x

Cube root function

t

−4

x=b

A Library of Functions 5.5 Sine function

(cont’d)

5.5 Cosine function

5.6 Tangent function y

y

y f (x) = sin x

1

−2π

2π x

π

−π

− 2π

π

−π

−1

1

f(x) = cos x

1

− 2π

2π x

2π x

π

−π −1

−1 f(x) = tan x

5.6 Cosecant function

−2π

5.6 Secant function

5.6 Cotangent function

y

y

y

1

1

1

2π x

π

−π

− 2π

π

−π

−1

2π x

− 2π

−1

f (x) = csc x

2π x

π

−π −1

f (x) = sec x

f(x) = cot x

Conic Sections 8.1 Parabolas y

y

y Focus

Directrix y=k−p

(h, k + p)

y Directrix x=h−p Vertex (h, k)

Vertex (h, k) Vertex (h, k)

x

x

x

(x – h) = 4p(y – k); p < 0

2

Vertex (h + a, k)

Vertex (h − a, k)

Focus (h − c, k)

y

Vertex (h, k + a)

y

Asymptote b (x − h) a

Focus

y−k=

Focus (h, k + c)

Center (h, k)

Vertex (h − a, k)

(h, k + c)

Vertex (h + a, k)

Center (h, k)

Focus

Focus (h + c, k)

a

b

2

Center (h, k) Asymptote a y − k = − (x − h) b x

Vertex (h, k − a)

(h + c, k)

x

Focus (h, k − c)

2

y−k=

Vertex (h, k − a)

= 1, a > b x−h b2

2

+

y−k a2

2

x−h = 1, a > b

Focus

Asymptote b y − k = − (x − h) a

x

y−k

Asymptote a (x − h) b

Vertex (h, k + a)

Focus

(h − c, k)

Center (h, k) x

2

(y – k) = 4p(x – h); p < 0

8.3 Hyperbolas y

y

+

2

(y – k) = 4p(x – h); p > 0

8.2 Ellipses

2

x

2

2

(x – h) = 4p(y – k); p > 0

x−h

Focus (h + p, k)

Focus (h + p, k)

Focus (h, k + p)

Directrix y=k−p

Directrix x=h−p

Vertex (h, k)

a2

2

−

y−k b2

(h, k − c)

2

y−k

=1

2

a

2

−

x −h b

2

2

=1

P.2 Properties of Exponents aman = am + n

am = am - n an

(amb n)p = ampb np

a

2.5 Graphing Concepts (am)n = amn

am p amp nb = b bnp

b-p =

1 bp

P.2 Properties of Radicals n

n

n

(1b)m = 1bm = bm/n n

n

1a n a = n Ab 1b m n

n

1a 1b = 1ab

mn

31b = 1 b

P.4 Factoring a 2 + 2ab + b2 = (a + b)2 a 2 - 2ab + b2 = (a - b)2 a 2 - b2 = (a + b)(a - b)

Odd Functions A function f is an odd function if f ( -x) = - f(x) for all x in the domain of f. The graph of an odd function is symmetric with respect to the origin. Even Functions A function is an even function if f( - x) = f(x) for all x in the domain of f. The graph of an even function is symmetric with respect to the y-axis. Vertical and Horizontal Translations If f is a function and c is a positive constant, then the graph of G

y = f (x) + c is the graph of y = f (x) shifted up vertically c units.

G

y = f (x) - c is the graph of y = f (x) shifted down vertically c units.

G

y = f (x + c) is the graph of y = f (x) shifted left horizontally c units.

G

y = f (x - c) is the graph of y = f (x) shifted right horizontally c units.

a + b = (a + b)(a - ab + b ) 3

3

2

2

a3 - b3 = (a - b)(a2 + ab + b2)

1.5 Properties of Absolute Value Inequalities ƒ x ƒ 6 c (c Ú 0) if and only if - c 6 x 6 c. ƒ x ƒ 7 c (c Ú 0) if and only if either x 7 c or x 6 - c.

2.2 Properties of Functions A function is a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates. If a and b are elements of an interval I that is a subset of the domain of a function f, then G f is an increasing function on I if f (a) 6 f (b) whenever a 6 b. G f is a decreasing function on I if f (a) 7 f (b) whenever a 6 b. G f is a constant function on I if f (a) = f (b) for all a and b. A one-to-one function satisfies the additional condition that given any y, there is one and only one x that can be paired with that given y.

Reflections If f is a function then the graph of G

y = - f(x) is the graph of y = f (x) reflected across the x-axis.

G

y = f( - x) is the graph of y = f (x) reflected across the y-axis.

Vertical Shrinking and Stretching G

G

G

If c 7 0 and the graph of y = f (x) contains the point (x, y), then the graph of y = c # f(x) contains the point (x, cy). If c 7 1, the graph of y = c # f(x) is obtained by stretching the graph of y = f (x) away from the x-axis by a factor of c. If 0 6 c 6 1, the graph of y = c # f(x) is obtained by shrinking the graph of y = f (x) toward the x-axis by a factor of c.

Horizontal Shrinking and Stretching G

If a 7 0 and the graph of y = f (x) contains the point (x, y), then the graph of y = f (ax) contains the 1 point a x, yb . a

G

If a 7 1, the graph of y = f (ax) is a horizontal shrinking of the graph of y = f (x).

G

If 0 6 a 6 1, the graph of y = f (ax) is a horizontal stretching of the graph of y = f (x).

Important Formulas

4.6 Compound Interest Formulas

The distance between P1(x1, y1) and P2(x2 , y2) is

P = principal invested, t = time in years, r = annual interest rate, A = balance:

d(P1, P2) = 2(x2 - x1)2 + ( y2 - y1)2 The slope m of a line through P1(x1, y1) and P2(x2, y2) is m =

y2 - y1 , x2 - x1

x1 Z x2

A = Pert (compounded continuously)

The slope-intercept form of a line with slope m and y-intercept b is y = mx + b The point-slope formula for a line with slope m passing through P1(x1, y1) is y - y1 = m(x - x1) Quadratic Formula If a Z 0, the solutions of ax 2 + bx + c = 0 are x =

-b 2b2 - 4ac 2a

Leg a

5.3 Definitions of Trigonometric Functions b r a cos u = r b tan u = a sin u =

r b r sec u = a a cot u = b

y

csc u =

P (a, b) b

θ a

x

where r = 2a2 + b2

Important Theorems Pythagorean Theorem c 2 = a2 + b2

r nt A = P a1 + n b (compounded n time per year)

Hypotenuse c

5.4 Definitions of Circular Functions sin t = y

Leg b

Remainder Theorem If a polynomial P(x) is divided by x - c, then the remainder is P(c). Factor Theorem A polynomial P(x) has a factor (x - c) if and only if P(c) = 0.

cos t = x y tan t = x

y

1 y 1 sec t = x x cot t = y

csc t =

t

P (x, y)

A(1, 0) r=1

x

Fundamental Theorem of Algebra If P is a polynomial of degree n Ú 1 with complex coefficients, then P has at least one complex zero.

6.1 Fundamental Trigonometric Identities

Binomial Theorem

sin2 u + cos2 u = 1

n n (a + b) = a + a ban - 1b + a b an - 2b 2 1 2 n + Á + a ban - kbk + Á + bn k n

n

4.4 Properties of Logarithms y = logb x if and only if b = x y

logb b = 1

logb 1 = 0

logb (b) p = p

blogb p = p

log x = log10 x

ln x = log e x

logb (MN ) = logb M + logb N logb (M>N) = logb M - logb N logb M p = p logb M

tan u =

sin u cos u

cot u =

cos u sin u

1 + tan2 u = sec2 u

1 + cot 2 u = csc2 u sin (- u) = - sin u tan (- u) = - tan u 1 csc u = sin u

cos ( - u) = cos u

sec u =

1 cos u

cot u =

6.2 Sum and Difference Identities sin (a + b) = sin a cos b + cos a sin b cos (a + b) = cos a cos b - sin a sin b tan (a + b) =

tan a + tan b 1 - tan a tan b

1 tan u

sin (a - b) = sin a cos b - cos a sin b

6.4 Sum-to-Product Identities

cos (a - b) = cos a cos b + sin a sin b

x - y x + y cos 2 2 x - y x + y sin sin x - sin y = 2 cos 2 2 x - y x + y cos cos x + cos y = 2 cos 2 2 x - y x + y sin cos x - cos y = - 2 sin 2 2

tan (a - b) =

tan a - tan b 1 + tan a tan b

6.2 Cofunction Identities sin a

p p - ub = cos u cos a - ub = sin u 2 2

tan a

p - ub = cot u 2

7.1–7.2 Formulas for Triangles For any triangle ABC, the following formula can be used.

6.3 Double-Angle Identities sin 2a = 2 sin a cos a cos 2a = cos a - sin a = 1 - 2 sin a = 2 cos2 a - 1 2

tan 2a =

sin x + sin y = 2 sin

2

2

2 tan a 1 - tan2 a

6.3 Power-Reducing Identities

Law of Sines b c a = = sin A sin B sin C

C

a

b

Law of Cosines

A

c 2 = a 2 + b 2 - 2ab cos C

c

Area of a Triangle K =

1 a2 sin B sin C ab sin C K = 2 2 sin A

K = 3s(s - a)(s - b)(s - c), where s =

a + b + c 2

1 - cos 2a sin2 a = 2 1 + cos 2a cos2 a = 2 1 - cos 2a 2 tan a = 1 + cos 2a

11.2 Arithmetic Sequences and Series

6.3 Half-Angle Identities sin

a 1 - cos a = 2 A 2

cos

1 + cos a a = 2 A 2

tan

sin a 1 - cos a a = = 2 1 + cos a sin a

Common difference

ai + 1 - ai = d

nth-term

an = a1 + (n - 1)d

Sum of n terms

Sn = =

Common ratio

ai + 1 = r ai

nth-term

an = a1r n - 1

Sum of n terms

Sn =

Sum of an infinite series

S =

2 cos a sin b = sin (a + b) - sin (a - b) 2 cos a cos b = cos (a + b) + cos (a - b) 2 sin a sin b = cos (a - b) - cos (a + b)

n [2a1 + (n - 1)d] 2

11.3 Geometric Sequences and Series

6.4 Product-to-Sum Identities 2 sin a cos b = sin (a + b) + sin (a - b)

n (a + an) 2 1

a1(1 - r n) 1 - r a1 , ƒrƒ 6 1 1 - r

B

College Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry

Algebra and Trigonometry