RESULTS FROM GEOMETRY
Triangle Area 21 bh
Circle Area πr 2 Circumference 2πr
h
r
b
Trapezoid Area 2h (a b)
Parallelogram Area bh
a h b
Rectangular Box Volume lwh Surface area 2wh 2lh 2wl
b
l
h
Right Circular Cylinder Volume πr 2h Lateral surface area 2πrh
w
h r
•
The sum of the angles in a triangle is 180°.
•
Two sides of a triangle are equal if and only if the angles opposite those sides are equal.
•
In a 30°-60° right triangle, the length of the side opposite the 30° angle is half the length of the hypotenuse.
THE PYTHAGOREAN THEOREM: In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. HERON’S FORMULA: If a, b, and c are the lengths of the sides of a triangle, then the area of the triangle is 1s1s a2 1s b2 1s c2 where s 12 1a b c2
h
Sphere Volume 43 πr 3 Surface area 4πr 2
Right Circular Cone Volume 13 πr 2h r 2 h2 Lateral surface area πr œ„„„„„„
r
h r
THEOREMS FOR PROVING THAT TWO TRIANGLES ARE CONGRUENT: 1. If two sides and the included angle of one triangle are equal respectively to two sides and the included angle of another triangle, then the triangles are congruent. 2. If two angles and the included side of one triangle are equal respectively to two angles and the included side of another triangle, then the triangles are congruent. 3. If the three sides of one triangle are equal respectively to the three sides of another triangle, then the triangles are congruent. THEOREMS FOR PROVING THAT TWO TRIANGLES ARE SIMILAR: 1. If two angles of one triangle are equal respectively to two angles of another triangle, then the triangles are similar. 2. If the corresponding sides of two triangles are proportional, then the triangles are similar.
GRAPHS y
y
y
slo pe m
r
(x 1, y1)
(h, k) x
x
(x-h)2+(y-k)2=r 2
y-y1=m(x-x1)
y
x
a
a
y=œ„„„„„„ a 2-x 2
y
y
x x
x
y=x 2
y=|x| y
y=x 3
y
y
x
x
x
y=œ„ x
1 y= x
y=[ x ]
y
y
y y=ex
(0, b) y=ln x b
(0, 1)
(a, 0)
(_a, 0)
x
x
x
a
(1, 0)
(0, _b)
y=´ and y=ln x
x2 a2
+
y2 b2
=1
x2 a2
-
y2 b2
=1
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Precalculus with Unit-Circle Trigonometry
Fourth Edition
David Cohen Late of University of California Los Angeles With
Theodore B. Lee City College of San Francisco
David Sklar San Francisco State University
Australia • Canada • Mexico • Singapore • Spain United Kingdom • United States
Precalculus: with Unit-Circle Trigonometry Fourth Edition David Cohen, Theodore B. Lee, David Sklar Acquisitions Editor: John-Paul Ramin Assistant Editor: Katherine Brayton Editorial Assistant: Leata Holloway Technology Project Manager: Earl Perry Senior Marketing Manager: Karin Sandberg Marketing Assistant: Erin Mitchell Managing Marketing Communications Manager: Bryan Vann Senior Project Manager, Editorial Production: Janet Hill Senior Art Director: Vernon Boes Senior Print /Media Buyer: Karen Hunt Permissions Editor: Kiely Sisk
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ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher. Printed in the United States of America 1 2 3 4 5 6 7 09 08 07 06 05 For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com. Any additional questions about permissions can be submitted by email to
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To our parents: Ruth and Ernest Cohen Lorraine and Robert Lee Helen and Rubin Sklar
Tribute to David Cohen Because he loved the part in “The Myth of Sisyphus,” where Camus re-envisions Sisyphus eternally condemned to the task of pushing a giant boulder to the summit of a mountain only to watch it roll back down again—descending the hill and smiling, I like to remember my dad in similar moments of inbetweenness. When the unknown and hope would combine to make anything seem attainable. When becoming conscious of a situation did not necessarily signify limitation by it, but instead, liberation from it. And so I imagine my dad, after hours of working at the computer, between a thoughtful sip of coffee and suddenly realizing the clearest way to word a problem. After years of early morning commutes to avoid 405 traffic, solitary interludes that began with waking to a still-dark sky and still-dreaming family, and ended with walking, transformed into a professor, into a UCLA lecture hall. After a lifetime of doing his best for those he cared about, intervals of good and poor health simply challenged him to give a little more. My dad endured his task bravely, from his decision 15 years ago to fight leukemia until May 2002, when the illness reminded us that even fathers and teachers are mortal, regardless of scheduled office hours or books left unwritten. My dad once told me the five most important things in life to him, and teaching and writing these math books were both on that list. I was never a “registered” student of his, but to anyone using this book, especially students, I pass on what I know he would have told me: Your best is always good enough. Enjoy. Emily Cohen •
•
•
My dad was gifted at teaching. He had an innate ability to explain things simply and relevantly. He excelled at combining his explanations with a patient ear and his selfdefined nerdy sense of humor. I remember when my dad taught me to read. I had become upset one night while we were out to dinner because I couldn’t read the menu. Frustrated, I expressed my dismay from my booster chair. My dad responded with the first of my reading lessons. He began by pointing out something I already knew, the letter “a.” He carefully explained that this letter was actually a word . . . every day after that he would write new words on one of his ubiquitous lined yellow notepads and teach them to me. Eventually we began to form simple sentences. He always kept my attention because he made up sentences that made me laugh. The frogs go jog, the cat loves the dog, so on and so forth. My dad’s sense of humor was quite captivating. I was lucky in high school to have someone to assist me with all of my mathematical questions. My math teachers seemed to have a special aptitude for making formulas, theories, and problems both complicated and boring. My dad was good at simplifying these matters for me. He was so good at explaining math to me that I would often remark afterward, with the hindsight of an enlightened one, “Oh, that’s all? Why didn’t they just say that in the first place?” I took comfort in knowing that math didn’t have to be complicated when it was explained well. iv
Tribute to David Cohen
v
Hemingway said that the key to immortality was to write a book. I feel lucky because my dad left behind several. Try reading one—you might find that you actually like it! Jennifer Cohen •
•
•
I first met Dave Cohen in the spring of 1981, when I signed up for an algebra course he was teaching as part of my graduate studies at UCLA. I quickly took a liking to his conversational style of teaching and the clear explanations he would use to illustrate an idea. Dave and I began an informal, weekly meeting to discuss solutions to various problems he would pose. Usually he would buy me a cup of coffee and present me with some new trigonometric identity he had discovered, or some conic section property he had come across in an old 1886 algebra book. We would discuss the problem (“Did it have a solution? Was it really an identity? Could we prove it?”), then go about our business. In the summer of 1981 I had the opportunity to TA a course called Precalculus, which used Dave’s notes (sometimes written by hand), rather than a traditional textbook. I had no idea what the term “precalculus” meant, and soon learned that these notes Dave had prepared were beginning to define, or at the least greatly expand, the subject. It was the beginning of a textbook. Two years later, after graduating UCLA, I was pleased to receive a copy of Precalculus, by David Cohen. Dave had scribbled a note to me saying: “Thanks for the great coffee breaks.” I also was pleased to see that some of our weekly problems appeared in that book (others showed up years later). More importantly, that book, and every text Dave has ever written, talked to you. Even now, when I read this book, I can hear Dave talking, and sometimes even sense him listening. Lecturing was never a part of Dave Cohen’s vocabulary. Since that time Dave has written books entitled College Algebra, Trigonometry, and Algebra and Trigonometry. Most have gone on to multiple editions, including this one. In each edition Dave sought the better explanation, the “cooler” problem, the more interesting data set. I’ve had the honor of working with him on all of these books, and we had developed a mathematical friendship. Perhaps some of you reading this have a great study partner; one who intuitively knows what the other is thinking. That was the relationship Dave and I had. I last saw Dave in November 2001, when we met at UCLA to (naturally) have a cup of coffee and discuss his latest textbook. He was excited about the quality of problems and interesting data sets he had found to use in the book and felt it was going to be the best book he had ever written. This present text is strongly based on that material. In May 2002, just after completing much of this manuscript, Dave Cohen passed away from complications caused by his leukemia. His legacy of fine textbooks and great teaching has influenced a generation of students and teachers. Everyone who had contact with Dave feels a bit richer from the experience. His total lack of ego, curiosity about the world, and respect for what others think, made him one of the finest human beings I have ever met. Ross Rueger
About the Co-Authors David Sklar was a longtime friend of David Cohen, whom he met while they were both attending graduate school at San Francisco State University. Because of his relationship with the author, he has followed the progress of this book since work started on the first edition. He brings a unique blend of teaching and professional experience to the table, having taught at San Francisco State University, Sonoma State University, Menlo College, and City College of San Francisco. At the same time, David is a researcher and consultant in the field of optics—you’ll notice that interest in some of the new group projects in the text. David is an active member of the American Mathematical Society, Mathematical Association of America, is past chairman of the Northern California section of the MAA, and serves on the mathematics department advisory board at San Francisco State University. David is joined by Theodore Lee, professor of mathematics at City College of San Francisco. Ted is a highly respected teacher who brings nearly 30 years of teaching experience to this project and provides a valuable perspective on teaching precalculus mathematics. On three separate occasions he has been honored with distinguished teaching awards from his colleagues at CCSF. Ted has also been honored by Alpha Sigma Gamma, CCSF’s student honor society, as a favorite teacher. A fourth-generation Californian, Ted received his bachelor’s degree from the University of California, Berkeley, and his master’s degree from the University of California, Los Angeles. Both David and Ted understand the factors that make the book so special to the people who use it—the clear writing, the conversational style, the variety of problems (including many challenging ones), and the thoughtful use of technology. They have made every effort to maintain the standard and quality of David Cohen’s work.
vi
Contents 4.5
1
Fundamentals 1.1 1.2 1.3 1.4 1.5 1.6 1.7
2
1
Sets of Real Numbers 1 Absolute Value 6 Solving Equations (Review and Preview) 10 Rectangular Coordinates. Visualizing Data 18 Graphs and Graphing Utilities 32 Equations of Lines 43 Symmetry and Graphs. Circles 57
Equations and Inequalities 2.1 2.2 2.3 2.4
3
Quadratic Equations: Theory and Examples 80 Other Types of Equations 91 Inequalities 103 More on Inequalities 113
Functions 3.1 3.2 3.3 3.4 3.5 3.6
4 4.1 4.2 4.3 4.4
4.6 4.7
5
5.6 5.7
6
Exponential Functions 327 The Exponential Function y e x 336 Logarithmic Functions 347 Properties of Logarithms 361 Equations and Inequalities with Logs and Exponents 371 Compound Interest 382 Exponential Growth and Decay 394
The Trigonometric Functions 423 6.1 6.2
129
The Definition of a Function 129 The Graph of a Function 145 Shapes of Graphs. Average Rate of Change 159 Techniques in Graphing 171 Methods of Combining Functions. Iteration 182 Inverse Functions 194
Exponential and Logarithmic Functions 325 5.1 5.2 5.3 5.4 5.5
80
Maximum and Minimum Problems 272 Polynomial Functions 287 Rational Functions 304
6.3 6.4 6.5
7
Radian Measure 423 Trigonometric Functions of Angles 437 Evaluating the Trigonometric Functions 448 Algebra and the Trigonometric Functions 460 Right-Triangle Trigonometry 470
Graphs of the Trigonometric Functions 495
Polynomial and Rational Functions. Applications to Optimization 213
7.1
Linear Functions 213 Quadratic Functions 229 Using Iteration to Model Population Growth (Optional Section) 246 Setting Up Equations That Define Functions 260
7.3
7.2
7.4 7.5
Trigonometric Functions of Real Numbers 496 Graphs of the Sine and Cosine Functions 508 Graphs of y A sin(Bx C) and y A cos(Bx C) 525 Simple Harmonic Motion 541 Graphs of the Tangent and the Reciprocal Functions 547 vii
viii
Contents
8
Analytical Trigonometry 8.1 8.2 8.3 8.4 8.5
563
The Addition Formulas 563 The Double-Angle Formulas 575 The Product-to-Sum and Sum-to-Product Formulas 585 Trigonometric Equations 593 The Inverse Trigonometric Functions 607
12 12.1 12.2 12.3 12.4 12.5 12.6
9
Additional Topics in Trigonometry 633 9.1 9.2 9.3 9.4 9.5 9.6 9.7
12.7 12.8
Right-Triangle Applications 633 The Law of Sines and the Law of Cosines 651 Vectors in the Plane: A Geometric Approach 667 Vectors in the Plane: An Algebraic Approach 677 Parametric Equations 690 Introduction to Polar Coordinates 700 Curves in Polar Coordinates 711
13 13.1 13.2 13.3 13.4 13.5
10 10.1 10.2 10.3 10.4 10.5 10.6 10.7
11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8
Systems of Equations
729
13.6
Systems of Two Linear Equations in Two Unknowns 729 Gaussian Elimination 743 Matrices 757 The Inverse of a Square Matrix 771 Determinants and Cramer’s Rule 784 Nonlinear Systems of Equations 797 Systems of Inequalities 805
The Conic Sections
The Complex Number System 899 Division of Polynomials 906 The Remainder Theorem and the Factor Theorem 913 The Fundamental Theorem of Algebra 922 Rational and Irrational Roots 934 Conjugate Roots and Descartes’s Rule of Signs 942 Introduction to Partial Fractions 949 More About Partial Fractions 956
Additional Topics in Algebra 973 Mathematical Induction 973 The Binomial Theorem 979 Introduction to Sequences and Series 989 Arithmetic Sequences and Series 1000 Geometric Sequences and Series 1010 DeMoivre’s Theorem 1015
Appendix
A.1 A.2 A.3
B
817
The Basic Equations 817 The Parabola 825 Tangents to Parabolas (Optional Section) 839 The Ellipse 841 The Hyperbola 856 The Focus–Directrix Property of Conics 867 The Conics in Polar Coordinates Rotation of Axes 880
A
Roots of Polynomial Equations 898
Significant Digits A-1 Properties of the Real Numbers 12 is Irrational A-7
Appendix B.1 B.2 B.3 B.4 B.5
Index
A-4
A-8
Review of Integer Exponents A-8 Review of nth Roots A-14 Review of Rational Exponents A-21 Review of Factoring A-25 Review of Fractional Expressions A-31
Answers 875
A-1
A-37 I-1
ILIST OF PROJECTS Section 1.4 1.5 1.6 1.7 2.1 2.2 2.2 2.3 2.4 3.1 3.2 3.4 3.5 3.6 4.1 4.2
4.2 4.4 4.5 4.6 4.7 5.1 5.2 5.3 5.6 5.7 6.3
Title and Page Number Discuss, Compute, Reassess 31 Drawing Conclusions from Visual Evidence 43 Thinking About Slope 57 Thinking About Symmetry 69 Put the Quadratic Equation in Its Place? 90 Flying the Flag 102 Specific or General? Whatever Works! 102 An Inequality for the Garden 112 Wind Power 121 A Prime Function 143 Implicit Functions: Batteries Required! 157 Correcting a Graphing Utility Display 181 A Graphical Approach to Composition of Functions 194 A Frequently Asked Question About Inverses 206 Who Are Better Runners, Men or Women? 228 How Do You Know that the Graph of a Quadratic Function Is Always Symmetric about a Vertical Line? 243 What’s Left in the Tank? 244 Group Work on Functions of Time 271 The Least-Squares Line 284 Finding Some Maximum Values Without Using Calculus 302 Finding Some Minimum Values Without Using Calculus 317 Using Differences to Compare Exponential and Polynomial Growth 336 Coffee Temperature 346 More Coffee 361 Loan Payments 392 A Variable Growth Constant? 417 A Linear Approximation for the Sine Function 457
Section 6.4 6.5 6.5 7.2 7.3 7.4 8.1 8.3 8.4 8.5 9.1 9.3 9.4 9.5 10.1 10.2 10.3 10.4 11.2 11.2 11.4 11.5 12.1 12.4 12.7 12.8 13.3 13.4
Title and Page Number Identities and Graphs 470 Transits of Venus and the Scale of the Solar System 482 Constructing a Regular Pentagon 486 Making Waves 523 Fourier Series 537 The Motion of a Piston 546 The Design of a Fresnel Lens 572 Superposition 591 Astigmatism and Eyeglass Lenses 607 Inverse Secant Functions 621 Snell’s Law and an Ancient Experiment 647 Vector Algebra Using Vector Geometry 675 Lines, Circles, and Ray Tracing with Vectors 685 Parameterizations for Lines and Circles 696 Geometry Workbooks on the Euler Line and the Nine-Point Circle 740 The Leontief Input-Output Model 754 Communications and Matrices 768 The Leontief Model Revisited 782 A Bridge with a Parabolic Arch 838 Constructing a Parabola 838 The Circumference of an Ellipse 855 Using Hyperbolas to Determine a Location 866 A Geometric Interpretation of Complex Roots 906 Two Methods for Solving Certain Cubic Equations 932 Checking a Partial Fraction Decomposition 955 An Unusual Partial Fractions Problem 965 Perspective and Alternate Scenarios for Example 5 999 More on Sums 1005
ix
Preface This text develops the elements of college algebra and trigonometry in a straightforward manner. As in the earlier editions, our goal has been to create a book that is accessible to the student. The presentation is student-oriented in three specific ways. First, we’ve tried to talk to, rather than lecture at, the student. Second, examples are consistently used to introduce, to explain, and to motivate concepts. And third, many of the initial exercises for each section are carefully coordinated with the worked examples in that section.
IAUDIENCE In writing Precalculus with Unit Circle Trigonometry, we have assumed that the students have been exposed to intermediate algebra, but that they have not necessarily mastered that subject. Also, for many college algebra students, there may be a gap of several years between their last mathematics course and the present one. Appendix B consists of review sections for such students, reviewing topics on integer exponents, nth roots, rational exponents, factoring, and fractional expressions. In Chapter 1, many references refer the reader to Appendix B for further practice.
ICURRICULUM REFORM This new edition of Precalculus reflects several of the major themes that have developed in the curriculum reform movement of the past decade. Graphs, visualization of data, and functions are now introduced much earlier and receive greater emphasis. Many sections now contain more examples and exercises involving applications and real-life data. In addition to the Writing Mathematics sections from the previous edition, there are now mini projects or projects at the ends of many sections. These give the students additional opportunities to discuss, explore, learn, and write mathematics, often using real-life data.
ITECHNOLOGY In the following discussion and throughout this text, the term graphing utility refers to either a graphing calculator or a computer with software for graphing and analyzing functions. Over the past decade, all of us in the mathematics teaching community have become increasingly aware of the graphing utility and its potential for making a positive impact on our students’ learning. We are also aware of the limitations of the graphing utility as a sole analysis device. The role of the graphing utility is expanded in this new edition: The existence of the graphing utility is taken for granted and some examples do make use of x
Preface
xi
this technology. However, just as with the previous edition, this remains a text in which the central focus is on mathematics and its applications. If the instructor chooses, the text can be used without reference to the graphing utility, but a scientific calculator will be required since the text no longer includes logarithmic or trigonometric tables. Students already familiar with a graphing utility will, at a minimum, need to read the page in Section 1.5 explaining how to specify the dimensions of a viewing rectangle, since that notation accompanies some figures in the text. Overall, the quality and relevance of the graphing utility exercises is vastly improved over the previous edition. Graphing utility exercises (identified by the symbol G ) are now integrated into the regular exercise sets.
ICHANGES IN THIS EDITION Comments and suggestions from students, instructors, and reviewers have helped us to revise this text in a number of ways that we believe will make the book more useful to the instructor and more accessible to the student. As previously mentioned, graphing utility exercises now appear in virtually every exercise set as well as in the text examples. Some of the major changes occur in the following areas.
Chapter 1 Some of the previous edition’s Chapter 1 review material has been shifted to two appendices in the new edition. Notes to students at appropriate places remind them to consult one of these appendices if the use of one of these topics seems unclear. Section 1.4 introduces graphs and data visualization and incorporates much of the material that appeared in Section 2.1 of the previous edition. Section 1.5 includes an introduction to graphing utilities and material formerly covered in Section 2.2; Sections 1.6 and 1.7 correspond to Sections 2.3 and 2.4 of earlier editions.
Chapter 2 Section 2.1 incorporates much of the material on quadratic equations that appeared in Section 1.4 of the former text, although graphing utilities are used more extensively in the examples and exercises. Section 2.2 covers equations that can be solved using some of the techniques used in solving linear and quadratic equations and includes a discussion of extraneous solutions. Sections 2.3 and 2.4 correspond to Sections 2.5 and 2.6 of the previous edition.
Chapter 3 Section 3.1 includes an expanded introduction to the function concept. Functions are introduced at length using algebraic, verbal, tabular, and graphical forms. Functions as models are introduced and used in examples and exercises. Implicit functions are introduced in the section project. In Section 3.3, material on the average rate of change of a function has been expanded, and there is an increased emphasis on applications. Sections 3.4, 3.5, and 3.6 correspond to Sections 3.3, 3.4, and 3.5 of the previous edition. Chapter 4 Section 4.1 includes additional examples on applications of the regression line. There is also a new discussion on the use of spreadsheets in generating scatter plots and regression lines. In Section 4.2, there is new material on modeling real-life data with quadratic functions. Also, first and second differences are introduced as a tool for determining whether a set of data points may be generated by a linear or quadratic function. Section 4.3 on iteration and population dynamics is now marked “optional,” and some exercises that don’t refer directly to the text
xii
Preface
exposition have been deleted. In Section 4.5, in order to better focus on problem solving rather than algebraic manipulation in the applied max/min problems, the vertex formula x b/2a is introduced as a simple tool for determining the vertex of a parabola. Instructors who require the use of completing the square rather than the vertex formula should find their students well prepared in view of the previous work on that topic in Sections 1.7, 2.1, and 4.2.
Chapter 5 Section 5.2 contains new material explaining in a careful, but studentfriendly, manner exactly why the use of e as the base for an exponential function indeed simplifies matters. In Section 5.3 the Richter magnitude scale is discussed as an example of a logarithmic scale. Historical background motivating the use of the logarithmic scale is provided. The Section 5.6 project explains how loan payments in an amortization schedule are calculated under compound interest. In Section 5.7 the concept of an average relative growth rate is explained in the text, and a followup exercise gives an empirical introduction to the instantaneous relative growth rate. These ideas are used to supply insight into the interpretation and use of the exponential growth constant k.
Chapter 6 In Section 6.3 we have clarified the explanations in the use of the now four-step procedure for evaluating trigonometric functions. Section 6.3 ends with a project on using a basic trigonometric inequality to construct a linear approximation for the sine fuction. A mini project at the end of Section 6.4 uses the graphing utility to help understand the nature of identities and provide a way to visualize them. We now present both right-triangle and trigonometric identity approaches in the solutions of several examples (see Examples 4, 5, and 6 in Section 6.5), and encourage students to use whichever method they prefer. Two projects follow Section 6.5. The first uses data gathered from observations of transits of Venus to calculate the distance from Earth to the Sun. The second project discusses the construction of regular polygons. Chapter 7 The exercises in this chapter make more extensive use of the graphing utility. A project on constructing square waves and other nontrigonometric wave forms follows Section 7.2. This material is used in a project on an introduction to Fourier series that follows Section 7.3. Section 7.3 includes a worked example using a trigonometric function to model weather data. Section 7.4 ends with a project developing a model for the motion of a piston in an internal combustion engine.
Chapter 8 The Section 8.1 project uses an addition formula for the sine function to design a Fresnel lens, and the project concluding Section 8.2 uses an addition formula for the cosine to derive a superposition formula useful in differential equations. The derivation leads to a more general formula for combining waveforms. The project at the end of Section 8.4 applies trigonometric identities and techniques for solving trigonometric equations to a measurement problem in the manufacture of eyeglass lenses. The last project in the chapter provides a careful development of the two versions of the inverse secant function most commonly encountered in first-year calculus. Chapter 9 The project at the end of Section 9.1 uses Snell’s Law to analyze the optical phenomenon of “lifting.” The project at the end of Section 9.3 uses the geometric approach to vectors to develop vector algebra, providing an alternative to the component approach to vector algebra presented in Section 9.4. In Exercise Set
Preface
xiii
9.4, the exercises on the dot product, defined using components, have been rewritten to emphasize the algebra, as opposed to the arithmetic, of the dot product. These exercises are extended to reveal an expression for the dot product in purely geometric terms. The algebraic and geometric approaches to vectors are combined in the Section 9.4 project to develop vector and scalar descriptions of lines and circles in the plane and to use these constructions to analyze an optics problem on ray tracing. This project leads naturally to the project after Section 9.5 that deals with using parametric equations for lines and circles in the plane to develop formulas for an important trigonometric substitution used in calculus.
Chapter 10 In Section 10.1 the work on simple linear systems is applied to supply and demand models in determining market equilibrium. The Section 10.2 project provides an introduction to the Leontief input-output model in economics. Section 10.3 discusses the use of graphing calculators and spreadsheets in computing matrix products. Students can apply this technology in an extended project on the use of matrices in the study of communication networks. Section 10.4 discusses the use of graphing calculators and spreadsheets in computing matrix inverses. Section 10.5 does likewise for determinants. Section 10.4 also includes new examples and exercises on using matrices to code and decode messages. In an extended project, students will solve a Leontief input-output problem involving a seven-sector model for the U.S. economy. Chapter 11 Section 11.2 defines focal length and focal ratio for a parabola in preparation for an example and exercises on radio telescopes and parabolic reflectors. Mini Project 2 at the end of the section discusses the classic string and T-square construction of the parabola. Section 11.4 includes new material on the perihelion and aphelion for an elliptical orbit. The use of hyperbolas in determining a location appears in the Section 11.5 project. Chapter 12 In Section 12.2 long division for polynomials is discussed just prior to synthetic division. Two methods for solving certain types of cubic equations are explained and applied in a project for Section 12.4. This complements the discussion in the text on the history of polynomial equations. Chapter 13 Section 13.3 now has an example using a recursive sequence to model population growth and a project to explore some variations to the model. Section 13.4 concludes with a project on sigma notation that develops some additional algebra for simplifying sums. The Accompanying CD Interactive Video Skillbuilder CD (0-534-40239-9) The Interactive Video Skillbuilder CD-ROM contains video instruction covering each chapter of the text. The problems worked during each video lesson are shown first, so that the students can try working them before watching the solution. To help students evaluate their progress, each section contains a 10-question Web quiz (the quiz results can be e-mailed to the instructor) and each chapter contains a chapter test, with the answer to each problem on each test. This CD-ROM also includes MathCue tutorial and answers with step-by-step explanations, a Quiz function that enables students to generate quiz problems keyed to problem types from each section of the book, a Chapter Test that provides many problems keyed to problem types from each chapter, and a Solution Finder that allows students to enter their own basic problems and receive step-by-step graphing calculator tutorial for precalculus and college algebra, featuring examples, exercises, and video tuto-
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rials. Also new, English/Spanish closed-caption translations can be selected to display along with the video instruction.
ISUPPLEMENTARY MATERIALS For the Instructor Instructor’s Edition This special version of the complete student text contains a Resource Integration Guide, an easy-to-use tool that helps you quickly compile a teaching and learning program that complements both the text and your personal teaching style. A complete set of answers is printed in the back of the text. ISBN 0-534-40231-3 Test Bank The Test Bank includes multiple tests per chapter as well as final exams. The tests are made up of a combination of multiple-choice, free-response, true/false, and fill-in-the-blank questions. ISBN 0-534-40238-0 Complete Solutions Manual The Complete Solutions Manual provides workedout solutions to all of the problems in the text. ISBN 0-534-40234-8 Text-Specific Videotapes These text-specific videotape sets, available at no charge to qualified adopters of the text, feature 10- to 20-minute problem-solving lessons that cover each section of every chapter. ISBN 0-534-40219-4 Instructor’s Resource CD-ROM This CD-Rom provides the instructor with dynamic media tools for teaching precalculus. PowerPoint lecture slides, combined with all of the instructor supplements in electronic format, are available on this CD-Rom. ISBN 0-534-40227-5 iLrn™ Instructor Version Efficient and versatile, iLrn gives you the power to transform the teaching and learning experience. iLrn Instructor Version is made up of two components, iLrn Testing and iLrn Tutorial. iLrn Testing is an internetready, text-specific testing suite that allows instructors to customize exams and track student progress in an accessible, browser-based format. iLrn offers full algorithmic generation of problems and free-response mathematics. iLrn Tutorial is a textspecific, interactive tutorial software program that is delivered via the web (at http://www.iLrn.com) and is offered in both student and instructor versions. Like iLrn Testing, it is browser-based, making it an intuitive mathematical guide even for students with little technological proficiency. iLrn Tutorial allows students to work with real math notation in real time, providing instant analysis and feedback. The tracking program built into the instructor version of the software enables instructors to carefully monitor student progress. The complete integration of the testing, tutorial, and course management components simplifies your routine tasks. Results flow automatically to your grade book, and you can easily communicate with individuals, sections, or entire courses. ISBN 0-534-28038-2 A personalized study plan can be generated by course-specific diagnostic built into iLrn. With a personalized study plan, students can focus their time where they need it the most, creating a positive learning environment and paving a pathway to success in their mathematics course. Additional features you’ll find when using iLrn: •
Manipulate and format your tests with the rich text format (RTF) conversion tool. RTF conversion allows instructors to open tests in most word processors, such as Microsoft Word, for further formatting and customization.
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iLrn offers more problem types to provide greater variety and more diverse challenges in your tests.
The iLrn Tutorial interface effectively engages students and helps them learn math concepts faster. WebTutor ToolBox for WebCT and Blackboard ISBN 0-534-27488-9 WebCT; ISBN 0-534-27489-7 Blackboard Preloaded with content and available free via access code when packaged with this text, WebTutor ToolBox for WebCT and Blackboard pairs all the content of this text’s rich Book Companion Website with all the sophisticated course management functionality of a WebCT or Blackboard product. You can assign materials (including online quizzes) and have the results flow automatically to your grade book. ToolBox is ready to use as soon as you log on—or, you can customize its preloaded content by uploading images and other resources, adding web links, or creating your own practice materials. Students only have access to student resources on the website. Instructors can enter an access code to reach password-protected Instructor Resources.
For the Student Student Solutions Manual The Student Solutions Manual provides worked-out solutions to the odd-numbered problems in the text. ISBN 0-534-40232-1 Thomson Brooks/Cole Mathematics Website http://mathematics.brookscole.com When a Thomson Brooks/Cole mathematics text is adopted, the instructor and students have access to everything from book-specific resources to newsgroups. It’s a great way to make teaching and learning an interactive and intriguing experience. iLrn™ Tutorial Student Version Free access to this text-specific, interactive, webbased tutorial system is included with the text. iLrn Tutorial Student Version is browser-based, making it an intuitive mathematical guide even for students with little technological proficiency. Simple to use, iLrn Tutorial allows students to work with real math notation in real time, providing instant analysis and feedback. The entire textbook is available in PDF format through iLrn Tutorial, as are sectionspecific video tutorials, unlimited practice problems, and additional student resources such as a glossary, web links, and more. And, when students get stuck on a particular problem or concept, they need only log on to vMentor™, accessed through iLrn Tutorial, where they can talk (using their own computer microphones) to vMentor tutors who will skillfully guide them through the problem using an interactive whiteboard for illustration.
IREVIEW BOARD We have worked diligently with our editor to ensure that we’re providing an updated version of the same Cohen textbook you know and trust. Brooks/Cole asked some of the longest-standing and most loyal users of the text to participate in a review board that compared the parts of this book we wrote to the parts David Cohen wrote. Their feedback has been very positive, and we’re confident that we’ve maintained the quality and approach of David Cohen’s work. We are grateful to the fol-
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lowing review board participants for their contributions and would like to acknowledge them. Donna J. Bailey, Truman State University Satish Bhatnagar, University of Nevada, Las Vegas M. Hilary Davies, University of Alaska, Anchorage Greg Dietrich, Florida Community College at Jacksonville John Gosselin, University of Georgia Johnny A. Johnson, University of Houston Richard Riggs, New Jersey City University Ross Rueger, College of the Sequoias Fred Schifando, Pennsylvania State University Jeffrey S. Snapp, Harvard–Westlake School Thomas J. Walters, University of California, Los Angeles (retired) Sandra Wray-McAfee, University of Michigan, Dearborn Loris I. Zucca, Kingwood College
IACKNOWLEDGMENTS A wonderful team of editors, accuracy checkers, and proofreaders has helped to eliminate many errors from the original manuscript. The remaining errors are those of the authors. Inspired by Donald Knuth, we would like to offer a reward of $5.00 to the first person to inform us of each remaining error. We can be reached through our editor whose e-mail address is
[email protected]. Many students and teachers from both colleges and high schools have made constructive suggestions about the text and exercises, and we thank them for that. We would also like to thank David Cohen’s cousin, Bruce Cohen, for helpful discussions on using technology in the classroom, and Tom Walters for suggesting a project on identities and graphs. We are particularly indebted to Eric Barkan for numerous discussions on the material and his detailed comments on each of the seemingly endless revisions of the applications-oriented projects as well as his help in preparing the manuscript. Special thanks to Ross Rueger who wrote the supplementary manuals and prepared the answer section for the text. Ross worked with David Cohen on many of David’s textbooks and we very much appreciate that he agreed to work with us on this new edition. Thanks to Charles Heuer for his careful work in checking the text and the exercise answers for accuracy. It has been a rare pleasure to work with Martha Emry on the production of the text, and we thank her for her patience and extraordinary ability to create order out of chaos and to keep us on track at all times. To John-Paul Ramin, Janet Hill, Katherine Brayton, Karin Sandberg, Vernon Boes, Leata Holloway, and the staff at Thomson Brooks/Cole, thank you for all your work and help in bringing this manuscript into print. Finally we want to thank David Cohen’s sister, Susan Cohen, and David’s wife, Annie Cohen, for their warm encouragement. Theodore B. Lee David Sklar
CHAPTER
1
1.1 Sets of Real Numbers 1.2 Absolute Value 1.3 Solving Equations (Review and Preview) 1.4 Rectangular Coordinates. Visualizing Data 1.5 Graphs and Graphing Utilities 1.6 Equations of Lines 1.7 Symmetry and Graphs. Circles
Natural numbers have been used since time immemorial; fractions were employed by the ancient Egyptians as early as 1700 B.C.; and the Pythagoreans, in ancient Greece, about 400 B.C., discovered numbers, like 12, which cannot be fractions.—Stefan Drobot in Real Numbers (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1964) What secrets lie hidden in decimals?—Stephan P. Richards in A Number for Your Thoughts (New Providence, N.J.: S. P. Richards, 1982)
Fundamentals Real numbers, equations, graphs—these topics set the stage for our work in precalculus. How much from previous courses should you remember about solving equations? Section 1.3 provides a review of the fundamentals. The rest of the chapter reviews and begins to extend what you’ve learned in previous courses about graphs and graphing. For example, we use graphs to visualize trends in • • • •
Spending by the television networks to broadcast the Olympic Games (Exercise 21, page 27) Internet usage (Exercise 23, pages 27–28) Carbon dioxide levels in the atmosphere (Example 5, page 25) U.S. population growth (Exercises 7 and 8 on page 53)
I1.1
SETS OF REAL NUMBERS
Here, as in your previous mathematics courses, most of the numbers we deal with are real numbers. These are the numbers used in everyday life, in the sciences, in industry, and in business. Perhaps the simplest way to define a real number is this: A real number is any number that can be expressed in decimal form. Some examples of real numbers are 7 1 7.000 . . .2
12 1 1.4142 . . .2
2/3 1 0.6 2
(Recall that the bar above the 6 in the decimal 0.6 indicates that the 6 repeats indefinitely.) Certain sets of real numbers are referred to often enough to be given special names. These are summarized in the box that follows. As you’ve seen in previous courses, the real numbers can be represented as points on a number line, as shown in Figure 1. As indicated in Figure 1, the point associated with the number zero is referred to as the origin. The fundamental fact here is that there is a one-to-one correspondence between the set of real numbers and the set of points on the line. This means that each real number is identified with exactly one point on the line; conversely, with each point on the line we identify exactly one real number. The real number associated with a given point is called the coordinate of the point. As a practical matter, we’re Origin
Figure 1
_4
_3
_2
_1
0
1
2
3
4
1
2
Chapter 1 Fundamentals
PROPERTY SUMMARY Sets of Real Numbers Name
Definition and Comments
Examples
Natural numbers
These are the ordinary counting numbers: 1, 2, 3, and so on.
1, 4, 29, 1066
Integers
These are the natural numbers along with their negatives and zero.
26, 0, 1, 1776
Rational numbers
As the name suggests, these are the real numbers that are ratios of two integers (with nonzero denominators). It can be proved that a real number is rational if and only if its decimal expansion terminates (e.g., 3.15) or repeats (e.g., 2.43).
Irrational numbers
These are the real numbers that are not rational. Section A.3 of the Appendix contains a proof of the fact that the number 12 is irrational. The proof that p is irrational is more difficult. The first person to prove that p is irrational was the Swiss mathematician J. H. Lambert (1728 –1777).
œ„ 2 _1
0
1
π 2
3
Figure 2
4 1 41 2 , 23 , 1.7 1 17 10 2 , 4.3, 4.173 12, 3 12, 3 12, p, 4 p, 4p
usually more interested in relative locations than precise locations on a number line. For instance, since p is approximately 3.1, we show p slightly to the right of 3 in Figure 2. Similarly, since 12 is approximately 1.4, we show 12 slightly less than halfway from 1 to 2 in Figure 2. It is often convenient to use number lines that show reference points other than the integers used in Figure 2. For instance, Figure 3(a) displays a number line with reference points that are multiples of p. In this case it is the integers that we then locate approximately. For example, in Figure 3(b) we show the approximate location of the number 1 on such a line. 1 _2π _π
a
b
b
(b) The closed interval [a, b] contains all real numbers from a to b, including a and b.
Figure 4
2π
3π
_2π _π
0
π
2π
3π
(b)
Two of the most basic relations for real numbers are less than and greater than, symbolized by and , respectively. For ease of reference, we review these and two related symbols in the box on page 3. In general, relationships involving real numbers and any of the four symbols , , , and are called inequalities. One of the simplest uses of inequalities is in defining certain sets of real numbers called intervals. Roughly speaking, any uninterrupted portion of the number line is referred to as an interval. In the definitions that follow, you’ll see notations such as a x b. This means that both of the inequalities a x and x b hold; in other words, the number x is between a and b.
DEFINITION a
π (a)
Figure 3
(a) The open interval (a, b) contains all real numbers from a to b, excluding a and b.
0
Open Intervals and Closed Intervals
The open interval (a, b) consists of all real numbers x such that a x b. See Figure 4(a). The closed interval [a, b] consists of all real numbers x such that a x b. See Figure 4(b).
1.1 Sets of Real Numbers
3
PROPERTY SUMMARY Notation for Less Than and Greater Than Notation
Definition
Examples
ab
a is less than b. On a number line, oriented as in Figure 1, 2, or 3, the point a lies to the left of b.
2 3; 4 1
ab
a is less than or equal to b.
2 3; 3 3
ba
b is greater than a. On a number line oriented as in Figure 1, 2, or 3, the point b lies to the right of a. (b a is equivalent to a b.)
3 2; 0 1
ba
b is greater than or equal to a.
3 2; 3 3
Note that the brackets in Figure 4(b) are used to indicate that the numbers a and b are included in the interval [a, b], whereas the parentheses in Figure 4(a) indicate that a and b are excluded from the interval (a, b). At times you’ll see notation such as [a, b). This stands for the set of all real numbers x such that a x b. Similarly, (a, b] denotes the set of all numbers x such that a x b.
EXAMPLE
1
Understanding interval notation
Show each interval on a number line, and specify inequalities describing the numbers x in each interval. 31, 24
11, 22
11, 2 4
31, 2 2
SOLUTION See Figure 5.
_1
2
_1
2
_1
2
_1
2
[_1, 2]
(_1, 2)
(_1, 2]
[_1, 2)
_1≤x≤2
_1<x
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