Calculus Early Transcendentals for AP

ROGAWSKI’S

CALCULUS for AP* EARLY TRANSCENDENTALS

SECOND EDITION

Director, BFW High School: Craig Bleyer Executive Editor: Ann Heath Publisher: Ruth Baruth Senior Acquisitions Editor: Terri Ward Development Editor: Tony Palermino Development Editor: Julie Z. Lindstrom, Andrew Sylvester Associate Editor: Katrina Wilhelm Assistant Editor: Dora Figueiredo Editorial Assistant: Tyler Holzer Market Development: Steven Rigolosi Executive Marketing Manager: Cindi Weiss Media Editor: Laura Capuano Assistant Media Editor: Catriona Kaplan Senior Media Acquisitions Editor: Roland Cheyney Photo Editor: Ted Szczepanski Photo Researcher: Julie Tesser Cover and Text Designer: Blake Logan Illustrations: Network Graphics and Techsetters, Inc. Illustration Coordinator: Bill Page Production Coordinator: Paul W. Rohloff Composition: Techsetters, Inc. Printing and Binding: RR Donnelley and Sons

Library of Congress Control Number ISBN-13: 978-1-4292-5074-0 ISBN-10: 1-4292-5074-7 © 2012 by W. H. Freeman and Company All rights reserved Printed in the United States of America First printing

W. H. Freeman and Company, 41 Madison Avenue, New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England www.whfreeman.com

ROGAWSKI’S

CALCULUS for AP* EARLY TRANSCENDENTALS

JON ROGAWSKI University of California, Los Angeles

RAY CANNON Baylor University, TX

W. H. FREEMAN AND COMPANY New York

SECOND EDITION

To Julie and To the AP Teachers

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CONTENTS ROGAWSKI’S CALCULUS for AP* Early Transcendentals Chapter 1 PRECALCULUS REVIEW 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Real Numbers, Functions, and Graphs Linear and Quadratic Functions The Basic Classes of Functions Trigonometric Functions Inverse Functions Exponential and Logarithmic Functions Technology: Calculators and Computers

Chapter 2 LIMITS

1 1 13 21 25 33 43 51

Chapter 4 APPLICATIONS OF THE DERIVATIVE 207 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Linear Approximation and Applications Extreme Values The Mean Value Theorem and Monotonicity The Shape of a Graph L’Hôpital’s Rule Graph Sketching and Asymptotes Applied Optimization Newton’s Method Antiderivatives Preparing for the AP Exam

59 Chapter 5 THE INTEGRAL

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Limits, Rates of Change, and Tangent Lines Limits: A Numerical and Graphical Approach Basic Limit Laws Limits and Continuity Evaluating Limits Algebraically Trigonometric Limits Limits at Infinity Intermediate Value Theorem The Formal Definition of a Limit Preparing for the AP Exam

59 67 77 81 90 95 100 106 110 AP2-1

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Approximating and Computing Area The Definite Integral The Fundamental Theorem of Calculus, Part I The Fundamental Theorem of Calculus, Part II Net Change as the Integral of a Rate Substitution Method Further Transcendental Functions Exponential Growth and Decay Preparing for the AP Exam

Chapter 6 APPLICATIONS OF THE INTEGRAL Chapter 3 DIFFERENTIATION 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

120

Definition of the Derivative 120 The Derivative as a Function 129 Product and Quotient Rules 143 Rates of Change 150 Higher Derivatives 159 Trigonometric Functions 165 The Chain Rule 169 Derivatives of Inverse Functions 178 Derivatives of General Exponential and Logarithmic Functions 182 3.10 Implicit Differentiation 188 3.11 Related Rates 195 Preparing for the AP Exam AP3-1 vi

207 215 226 234 241 248 257 269 275 AP4-1

6.1 6.2 6.3 6.4 6.5

286 299 309 316 322 328 336 341 AP5-1

357

Area Between Two Curves 357 Setting Up Integrals: Volume, Density, Average Value 365 Volumes of Revolution 375 The Method of Cylindrical Shells 384 Work and Energy 391 Preparing for the AP Exam AP6-1

Chapter 7 TECHNIQUES OF INTEGRATION 7.1 7.2 7.3 7.4

286

400

Integration by Parts 400 Trigonometric Integrals 405 Trigonometric Substitution 413 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions 420

C ONTE N TS

7.5 7.6 7.7 7.8

The Method of Partial Fractions Improper Integrals Probability and Integration Numerical Integration Preparing for the AP Exam

Chapter 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 8.1 8.2 8.3 8.4

Arc Length and Surface Area Fluid Pressure and Force Center of Mass Taylor Polynomials Preparing for the AP Exam

426 436 448 454 AP7-1

467 467 474 480 488 AP8-1

Chapter 9 INTRODUCTION TO DIFFERENTIAL EQUATIONS 502 9.1 9.2 9.3 9.4 9.5

Solving Differential Equations Models Involving y  = k(y − b) Graphical and Numerical Methods The Logistic Equation First-Order Linear Equations Preparing for the AP Exam

Chapter 10 INFINITE SERIES 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Sequences Summing an Infinite Series Convergence of Series with Positive Terms Absolute and Conditional Convergence The Ratio and Root Tests Power Series Taylor Series Preparing for the AP Exam

502 511 516 524 528 AP9-1

537 537 548 559 569 575 579 591 AP10-1

CALCULUS

vii

Chapter 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS 607 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Parametric Equations 607 Arc Length and Speed 620 Polar Coordinates 626 Area, Arc Length, and Slope in Polar Coordinates 634 Vectors in the Plane 641 Dot Product and the Angle between Two Vectors 653 Calculus of Vector-Valued Functions 660 Preparing for the AP Exam AP11-1

Chapter 12 DIFFERENTIATION IN SEVERAL VARIABLES 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Functions of Two or More Variables Limits and Continuity in Several Variables Partial Derivatives Differentiability and Tangent Planes The Gradient and Directional Derivatives The Chain Rule Optimization in Several Variables Lagrange Multipliers: Optimizing with a Constraint

672 672 684 692 703 711 723 731 745

APPENDICES A. The Language of Mathematics B. Properties of Real Numbers C. Induction and the Binomial Theorem D. Additional Proofs

A1 A1 A8 A13 A18

ANSWERS TO ODD-NUMBERED EXERCISES

A27

ANSWERS TO THE ODD-NUMBERED PREPARING FOR THE AP EXAM QUESTIONS

A104

REFERENCES

A109

PHOTO CREDITS

A112

INDEX

I1

ABOUT JON ROGAWSKI As a successful teacher for more than 30 years, Jon Rogawski has listened to and learned much from his own students. These valuable lessons have made an impact on his thinking, his writing, and his shaping of a calculus text. Jon Rogawski received his undergraduate and master’s degrees in mathematics simultaneously from Yale University, and he earned his Ph.D. in mathematics from Princeton University, where he studied under Robert Langlands. Before joining the Department of Mathematics at UCLA in 1986, where he is currently a full professor, he held teaching and visiting positions at the Institute for Advanced Study, the University of Bonn, and the University of Paris at Jussieu and at Orsay. Jon’s areas of interest are number theory, automorphic forms, and harmonic analysis on semisimple groups. He has published numerous research articles in leading mathematics journals, including the research monograph Automorphic Representations of Unitary Groups in Three Variables (Princeton University Press). He is the recipient of a Sloan Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of the AMS. Jon and his wife, Julie, a physician in family practice, have four children. They run a busy household and, whenever possible, enjoy family vacations in the mountains of California. Jon is a passionate classical music lover and plays the violin and classical guitar.

ABOUT RAY CANNON Ray Cannon received his B.A. in mathematics from the College of the Holy Cross and his Ph.D. from Tulane University, studying under Gail Young. He served on the faculties of Vanderbilt University, the University of North Carolina-Chapel Hill, and Stetson University before landing at Baylor University, where he has taught since 1980. He has spent time at the University of Michigan as an ONR Postdoctoral Fellow and at the United States Military Academy as a visiting professor. Ray has long been interested in the articulation between high school and college mathematics and has served the AP Calculus program in a variety of ways: as a Reader of the exams, as a Table Leader, as Exam Leader (both AB and BC), and, finall , through four years as Chief Reader. He has also served on the College Board’s Test Development Committee for AP Calculus. Ray is a frequent consultant for the College Board, presenting at workshops and leading week-long summer institutes. Additionally, Ray served on Mathematical Association of America (MAA) committees concerned with the issue of proper placement of students in precalculus and calculus courses. Ray has won numerous awards for his teaching and service, including university-wide teaching awards from the University of North Carolina and Baylor University. He was named a Piper Professor in the state of Texas in 1997 and has twice been given awards by the Southwestern region of the College Board for outstanding contributions to the Advanced Placement Program. Ray and his wife, Jo, have three grown daughters and enjoy traveling together.

PREFACE ABOUT ROGAWSKI’S CALCULUS for AP* by Jon Rogawski On Teaching Mathematics As a young instructor, I enjoyed teaching but I didn’t appreciate how difficul it is to communicate mathematics effectively. Early in my teaching career, I was confronted with a student rebellion when my efforts to explain epsilon-delta proofs were not greeted with the enthusiasm I anticipated. Experiences of this type taught me two basic principles: 1. We should try to teach students as much as possible, but not more. 2. As math teachers, how we say it is as important as what we say. The formal language of mathematics is intimidating to the uninitiated. By presenting concepts in everyday language, which is more familiar but not less precise, we open the way for students to understand the underlying ideas and integrate them into their way of thinking. Students are then in a better position to appreciate the need for formal definition and proofs and to grasp their logic.

On Writing a Calculus Text I began writing Calculus with the goal of creating a text in which exposition, graphics, and layout would work together to enhance all facets of a student’s calculus experience: mastery of basic skills, conceptual understanding, and an appreciation of the wide range of applications. I also wanted students to be aware, early in the course, of the beauty of the subject and the important role it will play, both in their further studies and in their understanding of the wider world. I paid special attention to the following aspects of the text: (a) Clear, accessible exposition that anticipates and addresses student difficulties (b) Layout and figure that communicate the flo of ideas. (c) Highlighted features in the text that emphasize concepts and mathematical reasoning: Conceptual Insight, Graphical Insight, Assumptions Matter, Reminder, and Historical Perspective. (d) A rich collection of examples and exercises of graduated difficult that teach basic skills, problem-solving techniques, reinforce conceptual understanding, and motivate calculus through interesting applications. Each section also contains exercises that develop additional insights and challenge students to further develop their skills. Encouraged by the enthusiastic response to the First Edition, I approached the new edition with the aim of further developing these strengths. Every section of text was carefully revised. During the revision process, I paid particular attention to feedback from adopters, reviewers, and students who have used the book. Their insights and creative suggestions brought numerous improvements to the text. Calculus has a deservedly central role in higher education. It is not only the key to the full range of quantitative disciplines; it is also a crucial component in a student’s intellectual development. I hope this new edition will continue to play a role in opening up for students the multifaceted world of calculus. My textbook follows a largely traditional organization, with a few exceptions. One such exception is the placement of Taylor polynomials in Chapter 8. ix

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P R EF A C E

Placement of Taylor Polynomials Taylor polynomials appear in Chapter 8, before infinit series in Chapter 10. My goal is to present Taylor polynomials as a natural extension of the linear approximation. When I teach infinit series, the primary focus is on convergence, a topic that many students fin challenging. After studying the basic convergence tests and convergence of power series, students are ready to tackle the issues involved in representing a function by its Taylor series. They can then rely on their previous work with Taylor polynomials and the Error Bound from Chapter 8. However, the section on Taylor polynomials is designed so that you can cover it together with the material on power series and Taylor series in Chapter 10 if you prefer this order.

CAREFUL, PRECISE DEVELOPMENT W. H. Freeman is committed to high quality and precise textbooks and supplements. From this project’s inception and throughout its development and production, quality and precision have been given significan priority. We have in place unparalleled procedures to ensure the accuracy of all facets of the text: • • • • •

Exercises and Examples Exposition Figures Editing Composition

Together, these procedures far exceed prior industry standards to safeguard the quality and precision of a calculus textbook.

New to the Second Edition The new edition of Rogawski’s Calculus for AP* builds on the strengths of the bestselling First Edition by incorporating the author’s own classroom experience, as well as extensive feedback from many in the mathematics community, including adopters, nonusers, reviewers, and students. Every section has been carefully revised in order to further polish a text that has been enthusiastically recognized for its meticulous pedagogy and its careful balance among the fundamental pillars of calculus instruction: conceptual understanding, skill development, problem solving, and innovative real-world applications. Enhanced Exercise Sets—with Approximately 25% New and Revised Problems The Second Edition features thousands of new and updated problems. Exercise sets were meticulously reviewed by users and nonusers to assist the author as he revised this cornerstone feature of the text. Rogawski carefully evaluated and rewrote exercise sets as needed to further refin quality, pacing, coverage, and quantity. The Second Edition also includes new AP-style multiple-choice and free-response questions (FRQ’s) written by former College Board AP Calculus Chief Reader, Ray Cannon. These questions, found at the end of each chapter, will help prepare students for the style and structure of questions on the AP exam. New and Larger Variety of Applications To show students how calculus directly relates to the real world, the Second Edition features many fresh and creative examples and exercises centered on innovative, contemporary applications from engineering, the life sciences, physical sciences, business, economics, medicine, and the social sciences. Updated Art Program—with Approximately 15% New Figures Throughout the Second Edition, there are numerous new and updated figure with refine labeling to enhance student understanding. The author takes special care to position the art with the related ex-

Preface

xi

position and provide multiple figure rather than a single one for increased visual support of the concepts. Key Content Changes Rogawski’s Second Edition includes several content changes in response to feedback from users and reviewers. The key revisions include the following: •

•

•

•

•

• • •

The topic “Limits at Infinity” has been moved up from Chapter 4 to Section 2.7 so all types of limits are introduced together (Chapter 2 Limits). Coverage of “Differentials” has been expanded in Section 4.1 and Section 12.4 for those who wish to emphasize differentials in their approach to Linear Approximation. “L’Hôpital’s Rule” has been moved up so this topic can support the section on graph sketching (Chapter 4 Applications of the Derivative). The section on “Numerical Integration” has been moved to the end of the chapter after the techniques of integration are presented (Techniques of Integration chapter). A section on “Probability and Integration” was added to allow students to explore new applications of integration important in the physical sciences as well as in business and the social sciences (Techniques of Integration chapter). A new example addressing the trapezoidal sum has been added to Section 7.8. Lagrange error bound for Taylor polynomials in Section 8.4. A new example covering the derivative of polar coordinates has been added to Section 11.4.

Multivariable Calculus: Recognized as especially strong in Rogawski’s Calculus, the multivariable chapter has been meticulously refine to enhance pedagogy and conceptual clarity. Exercise sets have been improved and rebalanced to fully support basic skill development, as well as conceptual and visual understanding.

xii

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TO THE AP INSTRUCTOR by Ray Cannon Through my fiv years as Chief Reader Designate and Chief Reader, and my time on the Test Development Committee for the AP Examinations, I have come to understand the challenge of preparing students for the AP exam. Enabling students to master the concepts required for a Calculus AB or BC course is only half of the equation. Students must also learn how to approach the AP exam questions, which can be different from the style of questions they’re accustomed to findin in their high school math texts. This text is designed to prepare students for these challenges by providing the relevant content at the proper level and in a clear and accessible manner. AP students are expected to deal with functions presented analytically, graphically, numerically, and verbally. This text uses all four representations throughout. In addition to the text coverage and style, beginning with Chapter 2, each chapter is followed by 20 multiple-choice questions, each with fiv distracters, and four multipart free response questions. These questions have been written in the style of the questions that appear on the AP Calculus exam, and they are designed to present a range of difficult from the routine to the challenging. Some observers may think all questions on the AP exam are challenging, but the test developers must make sure there is a mix of many different levels of difficult . When a chapter deals with BC-only material as well as other material, a “BC” icon next to a question number indicates that it tests a BC-only topic. Questions in chapters whose content is entirely BC will not carry this designation. The AP exam tests students on the cumulative knowledge they have gained in preparation for the rigors of a college-level Calculus course. However, because I have written questions to cover the content found within a given chapter, not all question sets will reflec the breadth of the questions on the AP exam (though questions naturally become more comprehensive in later chapters). The benefi of my approach is that instructors will be able to use these questions immediately to test material they have just covered. By gaining experience with the AP style of questions from the beginning of the course, students should feel more confiden in their ability to do well on the AP exam. The free response questions will help them understand the importance of showing their work and justifying their answers. Similarly, having course-long experience with the multiple-choice questions will equip students with the techniques for efficientl handling distracters and identifying correct answers. Furthermore, though questions are written in the style of AP questions, some questions at the end of a particular chapter may require knowledge of material from that chapter not specificall required by the AP course description. As the course progresses, instructors can assemble more general coverage in mock AP exams by choosing different questions from different chapters. To help AP instructors navigate the text, I have provided an overview for each chapter. These overviews briefl address how the chapter content fit the AP course descriptions. They also point out how topics that are not required per se can be integrated into the teaching of the AP content. Sometimes we work so hard on an individual topic that we lose sight of how it fit into the development of the course. The following overviews will allow instructors and students to develop a feel for the flo of the AP Calculus course. (For more detailed chapter overviews, please see the accompanying Teacher’s Resource Binder.)

Preface

xiii

CHAPTER OVERVIEWS •

Chapter 1: Precalculus Review provides a review of material the students should have encountered in preparation for their AP Calculus course. This chapter provides a useful reference for students as they proceed through the course.

•

Chapter 2: Limits introduces the concept of limit, the central concept that distinguishes a Calculus course from math courses the student may have taken in the past. The concept of limit underlies the ideas of continuity, derivatives, definit integrals, and series. All the material covered in this chapter is required for both the AB and BC exams, with the exception of Section 2.9: The Formal Definitio of a Limit, which is not required on either exam.

•

Chapter 3: Differentiation starts with the definitio of the derivative. The chapter then develops the theorems that allow students to compute derivatives of combinations of the elementary functions quickly, and ends with the closely related topics of implicit differentiation and related rates. All this material, except the hyperbolic functions, is covered on both the AB and BC exams.

•

Chapter 4: Applications of the Derivative presents applications of the derivative starting with a reminder that the tangent line is a local linearization of the function. Section 4.8: Newton’s Method covers material that neither exam requires. Section 4.5: L’Hôpital’s Rule is covered only in the BC course description.

•

Chapter 5: The Integral develops the definitio of the definit integral, leading to both versions of the Fundamental Theorem of Calculus. Section 5.5: Net Change as the Integral of a Rate gives special emphasis to an interpretation of the definit integral that has become central to both AP exams. All of the material in this chapter is found in both course descriptions.

•

Chapter 6: Applications of the Integral presents additional applications of the definit integral. The topics in the firs three sections are common to both course descriptions, except for density, which neither exam requires. The topics addressed in Sections 6.4 and 6.5 are also not demanded by either course description. However, both course descriptions stress that a wide variety of applications should be chosen, with the common theme of the setting up of an approximating Riemann sum and then using the limit to arrive at the definit integral. The text does that here, as well as in the applications presented in Chapter 8.

•

Chapter 7: Techniques of Integration deals with the techniques of antidifferentiation. The particular techniques required by only the BC exam are covered in Sections 1, 2, 3, 5, and 6. (The exam will deal only with denominators with nonrepeating linear factors from Section 5: Partial Fractions.) Sections 4 and 7 are optional. Section 8: Numerical Integration deals with trapezoidal sums as the average of the left-hand and right-hand Riemann sums; this is required for both exams. Simpson’s rule is optional.

•

Chapter 8: FurtherApplications of the Integral and Taylor Polynomials focuses on more applications. Of the applications presented in this chapter, only arc length is required by the BC course description. Section 8.4: Taylor Polynomials is also a required BC topic.

•

Chapter 9: Introduction to Differential Equations presents material that both courses require, although to different degrees. Sections 1 and 2 are common to both courses. Both courses require coverage of slope field found in Section 3, but only the BC course requires Euler’s method. The BC course also requires the logistics equation, presented in Section 4. Section 5: First-Order Linear Equations is optional.

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Chapter 10: Infinite Series is devoted to the BC-only topic of infinit series. Everything in this chapter is required by the exam, except for the Root Test in Section 5.

•

Chapter 11: Parametric Equations, Polar Coordinates, and Vector Functions treats the calculus that is associated with plane curves, whether given parametrically, in polar form, or in terms of vectors. Coverage of vector functions in this chapter focuses exclusively on two dimensions, as required by the exam. This material is BC-only.

•

Chapter 12: Differentiation in Several Variables provides students with an introduction to multivariable calculus. The content in this chapter is not required by either AP course, but its inclusion in the text allows teachers to look beyond the exam and address more advanced calculus topics that their students will face in the future.

•

Rogawski’s Calculus for AP* Teacher’s Resource Binder Lin McMullin, National Math and Science Initiative ET: 1-4292-8629-6 LT: 1-4292-8634-2 An invaluable resource for new and experienced teachers alike, the Teacher’s Resource Binder addresses a variety of approaches to the course with pacing guides, key points, lecture materials, discussion topics, activities and projects, worksheets, AP-style questions, and more. New chapter overviews by Ray Cannon provide a succinct look at each chapter and identify which concepts and topics are most important for students to master in preparation for the AP exam.

•

Teacher’s Resource CD ET: 1-4292-8884-1 LT: 1-4292-8885-X The contents of the Teacher’s Resource Binder, complete solutions, PowerPoint slides, images, and extra material from the book companion site are all included on this searchable CD.

•

Instructor’s Solutions Manual Brian Bradie, Christopher Newport University; Greg Dresden, Washington and Lee University; and Ray Cannon, Baylor University ET: 1-4292-8626-1 LT: 1-4292-8631-8 Complete worked-out solutions to all text exercises are provided to support teachers.

•

Printed Test Bank ET: 1-4292-8627-X LT: 1-4292-8632-6 The comprehensive test bank includes thousands of AP-style multiple-choice questions and short answer problems. Modeled on the types of questions students will see on the AP exam, formats include fiv distracters and questions based on figure or graphs. All questions may also be found in the ExamView® test bank.

•

ExamView® Assessment Suite ET: 1-4292-8625-3 LT: 1-4292-8630-X ExamView Test Generator guides teachers through the process of creating online or paper tests and quizzes quickly and easily. Users may select from our extensive bank of test questions or use the step-by-step tutorial to write their own questions. Tests

SUPPLEMENTS For Instructors

Preface

xv

may be printed in many different types of formats to provide maximum flexibilit or may be administered on-line using the ExamView Player. Results can flo into a number of different course management systems or be recorded and managed in the integrated Test Manager.

For Students

•

Student Solutions Manual Brian Bradie, Christopher Newport University; Greg Dresden, Washington and Lee University; and Ray Cannon, Baylor University ET: 1-4292-8628-8 LT: 1-4292-8633-4 Complete worked-out solutions to all odd-numbered text exercises.

•

Online eBook Both the LT and ET versions of Rogawski’s Calculus for AP* are available in eBook format. The eBook integrates the text with the student media. Each eBook offers a range of customization tools including bookmarking, highlighting, note-taking, and a convenient glossary.

•

Book Companion Site at www.whfreeman.com/rogawskiforAP This site serves as a FREE 24/7 interactive study guide with online quizzing, technology manuals, and other study tools. The password-protected teacher’s side offers a variety of presentation, assessment, and course management resources—including many of the valuable materials from the Teacher’s Resource Binder.

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FEATURES Conceptual Insights encourage students to develop a conceptual understanding of calculus by explaining important ideas clearly but informally.

Leibniz notation is widely used for several reasons. First, it reminds us that the derivative df/dx, although not itself a ratio, is in fact a limit of ratios . Second, the notation specifie the independent variable. This is useful when variables other than x are used. For example, if the independent variable is t, we write df/dt. Third, we often think of d/dx as an “operator” that performs differentiation on functions. In other words, we apply the operator d/dx to f to obtain the derivative df/dx. We will see other advantages of Leibniz notation when we discuss the Chain Rule in Section 3.7. CONCEPTUAL INSIGHT

Ch. 3, p. 130 Graphical Insights enhance students’ visual understanding by making the crucial connections between graphical properties and the underlying concepts.

Keep the graphical interpretation of limits in mind. In Figure 4(A), f (x) approaches L as x → c because for any 0, we can make the gap less than by taking δ sufficient y small. By contrast, the function in Figure 4(B) has a jump discontinuity at x  c. The gap cannot be made small, no matter how small δ is taken. Therefore, the limit does not exist. GRAPHICAL INSIGHT

Ch. 2, p. 114 Reminders are margin notes that link the current discussion to important concepts introduced earlier in the text to give students a quick review and make connections with related ideas.

y

y

y

B  (cos θ, sin θ)

C B

B

tan θ θ O

1

A

θ

x

O

1

1

A

1

Area of triangle  2 sin θ

θ

x

O

1

A

x

1

Area of sector  2 θ

Area of triangle  2 tan θ

FIGURE 5

Proof Assume firs that 0 < θ < the areas in Figure 5: Area of REMINDER Let’s recall why a sector of angle θ in a circle of radius r has area 1 2 r θ . A sector of angle θ represents a 2 θ fraction 2π of the entire circle. The circle has area π r 2 , so the sector has area θ π r 2  12 r 2 θ . In the unit circle 2π (r  1), the sector has area 12 θ .

Note: Our proof of Theorem 3 uses the formula 12 θ for the area of a sector, but this formula is based on the formula πr 2 for the area of a circle, a complete proof of which requires integral calculus.

π 2 . Our proof is based on the following relation between

OAB < area of sector BOA < area of

OAC

2

Let’s compute these three areas. First, OAB has base 1 and height sin θ, so its area is 1 1 2 sin θ. Next, recall that a sector of angle θ has area 2 θ. Finally, to compute the area of OAC, we observe that tan θ  Thus,

AC AC opposite side    AC adjacent side OA 1

OAC has base 1, height tan θ, and area 1 sin θ ≤ 2 Area

OAB

1 2

tan θ. We have shown, therefore, that

1 θ 2 Area of sector

≤

1 sin θ 2 cos θ Area

3

OAC

The firs inequality yields sin θ ≤ θ, and because θ > 0, we obtain sin θ ≤1 θ

4

Ch. 2, p. 97

FEATURES

Caution Notes warn students of common pitfalls they may encounter in understanding the material.

xvii

We make a few remarks before proceeding: CAUTION The Power Rule applies only to the power functions y  x n . It does not apply to exponential functions such as y  2x . The derivative of y  2x is not x2x−1 . We will study the derivatives of exponential functions later in this section.

•

It may be helpful to remember the Power Rule in words: To differentiate x n , “bring down the exponent and subtract one (from the exponent).” d exponent x  (exponent) x exponent−1 dx

•

The Power Rule is valid for all exponents, whether negative, fractional, or irrational: d −3/5 3 x  − x −8/5 , dx 5

d √2 √ √2−1 x  2x dx

Ch. 3, p. 131 Historical Perspectives are brief vignettes that place key discoveries and conceptual advances in their historical context. They give students a glimpse into some of the accomplishments of great mathematicians and an appreciation for their significance

HISTORICAL PERSPECTIVE Philosophy is written in this grand book—I mean the universe— which stands continually open to our gaze, but it cannot be understood unless one firs learns to comprehend the language … in which it is written. It is written in the language of mathematics … —Galileo Galilei, 1623

This statue of Isaac Newton in Cambridge University was described in The Prelude, a poem by William Wordsworth (1770–1850): “Newton with his prism and silent face, The marble index of a mind for ever Voyaging through strange seas of Thought, alone.”

The scientifi revolution of the sixteenth and seventeenth centuries reached its high point in the work of Isaac Newton (1643–1727), who was the firs scientist to show that the physical world, despite its complexity and diversity, is governed by a small number of universal laws. One of Newton’s great insights was that the universal laws are dynamical, describing how the world changes over time in response to forces, rather than how the world actually is at any given moment in time. These laws are expressed best in the language of calculus, which is the mathematics of change.

More than 50 years before the work of Newton, the astronomer Johannes Kepler (1571–1630) discovered his three laws of planetary motion, the most famous of which states that the path of a planet around the sun is an ellipse. Kepler arrived at these laws through a painstaking analysis of astronomical data, but he could not explain why they were true. According to Newton, the motion of any object—planet or pebble—is determined by the forces acting on it. The planets, if left undisturbed, would travel in straight lines. Since their paths are elliptical, some force—in this case, the gravitational force of the sun—must be acting to make them change direction continuously. In his magnum opus Principia Mathematica, published in 1687, Newton proved that Kepler’s laws follow from Newton’s own universal laws of motion and gravity. For these discoveries, Newton gained widespread fame in his lifetime. His fame continued to increase after his death, assuming a nearly mythic dimension and his ideas had a profound inf uence, not only in science but also in the arts and literature, as expressed in the epitaph by British poet Alexander Pope: “Nature and Nature’s Laws lay hid in Night. God said, Let Newton be! and all was Light.”

Ch. 2, p. 60

Assumptions Matter uses short explanations and well-chosen counterexamples to help students appreciate why hypotheses are needed in theorems. Section Summaries summarize a section’s key points in a concise and useful way and emphasize for students what is most important in each section. Section Exercise Sets offer a comprehensive set of exercises closely coordinated with the text. These exercises vary in difficult from routine, to moderate, to more challenging. Also included are icons indicating problems that require the student to give a written response

or require the use of technology

.

Chapter Review Exercises offer a comprehensive set of exercises closely coordinated with the chapter material to provide additional problems for self-study or assignments. Preparing for the AP Exam helps to ready students for the exam by providing AP-style multiple-choice and free response questions that are tied directly to the chapter material. These questions allow students to familiarize themselves with the AP question format from the beginning of the course so that they will be more comfortable and successful when they take the AP exam.

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ACKNOWLEDGMENTS Jon Rogawski and W. H. Freeman and Company are grateful to the many instructors from across the United States and Canada who have offered comments that assisted in the development and refinemen of this book. These contributions included class testing, manuscript reviewing, problems reviewing, and participating in surveys about the book and general course needs.

ALABAMA Tammy Potter, Gadsden State Community College; David Dempsey, Jacksonville State University; Douglas Bailer, Northeast Alabama Community College; Michael Hicks, Shelton State Community College; Patricia C. Eiland, Troy University, Montgomery Campus; James L. Wang, The University of Alabama; Stephen Brick, University of South Alabama; Joerg Feldvoss, University of South Alabama ALASKA Mark A. Fitch, University of Alaska Anchorage; Kamal Narang, University of Alaska Anchorage; Alexei Rybkin, University of Alaska Fairbanks; Martin Getz, University of Alaska Fairbanks ARIZONA Stefania Tracogna, Arizona State University; Bruno Welfert, Arizona State University; Light Bryant, Arizona Western College; Daniel Russow, Arizona Western College; Jennifer Jameson, Coconino College; George Cole, Mesa Community College; David Schultz, Mesa Community College; Michael Bezusko, Pima Community College, Desert Vista Campus; Garry Carpenter, Pima Community College, Northwest Campus; Paul Flasch, Pima County Community College; Jessica Knapp, Pima Community College, Northwest Campus; Roger Werbylo, Pima County Community College; Katie Louchart, Northern Arizona University; Janet McShane, Northern Arizona University; Donna M. Krawczyk, The University of Arizona ARKANSAS Deborah Parker, Arkansas Northeastern College; J. Michael Hall, Arkansas State University; Kevin Cornelius, Ouachita Baptist University; Hyungkoo Mark Park, Southern Arkansas University; Katherine Pinzon, University of Arkansas at Fort Smith; Denise LeGrand, University of Arkansas at Little Rock; John Annulis, University of Arkansas at Monticello; Erin Haller, University of Arkansas, Fayetteville; Daniel J. Arrigo, University of Central Arkansas CALIFORNIA Harvey Greenwald, California Polytechnic State University, San Luis Obispo; Charles Hale, California Polytechnic State University; John M. Alongi, California Polytechnic State University, San Luis Obispo; John Hagen, California Polytechnic State University, San Luis Obispo; Colleen Margarita Kirk, California Polytechnic State University, San Luis Obispo; Lawrence Sze, California Polytechnic State University, San Luis Obispo; Raymond Terry, California Polytechnic State University, San Luis Obispo; James R. McKinney, California State Polytechnic University, Pomona; Robin Wilson, California State Polytechnic University, Pomona; Charles Lam, California State University, Bakersfield ; David McKay, California State University, Long Beach; Melvin Lax, California State University, Long Beach; Wallace A. Etterbeek, California State University, Sacramento; Mohamed Allali, Chapman University; George Rhys, College of the Canyons; Janice Hector, DeAnza College; Isabelle Saber, Glendale Community College; Peter Stathis, Glendale Community College; Douglas B. Lloyd, Golden West College; Thomas Scardina, Golden West College; Kristin Hartford, Long Beach City College; Eduardo Arismendi-Pardi, Orange Coast College; Mitchell Alves, Orange Coast College; Yenkanh Vu, Orange Coast College; Yan Tian, Palomar College; Donna E. Nordstrom, Pasadena City College; Don L. Hancock, Pepperdine University; Kevin Iga, Pepperdine University; Adolfo J. Rumbos, Pomona College; Carlos de la Lama, San Diego City College; Matthias Beck, San Francisco State University; Arek Goetz, San Francisco State University; Nick Bykov, San Joaquin Delta College; Eleanor Lang Kendrick, San Jose City College; Elizabeth Hodes, Santa Barbara City College; William Konya, Santa Monica College; John Kennedy, Santa Monica College; Peter Lee, Santa Monica College; Richard Salome, Scotts Valley High School; Norman Feldman, Sonoma State University; Elaine McDonald, Sonoma State University; John D. Eggers, University of California, San Diego; Bruno Nachtergaele, University of California, Davis; Boumediene Hamzi, University of California, Davis; Richard Leborne, University of California, San Diego; Peter Stevenhagen, University of California, San Diego; Jeffrey Stopple, University of California, Santa Barbara; Guofang Wei, University of California, Santa Barbara; Rick A. Simon, University of La Verne; Mohamad A. Alwash, West Los Angeles College; Calder Daenzer, University of California, Berkeley; Jude Thaddeus Socrates, Pasadena City College; Cheuk Ying Lam, California State University Bakersfield ; Borislava Gutarts, California State University, Los Angeles; Daniel Rogalski, University of California, San Diego; Don Hartig, California Polytechnic State University; Anne Voth, Palomar College; Jay Wiestling, Palomar College; Lindsey Bramlett-Smith, Santa Barbara City College; Dennis Morrow, College of the Canyons; Sydney Shanks, College of the Canyons; Bob Tolar, College of the Canyons; Gene W. Majors, Fullerton College; Robert Diaz, Fullerton College; Gregory Nguyen, Fullerton College; Paul Sjoberg, Fullerton College; Deborah Ritchie, Moorpark College; Maya Rahnamaie, Moorpark College; Kathy Fink, Moorpark College; Christine Cole, Moorpark College; K. Di Passero, Moorpark College; Sid Kolpas, Glendale Community College; Miriam Castrconde, Irvine Valley College;

ACKNOWLEDGMENTS

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Ilkner Erbas-White, Irvine Valley College; Corey Manchester, Grossmont College; Donald Murray, Santa Monica College; Barbara McGee, Cuesta College; Marie Larsen, Cuesta College; Joe Vasta, Cuesta College; Mike Kinter, Cuesta College; Mark Turner, Cuesta College; G. Lewis, Cuesta College; Daniel Kleinfelter, College of the Desert; Esmeralda Medrano, Citrus College; James Swatzel, Citrus College; Mark Littrell, Rio Hondo College; Rich Zucker, Irvine Valley College; Cindy Torigison, Palomar College; Craig Chamberline, Palomar College; Lindsey Lang, Diablo Valley College; Sam Needham, Diablo Valley College; Dan Bach, Diablo Valley College; Ted Nirgiotis, Diablo Valley College; Monte Collazo, Diablo Valley College; Tina Levy, Diablo Valley College; Mona Panchal, East Los Angeles College; Ron Sandvick, San Diego Mesa College; Larry Handa, West Valley College; Frederick Utter, Santa Rose Junior College; Farshod Mosh, DeAnza College; Doli Bambhania, DeAnza College; Charles Klein, DeAnza College; Tammi Marshall, Cauyamaca College; Inwon Leu, Cauyamaca College; Michael Moretti, Bakersfield College; Janet Tarjan, Bakersfield College; Hoat Le, San Diego City College; Richard Fielding, Southwestern College; Shannon Gracey, Southwestern College; Janet Mazzarella, Southwestern College; Christina Soderlund, California Lutheran University; Rudy Gonzalez, Citrus College; Robert Crise, Crafton Hills College; Joseph Kazimir, East Los Angeles College; Randall Rogers, Fullerton College; Peter Bouzar, Golden West College; Linda Ternes, Golden West College; Hsiao-Ling Liu, Los Angeles Trade Tech Community College; Yu-Chung Chang-Hou, Pasadena City College; Guillermo Alvarez, San Diego City College; Ken Kuniyuki, San Diego Mesa College; Laleh Howard, San Diego Mesa College; Sharareh Masooman, Santa Barbara City College; Jared Hersh, Santa Barbara City College; Betty Wong, Santa Monica College; Brian Rodas, Santa Monica College COLORADO Tony Weathers, Adams State College; Erica Johnson, Arapahoe Community College; Karen Walters, Arapahoe Community College; Joshua D. Laison, Colorado College; G. Gustave Greivel, Colorado School of Mines; Jim Thomas, Colorado State University; Eleanor Storey, Front Range Community College; Larry Johnson, Metropolitan State College of Denver; Carol Kuper, Morgan Community College; Larry A. Pontaski, Pueblo Community College; Terry Chen Reeves, Red Rocks Community College; Debra S. Carney, University of Denver; Louis A. Talman, Metropolitan State College of Denver; Mary A. Nelson, University of Colorado at Boulder; J. Kyle Pula, University of Denver; Jon Von Stroh, University of Denver; Sharon Butz, University of Denver; Daniel Daly, University of Denver; Tracy Lawrence, Arapahoe Community College; Shawna Mahan, University of Colorado Denver; Adam Norris, University of Colorado at Boulder; Anca Radulescu, University of Colorado at Boulder; Mike Kawai, University of Colorado Denver; Janet Barnett, Colorado State University–Pueblo; Byron Hurley, Colorado State University–Pueblo; Jonathan Portiz, Colorado State University–Pueblo; Bill Emerson, Metropolitan State College of Denver; Suzanne Caulk, Regis University; Anton Dzhamay, University of Northern Colorado CONNECTICUT Jeffrey McGowan, Central Connecticut State University; Ivan Gotchev, Central Connecticut State University; Charles Waiveris, Central Connecticut State University; Christopher Hammond, Connecticut College; Kim Ward, Eastern Connecticut State University; Joan W. Weiss, Fairfield University; Theresa M. Sandifer, Southern Connecticut State University; Cristian Rios, Trinity College; Melanie Stein, Trinity College; Steven Orszag, Yale University DELAWARE Patrick F. Mwerinde, University of Delaware DISTRICT OF COLUMBIA Jeffrey Hakim, American University; Joshua M. Lansky, American University; James A. Nickerson, Gallaudet University FLORIDA Abbas Zadegan, Florida International University; Gerardo Aladro, Florida International University; Gregory Henderson, Hillsborough Community College; Pam Crawford, Jacksonville University; Penny Morris, Polk Community College; George Schultz, St. Petersburg College; Jimmy Chang, St. Petersburg College; Carolyn Kistner, St. Petersburg College; Aida Kadic-Galeb, The University of Tampa; Constance Schober, University of Central Florida; S. Roy Choudhury, University of Central Florida; Kurt Overhiser, Valencia Community College; Jiongmin Yong, University of Central Florida; Giray Okten, The Florida State University; Frederick Hoffman, Florida Atlantic University; Thomas Beatty, Florida Gulf Coast University; Witny Librun, Palm Beach Community College North; Joe Castillo, Broward County College; Joann Lewin, Edison College; Donald Ransford, Edison College; Scott Berthiaume, Edison College; Alexander Ambrioso, Hillsborough Community College; Jane Golden, Hillsborough Community College; Susan Hiatt, Polk Community College–Lakeland Campus; Li Zhou, Polk Community College–Winter Haven Campus; Heather Edwards, Seminole Community College; Benjamin Landon, Daytona State College; Tony Malaret, Seminole Community College; Lane Vosbury, Seminole Community College; William Rickman, Seminole Community College; Cheryl Cantwell,

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Seminole Community College; Michael Schramm, Indian River State College; Janette Campbell, Palm Beach Community College–Lake Worth GEORGIA Thomas T. Morley, Georgia Institute of Technology; Ralph Wildy, Georgia Military College; Shahram Nazari, Georgia Perimeter College; Alice Eiko Pierce, Georgia Perimeter College, Clarkson Campus; Susan Nelson, Georgia Perimeter College, Clarkson Campus; Laurene Fausett, Georgia Southern University; Scott N. Kersey, Georgia Southern University; Jimmy L. Solomon, Georgia Southern University; Allen G. Fuller, Gordon College; Marwan Zabdawi, Gordon College; Carolyn A. Yackel, Mercer University; Shahryar Heydari, Piedmont College; Dan Kannan, The University of Georgia; Abdelkrim Brania, Morehouse College; Ying Wang, Augusta State University; James M. Benedict, Augusta State University; Kouong Law, Georgia Perimeter College; Rob Williams, Georgia Perimeter College; Alvina Atkinson, Georgia Gwinnett College; Amy Erickson, Georgia Gwinnett College HAWAII Shuguang Li, University of Hawaii at Hilo; Raina B. Ivanova, University of Hawaii at Hilo IDAHO Charles Kerr, Boise State University; Otis Kenny, Boise State University; Alex Feldman, Boise State University; Doug Bullock, Boise State University; Ed Korntved, Northwest Nazarene University ILLINOIS Chris Morin, Blackburn College; Alberto L. Delgado, Bradley University; John Haverhals, Bradley University; Herbert E. Kasube, Bradley University; Marvin Doubet, Lake Forest College; Marvin A. Gordon, Lake Forest Graduate School of Management; Richard J. Maher, Loyola University Chicago; Joseph H. Mayne, Loyola University Chicago; Marian Gidea, Northeastern Illinois University; Miguel Angel Lerma, Northwestern University; Mehmet Dik, Rockford College; Tammy Voepel, Southern Illinois University Edwardsville; Rahim G. Karimpour, Southern Illinois University; Thomas Smith, University of Chicago; Laura DeMarco, University of Illinois; Jennifer McNeilly, University of Illinois at Urbana-Champaign; Manouchehr Azad, Harper College; Minhua Liu, Harper College; Mary Hill, College of DuPage; Arthur N. DiVito, Harold Washington College INDIANA Julie A. Killingbeck, Ball State University; John P. Boardman, Franklin College; Robert N. Talbert, Franklin College; Robin Symonds, Indiana University Kokomo; Henry L. Wyzinski, Indiana University Northwest; Melvin Royer, Indiana Wesleyan University; Gail P. Greene, Indiana Wesleyan University; David L. Finn, Rose-Hulman Institute of Technology IOWA Nasser Dastrange, Buena Vista University; Mark A. Mills, Central College; Karen Ernst, Hawkeye Community College; Richard Mason, Indian Hills Community College; Robert S. Keller, Loras College; Eric Robert Westlund, Luther College; Weimin Han, The University of Iowa KANSAS Timothy W. Flood, Pittsburg State University; Sarah Cook, Washburn University; Kevin E. Charlwood, Washburn University; Conrad Uwe, Cowley County Community College KENTUCKY Alex M. McAllister, Center College; Sandy Spears, Jefferson Community & Technical College; Leanne Faulkner, Kentucky Wesleyan College; Donald O. Clayton, Madisonville Community College; Thomas Riedel, University of Louisville; Manabendra Das, University of Louisville; Lee Larson, University of Louisville; Jens E. Harlander, Western Kentucky University; Philip McCartney, Northern Kentucky University; Andy Long, Northern Kentucky University; Omer Yayenie, Murray State University; Donald Krug, Northern Kentucky University LOUISIANA William Forrest, Baton Rouge Community College; Paul Wayne Britt, Louisiana State University; Galen Turner, Louisiana Tech University; Randall Wills, Southeastern Louisiana University; Kent Neuerburg, Southeastern Louisiana University; Guoli Ding, Louisiana State University; Julia Ledet, Louisiana State University MAINE Andrew Knightly, The University of Maine; Sergey Lvin, The University of Maine; Joel W. Irish, University of Southern Maine; Laurie Woodman, University of Southern Maine; David M. Bradley, The University of Maine; William O. Bray, The University of Maine MARYLAND Leonid Stern, Towson University; Mark E. Williams, University of Maryland Eastern Shore; Austin A. Lobo, Washington College; Supawan Lertskrai, Harford Community College; Fary Sami, Harford Community College; Andrew Bulleri, Howard Community College MASSACHUSETTS Sean McGrath, Algonquin Regional High School; Norton Starr, Amherst College; Renato Mirollo, Boston College; Emma Previato, Boston University; Richard H. Stout, Gordon College; Matthew P. Leingang, Harvard University; Suellen Robinson, North Shore Community College; Walter Stone, North Shore Community College; Barbara Loud, Regis College; Andrew B. Perry, Springfield College; Tawanda Gwena, Tufts University; Gary Simundza, Wentworth Institute of Technology; Mikhail Chkhenkeli, Western New England College; David Daniels, Western New England College; Alan Gorfin Western New England College; Saeed Ghahramani, Western New England College; Julian Fleron, Westfield State College; Brigitte Servatius, Worcester Polytechnic Institute; John Goulet, Worcester Polytechnic Institute; Alexander Martsinkovsky, Northeastern University; Marie Clote, Boston

ACKNOWLEDGMENTS

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College MICHIGAN Mark E. Bollman, Albion College; Jim Chesla, Grand Rapids Community College; Jeanne Wald, Michigan State University; Allan A. Struthers, Michigan Technological University; Debra Pharo, Northwestern Michigan College; Anna Maria Spagnuolo, Oakland University; Diana Faoro, Romeo Senior High School; Andrew Strowe, University of Michigan–Dearborn; Daniel Stephen Drucker, Wayne State University; Christopher Cartwright, Lawrence Technological University; Jay Treiman, Western Michigan University MINNESOTA Bruce Bordwell, Anoka-Ramsey Community College; Robert Dobrow, Carleton College; Jessie K. Lenarz, Concordia College–Moorhead Minnesota; Bill Tomhave, Concordia College; David L. Frank, University of Minnesota; Steven I. Sperber, University of Minnesota; Jeffrey T. McLean, University of St. Thomas; Chehrzad Shakiban, University of St. Thomas; Melissa Loe, University of St. Thomas; Nick Christopher Fiala, St. Cloud State University; Victor Padron, Normandale Community College; Mark Ahrens, Normandale Community College; Gerry Naughton, Century Community College; Carrie Naughton, Inver Hills Community College MISSISSIPPI Vivien G. Miller, Mississippi State University; Ted Dobson, Mississippi State University; Len Miller, Mississippi State University; Tristan Denley, The University of Mississippi MISSOURI Robert Robertson, Drury University; Gregory A. Mitchell, Metropolitan Community College–Penn Valley; Charles N. Curtis, Missouri Southern State University; Vivek Narayanan, Moberly Area Community College; Russell Blyth, Saint Louis University; Blake Thornton, Saint Louis University; Kevin W. Hopkins, Southwest Baptist University; Joe Howe, St. Charles Community College; Wanda Long, St. Charles Community College; Andrew Stephan, St. Charles Community College MONTANA Kelly Cline, Carroll College; Richard C. Swanson, Montana State University; Nikolaus Vonessen, The University of Montana NEBRASKA Edward G. Reinke Jr., Concordia University; Judith Downey, University of Nebraska at Omaha NEVADA Rohan Dalpatadu, University of Nevada, Las Vegas; Paul Aizley, University of Nevada, Las Vegas NEW HAMPSHIRE Richard Jardine, Keene State College; Michael Cullinane, Keene State College; Roberta Kieronski, University of New Hampshire at Manchester; Erik Van Erp, Dartmouth College NEW JERSEY Paul S. Rossi, College of Saint Elizabeth; Mark Galit, Essex County College; Katarzyna Potocka, Ramapo College of New Jersey; Nora S. Thornber, Raritan Valley Community College; Avraham Soffer, Rutgers, The State University of New Jersey; Chengwen Wang, Rutgers, The State University of New Jersey; Stephen J. Greenfield Rutgers, The State University of New Jersey; John T. Saccoman, Seton Hall University; Lawrence E. Levine, Stevens Institute of Technology; Barry Burd, Drew University; Penny Luczak, Camden County College; John Climent, Cecil Community College; Kristyanna Erickson, Cecil Community College; Eric Compton, Brookdale Community College; John Atsu-Swanzy, Atlantic Cape Community College NEW MEXICO Kevin Leith, Central New Mexico Community College; David Blankenbaker, Central New Mexico Community College; Joseph Lakey, New Mexico State University; Kees Onneweer, University of New Mexico; Jurg Bolli, The University of New Mexico NEW YORK Robert C. Williams, Alfred University; Timmy G. Bremer, Broome Community College State University of New York; Joaquin O. Carbonara, Buffalo State College; Robin Sue Sanders, Buffalo State College; Daniel Cunningham, Buffalo State College; Rose Marie Castner, Canisius College; Sharon L. Sullivan, Catawba College; Camil Muscalu, Cornell University; Maria S. Terrell, Cornell University; Margaret Mulligan, Dominican College of Blauvelt; Robert Andersen, Farmingdale State University of New York; Leonard Nissim, Fordham University; Jennifer Roche, Hobart and William Smith Colleges; James E. Carpenter, Iona College; Peter Shenkin, John Jay College of Criminal Justice/CUNY ; Gordon Crandall, LaGuardia Community College/CUNY ; Gilbert Traub, Maritime College, State University of New York; Paul E. Seeburger, Monroe Community College Brighton Campus; Abraham S. Mantell, Nassau Community College; Daniel D. Birmajer, Nazareth College; Sybil G. Shaver, Pace University; Margaret Kiehl, Rensselaer Polytechnic Institute; Carl V. Lutzer, Rochester Institute of Technology; Michael A. Radin, Rochester Institute of Technology; Hossein Shahmohamad, Rochester Institute of Technology; Thomas Rousseau, Siena College; Jason Hofstein, Siena College; Leon E. Gerber, St. Johns University; Christopher Bishop, Stony Brook University; James Fulton, Suffolk County Community College; John G. Michaels, SUNY Brockport; Howard J. Skogman, SUNY Brockport; Cristina Bacuta, SUNY Cortland ; Jean Harper, SUNY Fredonia; Kelly Black, Union College; Thomas W. Cusick, University at Buffalo/The State University of New York; Gino Biondini, University at Buffalo/The State University of New York; Robert Koehler, University at Buffalo/The State University of New York; Robert Thompson, Hunter College; Ed Grossman, The City College of New York NORTH CAROLINA Jeffrey Clark, Elon University; William L. Burgin, Gaston College; Manouchehr H. Misaghian, Johnson C. Smith

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University; Legunchim L. Emmanwori, North Carolina A&T State University; Drew Pasteur, North Carolina State University; Demetrio Labate, North Carolina State University; Mohammad Kazemi, The University of North Carolina at Charlotte; Richard Carmichael, Wake Forest University; Gretchen Wilke Whipple, Warren Wilson College; John Russell Taylor, University of North Carolina at Charlotte; Mark Ellis, Piedmont Community College NORTH DAKOTA Anthony J. Bevelacqua, The University of North Dakota; Richard P. Millspaugh, The University of North Dakota; Thomas Gilsdorf, The University of North Dakota; Michele Iiams, The University of North Dakota OHIO Christopher Butler, Case Western Reserve University; Pamela Pierce, The College of Wooster; Tzu-Yi Alan Yang, Columbus State Community College; Greg S. Goodhart, Columbus State Community College; Kelly C. Stady, Cuyahoga Community College; Brian T. Van Pelt, Cuyahoga Community College; David Robert Ericson, Miami University; Frederick S. Gass, Miami University; Thomas Stacklin, Ohio Dominican University; Vitaly Bergelson, The Ohio State University; Robert Knight, Ohio University; John R. Pather, Ohio University, Eastern Campus; Teresa Contenza, Otterbein College; Ali Hajjafar, The University of Akron; Jianping Zhu, The University of Akron; Ian Clough, University of Cincinnati Clermont College; Atif Abueida, University of Dayton; Judith McCrory, The University at Findlay; Thomas Smotzer, Youngstown State University; Angela Spalsbury, Youngstown State University; James Osterburg, The University of Cincinnati; Frederick Thulin, University of Illinois at Chicago; Weimin Han, The Ohio State University; Critchton Ogle, The Ohio State University; Jackie Miller, The Ohio State University; Walter Mackey, Owens Community College; Jonathan Baker, Columbus State Community College OKLAHOMA Michael McClendon, University of Central Oklahoma; Teri Jo Murphy, The University of Oklahoma; Shirley Pomeranz, University of Tulsa OREGON Lorna TenEyck, Chemeketa Community College; Angela Martinek, Linn-Benton Community College; Tevian Dray, Oregon State University; Mark Ferguson, Chemekata Community College; Andrew Flight, Portland State University PENNSYLVANIA John B. Polhill, Bloomsburg University of Pennsylvania; Russell C. Walker, Carnegie Mellon University; Jon A. Beal, Clarion University of Pennsylvania; Kathleen Kane, Community College of Allegheny County; David A. Santos, Community College of Philadelphia; David S. Richeson, Dickinson College; Christine Marie Cedzo, Gannon University; Monica Pierri-Galvao, Gannon University; John H. Ellison, Grove City College; Gary L. Thompson, Grove City College; Dale McIntyre, Grove City College; Dennis Benchoff, Harrisburg Area Community College; William A. Drumin, King’s College; Denise Reboli, King’s College; Chawne Kimber, Lafeyette College; David L. Johnson, Lehigh University; Zia Uddin, Lock Haven University of Pennsylvania; Donna A. Dietz, Mansfield University of Pennsylvania; Samuel Wilcock, Messiah College; Neena T. Chopra, The Pennsylvania State University; Boris A. Datskovsky, Temple University; Dennis M. DeTurck, University of Pennsylvania; Jacob Burbea, University of Pittsburgh; Mohammed Yahdi, Ursinus College; Timothy Feeman, Villanova University; Douglas Norton, Villanova University; Robert Styer, Villanova University; Peter Brooksbank, Bucknell University; Larry Friesen, Butler County Community College; Lisa Angelo, Bucks County College; Elaine Fitt, Bucks County College; Pauline Chow, Harrisburg Area Community College; Diane Benner, Harrisburg Area Community College; Emily B. Dryden, Bucknell University RHODE ISLAND Thomas F. Banchoff, Brown University; Yajni Warnapala-Yehiya, Roger Williams University; Carol Gibbons, Salve Regina University; Joe Allen, Community College of Rhode Island ; Michael Latina, Community College of Rhode Island SOUTH CAROLINA Stanley O. Perrine, Charleston Southern University; Joan Hoffacker, Clemson University; Constance C. Edwards, Coastal Carolina University; Thomas L. Fitzkee, Francis Marion University; Richard West, Francis Marion University; John Harris, Furman University; Douglas B. Meade, University of South Carolina; George Androulakis, University of South Carolina; Art Mark, University of South Carolina Aiken; Sherry Biggers, Clemson University; Mary Zachary Krohn, Clemson University; Andrew Incognito, Coastal Carolina University; Deanna Caveny, College of Charleston SOUTH DAKOTA Dan Kemp, South Dakota State University TENNESSEE Andrew Miller, Belmont University; Arthur A. Yanushka, Christian Brothers University; Laurie Plunk Dishman, Cumberland University; Beth Long, Pellissippi State Technical Community College; Judith Fethe, Pellissippi State Technical Community College; Andrzej Gutek, Tennessee Technological University; Sabine Le Borne, Tennessee Technological University; Richard Le Borne, Tennessee Technological University; Jim Conant, The University of Tennessee; Pavlos Tzermias, The University of Tennessee; Jo Ann W. Staples, Vanderbilt University; Dave Vinson, Pellissippi State Community College; Jonathan Lamb, Pellissippi State Community College TEXAS Sally

ACKNOWLEDGMENTS

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Haas, Angelina College; Michael Huff, Austin Community College; Scott Wilde, Baylor University and The University of Texas at Arlington; Rob Eby, Blinn College; Tim Sever, Houston Community College–Central; Ernest Lowery, Houston Community College–Northwest; Shirley Davis, South Plains College; Todd M. Steckler, South Texas College; Mary E. Wagner-Krankel, St. Mary’s University; Elise Z. Price, Tarrant County College, Southeast Campus; David Price, Tarrant County College, Southeast Campus; Michael Stecher, Texas A&M University; Philip B. Yasskin, Texas A&M University; Brock Williams, Texas Tech University; I. Wayne Lewis, Texas Tech University; Robert E. Byerly, Texas Tech University; Ellina Grigorieva, Texas Woman’s University; Abraham Haje, Tomball College; Scott Chapman, Trinity University; Elias Y. Deeba, University of Houston Downtown; Jianping Zhu, The University of Texas at Arlington; Tuncay Aktosun, The University of Texas at Arlington; John E. Gilbert, The University of Texas at Austin; Jorge R. Viramontes-Olivias, The University of Texas at El Paso; Melanie Ledwig, The Victoria College; Gary L. Walls, West Texas A&M University; William Heierman, Wharton County Junior College; Lisa Rezac, University of St. Thomas; Raymond J. Cannon, Baylor University; Kathryn Flores, McMurry University; Jacqueline A. Jensen, Sam Houston State University; James Galloway, Collin County College; Raja Khoury, Collin County College; Annette Benbow, Tarrant County College–Northwest; Greta Harland, Tarrant County College–Northeast; Doug Smith, Tarrant County College–Northeast; Marcus McGuff, Austin Community College; Clarence McGuff, Austin Community College; Steve Rodi, Austin Community College; Vicki Payne, Austin Community College; Anne Pradera, Austin Community College; Christy Babu, Laredo Community College; Deborah Hewitt, McLennan Community College; W. Duncan, McLennan Community College; Hugh Griffith Mt. San Antonio College UTAH Jason Isaac Preszler, The University of Utah; Ruth Trygstad, Salt Lake City Community College VERMONT David Dorman, Middlebury College; Rachel Repstad, Vermont Technical College VIRGINIA Verne E. Leininger, Bridgewater College; Brian Bradie, Christopher Newport University; Hongwei Chen, Christopher Newport University; John J. Avioli, Christopher Newport University; James H. Martin, Christopher Newport University; Mike Shirazi, Germanna Community College; Ramon A. Mata-Toledo, James Madison University; Adrian Riskin, Mary Baldwin College; Josephine Letts, Ocean Lakes High School; Przemyslaw Bogacki, Old Dominion University; Deborah Denvir, Randolph-Macon Woman’s College; Linda Powers, Virginia Tech; Gregory Dresden, Washington and Lee University; Jacob A. Siehler, Washington and Lee University; Nicholas Hamblet, University of Virginia; Lester Frank Caudill, University of Richmond WASHINGTON Jennifer Laveglia, Bellevue Community College; David Whittaker, Cascadia Community College; Sharon Saxton, Cascadia Community College; Aaron Montgomery, Central Washington University; Patrick Averbeck, Edmonds Community College; Tana Knudson, Heritage University; Kelly Brooks, Pierce College; Shana P. Calaway, Shoreline Community College; Abel Gage, Skagit Valley College; Scott MacDonald, Tacoma Community College; Martha A. Gady, Whitworth College; Wayne L. Neidhardt, Edmonds Community College; Simrat Ghuman, Bellevue College; Jeff Eldridge, Edmonds Community College; Kris Kissel, Green River Community College; Laura Moore-Mueller, Green River Community College; David Stacy, Bellevue College; Eric Schultz, Walla Walla Community College; Julianne Sachs, Walla Walla Community College WEST VIRGINIA Ralph Oberste-Vorth, Marshall University; Suda Kunyosying, Shepard University; Nicholas Martin, Shepherd University; Rajeev Rajaram, Shepherd University; Xiaohong Zhang, West Virginia State University; Sam B. Nadler, West Virginia University WISCONSIN Paul Bankston, Marquette University; Jane Nichols, Milwaukee School of Engineering; Yvonne Yaz, Milwaukee School of Engineering; Terry Nyman, University of Wisconsin–Fox Valley; Robert L. Wilson, University of Wisconsin–Madison; Dietrich A. Uhlenbrock, University of Wisconsin–Madison; Paul Milewski, University of Wisconsin–Madison; Donald Solomon, University of Wisconsin–Milwaukee; Kandasamy Muthuvel, University of Wisconsin–Oshkosh; Sheryl Wills, University of Wisconsin–Platteville; Kathy A. Tomlinson, University of Wisconsin–River Falls; Joy Becker, University of Wisconsin-Stout; Jeganathan Sriskandarajah , Madison Area Tech College; Wayne Sigelko, Madison Area Tech College WYOMING Claudia Stewart, Casper College; Pete Wildman, Casper College; Charles Newberg, Western Wyoming Community College; Lynne Ipina, University of Wyoming; John Spitler, University of Wyoming CANADA Don St. Jean, George Brown College; Len Bos, University of Calgary; Tony Ware, University of Calgary; Peter David Papez, University of Calgary; John O’Conner, Grant MacEwan University; Michael P. Lamoureux, University of Calgary; Yousry Elsabrouty, University of Calgary; Douglas Farenick, University of Regina

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AP REVIEWER PANEL Bill Compton, Montgomery Bell Academy (TN); Ben Cornelius, Oregon Institute of Technology; Ruth Dover, Illinois Mathematics and Science Academy; Leah Evans, Loveland High School (OH); Guy Mauldin, Science Hill High School (TN); Brendan Murphy, John Bapst Memorial High School (ME); Theresa Reardon Offerman, Mounds Park Academy (MN); Dixie Ross, Pflugerville High School (TX); Dr. Norma P. Royster, AP Calculus Consultant; Mary Anne Valch, Stanford University (CA)

It is a pleasant task to thank the many people whose guidance and support were crucial in bringing this new edition to fruition. I was fortunate that Tony Palermino continued on as my developmental editor. I am happy to thank him again for the wisdom and dedication he brought to the job, and for improvements too numerous to detail. I wish to thank the many mathematicians who generously shared valuable insights, constructive criticism, and innovative exercises. I am particularly grateful to Professors Elka Block, Brian Bradie, C. K. Cheung, Greg Dresden, Stephen Greenfield John Kennedy, Frank Purcell, and Jude Socrates, and to Frances Hammock, Don Larson, Nikki Meshkat, and Jane Sherman for invaluable assistance. I would also like to thank Ricardo Chavez and Professors Elena Galaktionova, Istvan Kovacs, and Jiri Lebl for helpful and perceptive comments. Warmest thanks go to Terri Ward for managing the Second Edition with great skill and grace, and to Julie Lindstrom for overseeing the revision process. I remain indebted to Craig Bleyer for signing this project and standing behind it through the years. I am grateful to Ruth Baruth for bringing her vast knowledge and publishing experience to the project, to Steve Rigolosi for expert market development, and to Katrina Wilhelm for editorial assistance. My thanks are also due to W. H. Freeman’s superb production team: Blake Logan, Bill Page, Paul Rohloff, Ted Szczepanski, and Vivien Weiss, as well as to John Rogosich and Carol Sawyer at Techsetters, Inc. for their expert composition, and to Ron Weickart at Network Graphics for his skilled and creative execution of the art program. To my dearest wife, Julie, I owe more than I can say. Thank you for everything. To our wonderful children Rivkah, Dvora, Hannah, and Akiva, thank you for putting up with the calculus book through all these years. And to my mother Elise, and my late father Alexander Rogawski, MD l¹z, thank you for your love and support from the beginning.

TO THE AP STUDENT

xxv

TO THE AP STUDENT Although I have taught calculus for more than 30 years, when I enter the classroom on the firs day of a new semester, I always have a feeling of excitement, as if a great drama is about to unfold. Does the word drama seem out of place in a discussion of mathematics? Most people would agree that calculus is useful—it is applied across the sciences and engineering to everything from space fligh and weather prediction to nanotechnology and financia modeling. But what is dramatic about it? For me, one part of the drama lies in the conceptual and logical development of calculus. Calculus is based on just a few fundamental concepts (such as limits, tangent lines, and approximations). But as the subject develops, we fin that these concepts are adequate to build, step-by-step, a mathematical discipline capable of solving innumerable problems of great practical importance. Along the way, there are high points and moments of suspense—for example, computing a derivative using limits for the firs time or learning from the Fundamental Theorem of Calculus that the two branches of calculus (differential and integral) are much more closely related than we might have expected. We also discover that calculus provides the right language for expressing our most fundamental and universal laws of nature, not just Newton’s laws of motion, but also the laws of electromagnetism and even the quantum laws of atomic structure. Another part of the drama is the learning process itself—the personal voyage of discovery. Certainly, one aspect of learning calculus is developing various technical skills. You will learn how to compute derivatives and integrals, solve optimization problems, and so on. These skills are necessary for applying calculus in practical situations, and they provide a foundation for further study of more advanced branches of mathematics. But perhaps more importantly, you will become acquainted with the fundamental ideas on which calculus is based. These ideas are central in the sciences and in all quantitative disciplines, and so they will open up for you a world of new opportunities. The distinguished mathematician I. M. Gelfand put it this way: “The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level.” This text is designed to develop both skills and conceptual understanding. In fact, the two go hand in hand. As you become proficien in problem solving, you will come to appreciate the underlying ideas. And it is equally true that a solid understanding of the concepts will make you a more effective problem solver. You are likely to devote much of your time to studying the examples in the text and working the exercises. However, the text also contains numerous down-to-earth explanations of the underlying concepts, ideas, and motivations (sometimes under the heading “Conceptual Insight” or “Graphical Insight”). I urge you to take the time to read these explanations and think about them. Learning calculus will always be a challenge, and it will always require effort. According to legend, Alexander the Great once asked the mathematician Menaechmus to show him an easy way to learn geometry. Menaechmus replied, “There is no royal road to geometry.” Even kings must work hard to learn geometry, and the same is true of calculus. One of the main challenges in writing this textbook was findin a way to present calculus as clearly as possible, in a style that students would fin comprehensible and interesting. While writing, I continually asked myself: Can it be made simpler? Have I assumed something the student may not be aware of? Can I explain the deeper significanc of an underlying concept without confusing a student who is learning the subject for the firs time? I hope my efforts have resulted in a textbook that is not only student friendly but also encourages you to see the big picture—the beautiful and elegant ideas that hold the entire structure of calculus together. Please let me know if you have any comments or suggestions for improving the text. I look forward to hearing from you. Best wishes and good luck! Jon Rogawski

xxvi

P R EF ACE

A Note from Ray Cannon Welcome to the wonderful world of calculus, one of the greatest constructs of the human mind! Professor Rogawski has provided a road map of this world for you, and your teacher will be your guide on the journey. There is much technical material to master, but there is more to calculus than mastering skills. This text has a large collection of exercises that will help you develop these skills and learn how to apply them, but be sure to pause and enjoy the “Historical Perspectives” and the “Conceptual Insights” also found within the chapters. This course culminates in the AP exam that you will take at the conclusion. It is important to familiarize yourself with the types of questions you will encounter. To help you do so, this text includes AP-style multiple-choice and free response questions at the end of each chapter. In addition to introducing you to the level of understanding you are expected to display on the AP exam, these questions will help you hone your test-taking abilities. The free response questions, in particular, are intended to help you, from the beginning of the course, become accustomed to writing out your complete solutions in a way that someone else can easily follow. Remember that on a written exam, it is not just what you know that matters, but also how well you communicate what you know. TheAP-style multiple-choice and free response questions in this text follow the format you’ll see on the exam of either allowing you to use a calculator or not. It is important to learn that just because you can use a calculator doesn’t always mean you should. When the calculator icon appears next to a question in the text, it means a calculator is allowed; it does not necessarily mean a calculator is required. The text begins with a chapter called Precalculus Review, which provides a succinct review of material that you may fin handy to reference at various times in the course. Familiarize yourself with the concepts covered in that chapter, and keep it handy when you begin your exam preparation. There is hard work ahead, but we all want you to succeed! Ray Cannon

1 PRECALCULUS REVIEW

alculus builds on the foundation of algebra, analytic geometry, and trigonometry. In this chapter, therefore, we review some concepts, facts, and formulas from precalculus that are used throughout the text. In the last section, we discuss ways in which technology can be used to enhance your visual understanding of functions and their properties.

C

1.1 Real Numbers, Functions, and Graphs

Functions are one of our most important tools for analyzing phenomena. Biologists have studied the antler weight of male red deer as a function of age (see p. 6).

We begin with a short discussion of real numbers. This gives us the opportunity to recall some basic properties and standard notation. A real number is a number represented by a decimal or “decimal expansion.” There are three types of decimal expansions: finite, repeating, and infinite but nonrepeating. For example, 1 = 0.142857142857 . . . = 0.142857 7

3 = 0.375, 8

π = 3.141592653589793 . . . The number 38 is represented by a finite decimal, whereas 17 is represented by a repeating or periodic decimal. The bar over 142857 indicates that this sequence repeats indefinitely. The decimal expansion of π is infinite but nonrepeating. The set of all real numbers is denoted by a boldface R. When there is no risk of confusion, we refer to a real number simply as a number. We also use the standard symbol ∈ for the phrase “belongs to.” Thus, a∈R Additional properties of real numbers are discussed in Appendix B.

−2

−1

0

1

2

FIGURE 1 The set of real numbers

represented as a line.

reads

“a belongs to R”

The set of integers is commonly denoted by the letter Z (this choice comes from the German word Zahl, meaning “number”). Thus, Z = {. . . , −2, −1, 0, 1, 2, . . . }. A whole number is a nonnegative integer—that is, one of the numbers 0, 1, 2, . . . . A real number is called rational if it can be represented by a fraction p/q, where p and q are integers with q  = 0. The set of rational √ numbers is denoted Q (for “quotient”). Numbers that are not rational, such as π and 2, are called irrational. We can tell whether a number is rational from its decimal expansion: Rational numbers have finite or repeating decimal expansions, and irrational numbers have infinite, nonrepeating decimal expansions. Furthermore, the decimal expansion of a number is unique, apart from the following exception: Every finite decimal is equal to an infinite decimal in which the digit 9 repeats. For example, 1 = 0.999 . . . ,

3 = 0.375 = 0.374999 . . . , 8

47 = 2.35 = 2.34999 . . . 20

We visualize real numbers as points on a line (Figure 1). For this reason, real numbers are often referred to as points. The point corresponding to 0 is called the origin. 1

2

CHAPTER 1

PRECALCULUS REVIEW

|a| 0

a

FIGURE 2 |a| is the distance from a to the

origin.

The absolute value of a real number a, denoted |a|, is defined by (Figure 2)  a if a ≥ 0 |a| = distance from the origin = −a if a < 0 For example, |1.2| = 1.2 and |−8.35| = 8.35. The absolute value satisfies |a| = |−a|,

|b − a| −2

−1 a 0

1

2

b

FIGURE 3 The distance from a to b is

|b − a|.

|ab| = |a| |b|

The distance between two real numbers a and b is |b − a|, which is the length of the line segment joining a and b (Figure 3). Two real numbers a and b are close to each other if |b − a| is small, and this is the case if their decimal expansions agree to many places. More precisely, if the decimal expansions of a and b agree to k places (to the right of the decimal point), then the distance |b − a| is at most 10−k . Thus, the distance between a = 3.1415 and b = 3.1478 is at most 10−2 because a and b agree to two places. In fact, the distance is exactly |3.1478 − 3.1415| = 0.0063. Beware that |a + b| is not equal to |a| + |b| unless a and b have the same sign or at least one of a and b is zero. If they have opposite signs, cancellation occurs in the sum a + b, and |a + b| < |a| + |b|. For example, |2 + 5| = |2| + |5| but |−2 + 5| = 3, which is less than |−2| + |5| = 7. In any case, |a + b| is never larger than |a| + |b| and this gives us the simple but important triangle inequality: |a + b| ≤ |a| + |b|

1

We use standard notation for intervals. Given real numbers a < b, there are four intervals with endpoints a and b (Figure 4). They all have length b − a but differ according to which endpoints are included.

FIGURE 4 The four intervals with endpoints

a and b.

a b Open interval (a, b) (endpoints excluded)

a b Closed interval [a, b] (endpoints included)

a b Half-open interval [a, b)

a b Half-open interval (a, b]

The closed interval [a, b] is the set of all real numbers x such that a ≤ x ≤ b: [a, b] = {x ∈ R : a ≤ x ≤ b} We usually write this more simply as {x : a ≤ x ≤ b}, it being understood that x belongs to R. The open and half-open intervals are the sets (a, b) = {x : a < x < b} ,    Open interval (endpoints excluded)

[a, b) = {x : a ≤ x < b},   

(a, b] = {x : a < x ≤ b}   

Half-open interval

Half-open interval

The infinite interval (−∞, ∞) is the entire real line R. A half-infinite interval is closed if it contains its finite endpoint and is open otherwise (Figure 5): [a, ∞) = {x : a ≤ x < ∞},

(−∞, b] = {x : −∞ < x ≤ b}

a FIGURE 5 Closed half-infinite intervals.

b [a, ∞)

(−∞, b]

Real Numbers, Functions, and Graphs

S E C T I O N 1.1

|x| < r −r

0

r

3

Open and closed intervals may be described by inequalities. For example, the interval (−r, r) is described by the inequality |x| < r (Figure 6):

FIGURE 6 The interval

|x| < r

(−r, r) = {x : |x| < r}.

⇔

−r < x < r

⇔

x ∈ (−r, r)

2

More generally, for an interval symmetric about the value c (Figure 7), r c−r

|x − c| < r

r c+r

c

FIGURE 7 (a, b) = (c − r, c + r), where

c=

a+b , 2

r=

b−a 2

⇔

c−r <x 4 in terms of intervals.

In Example 2 we use the notation ∪ to denote “union”: The union A ∪ B of sets A and B consists of all elements that belong to either A or B (or to both).

−2 0 14   FIGURE 9 The set S = x : 12 x − 3 > 4 .

Solution It is easier to consider the opposite inequality 12 x − 3 ≤ 4 first. By (2), 1 x − 3 ≤ 4 ⇔ −4 ≤ 1 x − 3 ≤ 4 2 2 1 x≤7 2 −2 ≤ x ≤ 14 −1 ≤

(add 3)

(multiply by 2) 1 Thus, 2 x − 3 ≤ 4 is satisfied when x belongs to [−2, 14]. The set S is the complement, consisting of all numbers x not in [−2, 14]. We can describe S as the union of two intervals: S = (−∞, −2) ∪ (14, ∞) (Figure 9).

Graphing

The term “Cartesian” refers to the French philosopher and mathematician René Descartes (1596–1650), whose Latin name was Cartesius. He is credited (along with Pierre de Fermat) with the invention of analytic geometry. In his great work La Géométrie, Descartes used the letters x, y, z for unknowns and a, b, c for constants, a convention that has been followed ever since.

Graphing is a basic tool in calculus, as it is in algebra and trigonometry. Recall that rectangular (or Cartesian) coordinates in the plane are defined by choosing two perpendicular axes, the x-axis and the y-axis. To a pair of numbers (a, b) we associate the point P located at the intersection of the line perpendicular to the x-axis at a and the line perpendicular to the y-axis at b [Figure 10(A)]. The numbers a and b are the x- and y-coordinates of P . The x-coordinate is sometimes called the “abscissa” and the y-coordinate the “ordinate.” The origin is the point with coordinates (0, 0). The axes divide the plane into four quadrants labeled I–IV, determined by the signs of the coordinates [Figure 10(B)]. For example, quadrant III consists of points (x, y) such that x < 0 and y < 0. The distance d between two points P1 = (x1 , y1 ) and P2 = (x2 , y2 ) is computed using the Pythagorean Theorem. In Figure 11, we see that P1 P2 is the hypotenuse of a right triangle with sides a = |x2 − x1 | and b = |y2 − y1 |. Therefore, d 2 = a 2 + b2 = (x2 − x1 )2 + (y2 − y1 )2 We obtain the distance formula by taking square roots.

4

CHAPTER 1

PRECALCULUS REVIEW

y

y

2 b 1 −2

P = (a, b)

II (−, +)

x

−1

1

2

x

a

−1

III (−, −)

−2

(A)

FIGURE 10 Rectangular coordinate system.

y

y2

| x 2 − x1|

x

x2

x1

(x2 − x1 )2 + (y2 − y1 )2

P2 = (x 2, y2)

FIGURE 11 Distance d is given by the

distance formula. y

Once we have the distance formula, we can derive the equation of a circle of radius r and center (a, b) (Figure 12). A point (x, y) lies on this circle if the distance from (x, y) to (a, b) is r:

(x − a)2 + (y − b)2 = r Squaring both sides, we obtain the standard equation of the circle:

(x, y)

(x − a)2 + (y − b)2 = r 2

r b

(B)

d=

d

| y2 − y1|

IV (+, −)

Distance Formula The distance between P1 = (x1 , y1 ) and P2 = (x2 , y2 ) is equal to

P1 = (x1, y1)

y1

I (+, +)

We now review some definitions and notation concerning functions.

(a, b)

a

x

DEFINITION A function f from a set D to a set Y is a rule that assigns, to each element x in D, a unique element y = f (x) in Y . We write f :D→Y

FIGURE 12 Circle with equation (x − a)2 + (y − b)2 = r 2 .

The set D, called the domain of f , is the set of “allowable inputs.” For x ∈ D, f (x) is called the value of f at x (Figure 13). The range R of f is the subset of Y consisting of all values f (x): R = {y ∈ Y : f (x) = y for some x ∈ D} A function f : D → Y is also called a “map.” The sets D and Y can be arbitrary. For example, we can define a map from the set of living people to the set of whole numbers by mapping each person to his or her year of birth. The range of this map is the set of years in which a living person was born. In multivariable calculus, the domain might be a set of points in three-dimensional space and the range a set of numbers, points, or vectors.

Informally, we think of f as a “machine” that produces an output y for every input x in the domain D (Figure 14).

x

f

f (x)

Domain D Y FIGURE 13 A function assigns an element

f (x) in Y to each x ∈ D.

x input

Machine “f ”

f (x) output

FIGURE 14 Think of f as a “machine” that

takes the input x and produces the output f (x).

S E C T I O N 1.1

Real Numbers, Functions, and Graphs

5

The first part of this text deals with numerical functions f , where both the domain and the range are sets of real numbers. We refer to such a function interchangeably as f or f (x). The letter x is used often to denote the independent variable that can take on any value in the domain D. We write y = f (x) and refer to y as the dependent variable (because its value depends on the choice of x). When f is defined by a formula, its natural domain is the set of√real numbers x for which the formula is meaningful. For example, the function f (x) = 9 − x has domain √ D = {x : x ≤ 9} because 9 − x is defined if 9 − x ≥ 0. Here are some other examples of domains and ranges:

y y = f (x) (a, f (a))

f (a)

Zero of f (x) x a FIGURE 15

c

f (x)

Domain D

Range R

x2 cos x 1 x+1

R R

{y : y ≥ 0} {y : −1 ≤ y ≤ 1}

{x : x  = −1}

{y : y  = 0}

The graph of a function y = f (x) is obtained by plotting the points (a, f (a)) for a in the domain D (Figure 15). If you start at x = a on the x-axis, move up to the graph and then over to the y-axis, you arrive at the value f (a). The absolute value |f (a)| is the distance from the graph to the x-axis. A zero or root of a function f (x) is a number c such that f (c) = 0. The zeros are the values of x where the graph intersects the x-axis. In Chapter 4, we will use calculus to sketch and analyze graphs. At this stage, to sketch a graph by hand, we can make a table of function values, plot the corresponding points (including any zeros), and connect them by a smooth curve. E X A M P L E 3 Find the roots and sketch the graph of f (x) = x 3 − 2x.

Solution First, we solve x 3 − 2x = x(x 2 − 2) = 0. √ The roots of f (x) are x = 0 and x = ± 2. To sketch the graph, we plot the roots and a few values listed in Table 1 and join them by a curve (Figure 16). y 4

TABLE 1 x −2 −1 0 1 2

x 3 − 2x −4 1 0 −1 4

1

−2 −2

−1

−1

1 2

x 2

−4

FIGURE 16 Graph of f (x) = x 3 − 2x.

Functions arising in applications are not always given by formulas. For example, data collected from observation or experiment define functions for which there may be no exact formula. Such functions can be displayed either graphically or by a table of values. Figure 17 and Table 2 display data collected by biologist Julian Huxley (1887–1975) in a study of the antler weight W of male red deer as a function of age t. We will see that many of the tools from calculus can be applied to functions constructed from data in this way.

6

CHAPTER 1

PRECALCULUS REVIEW

Antler weight W (kg)

y

8 7 6 5 4 3 2 1 0

TABLE 2

0

2

4 6 8 Age t (years)

10

12

t (years)

W (kg)

t (years)

W (kg)

1 2 3 4 5 6

0.48 1.59 2.66 3.68 4.35 4.92

7 8 9 10 11 12

5.34 5.62 6.18 6.81 6.21 6.1

1

x

−1

1 (1, −1)

−1

FIGURE 18 Graph of 4y 2 − x 3 = 3. This graph fails the Vertical Line Test, so it is not the graph of a function.

FIGURE 17 Male red deer shed their antlers

every winter and regrow them in the spring. This graph shows average antler weight as a function of age.

(1, 1)

We can graph not just functions but, more generally, any equation relating y and x. Figure 18 shows the graph of the equation 4y 2 − x 3 = 3; it consists of all pairs (x, y) satisfying the equation. This curve is not the graph of a function because some x-values are associated with two y-values. For example, x = 1 is associated with y = ±1. A curve is the graph of a function if and only if it passes the Vertical Line Test; that is, every vertical line x = a intersects the curve in at most one point. We are often interested in whether a function is increasing or decreasing. Roughly speaking, a function f (x) is increasing if its graph goes up as we move to the right and is decreasing if its graph goes down [Figures 19(A) and (B)]. More precisely, we define the notion of increase/decrease on an open interval: Increasing on (a, b) if f (x1 ) < f (x2 ) for all x1 , x2 ∈ (a, b) such that x1 < x2 Decreasing on (a, b) if f (x1 ) > f (x2 ) for all x1 , x2 ∈ (a, b) such that x1 < x2

• •

We say that f (x) is monotonic if it is either increasing or decreasing. In Figure 19(C), the function is not monotonic because it is neither increasing nor decreasing for all x. A function f (x) is called nondecreasing if f (x1 ) ≤ f (x2 ) for x1 < x2 (defined by ≤ rather than a strict inequality 0] and even [f (−x) = f (x)]. Therefore, the graph lies above the x-axis and is symmetric with respect to the y-axis. Furthermore, f (x) is decreasing for x ≥ 0 (because a larger value of x makes the denominator larger). We use this information and a short table of values (Table 3) to sketch the graph (Figure 21). Note that the graph approaches the x-axis as we move to the right or left because f (x) gets smaller as |x| increases. TABLE 3 x

1

0

1

±1

1 2 1 5

±2

y

x2 + 1

1 f (x) =

−2

1 x2 + 1

x

−1

1 FIGURE 21

2

8

PRECALCULUS REVIEW

CHAPTER 1

Two important ways of modifying a graph are translation (or shifting) and scaling. Translation consists of moving the graph horizontally or vertically: DEFINITION Translation (Shifting) Remember that f (x) + c and f (x + c) are different. The graph of y = f (x) + c is a vertical translation and y = f (x + c) a horizontal translation of the graph of y = f (x).

•

•

Vertical translation y = f (x) + c: shifts the graph by |c| units vertically, upward if c > 0 and c units downward if c < 0. Horizontal translation y = f (x + c): shifts the graph by |c| units horizontally, to the right if c < 0 and c units to the left if c > 0.

Figure 22 shows the effect of translating the graph of f (x) = 1/(x 2 + 1) vertically and horizontally. y

y

Shift one unit upward

2 1

−2

x

−1

1

(A) y = f (x) =

−2

2

1 x2 + 1

y

2

Shift one unit 2 to the left

1

1 x

−1

1

(B) y = f (x) + 1 =

−3

2

1 +1 x2 + 1

−2

−1

(C) y = f (x + 1) =

x 1 1 (x + 1)2 + 1

FIGURE 22

E X A M P L E 6 Figure 23(A) is the graph of f (x) = x 2 , and Figure 23(B) is a horizontal

and vertical shift of (A). What is the equation of graph (B)? y

y

−2

4

4

3

3

2

2

1

1

−1

1

2

−1

(A) f (x)

3

x

−2

−1

1

2

3

x

−1 = x2

(B)

FIGURE 23

Solution Graph (B) is obtained by shifting graph (A) one unit to the right and one unit down. We can see this by observing that the point (0, 0) on the graph of f (x) is shifted to (1, −1). Therefore, (B) is the graph of g(x) = (x − 1)2 − 1. y 2

Scaling (also called dilation) consists of compressing or expanding the graph in the vertical or horizontal directions:

y = f (x)

1 x

DEFINITION Scaling •

−2 −4

y = −2 f (x)

FIGURE 24 Negative vertical scale factor

k = −2.

•

Vertical scaling y = kf (x): If k > 1, the graph is expanded vertically by the factor k. If 0 < k < 1, the graph is compressed vertically. When the scale factor k is negative (k < 0), the graph is also reflected across the x-axis (Figure 24). Horizontal scaling y = f (kx): If k > 1, the graph is compressed in the horizontal direction. If 0 < k < 1, the graph is expanded. If k < 0, then the graph is also reflected across the y-axis.

S E C T I O N 1.1

Real Numbers, Functions, and Graphs

9

We refer to the vertical size of a graph as its amplitude. Thus, vertical scaling changes the amplitude by the factor |k|. Remember that kf (x) and f (kx) are different. The graph of y = kf (x) is a vertical scaling, and y = f (kx) a horizontal scaling, of the graph of y = f (x).

E X A M P L E 7 Sketch the graphs of f (x) = sin(π x) and its dilates f (3x) and 3f (x).

Solution The graph of f (x) = sin(π x) is a sine curve with period 2. It completes one cycle over every interval of length 2—see Figure 25(A). •

•

The graph of f (3x) = sin(3π x) is a compressed version of y = f (x), completing three cycles instead of one over intervals of length 2 [Figure 25(B)]. The graph of y = 3f (x) = 3 sin(π x) differs from y = f (x) only in amplitude: It is expanded in the vertical direction by a factor of 3 [Figure 25(C)].

y 3 y

2

y 1

1

1

x 1

2

3

x

4

−1

1

2

3

−1

One cycle

x

4

1

2

3

−1

Three cycles

−2 −3

(A) y = f (x) = sin (πx)

FIGURE 25 Horizontal and vertical scaling

(C) Vertical expansion: y = 3f (x) = 3sin (πx)

(B) Horizontal compression: y = f (3x) = sin (3πx)

of f (x) = sin(πx).

1.1 SUMMARY 

a −a

if a ≥ 0 if a < 0

•

Absolute value: |a| =

•

Triangle inequality: |a + b| ≤ |a| + |b| Four intervals with endpoints a and b:

•

(a, b), •

[a, b],

[a, b),

(a, b]

Writing open and closed intervals using inequalities: (a, b) = {x : |x − c| < r},

[a, b] = {x : |x − c| ≤ r}

where c = 12 (a + b) is the midpoint and r = 21 (b − a) is the radius. •

Distance d between (x1 , y1 ) and (x2 , y2 ):

d = (x2 − x1 )2 + (y2 − y1 )2

•

Equation of circle of radius r with center (a, b): (x − a)2 + (y − b)2 = r 2

•

A zero or root of a function f (x) is a number c such that f (c) = 0.

4

10

CHAPTER 1

PRECALCULUS REVIEW •

Vertical Line Test: A curve in the plane is the graph of a function if and only if each vertical line x = a intersects the curve in at most one point. Increasing: Nondecreasing: Decreasing: Nonincreasing:

•

• • •

f (x1 ) < f (x2 ) if x1 f (x1 ) ≤ f (x2 ) if x1 f (x1 ) > f (x2 ) if x1 f (x1 ) ≥ f (x2 ) if x1

< x2 < x2 < x2 < x2

Even function: f (−x) = f (x) (graph is symmetric about the y-axis). Odd function: f (−x) = −f (x) (graph is symmetric about the origin). Four ways to transform the graph of f (x):

f (x) + c

Shifts graph vertically |c| units (upward if c > 0, downward if c < 0)

f (x + c)

Shifts graph horizontally |c| units (to the right if c < 0, to the left if c > 0)

kf (x)

Scales graph vertically by factor k; if k < 0, graph is reflected across x-axis

f (kx)

Scales graph horizontally by factor k (compresses if k > 1); if k < 0, graph is reflected across y-axis

1.1 EXERCISES Preliminary Questions 1. Give an example of numbers a and b such that a < b and |a| > |b|. 2. Which numbers satisfy |a| = a? Which satisfy |a| = −a? What about |−a| = a? 3. Give an example of numbers a and b such that |a + b| < |a| + |b|. 4. What are the coordinates of the point lying at the intersection of the lines x = 9 and y = −4?

(a) (1, 4)

(b) (−3, 2)

(c) (4, −3)

(d) (−4, −1)

6. What is the radius of the circle with equation (x − 9)2 + (y − 9)2 = 9? 7. The equation f (x) = 5 has a solution if (choose one): (a) 5 belongs to the domain of f . (b) 5 belongs to the range of f . 8. What kind of symmetry does the graph have if f (−x) = −f (x)?

5. In which quadrant do the following points lie?

Exercises 1. Use a calculator to find a rational number r such that |r − π 2 | < 10−4 . 2. Which of (a)–(f) are true for a = −3 and b = 2? (a) a < b

(b) |a| < |b|

(c) ab > 0

(d) 3a < 3b

(e) −4a < −4b

(f)

1 1 < a b

In Exercises 3–8, express the interval in terms of an inequality involving absolute value. 3. [−2, 2]

4. (−4, 4)

5. (0, 4)

6. [−4, 0]

7. [1, 5]

8. (−2, 8)

In Exercises 9–12, write the inequality in the form a < x < b. 9. |x| < 8 11. |2x + 1| < 5

10. |x − 12| < 8 12. |3x − 4| < 2

In Exercises 13–18, express the set of numbers x satisfying the given condition as an interval. 13. |x| < 4

14. |x| ≤ 9

15. |x − 4| < 2

16. |x + 7| < 2

17. |4x − 1| ≤ 8

18. |3x + 5| < 1

In Exercises 19–22, describe the set as a union of finite or infinite intervals. 19. {x : |x − 4| > 2}

20. {x : |2x + 4| > 3}

21. {x : |x 2 − 1| > 2}

22. {x : |x 2 + 2x| > 2}

23. Match (a)–(f) with (i)–(vi). (a) a > 3 1 (c) a − < 5 3 (e) |a − 4| < 3

(b) |a − 5| < (d) |a| > 5 (f) 1 ≤ a ≤ 5

1 3

S E C T I O N 1.1

(i) a lies to the right of 3.

Real Numbers, Functions, and Graphs

39. Determine the domain and range of the function

(ii) a lies between 1 and 7. (iii) The distance from a to 5 is less than 13 . (iv) The distance from a to 3 is at most 2. (v) a is less than 5 units from 13 . (vi) a lies either to the left of −5 or to the right of 5.  x < 0 as an interval. 24. Describe x : x+1

f : {r, s, t, u} → {A, B, C, D, E} defined by f (r) = A, f (s) = B, f (t) = B, f (u) = E. 40. Give an example of a function whose domain D has three elements and whose range R has two elements. Does a function exist whose domain D has two elements and whose range R has three elements? In Exercises 41–48, find the domain and range of the function.

2x − 3.

43. f (x) = x 3

42. g(t) = t 4 √ 44. g(t) = 2 − t

26. Describe the set of real numbers satisfying |x − 3| = |x − 2| + 1 as a half-infinite interval.

45. f (x) = |x|

46. h(s) =

27. Show that if a > b, then b−1 > a −1 , provided that a and b have the same sign. What happens if a > 0 and b < 0?

1 47. f (x) = 2 x

48. g(t) = cos

25. Describe {x : x 2 + 2x < 3} as an interval. Hint: Plot y = x 2 +

28. Which x satisfy both |x − 3| < 2 and |x − 5| < 1? 29. Show that if |a − 5| < 12 and |b − 8| < 12 , then |(a + b) − 13| < 1. Hint: Use the triangle inequality. 30. Suppose that |x − 4| ≤ 1. (a) What is the maximum possible value of |x + 4|? (b) Show that |x 2 − 16| ≤ 9. 31. Suppose that |a − 6| ≤ 2 and |b| ≤ 3. (a) What is the largest possible value of |a + b|? (b) What is the smallest possible value of |a + b|? 32. Prove that |x| − |y| ≤ |x − y|. Hint: Apply the triangle inequality to y and x − y. 33. Express r1 = 0.27 as a fraction. Hint: 100r1 − r1 is an integer. Then express r2 = 0.2666 . . . as a fraction.

41. f (x) = −x

35. The text states: If the decimal expansions of numbers a and b agree to k places, then |a − b| ≤ 10−k . Show that the converse is false: For all k there are numbers a and b whose decimal expansions do not agree at all but |a − b| ≤ 10−k .

1 s 1 t

In Exercises 49–52, determine where f (x) is increasing. 49. f (x) = |x + 1|

50. f (x) = x 3

51. f (x) = x 4

1 52. f (x) = 4 x + x2 + 1

In Exercises 53–58, find the zeros of f (x) and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate. 53. f (x) = x 2 − 4

54. f (x) = 2x 2 − 4

55. f (x) = x 3 − 4x

56. f (x) = x 3

57. f (x) = 2 − x 3

58. f (x) =

1 (x − 1)2 + 1

59. Which of the curves in Figure 26 is the graph of a function? y

y x

34. Represent 1/7 and 4/27 as repeating decimals.

x (A)

(B)

y

36. Plot each pair of points and compute the distance between them: (a) (1, 4) and (3, 2) (b) (2, 1) and (2, 4) (c) (0, 0) and (−2, 3)

11

y x

x

(d) (−3, −3) and (−2, 3) (C)

37. Find the equation of the circle with center (2, 4): (a) with radius r = 3. (b) that passes through (1, −1). 38. Find all points with integer coordinates located at a distance 5 from the origin. Then find all points with integer coordinates located at a distance 5 from (2, 3).

(D) FIGURE 26

60. Determine whether the function is even, odd, or neither. (b) g(t) = t 3 − t 2 (a) f (x) = x 5 1 (c) F (t) = 4 t + t2

12

CHAPTER 1

PRECALCULUS REVIEW

61. Determine whether the function is even, odd, or neither. 1 1 (a) f (t) = 4 − (b) g(t) = 2t − 2−t t + t + 1 t4 − t + 1 (c) G(θ ) = sin θ + cos θ (d) H (θ ) = sin(θ 2 ) 62. Write f (x) = 2x 4 − 5x 3 + 12x 2 − 3x + 4 as the sum of an even and an odd function. 

1−x is an odd function. 63. Show that f (x) = ln 1+x 64. (a) (b) (c) (d)

State whether the function is increasing, decreasing, or neither. Surface area of a sphere as a function of its radius Temperature at a point on the equator as a function of time Price of an airline ticket as a function of the price of oil Pressure of the gas in a piston as a function of volume

In Exercises 65–70, let f (x) be the function shown in Figure 27.

y

y

3 2 1

y

3 2 1

x

−3 −2 −1 −1

1 2 3

x

−3 −2 −1 −1

y = f (x) = |x| + 1

1 2 3

y

y

x

−3 −2 −1 −1 −2 −3

(iii)

1 2 3

(ii) y

3 2 1 1 2 3

x

−3 −2 −1 −1

(i)

3 2 1 −3 −2 −1 −1 −2 −3

3 2 1

3 2 1

x 1 2 3

x

−3 −2 −1 −1 −2 −3

(iv)

1 2 3

(v)

FIGURE 28

65. Find the domain and range of f (x)? 66. Sketch the graphs of f (x + 2) and f (x) + 2.   67. Sketch the graphs of f (2x), f 12 x , and 2f (x). 68. Sketch the graphs of f (−x) and −f (−x). 69. Extend the graph of f (x) to [−4, 4] so that it is an even function.

  75. Sketch the graph of f (2x) and f 12 x , where f (x) = |x| + 1 (Figure 28). 76. Find the function f (x) whose graph is obtained by shifting the parabola y = x 2 three units to the right and four units down, as in Figure 29. y

70. Extend the graph of f (x) to [−4, 4] so that it is an odd function. y

y = x2

4 3 3

2 −4

1 0

x y = f (x)

FIGURE 29

x 1 2 3 FIGURE 27

4

71. Suppose that f (x) has domain [4, 8] and range [2, 6]. Find the domain and range of: (a) f (x) + 3 (b) f (x + 3) (c) f (3x) (d) 3f (x) 72. Let f (x) = x 2 . Sketch the graph over [−2, 2] of: (a) f (x + 1) (b) f (x) + 1 (c) f (5x) (d) 5f (x) 73. Suppose that the graph of f (x) = sin x is compressed horizontally by a factor of 2 and then shifted 5 units to the right. (a) What is the equation for the new graph? (b) What is the equation if you first shift by 5 and then compress by 2? (c) Verify your answers by plotting your equations. 74. Figure 28 shows the graph of f (x) = |x| + 1. Match the functions (a)–(e) with their graphs (i)–(v). (a) f (x − 1) (b) −f (x) (c) −f (x) + 2 (d) f (x − 1) − 2 (e) f (x + 1)

77. Define f (x) to be the larger of x and 2 − x. Sketch the graph of f (x). What are its domain and range? Express f (x) in terms of the absolute value function. 78. For each curve in Figure 30, state whether it is symmetric with respect to the y-axis, the origin, both, or neither. y

y x

x

(A)

(B)

y

y x

(C)

x

(D) FIGURE 30

S E C T I O N 1.2

Linear and Quadratic Functions

13

f (x) g(x)

79. Show that the sum of two even functions is even and the sum of two odd functions is odd.

(c) f (x) − g(x)

80. Suppose that f (x) and g(x) are both odd. Which of the following functions are even? Which are odd? (a) f (x)g(x) (b) f (x)3

81. Prove that the only function whose graph is symmetric with respect to both the y-axis and the origin is the function f (x) = 0.

(d)

Further Insights and Challenges 2 . Note that Use this to find the decimal expansion of r = 11

82. Prove the triangle inequality by adding the two inequalities −|a| ≤ a ≤ |a|,

−|b| ≤ b ≤ |b|

r=

83. Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if b = 2n 5m

for some n, m ≥ 0.

Hint: Observe that r has a finite decimal expansion when 10N r is an integer for some N ≥ 0 (and hence b divides 10N ). 84. Let p = p1 . . . ps be an integer with digits p1 , . . . , ps . Show that p = 0.p1 . . . ps 10s − 1

18 2 = 2 11 10 − 1

A function f (x) is symmetric with respect to the vertical 85. line x = a if f (a − x) = f (a + x). (a) Draw the graph of a function that is symmetric with respect to x = 2. (b) Show that if f (x) is symmetric with respect to x = a, then g(x) = f (x + a) is even. 86. Formulate a condition for f (x) to be symmetric with respect to the point (a, 0) on the x-axis.

1.2 Linear and Quadratic Functions Linear functions are the simplest of all functions, and their graphs (lines) are the simplest of all curves. However, linear functions and lines play an enormously important role in calculus. For this reason, you should be thoroughly familiar with the basic properties of linear functions and the different ways of writing an equation of a line. Let’s recall that a linear function is a function of the form

y y = mx + b y2

f (x) = mx + b

y

y1

x

b

m=

y-intercept

y x x

x1

x2

FIGURE 1 The slope m is the ratio “rise

over run.”

(m and b constants)

The graph of f (x) is a line of slope m, and since f (0) = b, the graph intersects the y-axis at the point (0, b) (Figure 1). The number b is called the y-intercept, and the equation y = mx + b for the line is said to be in slope-intercept form. We use the symbols x and y to denote the change (or increment) in x and y = f (x) over an interval [x1 , x2 ] (Figure 1): x = x2 − x1 ,

y = y2 − y1 = f (x2 ) − f (x1 )

The slope m of a line is equal to the ratio m=

y vertical change rise = = x horizontal change run

This follows from the formula y = mx + b: y y2 − y1 (mx2 + b) − (mx1 + b) m(x2 − x1 ) = = = =m x x2 − x 1 x2 − x 1 x2 − x 1

14

CHAPTER 1

PRECALCULUS REVIEW

The slope m measures the rate of change of y with respect to x. In fact, by writing y = mx we see that a one-unit increase in x (i.e., x = 1) produces an m-unit change y in y. For example, if m = 5, then y increases by five units per unit increase in x. The rate-of-change interpretation of the slope is fundamental in calculus. We discuss it in greater detail in Section 2.1. Graphically, the slope m measures the steepness of the line y = mx + b. Figure 2(A) shows lines through a point of varying slope m. Note the following properties: • • • • •

Steepness: The larger the absolute value |m|, the steeper the line. Negative slope: If m < 0, the line slants downward from left to right. f (x) = mx + b is increasing if m > 0 and decreasing if m < 0. The horizontal line y = b has slope m = 0 [Figure 2(B)]. A vertical line has equation x = c, where c is a constant. The slope of a vertical line is undefined. It is not possible to write the equation of a vertical line in slopeintercept form y = mx + b.

y

y −1

−2 −5

5

−0.5

2

x=c (slope undefined)

1 0.5 0

b

P

P

Profits (in millions)

x

y=b (slope 0)

x c

150

(A) Lines of varying slopes through P

(B) Horizontal and vertical lines through P

FIGURE 2

125

100 2000

2001

2002

Profits (in millions) 300 275 250 225 200 175 150 125 100 2000 2001 2002

2003

2004

CAUTION: Graphs are often plotted using different scales for the x- and y-axes. This is necessary to keep the sizes of graphs within reasonable bounds. However, when the scales are different, lines do not appear with their true slopes.

Scale is especially important in applications because the steepness of a graph depends on the choice of units for the x- and y-axes. We can create very different subjective impressions by changing the scale. Figure 3 shows the growth of company profits over a four-year period. The two plots convey the same information, but the upper plot makes the growth look more dramatic. Next, we recall the relation between the slopes of parallel and perpendicular lines (Figure 4): •

2003

2004

FIGURE 3 Growth of company profits.

•

Lines of slopes m1 and m2 are parallel if and only if m1 = m2 . Lines of slopes m1 and m2 are perpendicular if and only if m1 = −

1 m2

(or m1 m2 = −1).

S E C T I O N 1.2

Linear and Quadratic Functions

15

y

y

1 Slope = − m

Slope = m

Slope = m

Slope = m

x

x (B) Perpendicular lines

(A) Parallel lines

FIGURE 4 Parallel and perpendicular lines.

CONCEPTUAL INSIGHT

The increments over an interval [x1 , x2 ]: x = x2 − x1 ,

y = f (x2 ) − f (x1 )

are defined for any function f (x) (linear or not), but the ratio y/x may depend on the interval (Figure 5). The characteristic property of a linear function f (x) = mx + b is that y/x has the same value m for every interval. In other words, y has a constant rate of change with respect to x. We can use this property to test if two quantities are related by a linear equation.

y

y y

y

x x

x Linear function: the ratio y/x is the same over all intervals.

x Nonlinear function: the ratio y /x changes, depending on the interval.

FIGURE 5

E X A M P L E 1 Testing for a Linear Relationship Do the data in Table 1 suggest a linear relation between the pressure P and temperature T of a gas?

TABLE 1 Temperature (◦ C)

Pressure (kPa)

40 45 55 70 80

1365.80 1385.40 1424.60 1483.40 1522.60

16

PRECALCULUS REVIEW

CHAPTER 1

Solution We calculate P /T at successive data points and check whether this ratio is constant: Real experimental data are unlikely to reveal perfect linearity, even if the data points do essentially lie on a line. The method of “linear regression” is used to find the linear function that best fits the data.

Pressure (kPa)

(T1 , P1 )

P T

(T2 , P2 )

(40, 1365.80)

(45, 1385.40)

1385.40 − 1365.80 = 3.92 45 − 40

(45, 1385.40)

(55, 1424.60)

1424.60 − 1385.40 = 3.92 55 − 45

(55, 1424.60)

(70, 1483.40)

1483.40 − 1424.60 = 3.92 70 − 55

(70, 1483.40)

(80, 1522.60)

1522.60 − 1483.40 = 3.92 80 − 70

Because P /T has the constant value 3.92, the data points lie on a line with slope m = 3.92 (this is confirmed in the plot in Figure 6).

1550 1500 1450

As mentioned above, it is important to be familiar with the standard ways of writing the equation of a line. The general linear equation is

1400 1350 40

60

80

T (°C)

ax + by = c

FIGURE 6 Line through pressure-

1

temperature data points.

where a and b are not both zero. For b = 0, we obtain the vertical line ax = c. When b  = 0, we can rewrite Eq. (1) in slope-intercept form. For example, −6x + 2y = 3 can be rewritten as y = 3x + 32 . Two other forms we will use frequently are the point-slope and point-point forms. Given a point P = (a, b) and a slope m, the equation of the line through P with slope m is y − b = m(x − a). Similarly, the line through two distinct points P = (a1 , b1 ) and Q = (a2 , b2 ) has slope (Figure 7)

y a2 − a1 (a1, b1) b2 − b1

(a2, b2)

m=

b2 − b1 a2 − a 1

x

Therefore, we can write its equation as y − b1 = m(x − a1 ). FIGURE 7 Slope of the line between

P = (a1 , b1 ) and Q = (a2 , b2 ) is b − b1 . m= 2 a2 − a1

Equations for Lines 1. Point-slope form of the line through P = (a, b) with slope m: y − b = m(x − a) 2. Point-point form of the line through P = (a1 , b1 ) and Q = (a2 , b2 ): y − b1 = m(x − a1 )

where m =

b2 − b1 a2 − a 1

S E C T I O N 1.2

y y = − 23 x + 8

Solution In point-slope form: 2 y − 2 = − (x − 9) 3

P = (9, 2) x 9

17

E X A M P L E 2 Line of Given Slope Through a Given Point Find the equation of the line through (9, 2) with slope − 23 .

8

2

Linear and Quadratic Functions

12

FIGURE 8 Line through P = (9, 2) with slope m = − 23 .

In slope-intercept form: y = − 23 (x − 9) + 2 or y = − 23 x + 8. See Figure 8. E X A M P L E 3 Line Through Two Points Find the equation of the line through (2, 1) and (9, 5).

Solution The line has slope m=

5−1 4 = 9−2 7

Because (2, 1) lies on the line, its equation in point-slope form is y − 1 = 47 (x − 2). A quadratic function is a function defined by a quadratic polynomial f (x) = ax 2 + bx + c

(a, b, c, constants with a = 0)

The graph of f (x) is a parabola (Figure 9). The parabola opens upward if the leading coefficient a is positive and downward if a is negative. The discriminant of f (x) is the quantity D = b2 − 4ac The roots of f (x) are given by the quadratic formula (see Exercise 56): Roots of f (x) =

−b ±

√

√ −b ± D b2 − 4ac = 2a 2a

The sign of D determines whether or not f (x) has real roots (Figure 9). If D > 0, then f (x)√ has two real roots, and if D = 0, it has one real root (a “double root”). If D < 0, then D is imaginary and f (x) has no real roots. y

y

y

x

FIGURE 9 Graphs of quadratic functions f (x) = ax 2 + bx + c.

Two real roots a > 0 and D > 0

x

Double root a > 0 and D = 0

y

x

No real roots a > 0 and D < 0

x

Two real roots a < 0 and D > 0

When f (x) has two real roots r1 and r2 , then f (x) factors as f (x) = a(x − r1 )(x − r2 ) For example, f (x) = 2x 2 − 3x + 1 has discriminant D = b2 − 4ac = 9 − 8 = 1 > 0, and by the quadratic formula, its roots are (3 ± 1)/4 or 1 and 12 . Therefore,

 1 2 f (x) = 2x − 3x + 1 = 2(x − 1) x − 2

18

CHAPTER 1

PRECALCULUS REVIEW

The technique of completing the square consists of writing a quadratic polynomial as a multiple of a square plus a constant:  b 2 4ac − b2 ax 2 + bx + c = a x + +  2a   4a   Square term

2

Constant

It is not necessary to memorize this formula, but you should know how to carry out the process of completing the square. Cuneiform texts written on clay tablets show that the method of completing the square was known to ancient Babylonian mathematicians who lived some 4000 years ago.

E X A M P L E 4 Completing the Square Complete the square for the quadratic polynomial 4x 2 − 12x + 3.

Solution First factor out the leading coefficient:

 3 4x 2 − 12x + 3 = 4 x 2 − 3x + 4 Then complete the square for the term x 2 − 3x:

  b 2 b2 3 2 9 2 2 x + bx = x + − , − x − 3x = x − 2 4 2 4 Therefore,

 4x − 12x + 3 = 4 2

3 x− 2

2

9 3 − + 4 4



3 =4 x− 2

2 −6

The method of completing the square can be used to find the minimum or maximum value of a quadratic function. y

E X A M P L E 5 Finding the Minimum of a Quadratic Function Complete the square and find the minimum value of f (x) = x 2 − 4x + 9.

Solution We have

9

This term is ≥ 0

f (x) = x − 4x + 9 = (x − 2) − 4 + 9 = 2

5

2

   (x − 2)2 + 5

Thus, f (x) ≥ 5 for all x, and the minimum value of f (x) is f (2) = 5 (Figure 10). x 2 FIGURE 10 Graph of f (x) = x 2 − 4x + 9.

1.2 SUMMARY A linear function is a function of the form f (x) = mx + b. The general equation of a line is ax + by = c. The line y = c is horizontal and x = c is vertical. • Three convenient ways of writing the equation of a nonvertical line: • •

– Slope-intercept form: y = mx + b (slope m and y-intercept b) – Point-slope form: y − b = m(x − a) [slope m, passes through (a, b)] – Point-point form: The line through two points P = (a1 , b1 ) and Q = (a2 , b2 ) has b2 − b1 slope m = and equation y − b1 = m(x − a1 ). a2 − a 1 Two lines of slopes m1 and m2 are parallel if and only if m1 = m2 , and they are perpendicular if and only if m1 = −1/m2 .

•

Linear and Quadratic Functions

S E C T I O N 1.2

19

√ Quadratic function: f (x) = ax 2 + bx + c. The roots are x = (−b ± D)/(2a), where D = b2 − 4ac is the discriminant. The roots are real and distinct if D > 0, there is a double root if D = 0, and there are no real roots if D < 0. • Completing the square consists of writing a quadratic function as a multiple of a square plus a constant. •

1.2 EXERCISES Preliminary Questions 1. What is the slope of the line y = −4x − 9?

4. Suppose y = 3x + 2. What is y if x increases by 3?

2. Are the lines y = 2x + 1 and y = −2x − 4 perpendicular?

5. What is the minimum of f (x) = (x + 3)2 − 4?

3. When is the line ax + by = c parallel to the y-axis? To the x-axis?

6. What is the result of completing the square for f (x) = x 2 + 1?

Exercises In Exercises 1–4, find the slope, the y-intercept, and the x-intercept of the line with the given equation. 1. y = 3x + 12

2. y = 4 − x

3. 4x + 9y = 3

4. y − 3 = 12 (x − 6)

y

Perpendicular bisector (5, 4) Q (1, 2)

In Exercises 5–8, find the slope of the line. 5. y = 3x + 2

6. y = 3(x − 9) + 2

7. 3x + 4y = 12

8. 3x + 4y = −8

In Exercises 9–20, find the equation of the line with the given description.

x FIGURE 11

22. Intercept-Intercept Form Show that if a, b  = 0, then the line with x-intercept x = a and y-intercept y = b has equation (Figure 12) x y + =1 a b

9. Slope 3, y-intercept 8

y

10. Slope −2, y-intercept 3 11. Slope 3, passes through (7, 9)

b

12. Slope −5, passes through (0, 0) 13. Horizontal, passes through (0, −2) 14. Passes through (−1, 4) and (2, 7) 15. Parallel to y = 3x − 4, passes through (1, 1) 16. Passes through (1, 4) and (12, −3) 17. Perpendicular to 3x + 5y = 9, passes through (2, 3) 18. Vertical, passes through (−4, 9) 19. Horizontal, passes through (8, 4) 20. Slope 3, x-intercept 6 21. Find the equation of the perpendicular bisector of the segment joining (1, 2) and (5, 4) (Figure 11). Hint: The midpoint Q of the segment a+c b+d , . joining (a, b) and (c, d) is 2 2

a

x

FIGURE 12

23. Find an equation of the line with x-intercept x = 4 and y-intercept y = 3. 24. Find y such that (3, y) lies on the line of slope m = 2 through (1, 4). 25. Determine whether there exists a constant c such that the line x + cy = 1: (a) Has slope 4 (b) Passes through (3, 1) (c) Is horizontal (d) Is vertical 26. Assume that the number N of concert tickets that can be sold at a price of P dollars per ticket is a linear function N(P ) for 10 ≤ P ≤ 40. Determine N (P ) (called the demand function) if N(10) = 500 and N (40) = 0. What is the decrease N in the number of tickets sold if the price is increased by P = 5 dollars?

20

CHAPTER 1

PRECALCULUS REVIEW

27. Materials expand when heated. Consider a metal rod of length L0 at temperature T0 . If the temperature is changed by an amount T , then the rod’s length changes by L = αL0 T , where α is the thermal expansion coefficient. For steel, α = 1.24 × 10−5 ◦ C−1 . (a) A steel rod has length L0 = 40 cm at T0 = 40◦ C. Find its length at T = 90◦ C. (b) Find its length at T = 50◦ C if its length at T0 = 100◦ C is 65 cm. (c) Express length L as a function of T if L0 = 65 cm at T0 = 100◦ C. 28. Do the points (0.5, 1), (1, 1.2), (2, 2) lie on a line? 29. Find b such that (2, −1), (3, 2), and (b, 5) lie on a line. 30. Find an expression for the velocity v as a linear function of t that matches the following data. t (s) v (m/s)

0

2

4

6

39.2

58.6

78

97.4

31. The period T of a pendulum is measured for pendulums of several different lengths L. Based on the following data, does T appear to be a linear function of L? L (cm)

20

30

40

50

T (s)

0.9

1.1

1.27

1.42

46. Let f (x) be a quadratic function and c a constant. Which of the following statements is correct? Explain graphically. (a) There is a unique value of c such that y = f (x) − c has a double root. (b) There is a unique value of c such that y = f (x − c) has a double root. 47. Prove that x + x1 ≥ 2 for all x > 0. Hint: Consider (x 1/2 − x −1/2 )2 . √ 48. Let a, b > 0. Show that the geometric mean ab is not larger than the arithmetic mean (a + b)/2. Hint: Use a variation of the hint given in Exercise 47. 49. If objects of weights x and w1 are suspended from the balance in Figure 13(A), the cross-beam is horizontal if bx = aw1 . If the lengths a and b are known, we may use this equation to determine an unknown weight x by selecting w1 such that the cross-beam is horizontal. If a and b are not known precisely, we might proceed as follows. First balance x by w1 on the left as in (A). Then switch places and balance x by w2 on the right as in (B). The average x¯ = 12 (w1 + w2 ) gives an estimate for x. Show that x¯ is greater than or equal to the true weight x.

a

b

a

b

32. Show that f (x) is linear of slope m if and only if f (x + h) − f (x) = mh (for all x and h) 33. Find the roots of the quadratic polynomials: (b) x 2 − 2x − 1 (a) 4x 2 − 3x − 1 In Exercises 34–41, complete the square and find the minimum or maximum value of the quadratic function.

w1

x

w2

x

(A)

(B) FIGURE 13

34. y = x 2 + 2x + 5

35. y = x 2 − 6x + 9

50. Find numbers x and y with sum 10 and product 24. Hint: Find a quadratic polynomial satisfied by x.

36. y = −9x 2 + x

37. y = x 2 + 6x + 2

51. Find a pair of numbers whose sum and product are both equal to 8.

38. y = 2x 2 − 4x − 7

39. y = −4x 2 + 3x + 8

40. y = 3x 2 + 12x − 5

41. y = 4x − 12x 2

42. Sketch the graph of y = x 2 − 6x + 8 by plotting the roots and the minimum point.

of all points P such that 52. Show that the parabola y = x 2 consists  d1 = d2 , where d1 is the distance from P to 0, 14 and d2 is the distance from P to the line y = − 14 (Figure 14). y

43. Sketch the graph of y = x 2 + 4x + 6 by plotting the minimum point, the y-intercept, and one other point. 44. If the alleles A and B of the cystic fibrosis gene occur in a population with frequencies p and 1 − p (where p is a fraction between 0 and 1), then the frequency of heterozygous carriers (carriers with both alleles) is 2p(1 − p). Which value of p gives the largest frequency of heterozygous carriers? 45. For which values of c does f (x) = x 2 + cx + 1 have a double root? No real roots?

y = x2

1 4

d1

−1 4

FIGURE 14

P = (x, x 2 ) d2

x

The Basic Classes of Functions

S E C T I O N 1.3

21

Further Insights and Challenges 53. Show that if f (x) and g(x) are linear, then so is f (x) + g(x). Is the same true of f (x)g(x)? 54. Show that if f (x) and g(x) are linear functions such that f (0) = g(0) and f (1) = g(1), then f (x) = g(x).

57. Let a, c  = 0. Show that the roots of ax 2 + bx + c = 0

and

cx 2 + bx + a = 0

are reciprocals of each other. 58. Show, by completing the square, that the parabola

55. Show that y/x for the function f (x) = x 2 over the interval [x1 , x2 ] is not a constant, but depends on the interval. Determine the exact dependence of y/x on x1 and x2 . 56. Use Eq. (2) to derive the quadratic formula for the roots of ax 2 + bx + c = 0.

y = ax 2 + bx + c is congruent to y = ax 2 by a vertical and horizontal translation. 59. Prove Viète’s Formulas: The quadratic polynomial with α and β as roots is x 2 + bx + c, where b = −α − β and c = αβ.

1.3 The Basic Classes of Functions It would be impossible (and useless) to describe all possible functions f (x). Since the values of a function can be assigned arbitrarily, a function chosen at random would likely be so complicated that we could neither graph it nor describe it in any reasonable way. However, calculus makes no attempt to deal with all functions. The techniques of calculus, powerful and general as they are, apply only to functions that are sufficiently “wellbehaved” (we will see what well-behaved means when we study the derivative in Chapter 3). Fortunately, such functions are adequate for a vast range of applications. Most of the functions considered in this text are constructed from the following familiar classes of well-behaved functions: y

polynomials

rational functions

exponential functions

5

logarithmic functions −2

x

−1

1

2

algebraic functions

trigonometric functions inverse trigonometric functions

We shall refer to these as the basic functions. •

FIGURE 1 The polynomial y = x 5 − 5x 3 + 4x.

Polynomials: For any real number m, f (x) = x m is called the power function with exponent m. A polynomial is a sum of multiples of power functions with whole-number exponents (Figure 1): f (x) = x 5 − 5x 3 + 4x,

g(t) = 7t 6 + t 3 − 3t − 1

Thus, the function f (x) = x + x −1 is not a polynomial because it includes a power function x −1 with a negative exponent. The general polynomial in the variable x may be written

y 5

P (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 – – – –

x

−2

1

−3 •

FIGURE 2 The rational function

x+1 f (x) = 3 . x − 3x + 2

The numbers a0 , a1 , . . . , an are called coefficients. The degree of P (x) is n (assuming that an  = 0). The coefficient an is called the leading coefficient. The domain of P (x) is R.

A rational function is a quotient of two polynomials (Figure 2): f (x) =

P (x) Q(x)

[P (x) and Q(x) polynomials]

22

PRECALCULUS REVIEW

CHAPTER 1

The domain of f (x) is the set of numbers x such that Q(x) = 0. For example, f (x) = h(t) = y •

−2

domain {x : x = 0}

7t 6 + t 3 − 3t − 1 t2 − 1

domain {t : t = ±1}

Every polynomial is also a rational function [with Q(x) = 1]. An algebraic function is produced by taking sums, products, and quotients of roots of polynomials and rational functions (Figure 3): f (x) =

x

1 x2

 1 + 3x 2 − x 4 ,

√ g(t) = ( t − 2)−2 ,

h(z) =

2

FIGURE 3The algebraic function f (x) = 1 + 3x 2 − x 4 .

•

Any function that is not algebraic is called transcendental. Exponential and trigonometric functions are examples, as are the Bessel and gamma functions that appear in engineering and statistics. The term “transcendental” goes back to the 1670s, when it was used by Gottfried Wilhelm Leibniz (1646–1716) to describe functions of this type.

A number x belongs to the domain of f if each term in the formula is defined and the result √ does not involve division by zero. For example, g(t) is defined if t ≥ 0 and t  = 2, so the domain of g(t) is D = {t : t ≥ 0 and t = 4}. More generally, algebraic functions are defined by polynomial equations between x and y. In this case, we say that y is implicitly defined as a function of x. For example, the equation y 4 + 2x 2 y + x 4 = 1 defines y implicitly as a function of x. Exponential functions: The function f (x) = bx , where b > 0, is called the exponential function with base b. Some examples are x

f (x) = 2 ,

•

z + z−5/3 √ 5z3 − z

t

g(t) = 10 ,

x 1 h(x) = , 3

√ p(t) = ( 5)t

Exponential functions and their inverses, the logarithmic functions, are treated in greater detail in Section 1.6. Trigonometric functions are functions built from sin x and cos x. These functions and their inverses are discussed in the next two sections.

Constructing New Functions Given functions f and g, we can construct new functions by forming the sum, difference, product, and quotient functions: (f + g)(x) = f (x) + g(x), (f g)(x) = f (x) g(x),

(f − g)(x) = f (x) − g(x)

 f f (x) (x) = (where g(x) = 0) g g(x)

For example, if f (x) = x 2 and g(x) = sin x, then (f + g)(x) = x 2 + sin x, (f g)(x) = x 2 sin x,

(f − g)(x) = x 2 − sin x

 x2 f (x) = g sin x

We can also multiply functions by constants. A function of the form c1 f (x) + c2 g(x)

(c1 , c2 constants)

is called a linear combination of f (x) and g(x).

S E C T I O N 1.3

The Basic Classes of Functions

23

Composition is another important way of constructing new functions. The composition of f and g is the function f ◦ g defined by (f ◦ g)(x) = f (g(x)). The domain of f ◦ g is the set of values of x in the domain of g such that g(x) lies in the domain of f . E X A M P L E 1 Compute the composite functions f ◦ g and g ◦ f and discuss their domains, where √ g(x) = 1 − x f (x) = x, Example 1 shows that the composition of functions is not commutative: The functions f ◦ g and g ◦ f may be (and usually are) different.

Solution We have (f ◦ g)(x) = f (g(x)) = f (1 − x) =

√ 1−x

√ The square root 1 − x is defined if 1 − x ≥ 0 or x ≤ 1, so the domain of f ◦ g is {x : x ≤ 1}. On the other hand, √ √ (g ◦ f )(x) = g(f (x)) = g( x) = 1 − x The domain of g ◦ f is {x : x ≥ 0}.

Elementary Functions Inverse functions are discussed in Section 1.5.

As noted above, we can produce new functions by applying the operations of addition, subtraction, multiplication, division, and composition. It is convenient to refer to a function constructed in this way from the basic functions listed above as an elementary function. The following functions are elementary: f (x) =

√ 2x + sin x,

f (x) = 10

√

x

,

f (x) =

1 + x −1 1 + cos x

1.3 SUMMARY For m a real number, f (x) = x m is called the power function with exponent m. A polynomial P (x) is a sum of multiples of power functions x m , where m is a whole number:

•

P (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 This polynomial has degree n (assuming that an  = 0) and an is called the leading coefficient. • A rational function is a quotient P (x)/Q(x) of two polynomials. • An algebraic function is produced by taking sums, products, and nth roots of polynomials and rational functions. • Exponential function: f (x) = b x , where b > 0 (b is called the base). • The composite function f ◦ g is defined by (f ◦ g)(x) = f (g(x)). The domain of f ◦ g is the set of x in the domain of g such that g(x) belongs to the domain of f .

1.3 EXERCISES Preliminary Questions 1. Give an example of a rational function. 2. Is |x| a polynomial function? What about |x 2 + 1|? 3. What is unusual about the domain of the composite function f ◦ g

for the functions f (x) = x 1/2 and g(x) = −1 − |x|?  x 4. Is f (x) = 12 increasing or decreasing? 5. Give an example of a transcendental function.

24

PRECALCULUS REVIEW

CHAPTER 1

Exercises

2

25. Is f (x) = 2x a transcendental function?

In Exercises 1–12, determine the domain of the function. 1. f (x) = x 1/4

2. g(t) = t 2/3

3. f (x) = x 3 + 3x − 4

4. h(z) = z3 + z−3

5. g(t) =

1 t +2

1 7. G(u) = 2 u −4 9. f (x) = x −4 + (x − 1)−3 √

−1 11. g(y) = 10 y+y

26. Show that f (x) = x 2 + 3x −1 and g(x) = 3x 3 − 9x + x −2 are rational functions—that is, quotients of polynomials. In Exercises 27–34, calculate the composite functions f ◦ g and g ◦ f , and determine their domains. √ 27. f (x) = x, g(x) = x + 1

1 6. f (x) = 2 x +4 √ x 8. f (x) = 2 x −9 

s 10. F (s) = sin s+1 12. f (x) =

28. f (x) =

29. f (x) = 2x ,

x + x −1 (x − 3)(x + 4)

15. f (x) =

√ x

17. f (x) =

x2 x + sin x

19. f (x) =

2x 3 + 3x 9 − 7x 2

14. f (x) = x −4  16. f (x) = 1 − x 2

g(x) = x 3 + x 2

1 32. f (x) = 2 , g(x) = x −2 x +1

35. The population (in millions) of a country as a function of time t (years) is P (t) = 30.20.1t . Show that the population doubles every 10 years. Show more generally that for any positive constants a and k, the function g(t) = a2kt doubles after 1/k years.

3x − 9x −1/2 9 − 7x 2 x 22. f (x) = √ x+1 20. f (x) =

23. f (x) = x 2 + 3x −1

31. f (θ) = cos θ,

1 33. f (t) = √ , g(t) = −t 2 t √ 34. f (t) = t, g(t) = 1 − t 3

18. f (x) = 2x

21. f (x) = sin(x 2 )

g(x) = x 2

30. f (x) = |x|, g(θ) = sin θ

In Exercises 13–24, identify each of the following functions as polynomial, rational, algebraic, or transcendental. 13. f (x) = 4x 3 + 9x 2 − 8

1 , g(x) = x −4 x

x+1 36. Find all values of c such that f (x) = 2 has domain R. x + 2cx + 4

24. f (x) = sin(3x )

Further Insights and Challenges In Exercises 37–43, we define the first difference δf of a function f (x) by δf (x) = f (x + 1) − f (x). 37. Show that if f (x) = x 2 , then δf (x) = 2x + 1. Calculate δf for f (x) = x and f (x) = x 3 . 38. Show that δ(10x ) = 9 · 10x and, more generally, that δ(bx ) = (b − 1)bx . 39. Show that for any two functions f and g, δ(f + g) = δf + δg and δ(cf ) = cδ(f ), where c is any constant. 40. Suppose we can find a function P (x) such that δP = (x + 1)k and P (0) = 0. Prove that P (1) = 1k , P (2) = 1k + 2k , and, more generally, for every whole number n, P (n) = 1k + 2k + · · · + nk

1

satisfies δP = (x + 1). Then apply Exercise 40 to conclude that 1 + 2 + 3 + ··· + n =

42. Calculate δ(x 3 ), δ(x 2 ), and δ(x). Then find a polynomial P (x) of degree 3 such that δP = (x + 1)2 and P (0) = 0. Conclude that P (n) = 12 + 22 + · · · + n2 . 43. This exercise combined with Exercise 40 shows that for all whole numbers k, there exists a polynomial P (x) satisfying Eq. (1). The solution requires the Binomial Theorem and proof by induction (see Appendix C). (a) Show that δ(x k+1 ) = (k + 1) x k + · · · , where the dots indicate terms involving smaller powers of x. (b) Show by induction that there exists a polynomial of degree k + 1 with leading coefficient 1/(k + 1): P (x) =

41. First show that P (x) =

x(x + 1) 2

n(n + 1) 2

1 x k+1 + · · · k+1

such that δP = (x + 1)k and P (0) = 0.

S E C T I O N 1.4

Trigonometric Functions

25

1.4 Trigonometric Functions We begin our trigonometric review by recalling the two systems of angle measurement: radians and degrees. They are best described using the relationship between angles and rotation. As is customary, we often use the lowercase Greek letter θ (“theta”) to denote angles and rotations. Q

Q

θ O

1

θ= π 2

θ = 2π

P

O

1

P=Q

O

1

P

1

P

O

θ = −π 4 Q (A)

(B)

(C)

(D)

FIGURE 1 The radian measure θ of a counterclockwise rotation is the length along the unit circle of the arc traversed by P

as it rotates into Q.

θr θ O

r

FIGURE 2 On a circle of radius r, the arc

traversed by a counterclockwise rotation of θ radians has length θ r. TABLE 1 Rotation through

Radian measure

Two full circles Full circle Half circle Quarter circle One-sixth circle

4π 2π π 2π/4 = π/2 2π/6 = π/3

Radians

Degrees

0 π 6 π 4 π 3 π 2

0◦ 30◦ 45◦ 60◦ 90◦

Figure 1(A) shows a unit circle with radius OP rotating counterclockwise into radius OQ. The radian measure of this rotation is the length θ of the circular arc traversed by P as it rotates into Q. On a circle of radius r, the arc traversed by a counterclockwise rotation of θ radians has length θr (Figure 2). The unit circle has circumference 2π. Therefore, a rotation through a full circle has radian measure θ = 2π [Figure 1(B)]. The radian measure of a rotation through onequarter of a circle is θ = 2π/4 = π/2 [Figure 1(C)] and, in general, the rotation through one-nth of a circle has radian measure 2π/n (Table 1). A negative rotation (with θ < 0) is a rotation in the clockwise direction [Figure 1(D)]. The unit circle has circumference 2π (by definition of the number π ). The radian measure of an angle such as  P OQ in Figure 1(A) is defined as the radian measure of a rotation that carries OP to OQ. Notice, however, that the radian measure of an angle is not unique. The rotations through θ and θ + 2π both carry OP to OQ. Therefore, θ and θ + 2π represent the same angle even though the rotation through θ + 2π takes an extra trip around the circle. In general, two radian measures represent the same angle if the corresponding rotations differ by an integer multiple of 2π . For example, π/4, 9π/4, and −15π/4 all represent the same angle because they differ by multiples of 2π : 9π 15π π = − 2π = − + 4π 4 4 4 Every angle has a unique radian measure satisfying 0 ≤ θ < 2π. With this choice, the angle θ subtends an arc of length θr on a circle of radius r (Figure 2). Degrees are defined by dividing the circle (not necessarily the unit circle) into 360 1 of a circle. A rotation through θ degrees (denoted θ ◦ ) is equal parts. A degree is 360 a rotation through the fraction θ/360 of the complete circle. For example, a rotation 90 through 90◦ is a rotation through the fraction 360 , or 41 , of a circle. As with radians, the degree measure of an angle is not unique. Two degree measures represent that same angle if they differ by an integer multiple of 360. For example, the angles −45◦ and 675◦ coincide because 675 = −45 + 2(360). Every angle has a unique degree measure θ with 0 ≤ θ < 360. To convert between radians and degrees, remember that 2π rad is equal to 360◦ . Therefore, 1 rad equals 360/2π or 180/π degrees. • •

To convert from radians to degrees, multiply by 180/π . To convert from degrees to radians, multiply by π/180.

26

CHAPTER 1

PRECALCULUS REVIEW

Radian measurement is usually the better choice for mathematical purposes, but there are good practical reasons for using degrees. The number 360 has many divisors (360 = 8 · 9 · 5), and consequently, many fractional parts of the circle can be expressed as an integer number of degrees. For example, one-fifth of the circle is 72◦ , two-ninths is 80◦ , three-eighths is 135◦ , etc.

E X A M P L E 1 Convert

(a) 55◦ ×

π ≈ 0.9599 rad 180

Adjacent FIGURE 3

(b) 0.5 rad to degrees.

(b) 0.5 rad ×

180 ≈ 28.648◦ π

Convention: Unless otherwise stated, we always measure angles in radians. The trigonometric functions sin θ and cos θ can be defined in terms of right triangles. Let θ be an acute angle in a right triangle, and let us label the sides as in Figure 3. Then sin θ =

b Opposite

a

and

Solution

Hypotenuse c θ

(a) 55◦ to radians

opposite b = , c hypotenuse

cos θ =

a adjacent = c hypotenuse

A disadvantage of this definition is that it makes sense only if θ lies between 0 and π/2 (because an angle in a right triangle cannot exceed π/2). However, sine and cosine can be defined for all angles in terms of the unit circle. Let P = (x, y) be the point on the unit circle corresponding to the angle θ as in Figures 4(A) and (B), and define cos θ = x-coordinate of P ,

sin θ = y-coordinate of P

This agrees with the right-triangle definition when 0 < θ < π2 . On the circle of radius r (centered at the origin), the point corresponding to the angle θ has coordinates (r cos θ, r sin θ) Furthermore, we see from Figure 4(C) that sin θ is an odd function and cos θ is an even function: sin(−θ) = − sin θ,

cos(−θ) = cos θ

P = (cos θ, sin θ) 1

(x, y)

y

θ x

θ x

y

(x, −y)

P = (cos θ, sin θ)

FIGURE 4 The unit circle definition of sine

and cosine is valid for all angles θ .

θ −θ

(B)

(A)

(C)

Although we use a calculator to evaluate sine and cosine for general angles, the standard values listed in Figure 5 and Table 2 appear often and should be memorized.

( π/6 FIGURE 5 Four standard angles: The x- and

y-coordinates of the points are cos θ and sin θ.

3, 1 2 2

(

) π/4

2, 2 2 2

)

(

1 , 3 2 2

π/3

)

(0, 1)

π/2

Trigonometric Functions

S E C T I O N 1.4

27

TABLE 2 θ

0

sin θ

0

cos θ

1

π 4 √ 2 2 √ 2 2

π 6 1 2 √ 3 2

π 3 √ 3 2

π 2

2π 3 √ 3 2

1

1 2

−

0

3π 4 √ 2 2 √ 2 − 2

1 2

5π 6 1 2 √ −

3 2

π

0

−1

The graph of y = sin θ is the familiar “sine wave” shown in Figure 6. Observe how the graph is generated by the y-coordinate of the point P = (cos θ, sin θ ) moving around the unit circle. y

y 1

1 P θ

x

θ

θ

π

2π

FIGURE 6 The graph of y = sin θ is

generated as the point P = (cos θ, sin θ ) moves around the unit circle.

The graph of y = cos θ has the same shape but is shifted to the left π/2 units (Figure 7). The signs of sin θ and cos θ vary as P = (cos θ, sin θ) changes quadrant. Quadrant of unit circle

y I

II

1

−1

π 4

π 2

3π 4

FIGURE 7 Graphs of y = sin θ and

IV

5π 4

7π 4

π

3π 2

I

2π

θ

π 4

π 2

y = sin θ

y = cos θ over one period of length 2π.

We often write sin x and cos x , using x instead of θ . Depending on the application, we may think of x as an angle or simply as a real number.

y

III

III

3π 4

5π 4 π

IV

3π 7π 2 4

2π

θ

y = cos θ

A function f (x) is called periodic with period T if f (x + T ) = f (x) (for all x) and T is the smallest positive number with this property. The sine and cosine functions are periodic with period T = 2π (Figure 8) because the radian measures x and x + 2π k correspond to the same point on the unit circle for any integer k: sin x = sin(x + 2π k),

cos x = cos(x + 2π k)

y

y 1

1 x

−2π

FIGURE 8 Sine and cosine have period 2π.

II

2π y = sin x

4π

x

−2π

2π y = cos x

4π

28

PRECALCULUS REVIEW

CHAPTER 1

There are four other standard trigonometric functions, each defined in terms of sin x and cos x or as ratios of sides in a right triangle (Figure 9):

Hypotenuse c b Opposite x

Tangent:

tan x =

sin x b = , cos x a

Cotangent:

cot x =

cos x a = sin x b

Secant:

sec x =

1 c = , cos x a

Cosecant:

csc x =

1 c = sin x b

a Adjacent FIGURE 9

These functions are periodic (Figure 10): y = tan x and y = cot x have period π ; y = sec x and y = csc x have period 2π (see Exercise 55). y

1 −π

−

π 2

y

π 2 −1

1 x

π 3π 2π 2

−π

y = tan x

−

π 2

y

π 2 −1

1 π 3π 2π 5π 2 2

x

−π

−

π 2

y = cot x

y

π 2 −1

−

π 3π 2π 5π 2 2

x

−π

y = sec x

π 1 2 −1

3π 2 π 2

π

x 2π

y = csc x

FIGURE 10 Graphs of the standard trigonometric functions.

( 4π 3

1 , 3 2 2 π 3 1

(

1 3 P= − ,− 2 2

)

)

E X A M P L E 2 Computing Values of Trigonometric Functions Find the values of the six trigonometric functions at x = 4π/3.

Solution The point P on the unit circle corresponding to the angle x = 4π/3 lies opposite the point with angle π/3 (Figure 11). Thus, we see that (refer to Table 2) √ π 4π π 1 3 4π = − sin = − , cos = − cos = − sin 3 3 2 3 3 2 The remaining values are

√ sin 4π/3 − 3/2 √ 4π = = = 3, tan 3 cos 4π/3 −1/2

FIGURE 11

sec

4π 1 1 = = = −2, 3 cos 4π/3 −1/2

√ 4π cos 4π/3 3 cot = = 3 sin 4π/3 3 √ 4π 1 −2 3 csc = = 3 sin 4π/3 3

E X A M P L E 3 Find the angles x such that sec x = 2. π 3 π − 1 32

1

Solution Because sec x = 1/ cos x, we must solve cos x = 12 . From Figure 12 we see that x = π/3 and x = −π/3 are solutions. We may add any integer multiple of 2π , so the general solution is x = ±π/3 + 2π k for any integer k. E X A M P L E 4 Trigonometric Equation

FIGURE 12 cos x = 12 for x = ± π3

Solve sin 4x + sin 2x = 0 for x ∈ [0, 2π ).

Solution We must find the angles x such that sin 4x = − sin 2x. First, let’s determine when angles θ1 and θ2 satisfy sin θ2 = − sin θ1 . Figure 13 shows that this occurs if θ2 = −θ1 or θ2 = θ1 + π . Because the sine function is periodic with period 2π , sin θ2 = − sin θ1

⇔

θ2 = −θ1 + 2π k

or

θ2 = θ1 + π + 2π k

where k is an integer. Taking θ2 = 4x and θ1 = 2x, we see that sin 4x = − sin 2x

⇔

4x = −2x + 2π k

or

4x = 2x + π + 2π k

Trigonometric Functions

S E C T I O N 1.4

29

The first equation gives 6x = 2π k or x = (π/3)k and the second equation gives 2x = π + 2π k or x = π/2 + π k. We obtain eight solutions in [0, 2π ) (Figure 14): π , 3

x = 0,

2π , 3

π,

4π , 3

5π 3

x=

and

π , 2

3π 2

sin θ1 θ1 y θ1

y = sin 4x + sin 2x

1 0

θ2 = θ1 + π

θ2 = −θ1

5π 3

2π 3 π π 3

−1

π 2

4π 3

2π

x

3π 2

−sin θ1

FIGURE 13 sin θ2 = − sin θ1 when θ2 = −θ1 or θ2 = θ1 + π.

FIGURE 14 Solutions of sin 4x + sin 2x = 0.

   E X A M P L E 5 Sketch the graph of f (x) = 3 cos 2 x + π2 over [0, 2π ]. CAUTION To shift the graph of y = cos 2x to the left π/2 units,we must replace x by  x + π2 to obtain cos 2 x + π2 . It is



incorrect to take cos 2x +



3 π 2

•

Shift to the left π/2 units:

•

Expand vertically by a factor of 3:

Shift left π/2 units

3

x 2π

Compress horizontally by a factor of 2:

y

1 π

−1

•

.

Compress horizontally by a factor of 2

y

1

π 2

Solution The graph is obtained by scaling and shifting the graph of y = cos x in three steps (Figure 15):

−1

π 2

(A) y = cos x

Expand vertically by a factor of 3

y 3

π

x 2π

−1

π 2

x

π

2π

−3

x

π

−1

2π

−3

(

(B) y = cos 2x (periodic with period π)

y 3 1

1

−3

−3

y = cos 2x   π  y = cos 2 x + 2   π  y = 3 cos 2 x + 2

(C) y = cos 2 x +

π 2

)

(

(D) y = 3 cos 2 x +

π 2

)

FIGURE 15

Trigonometric Identities

The expression (sin x)k is usually denoted sink x . For example, sin2 x is the square of sin x . We use similar notation for the other trigonometric functions.

A key feature of trigonometric functions is that they satisfy a large number of identities. First and foremost, sine and cosine satisfy a fundamental identity, which is equivalent to the Pythagorean Theorem: sin2 x + cos2 x = 1

1

Equivalent versions are obtained by dividing Eq. (1) by cos2 x or sin2 x: tan2 x + 1 = sec2 x,

1 + cot 2 x = csc2 x

2

30

PRECALCULUS REVIEW

CHAPTER 1

Here is a list of some other commonly used identities. The identities for complementary angles are justified by Figure 16.

π−θ 2

c

b

Basic Trigonometric Identities π  π  Complementary angles: sin − x = cos x, cos − x = sin x 2 2 Addition formulas: sin(x + y) = sin x cos y + cos x sin y

θ a FIGURE 16 For complementary angles, the

sine of one is equal to the cosine of the other.

cos(x + y) = cos x cos y − sin x sin y Double-angle formulas:

sin2 x =

1 (1 − cos 2x), 2

cos2 x =

1 (1 + cos 2x) 2

cos 2x = cos2 x − sin2 x, sin 2x = 2 sin x cos x   π π sin x + = cos x, cos x + = − sin x 2 2

Shift formulas:

E X A M P L E 6 Suppose that cos θ = 25 . Calculate tan θ in the following two cases: π

(a) 0 < θ
S2 and S2 = 12 S. Observe that (b) contradicts (a), and conclude that S diverges.

25 ≈ 1.04 book lengths 24

1 2 3

1 8

1 4

1 6

...

4 1 2

1 1 1 1 25 + + + = 2 4 6 8 24

x

1 6

n n+1 1 2(n + 1) cn +1 cn

FIGURE 5

c3 c2

1 2

1 4

c1

0

FIGURE 6

Further Insights and Challenges 90. Let S =

∞ 

an , where an = (ln(ln n))− ln n .

n=2

(a) Show, by taking logarithms, that an = n− ln(ln(ln n)) . e2

(b) Show that ln(ln(ln n)) ≥ 2 if n > C, where C = ee . (c) Show that S converges.

91. Kummer’s Acceleration Method Suppose we wish to approx∞  1/n2 . There is a similar telescoping series whose value imate S = n=1

can be computed exactly (Example 1 in Section 10.2): ∞  n=1

1 =1 n(n + 1)

S E C T I O N 10.4

(a) Verify that S=

∞  n=1

1 + n(n + 1)

∞   n=1

1 1 − n(n + 1) n2

Thus for M large, S ≈1+

n=1

The series S =

6

Compute

n=1

1 , n2

1+

k −3 has been computed to more than

k=1

n=1 1000 

∞ 

100 million digits. The firs 30 digits are

1 n2 (n + 1)

(b) Explain what has been gained. Why is Eq. (6) a better approximaM  tion to S than is 1/n2 ? (c)

569

Which is a better approximation to S, whose exact value is π 2 /6?



92. M 

Absolute and Conditional Convergence

100  n=1

S = 1.202056903159594285399738161511 Approximate S using the Acceleration Method of Exercise 91 with ∞  (n(n + 1)(n + 2))−1 . AccordM = 100 and auxiliary series R = n=1

1 2 n (n + 1)

ing to Exercise 46 in Section 10.2, R is a telescoping series with the sum R = 14 .

10.4 Absolute and Conditional Convergence In the previous section, we studied positive series, but we still lack the tools to analyze series with both positive and negative terms. One of the keys to understanding such series is the concept of absolute convergence. DEFINITION Absolute Convergence The series  |an | converges.



an converges absolutely if

E X A M P L E 1 Verify that the series ∞  (−1)n−1 n=1

n2

=

1 1 1 1 − 2 + 2 − 2 + ··· 12 2 3 4

converges absolutely. Solution This series converges absolutely because the positive series (with absolute values) is a p-series with p = 2 > 1: ∞  n−1   1 1 1 1  (−1)    = 2 + 2 + 2 + 2 + ··· 2 n 1 2 3 4

(convergent p-series)

n=1

The next theorem tells us that if the series of absolute values converges, then the original series also converges. THEOREM 1 Absolute Convergence Implies Convergence  then an also converges.

If



|an | converges,

Proof We have −|an | ≤ an ≤ |an |. By adding |an | to all parts of the inequality, we get   0 ≤ |an | + an ≤ 2|an |. If |an | converges, then 2|an | also converges, and therefore,  (an + |an |) converges by the Comparison Test. Our original series converges because it is the difference of two convergent series:    an = (an + |an |) − |an |

570

C H A P T E R 10

INFINITE SERIES

E X A M P L E 2 Verify that S =

∞  (−1)n−1 n=1

n2

converges.

Solution We showed that S converges absolutely in Example 1. By Theorem 1, S itself converges. E X A M P L E 3 Does S =

lutely?

∞  (−1)n−1 1 1 1 = √ − √ + √ − · · · converge abso√ n 1 2 3 n=1

∞  1 √ is a p-series with p = 12 . It diverges because p < 1. n n=1 Therefore, S does not converge absolutely.

Solution The positive series

The series in the previous example does  not converge absolutely, but we still do not know whether or not it converges. A series an may converge without converging  absolutely. In this case, we say that an is conditionally convergent.

DEFINITION Conditional Convergence An infinit series   |an | diverges. tionally if an converges but

y

a1

−a2

a3

−a4

a5

−a6

x

an converges condi-

If a series is not absolutely convergent, how can we determine whether it is conditionally convergent? This is often a more difficul question, because we cannot use the Integral Test or the Comparison Test (they apply only to positive series). However, convergence is guaranteed in the particular case of an alternating series S=

FIGURE 1 An alternating series with

decreasing terms. The sum is the signed area, which is at most a1 .



∞ 

(−1)n−1 an = a1 − a2 + a3 − a4 + · · ·

n=1

where the terms an are positive and decrease to zero (Figure 1). THEOREM 2 Leibniz Test for Alternating Series Assume that {an } is a positive sequence that is decreasing and converges to 0: a1 > a2 > a3 > a4 > · · · > 0,

lim an = 0

n→∞

Then the following alternating series converges: Assumptions Matter The Leibniz Test is not valid if we drop the assumption that an is decreasing (see Exercise 35).

S=

∞ 

(−1)n−1 an = a1 − a2 + a3 − a4 + · · ·

n=1

Furthermore, 0 < S < a1

and

S2N < S < S2N+1

N ≥1

Absolute and Conditional Convergence

S E C T I O N 10.4

Proof We will prove that the partial sums zigzag above and below the sum S as in Figure 2. Note firs that the even partial sums are increasing. Indeed, the odd-numbered terms occur with a plus sign and thus, for example,

y S 1 S3

S5

S7

S9

S S4

S6

S8

571

S4 + a5 − a6 = S6

S10

S2

1 2 3 4 5 6 7 8 9 10

But a5 − a6 > 0 because an is decreasing, and therefore S4 < S6 . In general, x

FIGURE 2 The partial sums of an alternating

series zigzag above and below the limit. The odd partial sums decrease and the even partial sums increase.

S2N + (a2N+1 − a2N+2 ) = S2N+2 where a2n+1 − a2N+2 > 0. Thus S2N < S2N+2 and 0 < S2 < S4 < S6 < · · · Similarly, S2N−1 − (a2N − a2N+1 ) = S2N+1 Therefore S2N+1 < S2N−1 , and the sequence of odd partial sums is decreasing: · · · < S7 < S5 < S3 < S1 Finally, S2N < S2N + a2N+1 = S2N+1 . The picture is as follows: 0 < S2 < S4 < S6
1, then an diverges. (iii) If ρ = 1, the test is inconclusive (the series may converge or diverge). Proof The idea is to compare with a geometric series. If ρ < 1, we may choose a number r such that ρ < r < 1. Since |an+1 /an | converges to ρ, there exists a number M such that |an+1 /an | < r for all n ≥ M. Therefore,

576

C H A P T E R 10

INFINITE SERIES

|aM+1 | < r|aM | |aM+2 | < r|aM+1 | < r(r|aM |) = r 2 |aM | |aM+3 | < r|aM+2 | < r 3 |aM | In general, |aM+n | < r n |aM |, and thus, ∞ 

|an | =

∞ 

n=M

|aM+n | ≤

n=0

∞ 

|aM | r n = |aM |

n=0

∞ 

rn

n=0

The geometric series on the right converges because 0 < r < 1, so

∞ 

|an | converges by

 the Comparison Test and thus an converges absolutely. If ρ > 1, choose r such that 1 < r < ρ. Then there exists a number M such that |an+1 /an | > r for all n ≥ M. Arguing as before with the inequalities reversed, we fin that |aM+n | ≥  r n |aM |. Since r n tends to ∞, the terms aM+n do not tend to zero, and consequently, an diverges. Finally, Example 4 below shows that both convergence and divergence are possible when ρ = 1, so the test is inconclusive in this case. n=M

E X A M P L E 1 Prove that

∞ n  2 n=1

n!

converges.

Solution Compute the ratio and its limit with an = and thus

2n . Note that (n + 1)! = (n + 1)n! n!

an+1 n! 2n+1 n! 2n+1 2 = = = n n an (n + 1)! 2 2 (n + 1)! n+1    an+1  2  = lim ρ = lim  =0 n→∞ a  n→∞ n + 1 n

Since ρ < 1, the series

∞ n  2 n=1

E X A M P L E 2 Does

n!

∞  n2 n=1

2n

converges by the Ratio Test.

converge?

Solution Apply the Ratio Test with an =    an+1  (n + 1)2 2n    a  = 2n+1 n2 = n    an+1  = ρ = lim  n→∞ an 

n2 : 2n

   1 n2 + 2n + 1 2 1 = 1+ + 2 2 n n n2   1 1 2 1 lim 1 + + 2 = 2 n→∞ n n 2 1 2



Since ρ < 1, the series converges by the Ratio Test.

The Ratio and Root Tests

S E C T I O N 10.5

E X A M P L E 3 Does

∞ 

(−1)n

n=0

577

n! converge? 1000n

Solution This series diverges by the Ratio Test because ρ > 1:   n  an+1   = lim (n + 1)! 1000 = lim n + 1 = ∞ ρ = lim   n+1 n→∞ an n→∞ 1000 n→∞ 1000 n! E X A M P L E 4 Ratio Test Inconclusive Show that both convergence and divergence

are possible when ρ = 1 by considering

∞ 

n2 and

n=1

∞ 

n−2 .

n=1

Solution For an = n2 , we have     2 2  an+1   = lim (n + 1) = lim n + 2n + 1 = lim 1 + 2 + 1 = 1 ρ = lim  n→∞ an  n→∞ n→∞ n→∞ n n2 n2 n2 On the other hand, for bn = n−2 ,      bn+1     = lim  an  = ρ = lim    n→∞ bn n→∞ an+1  lim

n→∞

Thus, ρ = 1 in both cases, but

∞ 

n2 diverges and

n=1

∞ 

1   =1  an+1   an 

n−2 converges. This shows that

n=1

both convergence and divergence are possible when ρ = 1. √ Our next test is based on the limit of the nth roots n an rather than the ratios an+1 /an . Its proof, like that of the Ratio Test, is based on a comparison with a geometric series (see Exercise 57). THEOREM 2 Root Test Assume that the following limit exists:  L = lim n |an | n→∞

(i) If L < 1, then (ii) If L > 1, then

 

an converges absolutely. an diverges.

(iii) If L = 1, the test is inconclusive (the series may converge or diverge).

E X A M P L E 5 Does

∞   n=1

Solution We have L = lim verges by the Root Test.

n 2n + 3

n→∞

n converge?

√ n an = lim

n→∞

1 n = . Since L < 1, the series con2n + 3 2

578

C H A P T E R 10

INFINITE SERIES

10.5 SUMMARY •

    an+1    exists. Then an Ratio Test: Assume that ρ = lim  n→∞ an  – Converges absolutely if ρ < 1. – Diverges if ρ > 1. – Inconclusive if ρ = 1.

•

Root Test: Assume that L = lim

n→∞

 n

|an | exists. Then



an

– Converges absolutely if L < 1. – Diverges if L > 1. – Inconclusive if L = 1.

10.5 EXERCISES Preliminary Questions

    an+1   an   or lim  1. In the Ratio Test, is ρ equal to lim  n→∞ a  n→∞ a n

2. Is the Ratio Test conclusive for

  ? 

n+1 ∞ 

∞  1 ? Is it conclusive for 2n

n=1

n=1

3. Can the Ratio Test be used to show convergence if the series is only conditionally convergent?

1 ? n

Exercises In Exercises 1–20, apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. ∞  1 1. 5n n=1

3.

∞  n=1

5.

∞  n=1

∞  (−1)n−1 n 2. 5n n=1

1 nn

∞  3n + 2 4. 5n3 + 1

n 2 n +1

∞ n  2 6. n

∞  2n 7. n100

n=0

n=1

8.

n=1

9.

∞  10n n=1

11.

∞  n=1

13.

∞  n=0

15.

∞  n=2

2 2n

en nn n! 6n 1 n ln n

10.

∞  n=1

19.

∞  n=2

n2 (2n + 1)!

18.

1 2n + 1

20.

21. Show that

2 3n

∞ n  e n!

12.

n=1

14.

∞  n=1

16.

∞  n=1

n=1

∞  1 ln n

n=2

nk 3−n converges for all exponents k.

n=1

22. Show that

23. Show that

∞ 

n2 x n converges if |x| < 1.

∞ 

2n x n converges if |x| < 12 .

n=1

24. Show that

n=1 ∞ 

∞ 

∞  (n!)3 (3n)!

n=1

∞  n3 n=1

17.

∞ n  r converges for all r. n!

n=1

∞ n  r converges if |r| < 1. n

n40 n!

25. Show that

n! n9

26. Is there any value of k such that

1 (2n)!

n=1

∞ n  2 converges? nk

n=1

27. Show that

∞  n=1

  n! 1 n converges. Hint: Use lim = e. 1 + n→∞ nn n

Power Series

S E C T I O N 10.6

In Exercises 28–33, assume that |an+1 /an | converges to ρ = 13 . What can you say about the convergence of the given series? 28.

31.

∞  n=1 ∞ 

nan

29.

3n an

32.

n=1

∞  n=1 ∞ 

n3 an

30.

4n an

33.

n=1

∞  n=1 ∞ 

2n an

In Exercises 43–56, determine convergence or divergence using any method covered in the text so far. 43.

  ! −1 34. Assume that an+1 /an  converges to ρ = 4. Does ∞ n=1 an converge (assume that an = 0 for all n)? ∞  1 35. Is the Ratio Test conclusive for the p-series ? np

45.

47.

In Exercises 36–41, use the Root Test to determine convergence or divergence (or state that the test is inconclusive).

38.

∞  1 10n

n=0 ∞   k=0

k k k + 10

40.

∞   n=1

1 1+ n

42. Prove that

39.

49.

∞  1 nn

37.

n=1 ∞  

51.

n=4

46.

∞ 

1  3 − n2 n n=2 ∞ 

∞ 

 2 1 −n 1+ n

53.

∞ 

n−0.8

48.

∞ n2  2 2 diverges. Hint: Use 2n = (2n )n and n! ≤ nn . n!

55.

n=1

1 n(ln n)3

∞ 2  n + 4n 3n4 + 9

n=1

50.

∞ 

(0.8)−n n−0.8

n=1

4−2n+1

52.

∞  (−1)n−1 √ n

n=1

1 sin 2 n

54.

∞  (−2)n √ n

56.

n=1

∞  n=2

n=1

∞  

41.

∞ 3  n 5n

∞ 3  n n!

n=1

n=1

k k 3k + 1

k=0

−n

44.

n=1

n=1

36.

∞ n  2 + 4n 7n

n=1

an2

n=1

579

∞ 

(−1)n cos

n=1

n=1

∞   n=1

1 n

n n n + 12

Further Insights and Challenges

∞  57. Proof of the Root Test Let S = an be a positive √ series, and assume that L = lim n an exists. n=0

1 1 1 1 1 + + 3 + 4 + 5 + ··· 2 32 2 2 3

n→∞

(a) Show that S converges if L < 1. Hint: Choose R with L < R < 1 and show that an ≤ R n for n sufficientl large. Then compare with the  geometric series Rn . (b) Show that S diverges if L > 1. 58. Show that the Ratio Test does not apply, but verify convergence using the Comparison Test for the series

59. Let S =

∞ n  c n! , where c is a constant. nn

n=1

(a) Prove that S converges absolutely if |c| < e and diverges if |c| > e. √ en n! (b) It is known that lim n+1/2 = 2π. Verify this numerically. n→∞ n (c) Use the Limit Comparison Test to prove that S diverges for c = e.

10.6 Power Series A power series with center c is an infinit series F (x) =

∞ 

an (x − c)n = a0 + a1 (x − c) + a2 (x − c)2 + a3 (x − c)3 + · · ·

n=0

where x is a variable. For example, F (x) = 1 + (x − 2) + 2(x − 2)2 + 3(x − 2)3 + · · · is a power series with center c = 2.

1

580

C H A P T E R 10

INFINITE SERIES

Many functions that arise in applications can be represented as power series. This includes not only the familiar trigonometric, exponential, logarithm, and root functions, but also the host of “special functions” of physics and engineering such as Bessel functions and elliptic functions.

A power series F (x) =

∞ 

an (x − c)n converges for some values of x and may

n=0

diverge for others. For example, if we set x = 94 in the power series of Eq. (1), we obtain an infinit series that converges by the Ratio Test:     2 3   9 9 9 9 =1+ −2 +2 −2 +3 − 2 + ··· F 4 4 4 4  2  3   1 1 1 +2 +3 + ··· =1+ 4 4 4 On the other hand, the power series in Eq. (1) diverges for x = 3: F (3) = 1 + (3 − 2) + 2(3 − 2)2 + 3(3 − 2)3 + · · · = 1 + 1 + 2 + 3 + ··· There is a surprisingly simple way to describe the set of values x at which a power series F (x) converges. According to our next theorem, either F (x) converges absolutely for all values of x or there is a radius of convergence R such that F (x) converges absolutely when |x − c| < R and diverges when |x − c| > R. This means that F (x) converges for x in an interval of convergence consisting of the open interval (c − R, c + R) and possibly one or both of the endpoints c − R and c + R (Figure 1). Note that F (x) automatically converges at x = c because F (c) = a0 + a1 (c − c) + a2 (c − c)2 + a3 (c − c)3 + · · · = a0 We set R = 0 if F (x) converges only for x = c, and we set R = ∞ if F (x) converges for all values of x. Converges absolutely |x − c| < R

Diverges

c−R FIGURE 1 Interval of convergence of a

Diverges

c+R

c

x

Possible convergence at the endpoints

power series.

THEOREM 1 Radius of Convergence F (x) =

Every power series ∞ 

an (x − c)n

n=0

has a radius of convergence R, which is either a nonnegative number (R ≥ 0) or infinit (R = ∞). If R is finite F (x) converges absolutely when |x − c| < R and diverges when |x − c| > R. If R = ∞, then F (x) converges absolutely for all x. Proof We assume that c = 0 to simplify the notation. If F (x) converges only at x = 0, then R = 0. Otherwise, F (x) converges for some nonzero value x = B. We claim that F (x) must then converge absolutely for all |x| < |B|. To prove this, note that because ∞  F (B) = an B n converges, the general term an B n tends to zero. In particular, there n=0

exists M > 0 such that |an B n | < M for all n. Therefore,

S E C T I O N 10.6

∞ 

|an x n | =

n=0

Least Upper Bound Property: If S is a set of real numbers with an upper bound M (that is, x ≤ M for all x ∈ S ), then S has a least upper bound L. See Appendix B.

∞  n=0

Power Series

581

∞    x n   x n   |an B n |   < M   B B n=0

If |x| < |B|, then |x/B| < 1 and the series on the right is a convergent geometric series. By the Comparison Test, the series on the left also converges. This proves that F (x) converges absolutely if |x| < |B|. Now let S be the set of numbers x such that F (x) converges. Then S contains 0, and we have shown that if S contains a number B  = 0, then S contains the open interval (−|B|, |B|). If S is bounded, then S has a least upper bound L > 0 (see marginal note). In this case, there exist numbers B ∈ S smaller than but arbitrarily close to L, and thus S contains (−B, B) for all 0 < B < L. It follows that S contains the open interval (−L, L). The set S cannot contain any number x with |x| > L, but S may contain one or both of the endpoints x = ±L. So in this case, F (x) has radius of convergence R = L. If S is not bounded, then S contains intervals (−B, B) for B arbitrarily large. In this case, S is the entire real line R, and the radius of convergence is R = ∞. From Theorem 1, we see that there are two steps in determining the interval of convergence of F (x): Step 1. Find the radius of convergence R (using the Ratio Test, in most cases). Step 2. Check convergence at the endpoints (if R  = 0 or ∞). E X A M P L E 1 Using the Ratio Test Where does F (x) =

∞  xn n=0

2n

converge?

Solution Step 1. Find the radius of convergence. xn Let an = n and compute the ratio ρ of the Ratio Test: 2    n+1   n   an+1  x  2  1 1   = lim  n+1  ·  n  = lim |x| = |x| ρ = lim   n→∞ an n→∞ 2 n→∞ 2 x 2 We fin that ρ 1 if 12 |x| > 1, or |x| > 2. Thus F (x) converges if |x| > 2. Therefore, the radius of convergence is R = 2.

Converges absolutely

Step 2. Check the endpoints. The Ratio Test is inconclusive for x = ±2, so we must check these cases directly:

Diverges

0

Diverges

if

2 Diverges

x

F (2) =

n=0

FIGURE 2 The power series ∞  xn 2n

n=0

has interval of convergence (−2, 2).

∞ n  2

F (−2) =

2n

= 1 + 1 + 1 + 1 + 1 + 1···

∞  (−2)n n=0

2n

= 1 − 1 + 1 − 1 + 1 − 1···

Both series diverge. We conclude that F (x) converges only for |x| < 2 (Figure 2).

582

INFINITE SERIES

C H A P T E R 10

E X A M P L E 2 Where does F (x) =

∞  (−1)n n=1

Solution We compute ρ with an =

4n n

(x − 5)n converge?

(−1)n (x − 5)n : 4n n

     an+1   (x − 5)n+1 4n n     = lim  n+1 ρ = lim  n→∞ an  n→∞ 4 (n + 1) (x − 5)n      n   = |x − 5| lim  n→∞ 4(n + 1)  =

1 |x − 5| 4

We fin that ρ R. – F (x) may converge or diverge at the endpoints c − R and c + R. We set R = 0 if F (x) converges only for x = c and R = ∞ if F (x) converges for all x. • The interval of convergence of F (x) consists of the open interval (c − R, c + R) and possibly one or both endpoints c − R and c + R. • In many cases, the Ratio Test can be used to fin the radius of convergence R. It is necessary to check convergence at the endpoints separately. • If R > 0, then F (x) is differentiable on (c − R, c + R) and F (x) =

∞ 

nan (x − c)n−1 ,

F (x) dx = A +

n=1

∞  an (x − c)n+1 n+1 n=0

(A is any constant). These two power series have the same radius of convergence R. ∞  1 • The expansion = x n is valid for |x| < 1. It can be used to derive expansions 1−x n=0 of other related functions by substitution, integration, or differentiation.

10.6 EXERCISES Preliminary Questions 

1. Suppose that an x n converges for x = 5. Must it also converge for x = 4? What about x = −3?

 2. Suppose that an (x − 6)n converges for x = 10. At which of the points (a)–(d) must it also converge? (a) x = 8

(b) x = 11

(c) x = 3

(d) x = 0

3. What is the radius of convergence of F (3x) if F (x) is a power series with radius of convergence R = 12? 4. The power series F (x) =

∞  n=1

nx n has radius of convergence

R = 1. What is the power series expansion of F (x) and what is its radius of convergence?

Power Series

S E C T I O N 10.6

Exercises 1. Use the Ratio Test to determine the radius of convergence R of xn . Does it converge at the endpoints x = ±R? 2n

∞ 

21.

n=0

2. Use the Ratio Test to show that

∞  n=1

xn √ n has radius of convern2

23.

gence R = 2. Then determine whether it converges at the endpoints R = ±2.

25.

3. Show that the power series (a)–(c) have the same radius of convergence. Then show that (a) diverges at both endpoints, (b) converges at one endpoint but diverges at the other, and (c) converges at both endpoints. ∞ n ∞ ∞    x xn xn (a) (b) (c) 3n n3n n2 3n

27.

n=1

n=1

n=1

4. Repeat Exercise 3 for the following series: ∞ ∞   (x − 5)n (x − 5)n (a) (b) n 9 n9n n=1

∞  (x − 5)n (c) n2 9n

n=1

5. Show that

∞ 

n=1

n=0

6. For which values of x does

n!x n converge?

∞ 2n  x 7. Use has radius of convergence √ the Ratio Test to show that 3n R = 3. n=0 ∞ 3n+1  x has radius of convergence R = 4. 64n

n=0

In Exercises 9–34, find the interval of convergence. 9.

11.

∞ 

nx n

2n

n=1

∞ 

∞ 

n=1

13.

10.

∞ 

n=0

(−1)n

∞  n=4

x 2n+1 2n n

xn n5

∞  xn 15. (n!)2 n=0

∞  (2n)! n x 17. (n!)2

12.

n

n=2 ∞  n=1 ∞ 

ln n

24.

n(x − 3)n

26.

(−1)n n5 (x − 7)n

n=1 ∞ n  2 n=1 ∞ 

3n

(−5)n n!

n=0 ∞ 

28.

(x + 3)n

30.

(x + 10)n

32.

en (x − 2)n

34.

n=12

n (−1)n n x 2n 4

14.

∞ 

n7 x n

n=8 ∞ n  8 n x 16. n! n=0

18.

∞  n=0

20.

∞  n=0

∞  n=1 ∞  n=2 ∞  n=1 ∞  n=0 ∞  n=0 ∞ 

xn n − 4 ln n x 3n+2 ln n (−5)n (x − 3)n n2 27n (x − 1)3n+2 (x − 4)n n! n! (x + 5)n

n=10 ∞  n=2

1 1 − 3x 1 37. f (x) = 3−x 1 39. f (x) = 1 + x2 41. Use the equalities

(x + 4)n (n ln n)2

1 1 + 3x 1 38. f (x) = 4 + 3x 1 40. f (x) = 16 + 2x 3 36. f (x) =

− 13 1 1 = =   1−x −3 − (x − 4) 1 + x−4 3 to show that for |x − 4| < 3, ∞  1 (x − 4)n = (−1)n+1 n+1 1−x 3

xn

n=0

n=0

∞  (−1)n x n 19.  2 n=0 n + 1

33.

n=15 ∞  xn

35. f (x) =

n=0

8. Show that

31.

22.

In Exercises 35–40, use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence.

nn x n diverges for all x = 0. ∞ 

29.

∞  x 2n+1 3n + 1

589

n=0

42. Use the method of Exercise 41 to expand 1/(1 − x) in power series with centers c = 2 and c = −2. Determine the interval of convergence. 43. Use the method of Exercise 41 to expand 1/(4 − x) in a power series with center c = 5. Determine the interval of convergence. 44. Find a power series that converges only for x in [2, 6). 45. Apply integration to the expansion ∞  1 (−1)n x n = 1 − x + x 2 − x 3 + · · · = 1+x n=0

4n x 2n−1 (2n + 1)! xn 4 n +2

to prove that for −1 < x < 1, ln(1 + x) =

∞  x2 x3 x4 (−1)n−1 x n =x− + − + ··· n 2 3 4

n=1

590

C H A P T E R 10

INFINITE SERIES

46. Use the result of Exercise 45 to prove that ln

3 1 1 1 1 = − + − + ··· 3 2 2 2 · 22 3·2 4 · 24

Use your knowledge of alternating series to fin an N such that the partial sum SN approximates ln 32 to within an error of at most 10−3 . Confir using a calculator to compute both SN and ln 32 . 47. Let F (x) = (x + 1) ln(1 + x) − x. (a) Apply integration to the result of Exercise 45 to prove that for −1 < x < 1, F (x) =

∞ 

(−1)n+1

n=1

x n+1 n(n + 1)

53. Show that the following series converges absolutely for |x| < 1 and compute its sum: F (x) = 1 − x − x 2 + x 3 − x 4 − x 5 + x 6 − x 7 − x 8 + · · · Hint: Write F (x) as a sum of three geometric series with common ratio x3. 54. Show that for |x| < 1, 1 + 2x = 1 + x − 2x 2 + x 3 + x 4 − 2x 5 + x 6 + x 7 − 2x 8 + · · · 1 + x + x2 Hint: Use the hint from Exercise 53. 55. Find all values of x such that

(b) Evaluate at x = 12 to prove

∞ n2  x converges. n!

n=1

1 1 1 1 3 3 1 − + − + ··· ln − = 2 2 2 4 · 5 · 25 1 · 2 · 22 2 · 3 · 23 3 · 4 · 24 (c) Use a calculator to verify that the partial sum S4 approximates the left-hand side with an error no greater than the term a5 of the series.

56. Find all values of x such that the following series converges: F (x) = 1 + 3x + x 2 + 27x 3 + x 4 + 243x 5 + · · · ∞ 

an x n satisfying the differential

48. Prove that for |x| < 1,

57. Find a power series P (x) =

x9 dx x5 + − ··· = x − 5 9 x4 + 1 " 1/2 Use the firs two terms to approximate 0 dx/(x 4 + 1) numerically. Use the fact that you have an alternating series to show that the error in this approximation is at most 0.00022.

equation y = −y with initial condition y(0) = 1. Then use Theorem 1 of Section 5.8 to conclude that P (x) = e−x .

49. Use the result of Example 7 to show that F (x) =

x2 x4 x6 x8 − + − + ··· 1·2 3·4 5·6 7·8

is an antiderivative of f (x) = tan−1 x satisfying F (0) = 0. What is the radius of convergence of this power series? 50. Verify that function F (x) = x tan−1 x − 12 log(x 2 + 1) is an antiderivative of f (x) = tan−1 x satisfying F (0) = 0. Then use the result of Exercise 49 with x = π6 to show that 1 1 1 4 π 1 1 − + − + ··· √ − ln = 2 3 1 · 2(3) 3 · 4(3 ) 5 · 6(3 ) 7 · 8(34 ) 6 3 2 3 Use a calculator to compare the value of the left-hand side with the partial sum S4 of the series on the right. 51. Evaluate

∞  n . Hint: Use differentiation to show that 2n

n=1

(1 − x)−2 =

∞ 

n=0

x4 x6 x2 + − + ··· . 2! 4! 6! (a) Show that C(x) has an infinit radius of convergence.

58. Let C(x) = 1 −

(b) Prove that C(x) and f (x) = cos x are both solutions of y = −y with initial conditions y(0) = 1, y (0) = 0. This initial value problem has a unique solution, so we have C(x) = cos x for all x. 59. Use the power series for y = ex to show that 1 1 1 1 = − + − ··· e 2! 3! 4! Use your knowledge of alternating series to fin an N such that the partial sum SN approximates e−1 to within an error of at most 10−3 . Confir this using a calculator to compute both SN and e−1 .  60. Let P (x) = an x n be a power series solution to y = 2xy with n=0

initial condition y(0) = 1. (a) Show that the odd coefficient a2k+1 are all zero. (b) Prove that a2k = a2k−2 /k and use this result to determine the coefficient a2k . 61. Find a power series P (x) satisfying the differential equation

nx n−1

(for |x| < 1)

n=1

52. Use the power series for (1 + x 2 )−1 and differentiation to prove

that for |x| < 1,

∞  2x = (−1)n−1 (2n)x 2n−1 (x 2 + 1)2 n=1

y − xy + y = 0

9

with initial condition y(0) = 1, y (0) = 0. What is the radius of convergence of the power series? 62. Find a power series satisfying Eq. (9) with initial condition y(0) = 0, y (0) = 1.

Taylor Series

S E C T I O N 10.7

63. Prove that J2 (x) =

∞ 

(−1)k

k=0

22k+2 k! (k + 3)!

591

64. Why is it impossible to expand f (x) = |x| as a power series that converges in an interval around x = 0? Explain using Theorem 2.

x 2k+2

is a solution of the Bessel differential equation of order 2: x 2 y + xy + (x 2 − 4)y = 0

Further Insights and Challenges 65. Suppose that the coefficient of F (x) =

∞ 

an x n are periodic;

(b) Choose R1 with 0 < R1 < R. Show that the infinit series ∞  M= 2n|an |R1n converges. Hint: Show that n|an |R1n < |an |x n for

n=0

that is, for some whole number M > 0, we have aM+n = an . Prove that F (x) converges absolutely for |x| < 1 and that

n=0

all n sufficientl large if R1 < x < R. (c) Use Eq. (10) to show that if |x| < R1 and |y| < R1 , then |F (x) − F (y)| ≤ M|x − y|. (d) Prove that if |x| < R, then F (x) is continuous at x. Hint: Choose R1 such that |x| < R1 < R. Show that if  > 0 is given, then |F (x) − F (y)| ≤  for all y such that |x − y| < δ, where δ is any positive number that is less than /M and R1 − |x| (see Figure 6).

a + a1 x + · · · + aM−1 x M−1 F (x) = 0 1 − xM Hint: Use the hint for Exercise 53. ∞  66. Continuity of Power Series Let F (x) = an x n be a power series with radius of convergence R > 0. n=0 (a) Prove the inequality

|x n − y n | ≤ n|x − y|(|x|n−1 + |y|n−1 )

x−δ (

10

( 0

−R

Hint: x n − y n = (x − y)(x n−1 + x n−2 y + · · · + y n−1 ).

x+δ ) x

) R1 R

x

FIGURE 6 If x > 0, choose δ > 0 less than /M and R1 − x.

10.7 Taylor Series In this section we develop general methods for findin power series representations. Suppose that f (x) is represented by a power series centered at x = c on an interval (c − R, c + R) with R > 0: f (x) =

∞ 

an (x − c)n = a0 + a1 (x − c) + a2 (x − c)2 + · · ·

n=0

According to Theorem 2 in Section 10.6, we can compute the derivatives of f (x) by differentiating the series expansion term by term: f (x) =

a0 +

a1 (x − c) +

a2 (x − c)2 +

a3 (x − c)3 + · · ·

f (x) =

a1 +

2a2 (x − c) +

3a3 (x − c)2 +

4a4 (x − c)3 + · · ·

f (x) =

2a2 +

2 · 3a3 (x − c) +

3 · 4a4 (x − c)2 + 4 · 5a5 (x − c)3 + · · ·

f (x) = 2 · 3a3 + 2 · 3 · 4a4 (x − 2) + 3 · 4 · 5a5 (x − 2)2 +

···

In general,

f (k) (x) = k!ak + 2 · 3 · · · (k + 1) ak+1 (x − c) + · · · Setting x = c in each of these series, we fin that f (c) = a0 ,

f (c) = a1 ,

f (c) = 2a2 ,

f (c) = 2 · 3a2 ,

...,

f (k) (c) = k!ak ,

...

592

C H A P T E R 10

INFINITE SERIES

We see that ak is the kth coefficien of the Taylor polynomial studied in Section 8.4: ak =

f (k) (c) k!

1

Therefore f (x) = T (x), where T (x) is the Taylor series of f (x) centered at x = c: T (x) = f (c) + f (c)(x − c) +

f (c) f (c) (x − c)2 + (x − c)3 + · · · 2! 3!

This proves the next theorem. THEOREM 1 Taylor Series Expansion If f (x) is represented by a power series centered at c in an interval |x − c| < R with R > 0, then that power series is the Taylor series T (x) =

∞  f (n) (c) n=0

n!

(x − c)n

In the special case c = 0, T (x) is also called the Maclaurin series: f (x) =

∞  f (n) (0) n=0

n!

x n = f (0) + f (0)x +

f (0) 2 f (0) 3 f (4) (0) 4 x + x + x + ··· 2! 3! 4!

E X A M P L E 1 Find the Taylor series for f (x) = x −3 centered at c = 1.

Solution The derivatives of f (x) are f (x) = −3x −4 , f (x) = (−3)(−4)x −5 , and in general, f (n) (x) = (−1)n (3)(4) · · · (n + 2)x −3−n Note that (3)(4) · · · (n + 2) = 12 (n + 2)!. Therefore, 1 f (n) (1) = (−1)n (n + 2)! 2 Noting that (n + 2)! = (n + 2)(n + 1)n!, we write the coefficient of the Taylor series as: an =

(−1)n 21 (n + 2)! (n + 2)(n + 1) f (n) (1) = = (−1)n n! n! 2

The Taylor series for f (x) = x −3 centered at c = 1 is T (x) = 1 − 3(x − 1) + 6(x − 1)2 − 10(x − 1)3 + · · · =

∞  n=0

(−1)n

(n + 2)(n + 1) (x − 1)n 2

Theorem 1 tells us that if we want to represent a function f (x) by a power series centered at c, then the only candidate for the job is the Taylor series: T (x) =

∞  f (n) (c) n=0

n!

(x − c)n

S E C T I O N 10.7

See Exercise 92 for an example where a Taylor series T (x) converges but does not converge to f (x).

Taylor Series

593

However, there is no guarantee that T (x) converges to f (x), even if T (x) converges. To study convergence, we consider the kth partial sum, which is the Taylor polynomial of degree k: Tk (x) = f (c) + f (c)(x − c) +

f (c) f (k) (c) (x − c)2 + · · · + (x − c)k 2! k!

In Section 8.4, we define the remainder Rk (x) = f (x) − Tk (x) Since T (x) is the limit of the partial sums Tk (x), we see that The Taylor series converges to f (x) if and only if lim Rk (x) = 0. k→∞

There is no general method for determining whether Rk (x) tends to zero, but the following theorem can be applied in some important cases. REMINDER f (x) is called “infinitely differentiable” if f (n) (x) exists for all n.

THEOREM 2 Let I = (c − R, c + R), where R > 0. Suppose there exists K > 0 such that all derivatives of f are bounded by K on I : |f (k) (x)| ≤ K

k≥0

for all

and

x∈I

Then f (x) is represented by its Taylor series in I : f (x) =

∞  f (n) (c) n=0

n!

(x − c)n

x∈I

for all

Proof According to the Error Bound for Taylor polynomials (Theorem 2 in Section 8.4), |Rk (x)| = |f (x) − Tk (x)| ≤ K

|x − c|k+1 (k + 1)!

If x ∈ I , then |x − c| < R and |Rk (x)| ≤ K

Taylor expansions were studied throughout the seventeenth and eighteenth centuries by Gregory, Leibniz, Newton, Maclaurin, Taylor, Euler, and others. These developments were anticipated by the great Hindu mathematician Madhava (c. 1340–1425), who discovered the expansions of sine and cosine and many other results two centuries earlier.

R k+1 (k + 1)!

We showed in Example 9 of Section 10.1 that R k /k! tends to zero as k → ∞. Therefore, lim Rk (x) = 0 for all x ∈ (c − R, c + R), as required. k→∞

E X A M P L E 2 Expansions of Sine and Cosine Show that the following Maclaurin expansions are valid for all x.

sin x =

∞  n=0

(−1)n

x3 x5 x7 x 2n+1 =x− + − + ··· (2n + 1)! 3! 5! 7!

594

INFINITE SERIES

C H A P T E R 10

cos x =

∞ 

(−1)n

n=0

x2 x4 x6 x 2n =1− + − + ··· (2n)! 2! 4! 6!

Solution Recall that the derivatives of f (x) = sin x and their values at x = 0 form a repeating pattern of period 4: f (x) sin x 0

f (x) cos x 1

f (x) − sin x 0

f (x) − cos x −1

f (4) (x) sin x 0

··· ··· ···

In other words, the even derivatives are zero and the odd derivatives alternate in sign: f (2n+1) (0) = (−1)n . Therefore, the nonzero Taylor coefficient for sin x are a2n+1 =

(−1)n (2n + 1)!

For f (x) = cos x, the situation is reversed. The odd derivatives are zero and the even derivatives alternate in sign: f (2n) (0) = (−1)n cos 0 = (−1)n . Therefore the nonzero Taylor coefficient for cos x are a2n = (−1)n /(2n)!. We can apply Theorem 2 with K = 1 and any value of R because both sine and cosine satisfy |f (n) (x)| ≤ 1 for all x and n. The conclusion is that the Taylor series converges to f (x) for |x| < R. Since R is arbitrary, the Taylor expansions hold for all x. E X A M P L E 3 Taylor Expansion of f (x) = ex at x = c Find the Taylor series T (x) of

f (x) = ex at x = c.

Solution We have f (n) (c) = ec for all x, and thus T (x) =

∞ c  e n=0

n!

(x − c)n

Because ex

is increasing for all R > 0 we have |f (k) (x)| ≤ ec+R for x ∈ (c − R, c + R). Applying Theorem 2 with K = ec+R , we conclude that T (x) converges to f (x) for all x ∈ (c − R, c + R). Since R is arbitrary, the Taylor expansion holds for all x. For c = 0, we obtain the standard Maclaurin series ex = 1 + x +

x2 x3 + + ··· 2! 3!

Shortcuts to Finding Taylor Series There are several methods for generating new Taylor series from known ones. First of all, we can differentiate and integrate Taylor series term by term within its interval of convergence, by Theorem 2 of Section 10.6. We can also multiply two Taylor series or substitute one Taylor series into another (we omit the proofs of these facts). In Example 4, we can also write the Maclaurin series as ∞  x n+2 n=0

n!

E X A M P L E 4 Find the Maclaurin series for f (x) = x 2 ex .

Solution Multiply the known Maclaurin series for ex by x 2 .   x2 x3 x4 x5 + + + + ··· x 2 ex = x 2 1 + x + 2! 3! 4! 5! = x2 + x3 +

∞

 xn x5 x6 x7 x4 + + + + ··· = 2! 3! 4! 5! (n − 2)! n=2

S E C T I O N 10.7

Taylor Series

595

2

E X A M P L E 5 Substitution Find the Maclaurin series for e−x .

Solution Substitute −x 2 in the Maclaurin series for ex . 2

e−x =

n ∞  (−x 2 ) n=0

n!

=

∞  (−1)n x 2n n=0

n!

= 1 − x2 +

x4 x6 x8 − + − ··· 2! 3! 4!

2

The Taylor expansion of ex is valid for all x, so this expansion is also valid for all x. E X A M P L E 6 Integration Find the Maclaurin series for f (x) = ln(1 + x).

Solution We integrate the geometric series with common ratio −x (valid for |x| < 1): 1 = 1 − x + x2 − x3 + · · · 1+x ∞

ln(1 + x) =

 dx xn x2 x3 x4 (−1)n−1 =x− + − + ··· = 1+x 2 3 4 n n=1

The constant of integration on the right is zero because ln(1 + x) = 0 for x = 0. This expansion is valid for |x| < 1. It also holds for x = 1 (see Exercise 84). In many cases, there is no convenient general formula for the Taylor coefficients but we can still compute as many coefficient as desired. E X A M P L E 7 Multiplying Taylor Series Write out the terms up to degree fiv in the Maclaurin series for f (x) = ex cos x.

Solution We multiply the fifth-orde Taylor polynomials of ex and cos x together, dropping the terms of degree greater than 5:    x2 x3 x4 x5 x2 x4 1+x+ + + + 1− + 2 6 24 120 2 24 Distributing the term on the left (and ignoring terms of degree greater than 5), we obtain     2  4 x2 x3 x4 x5 x2 x3 x x 1+x+ + + + − 1+x+ + + (1 + x) 2 6 24 120 2 6 2 24 y

x3 x4 x5 =1+x− − − 3  6 30

1

Retain terms of degree ≤ 5

1

2

3

x

We conclude that the fift Maclaurin polynomial for f (x) = ex cos x is T5 (x) = 1 + x −

−1 FIGURE 1 Graph of T12 (x) for the power

series expansion of the antiderivative F (x) =

x 0

sin(t 2 ) dt

x4 x5 x3 − − 3 6 30

In the next example, we express the definit integral of sin(x 2 ) as an infinit series. This is useful because the integral cannot be evaluated explicitly. Figure 1 shows the graph of the Taylor polynomial T12 (x) of the Taylor series expansion of the antiderivative. E X A M P L E 8 Let J =

1 0

sin(x 2 ) dx.

(a) Express J as an infinit series. (b) Determine J to within an error less than 10−4 .

596

C H A P T E R 10

INFINITE SERIES

Solution (a) The Maclaurin expansion for sin x is valid for all x, so we have ∞  (−1)n 2n+1 x sin x = (2n + 1)!

∞  (−1)n 4n+2 sin(x ) = x (2n + 1)! 2

⇒

n=0

n=0

We obtain an infinit series for J by integration: J = =

1 0

sin(x 2 ) dx =

∞  (−1)n (2n + 1)!

1 0

n=0

x 4n+2 dx =

  ∞  (−1)n 1 (2n + 1)! 4n + 3 n=0

1 1 1 1 − + − + ··· 3 42 1320 75,600

3

(b) The infinit series for J is an alternating series with decreasing terms, so the sum of the firs N terms is accurate to within an error that is less than the (N + 1)st term. The absolute value of the fourth term 1/75,600 is smaller than 10−4 so we obtain the desired accuracy using the firs three terms of the series for J : J ≈ The error satisfie

1 1 1 − + ≈ 0.31028 3 42 1320

     1 J − 1 − 1 + 1 < ≈ 1.3 × 10−5  3 42 1320  75,600

The percentage error is less than 0.005% with just three terms.

Binomial Series Isaac Newton discovered an important generalization of the Binomial Theorem around 1665. For any number a (integer or not) and integer n ≥ 0, we defin the binomial coefficient:     a a(a − 1)(a − 2) · · · (a − n + 1) a = , =1 n n! 0 For example,   6 6·5·4 = = 20, 3 3·2·1

4 3

3

=

4 3

·

  · − 23 4 =− 3·2·1 81 1 3

Let f (x) = (1 + x)a The Binomial Theorem of algebra (see Appendix C) states that for any whole number a,       a a−1 a a−2 2 a (r + s)a = r a + r s+ r s + ··· + rs a−1 + s a 1 2 a−1 Setting r = 1 and s = x, we obtain the expansion of f (x):       a a 2 a a (1 + x) = 1 + x+ x + ··· + x a−1 + x a 1 2 a−1

Taylor Series

S E C T I O N 10.7

597

We derive Newton’s generalization by computing the Maclaurin series of f (x) without assuming that a is a whole number. Observe that the derivatives follow a pattern: f (x) = (1 + x)a

f (0) = 1

f (x) = a(1 + x)a−1

f (0) = a

f (x) = a(a − 1)(1 + x)a−2

f (0) = a(a − 1)

f (x) = a(a − 1)(a − 2)(1 + x)a−3

f (0) = a(a − 1)(a − 2)

In general, f (n) (0) = a(a − 1)(a − 2) · · · (a − n + 1) and   f (n) (0) a(a − 1)(a − 2) · · · (a − n + 1) a = = n! n! n Hence the Maclaurin series for f (x) = (1 + x)a is the binomial series

a 

When a is a whole number, n is zero for n > a , and in this case, the binomial series breaks off at degree n. The binomial series is an infinite series when a is not a whole number.

  ∞    a(a − 1) 2 a(a − 1)(a − 2) 3 a n a n x + ··· x + x + ··· + x = 1 + ax + n 2! 3! n n=0

The Ratio Test shows that this series has radius of convergence R = 1 (Exercise 86) and an additional argument (developed in Exercise 87) shows that it converges to (1 + x)a for |x| < 1 . THEOREM 3 The Binomial Series

For any exponent a and for |x| < 1,

  a a n a(a − 1) 2 a(a − 1)(a − 2) 3 (1 + x) = 1 + x + x + ··· x + x + ··· + n 1! 2! 3! a

E X A M P L E 9 Find the terms through degree four in the Maclaurin expansion of

f (x) = (1 + x)4/3   Solution The binomial coefficient an for a = 43 for 0 < n < 4 are 1,

 

4 3

4 1 3 3

4 = , 1! 3

2!

2 = , 9

 

4 1 3 3

− 3!

Therefore, (1 + x)4/3 ≈ 1 + 43 x + 29 x 2 −

2 3



4 3 81 x

 

4 1 3 3

4 =− , 81 +

5 4 243 x

− 23 4!



−

5 3

 =

+ ··· .

E X A M P L E 10 Find the Maclaurin series for

f (x) = √

1 1 − x2

Solution First, let’s fin the coefficient in the binomial series for (1 + x)−1/2 :      − 12 − 12 − 32 − 12 − 32 − 52 1 1·3 1·3·5 1, =− , = , = 1! 2 1·2 2·4 1·2·3 2·4·6 The general pattern is 

− 12 n



    − 12 − 32 − 52 · · · − = 1 · 2 · 3···n

2n−1 2

 = (−1)n

1 · 3 · 5 · · · (2n − 1) 2 · 4 · 6 · 2n

5 243

598

INFINITE SERIES

C H A P T E R 10

Thus, the following binomial expansion is valid for |x| < 1: √

1

=1+

1+x

∞ 

(−1)n

n=1

1 · 3 · 5 · · · (2n − 1) n 1·3 2 1 x =1− x+ x − ··· 2 · 4 · 6 · · · (2n) 2 2·4

If |x| < 1, then |x|2 < 1, and we can substitute −x 2 for x to obtain √

θ

FIGURE 2 Pendulum released at an angle θ .

1 1 − x2

=1+

∞  1 · 3 · 5 · · · (2n − 1) n=1

2 · 4 · 6 · · · 2n

E(k) =

8 Small-angle approximation

4 2 π 2

π

Angle θ

FIGURE 3 The period T of a 1-meter

pendulum as a function of the angle θ at which it is released.

4

Taylor series are particularly useful for studying the so-called special functions (such as Bessel and hypergeometric functions) that appear in a wide range of physics and engineering applications. One example is the following elliptic function of the first kind, define for |k| < 1:

Period T 6

1 1·3 4 x 2n = 1 + x 2 + x + ··· 2 2·4

π/2 0



dt 1 − k 2 sin2 t

This function is used in physics to compute the period T of pendulum of length L released √ from an angle θ (Figure 2). We can use the “small-angle approximation” T ≈ 2π L/g when θ is small, but this approximation breaks down for large angles (Figure 3). The exact √ value of the period is T = 4 L/gE(k), where k = sin 12 θ. E X A M P L E 11 Elliptic Function Find the Maclaurin series for E(k) and estimate E(k) for k = sin π6 .

Solution Substitute x = k sin t in the Taylor expansion (4): 

1·3 4 4 1·3·5 6 6 1 k sin t + k sin t + · · · = 1 + k 2 sin2 t + 2 2 · 4 2·4·6 1 − k 2 sin t 1

2

This expansion is valid because |k| < 1 and hence |x| = |k sin t| < 1. Thus E(k) is equal to π/2 0



dt 1 − k 2 sin2 t

=

π/2 0

dt +

∞  1 · 3 · · · (2n − 1) n=1



2 · 4 · (2n)

0

According to Exercise 78 in Section 7.2, π/2 0

sin

2n

 t dt =

1 · 3 · · · (2n − 1) 2 · 4 · (2n)



π 2

This yields E(k) =

2 ∞  π π  1 · 3 · · · (2n − 1)2 k 2n + 2 2 2 · 4 · · · (2n) n=1



π/2

sin

2n

t dt k 2n

Taylor Series

S E C T I O N 10.7

599

  We approximate E(k) for k = sin π6 = 12 using the firs fiv terms:        2  2  1 1 π 1 1·3 2 1 4 E ≈ 1+ + 2 2 2 2 2·4 2 

1·3·5 + 2·4·6

 2  6      1 1·3·5·7 2 1 8 + 2 2·4·6·8 2

≈ 1.68517 The value given by a computer algebra system to seven places is E

1 2

≈ 1.6856325.

TABLE 1 Function f (x)

Maclaurin series

ex

Converges to f (x) for

∞ n  x2 x3 x4 x =1+x+ + + + ··· n! 2! 3! 4!

All x

∞  (−1)n x 2n+1 x3 x5 x7 =x− + − + ··· (2n + 1)! 3! 5! 7!

All x

∞  x2 x4 x6 (−1)n x 2n =1− + − + ··· (2n)! 2! 4! 6!

All x

n=0

sin x

n=0

cos x

n=0 ∞ 

1 1−x

xn = 1 + x + x2 + x3 + x4 + · · ·

|x| < 1

(−1)n x n = 1 − x + x 2 − x 3 + x 4 − · · ·

|x| < 1

n=0 ∞ 

1 1+x

n=0

ln(1 + x)

∞  x2 x3 x4 (−1)n−1 x n =x− + − + ··· n 2 3 4

|x| < 1 and x = 1

∞  x3 x5 x7 (−1)n x 2n+1 =x− + − + ··· 2n + 1 3 5 7

|x| < 1 and x = 1

n=1

tan−1 x

n=0

∞    a

(1 + x)a

n=0

n

x n = 1 + ax +

a(a − 1) 2 a(a − 1)(a − 2) 3 x + x + ··· 2! 3!

10.7 SUMMARY •

Taylor series of f (x) centered at x = c: T (x) =

∞  f (n) (c) (x − c)n n! n=0

The partial sum Tk (x) is the kth Taylor polynomial. • Maclaurin series (c = 0): ∞  f (n) (0) n T (x) = x n! n=0

|x| < 1

600

C H A P T E R 10

INFINITE SERIES

•

If f (x) is represented by a power series

∞ 

an (x − c)n for |x − c| < R with R > 0,

n=0

then this power series is necessarily the Taylor series centered at x = c. • A function f (x) is represented by its Taylor series T (x) if and only if the remainder Rk (x) = f (x) − Tk (x) tends to zero as k → ∞. • Let I = (c − R, c + R) with R > 0. Suppose that there exists K > 0 such that |f (k) (x)| < K for all x ∈ I and all k. Then f (x) is represented by its Taylor series on I ; that is, f (x) = T (x) for x ∈ I . • A good way to fin the Taylor series of a function is to start with known Taylor series and apply one of the operations: differentiation, integration, multiplication, or substitution. • For any exponent a, the binomial expansion is valid for |x| < 1:   a(a − 1) 2 a(a − 1)(a − 2) 3 a n a (1 + x) = 1 + ax + x + x + ··· + x + ··· 2! 3! n

10.7 EXERCISES Preliminary Questions 1. Determine f (0) and f (0) for a function f (x) with Maclaurin series

4. Find the Taylor series for f (x) centered at c = 3 if f (3) = 4 and f (x) has a Taylor expansion

T (x) = 3 + 2x + 12x 2 + 5x 3 + · · ·

f (x) =

2. Determine f (−2) and f (4) (−2) for a function with Taylor series T (x) = 3(x + 2) + (x + 2)2 − 4(x + 2)3 + 2(x + 2)4 + · · · 3. What is the easiest way to fin the Maclaurin series for the function f (x) = sin(x 2 )?

n=1

5. Let T (x) be the Maclaurin series of f (x). Which of the following guarantees that f (2) = T (2)? (a) T (x) converges for x = 2. (b) The remainder Rk (2) approaches a limit as k → ∞. (c) The remainder Rk (2) approaches zero as k → ∞.

Exercises 1. Write out the firs four terms of the Maclaurin series of f (x) if f (0) = 2,

f (0) = 3,

f (0) = 4,

f (0) = 12

2. Write out the firs four terms of the Taylor series of f (x) centered at c = 3 if f (3) = 1,

f (3) = 2,

f (3) = 12,

f (3) = 3

In Exercises 3–18, find the Maclaurin series and find the interval on which the expansion is valid. 1 1 − 2x

∞  (x − 3)n n

15. f (x) = ln(1 − 5x)

16. f (x) = (x 2 + 2x)ex

17. f (x) = sinh x

18. f (x) = cosh x

In Exercises 19–28, find the terms through degree four of the Maclaurin series of f (x). Use multiplication and substitution as necessary. 19. f (x) = ex sin x 21. f (x) =

sin x 1−x

20. f (x) = ex ln(1 − x) 22. f (x) =

1 1 + sin x

x 1 − x4

23. f (x) = (1 + x)1/4

24. f (x) = (1 + x)−3/2

5. f (x) = cos 3x

6. f (x) = sin(2x)

25. f (x) = ex tan−1 x

26. f (x) = sin (x 3 − x)

7. f (x) = sin(x 2 )

8. f (x) = e4x

27. f (x) = esin x

x 28. f (x) = e(e )

3. f (x) =

9. f (x) = ln(1 − x 2 )

4. f (x) =

10. f (x) = (1 − x)−1/2

11. f (x) = tan−1 (x 2 )

12. f (x) = x 2 ex

13. f (x) = ex−2

14. f (x) =

2

1 − cos x x

In Exercises 29–38, find the Taylor series centered at c and find the interval on which the expansion is valid. 29. f (x) =

1 , c=1 x

30. f (x) = e3x ,

c = −1

Taylor Series

S E C T I O N 10.7

31. f (x) =

1 , 1−x

c=5

32. f (x) = sin x, c =

33. f (x) = x 4 + 3x − 1,

c=2

34. f (x) = x 4 + 3x − 1,

c=0

1 35. f (x) = 2 , x

c=4

1 , 37. f (x) = 1 − x2

36. f (x) =

c=3

π 2

39. Use the identity cos2 x = 12 (1 + cos 2x) to fin the Maclaurin series for cos2 x. 40. Show that for |x| < 1, tanh−1 x = x + Hint: Recall that

y F(x)

√ x, c = 4

1 , c = −1 38. f (x) = 3x − 2

T15(x) 1

2

x

FIGURE 4 The Maclaurin polynomial T15 (x) for F (t) =

48. Let F (x) =

x3 x5 + + ··· 3 5

x sin t dt

t

0

F (x) = x −

x 0

2

e−t dt.

. Show that

x3 x5 x7 + − + ··· 3 · 3! 5 · 5! 7 · 7!

Evaluate F (1) to three decimal places.

1 d tanh−1 x = . dx 1 − x2

41. Use the Maclaurin series for ln(1 + x) and ln(1 − x) to show that   x3 x5 1+x 1 ln =x+ + + ··· 2 1−x 3 5

In Exercises 49–52, express the definite integral as an infinite series and find its value to within an error of at most 10−4 . 1

49.

0 1

1

cos(x 2 ) dx

50.

0

tan−1 (x 2 ) dx

1

3

e−x dx

dx  0 x4 + 1

for |x| < 1. What can you conclude by comparing this result with that of Exercise 40?

51.

1 42. Differentiate the Maclaurin series for twice to fin the 1 − x 1 . Maclaurin series of (1 − x)3

In Exercises 53–56, express the integral as an infinite series.

1 , 43. Show, by integrating the Maclaurin series for f (x) =  1 − x2 that for |x| < 1, ∞  1 · 3 · 5 · · · (2n − 1) x 2n+1 sin−1 x = x + 2 · 4 · 6 · · · (2n) 2n + 1 n=1

53. 54.

45. How many terms of the Maclaurin series of f (x) = ln(1 + x) are needed to compute ln 1.2 to within an error of at most 0.0001? Make the computation and compare the result with the calculator value.

0

52.

x 1 − cos(t)

t

0

x t − sin t

t

0 x

55.

0

for all x

dt,

dt,

for all x

ln(1 + t 2 ) dt,

x

56.

44. Use the firs fiv terms of the Maclaurin series in Exercise 43 to approximate sin−1 12 . Compare the result with the calculator value.

dt ,  0 1 − t4

π3 π5 π7 + − + ··· 3! 5! 7!

converges to zero. How many terms must be computed to get within 0.01 of zero? 2

47. Use the Maclaurin expansion for e−t to express the function " 2 F (x) = 0x e−t dt as an alternating power series in x (Figure 4). (a) How many terms of the Maclaurin series are needed to approximate the integral for x = 1 to within an error of at most 0.001? Carry out the computation and check your answer using a (b) computer algebra system.

for |x| < 1

for |x| < 1

57. Which function has Maclaurin series

∞ 

(−1)n 2n x n ?

n=0

58. Which function has Maclaurin series ∞  (−1)k (x − 3)k ? 3k+1

46. Show that π−

601

k=0

For which values of x is the expansion valid? In Exercises 59–62, use Theorem 2 to prove that the f (x) is represented by its Maclaurin series on the interval I .   59. f (x) = ln(1 + x), I = − 12 , 12 60. f (x) = e−x ,

I = (−c, c) for all c > 0

61. f (x) = sinh x, I = R (see Exercise 17) 62. f (x) = (1 + x)100 ,

I =R

602

C H A P T E R 10

INFINITE SERIES

In Exercises 63–66, find the functions with the following Maclaurin series (refer to Table 1 on page 599). 63. 1 + x 3 +

x9 x 12 x6 + + + ··· 2! 3! 4!

20

rin series for f (x) = ex . How does the result show that f (k) (0) = 0 for 1 ≤ k ≤ 19?  74. Use the binomial series to fin f (8) (0) for f (x) = 1 − x 2 .

55 x 5 57 x 7 53 x 3 + − + ··· 3! 5! 7!

75. Does the Maclaurin series for f (x) = (1 + x)3/4 converge to f (x) at x = 2? Give numerical evidence to support your answer.

x 20 x 28 x 12 + − + ··· 3 5 7 In Exercises 67 and 68, let

66. x 4 −

1 f (x) = (1 − x)(1 − 2x) 67. Find the Maclaurin series of f (x) using the identity f (x) =

1 2 − 1−x 1 − 2x

68. Find the Taylor series for f (x) at c = 2. Hint: Rewrite the identity of Exercise 67 as 1 2 − f (x) = −3 − 2(x − 2) −1 − (x − 2) 69. When a voltage V is applied to a series circuit consisting of a resistor R and an inductor L, the current at time t is    V  I (t) = 1 − e−Rt/L R Expand I (t) in a Maclaurin series. Show that I (t) ≈

Use substitution to fin the firs three terms of the Maclau-

73.

64. 1 − 4x + 42 x 2 − 43 x 3 + 44 x 4 − 45 x 5 + · · · 65. 1 −

72. Find f (7) (0) and f (8) (0) for f (x) = tan−1 x using the Maclaurin series.

Vt for small t. L

70. Use the result of Exercise 69 and your knowledge of alternating series to show that   R Vt Vt 1− t ≤ I (t) ≤ (for all t) L 2L L 71. Find the Maclaurin series for f (x) = cos(x 3 ) and use it to determine f (6) (0).

76. Explain the steps required to verify that the Maclaurin series for f (x) = ex converges to f (x) for all x. √ 77. Let f (x) = 1 + x. (a) Use a graphing calculator to compare the graph of f with the graphs of the firs fiv Taylor polynomials for f . What do they suggest about the interval of convergence of the Taylor series? (b) Investigate numerically whether or not the Taylor expansion for f is valid for x = 1 and x = −1. 78. Use the firs fiv terms of the Maclaurin series for the elliptic function E(k) to estimate the period T of a 1-meter pendulum released at an angle θ = π4 (see Example 11). 79. Use Example 11 and the approximation sin x ≈ x to show that the period T of a pendulum released at an angle θ has the following second-order approximation:    θ2 L T ≈ 2π 1+ g 16 In Exercises 80–83, find the Maclaurin series of the function and use it to calculate the limit. 80. lim

2 cos x − 1 + x2

82. lim

81. lim

x4

x→0

x5

x→0

tan−1 x − x cos x − 16 x 3

x→0

3 sin x − x + x6

x5



sin(x 2 ) cos x 83. lim − 2 x→0 x4 x



Further Insights and Challenges 84. In this exercise we show that the Maclaurin expansion of f (x) = ln(1 + x) is valid for x = 1. (a) Show that for all x  = −1, N  1 (−1)N+1 x N+1 = (−1)n x n + 1+x 1+x n=0

(b) Integrate from 0 to 1 to obtain ln 2 =

N 1 x N+1 dx  (−1)n−1 + (−1)N+1 n 1+x 0 n=1

(c) Verify that the integral on the right tends to zero as N → ∞ by " showing that it is smaller than 01 x N+1 dx. (d) Prove the formula ln 2 = 1 −

1 1 1 + − + ··· 2 3 4

t 1 − . 1 + t2 1 + t2 1 π 1 g(t) dt = − ln 2. (a) Show that 4 2 0 (b) Show that g(t) = 1 − t − t 2 + t 3 − t 4 − t 5 + · · · . 85. Let g(t) =

(c) Evaluate S = 1 − 12 − 13 + 41 − 15 − 16 + · · · .

Chapter Review Exercises

y

In Exercises 86 and 87, we investigate the convergence of the binomial series Ta (x) =

b

∞    a n=0

n

xn a

86. Prove that Ta (x) has radius of convergence R = 1 if a is not a whole number. What is the radius of convergence if a is a whole number? 87. By Exercise 86, Ta (x) converges for |x| < 1, but we do not yet know whether Ta (x) = (1 + x)a . (a) Verify the identity a

      a a a =n + (n + 1) n n n+1

(b) Use (a) to show that y = Ta (x) satisfie the differential equation (1 + x)y = ay with initial condition y(0) = 1. (c) Prove that Ta (x) = (1 + x)a for |x| < 1 by showing that the Ta (x) derivative of the ratio is zero. (1 + x)a " π/2  1 − k 2 sin2 t dt is called an elliptic 88. The function G(k) = 0 function of the second kind. Prove that for |k| < 1, G(k) =

603

 ∞  π  1 · 3 · · · (2n − 1) 2 k 2n π − 2 2 2 · · · 4 · (2n) 2n − 1 n=1

89. Assume that a < b and let L be the arc length (circumference) of  2  2 the ellipse xa + yb = 1 shown in Figure 5. There is no explicit formula for L, but  it is known that L = 4bG(k), with G(k) as in Exercise 88 and k = 1 − a 2 /b2 . Use the firs three terms of the expansion of Exercise 88 to estimate L when a = 4 and b = 5.

FIGURE 5 The ellipse

x 2 a

x

+

y 2 b

= 1.

90. Use Exercise 88 to prove that if a < b and a/b is near 1 (a nearly circular ellipse), then π a2

L≈ 3b + 2 b Hint: Use the firs two terms of the series for G(k). 91. Irrationality of e Prove that e is an irrational number using the following argument by contradiction. Suppose that e = M/N , where M, N are nonzero integers. (a) Show that M! e−1 is a whole number. (b) Use the power series for ex at x = −1 to show that there is an integer B such that M! e−1 equals   1 1 B + (−1)M+1 − + ··· M + 1 (M + 1)(M + 2) (c) Use your knowledge of alternating series with decreasing terms to conclude that 0 < |M! e−1 − B| < 1 and observe that this contradicts (a). Hence, e is not equal to M/N . 92. Use the result of Exercise 73 in Section 4.5 to show that the Maclaurin series of the function  2 e−1/x for x  = 0 f (x) = 0 for x = 0 is T (x) = 0. This provides an example of a function f (x) whose Maclaurin series converges but does not converge to f (x) (except at x = 0).

CHAPTER REVIEW EXERCISES 1. Let an = each sequence.

n−3 and bn = an+3 . Calculate the firs three terms in n!

(a) an2

(b) bn

(c) an bn

(d) 2an+1 − 3an

2. Prove that lim

2n − 1

n→∞ 3n + 2

=

2 using the limit definition 3

In Exercises 3–8, compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞

3. lim (5an − 2an2 ) n→∞

1 n→∞ an

4. lim

5. lim ean

6. lim cos(π an )

7. lim (−1)n an

8. lim

n→∞

n→∞

n→∞

an + n n→∞ an + n2

In Exercises 9–22, determine the limit of the sequence or show that the sequence diverges. 10. an =

3n3 − n 1 − 2n3

11. an = 21/n

12. an =

10n n!

13. bm = 1 + (−1)m

14. bm =

1 + (−1)m m

9. an =

√ √ n+5− n+2 2

604

INFINITE SERIES

C H A P T E R 10

 n+2 n+5   17. bn = n2 + n − n2 + 1 15. bn = tan−1

18. cn =



  n2 + n − n2 − n

16. an =

3 + πn 100n − n! 5n

35. Evaluate S =

n=3

 19. bm = 1 +

1 m

3m

22. cn =

23. Use the Squeeze Theorem to show that lim

arctan(n2 ) √ n

= 0.

37.

39.

 26. Defin an+1 = an + 6 with a1 = 2. (a) Compute an for n = 2, 3, 4, 5.

n→∞

∞  n−2 . n2 + 2n

1 1 1 + − 3 + ··· . 4 42 4

4 8 16 32 + + + + ··· . 9 27 81 243 ∞  n  2 30. Find the sum . e n=2

.

∞  

 π b − tan−1 n2 diverges if b = . 2

∞ ∞   33. Give an example of divergent series an and bn such that ∞  n=1 n=1 (an + bn ) = 1. n=1

34. Let S =

 1 − . Compute SN for N = 1, 2, 3, 4. n n+2

∞   1 n=1

Find S by showing that SN =

1 (n + 2)(ln(n + 2))3

40.

∞ 

∞  n2 + 1 43. n3.5 − 2

44.

n=1

n2 (n3 + 1)1.01

∞  n3 4

n n=1 e

45.

47.

n  5 n=2 n + 5

n=1

∞  n=1

∞ 

∞ 

∞  n=1

n=2

46.

∞  n=1

n10 + 10n n11 + 11n

48.

1 √ n+n 1 n − ln n 1 3n − 2n

∞ 20  n + 21n n21 + 20n

n=1 ∞ 

2n + n using the Limit Com49. Determine the convergence of  2 n 3n − 2 parison Test with bn = 3 . n=1 ∞  ln n 50. Determine the convergence of n using the Limit Compar1.5 1 n=1 ison Test with bn = . 1.4n  ∞  51. Let an = 1 − 1 − n1 . Show that lim an = 0 and that an n→∞ 1 . diverges. Hint: Show that an ≥ 2n n=1    ∞  1 1 − 1 − 2 converges. 52. Determine whether n n=2

53. Let S =

∞  n=1

1 1 3 − − 2 N +1 N +2

∞  n=1

42.

29. Find the sum

n=1

38.

1 (n + 1)2

41.

n=1

32. Show that

n2 n3 + 1

In Exercises 41–48, use the Comparison or Limit Comparison Test to determine whether the infinite series converges.

(c) Prove that lim an exists and fin its value.

3n

∞  n=1

(b) Show that {an } is increasing and is bounded by 3.

n=−1

∞  n=1

an+1 1 1 , where an = 3n − 2n . n→∞ an 2 3

2n+3

1

1 2

In Exercises 37–40, use the Integral Test to determine whether the infinite series converges.

25. Calculate lim

27. Calculate the partial sums S4 and S7 of the series

1 4

FIGURE 1

24. Give an example of a divergent sequence {an } such that {sin an } is convergent.

∞ 

1 8

ln(n2 + 1) ln(n3 + 1)

n→∞

31. Find the sum

x

0

  21. bn = n ln(n + 1) − ln n

1 . n(n + 3)

36. Find the total area of the infinitel many circles on the interval [0, 1] in Figure 1.

  3 n 20. cn = 1 + n

28. Find the sum 1 −

∞ 

n . (n2 + 1)2

(a) Show that S converges. (b) Use Eq. (4) in Exercise 83 of Section 10.3 with M = 99 to approximate S. What is the maximum size of the error?

Chapter Review Exercises

In Exercises 54–57, determine whether the series converges absolutely. If it does not, determine whether it converges conditionally. 54.

∞  (−1)n √ 3 n + 2n

55.

n=1

∞  n=1

  ∞  cos π4 + πn 56. √ n

(−1)n 1.1 n ln(n + 1)

  ∞  cos π4 + 2πn 57. √ n

n=1

n=1

59. Catalan’s constant is define by K =

∞  k=0

(−1)k . (2k + 1)2

(a) How many terms of the series are needed to calculate K with an error of less than 10−6 ? Carry out the calculation.

(b)

∞  1 71. 4n

73.

∞ 

(c)

1

(d)

1 + an2 n=1

n=1 ∞ 

77.

n=1

81.

83.

87.

√ n a = 1 . Detern 2 n→∞ mine whether the following series converge or diverge: ∞ ∞ ∞    √ (a) 2an (b) 3n an (c) an n=1

n=1

In Exercises 63–70, apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. ∞ 5 ∞ √   n n+1 63. 64. 5n n8 n=1

65.

∞  n=1

n=1

1 n2n + n3

∞ 4  n 66. n! n=1

∞ n2  2 67. n!

∞  ln n 68. n3/2

∞

 n n 1 69. 2 n!

∞

 n n 1 70. 4 n!

n=1

n=1

n=4

n=1

∞  

cos

n=1

 3 1 n n

∞ 

n=1

e−0.02n

78.

n=1

(−1)n−1 √ √ n+ n+1 n=1

80.

∞  (−1)n ln n

82.

89.

84.

 ∞   1 1 √ −√ n n+1 n=1

86.

n=2

91.

∞ 

1 √ n+ n 1

n=1

1 n(ln n)3/2

∞ n  e n!

π n

∞ 

√ 3

n=1 ∞  

1 √ n(1 + n)

 ln n − ln(n + 1)

n=1

88.

∞  cos(π n) n2/3

n=2

90.

nln n sin2

∞  n=10

1 √ n n + ln n n=1

∞ 

ne−0.02n

n=1

∞ 

∞ 

∞  n=1

∞ 

n=1

62. Let {an } be a positive sequence such that lim

n=1

74.

n=1

85.

|an | n

n

In Exercises 75–92, determine convergence or divergence using any method covered in the text. ∞  n ∞   2 π 7n 76. 75. 3 e8n

n=1

the following series are convergent or divergent:  ∞  ∞   1 (a) (−1)n an an + 2 (b) n n=1 ∞ 

 ∞   3 n 4n

n=2

an be an absolutely convergent series. Determine whether

∞  n  2 n=1

n=1

n=1

61. Let

72.

n=1

79.

∞  an and 60. Give an example of conditionally convergent series ∞ ∞   n=1 bn such that (an + bn ) converges absolutely. n=1

In Exercises 71–74, apply the Root Test to determine convergence or divergence, or state that the Root Test is inconclusive.

n=1

Use a computer algebra system to approximate 58. ∞  (−1)n √ to within an error of at most 10−5 . n3 + n

605

92.

∞ 

1

ln3 n n=2 ∞ 2n  2 n!

n=0

In Exercises 93–98, find the interval of convergence of the power series. 93.

∞ n n  2 x n!

94.

n=0

95.

97.

∞ 

n6

n=0

n8 + 1

∞  n=0

(nx)n

∞  xn n+1

n=0

(x − 3)n

96.

∞ 

nx n

n=0

98.

∞  (2x − 3)n n ln n

n=0

2 as a power series centered at c = 0. De99. Expand f (x) = 4 − 3x termine the values of x for which the series converges.

606

INFINITE SERIES

C H A P T E R 10

105. f (x) = x 4 ,

100. Prove that ∞ 

ne−nx =

n=0

e−x (1 − e−x )2

106. f (x) = x 3 − x, 107. f (x) = sin x,

Hint: Express the left-hand side as the derivative of a geometric series. 109. f (x) =

∞  x 2k . 101. Let F (x) = 2k · k!

y(0) = 1,

102. Find a power series P (x) = differential equation n=0

c=π

y (0) = 0

an x n that satisfie the Laguerre

with initial condition satisfying P (0) = 1.

111. f (x) = ln

x , 2

113. f (x) = (x 2 − x)ex 1 1 + tan x

117. Calculate

2

c=0

104. f (x) = e2x ,

c = −1

c=2

114. f (x) = tan−1 (x 2 − x) √ 116. f (x) = (sin x) 1 + x

π7 π3 π π5 − 3 + 5 − 7 + ··· . 2 2 3! 2 5! 2 7!

In Exercises 103–112, find the Taylor series centered at c. 103. f (x) = e4x ,

c = −1

In Exercises 113–116, find the first three terms of the Maclaurin series of f (x) and use it to calculate f (3) (0).

115. f (x) =

xy + (1 − x)y − y = 0

108. f (x) = ex−1 ,

1 , c = −2 1 − 2x

110. f (x) =

(c) Plot the partial sums SN for N = 1, 3, 5, 7 on the same set of axes. ∞ 

c = −2

1 , c = −2 (1 − 2x)2 x

, c=0 112. f (x) = x ln 1 + 2

k=0

(a) Show that F (x) has infinit radius of convergence. (b) Show that y = F (x) is a solution of y = xy + y,

c=2

118. Find the Maclaurin series of the function F (x) =

x et − 1 0

t

dt.

CHAPTER 10 INFINITE SERIES PREPARING FOR THE AP EXAM

Multiple Choice Questions Use scratch paper as necessary to solve each problem. Pick the best answer from the choices provided. All questions cover BC topics. 1. If the radius of convergence for the series

∞  n=0

3. If the series

∞  n=0

cn x n converges for x = 6, then which of

the following must be true? I The series converges for x = −6.

n

cn (x + 4) is

II The series converges for x = −3.

5, then which of the following must be true? I The series converges for x = −8.

III The series diverges for x = 8. (A) I only

II The series converges for x = −1. III The series converges for x = 1. (A) I only

(B) II only

(B) II only

(D) II and III only

(C) I and III only

(C) I and II only

(E) I, II, and III

(D) II and III only (E) I, II, and III ∞  3 = 2. 5n n=2

(A) (B) (C) (D) (E)

3 20 9 20 9 10 5 2 15 4

4.

∞  2 + 3n

5n

n=0

(A)

15 4

(B)

25 6

(C)

9 2

is

(D) 5 (E) divergent AP10-1

AP10-2

5. If

CHAPTER 10 N 

ck =

k=0

INFINITE SERIES

∞  3N 2 + 7 ck is , then 2 2N + 5 k=0

N→∞

(A)

2 5

(B)

7 5

(B)

1 2

(C)

10 7

(C)

5 11

(D)

3 2

(D)

5 3

∞   1

k

k=2

−

1 k+1

(B)

1 6

(C)

1 2

 11. The series 2 −

is

(C)

1 1 − k k+2

e2 − 1 e2

e2 + 1 e2 (E) divergent

(D)

 is

(A) 0 (B)

12.

1 2

∞  2n + 5 converges only for np + 7 n=1

(C) 1

(A) p > 1

3 2

(B) p ≥ 2

(D)

(C) p > 2

(E) divergent 8. Let a1 = 2 and an+1 lim an = L, then L =

n→∞

(A) 2 (B) 4 (C) 6 (D) 8 (E) ∞  9. lim

n→∞

1−

1 n

1 e

  1 36 . Given that = an + 2 an

(D) p ≥ 3 (E) no values of p 1 1 − 14 + 64 − 16 + · · · + an + · · · , 13. For the series 14 − 12 + 16 ⎧ 1 ⎪ ⎨ n+1 if n is odd where an = 2 , which of the following ⎪ ⎩− 1 if n is even n is true? I lim an = 0 n→0

n

(A) 0 (B)

(−2)n 4 2 + + ··· + + · · · is 3 3 n!

1 − e2 e2 1 (B) 2 e

(E) divergent

k=1

is

(A)

(D) 1 ∞  

5k + 6

(E) nonexistent

(A) 0

7.

k=0

(A) 1

(E) divergent 6.

N  2k + 3

10. lim

II The series is an alternating series. is

III The series converges. (A) II only (B) I and II only

(C) 1

(C) II and III only

(D) e

(D) I and III only

(E) nonexistent

(E) I, II, and III

PREPARING FOR THE AP EXAM

14. The interval of convergence for the series

∞  x 3n is n8n n=1

(A) (−8, 8) (B) [−8, 8) (C) (−2, 2) (D) [−2, 2) (E) [−1, 1) ∞  cn x n , the Maclaurin series for f (x), has radius of 15. If n=0

convergence equal to R, which of the following must be true? ∞  cn x n converges absolutely on (−R, R). I The series n=0 ∞ 

II The series x = −R. III The series

cn x n converges conditionally for

n=0 ∞ 

ncn x n−1 equals f  (x) on (−R, R).

n=1

(A) (B) (C) (D) (E)

x

I only II only I and II only I and III only I, II, and III 1 16. dt = 4 0 1+t ∞  (A) x 4n

x 2n x2 x4 x6 18. If f (x) = 1 − + − + · · · + (−1)n + 5! 7! (2n + 1)!  π 3! · · · , then f = 2 (A) 0 (B)

2 π

(C) 1 (D) e−π/2 (E) eπ

2 /4

⎧ 1 ⎪ ⎪ ⎨ n 2 19. If an = ⎪ ⎪ ⎩3 5n 47 (A) 24 (B)

133 66

(C)

11 4

(D)

17 6

(E)

13 4

for n even , then

∞  n=0

for n odd

an =

n=0

(B)

∞  x 4n n=1

(C)

∞  (−x)4n n=1

(D)

∞ 

∞  n=0

n=0

∞ 

4n (−1)n

n=0

(E)

20. Which of the following are true statements? ∞  I If an converges conditionally, then

4n

(−1)n

(−1)n an converges.

n=0

x 4n+1 4n + 1

II If

∞  n=0

x 4n+1 (4n + 1)!

|an | converges, then

III If lim an = 0, then n→∞

1 = 17. 1 + 4x 2 (A) 1 + 4x + 8x 2 + 16x 3 + · · · for −1 < x < 1 (B) 1 + 4x 2 + 16x 4 + 64x 6 + · · · for −1 < x < 1 (C) 1 − 4x 2 + 16x 4 − 64x 6 + · · · for −2 < x < 2 (D) 1 − 4x 2 + 16x 4 − 64x 6 + · · ·

for − 12 < x
R in the parametric equations of Exercise 75. Describe the result.

d  dy  x  (t)y  (t) − y  (t)x  (t) = dt dx x  (t)2 Use this to prove the formula

77. Show that the line of slope t through (−1, 0) intersects the unit circle in the point with coordinates 1 − t2

2t y= 2 t +1

x= 2 , t +1

x  (t)y  (t) − y  (t)x  (t) d 2y = dx 2 x  (t)3

11

10

Conclude that these equations parametrize the unit circle with the point (−1, 0) excluded (Figure 22). Show further that t = y/(x + 1). y

82. The second derivative of y = x 2 is dy 2 /d 2 x = 2. Verify that Eq. (11) applied to c(t) = (t, t 2 ) yields dy 2 /d 2 x = 2. In fact, any parametrization may be used. Check that c(t) = (t 3 , t 6 ) and c(t) = (tan t, tan2 t) also yield dy 2 /d 2 x = 2. In Exercises 83–86, use Eq. (11) to find d 2 y/dx 2 .

(x, y)

Slope t

(−1, 0)

x

83. x = t 3 + t 2 ,

y = 7t 2 − 4, t = 2

84. x = s −1 + s, y = 4 − s −2 ,

s=1

85. x = 8t + 9, y = 1 − 4t, t = −3 86. x = cos θ,

y = sin θ,

θ = π4

FIGURE 22 Unit circle.

87. Use Eq. (11) to find the t-intervals on which c(t) = (t 2 , t 3 − 4t) is concave up.

78. The folium of Descartes is the curve with equation x 3 + y 3 = 3axy, where a = 0 is a constant (Figure 23).

88. Use Eq. (11) to find the t-intervals on which c(t) = (t 2 , t 4 − 4t) is concave up.

S E C T I O N 11.1

89. Area Under a Parametrized Curve Let c(t) = (x(t), y(t)), where y(t) > 0 and x  (t) > 0 (Figure 24). Show that the area A under c(t) for t0 ≤ t ≤ t1 is t1 A= y(t)x  (t) dt 12 t0

Hint: Because it is increasing, the function x(t) has an inverse t = g(x) and c(t) is the graph of y = y(g(x)). Apply the change-of-variables

x(t ) formula to A = x(t 1) y(g(x)) dx. 0

Parametric Equations

619

91. What does Eq. (12) say if c(t) = (t, f (t))? 92. Sketch the graph of c(t) = (ln t, 2 − t) for 1 ≤ t ≤ 2 and compute the area under the graph using Eq. (12). 93. Galileo tried unsuccessfully to find the area under a cycloid.Around 1630, Gilles de Roberval proved that the area under one arch of the cycloid c(t) = (Rt − R sin t, R − R cos t) generated by a circle of radius R is equal to three times the area of the circle (Figure 25). Verify Roberval’s result using Eq. (12).

y c(t) y R x(t 0)

x(t 1)

x

FIGURE 24

πR

90. Calculate the area under y = x 2 over [0, 1] using Eq. (12) with the parametrizations (t 3 , t 6 ) and (t 2 , t 4 ).

2π R

x

FIGURE 25 The area of one arch of the cycloid equals three times the

area of the generating circle.

Further Insights and Challenges 94. Prove the following generalization of Exercise 93: For all t > 0, the area of the cycloidal sector OP C is equal to three times the area of the circular segment cut by the chord P C in Figure 26.

has the following property: For all t, the segment from c(t) to (t, 0) is tangent to the curve and has length  (Figure 27). y

y

y

 c(t)

P O

t

P

R C = (Rt, 0)

(A) Cycloidal sector OPC

x

O

t



R C = (Rt, 0)

x

(B) Circular segment cut by the chord PC FIGURE 26

95. Derive the formula for the slope of the tangent line to a parametric curve c(t) = (x(t), y(t)) using a method different from that presented in the text. Assume that x  (t0 ) and y  (t0 ) exist and that x  (t0 ) = 0. Show that y(t0 + h) − y(t0 ) y  (t ) =  0 lim x (t0 ) h→0 x(t0 + h) − x(t0 ) Then explain why this limit is equal to the slope dy/dx. Draw a diagram showing that the ratio in the limit is the slope of a secant line. 96. Verify that the tractrix curve ( > 0)   t t c(t) = t −  tanh ,  sech  

t FIGURE 27 The tractrix c(t) =

x

 t t . t −  tanh ,  sech  



97. In Exercise 54 of Section 9.1, we described the tractrix by the differential equation y dy = − dx 2 − y 2 Show that the curve c(t) identified as the tractrix in Exercise 96 satisfies this differential equation. Note that the derivative on the left is taken with respect to x, not t. In Exercises 98 and 99, refer to Figure 28. 98. In the parametrization c(t) = (a cos t, b sin t) of an ellipse, t is not an angular parameter unless a = b (in which case the ellipse is a circle). However, t can be interpreted in terms of area: Show that if c(t) = (x, y), then t = (2/ab)A, where A is the area of the shaded region in Figure 28. Hint: Use Eq. (12).

620

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

y

99. Show that the parametrization of the ellipse by the angle θ is

b

(x, y) q

FIGURE 28 The parameter θ on the ellipse

ab cos θ x=  a 2 sin2 θ + b2 cos2 θ a

 x 2 a

ab sin θ y=  2 a sin2 θ + b2 cos2 θ

x

+

 y 2 b

= 1.

11.2 Arc Length and Speed We now derive a formula for the arc length of a curve in parametric form. Recall that in Section 8.1, arc length was defined as the limit of the lengths of polygonal approximations (Figure 1). y

y P5 = c(t5 )

P4 = c(t4 )

FIGURE 1 Polygonal approximations for

N = 5 and N = 10.

P1 = c(t1)

P3 = c(t3 )

P1 = c(t1) P0 = c(t0 )

P10 = c(t10)

P0 = c(t0 )

P2 = c(t2 ) x

N=5

N = 10

x

Given a parametrization c(t) = (x(t), y(t)) for a ≤ t ≤ b, we construct a polygonal approximation L consisting of the N segments by joining points P0 = c(t0 ),

P1 = c(t1 ),

...,

PN = c(tN )

corresponding to a choice of values t0 = a < t1 < t2 < · · · < tN = b. By the distance formula, 2  2  Pi−1 Pi = 1 x(ti ) − x(ti−1 ) + y(ti ) − y(ti−1 ) Now assume that x(t) and y(t) are differentiable. According to the Mean Value Theorem, there are values ti∗ and ti∗∗ in the interval [ti−1 , ti ] such that x(ti ) − x(ti−1 ) = x  (ti∗ )ti ,

y(ti ) − y(ti−1 ) = y  (ti∗∗ )ti

where ti = ti − ti−1 , and therefore, Pi−1 Pi = x  (ti∗ )2 ti2 + y  (ti∗∗ )2 ti2 = x  (ti∗ )2 + y  (ti∗∗ )2 ti The length of the polygonal approximation L is equal to the sum N

i=1

Pi−1 Pi =

N

i=1

x  (ti∗ )2 + y  (ti∗∗ )2 ti

2

 This is nearly a Riemann sum for the function x  (t)2 + y  (t)2 . It would be a true Riemann sum if the intermediate values ti∗ and ti∗∗ were equal. Although they are not necessarily equal, it can be shown (and we will take for granted) that if x  (t) and y  (t) are continuous,

Arc Length and Speed

S E C T I O N 11.2

621

then the sum in Eq. (2) still approaches the integral as the widths ti tend to 0. Thus, s = lim

N

Pi−1 Pi =

i=1

Because of the square root, the arc length integral cannot be evaluated explicitly except in special cases, but we can always approximate it numerically.

a

b



x  (t)2 + y  (t)2 dt

THEOREM 1 Arc Length Let c(t) = (x(t), y(t)), where x  (t) and y  (t) exist and are continuous. Then the arc length s of c(t) for a ≤ t ≤ b is equal to b x  (t)2 + y  (t)2 dt 3 s= a

The graph of a function y = f (x) has parametrization c(t) = (t, f (t)). In this case, x  (t)2 + y  (t)2 = 1 + f  (t)2 and Eq. (3) reduces to the arc length formula derived in Section 8.1. As mentioned above, the arc length integral can be evaluated explicitly only in special cases. The circle and the cycloid are two such cases. E X A M P L E 1 Use Eq. 3 to calculate the arc length of a circle of radius R.

Solution With the parametrization x = R cos θ , y = R sin θ, x  (θ )2 + y  (θ )2 = (−R sin θ)2 + (R cos θ)2 = R 2 (sin2 θ + cos2 θ ) = R 2 We obtain the expected result: s=

2π



x  (θ )2 + y  (θ )2 dθ =



0

t=π 2

Solution We use the parametrization of the cycloid in Eq. (6) of Section 1: t = 2π

2π

4π

x(t) = 2(t − sin t), x

FIGURE 2 One arch of the cycloid generated by a circle of radius 2. REMINDER

t 1 − cos t = sin2 2 2

R dθ = 2π R

0

E X A M P L E 2 Length of the Cycloid Calculate the length s of one arch of the cycloid generated by a circle of radius R = 2 (Figure 2).

y 4

2π



x (t) = 2(1 − cos t),

y(t) = 2(1 − cos t) y  (t) = 2 sin t

Thus, x  (t)2 + y  (t)2 = 22 (1 − cos t)2 + 22 sin2 t = 4 − 8 cos t + 4 cos2 t + 4 sin2 t = 8 − 8 cos t = 16 sin2

t 2

(Use the identity recalled in the margin.)

One arch of the cycloid is traced as t varies from 0 to 2π , and thus  2π 2π t 2π t  2  2 s= x (t) + y (t) dt = 4 sin dt = −8 cos  = −8(−1) + 8 = 16 2 2 0 0 0 Note that because sin 2t ≥ 0 for 0 ≤ t ≤ 2π , we did not need an absolute value when taking the square root of 16 sin2 2t .

622

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

In Section 7, we will discuss not just the speed but also the velocity of a particle moving along a curved path. Velocity is “speed plus direction” and is represented by a vector.

Now consider a particle moving along a path c(t). The distance traveled by the particle over the time interval [t0 , t] is given by the arc length integral: t x  (u)2 + y  (u)2 du s(t) = t0

On the other hand, speed is defined as the rate of change of distance traveled with respect to time, so by the Fundamental Theorem of Calculus, t d ds = x  (u)2 + y  (u)2 du = x  (t)2 + y  (t)2 Speed = dt dt t0 THEOREM 2 Speed Along a Parametrized Path The speed of c(t) = (x(t), y(t)) is Speed =



x  (t)2 + y  (t)2

The next example illustrates the difference between distance traveled along a path and displacement (also called net change in position). The displacement along a path is the distance between the initial point c(t0 ) and the endpoint c(t1 ). The distance traveled is greater than the displacement unless the particle happens to move in a straight line (Figure 3).

y Path

c(t 0 )

ds = dt

E X A M P L E 3 A particle travels along the path c(t) = (2t, 1 + t 3/2 ). Find:

c (t 1)

Displacement over [t 0 , t 1 ]

(a) The particle’s speed at t = 1 (assume units of meters and minutes). (b) The distance traveled s and displacement d during the interval 0 ≤ t ≤ 4.

x FIGURE 3 The distance along the path is

Solution We have

greater than or equal to the displacement.

x  (t) = 2, y

c(4) = (8, 9)

9

y  (t) =

3 1/2 t 2

The speed at time t is  9 s (t) = x  (t)2 + y  (t)2 = 4 + t m/min 4 (a) The particle’s speed at t = 1 is s  (1) = 4 + 94 = 2.5 m/min.

Displacement d 6



3

Path of length s c(0)

x 4

8

FIGURE 4 The path c(t) = (2t, 1 + t 3/2 ).

y (R cos ␻ t, R sin ␻ t) R

θ = ␻t

(b) The distance traveled in the first 4 min is    4  8 9 3/2 4 9 8  3/2 s= 4+ t 13 − 8 ≈ 11.52 m 4 + t dt = =  4 27 4 27 0 0 The displacement d is the distance from the initial point c(0) = (0, 1) to the endpoint c(4) = (8, 1 + 43/2 ) = (8, 9) (see Figure 4):  √ d = (8 − 0)2 + (9 − 1)2 = 8 2 ≈ 11.31 m

x

In physics, we often describe the path of a particle moving with constant speed along a circle of radius R in terms of a constant ω (lowercase Greek omega) as follows: FIGURE 5 A particle moving on a circle of

radius R with angular velocity ω has speed |ωR|.

c(t) = (R cos ωt, R sin ωt) The constant ω, called the angular velocity, is the rate of change with respect to time of the particle’s angle θ (Figure 5).

S E C T I O N 11.2

Arc Length and Speed

623

E X A M P L E 4 Angular Velocity Calculate the speed of the circular path of radius R and angular velocity ω. What is the speed if R = 3 m and ω = 4 rad/s?

Solution We have x = R cos ωt and y = R sin ωt, and x  (t) = −ωR sin ωt,

y  (t) = ωR cos ωt

The particle’s speed is  ds = x  (t)2 + y  (t)2 = (−ωR sin ωt)2 + (ωR cos ωt)2 dt = ω2 R 2 (sin2 ωt + cos2 ωt) = |ω|R Thus, the speed is constant with value |ω|R. If R = 3 m and ω = 4 rad/s, then the speed is |ω|R = 3(4) = 12 m/s. Consider the surface obtained by rotating a parametric curve c(t) = (x(t), y(t)) about the x-axis. The surface area is given by Eq. (4) in the next theorem. It can be derived in much the same way as the formula for a surface of revolution of a graph y = f (x) in Section 8.1. In this theorem, we assume that y(t) ≥ 0 so that the curve c(t) lies above the x-axis, and that x(t) is increasing so that the curve does not reverse direction. THEOREM 3 Surface Area Let c(t) = (x(t), y(t)), where y(t) ≥ 0, x(t) is increasing, and x  (t) and y  (t) are continuous. Then the surface obtained by rotating c(t) about the x-axis for a ≤ t ≤ b has surface area b y(t) x  (t)2 + y  (t)2 dt 4 S = 2π

y

a

1 c(t) = (t − tanh t, sech t)

E X A M P L E 5 Calculate the surface area of the surface obtained by rotating the tractrix c(t) = (t − tanh t, sech t) about the x-axis for 0 ≤ t < ∞. x

1

2

Solution Note that the surface extends infinitely to the right (Figure 6). We have

3

x  (t) =

d (t − tanh t) = 1 − sech2 t, dt

y  (t) =

d sech t = − sech t tanh t dt

Using the identities 1 − sech2 t = tanh2 t and sech2 t = 1 − tanh2 t, we obtain x  (t)2 + y  (t)2 = (1 − sech2 t)2 + (− sech t tanh t)2

FIGURE 6 Surface generated by revolving

the tractrix about the x-axis. REMINDER

2 1 = t sech t = cosh t e + e−t 1 − sech2 t = tanh2 t



d tanh t = sech2 t dt d sech t = − sech t tanh t dt sech t tanh t dt = − sech t + C

= (tanh2 t)2 + (1 − tanh2 t) tanh2 t = tanh2 t The surface area is given by an improper integral, which we evaluate using the integral formula recalled in the margin: S = 2π

∞

 sech t tanh2 t dt = 2π

0

0

∞

sech t tanh t dt = 2π lim

R→∞ 0

R

sech t tanh t dt

R  = 2π lim (− sech t) = 2π lim (sech 0 − sech R) = 2π sech 0 = 2π R→∞

0

Here we use that sech R =

R→∞

1 tends to zero (because eR → ∞ while e−R → 0). eR + e−R

624

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

11.2 SUMMARY •

Arc length of c(t) = (x(t), y(t)) for a ≤ t ≤ b: b x  (t)2 + y  (t)2 dt s = arc length = a

The arc length is the distance along the path c(t). The displacement is the distance from the starting point c(a) to the endpoint c(b). • Arc length integral: t s(t) = x  (u)2 + y  (u)2 du •

t0

•

Speed at time t: ds = dt



x  (t)2 + y  (t)2

Surface area of the surface obtained by rotating c(t) = (x(t), y(t)) about the x-axis for a ≤ t ≤ b: b y(t) x  (t)2 + y  (t)2 dt S = 2π

•

a

11.2 EXERCISES Preliminary Questions 1. What is the definition of arc length?  2. What is the interpretation of x  (t)2 + y  (t)2 for a particle following the trajectory (x(t), y(t))? 3. A particle travels along a path from (0, 0) to (3, 4). What is the

displacement? Can the distance traveled be determined from the information given? 4. Aparticle traverses the parabola y = x 2 with constant speed 3 cm/s. What is the distance traveled during the first minute? Hint: No computation is necessary.

Exercises In Exercises 1–10, use Eq. (3) to find the length of the path over the given interval. 1. (3t + 1, 9 − 4t), 0 ≤ t ≤ 2 2. (1 + 2t, 2 + 4t), 4. (3t, 4t 3/2 ),

3. (2t 2 , 3t 2 − 1),

1≤t ≤4

5. (3t 2 , 4t 3 ),

0≤t ≤1

6. (t 3 + 1, t 2 − 3), 7. (sin 3t, cos 3t),

0≤t ≤4

12. Find the length of the spiral c(t) = (t cos t, t sin t) for 0 ≤ t ≤ 2π to three decimal places (Figure 7). Hint: Use the formula    1  1  1 + t 2 dt = t 1 + t 2 + ln t + 1 + t 2 2 2

1≤t ≤4

y

0≤t ≤1 0≤t ≤π

5

8. (sin θ − θ cos θ, cos θ + θ sin θ ),

0≤θ ≤2

t = 2π t=0

−10

x 10

In Exercises 9 and 10, use the identity t 1 − cos t = sin2 2 2 0 ≤ t ≤ π2 0 ≤ θ ≤ 2π

9. (2 cos t − cos 2t, 2 sin t − sin 2t), 10. (5(θ − sin θ ), 5(1 − cos θ )),

11. Show that one arch of a cycloid generated by a circle of radius R has length 8R.

−10

FIGURE 7 The spiral c(t) = (t cos t, t sin t).

13. Find the length of the tractrix (see Figure 6) c(t) = (t − tanh(t), sech(t)),

0≤t ≤A

Arc Length and Speed

S E C T I O N 11.2

14. Find a numerical approximation to the length of c(t) = (cos 5t, sin 3t) for 0 ≤ t ≤ 2π (Figure 8). y 1

625

involute of the circle (Figure 9). Observe that P Q has length Rθ . Show that C is parametrized by   c(θ) = R(cos θ + θ sin θ), R(sin θ − θ cos θ ) Then find the length of the involute for 0 ≤ θ ≤ 2π . y

1

x

Q R

FIGURE 8

P = (x, y) x

θ

In Exercises 15–18, determine the speed s at time t (assume units of meters and seconds). 15. (t 3 , t 2 ),

16. (3 sin 5t, 8 cos 5t), t = π4

t =2

17. (5t + 1, 4t − 3),

t =9

18. (ln(t 2 + 1), t 3 ),

t =1

19. Find the minimum speed of a particle with trajectory c(t) = (t 3 − 4t, t 2 + 1) for t ≥ 0. Hint: It is easier to find the minimum of the square of the speed. 20. Find the minimum speed of a particle with trajectory c(t) = (t 3 , t −2 ) for t ≥ 0.5. 21. Find the speed of the cycloid c(t) = (4t − 4 sin t, 4 − 4 cos t) at points where the tangent line is horizontal.

FIGURE 9 Involute of a circle.

28. Let a > b and set  k=

b2 1− 2 a

Use a parametric representation to show that the ellipse  y 2   = 1 has length L = 4aG π2 , k , where b G(θ, k) =

 x 2 a

+

θ 1 − k 2 sin2 t dt 0

22. Calculate the arc length integral s(t) for the logarithmic spiral c(t) = (et cos t, et sin t).

is the elliptic integral of the second kind.

In Exercises 23–26, plot the curve and use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate its length.

In Exercises 29–32, use Eq. (4) to compute the surface area of the given surface.

23. c(t) = (cos t, esin t )

29. The cone generated by revolving c(t) = (t, mt) about the x-axis for 0 ≤ t ≤ A

for 0 ≤ t ≤ 2π

24. c(t) = (t − sin 2t, 1 − cos 2t) for 0 ≤ t ≤ 2π  x 2  y 2 + =1 25. The ellipse 5 3 26. x = sin 2t,

y = sin 3t

for 0 ≤ t ≤ 2π

27. If you unwind thread from a stationary circular spool, keeping the thread taut at all times, then the endpoint traces a curve C called the

30. A sphere of radius R 31. The surface generated by revolving one arch of the cycloid c(t) = (t − sin t, 1 − cos t) about the x-axis 32. The surface generated by revolving the astroid c(t) = (cos3 t, sin3 t) about the x-axis for 0 ≤ t ≤ π2

Further Insights and Challenges Let b(t) be the “Butterfly Curve”:   5  t cos t − 2 cos 4t − sin x(t) = sin t e 12   5  t cos t − 2 cos 4t − sin y(t) = cos t e 12

(b) Approximate the length b(t) for 0 ≤ t ≤ 10π . √ 2 ab . Show that the trochoid Let a ≥ b > 0 and set k = 34. a−b

(a) Use a computer algebra system to plot b(t) and the speed s  (t) for 0 ≤ t ≤ 12π.

35. A satellite orbiting at a distance R from the center of the earth follows the circular path x = R cos ωt, y = R sin ωt.

33.

x = at − b sin t, y = a − b cos t, 0 ≤ t ≤ T   has length 2(a − b)G T2 , k with G(θ, k) as in Exercise 28.

626

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

(a) Show that the period T (the time of one revolution) is T = 2π/ω. (b) According to Newton’s laws of motion and gravity, x x  (t) = −Gme 3 , R

36. The acceleration due to gravity on the surface of the earth is g=

y y  (t) = −Gme 3 R

Gme Re2

= 9.8 m/s2 ,

where Re = 6378 km

Use Exercise 35(b) to show that a√satellite orbiting at the earth’s surface would have period Te = 2π Re /g ≈ 84.5 min. Then estimate the distance Rm from the moon to the center of the earth. Assume that the period of the moon (sidereal month) is Tm ≈ 27.43 days.

where G is the universal gravitational constant and me is the mass of the earth. Prove that R 3 /T 2 = Gme /4π 2 . Thus, R 3 /T 2 has the same value for all orbits (a special case of Kepler’s Third Law).

11.3 Polar Coordinates Polar coordinates are appropriate when distance from the origin or angle plays a role. For example, the gravitational force exerted on a planet by the sun depends only on the distance r from the sun and is conveniently described in polar coordinates.

In polar coordinates, we label a point P by coordinates (r, θ ), where r is the distance to the origin O and θ is the angle between OP and the positive x-axis (Figure 1). By convention, an angle is positive if the corresponding rotation is counterclockwise. We call r the radial coordinate and θ the angular coordinate. y P=

y

(x, y) (rectangular) (r, θ) (polar)

Ray θ =

(

2π

P = 4, 3

y

2π 3

)

Circle r = 4 2π 3

4 r

y = r sin θ

θ O

O

x

x = r cos θ FIGURE 1

(

r=4

)

2π 3π 3 4

π 2

Circle centered at O

π 3

π 4

5π 6 π 7π 6

1

5π 4 4π 3

3π 2

FIGURE 2

  The point P in Figure 2 has polar coordinates (r, θ ) = 4, 2π 3 . It is located at distance r = 4 from the origin (so it lies on the circle of radius 4), and it lies on the ray of angle θ = 2π 3 . Figure 3 shows the two families of grid lines in polar coordinates:

y 5π Q = 3, 6

x

2

3

Ray starting at O π 6

4

7π 5π 4 3

x

11π 6

FIGURE 3 Grid lines in polar coordinates.

←→

r = constant

←→ θ = constant

Every point in the plane other than the origin lies at the intersection of the two grid lines and these two grid lines determine its polar coordinates. example, point Q in Figure  For 5π 3 lies on the circle r = 3 and the ray θ = 5π , so Q = 3, 6 6 in polar coordinates. Figure 1 shows that polar and rectangular coordinates are related by the equations x = r cos θ and y = r sin θ . On the other hand, r 2 = x 2 + y 2 by the distance formula, and tan θ = y/x if x  = 0. This yields the conversion formulas: Polar to Rectangular x = r cos θ

Rectangular to Polar r = x2 + y2

y = r sin θ

tan θ =

y x

(x  = 0)

Polar Coordinates

S E C T I O N 11.3

627

E X A M P L E 1 From Polar to Rectangular Coordinates Find the rectangular coordinates of point Q in Figure 3.   Solution The point Q = (r, θ ) = 3, 5π 6 has rectangular coordinates:  √  √   5π 3 3 3 x = r cos θ = 3 cos =3 − =− 6 2 2     5π 1 3 y = r sin θ = 3 sin =3 = 6 2 2 E X A M P L E 2 From Rectangular to Polar Coordinates Find the polar coordinates of point P in Figure 4.

y P = (3, 2)

2 r

1

Solution Since P = (x, y) = (3, 2),  √ r = x 2 + y 2 = 32 + 22 = 13 ≈ 3.6

2

θ x 1

2

3

FIGURE 4 The  polar coordinates of P satisfy r = 32 + 22 and tan θ = 23 .

tan θ =

y 2 = x 3

and because P lies in the first quadrant, y 2 = tan−1 ≈ 0.588 x 3 Thus, P has polar coordinates (r, θ ) ≈ (3.6, 0.588). θ = tan−1

A few remarks are in order before proceeding: •

By definition,

−

π π < tan−1 x < 2 2

If r > 0, a coordinate θ of P = (x, y) is

⎧ ⎪ tan−1 ⎪ ⎪ ⎪ ⎪ ⎨ θ = tan−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩± π 2

y if x > 0 x y + π if x < 0 x if x = 0

•

•

•

The angular coordinate is not unique because (r, θ ) and (r, θ + 2π n) label the same point for any integer n. For instance, point P in Figure 5 has radial coordinate r = 2, 3π 7π but its angular coordinate can be any one of π2 , 5π 2 , . . . or − 2 , − 2 , . . . . The origin O has no well-defined angular coordinate, so we assign to O the polar coordinates (0, θ) for any angle θ. By convention, we allow negative radial coordinates. By definition, (−r, θ) is the reflection of (r, θ ) through the origin (Figure 6). With this convention, (−r, θ) and (r, θ + π ) represent the same point. We may specify unique polar coordinates for points other than the origin by placing restrictions on r and θ. We commonly choose r > 0 and 0 ≤ θ < 2π . y (r, θ) y P = (0, 2) (rectangular)

θ+π θ

5π 2 π 2

x

x (−r, θ) or (r, θ + π)

FIGURE 5 The angular coordinate of P = (0, 2) is π2 or any angle π2 + 2πn, where n is an

integer.

FIGURE 6 Relation between (r, θ ) and

(−r, θ).

628

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

When determining the angular coordinate of a point P = (x, y), remember that there are two angles between 0 and 2π satisfying tan θ = y/x. You must choose θ so that (r, θ ) lies in the quadrant containing P .

y P = (−1, 1) 3 4

7 4

−

x 4

(1, −1) FIGURE 7

E X A M P L E 3 Choosing θ Correctly Find two polar representations of P = (−1, 1), one with r > 0 and one with r < 0.

Solution The point P = (x, y) = (−1, 1) has polar coordinates (r, θ ), where  √ y r = (−1)2 + 12 = 2, tan θ = tan = −1 x However, θ is not given by tan−1

y = tan−1 x



1 −1

 =−

π 4

because θ = − π4 would place P in the fourth quadrant (Figure 7). Since P is in the second quadrant, the correct angle is y π 3π +π =− +π = x 4 4 √ If we wish to use the negative radial coordinate r = − 2, then the angle becomes θ = − π4 or 7π 4 . Thus,     √ 3π √ 7π P = or − 2, 2, 4 4 θ = tan−1

A curve is described in polar coordinates by an equation involving r and θ , which we call a polar equation. By convention, we allow solutions with r < 0. A line through the origin O has the simple equation θ = θ0 , where θ0 is the angle between the line and the x-axis (Figure 8). Indeed, the points with θ = θ0 are (r, θ0 ), where r is arbitrary (positive, negative, or zero). E X A M P L E 4 Line Through the Origin Find the polar equation of the line through the origin of slope 32 (Figure 9).

Solution A line of slope m makes an angle θ0 with the x-axis, where m = tan θ0 . In our case, θ0 = tan−1 23 ≈ 0.98. The equation of the line is θ = tan−1 23 or θ ≈ 0.98. y

y r>0 θ0 O

3

(r, θ0)

x

θ0

r 0 ⎪ ⎪tan x ⎪ ⎪ ⎨ y θ = tan−1 + π if x < 0 ⎪ x ⎪ ⎪ ⎪ ⎪ ⎩± π if x = 0 2

•

Nonuniqueness: (r, θ ) and (r, θ + 2nπ ) represent the same point for all integers n. The origin O has polar coordinates (0, θ) for any θ . • Negative radial coordinates: (−r, θ ) and (r, θ + π ) represent the same point. • Polar equations: •

Curve

Polar equation

Circle of radius R, center at the origin

r=R

Line through origin of slope m = tan θ0

θ = θ0

Line on which P0 = (d, α) is the point closest to the origin Circle of radius a, center at (a, 0) (x − a)2 + y 2 = a 2 Circle of radius a, center at (0, a) x 2 + (y − a)2 = a 2

r = d sec(θ − α) r = 2a cos θ r = 2a sin θ

11.3 EXERCISES Preliminary Questions 1. Points P and Q with the same radial coordinate (choose the correct answer): (a) Lie on the same circle with the center at the origin. (b) Lie on the same ray based at the origin. 2. Give two polar representations for the point (x, y) = (0, 1), one with negative r and one with positive r.

3. Describe each of the following curves: (a) r = 2

(b) r 2 = 2

(c) r cos θ = 2

4. If f (−θ) = f (θ), then the curve r = f (θ ) is symmetric with respect to the (choose the correct answer): (a) x-axis

(b) y-axis

(c) origin

Exercises 1. Find polar coordinates for each of the seven points plotted in Figure 16. y 4

A

(x, y) = (2 3, 2) E

B

F 4

C

D G FIGURE 16

x

2. Plot the points with polar coordinates:       (a) 2, π6 (b) 4, 3π (c) 3, − π2 4 3. Convert from rectangular to polar coordinates. √ (b) (3, 3) (c) (−2, 2)

(a) (1, 0)

  (d) 0, π6

(d) (−1,

√ 3)

4. Convert from rectangular to polar coordinates using a calculator (make sure your choice of θ gives the correct quadrant). (a) (2, 3)

(b) (4, −7)

(c) (−3, −8)

5. Convert from polar to rectangular coordinates:       (a) 3, π6 (b) 6, 3π (c) 0, π5 4

(d) (−5, 2)   (d) 5, − π2

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C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

y

6. Which of the following are possible polar coordinates for the point P with rectangular coordinates (0, −2)?    π 7π (a) 2, (b) 2, 2 2     3π 7π (c) −2, − (d) −2, 2 2     π 7π (e) −2, − (f) 2, − 2 2

B

2

C

A x 4

2 −2

D

FIGURE 18 Plot of r = 4 cos θ .

7. Describe each shaded sector in Figure 17 by inequalities in r and θ.

y

y

23. Suppose that P = (x, y) has polar coordinates (r, θ ). Find the polar coordinates for the points: (a) (x, −y) (b) (−x, −y) (c) (−x, y) (d) (y, x)

y

24. Match each equation in rectangular coordinates with its equation in polar coordinates.

45° x 3 5

3 5

(A)

x

3 5

(B)

x

(C)

FIGURE 17

8. Find the equation in polar coordinates of the line through the origin with slope 12 .

(a) (b) (c) (d)

x2 + y2 = 4 x 2 + (y − 1)2 = 1 x2 − y2 = 4 x+y =4

(i) (ii) (iii) (iv)

r 2 (1 − 2 sin2 θ ) = 4 r(cos θ + sin θ ) = 4 r = 2 sin θ r=2

25. What are the polar equations of the lines parallel to the line  r cos θ − π3 = 1?   26. Show that the circle with center at 12 , 12 in Figure 26 has polar equation r = sin θ + cos θ and find the values of θ between 0 and π corresponding to points A, B, C, and D.

9. What is the slope of the line θ = 3π 5 ? 10. Which of r = 2 sec θ and r = 2 csc θ defines a horizontal line?

y A

D

In Exercises 11–16, convert to an equation in rectangular coordinates. 11. r = 7

12. r = sin θ

13. r = 2 sin θ

14. r = 2 csc θ

15. r =

1 cos θ − sin θ

16. r =

1 2 − cos θ

( 12 , 12 ) B

C

x

FIGURE 19 Plot of r = sin θ + cos θ .

In Exercises 17–20, convert to an equation in polar coordinates.

27. Sketch the curve r = 21 θ (the spiral of Archimedes) for θ between 0 and 2π by plotting the points for θ = 0, π4 , π2 , . . . , 2π.

17. x 2 + y 2 = 5

18. x = 5

28. Sketch r = 3 cos θ − 1 (see Example 8).

19. y = x 2

20. xy = 1

29. Sketch the cardioid curve r = 1 + cos θ. 30. Show that the cardioid of Exercise 29 has equation

21. Match each equation with its description. (a) (b) (c) (d)

r θ r r

=2 =2 = 2 sec θ = 2 csc θ

(i) (ii) (iii) (iv)

(x 2 + y 2 − x)2 = x 2 + y 2

Vertical line Horizontal line Circle Line through origin

in rectangular coordinates.

22. Find the values of θ in the plot of r = 4 cos θ corresponding to points A, B, C, D in Figure 18. Then indicate the portion of the graph traced out as θ varies in the following intervals: (a) 0 ≤ θ ≤ π2

(b) π2 ≤ θ ≤ π

(c) π ≤ θ ≤ 3π 2

31. Figure 20 displays the graphs of r = sin 2θ in rectangular coordinates and in polar coordinates, where it is a “rose with four petals.” Identify: (a) The points in (B) corresponding to points A–I in (A). (b) corresponding to the angle intervals  πThe  parts ofthe curve  in (B) 3π , 2π . 0, 2 , π2 , π , π, 3π , and 2 2

Polar Coordinates

S E C T I O N 11.3

r

38. Find an equation in rectangular coordinates for the curve r 2 = cos 2θ.

y B

F

A

C

E

π 2

G π

39. Show that cos 3θ = cos3 θ − 3 cos θ sin2 θ and use this identity to find an equation in rectangular coordinates for the curve r = cos 3θ .

I 2π

3π 2

θ

D H (A) Graph of r as a function of θ, where r = sin 2θ.

x

40. Use the addition formula for the cosine to show that the line L with polar equation r cos(θ − α) = d has the equation in rectangular coordinates (cos α)x + (sin α)y = d. Show that L has slope m = − cot α and y-intercept d/sin α.

(B) Graph of r = sin 2θ in polar coordinates.

In Exercises 41–44, find an equation in polar coordinates of the line L with the given description.   41. The point on L closest to the origin has polar coordinates 2, π9 .

FIGURE 20

32. Sketch the curve r = sin 3θ. First fill in the table of r-values below and plot the corresponding points of the curve. Notice  that  the three  petals of the curve correspond to the angle intervals 0, π3 , π3 , 2π 3 , π  and 3 , π . Then plot r = sin 3θ in rectangular coordinates and label the points on this graph corresponding to (r, θ) in the table. θ

0

π 12

π 6

π 4

π 3

5π 12

11π 12

···

π

42. The point on L closest to the origin has rectangular coordinates (−2, 2). √ 43. L is tangent to the circle r = 2 10 at the point with rectangular coordinates (−2, −6). 44. L has slope 3 and is tangent to the unit circle in the fourth quadrant. 45. Show that every line that does not pass through the origin has a polar equation of the form

r

r=

33. Plot the cissoid r = 2 sin θ tan θ and show that its equation in rectangular coordinates is y2 =

633

x3 2−x

b sin θ − a cos θ

where b  = 0. 46. By the Law of Cosines, the distance d between two points (Figure 22) with polar coordinates (r, θ) and (r0 , θ0 ) is d 2 = r 2 + r02 − 2rr0 cos(θ − θ0 )

34. Prove that r = 2a cos θ is the equation of the circle in Figure 21 using only the fact that a triangle inscribed in a circle with one side a diameter is a right triangle.

Use this distance formula to show that  π = 56 r 2 − 10r cos θ − 4 is the equation  of the circle of radius 9 whose center has polar coordinates 5, π4 .

y r

y θ 0

2a

(r, θ)

x

d r θ

FIGURE 21

(r0, θ0)

r0 θ0

x

FIGURE 22

35. Show that r = a cos θ + b sin θ is the equation of a circle passing through the origin. Express the radius and center (in rectangular coordinates) in terms of a and b. 36. Use the previous exercise to write the equation of the circle of radius 5 and center (3, 4) in the form r = a cos θ + b sin θ. 37. Use the identity cos 2θ = cos2 θ − sin2 θ to find a polar equation of the hyperbola x 2 − y 2 = 1.

47. For a > 0, a lemniscate curve is the set of points P such that the product of the distances from P to (a, 0) and (−a, 0) is a 2 . Show that the equation of the lemniscate is (x 2 + y 2 )2 = 2a 2 (x 2 − y 2 ) Then find the equation in polar coordinates. To obtain the simplest form of the equation, use the identity cos 2θ = cos2 θ − sin2 θ . Plot the lemniscate for a = 2 if you have a computer algebra system.

634

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

x = f (θ) cos θ,

48. Let c be a fixed constant. Explain the relationship between the graphs of: (a) y = f (x + c) and y = f (x) (rectangular) (b) r = f (θ + c) and r = f (θ ) (polar) (c) y = f (x) + c and y = f (x) (rectangular) (d) r = f (θ ) + c and r = f (θ ) (polar)

y = f (θ ) sin θ

Then apply Theorem 2 of Section 11.1 to prove dy f (θ) cos θ + f  (θ) sin θ = dx −f (θ) sin θ + f  (θ) cos θ

49. The Derivative in Polar Coordinates Show that a polar curve r = f (θ ) has parametric equations

2

where f  (θ) = df /dθ.

Further Insights and Challenges 51. Use a graphing utility to convince yourself that the polar equations r = f1 (θ) = 2 cos θ − 1 and r = f2 (θ ) = 2 cos θ + 1 have the same graph. Then explain why. Hint: Show that the points (f1 (θ + π), θ + π) and (f2 (θ), θ) coincide.

50. Let f (x) be a periodic function of period 2π —that is, f (x) = f (x + 2π). Explain how this periodicity is reflected in the graph of: (a) y = f (x) in rectangular coordinates (b) r = f (θ ) in polar coordinates

11.4 Area, Arc Length, and Slope in Polar Coordinates Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve as in Figure 1(A). Consider the region bounded by the curve r = f (θ ) and the two rays θ = α and θ = β with α < β. To derive a formula for the area, divide the region into N narrow sectors of angle θ = (β − α)/N corresponding to a partition of the interval [α, β]: θ0 = α < θ1 < θ2 < · · · < θN = β Recall that a circular sector of angle θ and radius r has area 12 r 2 θ (Figure 2). If θ is small, the j th narrow sector (Figure 3) is nearly a circular sector of radius rj = f (θj ), so its area is approximately 12 rj2 θ . The total area is approximated by the sum: Area of region ≈

N

1 j =1

2

1 f (θj )2 θ 2 N

rj2 θ =

1

j =1

1 β f (θ)2 dθ . If f (θ) is continuous, then the 2 α sum approaches the integral as N → ∞, and we obtain the following formula.

This is a Riemann sum for the integral

y

y r = f (θ )

θN = β rN

rj

θj r j −1

β FIGURE 1 Area bounded by the curve

r = f (θ ) and the two rays θ = α and θ = β.

α (A) Region α ≤ θ ≤ β

θ j −1 θ1 θ0 = α

r0 x

x (B) Region divided into narrow sectors

S E C T I O N 11.4

y

Area, Arc Length, and Slope in Polar Coordinates

635

THEOREM 1 Area in Polar Coordinates If f (θ ) is a continuous function, then the area bounded by a curve in polar form r = f (θ ) and the rays θ = α and θ = β (with α < β) is equal to

θ r

1 2



β

α

r 2 dθ =

1 2



β

f (θ)2 dθ

2

α

x FIGURE 2 The area of a circular sector is exactly 12 r 2 θ .

1 2

y

θj

E X A M P L E 1 Use Theorem 1 to compute the area of the right semicircle with equation r = 4 sin θ .

θ j−1

rj r j −1 Δθ x FIGURE 3 The area of the j th sector is approximately 12 rj2 θ .

REMINDER In Eq. (4), we use the identity

sin2 θ =

1 (1 − cos 2θ ) 2

We know that r = R defines a circle of radius R. By Eq. (2), the area is equal to 2π 1 R 2 dθ = R 2 (2π ) = π R 2 , as expected. 2 0



3

Solution The equation r = 4 sin θ defines a circle of radius 2 tangent to the x-axis at the origin. The right semicircle is “swept out” as θ varies from 0 to π2 as in Figure 4(A). By Eq. (2), the area of the right semicircle is π/2 1 π/2 2 1 π/2 2 r dθ = (4 sin θ) dθ = 8 sin2 θ dθ 4 2 0 2 0 0 π/2 1 =8 (1 − cos 2θ) dθ 2 0 π/2 π   =4 − 0 = 2π = (4θ − 2 sin 2θ) 2 0

y 2

CAUTION Keep in mind that the integral

1 β 2 r dθ does not compute the area 2 α under a curve as in Figure 4(B), but rather computes the area “swept out” by a radial segment as θ varies from α to β , as in Figure 4(A).

y

5 12 3

2

2

4 6

x

(A) The polar integral computes the area swept out by a radial segment.

x (B) The ordinary integral in rectangular coordinates computes the area underneath a curve.

FIGURE 4

E X A M P L E 2 Sketch r = sin 3θ and compute the area of one “petal.”

Solution To sketch the curve, we first graph r = sin 3θ in rectangular coordinates. Figure 5 shows that the radius r varies from 0 to 1 and back to 0 as θ varies from 0 to π3 . This gives petal A in Figure 6. Petal B is traced as θ varies from π3 to 2π 3 (with r ≤ 0), and 2π petal C is traced for 3 ≤ θ ≤ π . We find that the area of petal A (using Eq. (3) in the margin of the previous page to evaluate the integral) is equal to    π/3   1 π/3 1 1 1 π/3 1 − cos 6θ π 2 dθ = θ− sin 6θ  (sin 3θ) dθ = = 2 0 2 0 2 4 24 12 0

636

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

r=1 q=

y

2 3

5 6

3

C

r

A

r=1 q=

6

x A

C π 3

2π 3

B

π

B

θ

r = −1 q=

2

FIGURE 5 Graph of r = sin 3θ as a function

FIGURE 6 Graph of polar curve r = sin 3θ,

of θ.

a “rose with three petals.”

The area between two polar curves r = f1 (θ ) and r = f2 (θ ) with f2 (θ ) ≥ f1 (θ ), for α ≤ θ ≤ β, is equal to (Figure 7):

y r = f 2(θ )

1 Area between two curves = 2

r = f 1(θ )



β



α

 f2 (θ )2 − f1 (θ )2 dθ

5

β α

x

FIGURE 7 Area between two polar graphs in

E X A M P L E 3 Area Between Two Curves Find the area of the region inside the circle r = 2 cos θ but outside the circle r = 1 [Figure 8(A)].

Solution The two circles intersect at the points where (r, 2 cos θ ) = (r, 1) or in other words, when 2 cos θ = 1. This yields cos θ = 12 , which has solutions θ = ± π3 .

a sector.

3

r=1

−

regions (II) and (III).

(II)

(I) 1

FIGURE 8 Region (I) is the difference of

y

y

y

3

2

x

2

x

(III) 1

2

x

r = 2 cos θ

(A)

(C)

(B)

We see in Figure 8 that region (I) is the difference of regions (II) and (III) in Figures 8(B) and (C). Therefore,

REMINDER In Eq. (6), we use the identity

cos2 θ =

1 (1 + cos 2θ ) 2

Area of (I) = area of (II) − area of (III) 1 π/3 2 1 π/3 (2 cos θ)2 dθ − (1) dθ = 2 −π/3 2 −π/3 1 π/3 1 π/3 2 = (4 cos θ − 1) dθ = (2 cos 2θ + 1) dθ 2 −π/3 2 −π/3 √ π/3  3 π 1 = (sin 2θ + θ) = + ≈ 1.91 2 2 3 −π/3

6

We close this section by deriving a formula for arc length in polar coordinates. Observe that a polar curve r = f (θ) has a parametrization with θ as a parameter: x = r cos θ = f (θ ) cos θ,

y = r sin θ = f (θ ) sin θ

Area, Arc Length, and Slope in Polar Coordinates

S E C T I O N 11.4

637

Using a prime to denote the derivative with respect to θ, we have dx = −f (θ) sin θ + f  (θ ) cos θ dθ dy y  (θ ) = = f (θ) cos θ + f  (θ ) sin θ dθ

x  (θ ) =

Recall from Section 11.2 that arc length is obtained by integrating

x  (θ )2 + y  (θ )2 .

Straightforward algebra shows that x  (θ )2 + y  (θ )2 = f (θ)2 + f  (θ )2 , and thus Arc length s =

y

θ=

θ=

f (θ )2 + f  (θ )2 dθ

7

Solution In this case, f (θ) = 2a cos θ and θ = 0 or π x 2a

a

α



E X A M P L E 4 Find the total length of the circle r = 2a cos θ for a > 0.

π 4

π 2

β

θ = 3π 4 FIGURE 9 Graph of r = 2a cos θ.

f (θ)2 + f  (θ )2 = 4a 2 cos2 θ + 4a 2 sin2 θ = 4a 2 The total length of this circle of radius a has the expected value: π π f (θ )2 + f  (θ )2 dθ = (2a) dθ = 2π a 0

0

Note that the upper limit of integration is π rather than 2π because the entire circle is traced out as θ varies from 0 to π (see Figure 9). To find the slope of a polar curve r = f (θ ), remember that the curve is in the x-y dy plane, and so the slope is dx . Since x = r cos θ and y = r sin θ , we use the chain rule. dy/dθ dy = = dx dx/dθ

dr dθ dr dθ

sin θ + r cos θ cos θ − r sin θ

8

E X A M P L E 5 Find an equation of the line tangent to the polar curve r = sin 2θ when θ = 3π 4 .

Solution When θ =

= −1. Thus the point will be in the 4th quadrant.  √  √ 3π − 2 2 x = r cos θ = −1 cos = (−1) = 4 2 2 √  √ 3π 2 − 2 y = r sin θ = −1 sin = (−1) = 4 2 2 3π 4 ,

r = sin

3π 2

Next, dy (2 cos 2θ) sin θ + (sin 2θ)(cos θ) = dx (2 cos 2θ) cos θ − (sin 2θ)(sin θ) Substituting θ =

3π 4 ,

we have

dy dx

= 1. Finally, an equation of the line is  √  √ 2 2 =1 x− y+ 2 2

638

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

11.4 SUMMARY Area of the sector bounded by a polar curve r = f (θ ) and two rays θ = α and θ = β (Figure 10): 1 β Area = f (θ )2 dθ 2 α

•

•

Area between r = f1 (θ ) and r = f2 (θ ), where f2 (θ ) ≥ f1 (θ ) (Figure 11):  1 β Area = f2 (θ )2 − f1 (θ )2 dθ 2 α y

y

r = f (θ ) r = f 2(θ ) r = f 1(θ )

β α

β α

x

FIGURE 10 Region bounded by the polar

FIGURE 11 Region between two polar

curve r = f (θ) and the rays θ = α, θ = β. •

x

curves.

Arc length of the polar curve r = f (θ) for α ≤ θ ≤ β: β Arc length = f (θ)2 + f  (θ )2 dθ α

11.4 EXERCISES Preliminary Questions 1. Polar coordinates are suited to finding the area (choose one): (a) Under a curve between x = a and x = b. (b) Bounded by a curve and two rays through the origin.

y D 1

C y= 1

2. Is the formula for area in polar coordinates valid if f (θ) takes negative values? 3. The horizontal line y = 1 has polar equation r = csc θ. Which area 1 π/2 2 csc θ dθ (Figure 12)? is represented by the integral 2 π/6 (a) ABCD

(b) ABC

A

B 3

x

FIGURE 12

(c) ACD

Exercises

1. Sketch the area bounded by the circle r = 5 and the rays θ = π2 and θ = π , and compute its area as an integral in polar coordinates.

3. Calculate the area of the circle r = 4 sin θ as an integral in polar coordinates (see Figure 4). Be careful to choose the correct limits of integration.

2. Sketch the region bounded by the line r = sec θ and the rays θ = 0 and θ = π3 . Compute its area in two ways: as an integral in polar coordinates and using geometry.

4. Find the area of the shaded triangle in Figure 13 as an integral in polar coordinates. Then find the rectangular coordinates of P and Q and compute the area via geometry.

S E C T I O N 11.4

Area, Arc Length, and Slope in Polar Coordinates

639

y

y

r = sin 2θ P

(

r = 4 sec θ −

π

4

x

) x

Q

FIGURE 17 Four-petaled rose r = sin 2θ.

FIGURE 13

10. Find the area enclosed by one loop of the lemniscate with equation r 2 = cos 2θ (Figure 18). Choose your limits of integration carefully. 5. Find the area of the shaded region in Figure 14. Note that θ varies from 0 to π2 . 6. Which interval of θ -values corresponds to the the shaded region in Figure 15? Find the area of the region.

y

−1

x

1

FIGURE 18 The lemniscate r 2 = cos 2θ.

11. Sketch the spiral r = θ for 0 ≤ θ ≤ 2π and find the area bounded by the curve and the first quadrant.

y 8

12. Find the area of the intersection of the circles r = sin θ and r = cos θ.

r = θ 2 + 4θ y

13. Find the area of region A in Figure 19.

2

1

r = 3 −θ

x

2

y

3

FIGURE 14

x

FIGURE 15

7. Find the total area enclosed by the cardioid in Figure 16.

−1

r=1

A

−1

1

2

4

x

FIGURE 19

14. Find the area of the shaded region in Figure 20, enclosed by the circle r = 12 and a petal of the curve r = cos 3θ. Hint: Compute the area of both the petal and the region inside the petal and outside the circle.

y

−2

r = 4 cos θ

x

y

r = cos 3θ x

FIGURE 16 The cardioid r = 1 − cos θ.

r=

1 2

8. Find the area of the shaded region in Figure 16. FIGURE 20

9. Find the area of one leaf of the “four-petaled rose” r = sin 2θ (Figure 17). Then prove that the total area of the rose is equal to one-half the area of the circumscribed circle.

15. Find the area of the inner loop of the limaçon with polar equation r = 2 cos θ − 1 (Figure 21).

640

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

16. Find the area of the shaded region in Figure 21 between the inner and outer loop of the limaçon r = 2 cos θ − 1. y

In Exercises 25–30, compute the length of the polar curve. 25. The length of r = θ 2 for 0 ≤ θ ≤ π 26. The spiral r = θ for 0 ≤ θ ≤ A

1

27. The equiangular spiral r = eθ for 0 ≤ θ ≤ 2π 1

x

2

28. The inner loop of r = 2 cos θ − 1 in Figure 21

−1

29. The cardioid r = 1 − cos θ in Figure 16 30. r = cos2 θ

FIGURE 21 The limaçon r = 2 cos θ − 1.

17. Find the area of the part of the circle r = sin θ + cos θ in the fourth quadrant (see Exercise 26 in Section 11.3).   18. Find the area of the region inside the circle r = 2 sin θ + π4 and   above the line r = sec θ − π4 . 19. Find the area between the two curves in Figure 22(A).

y r = 2 + cos 2θ

r = 2 + sin 2θ

r = sin 2θ

x

31. r = (2 − cos θ)−1 , 32. r = sin3 t,

0 ≤ θ ≤ 2π

0 ≤ θ ≤ 2π

In Exercises 33–36, use a computer algebra system to calculate the total length to two decimal places.

20. Find the area between the two curves in Figure 22(B). y

In Exercises 31 and 32, express the length of the curve as an integral but do not evaluate it.

33.

The three-petal rose r = cos 3θ in Figure 20

34.

The curve r = 2 + sin 2θ in Figure 23

35.

The curve r = θ sin θ in Figure 24 for 0 ≤ θ ≤ 4π

x

y

r = sin 2θ (A)

(B)

10

FIGURE 22

21. Find the area inside both curves in Figure 23.

5

22. Find the area of the region that lies inside one but not both of the curves in Figure 23. 5 y

5

x

FIGURE 24 r = θ sin θ for 0 ≤ θ ≤ 4π.

2 + sin 2θ

x 2 + cos 2θ FIGURE 23

23. Calculate the total length of the circle r = 4 sin θ as an integral in polar coordinates. 24. Sketch the segment r = sec θ for 0 ≤ θ ≤ A. Then compute its length in two ways: as an integral in polar coordinates and using trigonometry.

36.

r=

√ θ,

0 ≤ θ ≤ 4π

37. Use Eq. (8) to find the slope of the tangent line to r = θ at θ = π2 and θ = π . 38. Use Eq. (8) to find the slope of the tangent line to r = sin θ at θ = π3 . 39. Find the polar coordinates of the points on the lemniscate r 2 = cos 2t in Figure 25 where the tangent line is horizontal. 40. Find the equation in rectangular coordinates of the tangent line to r = 4 cos 3θ at θ = π6 .

Vectors in the Plane

S E C T I O N 11.5

y

641

Then calculate the slopes of the tangent lines at points A, B, C in Figure 26.

r 2 = cos (2t)

y x

−1

1

A

D

( 12 , 12 )

FIGURE 25

B

x

C

FIGURE 26 Plot of r = sin θ + cos θ .

41. Use Eq. (8) to show that for r = sin θ + cos θ,

42. Find the polar coordinates of the points on the cardioid r = 1 + cos θ where the tangent line is horizontal (see Figure 27).

dy cos 2θ + sin 2θ = dx cos 2θ − sin 2θ

Further Insights and Challenges 43. Suppose that the polar coordinates of a moving particle at time t are (r(t), θ(t)). Prove that the particle’s speed is equal to (dr/dt)2 + r 2 (dθ/dt)2 .

Compute the speed at time t = 1 of a particle whose polar 44. coordinates at time t are r = t, θ = t (use Exercise 43). What would the speed be if the particle’s rectangular coordinates were x = t, y = t? Why is the speed increasing in one case and constant in the other?

(d) Use Eq. (8) to show that   dy b cos θ + cos 2θ =− csc θ dx b + 2 cos θ (e) Find the points where the tangent line is vertical. Note that there are three cases: 0 ≤ b < 2, b = 1, and b > 2. Do the plots constructed in (b) and (c) reflect your results? y

45. We investigate how the shape of the limaçon curve r = b + cos θ depends on the constant b (see Figure 27).

y

y

1

1

1

x

(a) Show that the constants b and −b yield the same curve.

1

2

3

x

x 1

2

3

1

2

3

(b) Plot the limaçon for b = 0, 0.2, 0.5, 0.8, 1 and describe how the curve changes. (c) Plot the limaçon for 1.2, 1.5, 1.8, 2, 2.4 and describe how the curve changes.

r = 1 + cos θ

r = 1.5 + cos θ

r = 2.3 + cos θ

FIGURE 27

11.5 Vectors in the Plane Vectors play a role in nearly all areas of mathematics and its applications. In physical settings, they are used to represent quantities that have both magnitude and direction, such as velocity and force. They also appear in such diverse fields as computer graphics, economics, and statistics. A two-dimensional vector v is determined by two points in the plane: an initial point P (also called the “tail” or basepoint) and a terminal point Q (also called the “head”). We write −→ v = PQ and we draw v as an arrow pointing from P to Q. This vector is said to be based at P . Figure 1(A) shows the vector with initial point P = (2, 2) and terminal point Q = (7, 5). The length or magnitude of v, denoted v, is the distance from P to Q.

642

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

NOTATION In this text, vectors are represented by boldface lowercase letters such as v, w, a, b, etc.

−→ The vector v = OR pointing from the origin to a point R is called the position vector of R. Figure 1(B) shows the position vector of the point R = (3, 5). y

y

6 5 4 3 2 1

6 5 4 3 2 1

Q = (7, 5)

P = (2, 2) 1

2

3

4

5

6

7

8

x

O

(A) The vector PQ

R = (3, 5)

1

2

3

4

5

x

(B) The position vector OR

FIGURE 1

We now introduce some vector terminology.

v

•

(A) Vectors parallel to v

•

w v

Two vectors v and w of nonzero length are called parallel if the lines through v and w are parallel. Parallel vectors point either in the same or in opposite directions [Figure 2(A)]. A vector v is said to undergo a translation when it is moved parallel to itself without changing its length or direction. The resulting vector w is called a translate of v [Figure 2(B)]. Translates have the same length and direction but different basepoints.

In many situations, it is convenient to treat vectors with the same length and direction as equivalent, even if they have different basepoints. With this in mind, we say that

(B) w is a translate of v

•

FIGURE 2

v and w are equivalent if w is a translate of v [Figure 3(A)].

Every vector can be translated so that its tail is at the origin [Figure 3(C)]. Therefore, Every vector v is equivalent to a unique vector v0 based at the origin. y v

v v v0 x

(A) Vectors equivalent to v (translates of v)

y Q = (a2, b2) v P = (a1, b1)

b

b = b2 − b1

a = a2 − a1

(C) v0 is the unique vector based at the origin and equivalent to v.

FIGURE 3

To work algebraically, we define the components of a vector (Figure 4). −→ DEFINITION Components of a Vector The components of v = P Q, where P = (a1 , b1 ) and Q = (a2 , b2 ), are the quantities

P0 = (a, b)

v0

(B) Inequivalent vectors

x a FIGURE 4 The vectors v and v0 have components a, b.

a = a2 − a1

(x-component),

The pair of components is denoted a, b.

b = b2 − b1

(y-component)

S E C T I O N 11.5 •

Vectors in the Plane

643

The length of a vector in terms of its components (by the distance formula, see Figure 4) is  −→ v = P Q = a 2 + b2

•

•

•

In this text, “angle brackets” are used to distinguish between the vector v = a, b and the point P = (a, b). Some textbooks denote both v and P by (a, b). When referring to vectors, we use the terms “length” and “magnitude” interchangeably. The term “norm” is also commonly used.

The zero vector (whose head and tail coincide) is the vector 0 = 0, 0 of length zero.

The components a, b determine the length and direction of v, but not its basepoint. Therefore, two vectors have the same components if and only if they are equivalent. Nevertheless, the standard practice is to describe a vector by its components, and thus we write v = a, b Although this notation is ambiguous (because it does not specify the basepoint), it rarely causes confusion in practice. To further avoid confusion, the following convention will be in force for the remainder of the text: We assume all vectors are based at the origin unless otherwise stated. −−−→

y

P1 = (3, 7)

P1 = (3, 7),

v1 P2 = (−1, 4)

5 4

Q2 = (2, 1) 2

3

6

x

FIGURE 5

v = 2, −3 2

1

−3

Q2 = (2, 1)

v2 = 2 − (−1), 1 − 4 = 3, −3

The components of v1 and v2 are not the same, so v1 and v2 are not equivalent. Since v1 = 3, −2, its magnitude is  √ v1  = 32 + (−2)2 = 13

Solution The vector v = 2, −3 based at P = (1, 4) has terminal point Q = (1 + 2, 4 − 3) = (3, 1), located two units to the right and three units down from P as shown in Figure 6. The vector v0 equivalent to v based at O has terminal point (2, −3).

P = (1, 4) 3

O

P2 = (−1, 4),

E X A M P L E 2 Sketch the vector v = 2, −3 based at P = (1, 4) and the vector v0 equivalent to v based at the origin.

y 4

and

Solution We can test for equivalence by computing the components (Figure 5): v1 = 6 − 3, 5 − 7 = 3, −2 ,

v2

−1

Q1 = (6, 5)

What is the magnitude of v1 ?

Q1 = (6, 5)

1

−−−→

E X A M P L E 1 Determine whether v1 = P1 Q1 and v2 = P2 Q2 are equivalent, where

1

2

Q = (3, 1) 3

x

v0 = 2, −3

FIGURE 6 The vectors v and v0 have the same components but different basepoints.

CAUTION Remember that the vector v − w points in the direction from the tip of w to the tip of v (not from the tip of v to the tip of w).

Vector Algebra We now define two basic vector operations: vector addition and scalar multiplication. The vector sum v + w is defined when v and w have the same basepoint: Translate w to the equivalent vector w whose tail coincides with the head of v. The sum v + w is the vector pointing from the tail of v to the head of w [Figure 7(A)]. Alternatively, we can use the Parallelogram Law: v + w is the vector pointing from the basepoint to the opposite vertex of the parallelogram formed by v and w [Figure 7(B)]. To add several vectors v1 , v2 , . . . , vn , translate the vectors to v1 = v1 , v2 , . . . , vn so that they lie head to tail as in Figure 8. The vector sum v = v1 + v2 + · · · + vn is the vector whose terminal point is the terminal point of vn . Vector subtraction v − w is carried out by adding −w to v as in Figure 9(A). Or, more simply, draw the vector pointing from w to v as in Figure 9(B), and translate it back to the basepoint to obtain v − w.

644

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

w

v

´

w

v

´

v+w

v

v

v+w

w

w

(A) The vector sum v + w

(B) Addition via the Parallelogram Law

FIGURE 7

y

y

´

v2

´

v4

v3

v3 v

´

v2

x

´

v1

v4

x

v1

FIGURE 8 The sum v = v1 + v2 + v3 + v4 .

v−w

v−w v

w

v

w

−w

(A) v − w equals v plus (−w) FIGURE 9 Vector subtraction.

NOTATION λ (pronounced “lambda") is the eleventh letter in the Greek alphabet. We use the symbol λ often (but not exclusively) to denote a scalar.

The term scalar is another word for “real number,” and we often speak of scalar versus vector quantities. Thus, the number 8 is a scalar, while 8, 2 is a vector. If λ is a scalar and v is a nonzero vector, the scalar multiple λv is defined as follows (Figure 10): • • •

y

P −v

2

3

λv has length |λ| v. It points in the same direction as v if λ > 0. It points in the opposite direction if λ < 0.

Note that 0v = 0 for all v, and

2v v

(B) More simply, v − w is the translate of the vector pointing from the tip of w to the tip of v.

λv = |λ| v

4

6

x FIGURE 10 Vectors v and 2v are based at P

but 2v is twice as long. Vectors v and −v have the same length but opposite directions.

In particular, −v has the same length as v but points in the opposite direction. A vector w is parallel to v if and only if w = λv for some nonzero scalar λ. Vector addition and scalar multiplication operations are easily performed using components. To add or subtract two vectors v and w, we add or subtract their components. This follows from the parallelogram law as indicated in Figure 11(A). Similarly, to multiply v by a scalar λ, we multiply the components of v by λ [Figures 11(B) and (C)]. Indeed, if v = a, b is nonzero, λa, λb has length |λ| v. It points in the same direction as a, b if λ > 0, and in the opposite direction if λ < 0.

Vectors in the Plane

S E C T I O N 11.5

y

y

y

v+w

b+d

λv = λa, λb v = a, b

w = c, d

d

b

a

a+c

x

x

λa (C)

(B)

If v = a, b and w = c, d, then:

Vector Operations Using Components (i) (ii) (iii) (iv)

x

a

(A)

components.

λb

b

v = a, b c

FIGURE 11 Vector operations using

645

v + w = a + c, b + d v − w = a − c, b − d λv = λa, λb v+0=0+v =v

We also note that if P = (a1 , b1 ) and Q = (a2 , b2 ), then components of the vector −→ v = P Q are conveniently computed as the difference −→ −−→ −→ P Q = OQ − OP = a2 , b2  − a1 , b1  = a2 − a1 , b2 − b1  E X A M P L E 3 For v = 1, 4, w = 3, 2, calculate

(a) v + w

(b) 5v

y v + w = 4, 6

6 v = 1, 4

Solution v + w = 1, 4 + 3, 2 = 1 + 3, 4 + 2 = 4, 6

4

5v = 5 1, 4 = 5, 20

2

The vector sum is illustrated in Figure 12.

w = 3, 2 x 1

3

4

Vector operations obey the usual laws of algebra.

FIGURE 12

THEOREM 1 Basic Properties of Vector Algebra scalars λ, Commutative Law: Associative Law: Distributive Law for Scalars:

For all vectors u, v, w and for all

v+w =w+v u + (v + w) = (u + v) + w λ(v + w) = λv + λw

These properties are verified easily using components. For example, we can check that vector addition is commutative: a, b + c, d = a + c, b + d = c + a, d + b = c, d + a, b    Commutativity of ordinary addition

A linear combination of vectors v and w is a vector rv + sw

646

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

where r and s are scalars. If v and w are not parallel, then every vector u in the plane can be expressed as a linear combination u = rv + sw [Figure 13(A)]. The parallelogram P whose vertices are the origin and the terminal points of v, w and v + w is called the parallelogram spanned by v and w [Figure 13(B)]. It consists of the linear combinations rv + sw with 0 ≤ r ≤ 1 and 0 ≤ s ≤ 1. y

y

v+w w w

sw

u = rv + sw

rv + sw

sw (0 ≤ s ≤ 1)

v x

v rv (0 ≤ r ≤ 1)

rv

x

(B) The parallelogram P spanned by v and w consists of all linear combinations rv + sw with 0 ≤ r, s ≤ 1.

(A) The vector u can be expressed as a linear combination u = rv + sw. In this figure, r < 0. FIGURE 13

E X A M P L E 4 Linear Combinations Express the vector u = 4, 4 in Figure 14 as a linear combination of v = 6, 2 and w = 2, 4.

y

Solution We must find r and s such that rv + sw = 4, 4, or

u = 4, 4

w = 2, 4

r 6, 2 + s 2, 4 = 6r + 2s, 2r + 4s = 4, 4

4 w 5

v = 6, 2 2 v 5

The components must be equal, so we have a system of two linear equations: 6r + 2s = 4

x

FIGURE 14

2r + 4s = 4 Subtracting the equations, we obtain 4r − 2s = 0 or s = 2r. Setting s = 2r in the first equation yields 6r + 4r = 4 or r = 25 , and then s = 2r = 45 . Therefore, u = 4, 4 =

4 2 6, 2 + 2, 4 5 5

In general, to write a vector u = e, f  as a linear combination of two other vectors v = a, b and w = c, d, we have to solve a system of two linear equations in two unknowns r and s:  ar + cs = e rv + sw = u ⇔ r a, b + s c, d = e, f  ⇔ br + ds = f

CONCEPTUAL INSIGHT

On the other hand, vectors give us a way of visualizing the system of equations geometrically. The solution is represented by a parallelogram as in Figure 14. This relation between vectors and systems of linear equations extends to any number of variables and is the starting point for the important subject of linear algebra.

S E C T I O N 11.5

y

Vectors in the Plane

647

A vector of length 1 is called a unit vector. Unit vectors are often used to indicate direction, when it is not necessary to specify length. The head of a unit vector e based at the origin lies on the unit circle and has components

e = cos q, sin q 

e = cos θ, sin θ

q x

1

where θ is the angle between e and the positive x-axis (Figure 15). We can always scale a nonzero vector v = a, b to obtain a unit vector pointing in the same direction (Figure 16):

FIGURE 15 The head of a unit vector lies on

ev =

the unit circle. y v = a, b

b

1 v v

Indeed, we can check that ev is a unit vector as follows:    1  1  v = v = 1 ev  =  v  v If v = a, b makes an angle θ with the positive x-axis, then

ev

v = a, b = vev = v cos θ, sin θ

q 1

1

x

a

E X A M P L E 5 Find the unit vector in the direction of v = 3, 5. FIGURE 16 Unit vector in the direction of v.

  √ √ 1 3 5 32 + 52 = 34, and thus ev = √ v = √ , √ . 34 34 34

It is customary to introduce a special notation for the unit vectors in the direction of the positive x- and y-axes (Figure 17):

y v = ai + bj

bj

Solution v =

i = 1, 0 ,

j = 0, 1

The vectors i and j are called the standard basis vectors. Every vector in the plane is a linear combination of i and j (Figure 17):

1 j i

x 1

v = a, b = ai + bj

ai

FIGURE 17

For example, 4, −2 = 4i − 2j. Vector addition is performed by adding the i and j coefficients. For example, (4i − 2j) + (5i + 7j) = (4 + 5)i + (−2 + 7)j = 9i + 5j

v2

It is often said that quantities such as force and velocity are vectors because they have both magnitude and direction, but there is more to this statement than meets the eye. A vector quantity must obey the law of vector addition (Figure 18), so if we say that force is a vector, we are really claiming that forces add according to the Parallelogram Law. In other words, if forces F1 and F2 act on an object, then the resultant force is the vector sum F1 + F2 . This is a physical fact that must be verified experimentally. It was well known to scientists and engineers long before the vector concept was introduced formally in the 1800s.

CONCEPTUAL INSIGHT v1 FIGURE 18 When an airplane traveling with

velocity v1 encounters a wind of velocity v2 , its resultant velocity is the vector sum v1 + v2 .

648

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

E X A M P L E 6 Find the forces on cables 1 and 2 in Figure 19(A). y 55°

30°

Cable 1

F1

Cable 2

125° F2 30°

55°

P

P

x

100 kg

Fg = 0, −980 (A)

(B) Force diagram

FIGURE 19

θ 125◦ 30◦

cos θ

sin θ

−0.573 0.866

0.819 0.5

Solution Three forces act on the point P in Figure 19(A): the force Fg due to gravity of 100g = 980 newtons (g = 9.8 m/s2 ) acting vertically downward, and two unknown forces F1 and F2 acting through cables 1 and 2, as indicated in Figure 19(B). Let f1 = F1  and f2 = F2 . Because F1 makes an angle of 125◦ (the supplement of 55◦ ) with the positive x-axis, and F2 makes an angle of 30◦ , we can use Eq. (1) and the table in the margin to write these vectors in component form: ! F1 =f1 cos 125◦ , sin 125◦ ≈ f1 −0.573, 0.819 ! F2 =f2 cos 30◦ , sin 30◦ ≈ f2 0.866, 0.5 Fg = 0, −980 Now, the point P is not in motion, so the net force on P is zero: F1 + F2 + Fg = 0 f1 −0.573, 0.819 + f2 0.866, 0.5 + 0, −980 = 0, 0 This gives us two equations in two unknowns: 0.819f1 + 0.5f2 − 980 = 0 −0.573f1 + 0.866f2 = 0,   By the first equation, f2 = 0.573 0.866 f1 . Substitution in the second equation yields   0.573 0.819f1 + 0.5 f1 − 980 ≈ 1.15f1 − 980 = 0 0.866 Therefore, the forces in newtons are 980 ≈ 852 N f1 ≈ 1.15

 and

f2 ≈

 0.573 852 ≈ 564 N 0.866

We close this section with the Triangle Inequality. Figure 20 shows the vector sum v + w for three different vectors w of the same length. Notice that the length v + w varies, depending on the angle between v and w. So in general, v + w is not equal to the sum v + w. What we can say is that v + w is at most equal to the sum v + w. This corresponds to the fact that the length of one side of a triangle is at most the sum of the lengths of the other two sides. A formal proof may be given using the dot product (see Exercise 42 in Section 11.6).

S E C T I O N 11.5

THEOREM 2 Triangle Inequality

Vectors in the Plane

649

For any two vectors v and w,

v + w ≤ v + w Equality holds only if v = 0 or w = 0, or if w = λv, where λ ≥ 0.

v+ w

v+ w

v+ w

w FIGURE 20 The length of v + w depends on the angle between v and w.

w v

v

w

v

11.5 SUMMARY −→ A vector v = P Q is determined by a basepoint P (the “tail”) and a terminal point Q (the “head”). −→ • Components of v = P Q where P = (a , b ) and Q = (a , b ): 1 1 2 2 •

v = a, b with a = a2 − a1 , b = b2 − b1 .√ Length or magnitude: v = a 2 + b2 . • The length v is the distance from P to Q. • The position vector of P = (a, b) is the vector v = a, b pointing from the origin O 0 to P0 . • Vectors v and w are equivalent if they are translates of each other: They have the same magnitude and direction, but possibly different basepoints. Two vectors are equivalent if and only if they have the same components. • We assume all vectors are based at the origin unless otherwise indicated. • The zero vector is the vector 0 = 0, 0 of length 0. • Vector addition is defined geometrically by the Parallelogram Law. In components, •

a1 , b1  + a2 , b2  = a1 + a2 , b1 + b2  Scalar multiplication: λv is the vector of length |λ| v in the same direction as v if λ > 0, and in the opposite direction if λ < 0. In components,

•

λ a, b = λa, λb • • • •

Nonzero vectors v and w are parallel if w = λv for some scalar λ. Unit vector making an angle θ with the positive x-axis: e = cos θ, sin θ . 1 v. Unit vector in the direction of v  = 0: ev = v If v = a, b makes an angle θ with the positive x-axis, then a = v cos θ,

• • •

b = v sin θ,

ev = cos θ, sin θ 

Standard basis vectors: i = 1, 0 and j = 0, 1. Every vector v = a, b is a linear combination v = ai + bj. Triangle Inequality: v + w ≤ v + w.

650

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

11.5 EXERCISES Preliminary Questions 1. (a) (b) (c) (d)

4. What are the components of the zero vector based at P = (3, 5)?

Answer true or false. Every nonzero vector is: Equivalent to a vector based at the origin. Equivalent to a unit vector based at the origin. Parallel to a vector based at the origin. Parallel to a unit vector based at the origin.

5. True or false? (a) The vectors v and −2v are parallel. (b) The vectors v and −2v point in the same direction.

2. What is the length of −3a if a = 5? 3. Suppose that v has components 3, 1. How, if at all, do the components change if you translate v horizontally two units to the left?

6. Explain the commutativity of vector addition in terms of the Parallelogram Law.

Exercises 1. Sketch the vectors v1 , v2 , v3 , v4 with tail P and head Q, and compute their lengths. Are any two of these vectors equivalent? v1

v2

v3

v4

P

(2, 4)

(−1, 3)

(−1, 3)

(4, 1)

Q

(4, 4)

(1, 3)

(2, 4)

(6, 3)

17. Sketch 2v, −w, v + w, and 2v − w for the vectors in Figure 23. y 5 4 3 2 1

2. Sketch the vector b = 3, 4 based at P = (−2, −1). 3. What is the terminal point of the vector a = 1, 3 based at P = (2, 2)? Sketch a and the vector a0 based at the origin and equivalent to a. −→ 4. Let v = P Q, where P = (1, 1) and Q = (2, 2). What is the head of the vector v equivalent to v based at (2, 4)? What is the head of the vector v0 equivalent to v based at the origin? Sketch v, v0 , and v . −→ In Exercises 5–8, find the components of P Q. 5. P = (3, 2),

Q = (2, 7)

6. P = (1, −4), Q = (3, 5)

7. P = (3, 5),

Q = (1, −4)

8. P = (0, 2),

Q = (5, 0)

In Exercises 9–14, calculate. 9. 2, 1 + 3, 4

w = 4, 1 x 1

2

3

4

5

6

FIGURE 23

18. Sketch v = 1, 3, w = 2, −2, v + w, v − w. 19. Sketch v = 0, 2, w = −2, 4, 3v + w, 2v − 2w. 20. Sketch v = −2, 1, w = 2, 2, v + 2w, v − 2w. 21. Sketch the vector v such that v + v1 + v2 = 0 for v1 and v2 in Figure 24(A). 22. Sketch the vector sum v = v1 + v2 + v3 + v4 in Figure 24(B).

10. −4, 6 − 3, −2

11. 5 6, 2 " # " # 13. − 12 , 53 + 3, 10 3

v = 2, 3

12. 4(1, 1 + 3, 2)

y

y

14. ln 2, e + ln 3, π

15. Which of the vectors (A)–(C) in Figure 21 is equivalent to v − w?

3 v2

v4

1

v2 x

x

−3

v

v3

v1

1

w

v1 (A)

(B)

(C) (A)

FIGURE 21

(B) FIGURE 24

16. Sketch v + w and v − w for the vectors in Figure 22. v w FIGURE 22

−→ 23. Let v = P Q, where P = (−2, 5), Q = (1, −2). Which of the following vectors with the given tails and heads are equivalent to v? (a) (−3, 3), (0, 4) (b) (0, 0), (3, −7) (c) (−1, 2), (2, −5) (d) (4, −5), (1, 4)

Vectors in the Plane

S E C T I O N 11.5

24. Which of the following vectors are parallel to v = 6, 9 and which point in the same direction? (a) 12, 18

(b) 3, 2

(c) 2, 3

(d) −6, −9

(e) −24, −27

(f) −24, −36

−→ −→ In Exercises 25–28, sketch the vectors AB and P Q, and determine whether they are equivalent. 25. A = (1, 1),

B = (3, 7),

26. A = (1, 4),

B = (−6, 3),

27. A = (−3, 2), 28. A = (5, 8),

B = (0, 0), B = (1, 8),

P = (0, 0), P = (1, 8),

Q = (6, 3) Q = (3, −2) Q = (−3, 8)

−→ −→ In Exercises 29–32, are AB and P Q parallel? And if so, do they point in the same direction? 29. A = (1, 1), 30. A = (−3, 2),

B = (3, 4), B = (0, 0),

46. What are the coordinates a and b in the parallelogram in Figure 25(B)?

y

P = (1, 1), P = (0, 0),

31. A = (2, 2),

B = (−6, 3), P = (9, 5),

32. A = (5, 8),

B = (2, 2),

P = (2, 2),

Q = (7, 10) Q = (3, 2) Q = (17, 4) Q = (−3, 8)

In Exercises 33–36, let R = (−2, 7). Calculate the following. −→ 33. The length of OR −→ 34. The components of u = P R, where P = (1, 2)

y

(7, 8) P (−1, b) (5, 4)

P = (4, −1), Q = (6, 5) P = (1, 4),

651

(2, 3) (−3, 2)

(2, 2) x

(a, 1)

x

(B)

(A) FIGURE 25

−→ −→ 47. Let v = AB and w = AC, where A, B, C are three distinct points in the plane. Match (a)–(d) with (i)–(iv). (Hint: Draw a picture.) (a) −w (b) −v (c) w − v (d) v − w −→ −→ −→ −→ (i) CB (ii) CA (iii) BC (iv) BA 48. Find the components and length of the following vectors: (a) 4i + 3j (b) 2i − 3j (c) i + j (d) i − 3j In Exercises 49–52, calculate the linear combination.   49. 3j + (9i + 4j) 50. − 32 i + 5 12 j − 12 i 51. (3i + j) − 6j + 2(j − 4i)

52. 3(3i − 4j) + 5(i + 4j)

53. For each of the position vectors u with endpoints A, B, and C in Figure 26, indicate with a diagram the multiples rv and sw such that −−→ u = rv + sw. A sample is shown for u = OQ.

−→ 35. The point P such that P R has components −2, 7

y

−→ 36. The point Q such that RQ has components 8, −3

A

In Exercises 37–42, find the given vector. w

37. Unit vector ev where v = 3, 4

B

C

38. Unit vector ew where w = 24, 7

rv

v x

39. Vector of length 4 in the direction of u = −1, −1

sw

40. Unit vector in the direction opposite to v = −2, 4 41. Unit vector e making an angle of 4π 7 with the x-axis 42. Vector v of length 2 making an angle of 30◦ with the x-axis 43. Find all scalars λ such that λ 2, 3 has length 1. 44. Find a vector v satisfying 3v + 5, 20 = 11, 17. 45. What are the coordinates of the point P in the parallelogram in Figure 25(A)?

Q

FIGURE 26

54. Sketch the parallelogram spanned by v = 1, 4 and w = 5, 2. Add the vector u = 2, 3 to the sketch and express u as a linear combination of v and w. In Exercises 55 and 56, express u as a linear combination u = rv + sw. Then sketch u, v, w, and the parallelogram formed by rv and sw. 55. u = 3, −1; v = 2, 1, w = 1, 3 56. u = 6, −2;

v = 1, 1, w = 1, −1

652

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

57. Calculate the magnitude of the force on cables 1 and 2 in Figure 27. 65°

25°

Cable 1

Cable 2

59. A plane flying due east at 200 km/h encounters a 40-km/h wind blowing in the north-east direction. The resultant velocity of the plane is the vector sum v = v1 + v2 , where v1 is the velocity vector of the plane and v2 is the velocity vector of the wind (Figure 29). The angle between v1 and v2 is π4 . Determine the resultant speed of the plane (the length of the vector v).

50 kg FIGURE 27

40 km/h

58. Determine the magnitude of the forces F1 and F2 in Figure 28, assuming that there is no net force on the object.

v2

v 200 km/h

F2

v1

20 kg FIGURE 29

45°

30° F1

FIGURE 28

Further Insights and Challenges In Exercises 60–62, refer to Figure 30, which shows a robotic arm consisting of two segments of lengths L1 and L2 . −→ 60. Find the components of the vector r = OP in terms of θ1 and θ2 . 61. Let L1 = 5 and L2 = 3. Find r for θ1 = π3 , θ2 = π4 . 62. Let L1 = 5 and L2 = 3. Show that the set of points reachable by the robotic arm with θ1 = θ2 is an ellipse.

64. Use vectors to prove that the segments joining the midpoints of opposite sides of a quadrilateral bisect each other (Figure 32). Hint: Show that the midpoints of these segments are the terminal points of 1 (2u + v + z) 4

1 (2v + w + u) 4

and

65. Prove that two vectors v = a, b and w = c, d are perpendicular if and only if

y

ac + bd = 0

L2 θ2

θ1

L1 θ1

P

z

w

r x

u

FIGURE 30

63. Use vectors to prove that the diagonals AC and BD of a parallelogram bisect each other (Figure 31). Hint: Observe that the midpoint of BD is the terminal point of w + 12 (v − w). 1 (v − 2

w) C

D 1 (v + 2

w

w)

A FIGURE 31

v

B

v FIGURE 32

S E C T I O N 11.6

Dot Product and the Angle between Two Vectors

653

11.6 Dot Product and the Angle between Two Vectors The dot product is one of the most important vector operations. It plays a role in nearly all aspects of multivariable calculus. DEFINITION Dot Product The dot product v · w of two vectors v = a1 , b1  ,

w = a2 , b2 

is the scalar defined by v · w = a 1 a2 + b 1 b2

Important concepts in mathematics often have multiple names or notations either for historical reasons or because they arise in more than one context. The dot product is also called the “scalar product” or “inner product” and in many texts, v · w is denoted (v, w) or v, w. The dot product appears in a very wide range of applications. To rank how closely a Web document matches a search input at Google, “We take the dot product of the vector of count-weights with the vector of typeweights to compute an IR score for the document.” From “The Anatomy of a Large-Scale Hypertextual Web Search Engine” by Google founders Sergey Brin and Lawrence Page.

In words, to compute the dot product, multiply the corresponding components and add. For example, 2, 3 · −4, 2 = 2(−4) + 3(2) = −8 + 6 = −2 We will see in a moment that the dot product is closely related to the angle between v and w. Before getting to this, we describe some elementary properties of dot products. First, the dot product is commutative: v · w = w · v, because the components can be multiplied in either order. Second, the dot product of a vector with itself is the square of the length: If v = a, b, then v · v = a · a + b · b = a 2 + b2 = v2 The dot product also satisfies a Distributive Law and a scalar property as summarized in the next theorem (see Exercises 38 and 39). THEOREM 1 Properties of the Dot Product (i) (ii) (iii) (iv)

0·v =v·0=0 Commutativity: v · w = w · v Pulling out scalars: (λv) · w = v · (λw) = λ(v · w) Distributive Law: u · (v + w) = u · v + u · w (v + w) · u = v · u + w · u

(v) Relation with length:

v · v = v2

E X A M P L E 1 Verify the Distributive Law u · (v + w) = u · v + u · w for

u = 4, 3 ,

v = 1, 2 ,

w = 3, −2

Solution We compute both sides and check that they are equal:   u · (v + w) = 4, 3 · 1, 2 + 3, −2 = 4, 3 · 4, 0 = 4(4) + 3(0) = 16 u · v + u · w = 4, 3 · 1, 2 + 4, 3 · 3, −2     = 4(1) + 3(2) + 4(3) + 3(−2) = 10 + 6 = 16

654

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

As mentioned above, the dot product v · w is related to the angle θ between v and w. This angle θ is not uniquely defined because, as we see in Figure 1, both θ and 2π − θ can serve as an angle between v and w. Furthermore, any multiple of 2π may be added to θ. All of these angles have the same cosine, so it does not matter which angle we use in the next theorem. However, we shall adopt the following convention:

v π w 2π − θ FIGURE 1 By convention, the angle θ

between two vectors is chosen so that 0 ≤ θ ≤ π.

The angle between two vectors is chosen to satisfy 0 ≤ θ ≤ π .

Let θ be the angle between two nonzero

THEOREM 2 Dot Product and the Angle vectors v and w. Then c

v · w = v w cos θ

a

θ

or

cos θ =

v·w v w

1

b

Proof According to the Law of Cosines, the three sides of a triangle satisfy (Figure 2) c2 = a 2 + b2 − 2ab cos θ If two sides of the triangle are v and w, then the third side is v − w, as in the figure, and the Law of Cosines gives

v−w v

θ

v − w2 = v2 + w2 − 2 cos θ v w

2

w

Now, by property (v) of Theorem 1 and the Distributive Law, FIGURE 2

v − w2 = (v − w) · (v − w) = v · v − 2v · w + w · w = v2 + w2 − 2v · w

3

Comparing Eq. (2) and Eq. (3), we obtain −2 cos θ v w = −2v · w, and Eq. (1) follows. By definition of the arccosine, the angle θ = cos−1 x is the angle in the interval [0, π ] satisfying cos θ = x. Thus, for nonzero vectors v and w, we have y

−1



θ = cos

v = 3, 4

v·w v w



w = 1, 3

E X A M P L E 2 Find the angle between v = 3, 4 and w = 1, 3.

θ

Solution Compute cos θ using the dot product

x FIGURE 3

The terms “orthogonal” and “perpendicular” are synonymous and are used interchangeably, although we usually use “orthogonal” when dealing with vectors.

v =

 √ 32 + 42 = 25 = 5,

w =

 √ 12 + 32 = 10

3, 4 · 1, 3 3·1+4·3 15 3 v·w = = = √ =√ √ √ vw 5 · 10 5 10 5 10 10   The angle itself is θ = cos−1 √3 ≈ 0.322 rad (Figure 3). cos θ =

10

Two nonzero vectors v and w are called perpendicular or orthogonal if the angle between them is π2 . In this case we write v ⊥ w.

S E C T I O N 11.6

Dot Product and the Angle between Two Vectors

655

We can use the dot product to test whether v and w are orthogonal. Because an angle between 0 and π satisfies cos θ = 0 if and only if θ = π2 , we see that π v · w = v w cos θ = 0 ⇔ θ = 2 and thus

y

j

v⊥w x i

if and only if

The standard basis vectors are mutually orthogonal and have length 1 (Figure 4). In terms of dot products, because i = 1, 0 and j = 0, 1, i · j = 0,

FIGURE 4 The standard basis vectors are

mutually orthogonal and have length 1.

v·w =0

i·i =j·j=1

E X A M P L E 3 Testing for Orthogonality Determine whether v = 2, 6 is orthogonal to u = 2, −1 or w = −3, 1.

Solution We test for orthogonality by computing the dot products (Figure 5): v · u = 2, 6 · 2, −1 = 2(2) + 6(−1) = −2

(not orthogonal)

v · w = 2, 6 · −3, 1 = 2(−3) + 6(1) = 0

(orthogonal)

E X A M P L E 4 Testing for Obtuseness Determine whether the angles between the vector v = 3, 1 and the vectors u = −2, 2 and w = 2, −1 are obtuse.

y v

Solution By definition, the angle θ between v and u is obtuse if π2 < θ ≤ π, and this is the case if cos θ < 0. Since v · u = v u cos θ and the lengths v and u are positive, we see that cos θ is negative if and only if v · u is negative. Thus,

w x

The angle θ between v and u is obtuse if v · u < 0.

u

We have

FIGURE 5 Vectors v, w, and u for

Example 3.

v · u = 3, 1 · −2, 2 = −6 + 2 = −4 < 0

(angle is obtuse)

v · w = 3, 1 · 2, −1 = 6 − 1 = 5 > 0

(angle is acute)

E X A M P L E 5 Using the Distributive Law Calculate the dot product v · w, where v = 4i − 3j and w = i + 2j.

Solution Use the Distributive Law and the orthogonality of i and j: v · w = (4i − 3j) · (i + 2j) = 4i · (i + 2j) − 3j · (i + 2j) = 4i · i − 3j · (2j) = 4 − 6 = −2 Another important use of the dot product is in finding the projection u|| of a vector u along a nonzero vector v. By definition, u|| is the vector obtained by dropping a perpendicular from u to the line through v as in Figures 6 and 7. In the next theorem, recall that ev = v/v is the unit vector in the direction of v. THEOREM 3 Projection Assume v  = 0. The projection of u along v is the vector u|| = (u · ev )ev

or

u|| =

u · v v·v

The scalar u · ev is called the component of u along v.

v

4

656

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

u

v

u ||

θ ev

FIGURE 6 The projection u|| of u along v has length u cos θ .

v

E X A M P L E 6 Find the projection of u = 5, 1 along v = 4, 4.

θ ev

u

Proof Referring to Figures 6 and 7, we see by trigonometry that u|| has length u| cos θ |. If θ is acute, then u|| is a positive multiple of ev and thus u|| = (u cos θ )ev since cos θ > 0. Similarly, if θ is obtuse, then u|| is a negative multiple of ev and u|| = (u cos θ)ev since cos θ < 0. The first formula for u|| now follows because u · ev = uev  cos θ = u cos θ. The second equality in Eq. (4) follows from the computation:   v v u|| = (u · ev )ev = u · v v   u · v u·v = v= v 2 v·v v

Solution It is convenient to use the second formula in Eq. (4):

u || FIGURE 7 When θ is obtuse, u|| and ev

u · v = 5, 1 · 4, 4 = 20 + 4 = 24, v · v = 42 + 42 = 32   u · v 24 4, 4 = 3, 3 u|| = v= v·v 32

point in opposite directions. u

v

We show now that if v  = 0, then every vector u can be written as the sum of the projection u|| and a vector u⊥ that is orthogonal to v (see Figure 8). In fact, if we set u⊥ = u − u||

u ||

θ u⊥

then we have

ev

FIGURE 8 Decomposition of u as a sum

u = u|| + u⊥ of vectors parallel and orthogonal to v.

u = u|| + u⊥

5

Eq. (5) is called the decomposition of u with respect to v. We must verify, however, that u⊥ is orthogonal to v. We do this by showing that the dot product is zero: u · v u · v v) · v = u · v − (v · v) = 0 u⊥ · v = (u − u|| ) · v = (u − v·v v·v E X A M P L E 7 Find the decomposition of u = 5, 1 with respect to v = 4, 4.

Solution In Example 6 we showed that u|| = 3, 3. The orthogonal vector is u⊥ = u − u|| = 5, 1 − 3, 3 = 2, −2 The decomposition of u with respect to v is u = 5, 1 = u|| + u⊥ =

90° − θ

θ

Projection along v

+

2, −2    Orthogonal to v

The decomposition into parallel and orthogonal vectors is useful in many applications.

F|| 90° − θ

E X A M P L E 8 What is the minimum force you must apply to pull a 20-kg wagon up a frictionless ramp inclined at an angle θ = 15◦ ?

θ Fg

F⊥

FIGURE 9 The angle between Fg and F|| is

90◦ − θ .

3, 3   

Solution Let Fg be the force on the wagon due to gravity. It has magnitude 20g newtons with g = 9.8. Referring to Figure 9, we decompose Fg as a sum Fg = F|| + F⊥

S E C T I O N 11.6

Dot Product and the Angle between Two Vectors

657

where F|| is the projection along the ramp and F⊥ is the “normal force” orthogonal to the ramp. The normal force F⊥ is canceled by the ramp pushing back against the wagon in the normal direction, and thus (because there is no friction), you need only pull against F|| . Notice that the angle between Fg and the ramp is the complementary angle 90◦ − θ. Since F|| is parallel to the ramp, the angle between Fg and F|| is also 90◦ − θ, or 75◦ , and F||  = Fg  cos(75◦ ) ≈ 20(9.8)(0.26) ≈ 51 N Since gravity pulls the wagon down the ramp with a 51-newton force, it takes a minimum force of 51 newtons to pull the wagon up the ramp. It seems that we are using the term “component” in two ways. We say that a vector u = a, b has components a and b. On the other hand, u · e is called the component of u along the unit vector e. In fact, these two notions of component are not different. The components a and b are the dot products of u with the standard unit vectors:

GRAPHICAL INSIGHT

u · i = a, b · 1, 0 = a u · j = a, b · 0, 1 = b and we have the decomposition [Figure 10(A)] u = ai + bj But any two orthogonal unit vectors e and f give rise to a rotated coordinate system, and we see in Figure 10(B) that u = (u · e)e + (u · f )f In other words, u · e and u · f really are the components when we express u relative to the rotated system. y

y

u = a, b

u = a, b

bj

(u · f )f

j i ai

x

f

(A)

(B)

FIGURE 10

11.6 SUMMARY •

The dot product of v = a1 , b1  and w = a2 , b2  is v · w = a 1 a2 + b 1 b 2

•

(u · e)e e

Basic Properties: – Commutativity: v · w = w · v – Pulling out scalars: (λv) · w = v · (λw) = λ(v · w)

x

658

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

– Distributive Law:

u · (v + w) = u · v + u · w (v + w) · u = v · u + w · u

–

v · v = v2

–

v · w = v w cos θ

where θ is the angle between v and w.

By convention, the angle θ is chosen to satisfy 0 ≤ θ ≤ π . Test for orthogonality: v ⊥ w if and only if v · w = 0. • The angle between v and w is acute if v · w > 0 and obtuse if v · w < 0. • Assume v  = 0. Every vector u has a decomposition u = u + u , where u is parallel || ⊥ || to v, and u⊥ is orthogonal to v (see Figure 11). The vector u|| is called the projection of u along v. v • Let e = . Then v v • •

u

θ u⊥

v

u ||

ev

u|| = (u · ev )ev =

FIGURE 11

•

u · v v·v

v,

u⊥ = u − u||

The coefficient u · ev is called the component of u along v: Component of u along v = u · ev = u cos θ

11.6 EXERCISES Preliminary Questions 1. Is the dot product of two vectors a scalar or a vector? 2. What can you say about the angle between a and b if a · b < 0? 3. Which property of dot products allows us to conclude that if v is orthogonal to both u and w, then v is orthogonal to u + w? 4. Which is the projection of v along v: (a) v or (b) ev ?

5. Let u|| be the projection of u along v. Which of the following is the projection u along the vector 2v and which is the projection of 2u along v? (a) 12 u|| (b) u|| (c) 2u|| 6. Which of the following is equal to cos θ , where θ is the angle between u and v? (a) u · v (b) u · ev (c) eu · ev

Exercises In Exercises 1–4, compute the dot product.

1 , 1 ! · 3, 1 ! 6 2 2

1. 3, 1 · 4, −7

2.

3. i · j

4. j · j

In Exercises 5 and 6, determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. 4! 1,−7! 6. 12, 6, 2, −4 5. 12 5 ,−5 , 2 4 In Exercises 7 and 8, find the angle between the vectors. Use a calculator if necessary. √ √ ! √ ! √ ! √ ! 8. 5, 3 , 3, 2 7. 2, 2 , 1 + 2, 1 − 2

In Exercises 9–12, simplify the expression. 9. (v − w) · v + v · w 10. (v + w) · (v + w) − 2v · w 11. (v + w) · v − (v + w) · w

12. (v + w) · v − (v − w) · w

In Exercises 13–16, use the properties of the dot product to evaluate the expression, assuming that u · v = 2, u = 1, and v = 3. 13. u · (4v)

14. (u + v) · v

15. 2u · (3u − v)

16. (u + v) · (u − v)

S E C T I O N 11.6

Dot Product and the Angle between Two Vectors

y

17. Find the angle between v and w if v · w = −v w.

u = 3, 5

18. Find the angle between v and w if v · w = 12 v w. 19. Assume that v = 3, w = 5 and that the angle between v and w is θ = π3 .

659

u⊥

(a) Use the relation v + w2 = (v + w) · (v + w) to show that v + w2 = 32 + 52 + 2v · w.

v = 8, 2 P

O

x

FIGURE 14

(b) Find v + w. 20. Assume that v = 2, w = 3, and the angle between v and w is 120◦ . Determine: (a) v · w (b) 2v + w (c) 2v − 3w

In Exercises 29 and 30, find the decomposition a = a|| + a⊥ with respect to b.

21. Show that if e and f are unit vectors such that e + f  = 32 , then

30. a = 2, −3, b = 5, 0

e − f  = 27 . Hint: Show that e · f = 18 .

31. Let eθ = cos θ, sin θ. Show that eθ · eψ = cos(θ − ψ) for any two angles θ and ψ.

√

22. Find 2e √ − 3f  assuming that e and f are unit vectors such that e + f  = 3/2. 23. Find the angle θ in the triangle in Figure 12.

y

29. a = 1, 0, b = 1, 1

Let v and w be vectors in the plane. 32. (a) Use Theorem 2 to explain why the dot product v · w does not change if both v and w are rotated by the same angle θ . "√ √ # (b) Sketch the vectors e1 = 1, 0 and e2 = 22 , 22 , and determine

the vectors e1 , e2 obtained by rotating e1 , e2 through an angle π4 . Verify that e1 · e2 = e1 · e2 .

(0, 10) (10, 8) θ

33. Let v and w be nonzero vectors and set u = ev + ew . Use the dot product to show that the angle between u and v is equal to the angle between u and w. Explain this result geometrically with a diagram.

(3, 2) x

34. Let v, w, and a be nonzero vectors such that v · a = w · a. Is it true that v = w? Either prove this or give a counterexample.

FIGURE 12

35. Calculate the force (in newtons) required to push a 40-kg wagon up a 10◦ incline (Figure 15).

24. Find all three angles in the triangle in Figure 13.

40 kg

y (2, 7) 10°

FIGURE 15

(6, 3)

(0, 0)

36. A force F is applied to each of two ropes (of negligible weight) attached to opposite ends of a 40-kg wagon and making an angle of 35◦ with the horizontal (Figure 16). What is the maximum magnitude of F (in newtons) that can be applied without lifting the wagon off the ground?

x FIGURE 13

In Exercises 25 and 26, find the projection of u along v. 25. u = 2, 5,

v = 1, 1

26. u = 2, −3,

F

v = 1, 2

F 35°

40 kg

27. Find the length of OP in Figure 14. 28. Find u⊥  in Figure 14.

FIGURE 16

35°

660

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

Incoming light

37. A light beam travels along the ray determined by a unit vector L, strikes a flat surface at point P , and is reflected along the ray determined by a unit vector R, where θ1 = θ2 (Figure 17). Show that if N is the unit vector orthogonal to the surface, then

Reflected light N

L

θ1

R = 2(L · N)N − L 38. Verify the Distributive Law:

R

θ2

P

u · (v + w) = u · v + u · w

FIGURE 17

39. Verify that (λv) · w = λ(v · w) for any scalar λ.

Further Insights and Challenges 40. Prove the Law of Cosines, c2 = a 2 + b2 − 2ab cos θ, by referring to Figure 18. Hint: Consider the right triangle PQR. R

a

v · w = v w cos θ

c

a sin θ

θ

43. This exercise gives another proof of the relation between the dot product and the angle θ between two vectors v = a1 , b1  and w = a2 , b2  in the plane. Observe that v = v cos θ1 , sin θ1  and w = w cos θ2 , sin θ2 , with θ1 and θ2 as in Figure 19. Then use the addition formula for the cosine to show that

P

Q

b

y

y

b − a cos θ

a2

FIGURE 18

41. In this exercise, we prove the Cauchy–Schwarz inequality: If v and w are any two vectors, then |v · w| ≤ v w

6

y w

w a1

b2 θ2 x

θ1

v

θ

b1

v

x

x θ = θ2 − θ1

(a) Let f (x) = xv + w2 for x a scalar. Show that f (x) = ax 2 + bx + c, where a = v2 , b = 2v · w, and c = w2 . (b) Conclude that b2 − 4ac ≤ 0. Hint: Observe that f (x) ≥ 0 for all x.

FIGURE 19

42. Use (6) to prove the Triangle Inequality 44. Let v = x, y and

v + w ≤ v + w Hint: First use the Triangle Inequality for numbers to prove |(v + w) · (v + w)| ≤ |(v + w) · v| + |(v + w) · w|

vθ = x cos θ + y sin θ, −x sin θ + y cos θ  Prove that the angle between v and vθ is θ.

11.7 Calculus of Vector-Valued Functions In this section, we revisit curves in the plane, using the language of vectors to deal with  them. Consider a particle moving in the plane whose coordinates at time t are x(t), y(t) . The particle’s path can be represented by the vector-valued function   r(t) = x(t), y(t) = x(t)i + y(t)j We now extend differentiation and integration to vector-valued functions. This is straightforward because the techniques of single-variable calculus carry over with little change. What is new and important, however, is the geometric interpretation of the derivative as a tangent vector. We describe this later in the section. The first step is to define limits of vector-valued functions.

Calculus of Vector-Valued Functions

S E C T I O N 11.7

661

DEFINITION Limit of a Vector-Valued Function A vector-valued function r(t) approaches the limit u (a vector) as t approaches t0 if lim r(t) − u = 0. In this case, t→t0

y

we write r(t)

lim r(t) = u

t→t0

We can visualize the limit of a vector-valued function as a vector r(t) “moving” toward the limit vector u (Figure 1). According to the next theorem, vector limits may be computed componentwise.

u

x FIGURE 1 The vector-valued function r(t)

approaches u as t → t0 .

THEOREM 1 Vector-Valued Limits Are Computed Componentwise A vector-valued function r(t) = x(t), y(t) approaches a limit as t → t0 if and only if each component approaches a limit, and in this case,   lim r(t) = lim x(t), lim y(t) 1 t→t0

The Limit Laws of scalar functions remain valid in the vector-valued case. They are verified by applying the Limit Laws to the components.

t→t0

t→t0

Proof Let u = a, b and consider the square of the length r(t) − u2 = (x(t) − a)2 + (y(t) − b)2

2

The term on the left approaches zero if and only if each term on the right approaches zero (because these terms are nonnegative). It follows that r(t) − u approaches zero if and only if |x(t) − a| and |y(t) − b| tend to zero. Therefore, r(t) approaches a limit u as t → t0 if and only if x(t) and y(t) converge to the components a and b. ! E X A M P L E 1 Calculate lim r(t), where r(t) = t 2 , 1 − t . t→3

Solution By Theorem 1, !





lim r(t) = lim t , 1 − t = lim t , lim (1 − t) = 9, −2 2

t→3

t→3

2

t→3

t→3

Continuity of vector-valued functions is defined in the same way as in the scalar case. A vector-valued function r(t) = x(t), y(t) is continuous at t0 if lim r(t) = r(t0 )

t→t0

By Theorem 1, r(t) is continuous at t0 if and only if the components x(t), y(t) are continuous at t0 . We define the derivative of r(t) as the limit of the difference quotient: r (t) =

d r(t + h) − r(t) r(t) = lim h→0 dt h

3

In Leibniz notation, the derivative is written dr/dt. We say that r(t) is differentiable at t if the limit in Eq. (3) exists. Notice that the components of the difference quotient are difference quotients:   r(t + h) − r(t) x(t + h) − x(t) y(t + h) − y(t) lim = lim , h→0 h→0 h h h and by Theorem 1, r(t) is differentiable if and only if the components are differentiable. In this case, r (t) is equal to the vector of derivatives x  (t), y  (t).

662

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

By Theorems 1 and 2, vector-valued limits and derivatives are computed “componentwise,” so they are not more difficult to compute than ordinary limits and derivatives.

THEOREM 2 Vector-Valued Derivatives Are Computed Componentwise A vectorvalued function r(t) = x(t), y(t) is differentiable if and only if each component is differentiable. In this case, r (t) =

! d r(t) = x  (t), y  (t) dt

Here are some vector-valued derivatives, computed componentwise: ! ! ! d 2 3! d t , t = 2t, 3t 2 , cos t, −1 = − sin t, 0 dt dt Higher-order derivatives are defined by repeated differentiation: r (t) =

d  r (t), dt

r (t) =

d  r (t), dt

...

E X A M P L E 2 Calculate r (3), where r(t) = ln t, t.

Solution We perform the differentiation componentwise: ! ! d ln t, t = t −1 , 1 dt ! d −1 ! r (t) = t , 1 = −t −2 , 0 dt r (t) =

! Therefore, r (3) = − 19 , 0 .

The differentiation rules of single-variable calculus carry over to the vector setting. Differentiation Rules Assume that r(t), r1 (t), and r2 (t) are differentiable. Then • • •

Sum Rule: (r1 (t) + r2 (t)) = r1 (t) + r2 (t) Constant Multiple Rule: For any constant c, (c r(t)) = c r (t). Product Rule: For any differentiable scalar-valued function f (t),  d f (t)r(t) = f (t)r (t) + f  (t)r(t) dt

•

Chain Rule: For any differentiable scalar-valued function g(t), d r(g(t)) = g  (t)r (g(t)) dt

Proof Each rule is proved by applying the differentiation rules to the components. For example, to prove the Product Rule, we write f (t)r(t) = f (t) x(t), y(t) = f (t)x(t), f (t)y(t) Now apply the Product Rule to each component:   d d d f (t)r(t) = f (t)x(t), f (t)y(t) dt dt dt

! = f  (t)x(t) + f (t)x  (t), f  (t)y(t) + f (t)y  (t) ! ! = f  (t)x(t), f  (t)y(t) + f (t)x  (t), f (t)y  (t) ! = f  (t) x(t), y(t) + f (t) x  (t), y  (t) = f  (t)r(t) + f (t)r (t)

The remaining proofs are left as exercises (Exercises 40–41).

Calculus of Vector-Valued Functions

S E C T I O N 11.7

663

! E X A M P L E 3 Let r(t) = t 2 , 5t and f (t) = e3t . Calculate: (a)

d f (t)r(t) dt

(b)

d r(f (t)) dt

Solution We have r (t) = 2t, 5 and f  (t) = 3e3t . (a) By the Product Rule, ! ! d f (t)r(t) = f (t)r (t) + f  (t)r(t) = e3t 2t, 5 + 3e3t t 2 , 5t dt ! = (3t 2 + 2t)e3t , (15t + 5)e3t (b) By the Chain Rule, ! ! d r(f (t)) = f  (t)r (f (t)) = 3e3t r (e3t ) = 3e3t 2e3t , 5 = 6e6t , 15e3t dt There is another Product Rule for vector-valued functions. In addition to the rule for the product of a scalar function f (t) and a vector-valued function r(t) stated above, there is a Product Rule for the dot product. THEOREM 3 Product Rule for Dot Product Assume that r1 (t) and r2 (t) are differentiable. Then  d 4 r1 (t) · r2 (t) = r1 (t) · r2 (t) + r1 (t) · r2 (t) dt Proof We verify Eq. (4) for vector-valued functions in the plane. If r1 (t) = x1 (t), y1 (t) and r2 (t) = x2 (t), y2 (t), then   d d r1 (t) · r2 (t) = x1 (t)x2 (t) + y1 (t)y2 (t) dt dt = x1 (t)x2 (t) + x1 (t)x2 (t) + y1 (t)y2 (t) + y1 (t)y2 (t)     = x1 (t)x2 (t) + y1 (t)y2 (t) + x1 (t)x2 (t) + y1 (t)y2 (t) = r1 (t) · r2 (t) + r1 (t) · r2 (t) Throughout this chapter, all vector-valued functions are assumed differentiable, unless otherwise stated.

The Derivative as a Tangent Vector The derivative vector r (t0 ) has an important geometric property: It points in the direction tangent to the path traced by r(t) at t = t0 . To understand why, consider the difference quotient, where r = r(t0 + h) − r(t0 ) and t = h with h  = 0: r(t0 + h) − r(t0 ) r = t h

5

The vector r points from the head of r(t0 ) to the head of r(t0 + h) as in Figure 2(A). The difference quotient r/t is a scalar multiple of r and therefore points in the same direction [Figure 2(B)].

664

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

r(t0 + h) − r(t0)

y

y

r(t0 + h) − r(t0) h

r(t0)

r(t0) r(t0 + h)

r(t0 + h) x

x FIGURE 2 The difference quotient points in

the direction of r = r(t0 + h) − r(t0 ).

(A)

(B)

r(t0 + h) − r(t0) h r(t0)

h tending to zero

limit as h

O

O (B)

(A)

FIGURE 3 The difference quotient converges to a vector r (t0 ), tangent to the curve.

As h = t tends to zero, r also tends to zero but the quotient r/t approaches a vector r (t0 ), which, if nonzero, points in the direction tangent to the curve. Figure 3 illustrates the limiting process. We refer to r (t0 ) as the tangent vector or the velocity vector at r(t0 ). The tangent vector r (t0 ) (if it is nonzero) is a direction vector for the tangent line to the curve. Therefore, the tangent line has vector parametrization: 6

E X A M P L E 4 Plotting Tangent Vectors Plot r(t) = cos t, sin t together with its tanπ gent vectors at t = π4 and 3π 2 . Find a parametrization of the tangent line at t = 4 .

r

t= π 4

(1, 0)

x



π  4

$ √ √ % 2 2 = − , , 2 2

r





3π 2

π 4

 = 1, 0

        based at r 3π Figure 4 shows a plot of r(t) with r π4 based at r π4 and r 3π 2 2 .  π  " √2 √2 # π At t = 4 , r 4 = 2 , 2 and thus the tangent line is parametrized by L(t) = r

FIGURE 4

L(t) = r(t0 ) + t r (t0 )

Tangent line at r(t0 ):

(0, 1)

t = 3π (0, −1) 2

(C)

Solution The derivative is r (t) = − sin t, cos t, and thus the tangent vectors at t = and 3π 2 are

y

(−1, 0)

r'(t0)

r(t0 + h)

O

Although it has been our convention to regard all vectors as based at the origin, the tangent vector r (t) is an exception; we visualize it as a vector based at the terminal point of r(t). This makes sense because r (t) then appears as a vector tangent to the curve (Figure 3).

0

π  4

+tr



π  4

$√ √ % $ √ √ % 2 2 2 2 = , +t − , 2 2 2 2

There are some important differences between vector- and scalar-valued derivatives. The tangent line to a plane curve y = f (x) is horizontal at x!0 if f  (x0 ) = 0. But in a vector parametrization, the tangent vector r (t0 ) = x  (t0 ), y  (t0 ) is horizontal and nonzero if y  (t0 ) = 0 but x  (t0 )  = 0.

S E C T I O N 11.7

Calculus of Vector-Valued Functions

665

E X A M P L E 5 Horizontal Tangent Vectors on the Cycloid The function

r(t) = t − sin t, 1 − cos t traces a cycloid. Find the points where: (a) r (t) is horizontal and nonzero.

(b) r (t) is the zero vector.

Solution The tangent vector is r (t) = 1 − cos t, sin t. The y-component of r (t) is zero if sin t = 0—that is, if t = 0, π, 2π, . . . . We have

y

x 2π



r(π ) = π, 2 ,

2 1 π

r (0) = 1 − cos 0, sin 0 = 0, 0

r(0) = 0, 0 ,

r'(π) horizontal r'(2π) = 0

3π

4π

r (π ) = 1 − cos π, sin π  = 2, 0

(zero vector) (horizontal)

By periodicity, we conclude that r (t) is nonzero and horizontal for t = π, 3π, 5π, . . . and r (t) = 0 for t = 0, 2π, 4π, . . . (Figure 5).

FIGURE 5 Points on the cycloid

CONCEPTUAL INSIGHT The cycloid in Figure 5 has sharp points called cusps at points where x = 0, 2π, 4π, . . . . If we represent the cycloid as the graph of a function y = f (x), then f  (x) does not exist at these points. By contrast, the vector derivative r (t) = 1 − cos t, sin t exists for all t, but r (t) = 0 at the cusps. In general, r (t) is a direction vector for the tangent line whenever it exists, but we get no information about the tangent line (which may or may not exist) at points where r (t) = 0.

r(t) = t − sin t, 1 − cos t where the tangent vector is horizontal.

The next example establishes an important property of vector-valued functions. E X A M P L E 6 Orthogonality of r and r When r Has Constant Length Prove that if r(t)

has constant length, then r(t) is orthogonal to r (t). Solution By the Product Rule for Dot Products,

 d d r(t)2 = r(t) · r(t) = r(t) · r (t) + r (t) · r(t) = 2r(t) · r (t) dt dt

y

This derivative is zero because r(t) is constant. Therefore r(t) · r (t) = 0, and r(t) is orthogonal to r (t) [or r (t) = 0].

´

r (t)

The result of Example 6 has a geometric explanation. A vector parametrization r(t) consisting of vectors of constant length R traces a curve on the circle of radius R with center at the origin (Figure 6). Thus r (t) is tangent to this circle. But any line that is tangent to a circle at a point P is orthogonal to the radial vector through P , and thus r(t) is orthogonal to r (t).

GRAPHICAL INSIGHT

r(t)

x

Vector-Valued Integration

FIGURE 6

The integral of a vector-valued function can be defined in terms of Riemann sums as in Chapter 5. We will define it more simply via componentwise integration (the two definitions are equivalent). In other words,  b  b b r(t) dt = x(t) dt, y(t) dt a

a

a

The integral exists if each of the components x(t), y(t) is integrable. For example,  π    π π 1 2 1, t dt = 1 dt, t dt = π, π 2 0 0 0 Vector-valued integrals obey the same linearity rules as scalar-valued integrals.

666

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

An antiderivative of r(t) is a vector-valued function R(t) such that R  (t) = r(t). In the single-variable case, two functions f1 (x) and f2 (x) with the same derivative differ by a constant. Similarly, two vector-valued functions with the same derivative differ by a constant vector (i.e., a vector that does not depend on t). This is proved by applying the scalar result to each component of r(t). THEOREM 4

If R1 (t) and R2 (t) are differentiable and R1 (t) = R2 (t), then R1 (t) = R2 (t) + c

for some constant vector c. The general antiderivative of r(t) is written r(t) dt = R(t) + c where c = c1 , c2  is an arbitrary constant vector. For example,     1 2 1 2 1, t dt = t, t + c = t + c1 , t + c2 2 2 Fundamental Theorem of Calculus for Vector-Valued Functions on [a, b], and R(t) is an antiderivative of r(t), then b r(t) dt = R(b) − R(a)

If r(t) is continuous

a

E X A M P L E 7 Finding Position via Vector-Valued Differential Equations The path of a particle satisfies   1 dr = 1 − 6 sin 3t, t dt 5

Find the particle’s location at t = 4 if r(0) = 4, 1. Solution The general solution is obtained by integration:

y

(7.69, 2.6) t=4 1

(4, 1) t=0 0

   1 1 1 − 6 sin 3t, t dt = t + 2 cos 3t, t 2 + c 5 10



3

2

4

6

9

FIGURE 7 Particle path 1 t 2 + 1! r(t) = t + 2 cos 3t + 2, 10

x

r(t) =

The initial condition r(0) = 4, 1 gives us r(0) = 2, 0 + c = 4, 1 Therefore, c = 2, 1 and (Figure 7)     1 2 1 2 r(t) = t + 2 cos 3t, t + 2, 1 = t + 2 cos 3t + 2, t + 1 10 10 The particle’s position at t = 4 is   1 2 r(4) = 4 + 2 cos 12 + 2, (4 ) + 1 ≈ 7.69, 2.6 10

S E C T I O N 11.7

Calculus of Vector-Valued Functions

667

11.7 SUMMARY •

Limits, differentiation, and integration of vector-valued functions are performed componentwise. • Differentation rules:

•

– Sum Rule: (r1 (t) + r2 (t)) = r1 (t) + r2 (t) – Constant Multiple Rule: (c r(t)) = c r (t) d – Chain Rule: r(g(t)) = g  (t)r (g(t)) dt Product Rules:  d Scalar times vector: f (t)r(t) = f (t)r (t) + f  (t)r(t) dt  d Dot product: r1 (t) · r2 (t) = r1 (t) · r2 (t) + r1 (t) · r2 (t) dt

The derivative r (t0 ) is called the tangent vector or velocity vector. If r (t0 ) is nonzero, then it points in the direction tangent to the curve at r(t0 ). The tangent line has vector parametrization

• •

L(t) = r(t0 ) + tr (t0 ) If R1 (t) = R2 (t), then R1 (t) = R2 (t) + c for some constant vector c. The Fundamental Theorem for vector-valued functions: If r(t) is continuous and R(t) is an antiderivative of r(t), then b r(t) dt = R(b) − R(a)

• •

a

11.7 EXERCISES Preliminary Questions 1. State two forms of the Product Rule for vector-valued functions. In Questions 2–5, indicate whether the statement is true or false, and if it is false, provide a correct statement. 2. The derivative of a vector-valued function is defined as the limit of the difference quotient, just as in the scalar-valued case. 3. There are two Chain Rules for vector-valued functions: one for the composite of two vector-valued functions and one for the composite of a vector-valued and a scalar-valued function.

4. The terms “velocity vector” and “tangent vector” for a path r(t) mean one and the same thing. 5. The derivative of a vector-valued function is the slope of the tangent line, just as in the scalar case. 6. State whether the following derivatives of vector-valued functions r1 (t) and r2 (t) are scalars or vectors:  d d (a) (b) r1 (t) r1 (t) · r2 (t) dt dt

Exercises In Exercises 1–6, evaluate the limit. # " 1. lim t 2 , 4t t→3

6. Evaluate lim 2. lim sin 2ti + cos tj t→π



et − 1 1 , t t→0 t→0 t + 1 " # r(t + h) − r(t) for r(t) = t −1 , sin t . 5. Evaluate lim h h→0 3. lim e2t i + ln(t + 1)j

4. lim



t→0

r(t) for r(t) = sin t, 1 − cos t. t

In Exercises 7–12, compute the derivative. ! √! 7. r(t) = t, t 2 8. r(t) = 7 − t, 4 t " # ! 10. b(t) = e3t−4 , e6−t 9. r(s) = e3s , e−s 11. c(t) = t −1 i

12. a(θ ) = (cos 3θ )i + (sin2 θ )j

668

C H A P T E R 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

! 13. Calculate r (t) and r (t) for r(t) = t, t 2 . ! 14. Sketch the curve r(t) = 1 − t 2 , t for −1 ≤ t ≤ 1. Compute the tangent vector at t = 1 and add it to the sketch. ! 15. Sketch the curve r1 (t) = t, t 2 together with its tangent vector at ! t = 1. Then do the same for r2 (t) = t 3 , t 6 . ! 16. Sketch the cycloid r(t) = t − sin t, 1 − cos t together with its tangent vectors at t = π3 and 3π 4 . In Exercises 17 and 18, evaluate

d r(g(t)) using the Chain Rule. dt

29.

dr = 1 − 2t, 4t, dt

  30. r (t) = sin 3t, sin 3t, r π2 = 2, 4 31. r (t) = 0, 2,

r(3) = 1, 1, r (3) = 0, 0 ! 32. r (t) = et , sin t , r(0) = 1, 0, r (0) = 0, 2 33. Find the location at t = 3 of a particle whose path (Figure 8) satisfies   1 dr = 2t − , 2t − 4 , r(0) = 3, 8 dt (t + 1)2

! 17. r(t) = t 2 , 1 − t , g(t) = et ! 18. r(t) = t 2 , t 3 , g(t) = sin t

y

In Exercises 19 and 20, find a parametrization of the tangent line at the point indicated. ! 19. r(t) = t 2 , t 4 , t = −2 ! 20. r(t) = cos 2t, sin 3t , t = π4 In Exercises 21–28, evaluate the integrals. 3 " 1 # 21. 22. 8t 2 − t, 6t 3 + t dt −1

23.

0

 s 1 , ds 1 + s2 1 + s2

2  3  u i + u5 j du −2

1

 2 te−t i + t ln(t 2 + 1)j dt

24. 0

1 25. 0

27.

2t, 4t dt

26.

4 

√  t −1 i + 4 t j dt

1

r(0) = 3, 1

28.

 1  1 1 , du 1/2 u2 u4 t  0

 3si + 6s 2 j ds

In Exercises 29–32, find both the general solution of the differential equation and the solution with the given initial condition.

10 5

(3, 8) t=0

5

t=3 10

15

20

25

x

FIGURE 8 Particle path.

34. Find the location and velocity at t = 4 of a particle whose path satisfies dr " −1/2 # = 2t ,6 , r(1) = 4, 9 dt 35. Find all solutions to r (t) = v with initial condition r(1) = w, where v and w are constant vectors in R 2 . 36. Let u be a constant vector in R 2 . Find the solution of the equation r (t) = (sin t)u satisfying r (0) = 0. 37. Find all solutions to r (t) = 2r(t) where r(t) is a vector-valued function. 38. Show that w(t) = sin(3t + 4), sin(3t − 2) satisfies the differential equation w (t) = −9w(t).

Further Insights and Challenges 39. Let r(t) = x(t), y(t) trace a plane curve C. Assume that x  (t0 )  = 0. Show that the slope of the tangent vector r (t0 ) is equal to the slope dy/dx of the curve at r(t0 ). 40. Verify the Sum and Product Rules for derivatives of vector-valued functions.

43. Prove the Substitution Rule (where g(t) is a differentiable scalar function): b a

r(g(t))g  (t) dt =

g −1 (b) g −1 (a)

r(u) du

41. Verify the Chain Rule for vector-valued functions. 42. Verify the linearity properties cr(t) dt = c r(t) dt

 r1 (t) + r2 (t) dt =





44. Prove that if r(t) ≤ K for t ∈ [a, b], then (c any constant)

r1 (t) dt +

r2 (t) dt

    b   r(t) dt  ≤ K(b − a)    a

Chapter Review Exercises

669

CHAPTER REVIEW EXERCISES 1. Which of the following curves pass through the point (1, 4)? (b) c(t) = (t 2 , t − 3) (a) c(t) = (t 2 , t + 3) (c) c(t) = (t 2 , 3 − t)

(d) c(t) = (t − 3, t 2 )

2. Find parametric equations for the line through P = (2, 5) perpendicular to the line y = 4x − 3. 3. Find parametric equations for the circle of radius 2 with center (1, 1). Use the equations to find the points of intersection of the circle with the x- and y-axes. 4. Find a parametrization c(t) of the line y = 5 − 2x such that c(0) = (2, 1). 5. Find a parametrization c(θ ) of the unit circle such that c(0) = (−1, 0). 6. Find a path c(t) that traces the parabolic arc y = x 2 from (0, 0) to (3, 9) for 0 ≤ t ≤ 1.

In Exercises 23 and 24, let c(t) = (e−t cos t, e−t sin t). 23. Show that c(t) for 0 ≤ t < ∞ has finite length and calculate its value. 24. Find the first positive value of t0 such that the tangent line to c(t0 ) is vertical, and calculate the speed at t = t0 . Plot c(t) = (sin 2t, 2 cos t) for 0 ≤ t ≤ π . Express the 25. length of the curve as a definite integral, and approximate it using a computer algebra system. 26. Convert the points (x, y) = (1, −3), (3, −1) from rectangular to polar coordinates.     27. Convert the points (r, θ) = 1, π6 , 3, 5π 4 from polar to rectangular coordinates. 28. Write (x + y)2 = xy + 6 as an equation in polar coordinates.

7. Find a path c(t) that traces the line y = 2x + 1 from (1, 3) to (3, 7) for 0 ≤ t ≤ 1.

29. Write r =

8. Sketch the graph c(t) = (1 + cos t, sin 2t) for 0 ≤ t ≤ 2π and draw arrows specifying the direction of motion.

30. Show that r =

2 cos θ as an equation in rectangular coordinates. cos θ − sin θ

In Exercises 9–12, express the parametric curve in the form y = f (x). 10. c(t) = (t 3 + 1, t 2 − 4)

9. c(t) = (4t − 3, 10 − t)   1 2 11. c(t) = 3 − , t 3 + t t

31.

Convert the equation 9(x 2 + y 2 ) = (x 2 + y 2 − 2y)2

12. x = tan t, y = sec t

In Exercises 13–16, calculate dy/dx at the point indicated. 13. c(t) = (t 3 + t, t 2 − 1),

t =3

14. c(θ ) = (tan2 θ, cos θ ),

θ = π4

15. c(t) = (et − 1, sin t),

t = 20

16. c(t) = (ln t, 3t 2 − t),

P = (0, 2)

4 is the polar equation of a line. 7 cos θ − sin θ

to polar coordinates, and plot it with a graphing utility. 32. Calculate the area of the circle r = 3 sin θ bounded by the rays θ = π3 and θ = 2π 3 . 33. Calculate the area of one petal of r = sin 4θ (see Figure 1).

Find the point on the cycloid c(t) = (t − sin t, 1 − cos t) 17. where the tangent line has slope 12 . 18. Find the points on (t + sin t, t − 2 sin t) where the tangent is vertical or horizontal.

34. The equation r = sin(nθ), where n ≥ 2 is even, is a “rose” of 2n petals (Figure 1). Compute the total area of the flower, and show that it does not depend on n.

y

y

y

19. Find the equation of the Bézier curve with control points x

P0 = (−1, −1),

P1 = (−1, 1),

P2 = (1, 1),

x

x

P3 (1, −1)

20. Find the speed at t = π4 of a particle whose position at time t seconds is c(t) = (sin 4t, cos 3t). 21. Find the speed (as a function of t) of a particle whose position at time t seconds is c(t) = (sin t + t, cos t + t). What is the particle’s maximal speed? 22. Find the length of (3et − 3, 4et + 7) for 0 ≤ t ≤ 1.

n = 2 (4 petals)

n = 4 (8 petals)

n = 6 (12 petals)

FIGURE 1 Plot of r = sin(nθ ).

35. Calculate the total area enclosed by the curve r 2 = cos θ esin θ (Figure 2).

670

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS

C H A P T E R 11

47. Find the vector with length 3 making an angle of 7π 4 with the positive x-axis.

y 1

48. Calculate 3 (i − 2j) − 6 (i + 6j). 49. Find the value of β for which w = −2, β is parallel to v = 4, −3.

x

−1

1

FIGURE 2 Graph of r 2 = cos θ esin θ .

36. Find the shaded area in Figure 3. y 1

−2

−1

r = 1 + cos 2θ

1

2

x

50. Let r1 (t) = v1 + tw1 and r2 (t) = v2 + tw2 be parametrizations of lines L1 and L2 . For each statement (a)–(e), provide a proof if the statement is true and a counterexample if it is false. (a) If L1 = L2 , then v1 = v2 and w1 = w2 . (b) If L1 = L2 and v1 = v2 , then w1 = w2 . (c) If L1 = L2 and w1 = w2 , then v1 = v2 . (d) If L1 is parallel to L2 , then w1 = w2 . (e) If L1 is parallel to L2 , then w1 = λw2 for some scalar λ. 51. Sketch the vector sum v = v1 − v2 + v3 for the vectors in Figure 5(A). y

−1

y

v2

FIGURE 3

v2

v3

v3

37. Find the area enclosed by the cardioid r = a(1 + cos θ), where a > 0. 38. Calculate the length of the curve with polar equation r = θ in Figure 4. y r=θ

v1

v1 x

x

(A)

(B) FIGURE 5

52. Sketch the sums v1 + v2 + v3 , v1 + 2v2 , and v2 − v3 for the vectors in Figure 5(B).

π 2

53. Use vectors to prove that the line connecting the midpoints of two sides of a triangle is parallel to the third side. x

π FIGURE 4

In Exercises 39–44, let v = −2, 5 and w = 3, −2. 39. Calculate 5w − 3v and 5v − 3w. 40. Sketch v, w, and 2v − 3w. 41. Find the unit vector in the direction of v. 42. Find the length of v + w. 43. Express i as a linear combination rv + sw. 44. Find a scalar α such that v + αw = 6. −→ 45. If P = (1, 4) and Q = (−3, 5), what are the components of P Q? −→ What is the length of P Q? 46. Let A = (2, −1), B = (1, 4), and P = (2, 3). Find the point Q −→ −→ −→ −→ such that P Q is equivalent to AB. Sketch P Q and AB.

54. Calculate the magnitude of the forces on the two ropes in Figure 6. A

30° Rope 1

45° P

B

Rope 2

10 kg FIGURE 6

55. A 50-kg wagon is pulled to the right by a force F1 making an angle of 30◦ with the ground. At the same time the wagon is pulled to the left by a horizontal force F2 . (a) Find the magnitude of F1 in terms of the magnitude of F2 if the wagon does not move. (b) What is the maximal magnitude of F1 that can be applied to the wagon without lifting it? 56. Find the angle between v and w if v + w = v = w. 57. Find e√− 4f , assuming that e and f are unit vectors such that e + f  = 3.

Chapter Review Exercises

π

671

! sin θ, θ dθ.

58. Find the area of the parallelogram spanned by vectors v and w such that v = w = 2 and v · w = 1.

68. Calculate

In Exercises 59–64, calculate the derivative indicated. ! 59. r (t), r(t) = 1 − t, t −2 ! 60. r (t), r(t) = t 3 , 4t 2 2! 61. r (0), r(t) = e2t , e−4t ! 62. r (−3), r(t) = t −2 , (t + 1)−1

69. A particle located at (1,!1) at time t = 0 follows a path whose velocity vector is v(t) = 1, t . Find the particle’s location at t = 2.

! d t e 1, t dt ! d 64. r(cos θ ), r(s) = s, 2s dθ In Exercises 65 and 66, calculate the derivative at t = 3, assuming that

63.

r1 (3) = 1, 1 ,

r2 (3) = 1, 1

r1 (3) = 0, 0 ,

r2 (3) = 0, 2

d (6r1 (t) − 4 · r2 (t)) 65. dt 3 ! 4t + 3, t 2 dt. 67. Calculate 0

 d  t e r2 (t) 66. dt

0

! 70. Find the vector-valued function r(t) = x(t), y(t) in R 2 satisfying  r (t) = −r(t) with initial conditions r(0) = 1, 2. 71. Calculate r(t) assuming that " # r (t) = 4 − 16t, 12t 2 − t ,

r (0) = 1, 0 ,

r(0) = 0, 1

" # 72. Solve r (t) = t 2 − 1, t + 1 subject to the initial conditions r(0) = 1, 0 and r (0) = −1, 1. 73. A projectile fired at an angle of 60◦ lands 400 m away. What was its initial speed? 74. A force F = 12t + 4, 8 − 24t (in newtons) acts on a 2-kg mass. Find the position of the mass at t = 2 s if it is located at (4, 6) at t = 0 and has initial velocity 2, 3 in m/s. ! 75. Find the unit tangent vector to r(t) = sin t, t at t = π .

CHAPTER 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS PREPARING FOR THE AP EXAM Multiple Choice Questions Use scratch paper as necessary to solve each problem. Pick the best answer from the choices provided. All questions cover BC topics. 1. A curve is given by the parametric equations x(t) = t 2 − 4t and y(t) = t 2 + 2t − 3. The line tangent to the curve at the point P is horizontal if P = (A) (−3, 0)

3.

C A curve is given by x(t) = t 2 + 3, y(t) = cos t. The length of the curve from the point (3, 1) to the point (7, cos 2) is given by  7 1 + cos2 t dt (A) 3



3



(B) (−3, −4)

7

(B) 2

(C) 

(C) (5, −4)

0

2

(D)

(D) (−2, −1)

0

 (E)

(E) (−6, 0)

2

1 + sin2 t dt 2

(t 2 + 3) + cos2 t dt 4t 2 + sin2 t dt 1 + sin2 t dt

0

2. If a curve is given by x(t) = + 4t + 3 and y(t) = t + 3, then an equation of the line tangent to the curve at the point (3, 5) is t3

− 4t 2

(A) y = 53 x (B) y = 5 (C) y = x + 2

4. Which of the following are parametrizations of the parabola y = x2? I x(t) = t 2 , y(t) = t 4 for −∞ < t < ∞. II x(t) = tan t, y(t) = tan2 t for − π2 < t < π2 . √ √ 3 III x(t) = 3 t, y(t) = t 2 for −∞ < t < ∞. (A) I only (B) III only (C) I and III only

(D) x = 3

(D) II and III only

(E) Not possible; there is no tangent line at (3, 5).

(E) I, II, and III

AP11-1

PREPARING FOR THE AP EXAM

5.

C Consider the polar curve r = 2 sin(3θ). The length of the loop of this curve that is in the first quadrant is given by  π/3  4 sin2 (3θ ) + 36 cos2 (3θ ) dθ (A) 

0

π/2 

(B) 

0

π/3 

(C) 

0

π/2 

(D) 

0

π/2 

(E)

9.

4 sin2 (3θ ) + 36 cos2 (3θ ) dθ 2 sin2 (3θ ) + 6 cos2 (3θ ) dθ 2 sin2 (3θ ) + 6 cos2 (3θ ) dθ 10. 4 sin2 (3θ ) − 36 cos2 (3θ ) dθ

0

6. The curve r = sin θ , 0 < θ < 2π, has a vertical tangent line when (A) θ = π2 and 3π 2 only (B) θ = (C) θ = (D) θ = (E) θ =

π 3π 5π 7π 4 , 4 , 4 , and 4 π 5π 4 and 4 only π 3π 2 , π , and 2 only π 2 and π only

III x(t) = 2 cos(t), (A) I only

y(t) = 3 sin(−t)

(B) II only

C Which of the following integrals represents the area inside the polar curve r = 2 cos θ ?  1 2π 4 cos2 θ dθ (A) 2 0  2π (B) 4 cos2 θ dθ 0



π/2

(C)

4 cos2 θ dθ

0

(D)

1 2 



2π

4 sin2 θ dθ

0 2π

(E)

4 sin2 θ dθ

0

11.

C A particle travels in the xy-plane with x(t) = t 3 + 3t 2 − 9t and y(t) = 2t 3 + 9t 2 . The particle is at rest when t= (A) t = 0 only (B) t = −3 only (C) t = 0 and −3 only

(C) I and II only

8.

C A vector valued function r(t) has as its velocity func  tion V (t) = sin t, 6e2t . If r(0) = 4, 7, then r(t) =   (A) cos t, 12e2t   (B) 4 cos t, 7e2t   (C) 3 + cos t, 3e2t + 4   (D) 5 − cos t, 7e2t   (E) 5 − cos t, 3e2t + 4

only

7. Which of the following is a parametrization of the ellipse 9x 2 + 4y 2 = 36? y2 x2 + =1 I 4 9 II x(t) = 2 cos t, y(t) = 3 sin t

(D) II and III only

(D) t = −3 and 1 only

(E) I, II, and III

(E) t = −3, 0, and 1 only

C If 6, 2 is the tangent vector to the vector curve r(t) at the point r(7) = 4, 5, then an equation of the line tangent to the curve is given by (A) y − 5 = 7(x − 4) 5x (B) y = 4 (C) y − 5 = 3(x − 4)

12. If a particle’s position is given by the vector function r(t) = sin(2t), cos(t) for the open interval 0 < t < 2π, then its velocity vector is pointing straight down for (A) t =

π 2

and

3π 2

only

(B) t =

π 4

and

3π 4

only

(C) t =

5π 4

(D) y − 5 = 13 (x − 4)

(D) t =

π 3π 5π 4, 4 , 4 ,

(E) y − 5 =

(E) no values of t

1 7 (x

AP11-2

− 4)

and

7π 4

only and

7π 4

only

AP11-3

CHAPTER 11

PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS

13. If a curve is given by x(t) = t 3 − 48t and y(t) = t 2 − 2t, then the time interval(s) for which the slope of the line tangent to the curve is positive are (A) t > 0 (B) t > 1 (C) t < −4 and t > 1 (D) t < −4 and t > 4 (E) −4 < t < 1 and t > 4 14. A vector function is given by r(t) = t 2 − 4t, t 2 + 6t. Its velocity vector is parallel to the x-axis when (A) t = −3 only (B) t = 0 and −6 only (C) t = 0 and 4 only (D) t = 2 only (E) t = −3 and 2 only 15. The slope of the polar curve r = 1 − cos θ at the point corresponding to θ = π2 is (A) −1 (B) − 12 (C) 0 (D) 12 (E) 1 16. The area enclosed by the polar graph r = sin θ is (A) π8 (B) (C)

1 2 π 4

(D) 1 (E) π

17.

18.

C Find the area inside the large loop of r = 1 + 2 cos θ , and outside the small loop. (A) 0.543 (B) 8.338 (C) 8.881 (D) 9.424 (E) 18.306 C Find the area outside the curve r = 3 cos θ and inside R = 1 + cos θ . (A) 0.204 (B) 0.393 (C) 0.596 (D) 0.785 (E) 1.570

19. If a particle has position given by r(t) = sin(2t), e3t , then its speed when t = 0 is (A) 1 (B) 32 (C)

9 4 √

(D) 13 (E) 5 20. A particle has velocity given by V (t) = 6 sin(2t), sec2 t and is at the point (1, 1) when t = 0. The position of the particle when t = π4 is (A) (4, 2) (B) (0, 0) (C) (−2, 1) (D) (1, 2) (E) (0, 1)

Free Response Questions Show all of your work and clearly label any functions, graphs, tables, or other objects that you use. On the AP, your work will be scored on the correctness and completeness of your methods, as well as your actual answer. You will usually not be given credit for answers that don’t include supporting work. All questions cover BC topics. 1. A particle is moving in the plane; at time t, its velocity vector is given by v(t) = 2t + 5, 4e2t and at time t = 0, the particle is at the point (−6, 2). (a) If a(t) is the acceleration vector, what is a(3)? (b) What is the speed of the particle when t = 0? (c) Find an equation of the line tangent to the path of the particle at the point (−6, 2). (d) Find all times t for which the particle is in the first quadrant. Justify your answer.

2. Consider the polar curve given by r = 1 + cos θ. (a) What integral gives the area of the region enclosed by the curve above the x-axis? (b) Find an equation for the line tangent to the curve at the point where the curve meets the positive y-axis. (c) Find the rectangular coordinates of the point(s) that minimize x.

PREPARING FOR THE AP EXAM

3. Two particles are traveling in the plane. Particle F is at the point 3, 4 at time t = −1, and its velocity vector is always 2 times its position vector. Particle G is at the point 9, 12 at time t = 0, and its velocity vector is always −3 times its position vector. In this problem t takes on all real values. (a) Find the vector functions F (t) and G(t). (b) Find all times, if any, when the particles have the same position. (c) Show that the points in the plane visited by F and the points visited by G are the same by giving a formula for these points in rectangular coordinates. (d) How far does G travel for t ≥ 0?

AP11-4

4. Consider the spiral in the xy-plane given by the polar curve 1 r= for θ ≥ 0. 2θ + 1 (a) Is the line tangent to the spiral when θ = π a vertical line? Explain. (b) Find the area enclosed by the curve below the y-axis. (c) Show that the spiral has infinite length. Answers to odd-numbered questions can be found in the back of the book.

12 DIFFERENTIATION

IN SEVERAL VARIABLES

n this chapter we extend the concepts and techniques of differential calculus to functions of several variables. As we will see, a function f that depends on two or more variables has not just one derivative but rather a set of partial derivatives, one for each variable. The partial derivatives are the components of the gradient vector, which provides valuable insight into the function’s behavior. In the last two sections, we apply the tools we have developed to optimization in several variables.

I

12.1 Functions of Two or More Variables The famous triple peaks Eiger, Monch, and Jungfrau in the Swiss alps. The steepness at a point in a mountain range is measured by the gradient, a concept defined in this chapter.

A familiar example of a function of two variables is the area A of a rectangle, equal to the product xy of the base x and height y. We write A(x, y) = xy or A = f (x, y), where f (x, y) = xy. An example in three variables is the distance from a point P = (x, y, z) to the origin:  g(x, y, z) = x 2 + y 2 + z2 An important but less familiar example is the density of seawater, denoted ρ, which is a function ρ(S, T ) of salinity S and temperature T (Figure 1). Although there is no simple formula for ρ(S, T ), scientists determine function values experimentally (Figure 2). According to Table 1, if S = 32 (in parts per thousand) and T = 10◦ C, then

FIGURE 1 The global climate is influenced

by the ocean “conveyer belt,” a system of deep currents driven by variations in seawater density.

ρ(32, 10) = 1.0246 kg/m3 TABLE 1 Seawater Density ρ (kg /m3 ) as a Function of Temperature and Salinity. Salinity (ppt)

FIGURE 2 A Conductivity-Temperature-

Depth (CDT) instrument is used to measure seawater variables such as density, temperature, pressure, and salinity.

◦C

32

32.5

33

5 10 15 20

1.0253 1.0246 1.0237 1.0224

1.0257 1.0250 1.0240 1.0229

1.0261 1.0254 1.0244 1.0232

A function of n variables is a function f (x1 , . . . , xn ) that assigns a real number to each n-tuple (x1 , . . . , xn ) in a domain in R n . Sometimes we write f (P ) for the value of f at a point P = (x1 , . . . , xn ). When f is defined by a formula, we usually take as domain the set of all n-tuples for which f (x1 , . . . , xn ) is defined. The range of f is the set of all values f (x1 , . . . , xn ) for (x1 , . . . , xn ) in the domain. Since we focus on functions of two or three variables, we shall often use the variables x, y, and z (rather than x1 , x2 , x3 ). 672

Functions of Two or More Variables

S E C T I O N 12.1

673

E X A M P L E 1 Sketch the domains of

(a) f (x, y) =

 9 − x2 − y

√ (b) g(x, y, z) = x y + ln(z − 1)

What are the ranges of these functions? Solution

 (a) f (x, y) = 9 − x 2 − y is defined only when 9 − x 2 − y ≥ 0, or y ≤ 9 − x 2 . Thus the domain consists of all points (x, y) lying below the parabola y = 9 − x 2 [Figure 3(A)]: D = {(x, y) : y ≤ 9 − x 2 } √ To determine the range, note that f is a nonnegative function and that f (0, y) = 9 − y. Since 9 − y can be any positive number, f (0, y) takes on all nonnegative values. Therefore the range of f is the infinite interval [0, ∞). √ √ (b) g(x, y, z) = x y + ln(z − 1) is defined only when both y and ln(z − 1) are defined. We must require that y ≥ 0 and z > 1, so the domain is {(x, y, z) : y ≥ 0, z > 1} [Figure 3(B)]. The range of g is the entire√real line R. Indeed, for the particular choices y = 1 and z = 2, we have g(x, 1, 2) = x 1 + ln 1 = x, and since x is arbitrary, we see that g takes on all values.

z y 9

3

1

x

x

(A) The domain of f (x, y) = 9 − x 2 − y is the set of all points lying below the parabola y = 9 − x 2.

y

(B) Domain of g(x, y, z) = x y + ln(z − 1) is the set of points with y ≥ 0 and z > 1. The domain continues out to infinity in the directions indicated by the arrows.

FIGURE 3

Graphing Functions of Two Variables In single-variable calculus, we use graphs to visualize the important features of a function. Graphs play a similar role for functions of two variables. The graph of f (x, y) consists of all points (a, b, f (a, b)) in R 3 for (a, b) in the domain D of f . Assuming that f is continuous (as defined in the next section), the graph is a surface whose height above or below the xy-plane at (a, b) is the function value f (a, b) [Figure 4]. We often write z = f (x, y) to stress that the z-coordinate of a point on the graph is a function of x and y. E X A M P L E 2 Sketch the graph of f (x, y) = 2x 2 + 5y 2 .

Solution The graph is a paraboloid (Figure 5). We sketch the graph using the fact that the horizontal cross section (called the horizontal “trace” below) at height z is the ellipse 2x 2 + 5y 2 = z.

674

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

z z y

(a, f (a)) (a, b, f (a, b))

x

a

(a, b)

y y

x (A) Graph of y = f (x)

(B) Graph of z = f (x, y)

x FIGURE 5 Graph of f (x, y) = 2x 2 + 5y 2

FIGURE 4

Plotting more complicated graphs by hand can be difficult. Fortunately, computer algebra systems eliminate the labor and greatly enhance our ability to explore functions graphically. Graphs can be rotated and viewed from different perspectives (Figure 6).

z

z

z

y

x y x

x

y

2 2 2 2 FIGURE 6 Different views of z = e−x −y − e−(x−1) −(y−1)

Traces and Level Curves One way of analyzing the graph of a function f (x, y) is to freeze the x-coordinate by setting x = a and examine the resulting curve z = f (a, y). Similarly, we may set y = b and consider the curve z = f (x, b). Curves of this type are called vertical traces. They are obtained by intersecting the graph with planes parallel to a vertical coordinate plane (Figure 7): •

•

Vertical trace in the plane x = a: Intersection of the graph with the vertical plane x = a, consisting of all points (a, y, f (a, y)). Vertical trace in the plane y = b: Intersection of the graph with the vertical plane y = b, consisting of all points (x, b, f (x, b)).

E X A M P L E 3 Describe the vertical traces of f (x, y) = x sin y.

Solution When we freeze the x-coordinate by setting x = a, we obtain the trace curve z = a sin y (see Figure 8). This is a sine curve located in the plane x = a. When we set y = b, we obtain a line z = (sin b)y of slope sin b, located in the plane y = b.

S E C T I O N 12.1

Functions of Two or More Variables

675

FIGURE 7

z

z z = x sin y

z = x sin y

y

x

(A) The traces in the planes x = a are the curves z = a(sin y).

FIGURE 8 Vertical traces of

f (x, y) = x sin y.

y

x

(B) The traces in the planes y = b are the lines z = (sin b) y.

E X A M P L E 4 Identifying Features of a Graph Match the graphs in Figure 9 with the following functions:

(i) f (x, y) = x − y 2

(ii) g(x, y) = x 2 − y

Solution Let’s compare vertical traces. The vertical trace of f (x, y) = x − y 2 in the plane x = a is a downward parabola z = a − y 2 . This matches (B). On the other hand, z

z Upward parabolas y = b, z = x 2 − b

Downward parabolas x = a, z = a − y 2

y

x y x Decreasing in positive y-direction FIGURE 9

(A)

Increasing in positive x-direction (B)

676

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

the vertical trace of g(x, y) in the plane y = b is an upward parabola z = x 2 − b. This matches (A). Notice also that f (x, y) = x − y 2 is an increasing function of x (that is, f (x, y) increases as x increases) as in (B), whereas g(x, y) = x 2 − y is a decreasing function of y as in (A).

Level Curves and Contour Maps In addition to vertical traces, the graph of f (x, y) has horizontal traces. These traces and their associated level curves are especially important in analyzing the behavior of the function (Figure 10):

Horizontal trace at z = c

c

•

z=c

•

z = f (x, y)

y x Level curve f (x, y) = c FIGURE 10 The level curve consists of all

points (x, y) where the function takes on the value c. On contour maps level curves are often referred to as contour lines.

Horizontal trace at height c: Intersection of the graph with the horizontal plane z = c, consisting of the points (x, y, f (x, y)) such that f (x, y) = c. Level curve: The curve f (x, y) = c in the xy-plane.

Thus the level curve consists of all points (x, y) in the plane where the function takes the value c. Each level curve is the projection onto the xy-plane of the horizontal trace on the graph that lies above it. A contour map is a plot in the xy-plane that shows the level curves f (x, y) = c for equally spaced values of c. The interval m between the values is called the contour interval. When you move from one level curve to next, the value of f (x, y) (and hence the height of the graph) changes by ±m. Figure 11 compares the graph of a function f (x, y) in (A) and its horizontal traces in (B) with the contour map in (C). The contour map in (C) has contour interval m = 100. It is important to understand how the contour map indicates the steepness of the graph. If the level curves are close together, then a small move from one level curve to the next in the xy-plane leads to a large change in height. In other words, the level curves are close together if the graph is steep (Figure 11). Similarly, the graph is flatter when the level curves are farther apart.

z

z

Level curves close together Steep part of graph

z = f (x, y)

0 – 10

500 300

0

z

–300

0

10

y

y

300

x x

(A)

Flatter part of graph

(B) Horizontal traces

Level curves farther apart (C) Contour map

FIGURE 11

E X A M P L E 5 Elliptic Paraboloid Sketch the contour map of f (x, y) = x 2 + 3y 2 and

comment on the spacing of the contour curves. Solution The level curves have equation f (x, y) = c, or x 2 + 3y 2 = c

Functions of Two or More Variables

S E C T I O N 12.1 • •

•

677

For c > 0, the level curve is an ellipse. For c = 0, the level curve is just the point (0, 0) because x 2 + 3y 2 = 0 only for (x, y) = (0, 0). The level curve is empty if c < 0 because f (x, y) is never negative.

The graph of f (x, y) is an elliptic paraboloid (Figure 12). As we move away from the origin, f (x, y) increases more rapidly. The graph gets steeper, and the level curves get closer together. E X A M P L E 6 Hyperbolic Paraboloid Sketch the contour map of g(x, y) = x 2 − 3y 2 .

Solution The level curves have equation g(x, y) = c, or x 2 − 3y 2 = c • •

For c  = 0, the level curve is the hyperbola x 2 − 3y 2 = c. √ For c = 0, the level curve consists of the two lines x = ± 3y because the equation g(x, y) = 0 factors as follows: √ √ x 2 − 3y 2 = 0 = (x − 3y)(x + 3y) = 0

The graph of g(x, y) is a hyperbolic paraboloid (Figure 13). When you stand at the origin, g(x, y) increases as you move along the x-axis in either direction and decreases as you move along the y-axis in either direction. Furthermore, the graph gets steeper as you move out from the origin, so the level curves get closer together. z

z

x

y

The hyperbolic paraboloid in Figure 13 is often called a “saddle” or “saddle-shaped surface.”

y x g (x, y) decreasing

c=0

10

30

50

y

x FIGURE 12 f (x, y) = x 2 + 3y 2 . Contour

interval m = 10.

c = 30 g(x, y) increasing

x

g (x, y) increasing

c = −30 g(x, y) y decreasing

FIGURE 13 g(x, y) = x 2 − 3y 2 . Contour

interval m = 10.

678

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

E X A M P L E 7 Contour Map of a Linear Function Sketch the graph of f (x, y) = 12 − 2x − 3y and the associated contour map with contour interval m = 4.

c = 20

z

c = 16 c = 12 12

c=8 c=4 c=0 4 c = −4

6 x

4 6 x

c = 20 c = 16 c = 12 c=8 c=4 c=0 c = −4

y

Solution To plot the graph, which is a plane, we find the intercepts with the axes (Figure 14). The graph intercepts the z-axis at z = f (0, 0) = 12. To find the x-intercept, we set y = z = 0 to obtain 12 − 2x − 3(0) = 0, or x = 6. Similarly, solving 12 − 3y = 0 gives y-intercept y = 4. The graph is the plane determined by the three intercepts. In general, the level curves of a linear function f (x, y) = qx + ry + s are the lines with equation qx + ry + s = c. Therefore, the contour map of a linear function consists of equally spaced parallel lines. In our case, the level curves are the lines 12 − 2x − 3y = c, or 2x + 3y = 12 − c (Figure 14). How can we measure steepness quantitatively? Let’s imagine the surface z = f (x, y) as a mountain range. In fact, contour maps (also called topographical maps) are used extensively to describe terrain (Figure 15). We place the xy-plane at sea level, so that f (a, b) is the height (also called altitude or elevation) of the mountain above sea level at the point (a, b) in the plane.

y

(Interval m = 4)

FIGURE 14 Graph and contour

map of f (x, y) = 12 − 2x − 3y.

FIGURE 15 Mount Whitney Range in

California, with contour map.

 and Figure 16 shows two points P and Q in the xy-plane, together with the points P  Q on the graph that lie above them. We define the average rate of change: Average rate of change from P to Q =

 altitude  horizontal

where  and Q   altitude = change in the height from P  horizontal = distance from P to Q E X A M P L E 8 Calculate the average rate of change of f (x, y) from P to Q for the function whose graph is shown in Figure 16.

Solution The segment P Q spans three level curves and the contour interval is 0.8 km,  to Q  is 3(0.8) = 2.4 km. From the horizontal scale of so the change in altitude from P the contour map, we see that the horizontal distance P Q is 2 km, so Average rate of change from P to Q =

 altitude 2.4 = = 1.2  horizontal 2

On average, your altitude gain is 1.2 times your horizontal distance traveled as you climb ˜ from P˜ to Q.

Functions of Two or More Variables

S E C T I O N 12.1

1100 1050

~ C

~ B

10000

~ Q Δ altitude

~ A

~ P

Δ horizontal

~ D

C

1100 1000

B

1050

679

D

A Q

Function does not change along the level curve

P A Contour interval: 0.8 km Horizontal scale: 2 km FIGURE 16

B 200 m

A

400 m

C

Contour interval: 50 m

FIGURE 17

We will discuss the idea that rates of change depend on direction when we come to directional derivatives in Section 12.5. In single-variable calculus, we measure the rate of change by the derivative f  (a). In the multivariable case, there is no single rate of change because the change in f (x, y) depends on the direction: The rate is zero along a level curve (because f (x, y) is constant along level curves), and the rate is nonzero in directions pointing from one level curve to the next (Figure 17). CONCEPTUAL INSIGHT

E X A M P L E 9 Average Rate of Change Depends on Direction Compute the average rate of change from A to the points B, C, and D in Figure 17.

Solution The contour interval in Figure 17 is m = 50 m. Segments AB and AC both span two level curves, so the change in altitude is 100 m in both cases. The horizontal scale shows that AB corresponds to a horizontal change of 200 m, and AC corresponds to a horizontal change of 400 m. On the other hand, there is no change in altitude from A to D. Therefore: Average rate of change from A to B =

100  altitude = = 0.5  horizontal 200

Average rate of change from A to C =

100  altitude = = 0.25  horizontal 400

Average rate of change from A to D =

 altitude =0  horizontal

We see here explicitly that the average rate varies according to the direction. When we walk up a mountain, the incline at each moment depends on the path we choose. If we walk “around” the mountain, our altitude does not change at all. On the other hand, at each point there is a steepest direction in which the altitude increases most rapidly.

680

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

A path of steepest descent is the same as a path of steepest ascent but in the opposite direction. Water flowing down a mountain follows a path of steepest descent.

On a contour map, the steepest direction is approximately the direction that takes us to the closest point on the next highest level curve [Figure 18(A)]. We say “approximately” because the terrain may vary between level curves. A path of steepest ascent is a path that begins at a point P and, everywhere along the way, points in the steepest direction. We can approximate the path of steepest ascent by drawing a sequence of segments that move as directly as possible from one level curve to the next. Figure 18(B) shows two paths from P to Q. The solid path is a path of steepest ascent, but the dashed path is not, because it does not move from one level curve to the next along the shortest possible segment. 10

10

20 30

20 30

40

40

Q

Approximate path of steepest ascent starting at P (A) Vectors pointing approximately in the direction of steepest ascent

P

Not a path of steepest ascent

(B)

FIGURE 18

z x2 + y2 + z2 = 1

x2 + y2 + z2 = 4

More Than Two Variables It is not possible to draw the graph of a function of more than two variables. The graph of a function f (x, y, z) would consist of the set of points (x, y, z, f (x, y, z)) in fourdimensional space R 4 . However, it is possible to draw the level surfaces of a function of three variables f (x, y, z). These are the surfaces with equation f (x, y, z) = c. For example, the level surfaces of f (x, y, z) = x 2 + y 2 + z2

y

are the spheres with equation x 2 + y 2 + z2 = c (Figure 19). For functions of four or more variables, we can no longer visualize the graph or the level surfaces. We must rely on intuition developed through the study of functions of two and three variables.

x x2 + y2 + z2 = 9 FIGURE 19 The level surfaces of f (x, y, z) = x 2 + y 2 + z2 are spheres.

E X A M P L E 10 Describe the level surfaces of g(x, y, z) = x 2 + y 2 − z2 .

Solution The level surface for c = 0 is the cone x 2 + y 2 − z2 = 0. For c = 0, the level surfaces are the hyperboloids x 2 + y 2 − z2 = c. The hyperboloid has one sheet if c > 0 and two sheets if c < 0 (Figure 20).

12.1 SUMMARY The domain D of a function f (x1 , . . . , xn ) of n variables is the set of n-tuples (a1 , . . . , an ) in R n for which f (a1 , . . . , an ) is defined. The range of f is the set of values taken by f . • The graph of a continuous real-valued function f (x, y) is the surface in R 3 consisting of the points (a, b, f (a, b)) for (a, b) in the domain D of f . • A vertical trace is a curve obtained by intersecting the graph with a vertical plane x = a or y = b. •

Functions of Two or More Variables

S E C T I O N 12.1

z

z

x

x

z

z

y

y

g(x, y, z) = c (c > 0)

y

y x

x

g(x, y, z) = 0

681

g(x, y, z) = c (c < 0)

FIGURE 20 Level surfaces of g(x, y, z) = x 2 + y 2 − z2 .

A level curve is a curve in the xy-plane defined by an equation f (x, y) = c. The level curve f (x, y) = c is the projection onto the xy-plane of the horizontal trace curve, obtained by intersecting the graph with the horizontal plane z = c. • A contour map shows the level curves f (x, y) = c for equally spaced values of c. The spacing m is called the contour interval. • When reading a contour map, keep in mind: •

– Your altitude does not change when you hike along a level curve. – Your altitude increases or decreases by m (the contour interval) when you hike from one level curve to the next. •

The spacing of the level curves indicates steepness: They are closer together where the graph is steeper. altitude • The average rate of change from P to Q is the ratio . horizontal • A direction of steepest ascent at a point P is a direction along which f (x, y) increases most rapidly. The steepest direction is obtained (approximately) by drawing the segment from P to the nearest point on the next level curve.

12.1 EXERCISES Preliminary Questions 1. What is the difference between a horizontal trace and a level curve? How are they related? 2. Describe the trace of f (x, y) = x 2 − sin(x 3 y) in the xz-plane. 3. Is it possible for two different level curves of a function to intersect? Explain.

Exercises In Exercises 1–4, evaluate the function at the specified points. 1. f (x, y) = x + yx 3 ,

(2, 2), (−1, 4)

4. Describe the contour map of f (x, y) = x with contour interval 1. 5. How will the contour maps of f (x, y) = x

and g(x, y) = 2x

with contour interval 1 look different?

y 2. g(x, y) = 2 , x + y2

(1, 3), (3, −2)

3. h(x, y, z) = xyz−2 ,

(3, 8, 2), (3, −2, −6)

682

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

    (y, z) = 2, π2 , − 2, π6 In Exercises 5–12, sketch the domain of the function.  5. f (x, y) = 12x − 5y 6. f (x, y) = 81 − x 2 4. Q(y, z) = y 2 + y sin z,

7. f (x, y) = ln(4x 2 − y)

8. h(x, t) =

(b) f (x, y) = cos(x − y) (c) f (x, y) =

2 2 (d) f (x, y) = cos(y 2 )e−0.1(x +y )

1 x+t

(e) f (x, y) =

y 10. f (x, y) = sin x

1 9. g(y, z) = z + y2 √ 11. F (I, R) = I R

−1 1 + 9x 2 + y 2

12. f (x, y) = cos−1 (x + y)

−1 1 + 9x 2 + 9y 2

2 2 (f) f (x, y) = cos(x 2 + y 2 )e−0.1(x +y )

In Exercises 13–16, describe the domain and range of the function. √ 14. f (x, y, z) = x y + zez/x 13. f (x, y, z) = xz + ey  15. P (r, s, t) = 16 − r 2 s 2 t 2 16. g(r, s) = cos−1 (rs)

z

z

17. Match graphs (A) and (B) in Figure 21 with the functions (i) f (x, y) = −x + y 2

(ii) g(x, y) = x + y 2

z

y x

z

y

y x

(A)

(B)

z

z

x y

y

x (A)

x

(B) FIGURE 21

(C)

18. Match each of graphs (A) and (B) in Figure 22 with one of the following functions:

z

x

y (D) z

(i) f (x, y) = (cos x)(cos y) (ii) g(x, y) = cos(x 2 + y 2 ) z

y

z

x (E)

(F) FIGURE 23

y

y x

x (A)

20. Match the functions (a)–(d) with their contour maps (A)–(D) in Figure 24. (B)

FIGURE 22

19. Match the functions (a)–(f) with their graphs (A)–(F) in Figure 23. (a) f (x, y) = |x| + |y|

(a) f (x, y) = 3x + 4y (b) g(x, y) = x 3 − y (c) h(x, y) = 4x − 3y (d) k(x, y) = x 2 − y

Functions of Two or More Variables

S E C T I O N 12.1

10

10

5

5

0

0

−5

−5

−10 −10 −5

0

5

10

−10 −10 −5

(A) 10

5

5

0

0

−5

−5

−10 −10 −5

0

y 2 1 −6 0

5

−3

3

−1

10

5

10

−10 −10 −5

(C)

x

6

−2

c=6 c=0

(B)

10

683

FIGURE 25 Contour map with contour interval m = 6

38. Use the contour map in Figure 26 to calculate the average rate of change: (a) From A to B. 0

5

(b) From A to C.

10

(D)

y

FIGURE 24

6 B

In Exercises 21–26, sketch the graph and describe the vertical and horizontal traces.  21. f (x, y) = 12 − 3x − 4y 22. f (x, y) = 4 − x 2 − y 2 23. f (x, y) = x 2 + 4y 2

24. f (x, y) = y 2

25. f (x, y) = sin(x − y)

1 26. f (x, y) = 2 x + y2 + 1

27. Sketch contour maps of f (x, y) = x + y with contour intervals m = 1 and 2. 28. Sketch the contour map of f (x, y) = x 2 + y 2 with level curves c = 0, 4, 8, 12, 16.

C A

4

c = −3 2

−6

−4

c=0

−2

2

4

6

x

FIGURE 26

39. Referring to Figure 27, answer the following questions: (a) At which of (A)–(C) is pressure increasing in the northern direction? (b) At which of (A)–(C) is temperature increasing in the easterly direction? (c) In which direction at (B) is temperature increasing most rapidly?

In Exercises 29–36, draw a contour map of f (x, y) with an appropriate contour interval, showing at least six level curves. 29. f (x, y) = x 2 − y 31. f (x, y) =

y x

1020

y 30. f (x, y) = 2 x 32. f (x, y) = xy

1032 1032

1024

1008

1028

1024 1020

1028 1024

1020

B

C 1012

1016

1016

1016

33. f (x, y) = x 2 + 4y 2

34. f (x, y) = x + 2y − 1

35. f (x, y) = x 2

36. f (x, y) = 3x 2 − y 2

A

1012

1008 1004 1006

37. Find the linear function whose contour map (with contour interval m = 6) is shown in Figure 25. What is the linear function if m = 3 (and the curve labeled c = 6 is relabeled c = 3)?

1004

1000

1012

1016 1016

1016

FIGURE 27 Atmospheric Pressure (in millibars) over the continental

U.S. on March 26, 2009

684

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

In Exercises 40–43, ρ(S, T ) is seawater density (kg/m3 ) as a function of salinity S (ppt) and temperature T (◦ C). Refer to the contour map in Figure 28.

In Exercises 44–47, refer to Figure 29.

25

230

44. Find the change in elevation from A and B.

1.0

35 1.02 0 4 0 1. 2 45 A 1.02 50 1.02 55 1.02 260 1.0

Temperature T °C

20 15 10

1.0

5 0 31.5

43. Does water density appear to be more sensitive to a change in temperature at point A or point B?

45. Estimate the average rate of change from A and B and from A to C. 46. Estimate the average rate of change from A to points i, ii, and iii.

C

B 32.5

33.0

0

27

1.0

32.0

47. Sketch the path of steepest ascent beginning at D.

265

33.5

34.0

34.5

C

Salinity (ppt)

540

B

FIGURE 28 Contour map of seawater density ρ(S, T ) (kg/m3 ).

500

i

40. Calculate the average rate of change of ρ with respect to T from B to A.

ii

D

400

iii A

41. Calculate the average rate of change of ρ with respect to S from B to C.

Contour interval = 20 m

42. At a fixed level of salinity, is seawater density an increasing or a decreasing function of temperature?

0

1

2 km

FIGURE 29

Further Insights and Challenges Temperature T

48. The function f (x, t) = t −1/2 e−x /t , whose graph is shown in Figure 30, models the temperature along a metal bar after an intense burst of heat is applied at its center point. (a) Sketch the vertical traces at times t = 1, 2, 3. What do these traces tell us about the way heat diffuses through the bar? (b) Sketch the vertical traces x = c for c = ±0.2, ±0.4. Describe how temperature varies in time at points near the center. 2

Time t 4 3 −0.4−0.2 0 0.2 0.4

49. Let x f (x, y) =  2 x + y2

for (x, y) = (0, 0)

Write f as a function f (r, θ ) in polar coordinates, and use this to find the level curves of f .

Metal bar

2 1 x

2 FIGURE 30 Graph of f (x, t) = t −1/2 e−x /t beginning shortly after

t = 0.

12.2 Limits and Continuity in Several Variables This section develops limits and continuity in the multivariable setting. We focus on functions of two variables, but similar definitions and results apply to functions of three or more variables. Recall that a number x is close to a if the distance |x − a| is small. In the plane, one point (x, y) is close to another point P = (a, b) if the distance between them is small.

Limits and Continuity in Several Variables

S E C T I O N 12.2

685

To express this precisely, we define the open disk of radius r and center P = (a, b) (Figure 1):

y D*(P, r) excludes P

D(P , r) = {(x, y) ∈ R 2 : (x − a)2 + (y − b)2 < r 2 }

r

The open punctured disk D ∗ (P , r) is the disk D(P , r) with its center point P removed. Thus D ∗ (P , r) consists of all points whose distance to P is less than r, other than P itself. Now assume that f (x, y) is defined near P but not necessarily at P itself. In other words, f (x, y) is defined for all (x, y) in some punctured disk D ∗ (P , r) with r > 0. We say that f (x, y) approaches the limit L as (x, y) approaches P = (a, b) if |f (x, y) − L| becomes arbitrarily small for (x, y) in a sufficiently small punctured disk centered at P [Figure 2(C)]. In this case, we write

P (x, y) Open disk D(P, r) x FIGURE 1 The open disk D(P , r) consists

of points (x, y) at distance < r from P . It does not include the boundary circle.

lim

(x,y)→P

f (x, y) =

lim

(x,y)→(a,b)

f (x, y) = L

Here is the formal definition. DEFINITION Limit Assume that f (x, y) is defined near P = (a, b). Then lim

(x,y)→P

f (x, y) = L

if, for any  > 0, there exists δ > 0 such that |f (x, y) − L| < 

(x, y) ∈ D ∗ (P , δ)

for all

This is similar to the limit definition in one variable, but there is an important difference. In a one-variable limit, we require that f (x) tend to L as x approaches a from the left or right [Figure 2(A)]. In a multivariable limit, f (x, y) must tend to L no matter how (x, y) approaches P [Figure 2(B)]. z

z

y

f (x, y) L + ⑀ L L−⑀

z = f(x, y) y = f(x)

L+⑀ L L−⑀

a

x

(A) In one variable, we can approach a from only two possible directions.

x

P = (a, b)

y

x

Open disk of radius

(a, b)

(x, y)

y

(C) | f (x, y) − L | < ⑀ for all (x, y) inside the disk

(B) In two variables, (x, y) can approach P = (a, b) along any direction or path.

FIGURE 2

E X A M P L E 1 Show that (a)

lim

(x,y)→(a,b)

x = a and (b)

lim

(x,y)→(a,b)

y = b.

Solution Let P = (a, b). To verify (a), let f (x, y) = x and L = a. We must show that for any  > 0, we can find δ > 0 such that |f (x, y) − L| = |x − a| < 

for all

(x, y) ∈ D ∗ (P , δ)

1

686

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

In fact, we can choose δ = , for if (x, y) ∈ D ∗ (P , ), then

z

(x − a)2 + (y − b)2 <  2

f (x, y) = y

b+⑀

⇒

(x − a)2 <  2

⇒

|x − a| < 

In other words, for any  > 0,

b

|x − a| < 

b−⑀

b a

y

x FIGURE 3 We have |f (x, y) − b| <  if

|y − b| < δ for δ = . Therefore, lim

(x, y) ∈ D ∗ (P , )

This proves (a). The limit (b) is similar (see Figure 3).

P = (a, b)

D*(P, )

(x,y)→(a,b)

for all

y=b

The following theorem lists the basic laws for limits. We omit the proofs, which are similar to the proofs of the single-variable Limit Laws. THEOREM 1 Limit Laws Assume that

f (x, y) and

lim

(x,y)→P

lim

(x,y)→P

g(x, y) exist.

Then: (i) Sum Law: lim

(x,y)→P

(f (x, y) + g(x, y)) =

lim

(x,y)→P

f (x, y) +

lim

(x,y)→P

g(x, y)

(ii) Constant Multiple Law: For any number k, lim

(x,y)→P

kf (x, y) = k

(iii) Product Law: lim

(x,y)→P

lim

(x,y)→P

f (x, y)

 f (x, y) g(x, y) =

(iv) Quotient Law: If

lim

(x,y)→P

 lim

(x,y)→P

f (x, y)

 lim

(x,y)→P

g(x, y)

g(x, y)  = 0, then

lim

(x,y)→P

f (x, y) = g(x, y)

lim

f (x, y)

lim

g(x, y)

(x,y)→P (x,y)→P

As in the single-variable case, we say that f is continuous at P = (a, b) if f (x, y) approaches the function value f (a, b) as (x, y) → (a, b). DEFINITION Continuity A function f (x, y) is continuous at P = (a, b) if lim

(x,y)→(a,b)

f (x, y) = f (a, b)

We say that f is continuous if it is continuous at each point (a, b) in its domain. The Limit Laws tell us that all sums, multiples, and products of continuous functions are continuous. When we apply them to f (x, y) = x and g(x, y) = y, which are continuous by Example 1, we find that the power functions f (x, y) = x m y n are continuous for all whole numbers m, n and that all polynomials are continuous. Furthermore, a rational function h(x, y)/g(x, y), where h and g are polynomials, is continuous at all points (a, b) where g(a, b)  = 0. As in the single-variable case, we can evaluate limits of continuous functions using substitution.

Limits and Continuity in Several Variables

S E C T I O N 12.2

z

E X A M P L E 2 Evaluating Limits by Substitution Show that

f (x, y) =

y

is continuous (Figure 4). Then evaluate

x

x2

3x + y + y2 + 1

lim

(x,y)→(1,2)

f (x, y).

Solution The function f (x, y) is continuous at all points (a, b) because it is a rational function whose denominator Q(x, y) = x 2 + y 2 + 1 is never zero. Therefore, we can evaluate the limit by substitution: lim

(x,y)→(1,2)

FIGURE 4 Top view of the graph

3x + y f (x, y) = 2 . x + y2 + 1

687

3x + y 3(1) + 2 5 = 2 = 6 x2 + y2 + 1 1 + 22 + 1

If f (x, y) is a product f (x, y) = h(x)g(y), where h(x) and g(y) are continuous, then the limit is a product of limits by the Product Law: 

 lim f (x, y) = lim h(x)g(y) = lim h(x) lim g(y) (x,y)→(a,b)

x→a

(x,y)→(a,b)

E X A M P L E 3 Product Functions Evaluate

lim

(x,y)→(3,0)

y→b

x3

sin y . y

Solution The limit is equal to a product of limits: lim

(x,y)→(3,0)

x

3 sin y

y



 =

lim x

sin y lim y→0 y

3

x→3

 = (33 )(1) = 27

Composition is another important way to build functions. If f (x, y) is a function of two variables and G(u) a function of one variable, then the composite G ◦ f is the function G(f (x, y)). According to the next theorem, a composite of continuous functions is again continuous. THEOREM 2 A Composite of Continuous Functions Is Continuous If f (x, y) is continuous at (a, b) and G(u) is continuous at c = f (a, b), then the composite function G(f (x, y)) is continuous at (a, b). 2 E X A M P L E 4 Write H (x, y) = e−x +2y as a composite function and evaluate

lim

(x,y)→(1,2)

H (x, y)

Solution We have H (x, y) = G ◦ f , where G(u) = eu and f (x, y) = −x 2 + 2y. Both f and G are continuous, so H is also continuous and lim

(x,y)→(1,2)

H (x, y) =

We know that if a limit

lim

lim

(x,y)→(1,2)

(x,y)→(a,b)

e−x

2 +2y

= e−(1)

2 +2(2)

= e3

f (x, y) exists and equals L, then f (x, y) tends to

L as (x, y) approaches (a, b) along any path. In the next example, we prove that a limit does not exist by showing that f (x, y) approaches different limits along lines through the origin.

688

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

E X A M P L E 5 Showing a Limit Does Not Exist Examine

cally. Then prove that the limit does not exist.

x2 numeri(x,y)→(0,0) x 2 + y 2 lim

Solution If the limit existed, we would expect the values of f (x, y) in Table 1 to get closer to a limiting value L as (x, y) gets close to (0, 0). But the table suggests that f (x, y) takes on all values between 0 and 1, no matter how close (x, y) gets to (0, 0). For example, f (0.1, 0) = 1,

f (0.1, 0.1) = 0.5,

f (0, 0.1) = 0

Thus, f (x, y) does not seem to approach any fixed value L as (x, y) → (0, 0). Now let’s prove that the limit does not exist by showing that f (x, y) approaches different limits along the x- and y-axes (Figure 5): x→0

y→0

These two limits are different and hence

x

y

0.5

z

x y

Values of f (x, y) =

02 = lim 0 = 0 y→0 02 + y 2 y→0

lim f (0, y) = lim

Limit along y-axis:

TABLE 1

x2 = lim 1 = 1 x→0 x 2 + 02 x→0

lim f (x, 0) = lim

Limit along x-axis:

lim

(x,y)→(0,0)

f (x, y) does not exist.

x2 x2 + y2

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.5

0.39

0.265

0.138

0.038

0

0.038

0.138

0.265

0.39

0.5

0.4

0.61

0.5

0.36

0.2

0.059

0

0.059

0.2

0.36

0.5

0.61

0.3

0.735

0.64

0.5

0.308

0.1

0

0.1

0.308

0.5

0.64

0.735

0.2

0.862

0.8

0.692

0.5

0.2

0

0.2

0.5

0.692

0.8

0.862

0.1

0.962

0.941

0.9

0.8

0.5

0

0.5

0.8

0.9

0.941

0.962

0

1

1

1

1

1

1

1

1

1

1

−0.1

0.962

0.941

0.9

0.8

0.5

0

0.5

0.8

0.9

0.941

0.962

−0.2

0.862

0.8

0.692

0.5

0.2

0

0.2

0.5

0.692

0.8

0.862

−0.3

0.735

0.640

0.5

0.308

0.1

0

0.1

0.308

0.5

0.640

0.735

−0.4

0.610

0.5

0.360

0.2

0.059

0

0.059

0.2

0.36

0.5

0.61

−0.5

0.5

0.39

0.265

0.138

0.038

0

0.038

0.138

0.265

0.390

0.5

The contour map in Figure 5 shows clearly that the function f (x, y) = x 2 /(x 2 + y 2 ) does not approach a limit as (x, y) approaches (0, 0). For nonzero c, the level curve f (x, y) = c is the line y = mx through the origin (with the origin deleted) where c = (m2 + 1)−1 :

GRAPHICAL INSIGHT

x c=1 c=0 0.9 0.7 0.5 0.3 0.1

y

x2 FIGURE 5 Graph of f (x, y) = . x2 + y2

f (x, mx) =

x2 1 = 2 2 2 x + (mx) m +1

(for x  = 0)

The level curve f (x, y) = 0 is the y-axis (with the origin deleted). As the slope m varies, f takes on all values between 0 and 1 in every disk around the origin (0, 0), no matter how small, so f cannot approach a limit. As we know, there is no single method for computing limits that always works. The next example illustrates two different approaches to evaluating a limit in a case where substitution cannot be used.

Limits and Continuity in Several Variables

S E C T I O N 12.2

z

E X A M P L E 6 Two Methods for Verifying a Limit Calculate

f (x, y) is defined for (x, y)  = (0, 0) by (Figure 6) f (x, y) =

lim

(x,y)→(0,0)

689

f (x, y) where

xy 2 + y2

x2

Solution x y FIGURE 6 Graph of f (x, y) =

xy 2 x2 + y2

.

First Method For (x, y)  = (0, 0), we have    y2  ≤1 0 ≤  2 x + y2  because the numerator is not greater than the denominator. Multiply by |x|:    xy 2   ≤ |x| 0 ≤  2 x + y2  and use the Squeeze Theorem (which is valid for limits in several variables):    xy 2   ≤ 0≤ lim lim |x| (x,y)→(0,0)  x 2 + y 2  (x,y)→(0,0) Because

lim

(x,y)→(0,0)

|x| = 0, we conclude that

lim

(x,y)→(0,0)

f (x, y) = 0 as desired.

Second Method Use polar coordinates: x = r cos θ,

y = r sin θ

Then x 2 + y 2 = r 2 and for r  = 0,      xy 2   (r cos θ )(r sin θ)2     = r|cos θ sin2 θ | ≤ r  = 0≤ 2  x + y2   r2 As (x, y) approaches (0, 0), the variable r also approaches 0, so again, the desired conclusion follows from the Squeeze Theorem:    xy 2    ≤ lim r = 0 0≤ lim r→0 (x,y)→(0,0)  x 2 + y 2 

12.2 SUMMARY •

The open disk of radius r centered at P = (a, b) is defined by D(P , r) = {(x, y) ∈ R 2 : (x − a)2 + (y − b)2 < r 2 }

The punctured disk D ∗ (P , r) is D(P , r) with P removed. • Suppose that f (x, y) is defined near P = (a, b). Then lim

(x,y)→(a,b)

f (x, y) = L

if, for any  > 0, there exists δ > 0 such that |f (x, y) − L| < 

for all

(x, y) ∈ D ∗ (P , δ)

690

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES •

The limit of a product f (x, y) = h(x)g(y) is a product of limits: 

 lim f (x, y) = lim h(x) lim g(y) x→a

(x,y)→(a,b)

•

y→b

A function f (x, y) is continuous at P = (a, b) if lim

(x,y)→(a,b)

f (x, y) = f (a, b)

12.2 EXERCISES Preliminary Questions 1. What is the difference between D(P , r) and D ∗ (P , r)? 2. Suppose that f (x, y) is continuous at (2, 3) and that f (2, y) = y 3 for y = 3. What is the value f (2, 3)?

(a) Q(x, y) is continuous for all (x, y). (b) Q(x, y) is continuous for (x, y)  = (0, 0). (c) Q(x, y)  = 0 for all (x, y).

3. Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). Which of the following statements are true?

4. Suppose that f (x, 0) = 3 for all x  = 0 and f (0, y) = 5 for all y  = 0. What can you conclude about lim f (x, y)? (x,y)→(0,0)

Exercises In Exercises 1–8, evaluate the limit using continuity 1. lim (x 2 + y) 2. lim

x 4 2 y (x,y)→( 9 , 9 )

(x,y)→(1,2)

3. 5.

lim

(xy − 3x 2 y 3 )

4.

tan x cos y

6.

(x,y)→(2,−1)

lim

(x,y)→( π4 ,0)

2x 2 (x,y)→(−2,1) 4x + y lim

lim

(x,y)→(2,3)

tan−1 (x 2 − y)

15. Prove that x

lim

(x,y)→(0,0) x 2 + y 2

does not exist by considering the limit along the x-axis. 16. Let f (x, y) = x 3 /(x 2 + y 2 ) and g(x, y) = x 2 /(x 2 + y 2 ). Using polar coordinates, prove that lim

(x,y)→(0,0)

ex − e−y x+y (x,y)→(1,1) 2

7.

2

lim

8.

lim

(x,y)→(1,0)

ln(x − y)

lim

9.

10.

12.

lim

lim

(x,y)→(2,5)

g(x, y) = 7



(x,y)→(2,5)

lim

f (x, y) = 3,

(x,y)→(2,5)

 g(x, y) − 2f (x, y)

f (x, y)2 g(x, y)

and that

lim

(x,y)→(0,0)

g(x, y) does not exist. Hint: Show that g(x, y) =

cos2 θ and observe that cos θ can take on any value between −1 and 1 as (x, y) → (0, 0).

In Exercises 9–12, assume that (x,y)→(2,5)

f (x, y) = 0

11.

lim

(x,y)→(2,5)

2 ef (x,y) −g(x,y)

 18. Evaluate

lim

(x,y)→(0,0)

tan x sin

 1 . |x| + |y|

In Exercises 19–32, evaluate the limit or determine that it does not exist.

f (x, y) (x,y)→(2,5) f (x, y) + g(x, y) lim

y2 exist? Explain. 13. Does lim (x,y)→(0,0) x 2 + y 2 14. Let f (x, y) = xy/(x 2 + y 2 ). Show that f (x, y) approaches zero along the x- and y-axes. Then prove that

17. Use the Squeeze Theorem to evaluate   1 lim (x 2 − 16) cos (x,y)→(4,0) (x − 4)2 + y 2

lim

(x,y)→(0,0)

19.

z4 cos(πw) ez+w (z,w)→(−2,1)

20.

21.

y−2  (x,y)→(4,2) x 2 − 4

22.

lim

lim

f (x, y) does not

exist by showing that the limit along the line y = x is nonzero.

23.

lim

(x,y)→(3,4)

1  2 x + y2

24.

lim

(z2 w − 9z)

(z,w)→(−1,2)

x2 + y2 (x,y)→(0,0) 1 + y 2 lim

lim

(x,y)→(0,0)

xy  2 x + y2

S E C T I O N 12.2

25. 27. 29.

lim

(x,y)→(1,−3)

ex−y ln(x − y)

26.

(x 2 y 3 + 4xy)

28.

lim

(x,y)→(−3,−2)

lim

(x,y)→(0,0)

tan(x 2 + y 2 ) tan−1



|x|

lim

(x,y)→(0,0) |x| + |y|

lim

(x,y)→(2,1)

1 2 x + y2

2 2 ex −y



2 2 (x + y + 2)e−1/(x +y ) 30. lim (x,y)→(0,0)

31.

lim

(x,y)→(0,0)



(c) Use the Squeeze Theorem to prove that 34. Let a, b ≥ 0. Show that

lim

lim

(x,y)→(0,0)

xa yb

(x,y)→(0,0) x 2 + y 2

691

f (x, y) = 0.

= 0 if a + b > 2 and

that the limit does not exist if a + b ≤ 2.

35. Figure 7 shows the contour maps of two functions. Explain why the limit lim f (x, y) does not exist. Does lim g(x, y) (x,y)→P

(x,y)→Q

appear to exist in (B)? If so, what is its limit?

x2 + y2 −1

x2 + y2 + 1 − 1

x2 + y2 − 2 (x,y)→(1,1) |x − 1| + |y − 1| Hint: Rewrite the limit in terms of u = x − 1 and v = y − 1. 32.

Limits and Continuity in Several Variables

lim

1

−3

30

3 −5

P

24

5

x3 + y3 33. Let f (x, y) = 2 . x + y2 (a) Show that |x 3 | ≤ |x|(x 2 + y 2 ),

18

Q

12 6

|y 3 | ≤ |y|(x 2 + y 2 )

0

(A) Contour map of f (x, y)

(b) Show that |f (x, y)| ≤ |x| + |y|.

(B) Contour map of g(x, y) FIGURE 7

Further Insights and Challenges 36. Evaluate

lim

(1 + x)y/x .

(x,y)→(0,2)

37. Is the following function continuous? x 2 + y 2 if x 2 + y 2 < 1 f (x, y) = 1 if x 2 + y 2 ≥ 1

(b) Calculate f (x, y) at the points (10−1 , 10−2 ), (10−5 , 10−10 ), (10−20 , 10−40 ). Do not use a calculator. f (x, y) does not exist. Hint: Compute the (c) Show that lim (x,y)→(0,0)

limit along the parabola y = x 2 . z

38. The function f (x, y) = sin(xy)/xy is defined for xy = 0. (a) Is it possible to extend the domain of f (x, y) to all of R 2 so that the result is a continuous function?

y

(b) Use a computer algebra system to plot f (x, y). Does the result support your conclusion in (a)? 39. Prove that the function ⎧ x ⎨ (2 − 1)(sin y) f (x, y) = xy ⎩ ln 2

x

if xy  = 0 if xy = 0

is continuous at (0, 0). y

40. Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F (x, y) = f (x)g(y) is continuous at (a, b). The function f (x, y) = x 2 y/(x 4 + y 2 ) provides an inter41. esting example where the limit as (x, y) → (0, 0) does not exist, even though the limit along every line y = mx exists and is zero (Figure 8). (a) Show that the limit along any line y = mx exists and is equal to 0.

x FIGURE 8 Graph of f (x, y) =

x2y . x4 + y2

692

C H A P T E R 12

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12.3 Partial Derivatives We have stressed that a function f of two or more variables does not have a unique rate of change because each variable may affect f in different ways. For example, the current I in a circuit is a function of both voltage V and resistance R given by Ohm’s Law: I (V , R) =

V R

The current I is increasing as a function of V but decreasing as a function of R. The partial derivatives are the rates of change with respect to each variable separately. A function f (x, y) of two variables has two partial derivatives, denoted fx and fy , defined by the following limits (if they exist):

fx (a, b) = lim

h→0

f (a + h, b) − f (a, b) , h

fy (a, b) = lim

k→0

f (a, b + k) − f (a, b) k

Thus, fx is the derivative of f (x, b) as a function of x alone, and fy is the derivative at f (a, y) as a function of y alone. The Leibniz notation for partial derivatives is The partial derivative symbol ∂ is a rounded “d.” The symbols ∂f/∂x and ∂f/∂y are read as follows: “dee-eff dee-ex” and “dee-eff dee-why.”

∂f = fx , ∂x  ∂f  = fx (a, b), ∂x (a,b)

∂f = fy ∂y  ∂f  = fy (a, b) ∂y  (a,b)

If z = f (x, y), then we also write ∂z/∂x and ∂z/∂y. Partial derivatives are computed just like ordinary derivatives in one variable with this difference: To compute fx , treat y as a constant, and to compute fy , treat x as a constant. E X A M P L E 1 Compute the partial derivatives of f (x, y) = x 2 y 5 .

Solution ∂  2 ∂  2 5 ∂f = x y = y5 x = y 5 (2x) = 2xy 5 ∂x ∂x ∂x    Treat y 5 as a constant

∂  2 5 ∂f ∂  5 = x y = x2 y = x 2 (5y 4 ) = 5x 2 y 4 ∂y ∂y ∂x    Treat x 2 as a constant

The partial derivatives at P = (a, b) are the slopes of the tangent lines to the vertical trace curves through the point (a, b, f (a, b)) in Figure 1(A). To compute fx (a, b), we set y = b and differentiate in the x-direction. This gives us the slope of the tangent line to the trace curve in the plane y = b [Figure 1(B)]. Similarly, fy (a, b) is the slope of the trace curve in the vertical plane x = a [Figure 1(C)]. GRAPHICAL INSIGHT

The differentiation rules from calculus of one variable (the Product, Quotient, and Chain Rules) are valid for partial derivatives.

S E C T I O N 12.3

z

z

Slope f x (a, b)

Slope f y (a, b)

P

a

b

x

(A)

P

The trace curve (a, y, f(a, y))

a

b

y

(a, b)

x

693

z The trace curve (x, b, f(x, b))

P = (a, b, f (a, b))

Partial Derivatives

y

Plane y = b

y Plane x = a

x

(B)

(C)

FIGURE 1 The partial derivatives are the slopes of the vertical trace curves.

z g (x, y) =

y2

E X A M P L E 2 Calculate gx (1, 3) and gy (1, 3), where g(x, y) =

(1 + x2)3

y2 . (1 + x 2 )3

Solution To calculate gx , treat y (and therefore y 2 ) as a constant:

y x

(

P = 1, 3, 9 8

)

FIGURE 2 The slopes of the tangent lines to

the trace curves are gx (1, 3) and gy (1, 3). CAUTION It is not necessary to use the Quotient Rule to compute the partial derivative in Eq. (1). The denominator does not depend on y , so we treat it as a constant when differentiating with respect to y .

gx (x, y) =

∂ −6xy 2 y2 2 ∂ 2 −3 = y ) = (1 + x ∂x (1 + x 2 )3 ∂x (1 + x 2 )4

gx (1, 3) =

−6(1)32 27 =− 8 (1 + 12 )4

To calculate gy , treat x (and therefore 1 + x 2 ) as a constant: gy (x, y) =

y2 ∂ 2 ∂ 1 2y y = = ∂y (1 + x 2 )3 (1 + x 2 )3 ∂y (1 + x 2 )3

gy (1, 3) =

2(3) 3 = 2 3 4 (1 + 1 )

1

  These partial derivatives are the slopes of the trace curves through the point 1, 3, 98 shown in Figure 2. We use the Chain Rule to compute partial derivatives of a composite function f (x, y) = F (g(x, y)), where F (u) is a function of one variable and u = g(x, y): ∂f dF ∂u = , ∂x du ∂x

∂f dF ∂u = ∂y du ∂y

E X A M P L E 3 Chain Rule for Partial Derivatives Compute

∂ sin(x 2 y 5 ). ∂x

Solution Write sin(x 2 y 5 ) = F (u), where F (u) = sin u and u = x 2 y 5 . Then we have dF = cos u and the Chain Rule give us du ∂ dF ∂u ∂ sin(x 2 y 5 ) = = cos(x 2 y 5 ) x 2 y 5 = 2xy 5 cos(x 2 y 5 ) ∂x du ∂x ∂x    Chain Rule

Partial derivatives are defined for functions of any number of variables. We compute the partial derivative with respect to any one of the variables by holding the remaining variables constant.

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C H A P T E R 12

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E X A M P L E 4 More Than Two Variables Calculate fz (0, 0, 1, 1), where

f (x, y, z, w) = In Example 4, the calculation

∂ xz+y = xexz+y e ∂z

exz+y z2 + w

Solution Use the Quotient Rule, treating x, y, and w as constants: ∂ ∂z

fz (x, y, z, w) =

follows from the Chain Rule, just like

d az+b e = aeaz+b dz



exz+y z2 + w

 =

∂ ∂ xz+y (z2 + w) ∂z e − exz+y ∂z (z2 + w)

(z2 + w)2

(z2 + w)xexz+y − 2zexz+y (z2 x + wx − 2z)exz+y = (z2 + w)2 (z2 + w)2

=

−2e0 1 =− 2 2 2 (1 + 1)

fz (0, 0, 1, 1) =

Because the partial derivative fx (a, b) is the derivative f (x, b), viewed as a function of x alone, we can estimate the change f when x changes from a to a + x as in the single-variable case. Similarly, we can estimate the change when y changes by y. For small x and y (just how small depends on f and the accuracy required): f (a + x, b) − f (a, b) ≈ fx (a, b)x f (a, b + y) − f (a, b) ≈ fy (a, b)y This applies to functions f in any number of variables. For example, f ≈ fw w if one of the variables w changes by w and all other variables remain fixed.

Chip

Solder ball of radius R

E X A M P L E 5 Testing Microchips A ball grid array (BGA) is a microchip joined to a circuit board by small solder balls of radius R mm separated by a distance L mm (Figure 3). Manufacturers test the reliability of BGAs by subjecting them to repeated cycles in which the temperature is varied from 0◦ C to 100◦ C over a 40-min period. According to one model, the average number N of cycles before the chip fails is

L

 N=

Circuit board FIGURE 3 A BGA package. Temperature

variations strain the BGA and may cause it to fail because the chip and board expand at different rates.

2200R Ld

1.9

where d is the difference between the coefficients of expansion of the chip and the board. Estimate the change N when R = 0.12, d = 10, and L is increased from 0.4 to 0.42. Solution We use the approximation N ≈

∂N L ∂L

with L = 0.42 − 0.4 = 0.02. Since R and d are constant, the partial derivative is ∂N ∂ = ∂L ∂L



2200R Ld

1.9

 =

2200R d

1.9

  ∂ −1.9 2200R 1.9 −2.9 L = −1.9 L ∂L d

S E C T I O N 12.3

Partial Derivatives

695

Now evaluate at L = 0.4, R = 0.12, and d = 10:    2200(0.12) 1.9 ∂N  = −1.9 (0.4)−2.9 ≈ −13,609 ∂L (L,R,d)=(0.4,0.12,10) 10 The decrease in the average number of cycles before a chip fails is N ≈

∂N L = −13,609(0.02) ≈ −272 cycles ∂L

In the next example, we estimate a partial derivative numerically. Since fx and fy are limits of difference quotients, we have the following approximations when h and k are “small”: fx (a, b) ≈

f f (a + h, b) − f (a, b) = x h

fy (a, b) ≈

f f (a, b + k) − f (a, b) = y k

A similar approximation is valid in any number of variables.

Temperature T (°C)

25

E X A M P L E 6 Estimating Partial Derivatives Using Contour Maps Seawater density ρ (kg/m3 ) depends on salinity S (ppt) and the temperature T (◦ C). Use Figure 4 to estimate ∂ρ/∂T and ∂ρ/∂S at A.

30

1.02

20 B A

15

35 1.02 40 0 . 1 2 45 1.02

C

Solution Point A has coordinates (S, T ) = (33, 15) and lies on the level curve ρ = 1.0245. We estimate ∂ρ/∂T at A in two steps.

50

1.02

260

1.0

10

265

1.0

5 0

32.5

33.0

33.5

34.0

34.5

Salinity S (ppt) FIGURE 4 Contour map of seawater density

Step 1. Move vertically from A. Since T varies in the vertical direction, we move up vertically from point A to point B on the next higher level curve, where ρ = 1.0240. Point B has coordinates (S, T ) = (33, 17). Note that in moving from A to B, we have kept S constant because both points have salinity S = 33. Step 2. Compute the difference quotient. ρ = 1.0240 − 1.0245 = −0.0005 kg/m3 T = 17 − 15 = 2◦ C

as a function of temperature and salinity.

This gives us the approximation For greater accuracy, we can estimate fx (a, b) by taking the average of the difference quotients for x and −x . A similar remark applies to fy (a, b).

 ρ −0.0005 ∂ρ  ≈ = = −0.00025 kg-m−3 /◦ C ∂T A T 2 We estimate ∂ρ/∂S in a similar way, by moving to the right horizontally to point C with coordinates (S, T ) ≈ (33.7, 15), where ρ = 1.0250:  ρ 1.0250 − 1.0245 0.0005 ∂ρ  ≈ = = ≈ 0.0007 kg-m−3 /ppt  ∂S A S 33.7 − 33 0.7

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C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

Higher-Order Partial Derivatives The higher-order partial derivatives are the derivatives of derivatives. The second-order partial derivatives of f are the partial derivatives of fx and fy . We write fxx for the x-derivative of fx and fyy for the y-derivative of fy : fxx =

∂ ∂x



∂f ∂x

 fyy =

,

∂ ∂y



∂f ∂y



We also have the mixed partials: fxy

∂ = ∂y



∂f ∂x

 ,

∂ = ∂x

fyx



∂f ∂y



The process can be continued. For example, fxyx is the x-derivative of fxy , and fxyy is the y-derivative of fxy (perform the differentiation in the order of the subscripts from left to right). The Leibniz notation for higher-order partial derivatives is fxx =

∂ 2f , ∂x 2

fxy =

∂ 2f , ∂y∂x

fyx =

∂ 2f , ∂x∂y

fyy =

∂ 2f ∂y 2

Higher partial derivatives are defined for functions of three or more variables in a similar manner. E X A M P L E 7 Calculate the second-order partials of f (x, y) = x 3 + y 2 ex .

Solution First, we compute the first-order partial derivatives: fx (x, y) =

∂ (x 3 + y 2 ex ) = 3x 2 + y 2 ex , ∂x

fy (x, y) =

∂ (x 3 + y 2 ex ) = 2yex ∂y

Then we can compute the second-order derivatives: fxx (x, y) =

∂ ∂ fx = (3x 2 + y 2 ex ) ∂x ∂x

fyy (x, y) =

= 6x + y 2 ex , fxy (x, y) =

= 2ex

∂ ∂fx = (3x 2 + y 2 ex ) ∂y ∂y

fyx (x, y) =

∂fy ∂ = 2yex ∂x ∂x

= 2yex

= 2yex , Remember how the subscripts are used in partial derivatives. The notation fxyy means “first differentiate with respect to x and then differentiate twice with respect to y .”

∂ ∂ fy = 2yex ∂y ∂y

E X A M P L E 8 Calculate fxyy for f (x, y) = x 3 + y 2 ex .

Solution By the previous example, fxy = 2yex . Therefore, fxyy =

∂ ∂ fxy = 2yex = 2ex ∂y ∂y

S E C T I O N 12.3

Partial Derivatives

697

Observe in Example 7 that fxy and fyx are both equal to 2yex . It is a pleasant circumstance that the equality fxy = fyx holds in general, provided that the mixed partials are continuous. See Appendix D for a proof of the following theorem named for the French mathematician Alexis Clairaut (Figure 5).

The hypothesis of Clairaut’s Theorem, that fxy and fyx are continuous, is almost always satisfied in practice, but see Exercise 84 for an example where the mixed partials are not equal.

THEOREM 1 Clairaut’s Theorem: Equality of Mixed Partials If fxy and fyx are both continuous functions on a disk D, then fxy (a, b) = fyx (a, b) for all (a, b) ∈ D. In other words, ∂ 2f ∂ 2f = ∂x ∂y ∂y ∂x

E X A M P L E 9 Check that

∂ 2W ∂ 2W = for W = eU/T . ∂U ∂T ∂T ∂U

Solution We compute both derivatives and observe that they are equal:     ∂W ∂ ∂ ∂W U U = eU/T = −U T −2 eU/T , = eU/T = T −1 eU/T ∂T ∂T T ∂U ∂U T ∂ ∂W = −T −2 eU/T − U T −3 eU/T , ∂U ∂T

∂ ∂W = −T −2 eU/T − U T −3 eU/T ∂T ∂U

Although Clairaut’s Theorem is stated for fxy and fyx , it implies more generally that partial differentiation may be carried out in any order, provided that the derivatives in question are continuous (see Exercise 75). For example, we can compute fxyxy by differentiating f twice with respect to x and twice with respect to y, in any order. Thus, FIGURE 5 Alexis Clairaut (1713–1765) was

a brilliant French mathematician who presented his first paper to the Paris Academy of Sciences at the age of 13. In 1752, Clairaut won a prize for an essay on lunar motion that Euler praised (surely an exaggeration) as “the most important and profound discovery that has ever been made in mathematics.”

fxyxy = fxxyy = fyyxx = fyxyx = fxyyx = fyxxy E X A M P L E 10 Choosing the Order Calculate the partial derivative gzzwx ,   Wisely

where g(x, y, z, w) = x 3 w 2 z2 + sin

xy . z2

Solution Let’s take advantage of the fact that the derivatives may be calculated in any order. If we differentiate with respect to w first, the second term disappears because it does not depend on w:    ∂ xy 3 2 2 gw = = 2x 3 wz2 x w z + sin ∂w z2 Next, differentiate twice with respect to z and once with respect to x: gwz =

∂ 3 2 2x wz = 4x 3 wz ∂z

gwzz =

∂ 3 4x wz = 4x 3 w ∂z

gwzzx =

∂ 4x 3 w = 12x 2 w ∂x

We conclude that gzzwx = gwzzx = 12x 2 w.

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A partial differential equation (PDE) is a differential equation involving functions of several variables and their partial derivatives. The heat equation in the next example is a PDE that models temperature as heat spreads through an object. There are infinitely many solutions, but the particular function in the example describes temperature at times t > 0 along a metal rod when the center point is given a burst of heat at t = 0 (Figure 6). Temperature T Time t

FIGURE 6 The plot of 2 1 u(x, t) = √ e−(x /4t) 2 πt

illustrates the diffusion of a burst of heat over time.

Metal bar

x

1 2 e−(x /4t) , defined for 2 πt

E X A M P L E 11 The Heat Equation Show that u(x, t) = √

t > 0, satisfies the heat equation

Solution First, compute

∂ 2u ∂x 2

∂u ∂ 2u = 2 ∂t ∂x

2

:

∂u 1 ∂ 1 2 2 = − √ xt −3/2 e−(x /4t) = √ t −1/2 e−(x /4t) ∂x ∂x 2 π 4 π   ∂ 2u 1 1 ∂ 1 2 2 −3/2 −(x 2 /4t) xt = − √ t −3/2 e−(x /4t) + √ x 2 t −5/2 e−(x /4t) − = e √ 2 ∂x ∂x 4 π 4 π 8 π Then compute ∂u/∂t and observe that it equals ∂ 2 u/∂x 2 as required:   ∂u 1 1 −1/2 −(x 2 /4t) 1 ∂ 2 2 = − √ t −3/2 e−(x /4t) + √ x 2 t −5/2 e−(x /4t) e = √ t ∂t 2 π ∂t 4 π 8 π

12.3 SUMMARY •

The partial derivatives of f (x, y) are defined as the limits  ∂f  f (a + h, b) − f (a, b) fx (a, b) = = lim  ∂x (a,b) h→0 h  f (a, b + k) − f (a, b) ∂f  fy (a, b) = = lim ∂y (a,b) k→0 k

Compute fx by holding y constant, and compute fy by holding x constant. fx (a, b) is the slope at x = a of the tangent line to the trace curve z = f (x, b). Similarly, fy (a, b) is the slope at y = b of the tangent line to the trace curve z = f (a, y). • For small changes x and y, • •

f (a + x, b) − f (a, b) ≈ fx (a, b)x f (a, b + y) − f (a, b) ≈ fy (a, b)y

S E C T I O N 12.3

Partial Derivatives

699

More generally, if f is a function of n variables and w is one of the variables, then f ≈ fw w if w changes by w and all other variables remain fixed. • The second-order partial derivatives are ∂2 f = fxx , ∂x 2

∂2 f = fxy , ∂y ∂x

∂2 f = fyx , ∂x ∂y

∂2 f = fyy ∂y 2

Clairaut’s Theorem states that mixed partials are equal—that is, fxy = fyx provided that fxy and fyx are continuous. • More generally, higher-order partial derivatives may be computed in any order. For example, fxyyz = fyxzy if f is a function of x, y, z whose fourth-order partial derivatives are continuous. •

HISTORICAL PERSPECTIVE The general heat equation, of which Eq. (2) is a special case, was first introduced in 1807 by French mathematician Jean Baptiste Joseph Fourier. As a young man, Fourier was unsure whether to enter the priesthood or pursue mathematics, but he must have been very ambitious. He wrote in a letter, “Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality.” In his twenties, Fourier got involved in the French Revolution and was imprisoned briefly in 1794 over an incident involving different factions. In 1798, he was summoned, along with more than 150 other scientists, to join Napoleon on his unsuccessful campaign in Egypt. Fourier’s true impact, however, lay in his mathematical contributions. The heat equation is applied throughout the physical sciences and engineering, from the study of heat flow through the earth’s oceans and atmosphere to the use of heat probes to destroy tumors and treat heart disease. Fourier also introduced a striking new technique—known as the Fourier transform— for solving his equation, based on the idea that a periodic function can be expressed as a (pos-

Joseph Fourier (1768–1830)

Adolf Fick (1829–1901)

sibly infinite) sum of sines and cosines. Leading mathematicians of the day, including Lagrange and Laplace, initially raised objections because this technique was not easy to justify rigorously. Nevertheless, the Fourier transform turned out to be one of the most important mathematical discoveries of the nineteenth century. A Web search on the term “Fourier transform” reveals its vast range of modern applications. In 1855, the German physiologist Adolf Fick showed that the heat equation describes not only heat conduction but also a wide range of diffusion processes, such as osmosis, ion transport at the cellular level, and the motion of pollutants through air or water. The heat equation thus became a basic tool in chemistry, molecular biology, and environmental science, where it is often called Fick’s Second Law.

12.3 EXERCISES Preliminary Questions 1. Patricia derived the following incorrect formula by misapplying the Product Rule: ∂ 2 2 (x y ) = x 2 (2y) + y 2 (2x) ∂x What was her mistake and what is the correct calculation?

2. Explain why it is not necessary to use the Quotient Rule to com  ∂ x+y pute . Should the Quotient Rule be used to compute ∂x y + 1   ∂ x+y ? ∂y y + 1

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C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

3. Which of the following partial derivatives should be evaluated without using the Quotient Rule? (a)

∂ xy ∂x y 2 + 1

(b)

∂ xy ∂y y 2 + 1

(c)

∂ y2 ∂x y 2 + 1

√

3 −1 y ? 4. What is fx , where f (x, y, z) = (sin yz)ez −z

5. Assuming the hypotheses of Clairaut’s Theorem are satisfied, which of the following partial derivatives are equal to fxxy ? (b) fyyx (c) fxyy (d) fyxx (a) fxyx

Exercises 1. Use the limit definition of the partial derivative to verify the formulas ∂ xy 2 = y 2 , ∂x

∂ xy 2 = 2xy ∂y

2. Use the Product Rule to compute

11. Starting at point B, in which compass direction (N, NE, SW, etc.) does f increase most rapidly? 12. At which of A, B, or C is fy smallest?

∂ 2 (x + y)(x + y 4 ). ∂y

70 50 30

y 4

∂ y . 3. Use the Quotient Rule to compute ∂y x + y

−30

2

∂ ln(u2 + uv). 4. Use the Chain Rule to compute ∂u

B

0

5. Calculate fz (2, 3, 1), where f (x, y, z) = xyz.

10

−10

0

30 50 70

10

−10

−2

C

A

−20

−4

6. Explain the relation between the following two formulas (c is a constant).

−4

∂ sin(xy) = y cos(xy) ∂x

d sin(cx) = c cos(cx), dx

7. The plane y = 1 intersects the surface z = x 4 + 6xy − y 4 in a certain curve. Find the slope of the tangent line to this curve at the point P = (1, 1, 6). 8. Determine whether the partial derivatives ∂f/∂x and ∂f/∂y are positive or negative at the point P on the graph in Figure 7.

FIGURE 7

y

14. z = x 4 y 3

15. z = x 4 y + xy −2

16. V = π r 2 h

17. z =

x y 

18. z = 9 − x2 − y2

9. Estimate fx and fy at point A. 10. Is fx positive or negative at B?

x y

x x−y

x 20. z =  2 x + y2 22. z = sin(u2 v) 24. S = tan−1 (wz)

25. z = ln(x 2 + y 2 )

26. A = sin(4θ − 9t)

27. W = er+s

28. Q = reθ

29. z = exy

30. R = e−v /k √ 2 2 32. P = e y +z

2 2 31. z = e−x −y

33. U = In Exercises 9–12, refer to Figure 8.

x

4

13. z = x 2 + y 2

23. z = tan P

2

In Exercises 13–40, compute the first-order partial derivatives.

21. z = (sin x)(sin y)

x

0

FIGURE 8 Contour map of f (x, y).

19. z =

z

−2

e−rt r

2

34. z = y x

35. z = sinh(x 2 y)

36. z = cosh(t − cos x)

37. w = xy 2 z3

38. w =

x y+z

Partial Derivatives

S E C T I O N 12.3

39. Q =

L −Lt/M e M

40. w =

x (x 2 + y 2 + z2 )3/2

In Exercises 41–44, compute the given partial derivatives. 41. f (x, y) = 3x 2 y + 4x 3 y 2 − 7xy 5 , 42. f (x, y) = sin(x 2 − y), 43. g(u, v) = u ln(u + v), 2 3 44. h(x, z) = exz−x z ,

fx (1, 2)

fy (0, π)

53. Use the contour map of f (x, y) in Figure 9 to explain the following statements. (a) fy is larger at P than at Q, and fx is smaller (more negative) at P than at Q. (b) fx (x, y) is decreasing as a function of y; that is, for any fixed value x = a, fx (a, y) is decreasing in y.

gu (1, 2)

y 20

hz (3, 0)

Q

16 14 10

Exercises 45 and 46 refer to Example 5. 45. Calculate N for L = 0.4, R = 0.12, and d = 10, and use the linear approximation to estimate N if d is increased from 10 to 10.4. 46. Estimate N if (L, R, d) = (0.5, 0.15, 8) and R is increased from 0.15 to 0.17. 47. The heat index I is a measure of how hot it feels when the relative humidity is H (as a percentage) and the actual air temperature is T (in degrees Fahrenheit). An approximate formula for the heat index that is valid for (T , H ) near (90, 40) is

8 6

P

4

x

FIGURE 9 Contour interval 2.

54. Estimate the partial derivatives at P of the function whose contour map is shown in Figure 10. y

21 18 15 12 9 6 3

4

I (T , H ) = 45.33 + 0.6845T + 5.758H − 0.00365T 2 − 0.1565H T + 0.001H T 2

P

2

(a) Calculate I at (T , H ) = (95, 50).

x 0

(b) Which partial derivative tells us the increase in I per degree increase in T when (T , H ) = (95, 50). Calculate this partial derivative. 48. The wind-chill temperature W measures how cold people feel (based on the rate of heat loss from exposed skin) when the outside temperature is T ◦ C (with T ≤ 10) and wind velocity is v m/s (with v ≥ 2): W = 13.1267 + 0.6215T − 13.947v 0.16 + 0.486T v 0.16 Calculate ∂W/∂v at (T , v) = (−10, 15) and use this value to estimate W if v = 2.

51. Calculate ∂W/∂E and ∂W/∂T , where W = e−E/kT , where k is a constant. 52. Calculate ∂P /∂T and ∂P /∂V , where pressure P , volume V , and temperature T are related by the ideal gas law, P V = nRT (R and n are constants).

2

4

6

8

FIGURE 10

55. Over most of the earth, a magnetic compass does not point to true (geographic) north; instead, it points at some angle east or west of true north. The angle D between magnetic north and true north is called the magnetic declination. Use Figure 11 to determine which of the following statements is true.     ∂D  ∂D  ∂D  ∂D  (a) > (b) >0 (c) >0 ∂y A ∂y B ∂x C ∂y C Note that the horizontal axis increases from right to left because of the way longitude is measured.

49. The volume of a right-circular cone of radius r and height h is V = π3 r 2 h. Suppose that r = h = 12 cm. What leads to a greater increase in V , a 1-cm increase in r or a 1-cm increase in h? Argue using partial derivatives. 50. Use the linear approximation to estimate the percentage change in volume of a right-circular cone of radius r = 40 cm if the height is increased from 40 to 41 cm.

701

Magnetic Declination for the U.S. 2004 50°N − 15

y

10

40°N

5

0 −5

B A

C 10

30°N 120°W

110°W

5

100°W

0

90°W x

80°W

FIGURE 11 Contour interval 1◦ .

70°W

702

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

56. Refer to Table 1. (a) Estimate ∂ρ/∂T and ∂ρ/∂S at the points (S, T ) = (34, 2) and (35, 10) by computing the average of left-hand and right-hand difference quotients. (b) For fixed salinity S = 33, is ρ concave up or concave down as a function of T ? Hint: Determine whether the quotients ρ/T are increasing or decreasing. What can you conclude about the sign of ∂ 2 ρ/∂T 2 ? TABLE 1 Seawater Density ρ as a Function of Temperature T and Salinity S S 30 31 32 33 34 35 36 T 12 22.75 23.51 24.27 25.07 25.82 26.6 27.36 10

23.07

23.85

24.62

25.42

26.17

26.99

27.73

8

23.36

24.15

24.93

25.73

26.5

27.28

29.09

6

23.62

24.44

25.22

26

26.77

27.55

28.35

4

23.85

24.62

25.42

26.23

27

27.8

28.61

2

24

24.78

25.61

26.38

27.18

28.01

28.78

0

24.11

24.92

25.72

26.5

27.34

28.12

28.91

2 68. u(x, t) = t −1/2 e−(x /4t) ,

69. F (θ, u, v) = sinh(uv + θ 2 ), Fuuθ u 70. R(u, v, w) = , Ruvw v+w  71. g(x, y, z) = x 2 + y 2 + z2 , gxyz 72. u(x, t) = sech2 (x − t),

uxxx

73. Find a function such that

∂f ∂f = 2xy and = x2. ∂x ∂y

Prove that there does not exist any function f (x, y) such ∂f ∂f = xy and = x 2 . Hint: Show that f cannot satisfy Clairaut’s that ∂x ∂y Theorem. 74.

75. Assume that fxy and fyx are continuous and that fyxx exists. Show that fxyx also exists and that fyxx = fxyx . 2 76. Show that u(x, t) = sin(nx) e−n t satisfies the heat equation for any constant n:

∂u ∂ 2u = ∂t ∂x 2

In Exercises 57–62, compute the derivatives indicated. 57. f (x, y) = 3x 2 y − 6xy 4 ,

∂ 2f ∂x 2

and

∂ 2f

xy ∂ 2g , x−y ∂x ∂y u 59. h(u, v) = , hvv (u, v) u + 4v

61. f (x, y) = x ln(y 2 ), 62. g(x, y) = xe−xy ,

78. The function 2 1 f (x, t) = √ e−x /4t 2 πt

describes the temperature profile along a metal rod at time t > 0 when a burst of heat is applied at the origin (see Example 11). A small bug sitting on the rod at distance x from the origin feels the temperature rise and fall as heat diffuses through the bar. Show that the bug feels the maximum temperature at time t = 12 x 2 .

hxy (x, y)

fyy (2, 3) gxy (−3, 2)

63. Compute fxyxzy for 

f (x, y, z) = y sin(xz) sin(x + z) + (x + z2 ) tan y + x tan

z + z−1 y − y −1



Hint: Use a well-chosen order of differentiation on each term. 64. Let f (x, y, u, v) =

3

77. Find all values of A and B such that f (x, t) = eAx+Bt satisfies Eq. (3).

∂y 2

58. g(x, y) =

60. h(x, y) = ln(x 3 + y 3 ),

uxx

x 2 + ey v 3y 2 + ln(2 + u2 )

In Exercises 79–82, the Laplace operator  is defined by f = fxx + fyy . A function u(x, y) satisfying the Laplace equation u = 0 is called harmonic. 79. Show that the following functions are harmonic: (a) u(x, y) = x (b) u(x, y) = ex cos y y (d) u(x, y) = ln(x 2 + y 2 ) (c) u(x, y) = tan−1 x 80. Find all harmonic polynomials u(x, y) of degree three, that is, u(x, y) = ax 3 + bx 2 y + cxy 2 + dy 3 .

What is the fastest way to show that fuvxyvu (x, y, u, v) = 0 for all (x, y, u, v)?

81. Show that if u(x, y) is harmonic, then the partial derivatives ∂u/∂x and ∂u/∂y are harmonic.

In Exercises 65–72, compute the derivative indicated.

82. Find all constants a, b such that u(x, y) = cos(ax)eby is harmonic.

65. f (u, v) = cos(u + v 2 ), 66. g(x, y, z) = x 4 y 5 z6 , 67. F (r, s, t) = r(s 2 + t 2 ),

fuuv

gxxyz Frst

83. Show that u(x, t) = sech2 (x − t) satisfies the Korteweg–deVries equation (which arises in the study of water waves): 4ut + uxxx + 12uux = 0

S E C T I O N 12.4

Differentiability and Tangent Planes

703

Further Insights and Challenges 84. Assumptions Matter This exercise shows that the hypotheses of Clairaut’s Theorem are needed. Let x2 − y2 f (x, y) = xy 2 x + y2

(b) Use the limit definition of the partial derivative to show that fx (0, 0) = fy (0, 0) = 0 and that fyx (0, 0) and fxy (0, 0) both exist but are not equal. (c) Show that for (x, y)  = (0, 0):

for (x, y) = (0, 0) and f (0, 0) = 0. (a) Verify for (x, y) = (0, 0):

fxy (x, y) = fyx (x, y) =

x 6 + 9x 4 y 2 − 9x 2 y 4 − y 6 (x 2 + y 2 )3

y(x 4 + 4x 2 y 2 − y 4 ) (x 2 + y 2 )2

Show that fxy is not continuous at (0, 0). Hint: Show that lim fxy (h, 0)  = lim fxy (0, h).

x(x 4 − 4x 2 y 2 − y 4 ) fy (x, y) = (x 2 + y 2 )2

(d) Explain why the result of part (b) does not contradict Clairaut’s Theorem.

fx (x, y) =

z

h→0

12.4 Differentiability and Tangent Planes

Tangent plane at P

P z = f (x, y)

x

h→0

In this section, we generalize two basic concepts from single-variable calculus: differentiability and the tangent line. The tangent line becomes the tangent plane for functions of two variables (Figure 1). Intuitively, we would like to say that a continuous function f (x, y) is differentiable if it is locally linear—that is, if its graph looks flatter and flatter as we zoom in on a point P = (a, b, f (a, b)) and eventually becomes indistinguishable from the tangent plane (Figure 2).

y

FIGURE 1 Tangent plane to the graph of

z = f (x, y). P

P

P

FIGURE 2 The graph looks flatter and flatter as we zoom in on a point P .

We can show that if the tangent plane at P = (a, b, f (a, b)) exists, then its equation must be z = L(x, y), where L(x, y) is the linearization at (a, b), defined by L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) Why must this be the tangent plane? Because it is the unique plane containing the tangent lines to the two vertical trace curves through P [Figure 3(A)]. Indeed, when we set y = b in z = L(x, y), the term fy (a, b)(y − b) drops out and we are left with the equation of the tangent line to the vertical trace z = f (x, b) at P : z = L(x, b) = f (a, b) + fx (a, b)(x − a) Similarly, z = L(a, y) is the tangent line to the vertical trace z = f (a, y) at P .

704

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

z z = L(x, y)

z = L(x, y)

P

P = (a, b, f (a, b))

z = f(x, y)

z = f (x, y)

| e(x, y) | = | f(x, y) − L(x, y) |

x

y

(a, b, 0)

(a, b, 0)

(A)

d

(B)

(x, y, 0)

FIGURE 3

Before we can say that the tangent plane exists, however, we must impose a condition on f (x, y) guaranteeing that the graph looks flat as we zoom in on P . Set e(x, y) = f (x, y) − L(x, y) As we see in Figure 3(B), |e(x, y)| is the vertical distance between the graph of f (x, y) and the plane z = L(x, y). This distance tends to zero as (x, y) approaches (a, b) because f (x, y) is continuous. To be locally linear, we require that the distance tend to zero faster than the distance from (x, y) to (a, b). We express this by the requirement lim

(x,y)→(a,b)



e(x, y) (x − a)2 + (y − b)2

=0

DEFINITION Differentiability Assume that f (x, y) is defined in a disk D containing (a, b) and that fx (a, b) and fy (a, b) exist. •

f (x, y) is differentiable at (a, b) if it is locally linear—that is, if f (x, y) = L(x, y) + e(x, y)

1

where e(x, y) satisfies

REMINDER

L(x, y) = f (a, b) + fx (a, b)(x − a)

lim

(x,y)→(a,b)

+ fy (a, b)(y − b) •



e(x, y) (x − a)2 + (y − b)2

=0

In this case, the tangent plane to the graph at (a, b, f (a, b)) is the plane with equation z = L(x, y). Explicitly, z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)

The definition of differentiability extends to functions of n-variables, and Theorem 1 holds in this setting: If all of the partial derivatives of f (x1 , . . . , xn ) exist and are continuous on an open domain D , then f (x1 , . . . , xn ) is differentiable on D.

2

If f (x, y) is differentiable at all points in a domain D, we say that f (x, y) is differentiable on D. It is cumbersome to check the local linearity condition directly (see Exercise 41), but fortunately, this is rarely necessary. The following theorem provides a criterion for differentiability that is easy to apply. It assures us that most functions arising in practice are differentiable on their domains. See Appendix D for a proof. THEOREM 1 Criterion for Differentiability If fx (x, y) and fy (x, y) exist and are continuous on an open disk D, then f (x, y) is differentiable on D.

S E C T I O N 12.4

Differentiability and Tangent Planes

705

E X A M P L E 1 Show that f (x, y) = 5x + 4y 2 is differentiable (Figure 4). Find the

z

equation of the tangent plane at (a, b) = (2, 1). z = −4 + 5x + 8y

Solution The partial derivatives exist and are continuous functions: f (x, y) = 5x + 4y 2 ,

fy (x, y) = 8y

Therefore, f (x, y) is differentiable for all (x, y) by Theorem 1. To find the tangent plane, we evaluate the partial derivatives at (2, 1):

P

f (2, 1) = 14,

fx (2, 1) = 5,

fy (2, 1) = 8

The linearization at (2, 1) is

y x

fx (x, y) = 5,

(2, 1, 0)

L(x, y) =

FIGURE 4 Graph of f (x, y) = 5x + 4y 2

and the tangent plane at P = (2, 1, 14).

14 + 5(x − 2) + 8(y − 1)   

= −4 + 5x + 8y

f (a,b)+fx (a,b)(x−a)+fy (a,b)(y−b)

The tangent plane through P = (2, 1, 14) has equation z = −4 + 5x + 8y. Assumptions Matter Local linearity plays a key role, and although most reasonable functions are locally linear, the mere existence of the partial derivatives does not guarantee local linearity. This is in contrast to the one-variable case, where f (x) is automatically locally linear at x = a if f  (a) exists (Exercise 44). The function g(x, y) in Figure 5(A) shows what can go wrong. The graph contains the x- and y-axes—in other words, g(x, y) = 0 if x or y is zero—and therefore, the partial derivatives gx (0, 0) and gy (0, 0) are both zero. The tangent plane at the origin (0, 0), if it existed, would have to be the xy-plane. However, Figure 5(B) shows that the graph also contains lines through the origin that do not lie in the xy-plane (in fact, the graph is composed entirely of lines through the origin). As we zoom in on the origin, these lines remain at an angle to the xy-plane, and the surface does not get any flatter. Thus g(x, y) cannot be locally linear at (0, 0), and the tangent plane does not exist. In particular, g(x, y) cannot satisfy the assumptions of Theorem 1, so the partial derivatives gx (x, y) and gy (x, y) cannot be continuous at the origin (see Exercise 45 for details).

Local linearity is used in the next section to prove the Chain Rule for Paths, upon which the fundamental properties of the gradient are based.

z

y x

FIGURE 5 Graphs of

g(x, y) =

2xy(x + y) . x2 + y2

(A) The horizontal trace at z = 0 consists of the x and y axes.

(B) But the graph also contains non-horizontal lines through the origin.

(C) So the graph does not appear any flatter as we zoom in on the origin.

706

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

E X A M P L E 2 Where is h(x, y) =

 x 2 + y 2 differentiable?

Solution The partial derivatives exist and are continuous for all (x, y) = (0, 0):

z

hx (x, y) = 

h(x, y) = x 2 + y 2 h (x, y) is not differentiable at the origin y

x x2

+ y2

hy (x, y) = 

,

y x2

+ y2

However, the partial √ derivatives do not exist at (0, 0). Indeed, hx (0, 0) does not exist because h(x, 0) = x 2 = |x| is not differentiable at x = 0. Similarly, hy (0, 0) does not exist. By Theorem 1, h(x, y) is differentiable except at (0, 0) (Figure 6). E X A M P L E 3 Find a tangent plane of the graph of f (x, y) = xy 3 + x 2 at (2, −2).

x

Solution The partial derivatives are continuous, so f (x, y) is differentiable:

FIGURE 6 The  function h(x, y) = x 2 + y 2 is differentiable

except at the origin. z

fx (x, y) = y 3 + 2x,

fx (2, −2) = −4

fy (x, y) = 3xy 2 ,

fy (2, −2) = 24

Since f (2, −2) = −12, the tangent plane through (2, −2, −12) has equation

y

z = −12 − 4(x − 2) + 24(y + 2) This can be rewritten as z = 44 − 4x + 24y (Figure 7). x

P = (2, −2, −12)

Linear Approximation and Differentials By definition, if f (x, y) is differentiable at (a, b), then it is locally linear and the linear approximation is f (x, y) ≈ L(x, y)

FIGURE 7 Tangent plane to the surface f (x, y) = xy 3 + x 2 passing through

P = (2, −2, −12).

for (x, y) near (a, b)

where L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) We shall rewrite this in several useful ways. First, set x = a + h and y = b + k. Then f (a + h, b + k) ≈ f (a, b) + fx (a, b)h + fy (a, b)k

3

We can also write the linear approximation in terms of the change in f :

z

f = f (x, y) − f (a, b), z = L(x, y)

df

x = x − a,

y = y − b

f ≈ fx (a, b)x + fy (a, b)y

4

z = f (x, y)

Δf

Finally, the linear approximation is often expressed in terms of differentials: df = fx (x, y) dx + fy (x, y) dy =

dy = Δy

dx = Δ x

FIGURE 8 The quantity df is the change in

height of the tangent plane.

∂f ∂f dx + dy ∂x ∂y

As shown in Figure 8, df represents the change in height of the tangent plane for given changes dx and dy in x and y (when we work with differentials, we call them dx and dy instead of x and y), whereas f is the change in the function itself. The linear approximation tells us that the two changes are approximately equal: f ≈ df

S E C T I O N 12.4

Differentiability and Tangent Planes

707

These approximations apply in any number of variables. In three variables, f (a + h, b + k, c + ) ≈ f (a, b, c) + fx (a, b, c)h + fy (a, b, c)k + fz (a, b, c) or in terms of differentials, f ≈ df , where df = fx (x, y, z) dx + fy (x, y, z) dy + fz (x, y, z) dz E X A M P L E 4 Use the linear approximation to estimate

(3.99)3 (1.01)4 (1.98)−1 REMINDER The percentage error is equal to

  

 error   × 100% actual value

Then use a calculator to find the percentage error. Solution Think of (3.99)3 (1.01)4 (1.98)−1 as a value of f (x, y, z) = x 3 y 4 z−1 : f (3.99, 1.01, 1.98) = (3.99)3 (1.01)4 (1.98)−1 Then it makes sense to use the linear approximation at (4, 1, 2): f (x, y, z) = x 3 y 4 z−1 ,

f (4, 1, 2) = (43 )(14 )(2−1 ) = 32

fx (x, y, z) = 3x 2 y 4 z−1 , 3 3 −1

fy (x, y, z) = 4x y z

fx (4, 1, 2) = 24

,

fy (4, 1, 2) = 128

fz (x, y, z) = −x 3 y 4 z−2 ,

fz (4, 1, 2) = −16

The linear approximation in three variables stated above, with a = 4, b = 1, c = 2, gives us (4 + h)3 (1 + k)4 (2 + )−1 ≈ 32 + 24h + 128k − 16    f (4+h,1+k,2+ )

For h = −0.01, k = 0.01, and = −0.02, we obtain the desired estimate (3.99)3 (1.01)4 (1.98)−1 ≈ 32 + 24(−0.01) + 128(0.01) − 16(−0.02) = 33.36 The calculator value is (3.99)3 (1.01)4 (1.98)−1 ≈ 33.384, so the error in our estimate is less than 0.025. The percentage error is Percentage error ≈

|33.384 − 33.36| × 100 ≈ 0.075% 33.384

E X A M P L E 5 Body Mass Index A person’s BMI is I = W/H 2 , where W is the body BMI is one factor used to assess the risk of certain diseases such as diabetes and high blood pressure. The range 18.5 ≤ I ≤ 24.9 is considered normal for adults over 20 years of age.

weight (in kilograms) and H is the body height (in meters). Estimate the change in a child’s BMI if (W, H ) changes from (40, 1.45) to (41.5, 1.47). Solution Step 1. Compute the differential at (W, H ) = (40, 1.45). ∂ ∂I = ∂W ∂W



W H2



1 = 2, H

∂ ∂I = ∂H ∂H



W H2

 =−

2W H3

At (W, H ) = (40, 1.45), we have  1 ∂I  = ≈ 0.48,  ∂W (40,1.45) 1.452

 ∂I  2(40) =− ≈ −26.24  ∂H (40,1.45) 1.453

Therefore, the differential at (40, 1.45) is dI ≈ 0.48 dW − 26.24 dH

708

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

Step 2. Estimate the change. We have shown that the differential dI at (40, 1.45) is 0.48 dW − 26.24 dH . If (W, H ) changes from (40, 1.45) to (41.5, 1.47), then dW = 41.5 − 40 = 1.5,

dH = 1.47 − 1.45 = 0.02

Therefore, I ≈ dI = 0.48 dW − 26.24 dH = 0.48(1.5) − 26.24(0.02) ≈ 0.2 We find that BMI increases by approximately 0.2.

12.4 SUMMARY •

The linearization in two and three variables: L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) L(x, y, z) = f (a, b, c) + fx (a, b, c)(x − a) + fy (a, b, c)(y − b) + fz (a, b, c)(z − c)

•

f (x, y) is differentiable at (a, b) if fx (a, b) and fy (a, b) exist and f (x, y) = L(x, y) + e(x, y)

where e(x, y) is a function such that lim

(x,y)→(a,b)



e(x, y) (x − a)2 + (y − b)2

=0

Result used in practice: If fx (x, y) and fy (x, y) exist and are continuous in a disk D containing (a, b), then f (x, y) is differentiable at (a, b). • Equation of the tangent plane to z = f (x, y) at (a, b): •

z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) •

Equivalent forms of the linear approximation: f (x, y) ≈ f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) f (a + h, b + k) ≈ f (a, b) + fx (a, b)h + fy (a, b)k f ≈ fx (a, b) x + fy (a, b) y

•

In differential form, f ≈ df , where df = fx (x, y) dx + fy (x, y) dy =

∂f ∂f dx + dy ∂x ∂y

df = fx (x, y, z) dx + fy (x, y, z) dy + fz (x, y, z) dz =

∂f ∂f ∂f dx + dy + dz ∂x ∂y ∂z

12.4 EXERCISES Preliminary Questions 1. How is the linearization of f (x, y) at (a, b) defined? 2. Define local linearity for functions of two variables.

In Exercises 3–5, assume that f (2, 3) = 8,

fx (2, 3) = 5,

fy (2, 3) = 7

S E C T I O N 12.4

3. Which of (a)–(b) is the linearization of f at (2, 3)? (a) L(x, y) = 8 + 5x + 7y (b) L(x, y) = 8 + 5(x − 2) + 7(y − 3) 4. Estimate f (2, 3.1).

Exercises 1. Use Eq. (2) to find an equation of the tangent plane to the graph of f (x, y) = 2x 2 − 4xy 2 at (−1, 2). 2. Find the equation of the tangent plane in Figure 9. The point of tangency is (a, b) = (1, 0.8, 0.34). z

5. Estimate f at (2, 3) if x = −0.3 and y = 0.2. 6. Which theorem allows us to conclude that f (x, y) = x 3 y 8 is differentiable?

15. Let f (x, y) = x 3 y −4 . Use Eq. (4) to estimate the change f = f (2.03, 0.9) − f (2, 1) 16. Use √ the linear approximation to f (x, y) = mate 9.1/3.9.

FIGURE 9 Graph of f (x, y) = 0.2x 4 + y 6 − xy.

18. Let f (x, y) = x 2 /(y 2 + 1). Use the linear approximation at an appropriate point (a, b) to estimate f (4.01, 0.98). √ 19. Find the linearization of f (x, y, z) = z x + y at (8, 4, 5). 20. Find the linearization to f (x, y, z) = xy/z at the point (2, 1, 2). Use it to estimate f (2.05, 0.9, 2.01) and compare with the value obtained from a calculator. 21. Estimate f (2.1, 3.8) assuming that f (2, 4) = 5,

In Exercises 3–10, find an equation of the tangent plane at the given point. 3. f (x, y) = x 2 y + xy 3 , (2, 1) x 4. f (x, y) = √ , (4, 4) y (4, 1) π  6 ,1

7. F (r, s) = r 2 s −1/2 + s −3 , 8. g(x, y) = ex/y ,

(2, 1)

10. f (x, y) = ln(4x 2 − y 2 ),

(ln 4, ln 2)

f (1, 0, 0) = −3, fy (1, 0, 0) = 4,

12. Find the points on the graph of z = xy 3 + 8y −1 where the tangent plane is parallel to 2x + 7y + 2z = 0. 13. Find the linearization L(x, y) of f (x, y) = x 2 y 3 at (a, b) = (2, 1). Use it to estimate f (2.01, 1.02) and f (1.97, 1.01) and compare with values obtained using a calculator. 14. Write the linear approximation to f (x, y) = x(1 + y)−1 at (a, b) = (8, 1) in the form f (a + h, b + k) ≈ f (a, b) + fx (a, b)h + fy (a, b)k Use it to estimate 7.98 2.02 and compare with the value obtained using a calculator.

fy (2, 4) = −0.2

fx (1, 0, 0) = −2, fz (1, 0, 0) = 2

In Exercises 23–28, use the linear approximation to estimate the value. Compare with the value given by a calculator. 23. (2.01)3 (1.02)2 25.

(1, 1)

11. Find the points on the graph of z = 3x 2 − 4y 2 at which the vector n = 3, 2, 2 is normal to the tangent plane.

fx (2, 4) = 0.3,

22. Estimate f (1.02, 0.01, −0.03) assuming that

(2, 1)

9. f (x, y) = sech(x − y),

√ x/y at (9, 4) to esti2

(a, b) x

6. G(u, w) = sin(uw),

709

17. Use the linear approximation of f (x, y) = ex +y at (0, 0) to estimate f (0.01, −0.02). Compare with the value obtained using a calculator.

y

5. f (x, y) = x 2 + y −2 ,

Differentiability and Tangent Planes

27.



3.012 + 3.992

√

(1.9)(2.02)(4.05)

24.

4.1 7.9

26.

0.982 2.013 + 1

28. √

8.01 (1.99)(2.01)

29. Find an equation of the tangent plane to z = f (x, y) at P = (1, 2, 10) assuming that f (1, 2) = 10,

f (1.1, 2.01) = 10.3,

f (1.04, 2.1) = 9.7

30. Suppose that the plane tangent to z = f (x, y) at (−2, 3, 4) has equation 4x + 2y + z = 2. Estimate f (−2.1, 3.1). In Exercises 31–34, let I = W/H 2 denote the BMI described in Example 5. 31. A boy has weight W = 34 kg and height H = 1.3 m. Use the linear approximation to estimate the change in I if (W, H ) changes to (36, 1.32).

710

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

32. Suppose that (W, H ) = (34, 1.3). Use the linear approximation to estimate the increase in H required to keep I constant if W increases to 35.

Estimate: (a) The change in monthly payment per $1000 increase in loan principal.

33. (a) Show that I ≈ 0 if H /W ≈ H /2W . (b) Suppose that (W, H ) = (25, 1.1). What increase in H will leave I (approximately) constant if W is increased by 1 kg?

(b) The change in monthly payment if the interest rate increases to r = 6.5% and r = 7%.

34. Estimate the change in height that will decrease I by 1 if (W, H ) = (25, 1.1), assuming that W remains constant.

(c) The change in monthly payment if the length of the loan increases to 24 years.

35. A cylinder of radius r and height h has volume V = πr 2 h. (a) Use the linear approximation to show that V 2r h ≈ + V r h (b) Estimate the percentage increase in V if r and h are each increased by 2%. (c) The volume of a certain cylinder V is determined by measuring r and h. Which will lead to a greater error in V : a 1% error in r or a 1% error in h? 36. Use the linear approximation to show that if I = x a y b , then x y I ≈a +b I x y 37. The monthly payment for a home loan is given by a function f (P , r, N ), where P is the principal (initial size of the loan), r the interest rate, and N is the length of the loan in months. Interest rates are expressed as a decimal: A 6% interest rate is denoted by r = 0.06. If P = $100,000, r = 0.06, and N = 240 (a 20-year loan), then the monthly payment is f (100,000, 0.06, 240) = 716.43. Furthermore, at these values, we have ∂f = 0.0071, ∂P

∂f = 5769, ∂r

∂f = −1.5467 ∂N

38. Automobile traffic passes a point P on a road of width w ft at an average rate of R vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time T until there is a gap in traffic of at least t seconds is approximately T = teRt seconds. A pedestrian walking at a speed of 3.5 ft/s (5.1 mph) requires t = w/3.5 s to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is f (w, R) = (w/3.5)ewR/3.5 s. (a) What is the pedestrian’s average waiting time if w = 25 ft and R = 0.2 vehicle per second? (b) Use the linear approximation to estimate the increase in waiting time if w is increased to 27 ft. (c) Estimate the waiting time if the width is increased to 27 ft and R decreases to 0.18. (d) What is the rate of increase in waiting time per 1-ft increase in width when w = 30 ft and R = 0.3 vehicle per second? 39. The volume V of a right-circular cylinder is computed using the values 3.5 m for diameter and 6.2 m for height. Use the linear approximation to estimate the maximum error in V if each of these values has a possible error of at most 5%. Recall that V = 13 π r 2 h.

Further Insights and Challenges 40. Show that if f (x, y) is differentiable at (a, b), then the function of one variable  f (x, b) is differentiable at x = a. Use this to prove that f (x, y) = x 2 + y 2 is not differentiable at (0, 0). 41. This exercise shows directly (without using Theorem 1) that the function f (x, y) = 5x + 4y 2 from Example 1 is locally linear at (a, b) = (2, 1). (a) Show that f (x, y) = L(x, y) + e(x, y) with e(x, y) = 4(y − 1)2 . (b) Show that 0≤ 

e(x, y) (x − 2)2 + (y − 1)2

≤ 4|y − 1|

(c) Verify that f (x, y) is locally linear. 42. Show directly, as in Exercise 41, that f (x, y) = xy 2 is differentiable at (0, 2). 43. Differentiability Implies Continuity Use the definition of differentiability to prove that if f is differentiable at (a, b), then f is continuous at (a, b). 44. Let f (x) be a function of one variable defined near x = a. Given a number M, set L(x) = f (a) + M(x − a),

e(x) = f (x) − L(x)

Thus f (x) = L(x) + e(x). We say that f is locally linear at x = a if e(x) M can be chosen so that lim = 0. x→a |x − a| (a) Show that if f (x) is differentiable at x = a, then f (x) is locally linear with M = f  (a). (b) Show conversely that if f is locally linear at x = a, then f (x) is differentiable and M = f  (a). 45. Assumptions Matter Define g(x, y) = 2xy(x + y)/(x 2 + y 2 ) for (x, y)  = 0 and g(0, 0) = 0. In this exercise, we show that g(x, y) is continuous at (0, 0) and that gx (0, 0) and gy (0, 0) exist, but g(x, y) is not differentiable at (0, 0). (a) Show using polar coordinates that g(x, y) is continuous at (0, 0). (b) Use the limit definitions to show that gx (0, 0) and gy (0, 0) exist and that both are equal to zero. (c) Show that the linearization of g(x, y) at (0, 0) is L(x, y) = 0. (d) Show that if g(x, y) were locally linear at (0, 0), we would have g(h, h) = 0. Then observe that this is not the case because lim h h→0 g(h, h) = 2h. This shows that g(x, y) is not locally linear at (0, 0) and, hence, not differentiable at (0, 0).

S E C T I O N 12.5

The Gradient and Directional Derivatives

711

12.5 The Gradient and Directional Derivatives We have seen that the rate of change of a function f of several variables depends on a choice of direction. Since directions are indicated by vectors, it is natural to use vectors to describe the derivative of f in a specified direction. To do this, we introduce the gradient ∇fP , which is the vector whose components are the partial derivatives of f at P . The gradient of a function of n variables is the vector



∇f =

∂f ∂f ∂f , ,..., ∂x1 ∂x2 ∂xn



DEFINITION The Gradient The gradient of a function f (x, y) at a point P = (a, b) is the vector   ∇fP = fx (a, b), fy (a, b) In three variables, if P = (a, b, c),   ∇fP = fx (a, b, c), fy (a, b, c), fz (a, b, c)

The symbol ∇ , called “del,” is an upside-down Greek delta. It was popularized by the Scottish physicist P. G. Tait (1831–1901), who called the symbol “nabla,” because of its resemblance to an ancient Assyrian harp. The great physicist James Clerk Maxwell was reluctant to adopt this term and would refer to the gradient simply as the “slope.” He wrote jokingly to his friend Tait in 1871, “Still harping on that nabla?”

We also write ∇f(a,b) or ∇f (a, b) for the gradient. Sometimes, we omit reference to the point P and write     ∂f ∂f ∂f ∂f ∂f ∇f = , or ∇f = , , ∂x ∂y ∂x ∂y ∂z The gradient ∇f “assigns” a vector ∇fP to each point in the domain of f , as in Figure 1. E X A M P L E 1 Drawing Gradient Vectors Let f (x, y) = x 2 + y 2 . Calculate the gradi-

ent ∇f , draw several gradient vectors, and compute ∇fP at P = (1, 1).

Solution The partial derivatives are fx (x, y) = 2x and fy (x, y) = 2y, so y

∇f = 2x, 2y ∇f(1, 1) = 2, 2

∇f(− 1 , 1 ) = −1, 1 1 2 2

(− 12 ,

The gradient attaches the vector 2x, 2y to the point (x, y). As we see in Figure 1, these vectors point away from the origin. At the particular point (1, 1),

(1, 1)

1 2)

1 (1, − 1 )

x

2

∇f(1, − 1 ) = 2, −1

∇fP = ∇f (1, 1) = 2, 2 E X A M P L E 2 Gradient in Three Variables Calculate ∇f(3,−2,4) , where

2

f (x, y, z) = ze2x+3y Solution The partial derivatives and the gradient are FIGURE 1 Gradient vectors of f (x, y) = x 2 + y 2 at several points

(vectors not drawn to scale).

∂f ∂f = 2ze2x+3y , = 3ze2x+3y , ∂x ∂y   ∇f = 2ze2x+3y , 3ze2x+3y , e2x+3y

∂f = e2x+3y ∂z

  Therefore, ∇f(3,−2,4) = 2 · 4e0 , 3 · 4e0 , e0 = 8, 12, 1. The following theorem lists some useful properties of the gradient. The proofs are left as exercises (see Exercises 62–64).

712

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

THEOREM 1 Properties of the Gradient If f (x, y, z) and g(x, y, z) are differentiable and c is a constant, then (i) (ii) (iii) (iv)

∇(f + g) = ∇f + ∇g ∇(cf ) = c∇f Product Rule for Gradients: ∇(f g) = f ∇g + g∇f Chain Rule for Gradients: If F (t) is a differentiable function of one variable, then ∇(F (f (x, y, z))) = F  (f (x, y, z))∇f

1

E X A M P L E 3 Using the Chain Rule for Gradients Find the gradient of

g(x, y, z) = (x 2 + y 2 + z2 )8 Solution The function g is a composite g(x, y, z) = F (f (x, y, z)) with F (t) = t 8 and f (x, y, z) = x 2 + y 2 + z2 and apply Eq. (1):   ∇g = ∇ (x 2 + y 2 + z2 )8 = 8(x 2 + y 2 + z2 )7 ∇(x 2 + y 2 + z2 ) = 8(x 2 + y 2 + z2 )7 2x, 2y, 2z = 16(x 2 + y 2 + z2 )7 x, y, z

The Chain Rule for Paths z

Our first application of the gradient is the Chain Rule for Paths. A path will be represented by a function c(t) = (x(t), y(t), z(t)). We think of c(t) as a moving point (Figure 2). By definition, c (t) is the vector of derivatives: c(t) = (x(t), y(t), z(t)), Tangent vector c'(t)

x

y c(t) = (x(t), y(t), z(t))

FIGURE 2 Tangent vector c (t) to a path

c(t) = (x(t), y(t), z(t)). y

  c (t) = x  (t), y  (t), z (t)

As we saw for paths in R 2 , c (t) is the tangent or “velocity” vector that is tangent to the path and points in the direction of motion. The Chain Rule for Paths deals with composite functions of the type f (c(t)). What is the idea behind a composite function of this type? As an example, suppose that T (x, y) is the temperature at location (x, y) (Figure 3). Now imagine a biker—we’ll call her Chloe—riding along a path c(t). We suppose that Chloe carries a thermometer with her and checks it as she rides. Her location at time t is c(t), so her temperature reading at time t is the composite function

∇T(x, y) c(t)

T (c(t)) = Chloe’s temperature at time t

c'(t)

The temperature reading varies as Chloe’s location changes, and the rate at which it changes is the derivative

x FIGURE 3 Chloe’s temperature changes at the rate ∇Tc(t) · c (t).

d T (c(t)) dt The Chain Rule for Paths tells us that this derivative is simply the dot product of the temperature gradient ∇T evaluated at c(t) and Chloe’s velocity vector c (t).

S E C T I O N 12.5

CAUTION Do not confuse the Chain Rule for Paths with the more elementary Chain Rule for Gradients stated in Theorem 1 above.

THEOREM 2 Chain Rule for Paths

The Gradient and Directional Derivatives

713

If f and c(t) are differentiable, then

d f (c(t)) = ∇fc(t) · c (t) dt Explicitly, in the case of two variables, if c(t) = (x(t), y(t)), then    ∂f dx d ∂f ∂f   ∂f dy f (c(t)) = , · x (t), y  (t) = + dt ∂x ∂y ∂x dt ∂y dt

Proof By definition, f (x(t + h), y(t + h)) − f (x(t), y(t)) d f (c(t)) = lim h→0 dt h To calculate this derivative, set f = f (x(t + h), y(t + h)) − f (x(t), y(t)) x = x(t + h) − x(t),

y = y(t + h) − y(t)

The proof is based on the local linearity of f . As in Section 12.4, we write f = fx (x(t), y(t))x + fy (x(t), y(t))y + e(x(t + h), y(t + h)) Now set h = t and divide by t: e(x(t + t), y(t + t)) x y f = fx (x(t), y(t)) + fy (x(t), y(t)) + t t t t Suppose for a moment that the last term tends to zero as t → 0. Then we obtain the desired result: f d f (c(t)) = lim t→0 t dt x y + fy (x(t), y(t)) lim t→0 t t dx dy + fy (x(t), y(t)) = fx (x(t), y(t)) dt dt

= fx (x(t), y(t)) lim

t→0

= ∇fc(t) · c (t) We verify that the last term tends to zero as follows:   e(x(t + t), y(t + t)) e(x(t + t), y(t + t)) (x)2 + (y)2 = lim lim  t→0 t→0 t t (x)2 + (y)2 ⎛ ⎞        e(x(t + t), y(t + t)) x 2 y 2 ⎠ = lim =0 lim ⎝ +  t→0 t→0 t t (x)2 + (y)2    Zero

The first limit is zero because  a differentiable function is locally linear (Section 12.4). The second limit is equal to x  (t)2 + y  (t)2 , so the product is zero.

714

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

y

2 2 E X A M P L E 4 The temperature at location (x, y) is T (x, y) = 20 + 10e−0.3(x +y )◦ C.

P = c(0.6)

c'(0.6)

A bug carries a tiny thermometer along the path c(t) = (cos(t − 2), sin 2t)

1 ∇T

(t in seconds) as in Figure 4. How fast is the temperature changing at t = 0.6 s? 1

x

FIGURE 4 Gradient vectors ∇T and the

path c(t) = (cos(t − 2), sin 2t).

Solution At t = 0.6 s, the bug is at location c(0.6) = (cos(−1.4), sin 0.6) ≈ (0.170, 0.932) By the Chain Rule for Paths, the rate of change of temperature is the dot product  dT  = ∇Tc(0.6) · c (0.6) dt t=0.6 We compute the vectors

 2 2 2 2 ∇T = −6xe−0.3(x +y ) , −6ye−0.3(x +y ) c (t) = − sin(t − 2), 2 cos 2t

and evaluate at c(0.6) = (0.170, 0.932) using a calculator: ∇Tc(0.6) ≈ −0.779, −4.272 c (0.6) ≈ 0.985, 0.725 Therefore, the rate of change is  dT  ∇Tc(0.6) · c (t) ≈ −0.779, −4.272 · 0.985, 0.725 ≈ −3.87◦ C/s dt t=0.6 In the next example, we apply the Chain Rule for Paths to a function of three variables. In general, if f (x1 , . . . , xn ) is a differentiable function of n variables and c(t) = (x1 (t), . . . , xn (t)) is a differentiable path, then d ∂f dx1 ∂f dx2 ∂f dxn + + ··· + f (c(t)) = ∇f · c (t) = ∂x1 dt ∂x2 dt ∂xn dt dt E X A M P L E 5 Calculate

  d , where f (c(t)) dt t=π/2

f (x, y, z) = xy + z2 and c(t) = (cos t, sin t, t) π      π π π Solution We have c 2 = cos 2 , sin 2 , 2 = 0, 1, π2 . Compute the gradient:   ∂f ∂f ∂f π

∇f = , , = y, x, 2z , ∇fc(π/2) = ∇f 0, 1, = 1, 0, π  ∂x ∂y ∂z 2 Then compute the tangent vector: c (t) = − sin t, cos t, 1 , By the Chain Rule,

c

π

2

 π π = − sin , cos , 1 = −1, 0, 1 2 2

 π

 d = 1, 0, π  · −1, 0, 1 = π − 1 = ∇fc(π/2) · c f (c(t)) 2 dt t=π/2

S E C T I O N 12.5

The Gradient and Directional Derivatives

715

Directional Derivatives We come now to one of the most important applications of the Chain Rule for Paths. Consider a line through a point P = (a, b) in the direction of a unit vector u = h, k (see Figure 5):

c(t) = (a + th, b + tk) u = h, k

y

c(t) = (a + th, b + tk) (a, b)

The derivative of f (c(t)) at t = 0 is called the directional derivative of f with respect to u at P, and is denoted Du f (P ) or Du f (a, b):   d f (a + th, b + tk) − f (a, b) Du f (a, b) = f (c(t)) = lim t→0 dt t t=0 x

Contour map of f (x, y) FIGURE 5 The directional derivative

Du f (a, b) is the rate of change of f along the linear path through P with direction vector u.

Directional derivatives of functions of three or more variables are defined in a similar way. DEFINITION Directional Derivative The directional derivative in the direction of a unit vector u = h, k is the limit (assuming it exists) f (a + th, b + tk) − f (a, b) t→0 t

Du f (P ) = Du f (a, b) = lim

Note that the partial derivatives are the directional derivatives with respect to the standard unit vectors i = 1, 0 and j = 0, 1. For example, f (a + t (1), b + t (0)) − f (a, b) f (a + t, b) − f (a, b) = lim t→0 t t

Di f (a, b) = lim

t→0

= fx (a, b) Thus we have fx (a, b) = Di f (a, b),

fy (a, b) = Dj f (a, b)

The directional derivative Du f (P ) is the rate of change of f per unit change in the horizontal direction of u at P (Figure 6). This is the slope of the tangent line at Q to the trace curve obtained when we intersect the graph with the vertical plane through P in the direction u.

CONCEPTUAL INSIGHT

z

Q = (a, b, f (a, b))

FIGURE 6 Du f (a, b) is the slope of the

tangent line to the trace curve through Q in the vertical plane through P in the direction u.

P = (a, b, 0) x

u

y

716

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

To evaluate directional derivatives, it is convenient to define Dv f (a, b) even when v = h, k is not a unit vector:   f (a + th, b + tk) − f (a, b) d = lim Dv f (a, b) = f (c(t)) t→0 dt t t=0 We call Dv f the derivative with respect to v. If we set c(t) = (a + th, b + tk), then Dv f (a, b) is the derivative at t = 0 of the composite function f (c(t)), where c(t) = (a + th, b + tk), and we can evaluate it using the Chain Rule for Paths. We have c (t) = h, k = v, so Dv f (a, b) = ∇f(a,b) · c (0) = ∇f(a,b) · v This yields the basic formula: Dv f (a, b) = ∇f(a,b) · v

2

Similarly, in three variables, Dv f (a, b, c) = ∇f(a,b,c) · v. For any scalar λ, Dλv f (P ) = ∇fP · (λv) = λ∇fP · v. Therefore, Dλv f (P ) = λDv f (P )

3

1 v is a unit vector in the direction of v. Applying Eq. (3) with v λ = 1/u gives us a formula for the directional derivative Du f (P ) in terms of Dv f (P ).

If v  = 0, then u =

THEOREM 3 Computing the Directional Derivative If v  = 0, then u = v/v is the unit vector in the direction of v, and the directional derivative is given by Du f (P ) =

1 ∇fP · v v

4

E X A M P L E 6 Let f (x, y) = xey , P = (2, −1), and v = 2, 3.

(a) Calculate Dv f (P ). (b) Then calculate the directional derivative in the direction of v. Solution (a) First compute the gradient at P = (2, −1):      ∂f ∂f ∇f = ⇒ ∇fP = ∇f(2,−1) = e−1 , 2e−1 , = ey , xey ∂x ∂y Then use Eq. (2):

 Dv f (P ) = ∇fP · v = e−1 , 2e−1 · 2, 3 = 8e−1 ≈ 2.94

(b) The directional derivative is Du f (P ), where u = v/v. By Eq. 4, Du f (P ) =

1 8e−1 8e−1 Dv f (P ) = √ = √ ≈ 0.82 v 13 22 + 3 2

E X A M P L E 7 Find the rate of change of pressure at the point Q = (1, 2, 1) in the direction of v = 0, 1, 1, assuming that the pressure (in millibars) is given by

f (x, y, z) = 1000 + 0.01(yz2 + x 2 z − xy 2 )

(x, y, z in kilometers)

The Gradient and Directional Derivatives

S E C T I O N 12.5

717

Solution First compute the gradient at Q = (1, 2, 1):  ∇f = 0.01 2xz − y 2 , z2 − 2xy, 2yz + x 2 ∇fQ = ∇f(1,2,1) = −0.02, −0.03, 0.05 Then use Eq. (2) to compute the derivative with respect to v: Dv f (Q) = ∇fQ · v = −0.02, −0.03, 0.05 · 0, 1, 1 = 0.01(−3 + 5) = 0.02 The rate of change per kilometer is the √ directional derivative. The unit vector in the direction of v is u = v/v. Since v = 2, Eq. (4) yields Du f (Q) =

1 0.02 Dv f (Q) = √ ≈ 0.014 mb/km v 2

Properties of the Gradient REMINDER For any vectors u and v,

v · u = vu cos θ where θ is the angle between v and u. If u is a unit vector, then

v · u = v cos θ

We are now in a position to draw some interesting and important conclusions about the gradient. First, suppose that ∇fP  = 0 and let u be a unit vector (Figure 7). By the properties of the dot product, Du f (P ) = ∇fP · u = ∇fP  cos θ

5

where θ is the angle between ∇fP and u. In other words, the rate of change in a given direction varies with the cosine of the angle θ between the gradient and the direction. Because the cosine takes values between −1 and 1, we have −∇fP  ≤ Du f (P ) ≤ ∇fP 

∇f P

u P

Unit vector

FIGURE 7 Du f (P ) = ∇fP  cos θ.

Since cos 0 = 1, the maximum value of Du f (P ) occurs for θ = 0—that is, when u points in the direction of ∇fP . In other words the gradient vector points in the direction of the maximum rate of increase, and this maximum rate is ∇fP . Similarly, f decreases most rapidly in the opposite direction, −∇fP , because cos θ = −1 for θ = π. The rate of maximum decrease is −∇fP . The directional derivative is zero in directions orthogonal to the gradient because cos π2 = 0. In the earlier scenario where the biker Chloe rides along a path (Figure 8), the temperature T changes at a rate that depends on the cosine of the angle θ between ∇T and the direction of motion.

y

Maximum temperature increase in the gradient direction.

In this direction, temperature changes at the rate ||∇T || cos .

∇T(x, y)

Temperature rate of change is zero in direction orthogonal to ∇T(x, y). x FIGURE 8

718

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

Another key property is that gradient vectors are normal to level curves (Figure 9). To prove this, suppose that P lies on the level curve f (x, y) = k. We parametrize this level curve by a path c(t) such that c(0) = P and c (0)  = 0 (this is possible whenever ∇fP  = 0). Then f (c(t)) = k for all t, so by the Chain Rule,   d d = k=0 ∇fP · c (0) = f (c(t)) dt dt t=0

REMINDER •

The words “normal” and “orthogonal” both mean “perpendicular.” We say that a vector is normal to a curve at a point P if it is normal to the tangent line to the curve at P .

•

This proves that ∇fP is orthogonal to c (0), and since c (0) is tangent to the level curve, we conclude that ∇fP is normal to the level curve (Figure 9). For functions of three variables, a similar argument shows that ∇fP is normal to the level surface f (x, y, z) = k through P .

y c'(0) ∇f P

THEOREM 4 Interpretation of the Gradient Assume that ∇fP = 0. Let u be a unit vector making an angle θ with ∇fP . Then

P 40

Du f (P ) = ∇fP  cos θ

80 •

120

x

•

FIGURE 9 Contour map of f (x, y). The

•

6

∇fP points in the direction of maximum rate of increase of f at P . −∇fP points in the direction of maximum rate of decrease at P . ∇fP is normal to the level curve (or surface) of f at P .

gradient at P is orthogonal to the level curve through P .

At each point P , there is a unique direction in which f (x, y) increases most rapidly (per unit distance). Theorem 4 tells us that this chosen direction is perpendicular to the level curves and that it is specified by the gradient vector (Figure 10). For most functions, however, the direction of maximum rate of increase varies from point to point.

GRAPHICAL INSIGHT y ∇f P

P

E X A M P L E 8 Let f (x, y) = x 4 y −2 and P = (2, 1). Find the unit vector that points

in the direction of maximum rate of increase at P . Solution The gradient points in the direction of maximum rate of increase, so we evaluate the gradient at P :  ∇f(2,1) = 32, −32 ∇f = 4x 3 y −2 , −2x 4 y −3 ,

Level curve of f (x, y) x FIGURE 10 The gradient points in the

The unit vector in this direction is

direction of maximum increase.

32, −32 32, −32 = = u= √ 32, −32 32 2

y 3

!√ √ " 2 2 ,− 2 2

E X A M P L E 9 The altitude of a mountain at (x, y) is

2

f (x, y) = 2500 + 100(x + y 2 )e−0.3y

1

2

where x, y are in units of 100 m.

0

2400

P

−1 −2

2500

2600

2700

(a) Find the directional derivative of f at P = (−1, −1) in the direction of unit vector u making an angle of θ = π4 with the gradient (Figure 11). (b) What is the interpretation of this derivative?

u

∇f P −2

−1

0

Solution First compute ∇fP : 1

2

3

FIGURE 11 Contour map of the function

f (x, y) in Example 9.

x

fx (x, y) = 100e−0.3y , 2

fx (−1, −1) = 100e−0.3 ≈ 74,

fy (x, y) = 100y(2 − 0.6x − 0.6y 2 )e−0.3y fy (−1, −1) = −200e−0.3 ≈ −148

2

The Gradient and Directional Derivatives

S E C T I O N 12.5

Hence, ∇fP ≈ 74, −148 and ∇fP  ≈

719

 742 + (−148)2 ≈ 165.5

Apply Eq. (6) with θ = π/4:

√  2 ≈ 117 Du f (P ) = ∇fP  cos θ ≈ 165.5 2

Recall that x and y are measured in units of 100 meters. Therefore, the interpretation is: If you stand on the mountain at the point lying above (−1, −1) and begin climbing so that your horizontal displacement is in the direction of u, then your altitude increases at a rate of 117 meters per 100 meters of horizontal displacement, or 1.17 meters per meter of horizontal displacement.

The symbol ψ (pronounced “p-sigh” or “p-see”) is the lowercase Greek letter psi.

CONCEPTUAL INSIGHT The directional derivative is related to the angle of inclination ψ in Figure 12. Think of the graph of z = f (x, y) as a mountain lying over the xy-plane. Let Q be the point on the mountain lying above a point P = (a, b) in the xy-plane. If you start moving up the mountain so that your horizontal displacement is in the direction of u, then you will actually be moving up the mountain at an angle of inclination ψ defined by

tan ψ = Du f (P )

7

The steepest direction up the mountain is the direction for which the horizontal displacement is in the direction of ∇fP .

z z = f (x, y) Du f (P) Q

x

P

u

u

y

FIGURE 12

E X A M P L E 10 Angle of Inclination You are standing on the side of a mountain in the shape z = f (x, y), at a point Q = (a, b, f (a, b)), where ∇f(a,b) = 0.4, 0.02. Find the angle of inclination in a direction making an angle of θ = π3 with the gradient.  Solution The gradient has length ∇f(a,b)  = (0.4)2 + (0.02)2 ≈ 0.4. If u is a unit vector making an angle of θ = π3 with ∇f(a,b) , then

Du f (a, b) = ∇f(a,b)  cos

π ≈ (0.4)(0.5) = 0.2 3

The angle of inclination at Q in the direction of u satisfies tan ψ = 0.2. It follows that ψ ≈ tan−1 0.2 ≈ 0.197 rad or approximately 11.3◦ .

720

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

Another use of the gradient is in finding normal vectors on a surface with equation F (x, y, z) = k, where k is a constant. Let P = (a, b, c) and assume that ∇FP = 0. Then ∇FP is normal to the level surface F (x, y, z) = k by Theorem 4. The tangent plane at P has equation ∇FP · x − a, y − b, z − c = 0 Expanding the dot product, we obtain Fx (a, b, c)(x − a) + Fy (a, b, c)(y − b) + Fz (a, b, c)(z − c) = 0 P = (2, 1, 3) FP

E X A M P L E 11 Normal Vector and Tangent Plane Find an equation of the tangent plane to the surface 4x 2 + 9y 2 − z2 = 16 at P = (2, 1, 3).

Solution Let F (x, y, z) = 4x 2 + 9y 2 − z2 . Then ∇F = 8x, 18y, −2z ,

FIGURE 13 The gradient vector ∇FP is

∇FP = ∇F(2,1,3) = 16, 18, −6

The vector 16, 18, −6 is normal to the surface F (x, y, z) = 16 (Figure 13), so the tangent plane at P has equation 16(x − 2) + 18(y − 1) − 6(z − 3) = 0

normal to the surface at P .

or

16x + 18y − 6z = 32

12.5 SUMMARY •

The gradient of a function f is the vector of partial derivatives:     ∂f ∂f ∂f ∂f ∂f ∇f = , or ∇f = , , ∂x ∂y ∂x ∂y ∂z

•

Chain Rule for Paths:

•

d f (c(t)) = ∇fc(t) · c (t) dt Derivative of f with respect to v = h, k: Dv f (a, b) = lim

t→0

f (a + th, b + tk) − f (a, b) t

This definition extends to three or more variables. • Formula for the derivative with respect to v: D f (a, b) = ∇f v (a,b) · v. • For u a unit vector, D f is called the directional derivative. u v 1 – If u = , then Du f (a, b) = Dv f (a, b). v v – Du f (a, b) = ∇f(a,b)  cos θ , where θ is the angle between ∇f(a,b) and u. •

Basic geometric properties of the gradient (assume ∇fP  = 0): – ∇fP points in the direction of maximum rate of increase. The maximum rate of increase is ∇fP . – −∇fP points in the direction of maximum rate of decrease. The maximum rate of decrease is −∇fP . – ∇fP is orthogonal to the level curve (or surface) through P .

•

Equation of the tangent plane to the level surface F (x, y, z) = k at P = (a, b, c): ∇FP · x − a, y − b, z − c = 0 Fx (a, b, c)(x − a) + Fy (a, b, c)(y − b) + Fz (a, b, c)(z − c) = 0

The Gradient and Directional Derivatives

S E C T I O N 12.5

721

12.5 EXERCISES Preliminary Questions 1. Which of the following is a possible value of the gradient ∇f of a function f (x, y) of two variables? (a) 5 (b) 3, 4 (c) 3, 4, 5 2. True or false? A differentiable function increases at the rate ∇fP  in the direction of ∇fP . 3. Describe the two main geometric properties of the gradient ∇f .

4. You are standing at a point where the temperature gradient vector is pointing in the northeast (NE) direction. In which direction(s) should you walk to avoid a change in temperature? (a) NE (b) NW (c) SE (d) SW 5. What is the rate of change of f (x, y) at (0, 0) in the direction making an angle of 45◦ with the x-axis if ∇f (0, 0) = 2, 4?

Exercises

  1. Let f (x, y) = xy 2 and c(t) = 12 t 2 , t 3 . (a) Calculate ∇f and c (t).

(b) Use the Chain Rule for Paths to evaluate t = −1.

d f (c(t)) at t = 1 and dt

2. Let f (x, y) = exy and c(t) = (t 3 , 1 + t). (a) Calculate ∇f and c (t). d (b) Use the Chain Rule for Paths to calculate f (c(t)). dt (c) Write out the composite f (c(t)) as a function of t and differentiate. Check that the result agrees with part (b). 3. Figure 14 shows the level curves of a function f (x, y) and a path c(t), traversed in the direction indicated. State whether the derivative d f (c(t)) is positive, negative, or zero at points A–D. dt

9. f (x, y) = 3x − 7y,

D

0

A

B

t =0 t = π2

c(t) = (cos t, sin t),

13. f (x, y) = sin(xy), c(t) = (e2t , e3t ), c(t) = (et , e2t ),

14. f (x, y) = cos(y − x), 15. f (x, y) = x − xy, 16. f (x, y) = xey ,

t =0 t = ln 3

c(t) = (t 2 , t 2 − 4t),

c(t) = (t 2 , t 2 − 4t),

t =4

t =0

c(t) = (cos t, t 2 ),

t = π4

c(t) = (t 2 , t 3 , t − 1),

t =1

17. f (x, y) = ln x + ln y,

c(t) = (et , t, t 2 ),

t =1

c(t) = (t 2 , t 3 , t, t−2),

In Exercises 21–30, calculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector or use Eq. (4).

0 10 20 30

21. f (x, y) = x 2 + y 3 ,

−4 −4

t =2

12. f (x, y) = x 2 − 3xy,

20. g(x, y, z, w) = x + 2y + 3z + 5w, t =1

−20

C −10

c(t) = (t 2 , t 3 ),

t =0

11. f (x, y) = x 2 − 3xy, c(t) = (cos t, sin t),

19. g(x, y, z) = xyz−1 ,

8

c(t) = (cos t, sin t),

10. f (x, y) = 3x − 7y,

18. g(x, y, z) = xyez ,

y

4

d f (c(t)). dt

In Exercises 9–20, use the Chain Rule to calculate

0

4

8

x

FIGURE 14

4. Let f (x, y) = x 2 + y 2 and c(t) = (cos t, sin t). d (a) Find f (c(t)) without making any calculations. Explain. dt (b) Verify your answer to (a) using the Chain Rule. In Exercises 5–8, calculate the gradient.

v = 4, 3,

P = (1, 2)

22. f (x, y) = x 2 y 3 ,

v = i + j, P = (−2, 1)   23. f (x, y) = x 2 y 3 , v = i + j, P = 16 , 3   24. f (x, y) = sin(x − y), v = 1, 1, P = π2 , π6 25. f (x, y) = tan−1 (xy), v = 1, 1,

P = (3, 4)

xy−y 2

P = (2, 2)

26. f (x, y) = e

,

v = 12, −5,

27. f (x, y) = ln(x 2 + y 2 ),

v = 3i − 2j, v = −1, 2, 2,

5. f (x, y) = cos(x 2 + y)

x 6. g(x, y) = 2 x + y2

28. g(x, y, z) = z2 − xy 2 ,

7. h(x, y, z) = xyz−3

8. r(x, y, z, w) = xzeyw

30. g(x, y, z) = x ln(y + z),

29. g(x, y, z) = xe−yz ,

v = 1, 1, 1,

P = (1, 0) P = (2, 1, 3)

P = (1, 2, 0)

v = 2i − j + k,

P = (2, e, e)

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C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

31. Find the directional derivative of f (x, y) = x 2 + 4y 2 at P = (3, 2) in the direction pointing to the origin.

z

32. Find the directional derivative of f (x, y, z) = xy + z3 at P = (3, −2, −1) in the direction pointing to the origin. 33. A bug located at (3, 9, 4) begins walking in a straight line toward (5, 7, 3). At what rate is the bug’s temperature changing if the temperature is T (x, y, z) = xey−z ? Units are in meters and degrees Celsius.

y x

34. The temperature at location (x, y) is T (x, y) = 20 + 0.1(x 2 − xy) (degrees Celsius). Beginning at (200, 0) at time t = 0 (seconds), a bug travels along a circle of radius 200 cm centered at the origin, at a speed of 3 cm/s. How fast is the temperature changing at time t = π/3? 35. Suppose that ∇fP = 2, −4, 4. Is f increasing or decreasing at P in the direction v = 2, 1, 3? 2 36. Let f (x, y) = xex −y and P = (1, 1). (a) Calculate ∇fP .

(b) Find the rate of change of f in the direction ∇fP . (c) Find the rate of change of f in the direction of a vector making an angle of 45◦ with ∇fP . 37. Let f (x, y, z) = sin(xy + z) and P = (0, −1, π). Calculate Du f (P ), where u is a unit vector making an angle θ = 30◦ with ∇fP . 38. Let T (x, y) be the temperature at location (x, y). Assume that ∇T = y − 4, x + 2y. Let c(t) = (t 2 , t) be a path in the plane. Find the values of t such that d T (c(t)) = 0 dt 39. Find a vector normal to the surface x 2 + y 2 − z2 = 6 at P = (3, 1, 2). 40. Find a vector normal to the surface 3z3 + x 2 y − y 2 x = 1 at

P = (1, −1, 1).

FIGURE 15 Graph of x 2 + y 2 − z2 = 0.

47. Use a computer algebra system to produce a contour plot of f (x, y) = x 2 − 3xy + y − y 2 together with its gradient vector field on the domain [−4, 4] × [−4, 4]. 48. Find a function f (x, y, z) such that ∇f is the constant vector 1, 3, 1. 49. Find a function f (x, y, z) such that ∇f = 2x, 1, 2.   50. Find a function f (x, y, z) such that ∇f = x, y 2 , z3 . 51. Find a function f (x, y, z) such that ∇f = z, 2y, x. 52. Find a function f (x, y) such that ∇f = y, x. that there does not exist a function f (x, y) such that ∇f =  53.2 Show y , x . Hint: Use Clairaut’s Theorem fxy = fyx . 54. Let f = f (a + h, b + k) − f (a, b) be the change in f at P = (a, b). Set v = h, k. Show that the linear approximation can be written f ≈ ∇fP · v

8

55. Use Eq. (8) to estimate f = f (3.53, 8.98) − f (3.5, 9) assuming that ∇f(3.5,9) = 2, −1.

41. Find the two points on the ellipsoid x2 4

+

y2 9

+ z2 = 1

where the tangent plane is normal to v = 1, 1, −2. In Exercises 42–45, find an equation of the tangent plane to the surface at the given point. 42. x 2 + 3y 2 + 4z2 = 20, P = (2, 2, 1) 43. xz + 2x 2 y + y 2 z3 = 11,

P = (2, 1, 1)   3 2 2 y−x 44. x + z e = 13, P = 2, 3, √ e 45. ln[1 + 4x 2 + 9y 4 ] − 0.1z2 = 0,

P = (3, 1, 6.1876)

46. Verify what is clear from Figure 15: Every tangent plane to the cone x 2 + y 2 − z2 = 0 passes through the origin.

56. Find a unit vector n that is normal to the surface z2 − 2x 4 − y 4 = 16 at P = (2, 2, 8) that points in the direction of the xy-plane (in other words, if you travel in the direction of n, you will eventually cross the xy-plane). 57. Suppose, in the previous exercise, that a particle located at the point P = (2, 2, 8) travels toward the xy-plane in the direction normal to the surface. (a) Through which point Q on the xy-plane will the particle pass? (b) Suppose the axes are calibrated in centimeters. Determine the path c(t) of the particle if it travels at a constant speed of 8 cm/s. How long will it take the particle to reach Q? !√ √ " x 2 2 58. Let f (x, y) = tan−1 and u = , . y 2 2 (a) Calculate the gradient of f . √ (b) Calculate Du f (1, 1) and Du f ( 3, 1). (c) Show that the lines y = mx for m  = 0 are level curves for f .

S E C T I O N 12.6

(d) Verify that ∇fP is orthogonal to the level curve through P for P = (x, y) = (0, 0). Suppose that the intersection of two surfaces F (x, y, z) = 59. 0 and G(x, y, z) = 0 is a curve C, and let P be a point on C. Explain why the vector v = ∇FP × ∇GP is a direction vector for the tangent line to C at P . 60. Let C be the curve of intersection of the spheres x 2 + y 2 + z2 = 3 and (x − 2)2 + (y − 2)2 + z2 = 3. Use the result of Exercise 59 to find parametric equations of the tangent line to C at P = (1, 1, 1).

The Chain Rule

723

61. Let C be the curve obtained by intersecting the two surfaces x 3 + 2xy + yz = 7 and 3x 2 − yz = 1. Find the parametric equations of the tangent line to C at P = (1, 2, 1). 62. Verify the linearity relations for gradients: (a) ∇(f + g) = ∇f + ∇g (b) ∇(cf ) = c∇f 63. Prove the Chain Rule for Gradients (Theorem 1). 64. Prove the Product Rule for Gradients (Theorem 1).

Further Insights and Challenges 65. Let u be a unit vector. Show that the directional derivative Du f is equal to the component of ∇f along u. 66. Let f (x, y) = (xy)1/3 . (a) Use the limit definition to show that fx (0, 0) = fy (0, 0) = 0. (b) Use the limit definition to show that the directional derivative Du f (0, 0) does not exist for any unit vector u other than i and j. (c) Is f differentiable at (0, 0)?

In Exercises 71–73, a path c(t) = (x(t), y(t)) follows the gradient of a function f (x, y) if the tangent vector c (t) points in the direction of ∇f for all t. In other words, c (t) = k(t)∇fc(t) for some positive function k(t). Note that in this case, c(t) crosses each level curve of f (x, y) at a right angle. 71. Show that if the path c(t) = (x(t), y(t)) follows the gradient of f (x, y), then fy y  (t) =  x (t) fx

67. Use the definition of differentiability to show that if f (x, y) is differentiable at (0, 0) and

72. Find a path of the form c(t) = (t, g(t)) passing through (1, 2) that follows the gradient of f (x, y) = 2x 2 + 8y 2 (Figure 16). Hint: Use Separation of Variables.

f (0, 0) = fx (0, 0) = fy (0, 0) = 0 then lim

(x,y)→(0,0)

f (x, y) =0  x2 + y2

9

68. This exercise shows that there exists a function that is not differentiable at (0, 0) even though all directional derivatives at (0, 0) exist. Define f (x, y) = x 2 y/(x 2 + y 2 ) for (x, y) = 0 and f (0, 0) = 0. (a) Use the limit definition to show that Dv f (0, 0) exists for all vectors v. Show that fx (0, 0) = fy (0, 0) = 0. (b) Prove that f is not differentiable at (0, 0) by showing that Eq. (9) does not hold. 69. Prove that if f (x, y) is differentiable and ∇f(x,y) = 0 for all (x, y), then f is constant. 70. Prove the following Quotient Rule, where f, g are differentiable:   f g∇f − f ∇g ∇ = g g2

y 2 1 1

x

FIGURE 16 The path c(t) is orthogonal to the level curves of f (x, y) = 2x 2 + 8y 2 .

Find the curve y = g(x) passing through (0, 1) that 73. crosses each level curve of f (x, y) = y sin x at a right angle. If you have a computer algebra system, graph y = g(x) together with the level curves of f .

12.6 The Chain Rule The Chain Rule for Paths that we derived in the previous section can be extended to general composite functions. Suppose, for example, that x, y, z are differentiable functions of s and t—say x = x(s, t), y = y(s, t), and z = z(s, t). The composite f (x(s, t), y(s, t), z(s, t)) is then a function of s and t. We refer to s and t as the independent variables.

1

724

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

E X A M P L E 1 Find the composite function where f (x, y, z) = xy + z and x = s 2 ,

y = st, z = t 2 .

Solution The composite function is f (x(s, t), y(s, t), z(s, t)) = xy + z = (s 2 )(st) + t 2 = s 3 t + t 2 The Chain Rule expresses the derivatives of f with respect to the independent variables. For example, the partial derivatives of f (x(s, t), y(s, t), z(s, t)) are ∂f ∂x ∂f ∂y ∂f ∂z ∂f = + + ∂s ∂x ∂s ∂y ∂s ∂z ∂s

2

∂f ∂f ∂x ∂f ∂y ∂f ∂z = + + ∂t ∂x ∂t ∂y ∂t ∂z ∂t

3

To prove these formulas, we observe that ∂f/∂s, when evaluated at a point (s0 , t0 ), is equal to the derivative with respect to the path c(s) = (x(s, t0 ), y(s, t0 ), z(s, t0 )) In other words, we fix t = t0 and take the derivative with respect to s:   d ∂f (s0 , t0 ) = f (c(s)) ∂s ds s=s0 The tangent vector is 



∂x ∂y ∂z (s, t0 ), (s, t0 ), (s, t0 ) c (s) = ∂s ∂s ∂s 

Therefore, by the Chain Rule for Paths,    d ∂f ∂x ∂f ∂y ∂f ∂z ∂f  = = ∇f · c (s0 ) = f (c(s)) + + ∂s (s0 ,t0 ) ds ∂x ∂s ∂y ∂s ∂z ∂s s=s0 The derivatives on the right are evaluated at (s0 , t0 ). This proves Eq. (2).Asimilar argument proves Eq. (3), as well as the general case of a function f (x1 , . . . , xn ), where the variables xi depend on independent variables t1 , . . . , tm . THEOREM 1 General Version of Chain Rule Let f (x1 , . . . , xn ) be a differentiable function of n variables. Suppose that each of the variables x1 , . . . , xn is a differentiable function of m independent variables t1 , . . . , tm . Then, for k = 1, . . . , m, ∂f ∂f ∂x1 ∂f ∂x2 ∂f ∂xn = + + ··· + ∂tk ∂x1 ∂tk ∂x2 ∂tk ∂xn ∂tk

4

As an aid to remembering the Chain Rule, we will refer to ∂f , ∂x1 The term “primary derivative” is not standard. We use it in this section only, to clarify the structure of the Chain Rule.

...,

∂f ∂xn

as the primary derivatives. They are the components of the gradient ∇f . By Eq. (4), the derivative of f with respect to the independent variable tk is equal to a sum of n terms: j th term:

∂f ∂xj ∂xj ∂tk

for j = 1, 2, . . . , n

S E C T I O N 12.6

The Chain Rule

725

Note that we can write Eq. (4) as a dot product:     ∂f ∂x1 ∂x2 ∂f ∂f ∂f ∂xn · = , ,..., , ,..., ∂tk ∂x1 ∂x2 ∂xn ∂tk ∂tk ∂tk

5

E X A M P L E 2 Using the Chain Rule Let f (x, y, z) = xy + z. Calculate ∂f/∂s, where

x = s2,

y = st,

z = t2

Solution Step 1. Compute the primary derivatives. ∂f = y, ∂x

∂f = x, ∂y

∂f =1 ∂z

Step 2. Apply the Chain Rule. ∂f ∂f ∂x ∂f ∂y ∂f ∂z ∂ ∂ ∂ = + + = y (s 2 ) + x (st) + (t 2 ) ∂s ∂x ∂s ∂y ∂s ∂z ∂s ∂s ∂s ∂s = (y)(2s) + (x)(t) + 0 = 2sy + xt This expresses the derivative in terms of both sets of variables. If desired, we can substitute x = s 2 and y = st to write the derivative in terms of s and t: ∂f = 2ys + xt = 2(st)s + (s 2 )t = 3s 2 t ∂s To check this result, recall that in Example 1, we computed the composite function: f (x(s, t), y(s, t), z(s, t)) = f (s 2 , st, t 2 ) = s 3 t + t 2 From this we see directly that ∂f/∂s = 3s 2 t, confirming our result. E X A M P L E 3 Evaluating the Derivative Let f (x, y) = exy . Evaluate ∂f/∂t at

(s, t, u) = (2, 3, −1), where x = st, y = s − ut 2 .

Solution We can use either Eq. (4) or Eq. (5). We’ll use the dot product form in Eq. (5). We have       xy ∂f ∂f ∂x ∂y xy ∇f = , = ye , xe , , = s, −2ut ∂x ∂y ∂t ∂t and the Chain Rule gives us

    ∂f ∂x ∂y , = yexy , xexy · s, −2ut = ∇f · ∂t ∂t ∂t = yexy (s) + xexy (−2ut) = (ys − 2xut)exy

To finish the problem, we do not have to rewrite ∂f/∂t in terms of s, t, u. For (s, t, u) = (2, 3, −1), we have x = st = 2(3) = 6,

y = s − ut 2 = 2 − (−1)(32 ) = 11

With (s, t, u) = (2, 3, −1) and (x, y) = (6, 11), we have      ∂f  xy  = (ys − 2xut)e = (11)(2) − 2(6)(−1)(3) e6(11) = 58e66  ∂t (2,3,−1) (2,3,−1)

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E X A M P L E 4 Polar Coordinates Let f (x, y) be a function of two variables, and let (r, θ ) be polar coordinates.

(a) Express ∂f/∂θ in terms of ∂f/∂x and ∂f/∂y. (b) Evaluate ∂f/∂θ at (x, y) = (1, 1) for f (x, y) = x 2 y. Solution (a) Since x = r cos θ and y = r sin θ , ∂x = −r sin θ, ∂θ

∂y = r cos θ ∂θ

By the Chain Rule, ∂f ∂x ∂f ∂y ∂f ∂f ∂f = + = −r sin θ + r cos θ ∂θ ∂x ∂θ ∂y ∂θ ∂x ∂y If you have studied quantum mechanics, you may recognize the right-hand side of Eq. (6) as the angular momentum operator (with respect to the z-axis).

Since x = r cos θ and y = r sin θ, we can write ∂f/∂θ in terms of x and y alone: ∂f ∂f ∂f =x −y ∂θ ∂y ∂x

6

(b) Apply Eq. (6) to f (x, y) = x 2 y: ∂ ∂ ∂f =x (x 2 y) − y (x 2 y) = x 3 − 2xy 2 ∂θ ∂y ∂x   ∂f  = 13 − 2(1)(12 ) = −1 ∂θ (x,y)=(1,1)

Implicit Differentiation In single-variable calculus, we used implicit differentiation to compute dy/dx when y is defined implicitly as a function of x through an equation f (x, y) = 0. This method also works for functions of several variables. Suppose that z is defined implicitly by an equation F (x, y, z) = 0 Thus z = z(x, y) is a function of x and y. We may not be able to solve explicitly for z(x, y), but we can treat F (x, y, z) as a composite function with x and y as independent variables, and use the Chain Rule to differentiate with respect to x: ∂F ∂y ∂F ∂z ∂F ∂x + + =0 ∂x ∂x ∂y ∂x ∂z ∂x We have ∂x/∂x = 1, and also ∂y/∂x = 0 since y does not depend on x. Thus ∂z ∂F ∂z ∂F + = Fx + Fz =0 ∂x ∂z ∂x ∂x If Fz  = 0, we may solve for ∂z/∂x (we compute ∂z/∂y similarly): ∂z Fx =− , ∂x Fz

Fy ∂z =− ∂y Fz

7

S E C T I O N 12.6

The Chain Rule

727

E X A M P L E 5 Calculate ∂z/∂x and ∂z/∂y at P = (1, 1, 1), where

z

F (x, y, z) = x 2 + y 2 − 2z2 + 12x − 8z − 4 = 0 What is the graphical interpretation of these partial derivatives? P = (1, 1, 1)

Solution We have

y

Fx = 2x + 12,

x

Fy = 2y,

Fz = −4z − 8

and hence, Fx 2x + 12 ∂z =− , = ∂x Fz 4z + 8 FIGURE 1 The surface x 2 + y 2 − 2z2 + 12x − 8z − 4 = 0.

A small patch of the surface around P can be represented as the graph of a function of x and y.

z

1

y

The derivatives at P = (1, 1, 1) are  2(1) + 12 14 7 ∂z  = = = , ∂x (1,1,1) 4(1) + 8 12 6

 2(1) ∂z  2 1 = = = ∂y (1,1,1) 4(1) + 8 12 6

Figure 1 shows the surface F (x, y, z) = 0. The surface as a whole is not the graph of a function because it fails the Vertical Line Test. However, a small patch near P may be represented as a graph of a function z = f (x, y), and the partial derivatives ∂z/∂x and ∂z/∂y are equal to fx and fy . Implicit differentiation has enabled us to compute these partial derivatives without finding f (x, y) explicitly. Assumptions Matter Implicit differentiation is based on the assumption that we can solve the equation F (x, y, z) = 0 for z in the form z = f (x, y). Otherwise, the partial derivatives ∂z/∂x and ∂z/∂y would have no meaning. The Implicit Function Theorem of advanced calculus guarantees that this can be done (at least near a point P ) if F has continuous partial derivatives and Fz (P )  = 0. Why is this  condition necessary? Recall that the gradient vector ∇FP = Fx (P ), Fy (P ), Fz (P ) is normal to the surface at P , so Fz (P ) = 0 means that the tangent plane at P is vertical. To see what can go wrong, consider the cylinder (shown in Figure 2):

x

FIGURE 2 Graph of the cylinder x 2 + y 2 − 1 = 0.

Fy ∂z 2y =− = ∂y Fz 4z + 8

F (x, y, z) = x 2 + y 2 − 1 = 0 In this extreme case, Fz = 0. The z-coordinate on the cylinder does not depend on x or y, so it is impossible to represent the cylinder as a graph z = f (x, y) and the derivatives ∂z/∂x and ∂z/∂y do not exist.

12.6 SUMMARY If f (x, y, z) is a function of x, y, z, and if x, y, z depend on two other variables, say s and t, then

•

f (x, y, z) = f (x(s, t), y(s, t), z(s, t)) is a composite function of s and t. We refer to s and t as the independent variables. The Chain Rule expresses the partial derivatives with respect to the independent variables s and t in terms of the primary derivatives: •

∂f , ∂x

∂f , ∂y

∂f ∂z

Namely, ∂f ∂y ∂f ∂z ∂f ∂x ∂f + + , = ∂y ∂s ∂z ∂s ∂s ∂x ∂s

∂f ∂f ∂x ∂f ∂y ∂f ∂z = + + ∂t ∂x ∂t ∂y ∂t ∂z ∂t

728

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES • In general, if f (x , . . . , x ) is a function of n variables and if x , . . . , x depend on the 1 n 1 n independent variables t1 , . . . , tm , then

∂f ∂f ∂x1 ∂f ∂x2 ∂f ∂xn = + + ··· + ∂tk ∂x1 ∂tk ∂x2 ∂tk ∂xn ∂tk •

The Chain Rule can be expressed as a dot product:     ∂f ∂f ∂f ∂x1 ∂x2 ∂xn ∂f = , ,..., · , ,..., ∂x ∂x2 ∂xn ∂tk ∂tk ∂tk ∂tk  1   ∇f

Implicit differentiation is used to find the partial derivatives ∂z/∂x and ∂z/∂y when z is defined implicitly by an equation F (x, y, z) = 0:

•

Fx ∂z =− , ∂x Fz

Fy ∂z =− ∂y Fz

12.6 EXERCISES Preliminary Questions (b)

∂f ∂x ∂f ∂y + ∂x ∂r ∂y ∂r

In Questions 2 and 3, suppose that f (u, v) = uev , where u = rs and v = r + s.

(c)

∂f ∂r ∂f ∂s + ∂r ∂x ∂s ∂x

2. The composite function f (u, v) is equal to: (a) rser+s (b) res

5. Suppose that x, y, z are functions of the independent variables u, v, w. Which of the following terms appear in the Chain Rule expression for ∂f/∂w? ∂f ∂x ∂f ∂w ∂f ∂z (a) (b) (c) ∂v ∂v ∂w ∂x ∂z ∂w

1. Let f (x, y) = xy, where x = uv and y = u + v. (a) What are the primary derivatives of f ? (b) What are the independent variables?

(c) rsers

3. What is the value of f (u, v) at (r, s) = (1, 1)? 4. According to the Chain Rule, ∂f/∂r is equal to (choose the correct answer): ∂f ∂x ∂f ∂x (a) + ∂x ∂r ∂x ∂s

6. With notation as in the previous question, does ∂x/∂v appear in the Chain Rule expression for ∂f/∂u?

Exercises 1. Let f (x, y, z) = x 2 y 3 + z4 and x = s 2 , y = st 2 , and z = s 2 t. ∂f ∂f ∂f , , . (a) Calculate the primary derivatives ∂x ∂y ∂z ∂x ∂y ∂z , , . (b) Calculate ∂s ∂s ∂s ∂f using the Chain Rule: (c) Compute ∂s ∂f ∂f ∂x ∂f ∂y ∂f ∂z = + + ∂s ∂x ∂s ∂y ∂s ∂z ∂s

(b) Use the Chain Rule to calculate ∂f/∂v. Leave the answer in terms of both the dependent and the independent variables. (c) Determine (x, y) for (u, v) = (2, 1) and evaluate ∂f/∂v at (u, v) = (2, 1). In Exercises 3–10, use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. 3.

∂f ∂f , ; f (x, y, z) = xy + z2 , x = s 2 , y = 2rs, z = r 2 ∂s ∂r

4.

∂f ∂f , ; f (x, y, z) = xy + z2 , x = r + s − 2t, y = 3rt, z = s 2 ∂r ∂t

5.

∂g ∂g , ; g(x, y) = cos(x − y), x = 3u − 5v, y = −7u + 15v ∂u ∂v

6.

∂R ∂R , ; R(x, y) = (3x + 4y)5 , x = u2 , y = uv ∂u ∂v

Express the answer in terms of the independent variables s, t. 2. Let f (x, y) = x cos(y) and x = u2 + v 2 and y = u − v. (a) Calculate the primary derivatives

∂f ∂f , . ∂x ∂y

S E C T I O N 12.6

The Chain Rule

729

7.

∂F ; F (u, v) = eu+v , u = x 2 , v = xy ∂y

(b) Suppose that a = 10, b = 16, c = 22. Estimate the change in θ if a and b are increased by 1 and c is increased by 2.

8.

∂f ; f (x, y) = x 2 + y 2 , x = eu+v , y = u + v ∂u

19. Let u = u(x, y), and let (r, θ) be polar coordinates. Verify the relation

9.

x ∂h ; h(x, y) = , x = t1 t2 , y = t12 t2 ∂t2 y

10.

1 ∇u2 = u2r + 2 u2θ r

∂f ; f (x, y, z) = xy − z2 , x = r cos θ , y = cos2 θ, z = r ∂θ

8

Hint: Compute the right-hand side by expressing uθ and ur in terms of ux and uy .

In Exercises 11–16, use the Chain Rule to evaluate the partial derivative at the point specified.

20. Let u(r, θ) = r 2 cos2 θ. Use Eq. (8) to compute ∇u2 . Then compute ∇u2 directly by observing that u(x, y) = x 2 , and compare.

11. ∂f/∂u and ∂f/∂v at (u, v) = (−1, −1), where f (x, y, z) = x 3 + yz2 , x = u2 + v, y = u + v 2 , z = uv.

21. Let x = s + t and y = s − t. Show that for any differentiable function f (x, y),

12. ∂f/∂s at (r, s) = (1, 0), where f (x, y) = ln(xy), x = 3r + 2s, y = 5r + 3s.   √ 13. ∂g/∂θ at (r, θ) = 2 2, π4 , where g(x, y) = 1/(x + y 2 ), x = r sin θ , y = r cos θ . 14. ∂g/∂s at s = 4, where g(x, y) = x 2 − y 2 , x = s 2 + 1, y = 1 − 2s. 15. ∂g/∂u at (u, v) = (0, 1), where g(x, y) = x 2 − y 2 , x = eu cos v, y = eu sin v. 16.

∂h at (q, r) = (3, 2), where h(u, v) = uev , u = q 3 , v = qr 2 . ∂q

17. Jessica and Matthew are running toward the point P along the straight paths that make a fixed angle of θ (Figure 3). Suppose that Matthew runs with velocity va m/s and Jessica with velocity vb m/s. Let f (x, y) be the distance from Matthew to Jessica when Matthew is x meters from P and Jessica is y meters from P .  (a) Show that f (x, y) = x 2 + y 2 − 2xy cos θ.

  2 ∂f 2 ∂f ∂f ∂f − = ∂x ∂y ∂s ∂t

22. Express the derivatives ∂f ∂f ∂f , , ∂ρ ∂θ ∂φ

where (ρ, θ, φ) are spherical coordinates. 23. Suppose that z is defined implicitly as a function of x and y by the equation F (x, y, z) = xz2 + y 2 z + xy − 1 = 0. (a) Calculate Fx , Fy , Fz . ∂z ∂z (b) Use Eq. (7) to calculate and . ∂x ∂y 24. Calculate ∂z/∂x and ∂z/∂y at the points (3, 2, 1) and (3, 2, −1), where z is defined implicitly by the equation z4 + z2 x 2 − y − 8 = 0. In Exercises 25–30, calculate the partial derivative using implicit differentiation. ∂z , ∂x

26.

∂w , x 2 w + w 3 + wz2 + 3yz = 0 ∂z

P

27.

∂z , ∂y

θ

28.

∂t ∂r and , r 2 = te s/r ∂t ∂r

29.

∂w , ∂y

vb

x y

∂f ∂f ∂f , , ∂x ∂y ∂z

in terms of

25.

(b) Assume that θ = π/3. Use the Chain Rule to determine the rate at which the distance between Matthew and Jessica is changing when x = 30, y = 20, va = 4 m/s, and vb = 3 m/s.

va



x 2 y + y 2 z + xz2 = 10

exy + sin(xz) + y = 0

1 1 + 2 = 1 at (x, y, w) = (1, 1, 1) w2 + x 2 w + y2

30. ∂U/∂T and ∂T /∂U , (T U − V )2 ln(W − U V ) = 1 at (T , U, V , W ) = (1, 1, 2, 4)

A B FIGURE 3

18. The Law of Cosines states that c2 = a 2 + b2 − 2ab cos θ, where a, b, c are the sides of a triangle and θ is the angle opposite the side of length c. (a) Compute ∂θ/∂a, ∂θ/∂b, and ∂θ/∂c using implicit differentiation.

31. Let r = x, y, z and er = r/r. Show that if a function f (x, y,z) = F (r) depends only on the distance from the origin r = r = x 2 + y 2 + z2 , then ∇f = F  (r)er

9

32. Let f (x, y, z) = e−x −y −z = e−r , with r as in Exercise 31. Compute ∇f directly and using Eq. (9). 2

2

2

2

730

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

33. Use Eq. (9) to compute ∇

  1 . r

(a) Show that dy/dx = −4y/(3y 2 + 4x). (b) Let g(x) = f (x, r(x)), where f (x, y) is a function satisfying

34. Use Eq. (9) to compute ∇(ln r).

fx (1, 2) = 8,

35. Figure 4 shows the graph of the equation

fy (1, 2) = 10

Use the Chain Rule to calculate g  (1). Note that r(1) = 2 because (x, y) = (1, 2) satisfies y 3 + 4xy = 16.

F (x, y, z) = x 2 + y 2 − z2 − 12x − 8z − 4 = 0 (a) Use the quadratic formula to solve for z as a function of x and y. This gives two formulas, depending on the choice of sign. (b) Which formula defines the portion of the surface satisfying z ≥ −4? Which formula defines the portion satisfying z ≤ −4? (c) Calculate ∂z/∂x using the formula z = f (x, y) (for both choices of sign) and again via implicit differentiation. Verify that the two answers agree.

37. The pressure P , volume V , and temperature T of a van der Waals gas with n molecules (n constant) are related by the equation   an2 P + 2 (V − nb) = nRT V where a, b, and R are constant. Calculate ∂P /∂T and ∂V /∂P . 38. When x, y, and z are related by an equation F (x, y, z) = 0, we sometimes write (∂z/∂x)y in place of ∂z/∂x to indicate that in the differentiation, z is treated as a function of x with y held constant (and similarly for the other variables). (a) Use Eq. (7) to prove the cyclic relation       ∂z ∂x ∂y = −1 10 ∂x y ∂y z ∂z x

z

y x z = −4

(b) Verify Eq. (10) for F (x, y, z) = x + y + z = 0. (c) Verify the cyclic relation for the variables P , V , T in the ideal gas law P V − nRT = 0 (n and R are constants). FIGURE 4 Graph of x 2 + y 2 − z2 − 12x − 8z − 4 = 0.

39. Show that if f (x) is differentiable and c  = 0 is a constant, then u(x, t) = f (x − ct) satisfies the so-called advection equation

36. For all x > 0, there is a unique value y = r(x) that solves the equation y 3 + 4xy = 16.

∂u ∂u +c =0 ∂t ∂x

Further Insights and Challenges In Exercises 40–43, a function f (x, y, z) is called homogeneous of degree n if f (λx, λy, λz) = λn f (x, y, z) for all λ ∈ R. 40. Show that the following functions are homogeneous and determine their degree. (a) f (x, y, z) = x 2 y + xyz   xy (c) f (x, y, z) = ln z2

(b) f (x, y, z) = 3x + 2y − 8z

44. Suppose that x = g(t, s), y = h(t, s). Show that ftt is equal to  fxx

45. Let r = formulas

∂ 2x ∂ 2y + f y ∂t 2 ∂t 2

11

Hint: Let F (t) = f (tx, ty, tz) and calculate F  (1) using the Chain Rule.

12

 x12 + · · · + xn2 and let g(r) be a function of r. Prove the

∂g xi = gr , ∂xi r

42. Prove that if f (x, y, z) is homogeneous of degree n, then ∂f ∂f ∂f +y +z = nf ∂x ∂y ∂z

     2 ∂y ∂x 2 ∂x ∂y + fyy + 2fxy ∂t ∂t ∂t ∂t

+ fx

(d) f (x, y, z) = z4

41. Prove that if f (x, y, z) is homogeneous of degree n, then fx (x, y, z) is homogeneous of degree n − 1. Hint: Either use the limit definition or apply the Chain Rule to f (λx, λy, λz).

x

43. Verify Eq. (11) for the functions in Exercise 40.

∂ 2g ∂xi2

x2 r 2 − xi2 = i2 grr + gr r r3

46. Prove that if g(r) is a function of r as in Exercise 45, then ∂ 2g ∂x12

+ ··· +

∂ 2g ∂xn2

= grr +

n−1 gr r

S E C T I O N 12.7

731

50. Verify that f (x, y) = tan−1 yx is harmonic using both the rectangular and polar expressions for f .

In Exercises 47–51, the Laplace operator is defined by f = fxx + fyy . A function f (x, y) satisfying the Laplace equation f = 0 is called harmonic.  A function f (x, y) is called radial if f (x, y) = g(r), where r = x 2 + y 2 .

51. Use the Product Rule to show that

47. Use Eq. (12) to prove that in polar coordinates (r, θ), 1 1 f = frr + 2 fθθ + fr r r

Optimization in Several Variables

1 ∂ frr + fr = r −1 r ∂r

13

48. Use Eq. (13) to show that f (x, y) = ln r is harmonic.

 r

∂f ∂r



Use this formula to show that if f is a radial harmonic function, then rfr = C for some constant C. Conclude that f (x, y) = C ln r + b for some constant b.

49. Verify that f (x, y) = x and f (x, y) = y are harmonic using both the rectangular and polar expressions for f .

12.7 Optimization in Several Variables z

Local and global maximum

Local maximum

Local and global minimum x Disk D(P, r)

Recall that optimization is the process of finding the extreme values of a function. This amounts to finding the highest and lowest points on the graph over a given domain. As we saw in the one-variable case, it is important to distinguish between local and global extreme values. A local extreme value is a value f (a, b) that is a maximum or minimum in some small open disk around (a, b) (Figure 1). DEFINITION Local Extreme Values A function f (x, y) has a local extremum at P = (a, b) if there exists an open disk D(P , r) such that: • •

Local maximum: f (x, y) ≤ f (a, b) for all (x, y) ∈ D(P , r) Local minimum: f (x, y) ≥ f (a, b) for all (x, y) ∈ D(P , r)

y

FIGURE 1 f (x, y) has a local maximum

at P .

REMINDER The term “extremum” (the plural is “extrema”) means a minimum or maximum value.

Fermat’s Theorem states that if f (a) is a local extreme value, then a is a critical point and thus the tangent line (if it exists) is horizontal at x = a. We can expect a similar result for functions of two variables, but in this case, it is the tangent plane that must be horizontal (Figure 2). The tangent plane to z = f (x, y) at P = (a, b) has equation z = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) Thus, the tangent plane is horizontal if fx (a, b) = fy (a, b) = 0—that is, if the equation reduces to z = f (a, b). This leads to the following definition of a critical point, where we take into account the possibility that one or both partial derivatives do not exist.

y

z Local maximum

Local maximum

x

x

FIGURE 2 The tangent line or plane is

horizontal at a local extremum.

(A)

y (B)

732 •

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

More generally, (a1 , . . . , an ) is a critical point of f (x1 , . . . , xn ) if each partial derivative satisfies

•

fxj (a1 , . . . , an ) = 0 •

DEFINITION Critical Point A point P = (a, b) in the domain of f (x, y) is called a critical point if: •

or does not exist. Theorem 1 holds in any number of variables: Local extrema occur at critical points.

fx (a, b) = 0 or fx (a, b) does not exist, and fy (a, b) = 0 or fy (a, b) does not exist.

As in the single-variable case, we have THEOREM 1 Fermat’s Theorem If f (x, y) has a local minimum or maximum at P = (a, b), then (a, b) is a critical point of f (x, y). Proof If f (x, y) has a local minimum at P = (a, b), then f (x, y) ≥ f (a, b) for all (x, y) near (a, b). In particular, there exists r > 0 such that f (x, b) ≥ f (a, b) if |x − a| < r. In other words, g(x) = f (x, b) has a local minimum at x = a. By Fermat’s Theorem for functions of one variable, either g  (a) = 0 or g  (a) does not exist. Since g  (a) = fx (a, b), we conclude that either fx (a, b) = 0 or fx (a, b) does not exist. Similarly, fy (a, b) = 0 or fy (a, b) does not exist. Therefore, P = (a, b) is a critical point. The case of a local maximum is similar. Usually, we deal with functions whose partial derivatives exist. In this case, finding the critical points amounts to solving the simultaneous equations fx (x, y) = 0 and fy (x, y) = 0. E X A M P L E 1 Show that f (x, y) = 11x 2 − 2xy + 2y 2 + 3y has one critical point.

z

Use Figure 3 to determine whether it corresponds to a local minimum or maximum. Solution Set the partial derivatives equal to zero and solve: fx (x, y) = 22x − 2y = 0 fy (x, y) = −2x + 4y + 3 = 0 By the first equation, y = 11x. Substituting y = 11x in the second equation gives y

x FIGURE 3 Graph of f (x, y) = 11x 2 − 2xy + 2y 2 + 3y.

z

−2x + 4y + 3 = −2x + 4(11x) + 3 = 42x + 3 = 0   1 11 1 11 and y = − 14 . There is just one critical point, P = − 14 , − 14 . Figure 3 Thus x = − 14 shows that f (x, y) has a local minimum at P . It is not always possible to find the solutions exactly, but we can use a computer to find numerical approximations. EXAMPLE 2 Numerical Example Use a computer algebra system to approximate the critical points of

f (x, y) =

2x 2

x−y + 8y 2 + 3

Are they local minima or maxima? Refer to Figure 4. x

Solution We use a CAS to compute the partial derivatives and solve y

fx (x, y) =

−2x 2 + 8y 2 + 4xy + 3 =0 (2x 2 + 8y 2 + 3)2

fy (x, y) =

−2x 2 + 8y 2 − 16xy − 3 =0 (2x 2 + 8y 2 + 3)2

FIGURE 4 Graph of

f (x, y) =

x−y . 2x 2 + 8y 2 + 3

S E C T I O N 12.7

Optimization in Several Variables

733

To solve these equations, set the numerators equal to zero. Figure 4 suggests that f (x, y) has a local max with x > 0 and a local min with x < 0. The following Mathematica command searches for a solution near (1, 0): FindRoot[{-2xˆ2+8yˆ2+4xy+3 == 0, -2xˆ2+8yˆ2-16xy-3 == 0}, {{x,1},{y,0}}] The result is {x -> 1.095, y -> -0.274} Thus, (1.095, −0.274) is an approximate critical point where, by Figure 4, f takes on a local maximum.Asecond search near (−1, 0) yields (−1.095, 0.274), which approximates the critical point where f (x, y) takes on a local minimum. We know that in one variable, a function f (x) may have a point of inflection rather than a local extremum at a critical point. A similar phenomenon occurs in several variables. Each of the functions in Figure 5 has a critical point at (0, 0). However, the function in Figure 5(C) has a saddle point, which is neither a local minimum nor a local maximum. If you stand at the saddle point and begin walking, some directions take you uphill and other directions take you downhill.

z

y x

y

y x

(A) Local maximum

z

z

x (B) Local minimum

(C) Saddle

FIGURE 5

The discriminant is also referred to as the “Hessian determinant.”

As in the one-variable case, there is a Second Derivative Test for determining the type of a critical point (a, b) of a function f (x, y) in two variables. This test relies on the sign of the discriminant D = D(a, b), defined as follows: 2 D = D(a, b) = fxx (a, b)fyy (a, b) − fxy (a, b)

If D > 0, then fxx (a, b) and fyy (a, b) must have the same sign, so the sign of fyy (a, b) also determines whether f (a, b) is a local minimum or a local maximum.

THEOREM 2 Second Derivative Test Let P = (a, b) be a critical point of f (x, y). Assume that fxx , fyy , fxy are continuous near P . Then: (i) (ii) (iii) (iv)

If D If D If D If D

> 0 and fxx (a, b) > 0, then f (a, b) is a local minimum. > 0 and fxx (a, b) < 0, then f (a, b) is a local maximum. < 0, then f has a saddle point at (a, b). = 0, the test is inconclusive.

A proof of this theorem is discussed at the end of this section.

734

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

E X A M P L E 3 Applying the Second Derivative Test Find the critical points of

f (x, y) = (x 2 + y 2 )e−x and analyze them using the Second Derivative Test. Solution Step 1. Find the critical points. Set the partial derivatives equal to zero and solve: fx (x, y) = −(x 2 + y 2 )e−x + 2xe−x = (2x − x 2 − y 2 )e−x = 0 fy (x, y) = 2ye−x = 0

⇒

y=0

Substituting y = 0 in the first equation then gives

z

(2x − x 2 − y 2 )e−x = (2x − x 2 )e−x = 0

⇒

x = 0, 2

The critical points are (0, 0) and (2, 0) [Figure 6]. Step 2. Compute the second-order partials.

Saddle point

 ∂  (2x − x 2 − y 2 )e−x = (2 − 4x + x 2 + y 2 )e−x ∂x ∂ (2ye−x ) = 2e−x fyy (x, y) = ∂y

fxx (x, y) =

x Local minimum FIGURE 6 Graph of f (x, y) = (x 2 + y 2 )e−x .

fxy (x, y) = fyx (x, y) =

y

∂ (2ye−x ) = −2ye−x ∂x

Step 3. Apply the Second Derivative Test. Critical Point

fxx

fyy

fxy

Discriminant 2 D = fxx fyy − fxy

Type

(0, 0)

2

2

0

2(2) − 02 = 4

(2, 0)

−2e−2

2e−2

0

−2e−2 (2e−2 ) − 02 = −4e−4

Local minimum since D > 0 and fxx > 0 Saddle since D 0

f (x, y) = x + y

GRAPHICAL INSIGHT A graph can take on a variety of different shapes at a saddle point. The graph of h(x, y) in Figure 8 is called a “monkey saddle” (because a monkey can sit on this saddle with room for his tail in the back).

D1

y

1

1 x

(1, 1) y

Maximum of f (x, y) = x + y on D1 occurs at (1, 1)

1 D1 x 1 FIGURE 9

Global Extrema Often we are interested in finding the minimum or maximum value of a function f on a given domain D. These are called global or absolute extreme values. However, global extrema do not always exist. The function f (x, y) = x + y has a maximum value on the unit square D1 in Figure 9 (the max is f (1, 1) = 2), but it has no maximum value on the entire plane R 2 . To state conditions that guarantee the existence of global extrema, we need a few definitions. First, we say that a domain D is bounded if there is a number M > 0 such that D is contained in a disk of radius M centered at the origin. In other words, no point of D is more than a distance M from the origin [Figures 11(A) and 11(B)]. Next, a point P is called: •

Interior point

•

a

x

b Boundary point

a

b

FIGURE 10 Interior and boundary points of

an interval [a, b].

An interior point of D if D contains some open disk D(P , r) centered at P . A boundary point of D if every disk centered at P contains points in D and points not in D.

To understand the concept of interior and boundary points, think of the familiar case of an interval I = [a, b] in the real line R (Figure 10). Every point x in the open interval (a, b) is an interior point of I (because there exists a small open interval around x entirely contained in I ). The two endpoints a and b are boundary points (because every open interval containing a or b also contains points not in I ).

CONCEPTUAL INSIGHT

736

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

The interior of D is the set of all interior points, and the boundary of D is the set of all boundary points. In Figure 11(C), the boundary is the curve surrounding the domain. The interior consists of all points in the domain not lying on the boundary curve. A domain D is called closed if D contains all its boundary points (like a closed interval in R). A domain D is called open if every point of D is an interior point (like an open interval in R). The domain in Figure 11(A) is closed because the domain includes its boundary curve. In Figure 11(C), some boundary points are included and some are excluded, so the domain is neither open nor closed. y

y Interior point

y

Boundary point

x

x

(A) This domain is bounded and closed (contains all boundary points).

FIGURE 11 Domains in R 2 .

Boundary point not in D

(B) An unbounded domain (contains points arbitrarily far from the origin).

x (C) A nonclosed domain (contains some but not all boundary points).

In Section 4.2, we stated two basic results. First, a continuous function f (x) on a closed, bounded interval [a, b] takes on both a minimum and a maximum value on [a, b]. Second, these extreme values occur either at critical points in the interior (a, b) or at the endpoints. Analogous results are valid in several variables. THEOREM 3 Existence and Location of Global Extrema Let f (x, y) be a continuous function on a closed, bounded domain D in R 2 . Then: (i) f (x, y) takes on both a minimum and a maximum value on D. (ii) The extreme values occur either at critical points in the interior of D or at points on the boundary of D.

z f(x, y) = 2x + y − 3xy

E X A M P L E 5 Find the maximum value of f (x, y) = 2x + y − 3xy on the unit square D = {(x, y) : 0 ≤ x, y ≤ 1}.

Solution By Theorem 3, the maximum occurs either at a critical point or on the boundary of the square (Figure 12).

y

Step 1. Examine the critical points. Set the partial derivatives equal to zero and solve:

(1, 1, 0)

x

(1, 0, 0)

(0, 1)

Edge y = 1 P

FIGURE 12

(1, 1)

D

Edge x = 1 (1, 0)

x

⇒

y=

2 , 3

fy (x, y) = 1 − 3x = 0

  There is a unique critical point P = 13 , 23 and          1 2 1 2 1 2 2 f (P ) = f , =2 + −3 = 3 3 3 3 3 3 3

y

Edge x = 0

Edge y = 0

fx (x, y) = 2 − 3y = 0

⇒

x=

1 3

Step 2. Check the boundary. We do this by checking each of the four edges of the square separately. The bottom edge is described by y = 0, 0 ≤ x ≤ 1. On this edge, f (x, 0) = 2x, and the maximum value occurs at x = 1, where f (1, 0) = 2. Proceeding in a similar fashion with the other edges, we obtain

Optimization in Several Variables

S E C T I O N 12.7

Edge

Restriction of f (x, y) to Edge

Maximum of f (x, y) on Edge

Lower: y = 0, 0 ≤ x ≤ 1 Upper: y = 1, 0 ≤ x ≤ 1 Left: x = 0, 0 ≤ y ≤ 1 Right: x = 1, 0 ≤ y ≤ 1

f (x, 0) = 2x f (x, 1) = 1 − x f (0, y) = y f (1, y) = 2 − 2y

f (1, 0) = 2 f (0, 1) = 1 f (0, 1) = 1 f (1, 0) = 2

737

Step 3. Compare. The maximum of f on the boundary is f (1, 0) = 2. This is larger than the value f (P ) = 23 at the critical point, so the maximum of f on the unit square is 2. E X A M P L E 6 Box of Maximum Volume Find the maximum volume of a box inscribed in the tetrahedron bounded by the coordinate planes and the plane 13 x + y + z = 1.

Solution Step 1. Find a function to be maximized. Let P = (x, y, z) be the corner of the box lying on the front face of the tetrahedron (Figure 13). Then the box has sides of lengths x, y, z and volume V = xyz. Using 1 1 3 x + y + z = 1, or z = 1 − 3 x − y, we express V in terms of x and y:   1 1 V (x, y) = xyz = xy 1 − x − y = xy − x 2 y − xy 2 3 3

z C = (0, 0, 1) P = (x, y, z) z x

y

B = (0, 1, 0)

D

x

A = (3, 0, 0)

FIGURE 13 The shaded triangle is the

domain of V (x, y).

y

Our problem is to maximize V , but which domain D should we choose? We let D be the shaded triangle OAB in the xy-plane in Figure 13. Then the corner point P = (x, y, z) of each possible box lies above a point (x, y) in D. Because D is closed and bounded, the maximum occurs at a critical point inside D or on the boundary of D. Step 2. Examine the critical points. First, set the partial derivatives equal to zero and solve:   2 2 ∂V 2 = y − xy − y = y 1 − x − y = 0 ∂x 3 3   ∂V 1 1 = x − x 2 − 2xy = x 1 − x − 2y = 0 ∂y 3 3 If x = 0 or y = 0, then (x, y) lies on the boundary of D, so assume that x and y are both nonzero. Then the first equation gives us 2 1− x−y =0 3

⇒

2 y =1− x 3

The second equation yields

  1 2 1 1 − x − 2y = 1 − x − 2 1 − x = 0 3 3 3

⇒

x−1=0

⇒

x=1

  For x = 1, we have y = 1 − 23 x = 13 . Therefore, 1, 13 is a critical point, and V

  2  1 1 1 1 1 1 = (1) − (1)2 − (1) = 1, 3 3 3 3 3 9

Step 3. Check the boundary. We have V (x, y) = 0 for all points on the boundary of D (because the three edges of the boundary are defined by x = 0, y = 0, and 1 − 13 x − y = 0). Clearly, then, the maximum occurs at the critical point, and the maximum volume is 19 .

738

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

Proof of the Second Derivative Test The proof is based on “completing the square” for quadratic forms. A quadratic form is a function Q(h, k) = ah2 + 2bhk + ck 2 where a, b, c are constants (not all zero). The discriminant of Q is the quantity D = ac − b2 Some quadratic forms take on only positive values for (h, k) = (0, 0), and others take on both positive and negative values. According to the next theorem, the sign of the discriminant determines which of these two possibilities occurs.

To illustrate Theorem 4, consider

Q(h, k) = h2 + 2hk + 2k 2

THEOREM 4 With Q(h, k) and D as above:

It has a positive discriminant

D = (1)(2) − 1 = 1 We can see directly that Q(h, k) takes on only positive values for (h, k)  = (0, 0) by writing Q(h, k) as

(i) If D > 0 and a > 0, then Q(h, k) > 0 for (h, k)  = (0, 0). (ii) If D > 0 and a < 0, then Q(h, k) < 0 for (h, k)  = (0, 0). (iii) If D < 0, then Q(h, k) takes on both positive and negative values. Proof Assume first that a  = 0 and rewrite Q(h, k) by “completing the square”:

Q(h, k) = (h + k)2 + k 2



b Q(h, k) = ah + 2bhk + ck = a h + k a  2 b D = a h + k + k2 a a 2



b2 + c− a

 k2

h, k (a + th, a + tk) r x

Now assume that f (x, y) has a critical point at P = (a, b). We shall analyze f by considering the restriction of f (x, y) to the line (Figure 14) through P = (a, b) in the direction of a unit vector h, k: F (t) = f (a + th, b + tk)

FIGURE 14 Line through P in the direction

of h, k.

1

If D > 0 and a > 0, then D/a > 0 and both terms in Eq. (1) are nonnegative. Furthermore, if Q(h, k) = 0, then each term in Eq. (1) must equal zero. Thus k = 0 and h + ab k = 0, and then, necessarily, h = 0. This shows that Q(h, k) > 0 if (h, k) = 0, and (i) is proved. Part (ii) follows similarly. To prove (iii), note that if a  = 0 and D < 0, then the coefficients of the squared terms in Eq. (1) have opposite signs and Q(h, k) takes on both positive and negative values. Finally, if a = 0 and D < 0, then Q(h, k) = 2bhk + ck 2 with b = 0. In this case, Q(h, k) again takes on both positive and negative values.

y

(a, b) P

2

2

Then F (0) = f (a, b). By the Chain Rule, F  (t) = fx (a + th, b + tk)h + fy (a + th, b + tk)k Because P is a critical point, we have fx (a, b) = fy (a, b) = 0, and therefore, F  (0) = fx (a, b)h + fy (a, b)k = 0 Thus t = 0 is a critical point of F (t).

S E C T I O N 12.7

Optimization in Several Variables

739

Now apply the Chain Rule again:

d fx (a + th, b + tk)h + fy (a + th, b + tk)k dt

= fxx (a + th, b + tk)h2 + fxy (a + th, b + tk)hk

+ fyx (a + th, b + tk)kh + fyy (a + th, b + tk)k 2

F  (t) =

= fxx (a + th, b + tk)h2 + 2fxy (a + th, b + tk)hk + fyy (a + th, b + tk)k 2 2 We see that F  (t) is the value at (h, k) of a quadratic form whose discriminant is equal to D(a + th, b + tk). Here, we set D(r, s) = fxx (r, s)fyy (r, s) − fxy (r, s)2 Note that the discriminant of f (x, y) at the critical point P = (a, b) is D = D(a, b). Case 1: D(a, b) > 0 and fxx (a, b) > 0. We must prove that f (a, b) is a local minimum. Consider a small disk of radius r around P (Figure 14). Because the second derivatives are continuous near P , we can choose r > 0 so that for every unit vector h, k, D(a + th, b + tk) > 0

for |t| < r

fxx (a + th, b + tk) > 0

for |t| < r

Then F  (t) is positive for |t| < r by Theorem 4(i). This tells us that F (t) is concave up, and hence F (0) < F (t) if 0 < |t| < |r| (see Exercise 64 in Section 4.4). Because F (0) = f (a, b), we may conclude that f (a, b) is the minimum value of f along each segment of radius r through (a, b). Therefore, f (a, b) is a local minimum value of f as claimed. The case that D(a, b) > 0 and fxx (a, b) < 0 is similar. Case 2: D(a, b) < 0. For t = 0, Eq. (2) yields F  (0) = fxx (a, b)h2 + 2fxy (a, b)hk + fyy (a, b)k 2 Since D(a, b) < 0, this quadratic form takes on both positive and negative values by Theorem 4(iii). Choose h, k for which F  (0) > 0. By the Second Derivative Test in one variable, F (0) is a local minimum of F (t), and hence, there is a value r > 0 such that F (0) < F (t) for all 0 < |t| < r. But we can also choose h, k so that F  (0) < 0, in which case F (0) > F (t) for 0 < |t| < r for some r > 0. Because F (0) = f (a, b), we conclude that f (a, b) is a local min in some directions and a local max in other directions. Therefore, f has a saddle point at P = (a, b).

12.7 SUMMARY •

We say that P = (a, b) is a critical point of f (x, y) if – fx (a, b) = 0 or fx (a, b) does not exist, and – fy (a, b) = 0 or fy (a, b) does not exist.

In n-variables, P = (a1 , . . . , an ) is a critical point of f (x1 , . . . , xn ) if each partial derivative fxj (a1 , . . . , an ) either is zero or does not exist.

740

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES • •

The local minimum or maximum values of f occur at critical points. The discriminant of f (x, y) at P = (a, b) is the quantity 2 (a, b) D(a, b) = fxx (a, b)fyy (a, b) − fxy

•

Second Derivative Test: If P = (a, b) is a critical point of f (x, y), then D(a, b) > 0, fxx (a, b) > 0

⇒

f (a, b) is a local minimum

D(a, b) > 0, fxx (a, b) < 0

⇒

f (a, b) is a local maximum

D(a, b) < 0

⇒

saddle point

D(a, b) = 0

⇒

test inconclusive

A point P is an interior point of a domain D if D contains some open disk D(P , r) centered at P . A point P is a boundary point of D if every open disk D(P , r) contains points in D and points not in D. The interior of D is the set of all interior points, and the boundary is the set of all boundary points. A domain is closed if it contains all of its boundary points and open if it is equal to its interior. • Existence and location of global extrema: If f is continuous and D is closed and bounded, then •

– f takes on both a minimum and a maximum value on D. – The extreme values occur either at critical points in the interior of D or at points on the boundary of D. To determine the extreme values, first find the critical points in the interior of D. Then compare the values of f at the critical points with the minimum and maximum values of f on the boundary.

12.7 EXERCISES Preliminary Questions 1. The functions f (x, y) = x 2 + y 2 and g(x, y) = x 2 − y 2 both have a critical point at (0, 0). How is the behavior of the two functions at the critical point different? 2. Identify the points indicated in the contour maps as local minima, local maxima, saddle points, or neither (Figure 15).

−3

1 −1

3

0 −1

−3

1 3 0

−3 −1

−10

6 −2

0

0 FIGURE 15

2

0

(a) If D is closed and bounded, then f takes on a maximum value on D. (b) If D is neither closed nor bounded, then f does not take on a maximum value of D. (c) f (x, y) need not have a maximum value on the domain D defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.

10

−6

1 3

3. Let f (x, y) be a continuous function on a domain D in R 2 . Determine which of the following statements are true:

(d) A continuous function takes on neither a minimum nor a maximum value on the open quadrant {(x, y) : x > 0, y > 0}

Optimization in Several Variables

S E C T I O N 12.7

Exercises

y

1. Let P = (a, b) be a critical point of f (x, y) = x 2 + y 4 − 4xy. (a) First use fx (x, y) = 0 to show that a = 2b. Then use fy (x, y) = 0 √ √ √ √ to show that P = (0, 0), (2 2, 2), or (−2 2, − 2). (b) Referring to Figure 16, determine the local minima and saddle points of f (x, y) and find the absolute minimum value of f (x, y).

741

1

−0.3 −0.2 −0.1

0

0

0.1 0.2 0.3

−0.1 −0.2

z −1 −1

0

1

x

FIGURE 18 Contour map of f (x, y) = 8y 4 + x 2 + xy − 3y 2 − y 3 .

4. Use the contour map in Figure 19 to determine whether the critical points A, B, C, D are local minima, local maxima, or saddle points. x

y

y 2

FIGURE 16

1

A

−3 −2 −1 0 0

0

2. Find the critical points of the functions f (x, y) = x 2 + 2y 2 − 4y + 6x,

Use the Second Derivative Test to determine the local minimum, local maximum, and saddle points. Match f (x, y) and g(x, y) with their graphs in Figure 17.

3

C

−2 −2

0

2

x

FIGURE 19

y(y − 2x + 1) = 0,

x x y

(A)

2

5. Let f (x, y) = y 2 x − yx 2 + xy. (a) Show that the critical points (x, y) satisfy the equations

z

z

1

−1

D

g(x, y) = x 2 − 12xy + y

B

y (B)

FIGURE 17

x(2y − x + 1) = 0

(b) Show that f has three critical points. (c) Use the second derivative to determine the nature of the critical points.  6. Show that f (x, y) = x 2 + y 2 has one critical point P and that f is nondifferentiable at P . Does f take on a minimum, maximum, or saddle point at P ? In Exercises 7–23, find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). 7. f (x, y) = x 2 + y 2 − xy + x

3. Find the critical points of f (x, y) = 8y 4 + x 2 + xy − 3y 2 − y 3 Use the contour map in Figure 18 to determine their nature (local minimum, local maximum, or saddle point).

9. f (x, y) = x 3 + 2xy − 2y 2 − 10x 10. f (x, y) = x 3 y + 12x 2 − 8y 11. f (x, y) = 4x − 3x 3 − 2xy 2 12. f (x, y) = x 3 + y 4 − 6x − 2y 2

8. f (x, y) = x 3 − xy + y 3

742

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

13. f (x, y) = x 4 + y 4 − 4xy

2 2 14. f (x, y) = ex −y +4y

2 2 15. f (x, y) = xye−x −y

16. f (x, y) = ex − xey

17. f (x, y) = sin(x + y) − cos x

18. f (x, y) = x ln(x + y)

28. Which of the following domains are closed and which are bounded? (a) {(x, y) ∈ R 2 : x 2 + y 2 ≤ 1} (b) {(x, y) ∈ R 2 : x 2 + y 2 < 1} (c) {(x, y) ∈ R 2 : x ≥ 0}

19. f (x, y) = ln x + 2 ln y − x − 4y

(d) {(x, y) ∈ R 2 : x > 0, y > 0}

20. f (x, y) = (x + y) ln(x 2 + y 2 )

(e) {(x, y) ∈ R 2 : 1 ≤ x ≤ 4, 5 ≤ y ≤ 10} 2 2 22. f (x, y) = (x − y)ex −y

21. f (x, y) = x − y 2 − ln(x + y) 23. f (x, y) = (x + 3y)ey−x

2

24. Show that f (x, y) = x 2 has infinitely many critical points (as a function of two variables) and that the Second Derivative Test fails for all of them. What is the minimum value of f ? Does f (x, y) have any local maxima? 25. Prove that the function f (x, y) = 13 x 3 + 23 y 3/2 − xy satisfies f (x, y) ≥ 0 for x ≥ 0 and y ≥ 0. (a) First, verify that the set of critical points of f is the parabola y = x 2 and that the Second Derivative Test fails for these points. (b) Show that for fixed b, the function g(x) = f (x, b) is concave up for x > 0 with a critical point at x = b1/2 . (c) Conclude that f (a, b) ≥ f (b1/2 , b) = 0 for all a, b ≥ 0. Let f (x, y) = (x 2 + y 2 )e−x −y . 26. (a) Where does f take on its minimum value? Do not use calculus to answer this question. (b) Verify that the set of critical points of f consists of the origin (0, 0) and the unit circle x 2 + y 2 = 1. (c) The Second Derivative Test fails for points on the unit circle (this can be checked by some lengthy algebra). Prove, however, that f takes on its maximum value on the unit circle by analyzing the function g(t) = te−t for t > 0. 2

2

Use a computer algebra system to find a numerical ap27. proximation to the critical point of 2

f (x, y) = (1 − x + x 2 )ey + (1 − y + y 2 )ex

2

Apply the Second Derivative Test to confirm that it corresponds to a local minimum as in Figure 20.

(f) {(x, y) ∈ R 2 : x > 0, x 2 + y 2 ≤ 10} In Exercises 29–32, determine the global extreme values of the function on the given set without using calculus. 29. f (x, y) = x + y,

0 ≤ x ≤ 1,

30. f (x, y) = 2x − y,

0≤y≤1

0 ≤ x ≤ 1,

31. f (x, y) = (x 2 + y 2 + 1)−1 , 2 2 32. f (x, y) = e−x −y ,

0≤y≤3

0 ≤ x ≤ 3,

0≤y≤5

x2 + y2 ≤ 1

33. Assumptions Matter Show that f (x, y) = xy does not have a global minimum or a global maximum on the domain D = {(x, y) : 0 < x < 1, 0 < y < 1} Explain why this does not contradict Theorem 3. 34. Find a continuous function that does not have a global maximum on the domain D = {(x, y) : x + y ≥ 0, x + y ≤ 1}. Explain why this does not contradict Theorem 3. 35. Find the maximum of f (x, y) = x + y − x 2 − y 2 − xy on the square, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 (Figure 21). (a) First, locate the critical point of f in the square, and evaluate f at this point. (b) On the bottom edge of the square, y = 0 and f (x, 0) = x − x 2 . Find the extreme values of f on the bottom edge. (c) Find the extreme values of f on the remaining edges. (d) Find the largest among the values computed in (a), (b), and (c).

z

y

f (x, 2) = −2 − x − x 2 Edge y = 2

2

Edge x = 2 f(2, y) = −2 − y − y 2

Edge x = 0 f (0, y) = y − y 2

2

x

Edge y = 0 f (x, 0) = x − x 2

x y

2 2 FIGURE 20 Plot of f (x, y) = (1 − x + x 2 )ey + (1 − y + y 2 )ex .

FIGURE 21 The function f (x, y) = x + y − x 2 − y 2 − xy on the

boundary segments of the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.

Optimization in Several Variables

S E C T I O N 12.7

36. Find the maximum of f (x, y) = y 2 + xy − x 2 on the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.

743

48. Show that the rectangular box (including the top and bottom) with fixed volume V = 27 m3 and smallest possible surface area is a cube (Figure 23).

In Exercises 37–43, determine the global extreme values of the function on the given domain. 37. f (x, y) = x 3 − 2y,

0 ≤ x ≤ 1,

0≤y≤1

38. f (x, y) = 5x − 3y,

y ≥ x − 2,

y ≥ −x − 2, y ≤ 3

39. f (x, y) = x 2 + 2y 2 ,

0 ≤ x ≤ 1,

z

FIGURE 23 Rectangular box with sides x, y, z.

40. f (x, y) = x 3 + x 2 y + 2y 2 ,

x, y ≥ 0,

x+y ≤1

41. f (x, y) = x 3 + y 3 − 3xy,

0 ≤ x ≤ 1,

0≤y≤1

42. f (x, y) = x 2 + y 2 − 2x − 4y,

x ≥ 0,

2 2 43. f (x, y) = (4y 2 − x 2 )e−x −y ,

x2 + y2 ≤ 2

y

x

0≤y≤1

0 ≤ y ≤ 3, y ≥ x

44. Find the maximum volume of a box inscribed in the tetrahedron bounded by the coordinate planes and the plane

49. Consider a rectangular box B that has a bottom and sides but no top and has minimal surface area among all boxes with fixed volume V . (a) Do you think B is a cube as in the solution to Exercise 48? If not, how would its shape differ from a cube? (b) Find the dimensions of B and compare with your response to (a). 50. Given n data points (x1 , y1 ), . . . , (xn , yn ), the linear leastsquares fit is the linear function

1 1 x+ y+ z=1 2 3

f (x) = mx + b that minimizes the sum of the squares (Figure 24):

45. Find the maximum volume of the largest box of the type shown in Figure 22, with one corner at the origin and the opposite corner at a point P = (x, y, z) on the paraboloid z=1−

x2 4

−

y2 9

n #

E(m, b) =

(yj − f (xj ))2

j =1

Show that the minimum value of E occurs for m and b satisfying the two equations ⎛ ⎞ n n # # m⎝ xj ⎠ + bn = yj

with x, y, z ≥ 0

z

j =1

1

m

n # j =1

xj2 + b

j =1 n #

xj =

j =1

n #

xj yj

j =1

y

P

(xn , yn) y

(x2, y2) (x1, y1)

x

y = mx + b

(xj , yj)

FIGURE 22

46. Find the point on the plane z=x+y+1 closest to the point P = (1, 0, 0). Hint: Minimize the square of the distance. 47. Show that the sum of the squares of the distances from a point P = (c, d) to n fixed points (a1 , b1 ), . . . ,(an , bn ) is minimized when c is the average of the x-coordinates ai and d is the average of the y-coordinates bi .

x FIGURE 24 The linear least-squares fit minimizes the sum of the

squares of the vertical distances from the data points to the line. 51. The power (in microwatts) of a laser is measured as a function of current (in milliamps). Find the linear least-squares fit (Exercise 50) for the data points. Current (mA)

1.0

1.1

1.2

1.3

1.4

1.5

Laser power (μW)

0.52

0.56

0.82

0.78

1.23

1.50

744

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

52. Let A = (a, b) be a fixed point in the plane, and let fA (P ) be the distance from A to the point P = (x, y). For P = A, let eAP be the unit vector pointing from A to P (Figure 25): −→ AP eAP = −→ AP 

y Distance fA (x, y)

e AP P = (x, y)

A = (a, b)

Show that x

∇fA (P ) = eAP Note that we can derive this result without calculation: Because ∇fA (P ) points in the direction of maximal increase, it must point directly away from A at P , and because the distance fA (x, y) increases at a rate of one as you move away from A along the line through A and P , ∇fA (P ) must be a unit vector.

FIGURE 25 The distance from A to P increases most rapidly in the

direction eAP .

Further Insights and Challenges 53. In this exercise, we prove that for all x, y ≥ 0:

the plane, find the point P = (x, y) that minimizes the sum of the distances

1 α 1 x + x β ≥ xy α β

f (x, y) = AP + BP + CP

where α ≥ 1 and β ≥ 1 are numbers such that α −1 + β −1 = 1. To do this, we prove that the function f (x, y) = α −1 x α + β −1 y β − xy satisfies f (x, y) ≥ 0 for all x, y ≥ 0. (a) Show that the set of critical points of f (x, y) is the curve y = x α−1 (Figure 26). Note that this curve can also be described as x = y β−1 . What is the value of f (x, y) at points on this curve? (b) Verify that the Second Derivative Test fails. Show, however, that for fixed b > 0, the function g(x) = f (x, b) is concave up with a critical point at x = bβ−1 . (c) Conclude that for all x > 0, f (x, b) ≥ f (bβ−1 , b) = 0. y

b

y = xa − 1 (b b − 1, b ) inc

Let e, f , g be the unit vectors pointing from P to the points A, B, C as in Figure 27. (a) Use Exercise 52 to show that the condition ∇f (P ) = 0 is equivalent to e+f +g =0

3

(b) Show that f (x, y) is differentiable except at points A, B, C. Conclude that the minimum of f (x, y) occurs either at a point P satisfying Eq. (3) or at one of the points A, B, or C. (c) Prove that Eq. (3) holds if and only if P is the Fermat point, defined as the point P for which the angles between the segments AP , BP , CP are all 120◦ (Figure 27). (d) Show that the Fermat point does not exist if one of the angles in ABC is > 120◦ . Where does the minimum occur in this case? B A

inc

Critical points of f (x, y)

x

FIGURE 26 The critical points of f (x, y) = α −1 x α + β −1 y β − xy form a curve y = x α−1 .

The following problem was posed by Pierre de Fermat: 54. Given three points A = (a1 , a2 ), B = (b1 , b2 ), and C = (c1 , c2 ) in

f

e g

B

P

C (A) P is the Fermat point (the angles between e, f, and g are all 120°).

140° A (B) Fermat point does not exist.

FIGURE 27

C

S E C T I O N 12.8

Lagrange Multipliers: Optimizing with a Constraint

745

12.8 Lagrange Multipliers: Optimizing with a Constraint y

2 1

Some optimization problems involve finding the extreme values of a function f (x, y) subject to a constraint g(x, y) = 0. Suppose that we want to find the point on the line 2x + 3y = 6 closest to the origin (Figure 1). The distance from (x, y) to the origin is f (x, y) = x 2 + y 2 , so our problem is

Constraint g(x, y) = 2x + 3y − 6 = 0 Point on the line closest to the origin

P

Minimize f (x, y) = 1

2

FIGURE 1 Finding the minimum of

f (x, y) =



3

x

 x2 + y2

g(x, y) = 2x + 3y − 6 = 0

subject to

We are not seeking the minimum value of f (x, y) (which is 0), but rather the minimum among all points (x, y) that lie on the line. The method of Lagrange multipliers is a general procedure for solving optimization problems with a constraint. Here is a description of the main idea. Imagine standing at point Q in Figure 2(A). We want to increase the value of f while remaining on the constraint curve. The gradient vector ∇fQ points in the direction of maximum increase, but we cannot move in the gradient direction because that would take us off the constraint curve. However, the gradient points to the right, and so we can still increase f somewhat by moving to the right along the constraint curve. We keep moving to the right until we arrive at the point P , where ∇fP is orthogonal to the constraint curve [Figure 2(B)]. Once at P , we cannot increase f further by moving either to the right or to the left along the constraint curve. Thus f (P ) is a local maximum subject to the constraint. Now, the vector ∇gP is also orthogonal to the constraint curve, so ∇fP and ∇gP must point in the same or opposite directions. In other words, ∇fP = λ∇gP for some scalar λ (called a Lagrange multiplier). Graphically, this means that a local max subject to the constraint occurs at points P where the level curves of f and g are tangent. GRAPHICAL INSIGHT

x2 + y2

on the line 2x + 3y = 6.

y

y ∇f Q Q

P

Tangent line at P ∇gP

Level curves of f(x, y)

∇f P

4 3 2 1

P

4 3 2 1

Constraint curve g(x, y) = 0 x (A) f increases as we move to the right along the constraint curve.

x (B) The local maximum of f on the constraint curve occurs where ∇f P and ∇gP are parallel.

FIGURE 2

THEOREM 1 Lagrange Multipliers Assume that f (x, y) and g(x, y) are differentiable functions. If f (x, y) has a local minimum or a local maximum on the constraint curve g(x, y) = 0 at P = (a, b), and if ∇gP  = 0, then there is a scalar λ such that ∇fP = λ∇gP

1

746

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

In Theorem 1, the assumption ∇gP  = 0 guarantees (by the Implicit Function Theorem of advanced calculus) that we can parametrize the curve g(x, y) = 0 near P by a path c such that c(0) = P and c (0) = 0.

Proof Let c(t) be a parametrization of the constraint curve g(x, y) = 0 near P , chosen so that c(0) = P and c (0)  = 0. Then f (c(0)) = f (P ), and by assumption, f (c(t)) has a local min or max at t = 0. Thus, t = 0 is a critical point of f (c(t)) and   d f (c(t)) = ∇fP · c (0) = 0 dt t=0    Chain Rule

This shows that ∇fP is orthogonal to the tangent vector c (0) to the curve g(x, y) = 0. The gradient ∇gP is also orthogonal to c (0) (because ∇gP is orthogonal to the level curve g(x, y) = 0 at P ). We conclude that ∇fP and ∇gP are parallel, and hence ∇fP is a multiple of ∇gP as claimed. REMINDER Eq. (1) states that if a local min or max of f (x, y) subject to a constraint g(x, y) = 0 occurs at P = (a, b), then

We refer to Eq. (1) as the Lagrange condition. When we write this condition in terms of components, we obtain the Lagrange equations:

∇fP = λ∇gP

fy (a, b) = λgy (a, b)

provided that ∇gP  = 0.

fx (a, b) = λgx (a, b) A point P = (a, b) satisfying these equations is called a critical point for the optimization problem with constraint and f (a, b) is called a critical value. E X A M P L E 1 Find the extreme values of f (x, y) = 2x + 5y on the ellipse

x 2 4

+

y 2 3

=1

Solution Step 1. Write out the Lagrange equations. The constraint curve is g(x, y) = 0, where g(x, y) = (x/4)2 + (y/3)2 − 1. We have   x 2y ∇f = 2, 5 , ∇g = , 8 9 The Lagrange equations ∇fP = λ∇gP are:   x 2y λx 2, 5 = λ , ⇒ 2= , 8 9 8

5=

λ(2y) 9

2

Step 2. Solve for λ in terms of x and y. Eq. (2) gives us two equations for λ: λ=

16 , x

λ=

45 2y

3

To justify dividing by x and y, note that x and y must be nonzero, because x = 0 or y = 0 would violate Eq. (2). Step 3. Solve for x and y using the constraint. 16 45 45 The two expressions for λ must be equal, so we obtain = or y = x. Now x 2y 32 substitute this in the constraint equation and solve for x:  2 45 x 2 32 x + =1 4 3     1 289 225 2 2 x + =x =1 16 1024 1024

S E C T I O N 12.8

Level curve of f(x, y) = 2x + 5y Constraint curve g(x, y) = 0

 Thus x = ± 1024 = ± 32 17 , and since y =   32 289 45 Q = − 17 , − 17 .

y fP gP

3

45x 32 , the critical points are P

=

 32

45 17 , 17

747



and

Step 4. Calculate the critical values.

P



17 fQ

Lagrange Multipliers: Optimizing with a Constraint

x

4 0

Q gQ −17

FIGURE 3 The min and max occur where a

level curve of f is tangent to the constraint curve x 2 y 2 g(x, y) = + −1=0 4 3

f (P ) = f

32 45 , 17 17



 =2

32 17



 +5

45 17

 = 17

and f (Q) = −17. We conclude that the maximum of f (x, y) on the ellipse is 17 and the minimum is −17 (Figure 3). Assumptions Matter According to Theorem 3 in Section 12.7, a continuous function on a closed, bounded domain takes on extreme values. This tells us that if the constraint curve is bounded (as in the previous example, where the constraint curve is an ellipse), then every continuous function f (x, y) takes on both a minimum and a maximum value subject to the constraint. Be aware, however, that extreme values need not exist if the constraint curve is not bounded. For example, the constraint x − y = 0 is an unbounded line. The function f (x, y) = x has neither a minimum nor a maximum subject to x − y = 0 because P = (a, a) satisfies the constraint, yet f (a, a) = a can be arbitrarily large or small. E X A M P L E 2 Cobb–Douglas Production Function By investing x units of labor and y units of capital, a low-end watch manufacturer can produce P (x, y) = 50x 0.4 y 0.6 watches. (See Figure 4.) Find the maximum number of watches that can be produced on a budget of $20,000 if labor costs $100 per unit and capital costs $200 per unit.

Solution The total cost of x units of labor and y units of capital is 100x + 200y. Our task is to maximize the function P (x, y) = 50x 0.4 y 0.6 subject to the following budget constraint (Figure 5): FIGURE 4 Economist Paul Douglas,

working with mathematician Charles Cobb, arrived at the production functions P (x, y) = Cx a y b by fitting data gathered on the relationships between labor, capital, and output in an industrial economy. Douglas was a professor at the University of Chicago and also served as U.S. senator from Illinois from 1949 to 1967.

y (capital) Increasing output

120

A

60

40

80

Budget constraint

120

x (labor)

FIGURE 5 Contour plot of the

Cobb–Douglas production function P (x, y) = 50x 0.4 y 0.6 . The level curves of a production function are called isoquants.

g(x, y) = 100x + 200y − 20,000 = 0

4

Step 1. Write out the Lagrange equations. Px (x, y) = λgx (x, y) :

20x −0.6 y 0.6 = 100λ

Py (x, y) = λgy (x, y) :

30x 0.4 y −0.4 = 200λ

Step 2. Solve for λ in terms of x and y. These equations yield two expressions for λ that must be equal: 3 y −0.4 1 y 0.6 = λ= 5 x 20 x

5

Step 3. Solve for x and y using the constraint. Multiply Eq. (5) by 5(y/x)0.4 to obtain y/x = 15/20, or y = 34 x. Then substitute in Eq. (4):   3 x = 20,000 ⇒ 250x = 20,000 100x + 200y = 100x + 200 4 3 We obtain x = 20,000 250 = 80 and y = 4 x = 60. The critical point is A = (80, 60). Step 4. Calculate the critical values. Since P (x, y) is increasing as a function of x and y, ∇P points to the northeast, and it is clear that P (x, y) takes on a maximum value at A (Figure 5). The maximum is P (80, 60) = 50(80)0.4 (60)0.6 = 3365.87, or roughly 3365 watches, with a cost per watch of 20,000 3365 or about $5.94.

748

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

In an ordinary optimization problem without constraint, the global maximum value is the height of the highest point on the surface z = f (x, y) (point Q in Figure 6). When a constraint is given, we restrict our attention to the curve on the surface lying above the constraint curve g(x, y) = 0 . The maximum value subject to the constraint is the height of the highest point on this curve. Figure 6(B) shows the optimization problem solved in Example 1.

GRAPHICAL INSIGHT

z

Global maximum Q Maximum on the constraint curve

x

z

z = f (x, y)

f(x, y) = 2x + 5y

( x4 )2 + ( 3y )2 = 1

P

Constrained max occurs here g(x, y) = 0

P

y

x

y

(A)

(B)

FIGURE 6

The method of Lagrange multipliers is valid in any number of variables. In the next example, we consider a problem in three variables. E X A M P L E 3 Lagrange Multipliers in Three Variables Find the point on the plane x y z + + = 1 closest to the origin in R 3 . 2 4 4  Solution Our task is to minimize the distance d = x 2 + y 2 + z2 subject to the conx y z straint + + = 1. But finding the minimum distance d is the same as finding the 2 4 4 minimum square of the distance d 2 , so our problem can be stated:

Minimize f (x, y, z) = x 2 + y 2 + z2 The Lagrange condition is

subject to

 1 1 1 2x, 2y, 2z = λ , ,    2 4 4    ∇f

λ = 4x = 8y = 8z

⇒

z=y=

x 2

Substituting in the constraint equation, we obtain

P (0, 4, 0) y

FIGURE 7 Point P closest to the origin on

the plane.



This yields

(0, 0, 4)

x

y z x + + −1=0 2 4 4

∇g

z

(2, 0, 0)

g(x, y, z) =

x y z 2z z z 3z + + = + + = =1 2 4 4 2 4 4 2

⇒

z=

2 3

Thus, x = 2z = 43 and y = z = 23 . This critical point must correspond to the minimum of f (because f has no maximum on the constraint plane). Hence, the point on the plane  closest to the origin is P = 43 , 23 , 23 (Figure 7).

Lagrange Multipliers: Optimizing with a Constraint

S E C T I O N 12.8

749

The method of Lagrange multipliers can be used when there is more than one constraint equation, but we must add another multiplier for each additional constraint. For example, if the problem is to minimize f (x, y, z) subject to constraints g(x, y, z) = 0 and h(x, y, z) = 0, then the Lagrange condition is ∇f = λ∇g + μ∇h

The intersection of a sphere with a plane through its center is called a great circle.

E X A M P L E 4 Lagrange Multipliers with Multiple Constraints The intersection of the plane x + 12 y + 13 z = 0 with the unit sphere x 2 + y 2 + z2 = 1 is a great circle (Figure 8). Find the point on this great circle with the largest x coordinate.

Solution Our task is to maximize the function f (x, y, z) = x subject to the two constraint equations

z

x+

y z + =1 2 3

1 1 g(x, y, z) = x + y + z = 0, 2 3

y

h(x, y, z) = x 2 + y 2 + z2 − 1 = 0

The Lagrange condition is x2 + y2 + z2 = 1

∇f = λ∇g + μ∇h   1 1 1, 0, 0 = λ 1, , + μ 2x, 2y, 2z 2 3

x Q

  Note that μ cannot be zero. The Lagrange condition would become 1, 0, 0 = λ 1, 12 , 13 , and this equation is not satisfed for any value of λ. Now, the Lagrange condition gives us three equations:

FIGURE 8 The plane intersects the sphere in

a great circle. Q is the point on this great circle with the largest x-coordinate.

λ + 2μx = 1,

1 λ + 2μy = 0, 2

1 λ + 2μz = 0 3

The last two equations yield λ = −4μy and λ = −6μz. Because μ = 0, −4μy = −6μz

⇒

y=

3 z 2

Now use this relation in the first constraint equation: 1 1 1 x+ y+ z=x+ 2 3 2



 3 1 z + z=0 2 3

⇒

x=−

13 z 12

Finally, we can substitute in the second constraint equation:   2  13 2 3 x 2 + y 2 + z2 − 1 = − z + z + z2 − 1 = 0 12 2 to obtain

637 2 144 z

= 1 or z = ±

12 √ . 7 13

3 Since x = − 13 12 z and y = 2 z, the critical points are

 √  13 18 12 P = − , √ , √ , 7 7 13 7 13

√  18 13 12 ,− √ ,− √ Q= 7 7 13 7 13

The critical point with the largest x-coordinate (the maximum value of f (x, y, z)) is Q √ with x-coordinate 713 ≈ 0.515.

750

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

12.8 SUMMARY Method of Lagrange multipliers: The local extreme values of f (x, y) subject to a constraint g(x, y) = 0 occur at points P (called critical points) satisfying the Lagrange condition ∇fP = λ∇gP . This condition is equivalent to the Lagrange equations

•

fx (x, y) = λgx (x, y),

fy (x, y) = λgy (x, y)

• If the constraint curve g(x, y) = 0 is bounded [e.g., if g(x, y) = 0 is a circle or ellipse], then global minimum and maximum values of f subject to the constraint exist. • Lagrange condition for a function of three variables f (x, y, z) subject to two constraints g(x, y, z) = 0 and h(x, y, z) = 0:

∇f = λ∇g + μ∇h

12.8 EXERCISES Preliminary Questions 1. Suppose that the maximum of f (x, y) subject to the constraint g(x, y) = 0 occurs at a point P = (a, b) such that ∇fP  = 0. Which of the following statements is true? (a) ∇fP is tangent to g(x, y) = 0 at P . (b) ∇fP is orthogonal to g(x, y) = 0 at P . 2. Figure 9 shows a constraint g(x, y) = 0 and the level curves of a function f . In each case, determine whether f has a local minimum, a local maximum, or neither at the labeled point. ∇f

A

1 4 3 2 1

g(x, y) = 0

2

3

(a) Identify the points where ∇f = λ∇g for some scalar λ. (b) Identify the minimum and maximum values of f (x, y) subject to g(x, y) = 0.

−6 −2

y

2

6

g (x, y) = 0

4

∇f

x

B

g(x, y) = 0

FIGURE 9

3. On the contour map in Figure 10:

6

2 −2 −6

Contour plot of f (x, y) (contour interval 2) FIGURE 10 Contour map of f (x, y); contour interval 2.

Exercises In this exercise set, use the method of Lagrange multipliers unless otherwise stated. 1. Find the extreme values of the function f (x, y) = 2x + 4y subject to the constraint g(x, y) = x 2 + y 2 − 5 = 0. (a) Show that the Lagrange equation ∇f = λ∇g gives λx = 1 and λy = 2. (b) Show that these equations imply λ = 0 and y = 2x. (c) Use the constraint equation to determine the possible critical points (x, y).

(d) Evaluate f (x, y) at the critical points and determine the minimum and maximum values. 2. Find the extreme values of f (x, y) = x 2 + 2y 2 subject to the constraint g(x, y) = 4x − 6y = 25. (a) Show that the Lagrange equations yield 2x = 4λ, 4y = −6λ. (b) Show that if x = 0 or y = 0, then the Lagrange equations give x = y = 0. Since (0, 0) does not satisfy the constraint, you may assume that x and y are nonzero. (c) Use the Lagrange equations to show that y = − 34 x.

Lagrange Multipliers: Optimizing with a Constraint

S E C T I O N 12.8

y

(d) Substitute in the constraint equation to show that there is a unique critical point P . (e) Does P correspond to a minimum or maximum value of f ? Refer to Figure 11 to justify your answer. Hint: Do the values of f (x, y) increase or decrease as (x, y) moves away from P along the line g(x, y) = 0?

2 P 0 −1

y

24 36 6 12

0

g(x, y) = 0

−4 0

−4

4

8

x

FIGURE 11 Level curves of f (x, y) = x 2 + 2y 2 and graph of the

constraint g(x, y) = 4x − 6y − 25 = 0.

3. Apply the method of Lagrange multipliers to the function f (x, y) = (x 2 + 1)y subject to the constraint x 2 + y 2 = 5. Hint: First show that y = 0; then treat the cases x = 0 and x  = 0 separately. In Exercises 4–13, find the minimum and maximum values of the function subject to the given constraint. 4. f (x, y) = 2x + 3y,

x2 + y2 = 4

5. f (x, y) = x 2 + y 2 ,

2x + 3y = 6

6. f (x, y) = 4x 2 + 9y 2 ,

xy = 4

4x 2 + 9y 2 = 32

8. f (x, y) = x 2 y + x + y, 9. f (x, y) = x 2 + y 2 , 10. f (x, y) = x 2 y 4 ,

xy = 4

−2

16. Find the rectangular box of maximum volume if the sum of the lengths of the edges is 300 cm. 17. The surface area of a right-circular cone of radius r and height h  2 is S = πr r + h2 , and its volume is V = 13 π r 2 h. (a) Determine the ratio h/r for the cone with given surface area S and maximum volume V . (b) What is the ratio h/r for a cone with given volume V and minimum surface area S? (c) Does a cone with given volume V and maximum surface area exist? 18. In Example 1, we found the maximum of f (x, y) = 2x + 5y on the ellipse (x/4)2 + (y/3)2 = 1. Solve this problem again without using Lagrange multipliers. First, show that the ellipse is parametrized by x = 4 cos t, y = 3 sin t. Then find the maximum value of f (4 cos t, 3 sin t) using single-variable calculus. Is one method easier than the other? 19. Find the point on the ellipse x 2 + 6y 2 + 3xy = 40

x 2 + 2y 2 + 6z2 = 1

y 4

x2 − y2 + z = 0

13. f (x, y, z) = xy + 3xz + 2yz,

x

2

with largest x-coordinate (Figure 13).

x 2 + 2y 2 = 6

12. f (x, y, z) = x 2 − y − z,

0

15. Find the point (a, b) on the graph of y = ex where the value ab is as small as possible.

x4 + y4 = 1

11. f (x, y, z) = 3x + 2y + 4z,

5x + 9y + z = 10 −8

14.

−5

FIGURE 12 Contour map of f (x, y) = x 3 + xy + y 3 and graph of the constraint g(x, y) = x 3 − xy + y 3 = 1.

P

7. f (x, y) = xy,

0

5 3 1

−3

−2

4

751

−4

4

8

x

Let f (x, y) = x 3 + xy + y 3 ,

g(x, y) = x 3 − xy + y 3

(a) Show that there is a unique point P = (a, b) on g(x, y) = 1 where ∇fP = λ∇gP for some scalar λ. (b) Refer to Figure 12 to determine whether f (P ) is a local minimum or a local maximum of f subject to the constraint. (c) Does Figure 12 suggest that f (P ) is a global extremum subject to the constraint?

−4 FIGURE 13 Graph of x 2 + 6y 2 + 3xy = 40

20. Find the maximum area of a rectangle inscribed in the ellipse (Figure 14): x2 y2 + 2 =1 2 a b

752

C H A P T E R 12

DIFFERENTIATION IN SEVERAL VARIABLES

y (−x, y)

(x, y) x

(−x, −y)

30. In a contest, a runner starting at A must touch a point P along a river and then run to B in the shortest time possible (Figure 16). The runner should choose the point P that minimizes the total length of the path. (a) Define a function f (x, y) = AP + P B,

(x, −y)

FIGURE 14 Rectangle inscribed in the ellipse

x2 y2 + 2 = 1. 2 a b

21. Find the point (x0 , y0 ) on the line 4x + 9y = 12 that is closest to the origin. 22. Show that the point (x0 , y0 ) closest to the origin on the line ax + by = c has coordinates ac , x0 = 2 a + b2

bc y0 = 2 a + b2

where P = (x, y)

Rephrase the runner’s problem as a constrained optimization problem, assuming that the river is given by an equation g(x, y) = 0. (b) Explain why the level curves of f (x, y) are ellipses. (c) Use Lagrange multipliers to justify the following statement: The ellipse through the point P minimizing the length of the path is tangent to the river. (d) Identify the point on the river in Figure 16 for which the length is minimal.

y

23. Find the maximum value of f (x, y) = x a y b for x ≥ 0, y ≥ 0 on the line x + y = 1, where a, b > 0 are constants.

River P

2 3 24. Show  that the maximum value of f (x, y) = x y on the unit circle 3 6 is 25 5 .

A

25. Find the maximum value of f (x, y) = x a y b for x ≥ 0, y ≥ 0 on the unit circle, where a, b > 0 are constants. 26. Find the maximum value of f (x, y, z) = x a y b zc for x, y, z ≥ 0 on the unit sphere, where a, b, c > 0 are constants.

B

x FIGURE 16

27. Show that the minimum distance from the origin to a point on the plane ax + by + cz = d is 

|d| a 2 + b2 + d 2

28. Antonio has $5.00 to spend on a lunch consisting of hamburgers ($1.50 each) and French fries ($1.00 per order). Antonio’s satisfaction from eating x1 hamburgers and x2 orders of French fries is measured √ by a function U (x1 , x2 ) = x1 x2 . How much of each type of food should he purchase to maximize his satisfaction? (Assume that fractional amounts of each food can be purchased.) 29. Let Q be the point on an ellipse closest to a given point P outside the ellipse. It was known to the Greek mathematician Apollonius (third century bce) that P Q is perpendicular to the tangent to the ellipse at Q (Figure 15). Explain in words why this conclusion is a consequence of the method of Lagrange multipliers. Hint: The circles centered at P are level curves of the function to be minimized.

P

Q

In Exercises 31 and 32, let V be the volume of a can of radius r and height h, and let S be its surface area (including the top and bottom). 31. Find r and h that minimize S subject to the constraint V = 54π . Show that for both of the following two problems, P = 32. (r, h) is a Lagrange critical point if h = 2r: • •

Minimize surface area S for fixed volume V . Maximize volume V for fixed surface area S.

Then use the contour plots in Figure 17 to explain why S has a minimum for fixed V but no maximum and, similarly, V has a maximum for fixed S but no minimum. h

Increasing S Level curves of S

Critical point P = (r, h) Level curve of V Increasing V r

FIGURE 15

FIGURE 17

Lagrange Multipliers: Optimizing with a Constraint

S E C T I O N 12.8

x y z + + = 1 (a, b, c > 0) together with a b c the positive coordinate planes forms a tetrahedron of volume V = 16 abc (Figure 18). Find the minimum value of V among all planes passing through the point P = (1, 1, 1). 33. A plane with equation

753

(a) Use Lagrange multipliers to show that L = (h2/3 + b2/3 )3/2 (Figure 19). Hint: Show that the problem amounts to minimizing f (x, y) = (x + b)2 + (y + h)2 subject to y/b = h/x or xy = bh. (b) Show that the value of L is also equal to the radius of the circle with center (−b, −h) that is tangent to the graph of xy = bh.

z y C = (0, 0, c)

xy = bh

y

L L

Wall

P

h

B = (0, b, 0) A = (a, 0, 0)

y

b

x

(−b, −h)

Ladder x Fence

FIGURE 19

x FIGURE 18

34. With the same set-up as in the previous problem, find the plane that minimizes V if the plane is constrained to pass through a point P = (α, β, γ ) with α, β, γ > 0. 35. Show that the Lagrange equations for f (x, y) = x + y subject to the constraint g(x, y) = x + 2y = 0 have no solution. What can you conclude about the minimum and maximum values of f subject to g = 0? Show this directly.

38. Find the maximum value of f (x, y, z) = xy + xz + yz − xyz subject to the constraint x + y + z = 1, for x ≥ 0, y ≥ 0, z ≥ 0. 39. Find the point lying on the intersection of the plane x + 12 y + 14 z = 0 and the sphere x 2 + y 2 + z2 = 9 with the largest z-coordinate. 40. Find the maximum of f (x, y, z) = x + y + z subject to the two constraints x 2 + y 2 + z2 = 9 and 14 x 2 + 14 y 2 + 4z2 = 9. 41. The cylinder x 2 + y 2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on that ellipse that is farthest from the origin.

36. Show that the Lagrange equations for f (x, y) = 2x + y subject to the constraint g(x, y) = x 2 − y 2 = 1 have a solution but that f has no min or max on the constraint curve. Does this contradict Theorem 1?

42. Find the minimum and maximum of f (x, y, z) = y + 2z subject to two constraints, 2x + z = 4 and x 2 + y 2 = 1.

37. Let L be the minimum length of a ladder that can reach over a fence of height h to a wall located a distance b behind the wall.

43. Find the minimum value of f (x, y, z) = x 2 + y 2 + z2 subject to two constraints, x + 2y + z = 3 and x − y = 4.

Further Insights and Challenges 44. Suppose that both f (x, y) and the constraint function g(x, y) are linear. Use contour maps to explain why f (x, y) does not have a maximum subject to g(x, y) = 0 unless g = af + b for some constants a, b.

utility is equal to the ratio of prices:

45. Assumptions Matter Consider the problem of minimizing f (x, y) = x subject to g(x, y) = (x − 1)3 − y 2 = 0. (a) Show, without using calculus, that the minimum occurs at P = (1, 0).

47. Consider the utility function U (x1 , x2 ) = x1 x2 with budget constraint p1 x1 + p2 x2 = c. (a) Show that the maximum of U (x1 , x2 ) subject to the budget constraint is equal to c2 /(4p1 p2 ).

(b) Show that the Lagrange condition ∇fP = λ∇gP is not satisfied for any value of λ.

(b) Calculate the value of the Lagrange multiplier λ occurring in (a).

Ux1 (a, b) Marginal utility of good 1 p = = 1 p2 Marginal utility of good 2 Ux2 (a, b)

(c) Does this contradict Theorem 1?

(c) Prove the following interpretation: λ is the rate of increase in utility per unit increase in total budget c.

46. Marginal Utility Goods 1 and 2 are available at dollar prices of p1 per unit of good 1 and p2 per unit of good 2. A utility function U (x1 , x2 ) is a function representing the utility or benefit of consuming xj units of good j . The marginal utility of the j th good is ∂U/∂xj , the rate of increase in utility per unit increase in the j th good. Prove the following law of economics: Given a budget of L dollars, utility is maximized at the consumption level (a, b) where the ratio of marginal

48. This exercise shows that the multiplier λ may be interpreted as a rate of change in general. Assume that the maximum of f (x, y) subject to g(x, y) = c occurs at a point P . Then P depends on the value of c, so we may write P = (x(c), y(c)) and we have g(x(c), y(c)) = c. (a) Show that   ∇g(x(c), y(c)) · x  (c), y  (c) = 1

754

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

Hint: Differentiate the equation g(x(c), y(c)) = c with respect to c using the Chain Rule. (b) Use the Chain Rule and the Lagrange condition ∇fP = λ∇gP to show that

51. Given constants E, E1 , E2 , E3 , consider the maximum of S(x1 , x2 , x3 ) = x1 ln x1 + x2 ln x2 + x3 ln x3 subject to two constraints:

d f (x(c), y(c)) = λ dc

x1 + x2 + x3 = N,

E1 x1 + E2 x2 + E3 x3 = E

(c) Conclude that λ is the rate of increase in f per unit increase in the “budget level” c.

Show that there is a constant μ such that xi = A−1 eμEi for i = 1, 2, 3, where A = N −1 (eμE1 + eμE2 + eμE3 ).

49. Let B > 0. Show that the maximum of

52. Boltzmann Distribution Generalize Exercise 51 to n variables: Show that there is a constant μ such that the maximum of

f (x1 , . . . , xn ) = x1 x2 · · · xn subject to the constraints x1 + · · · + xn = B and xj ≥ 0 for j = 1, . . . , n occurs for x1 = · · · = xn = B/n. Use this to conclude that

S = x1 ln x1 + · · · + xn ln xn subject to the constraints

a + · · · + an (a1 a2 · · · an )1/n ≤ 1 n for all positive numbers a1 , . . . , an . 50. Let B > 0. Show that the maximum√of f (x1 , . . . , xn ) = x1 + · · · + xn subject to x12 + · · · + xn2 = B 2 is nB. Conclude that |a1 | + · · · + |an | ≤

√

n(a12 + · · · + an2 )1/2

for all numbers a1 , . . . , an .

x1 + · · · + xn = N,

E1 x1 + · · · + En xn = E

occurs for xi = A−1 eμEi , where A = N −1 (eμE1 + · · · + eμEn ) This result lies at the heart of statistical mechanics. It is used to determine the distribution of velocities of gas molecules at temperature T ; xi is the number of molecules with kinetic energy Ei ; μ = −(kT )−1 , where k is Boltzmann’s constant. The quantity S is called the entropy.

CHAPTER REVIEW EXERCISES 

z

x2 − y2 : x+3 (a) Sketch the domain of f . 1. Given f (x, y) =

z

(b) Calculate f (3, 1) and f (−5, −3). (c) Find a point satisfying f (x, y) = 1. y x

2. Find the domain and range of: √ √ (a) f (x, y, z) = x − y + y − z

y x

(b) f (x, y) = ln(4x 2 − y)

(A)

3. Sketch the graph f (x, y) = x 2 − y + 1 and describe its vertical and horizontal traces.

z

(B)

z

4. Use a graphing utility to draw the graph of the func2 tion cos(x + y 2 )e1−xy in the domains [−1, 1] × [−1, 1], [−2, 2] × [−2, 2], and [−3, 3] × [−3, 3], and explain its behavior. 5. Match the functions (a)–(d) with their graphs in Figure 5.

y

(a) f (x, y) = x 2 + y

x

(b) f (x, y) = x 2 + 4y 2 2 2 (c) f (x, y) = sin(4xy)e−x −y 2 2 (d) f (x, y) = sin(4x)e−x −y

y x (C)

(D) FIGURE 1

Chapter Review Exercises

6. Referring to the contour map in Figure 2: (a) Estimate the average rate of change of elevation from A to B and from A to D. (b) Estimate the directional derivative at A in the direction of v. (c) What are the signs of fx and fy at D?

650

v D

Contour interval = 50 meters

(x, y) = (0, 0)

17. f (x, y) = 2x + y 2

18. f (x, y) = 4xy 3

19. f (x, y) = sin(xy)e−x−y

20. f (x, y) = ln(x 2 + xy 2 )

21. Calculate fxxyz for f (x, y, z) = y sin(x + z).

400

A

(x, y)  = (0, 0)

In Exercises 17–20, compute fx and fy .

C 750

⎧ p ⎨ (xy) 4 f (x, y) = x + y 4 ⎩ 0

Use polar coordinates to show that f (x, y) is continuous at all (x, y) if p > 2 but is discontinuous at (0, 0) if p ≤ 2.  16. Calculate fx (1, 3) and fy (1, 3) for f (x, y) = 7x + y 2 .

(d) At which of the labeled points are both fx and fy negative?

B

15. Let

755

22. Fix c > 0. Show that for any constants α, β, the function u(t, x) = sin(αct + β) sin(αx) satisfies the wave equation 0

1

∂ 2u ∂ 2u = c2 2 2 ∂t ∂x

2 km

FIGURE 2

23. Find an equation of the tangent plane to the graph of f (x, y) = xy 2 − xy + 3x 3 y at P = (1, 3). 7. Describe the level curves of: (a) f (x, y) = e4x−y

(b) f (x, y) = ln(4x − y)

(c) f (x, y) = 3x 2 − 4y 2

(d) f (x, y) = x + y 2

8. Match each function (a)–(c) with its contour graph (i)–(iii) in Figure 3: (a) f (x, y) = xy (b) f (x, y) = exy (c) f (x, y) = sin(xy) y

y

y

x

(i)

x

x

(ii)

11. 13.

(x,y)→(1,−3)

(xy + y 2 )

xy + xy 2 (x,y)→(0,0) x 2 + y 2 lim

lim

(x,y)→(1,−3)

(2x + y)e−x+y

(ex − 1)(ey − 1) 14. lim x (x,y)→(0,2)

27. Figure 4 shows the contour map of a function f (x, y) together with a path c(t) in the counterclockwise direction. The points c(1), c(2), and c(3) are indicated on the path. Let g(t) = f (c(t)). Which of statements (i)–(iv) are true? Explain. (i) g  (1) > 0. (ii) g(t) has a local minimum for some 1 ≤ t ≤ 2. (iii) g  (2) = 0. (iv) g  (3) = 0.

c(t)

In Exercises 9–14, evaluate the limit or state that it does not exist. lim

26. The plane z = 2x − y − 1 is tangent to the graph of z = f (x, y) at P = (5, 3). (a) Determine f (5, 3), fx (5, 3), and fy (5, 3). (b) Approximate f (5.2, 2.9).

(iii)

FIGURE 3

9.

24. Suppose that f (4, 4) = 3 and fx (4, 4) = fy (4, 4) = −1. Use the linear approximation to estimate f (4.1, 4) and f (3.88, 4.03).  x 2 + y 2 + z to esti25. Use a linear approximation of f (x, y, z) =  2 2 mate 7.1 + 4.9 + 69.5. Compare with a calculator value.

10.

12.

lim

(x,y)→(1,−3)

−4 −2

4 2

0

ln(3x + y)

x3y2 + x2y3 (x,y)→(0,0) x4 + y4

0 0

lim

c(1)

−2 −4

−6

FIGURE 4

c(3) c(2)

4 2

756

DIFFERENTIATION IN SEVERAL VARIABLES

C H A P T E R 12

 c 1.5 dollars per month at a used 28. Jason earns S(h, c) = 20h 1 + 100 car lot, where h is the number of hours worked and c is the number of cars sold. He has already worked 160 hours and sold 69 cars. Right now Jason wants to go home but wonders how much more he might earn if he stays another 10 minutes with a customer who is considering buying a car. Use the linear approximation to estimate how much extra money Jason will earn if he sells his 70th car during these 10 minutes. In Exercises 29–32, compute 29. f (x, y) = x + ey ,

d f (c(t)) at the given value of t. dt

c(t) = (3t − 1, t 2 ) at t = 2

30. f (x, y, z) = xz − y 2 ,

c(t) = (t, t 3 , 1 − t) at t = −2

31. f (x, y) = xe3y − ye3x , 32. f (x, y) = tan−1 yx ,

c(t) = (et , ln t) at t = 1

c(t) = (cos t, sin t), t = π3

In Exercises 33–36, compute the directional derivative at P in the direction of v. 33. f (x, y) = x 3 y 4 ,

P = (3, −1),

v = 2i + j

P = (1, 1, 1), v = 2, −1, 2 √ √  2 2 , , v = 3, −4 P = 2 2

34. f (x, y, z) = zx − xy 2 , 2 2 35. f (x, y) = ex +y ,

36. f (x, y, z) = sin(xy + z),

P = (0, 0, 0),

v =j+k

43. Let g(u, v) = f (u3 − v 3 , v 3 − u3 ). Prove that v2

∂g ∂g − u2 =0 ∂u ∂v

44. Let f (x, y) = g(u), where u = x 2 + y 2 and g(u) is differentiable. Prove that 

  2  2 ∂f 2 ∂f dg + = 4u ∂x ∂y du

45. Calculate ∂z/∂x, where xez + zey = x + y. 46. Let f (x, y) = x 4 − 2x 2 + y 2 − 6y. (a) Find the critical points of f and use the Second Derivative Test to determine whether they are a local minima or a local maxima. (b) Find the minimum value of f without calculus by completing the square. In Exercises 47–50, find the critical points of the function and analyze them using the Second Derivative Test. 47. f (x, y) = x 4 − 4xy + 2y 2 48. f (x, y) = x 3 + 2y 3 − xy 49. f (x, y) = ex+y − xe2y 1 50. f (x, y) = sin(x + y) − (x + y 2 ) 2

37. Find the unit vector e at P = (0, 0, 1) pointing in the direction 2 along which f (x, y, z) = xz + e−x +y increases most rapidly.

51. Prove that f (x, y) = (x + 2y)exy has no critical points.

38. Find an equation of the tangent plane at P = (0, 3, −1) to the surface with equation

52. Find the global extrema of f (x, y) = x 3 − xy − y 2 + y on the square [0, 1] × [0, 1].

zex + ez+1 = xy + y − 3

53. Find the global extrema of f (x, y) = 2xy − x − y on the domain {y ≤ 4, y ≥ x 2 }.

39. Let n = 0 be an integer and r an arbitrary constant. Show that the tangent plane to the surface x n + y n + zn = r at P = (a, b, c) has equation

54. Find the maximum of f (x, y, z) = xyz subject to the constraint g(x, y, z) = 2x + y + 4z = 1.

a n−1 x + bn−1 y + cn−1 z = r

55. Use Lagrange multipliers to find the minimum and maximum values of f (x, y) = 3x − 2y on the circle x 2 + y 2 = 4.

40. Let f (x, y) = (x − y)ex . Use the Chain Rule to calculate ∂f/∂u and ∂f/∂v (in terms of u and v), where x = u − v and y = u + v.

56. Find the minimum value of f (x, y) = xy subject to the constraint 5x − y = 4 in two ways: using Lagrange multipliers and setting y = 5x − 4 in f (x, y).

41. Let f (x, y, z) = x 2 y + y 2 z. Use the Chain Rule to calculate ∂f/∂s and ∂f/∂t (in terms of s and t), where

57. Find the minimum and maximum values of f (x, y) = x 2 y on the ellipse 4x 2 + 9y 2 = 36.

x = s + t,

y = st,

z = 2s − t

  42. Let P have spherical coordinates (ρ, θ, φ) = 2, π4 , π4 . Calculate  ∂f  ∂φ  assuming that P

fx (P ) = 4,

fy (P ) = −3,

fz (P ) = 8

Recall that x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ.

58. Find the point in the first quadrant on the curve y = x + x −1 closest to the origin. 59. Find the extreme values of f (x, y, z) = x + 2y + 3z subject to the two constraints x + y + z = 1 and x 2 + y 2 + z2 = 1. 60. Find the minimum and maximum values of f (x, y, z) = x − z on the intersection of the cylinders x 2 + y 2 = 1 and x 2 + z2 = 1 (Figure 5).

Chapter Review Exercises

z

757

z

y y

x FIGURE 6

x FIGURE 5

61. Use Lagrange multipliers to find the dimensions of a cylindrical can with a bottom but no top, of fixed volume V with minimum surface area. 62. Find the dimensions of the box of maximum volume with its sides parallel to the coordinate planes that can be inscribed in the ellipsoid (Figure 6) x 2 y 2 z 2 + + =1 a b c

63. Given n nonzero numbers σ1 , . . . , σn , show that the minimum value of f (x1 , . . . , xn ) = x12 σ12 + · · · + xn2 σn2 ⎛ subject to x1 + · · · + xn = 1 is c, where c = ⎝

n #

j =1

⎞−1 σj−2 ⎠

.

1019763_FM_VOL-I.qxp

9/17/07

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1st Pass Pages

A THE LANGUAGE

OF MATHEMATICS

One of the challenges in learning calculus is growing accustomed to its precise language and terminology, especially in the statements of theorems. In this section, we analyze a few details of logic that are helpful, and indeed essential, in understanding and applying theorems properly. Many theorems in mathematics involve an implication. If A and B are statements, then the implication A ⇒ B is the assertion that A implies B: A ⇒ B :

If A is true, then B is true.

Statement A is called the hypothesis (or premise) and statement B the conclusion of the implication. Here is an example: If m and n are even integers, then m + n is an even integer. This statement may be divided into a hypothesis and conclusion: m and n are even integers   

⇒

m + n is an even integer   

A

B

In everyday speech, implications are often used in a less precise way. An example is: If you work hard, then you will succeed. Furthermore, some statements that do not initially have the form A ⇒ B may be restated as implications. For example, the statement “Cats are mammals” can be rephrased as follows: Let X be an animal.

X is a cat   

⇒

A

X is a mammal    B

When we say that an implication A ⇒ B is true, we do not claim that A or B is necessarily true. Rather, we are making the conditional statement that if A happens to be true, then B is also true. In the above, if X does not happen to be a cat, the implication tells us nothing. The negation of a statement A is the assertion that A is false and is denoted ¬A. Statement A

Negation ¬A

X lives in California.

X does not live in California.

ABC is a right triangle.

ABC is not a right triangle.

The negation of the negation is the original statement: ¬(¬A) = A. To say that X does not not live in California is the same as saying that X lives in California. E X A M P L E 1 State the negation of each statement.

(a) The door is open and the dog is barking. (b) The door is open or the dog is barking (or both). Solution (a) The first statement is true if two conditions are satisfied (door open and dog barking), and it is false if at least one of these conditions is not satisfied. So the negation is Either the door is not open OR the dog is not barking (or both). A1

A2

APPENDIX A

THE LANGUAGE OF MATHEMATICS

(b) The second statement is true if at least one of the conditions (door open or dog barking) is satisfied, and it is false if neither condition is satisfied. So the negation is The door is not open AND the dog is not barking.

Contrapositive and Converse

Keep in mind that when we form the contrapositive, we reverse the order of A and B . The contrapositive of A ⇒ B is NOT ¬A ⇒ ¬B .

Two important operations are the formation of the contrapositive and the formation of the converse of a statement. The contrapositive of A ⇒ B is the statement “If B is false, then A is false”: The contrapositive of

A ⇒ B

is

¬B ⇒ ¬A.

Here are some examples: Statement

Contrapositive

If X is a cat, then X is a mammal.

If X is not a mammal, then X is not a cat.

If you work hard, then you will succeed.

If you did not succeed, then you did not work hard.

If m and n are both even, then m + n is even.

If m + n is not even, then m and n are not both even.

A key observation is this: The contrapositive and the original implication are equivalent. The fact that A ⇒ B is equivalent to its contrapositive ¬B ⇒ ¬A is a general rule of logic that does not depend on what A and B happen to mean. This rule belongs to the subject of “formal logic,” which deals with logical relations between statements without concern for the actual content of these statements.

In other words, if an implication is true, then its contrapositive is automatically true, and vice versa. In essence, an implication and its contrapositive are two ways of saying the same thing. For example, the contrapositive “If X is not a mammal, then X is not a cat” is a roundabout way of saying that cats are mammals. The converse of A ⇒ B is the reverse implication B ⇒ A: Implication: A ⇒ B If A is true, then B is true.

Converse B ⇒ A If B is true, then A is true.

The converse plays a very different role than the contrapositive because the converse is NOT equivalent to the original implication. The converse may be true or false, even if the original implication is true. Here are some examples: True Statement

Converse

Converse True or False?

If X is a cat, then X is a mammal.

If X is a mammal, then X is a cat.

False

If m is even, then m2 is even.

If m2 is even, then m is even.

True

THE LANGUAGE OF MATHEMATICS

APPENDIX A

A3

E X A M P L E 2 An Example Where the Converse Is False Show that the converse of “If m and n are even, then m + n is even” is false. A counterexample is an example that satisfies the hypothesis but not the conclusion of a statement. If a single counterexample exists, then the statement is false. However, we cannot prove that a statement is true merely by giving an example.

A c

E X A M P L E 3 An Example Where the Converse Is True State the contrapositive and converse of the Pythagorean Theorem. Are either or both of these true?

Solution Consider a triangle with sides a, b, and c, and let θ be the angle opposite the side of length c as in Figure 1. The Pythagorean Theorem states that if θ = 90◦ , then a 2 + b2 = c2 . Here are the contrapositive and converse:

b θ

B

Solution The converse is “If m + n is even, then m and n are even.” To show that the converse is false, we display a counterexample. Take m = 1 and n = 3 (or any other pair of odd numbers). The sum is even (since 1 + 3 = 4) but neither 1 nor 3 is even. Therefore, the converse is false.

a

C

Pythagorean Theorem

θ = 90◦ ⇒ a 2 + b2 = c2

Contrapositive

a2

Converse

a 2 + b2 = c2 ⇒ θ = 90◦

FIGURE 1

+ b2

=

c2

⇒ θ  =

90◦

True Automatically true True (but not automatic)

The contrapositive is automatically true because it is just another way of stating the original theorem. The converse is not automatically true since there could conceivably exist a nonright triangle that satisfies a 2 + b2 = c2 . However, the converse of the Pythagorean Theorem is, in fact, true. This follows from the Law of Cosines (see Exercise 38). When both a statement A ⇒ B and its converse B ⇒ A are true, we write A ⇐⇒ B. In this case, A and B are equivalent. We often express this with the phrase A ⇐⇒ B

A is true if and only if B is true.

For example, a 2 + b2 = c2

if and only if

θ = 90◦

It is morning

if and only if

the sun is rising.

We mention the following variations of terminology involving implications that you may come across: Statement

Is Another Way of Saying

A is true if B is true.

B ⇒ A

A is true only if B is true.

A ⇒ B (A cannot be true unless B is also true.)

For A to be true, it is necessary that B be true.

A ⇒ B (A cannot be true unless B is also true.)

For A to be true, it is sufficient that B be true.

B ⇒ A

For A to be true, it is necessary and sufficient that B be true.

B ⇐⇒ A

A4

APPENDIX A

THE LANGUAGE OF MATHEMATICS

Analyzing a Theorem y

To see how these rules of logic arise in calculus, consider the following result from Section 4.2:

Maximum value

THEOREM 1 Existence of a Maximum on a Closed Interval If f (x) is a continuous function on a closed (bounded) interval I = [a, b], then f (x) takes on a maximum value on I (Figure 2).

x a

b

FIGURE 2 A continuous function on a closed interval I = [a, b] has a maximum value.

To analyze this theorem, let’s write out the hypotheses and conclusion separately: Hypotheses A:

f (x) is continuous and I is closed.

Conclusion B:

f (x) takes on a maximum value on I .

A first question to ask is: “Are the hypotheses necessary?” Is the conclusion still true if we drop one or both assumptions? To show that both hypotheses are necessary, we provide counterexamples: •

•

The continuity of f (x) is a necessary hypothesis. Figure 3(A) shows the graph of a function on a closed interval [a, b] that is not continuous. This function has no maximum value on [a, b], which shows that the conclusion may fail if the continuity hypothesis is not satisfied. The hypothesis that I is closed is necessary. Figure 3(B) shows the graph of a continuous function on an open interval (a, b). This function has no maximum value, which shows that the conclusion may fail if the interval is not closed.

We see that both hypotheses in Theorem 1 are necessary. In stating this, we do not claim that the conclusion always fails when one or both of the hypotheses are not satisfied. We claim only that the conclusion may fail when the hypotheses are not satisfied. Next, let’s analyze the contrapositive and converse: •

•

Contrapositive ¬B ⇒ ¬A (automatically true): If f (x) does not have a maximum value on I , then either f (x) is not continuous or I is not closed (or both). Converse B ⇒ A (in this case, false): If f (x) has a maximum value on I , then f (x) is continuous and I is closed. We prove this statement false with a counterexample [Figure 3(C)].

y

y

y Maximum value

x a

b

(A) The interval is closed but the function is not continuous. The function has no maximum value. FIGURE 3

x a

b

x a

b

(B) The function is continuous (C) This function is not continuous but the interval is open. The and the interval is not closed, function has no maximum value. but the function does have a maximum value.

APPENDIX A

The technique of proof by contradiction is also known by its Latin name reductio ad absurdum or “reduction to the absurd.” The ancient Greek mathematicians used proof by contradiction as early as the fifth century BC, and Euclid (325–265 BC) employed it in his classic treatise on geometry entitled The Elements. A famous √ example is the proof that 2 is irrational in Example 4. The philosopher Plato (427–347 BC) wrote: “He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.”

2

1

1 FIGURE 4 The √ diagonal of the unit square

has length

THE LANGUAGE OF MATHEMATICS

A5

As we know, the contrapositive is merely a way of restating the theorem, so it is automatically true. The converse is not automatically true, and in fact, in this case it is false. The function in Figure 3(C) provides a counterexample to the converse: f (x) has a maximum value on I = (a, b), but f (x) is not continuous and I is not closed. Mathematicians have devised various general strategies and methods for proving theorems. The method of proof by induction is discussed inAppendix C.Another important method is proof by contradiction, also called indirect proof. Suppose our goal is to prove statement A. In a proof by contradiction, we start by assuming that A is false, and then show that this leads to a contradiction. Therefore, A must be true (to avoid the contradiction). E X A M P L E 4 Proof by Contradiction The number

√

2 is irrational (Figure 4). √ Solution Assume that the theorem is false, namely that 2 = p/q, where p and q are whole numbers. We may assume that p/q is in lowest terms, and therefore, at most one of p and q is even. Note that if the square m2 of a whole number is even, then m itself must be even. √ The relation 2 = p/q implies that 2 = p2 /q 2 or p2 = 2q 2 . This shows that p must be even. But if p is even, then p = 2m for some whole number m, and p2 = 4m2 . Because p 2 = 2q 2 , we obtain 4m2 = 2q 2 , or q 2 = 2m2 . This shows that q is also even. But we chose p and q so that at√ most one of them is even. This contradiction shows that our √ original assumption, that 2 = p/q, must be false. Therefore, 2 is irrational.

2.

The hallmark of mathematics is precision and rigor. A theorem is established, not through observation or experimentation, but by a proof that consists of a chain of reasoning with no gaps. This approach to mathematics comes down to us from the ancient Greek mathematicians, especially Euclid, and it remains the standard in contemporary research. In recent decades, the computer has become a powerful tool for mathematical experimentation and data analysis. Researchers may use experimental data to discover potential new mathematical facts, but the title “theorem” is not bestowed until someone writes down a proof. This insistence on theorems and proofs distinguishes mathematics from the other sciences. In the natural sciences, facts are established through experiment and are subject to change or modification as more knowledge is acquired. In mathematics, theories are also developed and expanded, but previous results are not invalidated. The Pythagorean Theorem was discovered in antiquity and is a cornerstone of plane geometry. In the nineteenth century, mathematicians began to study more general types of geometry (of the type that eventually led to Einstein’s four-dimensional space-time geometry in the Theory of Relativity). The Pythagorean Theorem does not hold in these more general geometries, but its status in plane geometry is unchanged. CONCEPTUAL INSIGHT

One of the most famous problems in mathematics is known as “Fermat’s Last Theorem.” It states that the equation

x n + y n = zn has no solutions in positive integers if n ≥ 3. In a marginal note written around 1630, Fermat claimed to have a proof, and over the centuries, that assertion was verified for many values of the exponent n. However, only in 1994 did the BritishAmerican mathematician Andrew Wiles, working at Princeton University, find a complete proof.

A. SUMMARY The implication A ⇒ B is the assertion “If A is true, then B is true.” The contrapositive of A ⇒ B is the implication ¬B ⇒ ¬A, which says “If B is false, then A is false.” An implication and its contrapositive are equivalent (one is true if and only if the other is true). • The converse of A ⇒ B is B ⇒ A. An implication and its converse are not necessarily equivalent. One may be true and the other false. • A and B are equivalent if A ⇒ B and B ⇒ A are both true. • •

A6

APPENDIX A

THE LANGUAGE OF MATHEMATICS • In a proof by contradiction (in which the goal is to prove statement A), we start by assuming that A is false and show that this assumption leads to a contradiction.

A. EXERCISES Preliminary Questions 1. Which is the contrapositive of A ⇒ B? (a) B ⇒ A (b) ¬B ⇒ A (c) ¬B ⇒ ¬A (d) ¬A ⇒ ¬B 2. Which of the choices in Question 1 is the converse of A ⇒ B?

3. Suppose that A ⇒ B is true. Which is then automatically true, the converse or the contrapositive? 4. Restate as an implication: “A triangle is a polygon.”

Exercises 1. Which is the negation of the statement “The car and the shirt are both blue”? (a) Neither the car nor the shirt is blue. (b) The car is not blue and/or the shirt is not blue. 2. Which is the contrapositive of the implication “If the car has gas, then it will run”? (a) If the car has no gas, then it will not run. (b) If the car will not run, then it has no gas. In Exercises 3–8, state the negation. 3. The time is 4 o’clock. 4. ABC is an isosceles triangle. 5. m and n are odd integers. 6. Either m is odd or n is odd. 7. x is a real number and y is an integer. 8. f (x) is a linear function. In Exercises 9–14, state the contrapositive and converse. 9. If m and n are odd integers, then mn is odd. 10. If today is Tuesday, then we are in Belgium. 11. If today is Tuesday, then we are not in Belgium. 12. If x > 4, then x 2 > 16.

19. If x > 4 and y > 4, then x + y > 8. 20. If x > 4, then x 2 > 16. 21. If |x| > 4, then x 2 > 16. 22. If m and n are even, then mn is even. In Exercises 23 and 24, state the contrapositive and converse (it is not necessary to know what these statements mean). 23. If f (x) and g(x) are differentiable, then f (x)g(x) is differentiable. 24. If the force field is radial and decreases as the inverse square of the distance, then all closed orbits are ellipses. In Exercises 25–28, the inverse of A ⇒ B is the implication ¬A ⇒ ¬B. 25. Which of the following is the inverse of the implication “If she jumped in the lake, then she got wet”? (a) If she did not get wet, then she did not jump in the lake. (b) If she did not jump in the lake, then she did not get wet. Is the inverse true? 26. (a) (b) (c)

State the inverses of these implications: If X is a mouse, then X is a rodent. If you sleep late, you will miss class. If a star revolves around the sun, then it’s a planet.

13. If m2 is divisible by 3, then m is divisible by 3.

27.

Explain why the inverse is equivalent to the converse.

14. If x 2 = 2, then x is irrational.

28.

State the inverse of the Pythagorean Theorem. Is it true?

In Exercise 15–18, give a counterexample to show that the converse of the statement is false. 15. If m is odd, then 2m + 1 is also odd. 16. If ABC is equilateral, then it is an isosceles triangle.

29. Theorem 1 in Section 2.4 states the following: “If f (x) and g(x) are continuous functions, then f (x) + g(x) is continuous.” Does it follow logically that if f (x) and g(x) are not continuous, then f (x) + g(x) is not continuous?

18. If m is odd, then m3 − m is divisible by 3.

30. Write out a proof by contradiction for this fact: There is no smallest positive rational number. Base your proof on the fact that if r > 0, then 0 < r/2 < r.

In Exercise 19–22, determine whether the converse of the statement is false.

31. Use proof by contradiction to prove that if x + y > 2, then x > 1 or y > 1 (or both).

17. If m is divisible by 9 and 4, then m is divisible by 12.

APPENDIX A

In Exercises 32–35, use proof by contradiction to show that the number is irrational.  √ √ √ 33. 3 34. 3 2 35. 4 11 32. 12 36. An isosceles triangle is a triangle with two equal sides. The following theorem holds: If  is a triangle with two equal angles, then  is an isosceles triangle. (a) What is the hypothesis? (b) Show by providing a counterexample that the hypothesis is necessary.

Further Insights and Challenges 38. Let a, b, and c be the sides of a triangle and let θ be the angle opposite c. Use the Law of Cosines (Theorem 1 in Section 1.4) to prove the converse of the Pythagorean Theorem. 39. √ Carry out the details of the following proof by√contradiction that 2 is irrational (This proof is due to R. Palais). If 2 is rational, then √ n 2 is a whole number for some whole √ number n. Let n be the smallest such whole number and let m = n 2 − n. (a) Prove that m < n. √ (b) Prove that m 2 is a whole number. √ Explain why (a) and (b) imply that 2 is irrational. √ 40. Generalize the argument of Exercise 39 to prove that A is irrational if A is a whole number but not a perfect square. Hint: Choose n

THE LANGUAGE OF MATHEMATICS

A7

(c) What is the contrapositive? (d) What is the converse? Is it true? 37. Consider the following theorem: Let f (x) be a quadratic polynomial with a positive leading coefficient. Then f (x) has a minimum value. (a) What are the hypotheses? (b) What is the contrapositive? (c) What is the converse? Is it true?

√ √ as before and let m = n A − n[ A], where [x] is the greatest integer function. 41. Generalize further and show that for any whole number√r, the rth √ root r A is irrational unless A is an rth power. Hint: Let x = r A. Show that if x is rational, then we may choose a smallest whole number n such that nx j is a whole number for j = 1, . . . , r − 1. Then consider m = nx − n[x] as before. Given a finite list of prime numbers p1 , . . . , pN , let 42. M = p1 · p2 · · · pN + 1. Show that M is not divisible by any of the primes p1 , . . . , pN . Use this and the fact that every number has a prime factorization to prove that there exist infinitely many prime numbers. This argument was advanced by Euclid in The Elements.

B PROPERTIES OF REAL NUMBERS

“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.” —Pierre-Simon Laplace, one of the great French mathematicians of the eighteenth century

−3

−2

−1

0

1

2

FIGURE 1 The real number line.

3

R

In this appendix, we discuss the basic properties of real numbers. First, let us recall that a real number is a number that may be represented by a finite or infinite decimal (also called a decimal expansion). The set of all real numbers is denoted R and is often visualized as the “number line” (Figure 1). Thus, a real number a is represented as a = ±n.a1 a2 a3 a4 . . . , where n is any whole number and each digit aj is a whole number between 0 and 9. For example, 10π = 31.41592 . . . . Recall that a is rational if its expansion is finite or repeating, and is irrational if its expansion is nonrepeating. Furthermore, the decimal expansion is unique apart from the following exception: Every finite expansion is equal ¯ to an expansion in which the digit 9 repeats. For example, 0.5 = 0.4999 · · · = 0.49. We shall take for granted that the operations of addition and multiplication are defined on R—that is, on the set of all decimals. Roughly speaking, addition and multiplication of infinite decimals are defined in terms of finite decimals. For d ≥ 1, define the dth truncation of a = n.a1 a2 a3 a4 . . . to be the finite decimal a(d) = a.a1 a2 . . . ad obtained by truncating at the dth place. To form the sum a + b, assume that both a and b are infinite (possibly ending with repeated nines). This eliminates any possible ambiguity in the expansion. Then the nth digit of a + b is equal to the nth digit of a(d) + b(d) for d sufficiently large (from a certain point onward, the nth digit of a(d) + b(d) no longer changes, and this value is the nth digit of a + b). Multiplication is defined similarly. Furthermore, the Commutative, Associative, and Distributive Laws hold (Table 1). TABLE 1

Algebraic Laws

Commutative Laws: Associative Laws: Distributive Law:

a + b = b + a, ab = ba (a + b) + c = a + (b + c), a(b + c) = ab + ac

(ab)c = a(bc)

Every real number x has an additive inverse −x such that x + (−x) = 0, and every nonzero real number x has a multiplicative inverse x −1 such that x(x −1 ) = 1. We do not regard subtraction and division as separate algebraic operations because they are defined in terms of inverses. By definition, the difference x − y is equal to x + (−y), and the quotient x/y is equal to x(y −1 ) for y  = 0. In addition to the algebraic operations, there is an order relation on R: For any two real numbers a and b, precisely one of the following is true: Either

a = b,

or

a < b,

or

a>b

To distinguish between the conditions a ≤ b and a < b, we often refer to a < b as a strict inequality. Similar conventions hold for > and ≥. The rules given in Table 2 allow us to manipulate inequalities. The last order property says that an inequality reverses direction when multiplied by a negative number c. For example, −2 < 5 A8

but

(−3)(−2) > (−3)5

APPENDIX B

TABLE 2 If a If a If a If a

PROPERTIES OF REAL NUMBERS

A9

Order Properties

< b and b < c, < b and c < d, < b and c > 0, < b and c < 0,

then a < c. then a + c < b + d. then ac < bc. then ac > bc.

The algebraic and order properties of real numbers are certainly familiar. We now discuss the less familiar Least Upper Bound (LUB) Property of the real numbers. This property is one way of expressing the so-called completeness of the real numbers. There are other ways of formulating completeness (such as the so-called nested interval property discussed in any book on analysis) that are equivalent to the LUB Property and serve the same purpose. Completeness is used in calculus to construct rigorous proofs of basic theorems about continuous functions, such as the Intermediate Value Theorem, (IVT) or the existence of extreme values on a closed interval. The underlying idea is that the real number line “has no holes.” We elaborate on this idea below. First, we introduce the necessary definitions. Suppose that S is a nonempty set of real numbers. A number M is called an upper bound for S if x≤M L −3

−2

−1

0

M

1

2

x

3

FIGURE 2 M = 3 is an upper bound for the

set S = (−2, 1). The LUB is L = 1.

for all x ∈ S

If S has an upper bound, we say that S is bounded above. A least upper bound L is an upper bound for S such that every other upper bound M satisfies M ≥ L. For example (Figure 2), • •

M = 3 is an upper bound for the open interval S = (−2, 1). L = 1 is the LUB for S = (−2, 1).

We now state the LUB Property of the real numbers. THEOREM 1 Existence of a Least Upper Bound Let S be a nonempty set of real numbers that is bounded above. Then S has an LUB. In a similar fashion, we say that a number B is a lower bound for S if x ≥ B for all x ∈ S. We say that S is bounded below if S has a lower bound. A greatest lower bound (GLB) is a lower bound M such that every other lower bound B satisfies B ≤ M. The set of real numbers also has the GLB Property: If S is a nonempty set of real numbers that is bounded below, then S has a GLB. This may be deduced immediately from Theorem 1. For any nonempty set of real numbers S, let −S be the set of numbers of the form −x for x ∈ S. Then −S has an upper bound if S has a lower bound. Consequently, −S has an LUB L by Theorem 1, and −L is a GLB for S. Theorem 1 may appear quite reasonable, but perhaps it is not clear why it is useful. We suggested above that the LUB Property expresses the idea that R is “complete” or “has no holes.” To illustrate this idea, let’s compare R to the set of rational numbers, denoted Q. Intuitively, Q is not complete because the irrational √ numbers are missing. For example, Q has a “hole” where the irrational number 2 should be located (Figure 3). This hole divides Q into two halves √ that are not connected to each other (the half to the left and the half to the right of 2). Furthermore, the half on the left is bounded above but no rational number is an LUB, and the half on the right is bounded below but no√rational number is a GLB. The LUB and GLB are both equal to the irrational number 2, which exists in only R but not Q. So unlike R, the rational numbers Q do not have the LUB property. CONCEPTUAL INSIGHT

2

−3

−2

−1

x 0

1

2

3

FIGURE 3 The rational√ numbers have a

“hole” at the location

2.

A10

APPENDIX B

PROPERTIES OF REAL NUMBERS

E X A M P L E 1 Show that 2 has a square root by applying the LUB Property to the set

S = {x : x 2 < 2} Solution First, we note that S is bounded with the upper bound M = 2. Indeed, if x > 2, then x satisfies x 2 > 4, and hence x does not belong √ to S. By the LUB Property, S has a least upper bound. Call it L. We claim that L = 2, or, equivalently, that L2 = 2. We prove this by showing that L2 ≥ 2 and L2 ≤ 2. If L2 < 2, let b = L + h, where h > 0. Then b2 = L2 + 2Lh + h2 = L2 + h(2L + h)

1

We can make the quantity h(2L + h) as small as desired by choosing h > 0 small enough. In particular, we may choose a positive h so that h(2L + h) < 2 − L2 . For this choice, b2 < L2 + (2 − L2 ) = 2 by Eq. (1). Therefore, b ∈ S. But b > L since h > 0, and thus L is not an upper bound for S, in contradiction to our hypothesis on L. We conclude that L2 ≥ 2. If L2 > 2, let b = L − h, where h > 0. Then b2 = L2 − 2Lh + h2 = L2 − h(2L − h) Now choose h positive but small enough so that 0 < h(2L − h) < L2 − 2. Then b2 > L2 − (L2 − 2) = 2. But b < L, so b is a smaller lower bound for S. Indeed, if x ≥ b, then x 2 ≥ b2 > 2, and x does not belong to S. This contradicts our hypothesis that L is the LUB. We conclude that L2 ≤ 2, and since we have already shown that L2 ≥ 2, we have L2 = 2 as claimed. We now prove three important theorems, the third of which is used in the proof of the LUB Property below. THEOREM 2 Bolzano–Weierstrass Theorem Let S be a bounded, infinite set of real numbers. Then there exists a sequence of distinct elements {an } in S such that the limit L = lim an exists. n→∞

Proof For simplicity of notation, we assume that S is contained in the unit interval [0, 1] (a similar proof works in general). If k1 , k2 , . . . , kn is a sequence of n digits (that is, each kj is a whole number and 0 ≤ kj ≤ 9), let S(k1 , k2 , . . . , kn ) be the set of x ∈ S whose decimal expansion begins 0.k1 k2 . . . kn . The set S is the union of the subsets S(0), S(1), . . . , S(9), and since S is infinite, at least one of these subsets must be infinite. Therefore, we may choose k1 so that S(k1 ) is infinite. In a similar fashion, at least one of the set S(k1 , 0), S(k2 , 1), . . . , S(k1 , 9) must be infinite, so we may choose k2 so that S(k1 , k2 ) is infinite. Continuing in this way, we obtain an infinite sequence {kn } such that S(k1 , k2 , . . . , kn ) is infinite for all n. We may choose a sequence of elements an ∈ S(k1 , k2 , . . . , kn ) with the property that an differs from a1 , . . . , an−1 for all n. Let L be the infinite decimal 0.k1 k2 k3 . . . . Then lim an = L since |L − an | < 10−n for all n. n→∞

We use the Bolzano–Weierstrass Theorem to prove two important results about sequences {an }. Recall that an upper bound for {an } is a number M such that aj ≤ M for all j . If an upper bound exists, {an } is said to be bounded from above. Lower bounds are defined similarly and {an } is said to be bounded from below if a lower bound exists.

APPENDIX B

PROPERTIES OF REAL NUMBERS

A11

A sequence is bounded if it is bounded from above and below. A subsequence of {an } is a sequence of elements an1 , an2 , an3 , . . . , where n1 < n2 < n3 < · · · . Now consider a bounded sequence {an }. If infinitely many of the an are distinct, the Bolzano–Weierstrass Theorem implies that there exists a subsequence {an1 , an2 , . . . } such that lim ank exists. Otherwise, infinitely many of the an must coincide, and these terms n→∞

form a convergent subsequence. This proves the next result. Section 10.1

THEOREM 3

Every bounded sequence has a convergent subsequence.

THEOREM 4 Bounded Monotonic Sequences Converge •

If {an } is increasing and an ≤ M for all n, then {an } converges and lim an ≤ M.

•

If {an } is decreasing and an ≥ M for all n, then {an } converges and lim an ≥ M.

n→∞

n→∞

Proof Suppose that {an } is increasing and bounded above by M. Then {an } is automatically bounded below by m = a1 since a1 ≤ a2 ≤ a3 · · · . Hence, {an } is bounded, and by Theorem 3, we may choose a convergent subsequence an1 , an2 , . . . . Let L = lim ank k→∞

Observe that an ≤ L for all n. For if not, then an > L for some n and then ank ≥ an > L for all k such that nk ≥ n. But this contradicts that ank → L. Now, by definition, for any  > 0, there exists N > 0 such that |ank − L| < 

if nk > N

Choose m such that nm > N . If n ≥ nm , then anm ≤ an ≤ L, and therefore, |an − L| ≤ |anm − L| < 

for all n ≥ nm

This proves that lim an = L as desired. It remains to prove that L ≤ M. If L > M, let n→∞

 = (L − M)/2 and choose N so that |an − L| < 

if k > N

Then an > L −  = M + . This contradicts our assumption that M is an upper bound for {an }. Therefore, L ≤ M as claimed. Proof of Theorem 1 We now use Theorem 4 to prove the LUB Property (Theorem 1). As above, if x is a real number, let x(d) be the truncation of x of length d. For example, If x = 1.41569, then x(3) = 1.415 We say that x is a decimal of length d if x = x(d). Any two distinct decimals of length d differ by at least 10−d . It follows that for any two real numbers A < B, there are at most finitely many decimals of length d between A and B. Now let S be a nonempty set of real numbers with an upper bound M. We shall prove that S has an LUB. Let S(d) be the set of truncations of length d: S(d) = {x(d) : x ∈ S} We claim that S(d) has a maximum element. To verify this, choose any a ∈ S. If x ∈ S and x(d) > a(d), then a(d) ≤ x(d) ≤ M

A12

APPENDIX B

PROPERTIES OF REAL NUMBERS

Thus, by the remark of the previous paragraph, there are at most finitely many values of x(d) in S(d) larger than a(d). The largest of these is the maximum element in S(d). For d = 1, 2, . . . , choose an element xd such that xd (d) is the maximum element in S(d). By construction, {xd (d)} is an increasing sequence (since the largest dth truncation cannot get smaller as d increases). Furthermore, xd (d) ≤ M for all d. We now apply Theorem 4 to conclude that {xd (d)} converges to a limit L. We claim that L is the LUB of S. Observe first that L is an upper bound for S. Indeed, if x ∈ S, then x(d) ≤ L for all d and thus x ≤ L. To show that L is the LUB, suppose that M is an upper bound such that M < L. Then xd ≤ M for all d and hence xd (d) ≤ M for all d. But then L = lim xd (d) ≤ M d→∞

This is a contradiction since M < L. Therefore, L is the LUB of S. As mentioned above, the LUB Property is used in calculus to establish certain basic theorems about continuous functions. As an example, we prove the IVT. Another example is the theorem on the existence of extrema on a closed interval (see Appendix D). THEOREM 5 Intermediate Value Theorem If f (x) is continuous on a closed interval [a, b] and f (a)  = f (b), then for every value M between f (a) and f (b), there exists at least one value c ∈ (a, b) such that f (c) = M. Proof Assume first that M = 0. Replacing f (x) by −f (x) if necessary, we may assume that f (a) < 0 and f (b) > 0. Now let S = {x ∈ [a, b] : f (x) < 0} Then a ∈ S since f (a) < 0 and thus S is nonempty. Clearly, b is an upper bound for S. Therefore, by the LUB Property, S has an LUB L. We claim that f (L) = 0. If not, set r = f (L). Assume first that r > 0. Since f (x) is continuous, there exists a number δ > 0 such that |f (x) − f (L)| = |f (x) − r|
0

if

L−δ <x L, and thus f (x) ≥ 0 for all x ∈ [a, b] such that x > L − δ. Thus, L − δ is an upper bound for S. This is a contradiction since L is the LUB of S, and it follows that r = f (L) cannot satisfy r > 0. Similarly, r cannot satisfy r < 0. We conclude that f (L) = 0 as claimed. Now, if M is nonzero, let g(x) = f (x) − M. Then 0 lies between g(a) and g(b), and by what we have proved, there exists c ∈ (a, b) such that g(c) = 0. But then f (c) = g(c) + M = M, as desired.

C INDUCTION AND THE BINOMIAL THEOREM

The Principle of Induction is a method of proof that is widely used to prove that a given statement P (n) is valid for all natural numbers n = 1, 2, 3, . . . . Here are two statements of this kind: P (n): The sum of the first n odd numbers is equal to n2 . d n • P (n): x = nx n−1 . dx The first statement claims that for all natural numbers n, •

1 + 3 + · · · + (2n − 1) = n2   

1

Sum of first n odd numbers

We can check directly that P (n) is true for the first few values of n: P (1) is the equality:

1 = 12

(true)

P (2) is the equality:

1 + 3 = 22

(true)

P (3) is the equality:

1 + 3 + 5 = 32

(true)

The Principle of Induction may be used to establish P (n) for all n. The Principle of Induction applies if P (n) is an assertion defined for n ≥ n0 , where n0 is a fixed integer. Assume that (i) Initial step: P (n0 ) is true. (ii) Induction step: If P (n) is true for n = k , then P (n) is also true for n = k + 1. Then P (n) is true for all n ≥ n0 .

THEOREM 1 Principle of Induction natural number n. Assume that

Let P (n) be an assertion that depends on a

(i) Initial step: P (1) is true. (ii) Induction step: If P (n) is true for n = k, then P (n) is also true for n = k + 1. Then P (n) is true for all natural numbers n = 1, 2, 3, . . . . E X A M P L E 1 Prove that 1 + 3 + · · · + (2n − 1) = n2 for all natural numbers n.

Solution As above, we let P (n) denote the equality P (n) :

1 + 3 + · · · + (2n − 1) = n2

Step 1. Initial step: Show that P (1) is true. We checked this above. P (1) is the equality 1 = 12 . Step 2. Induction step: Show that if P (n) is true for n = k, then P (n) is also true for n = k + 1. Assume that P (k) is true. Then 1 + 3 + · · · + (2k − 1) = k 2 Add 2k + 1 to both sides:   1 + 3 + · · · + (2k − 1) + (2k + 1) = k 2 + 2k + 1 = (k + 1)2 1 + 3 + · · · + (2k + 1) = (k + 1)2 A13

A14

INDUCTION AND THE BINOMIAL THEOREM

APPENDIX C

This is precisely the statement P (k + 1). Thus, P (k + 1) is true whenever P (k) is true. By the Principle of Induction, P (k) is true for all k. The intuition behind the Principle of Induction is the following. If P (n) were not true for all n, then there would exist a smallest natural number k such that P (k) is false. Furthermore, k > 1 since P (1) is true. Thus P (k − 1) is true [otherwise, P (k) would not be the smallest “counterexample”]. On the other hand, if P (k − 1) is true, then P (k) is also true by the induction step. This is a contradiction. So P (k) must be true for all k. E X A M P L E 2 Use Induction and the Product Rule to prove that for all whole numbers n,

d n x = nx n−1 dx d n x = nx n−1 . dx Step 1. Initial step: Show that P (1) is true. We use the limit definition to verify P (1):

Solution Let P (n) be the formula

(x + h) − x h d x = lim = lim = lim 1 = 1 h→0 h→0 h h→0 dx h Step 2. Induction step: Show that if P (n) is true for n = k, then P (n) is also true for n = k + 1. d k To carry out the induction step, assume that x = kx k−1 , where k ≥ 1. Then, by dx the Product Rule,

In Pascal’s Triangle, the nth row displays the coefficients in the expansion of (a + b)n :

n 0 1 2 3 4 5 6

d d d d k+1 = x (x · x k ) = x x k + x k x = x(kx k−1 ) + x k dx dx dx dx = kx k + x k = (k + 1)x k This shows that P (k + 1) is true.

1 1 1 1 1 1 1

5 6

2 3

4

By the Principle of Induction, P (n) is true for all n ≥ 1.

1 1 3 6

1 4

1

10 10 5 1 15 20 15 6 1

The triangle is constructed as follows: Each entry is the sum of the two entries above it in the previous line. For example, the entry 15 in line n = 6 is the sum 10 + 5 of the entries above it in line n = 5. The recursion relation guarantees that the entries in the triangle are the binomial coefficients.

As another application of induction, we prove the Binomial Theorem, which describes the expansion of the binomial (a + b)n . The first few expansions are familiar: (a + b)1 = a + b (a + b)2 = a 2 + 2ab + b2 (a + b)3 = a 3 + 3a 2 b + 3ab2 + b3 In general, we have an expansion





n n−1 n n−2 2 n n−3 3 n n a b+ a b + a b (a + b) = a + 1 2 3 2

n n−1 n + ··· + ab +b n−1

n n−k k where the coefficient of x x , denoted , is called the binomial coefficient. Note k that the first term in Eq. (2) corresponds to k = 0 and the last term to k = n; thus,

INDUCTION AND THE BINOMIAL THEOREM

APPENDIX C

A15



n n = = 1. In summation notation, 0 n n

 n k n−k (a + b) = a b k n

k=0

Pascal’s Triangle (described in the marginal note on page A14) can be used to compute binomial coefficients if n and k are not too large. The Binomial Theorem provides the following general formula:

n n! n(n − 1)(n − 2) · · · (n − k + 1) = = k k! (n − k)! k(k − 1)(k − 2) · · · 2 · 1

3

Before proving this formula, we prove a recursion relation for binomial coefficients. Note, however, that Eq. (3) is certainly correct for k = 0 and k = n (recall that by convention, 0! = 1):



n! n n! n! n! n = = = 1, = = =1 0 (n − 0)! 0! n! n (n − n)! n! n! THEOREM 2 Recursion Relation for Binomial Coefficients



n n−1 n−1 = + k k k−1

for 1 ≤ k ≤ n − 1

Proof We write (a + b)n as (a + b)(a + b)n−1 and expand in terms of binomial coefficients: (a + b)n = (a + b)(a + b)n−1

n

n−1   n n−k k n − 1 n−1−k k a b = (a + b) a b k k k=0

k=0

=a

n−1  k=0

=



n−1  n − 1 n−1−k k n − 1 n−1−k k a a b +b b k k k=0

n−1  n−1 k

k=0

a n−k bk +

n−1  n−1 k

k=0

a n−(k+1) bk+1

Replacing k by k − 1 in the second sum, we obtain



n−1 n n

 n n−k k  n − 1 n−k k  n − 1 n−k k a b = a b + a b k k k−1 k=0

k=0

k=1

On the right-hand side, the first term in the first sum is a n and the last term in the second sum is bn . Thus, we have

n−1



n

  n n−k k n−1 n−1 n n−k k a b =a + + a b + bn k k k−1 k=0

k=1

The recursion relation follows because the coefficients of a n−k bk on the two sides of the equation must be equal.

A16

APPENDIX C

INDUCTION AND THE BINOMIAL THEOREM

We now use induction to prove Eq. (3). Let P (n) be the claim

n n! = for 0 ≤ k ≤ n k k! (n − k)!

1 1 We have = = 1 since (a + b)1 = a + b, so P (1) is true. Furthermore, 0 1

n n = = 1 as observed above, since a n and bn have coefficient 1 in the exn 0 pansion of (a + b)n . For the inductive step, assume that P (n) is true. By the recursion relation, for 1 ≤ k ≤ n, we have



n+1 n n n! n! = + = + k k k−1 k! (n − k)! (k − 1)! (n − k + 1)!



n+1−k k n+1 = n! + = n! k! (n + 1 − k)! k! (n + 1 − k)! k! (n + 1 − k)! =

(n + 1)! k! (n + 1 − k)!

Thus, P (n + 1) is also true and the Binomial Theorem follows by induction. E X A M P L E 3 Use the Binomial Theorem to expand (x + y)5 and (x + 2)3 .

Solution The fifth row in Pascal’s Triangle yields (x + y)5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y 5 The third row in Pascal’s Triangle yields (x + 2)3 = x 3 + 3x 2 (2) + 3x(2)2 + 23 = x 3 + 6x 2 + 12x + 8

C. EXERCISES In Exercises 1–4, use the Principle of Induction to prove the formula for all natural numbers n. n(n + 1) 1. 1 + 2 + 3 + · · · + n = 2

The first few terms are 1, 1, 2, 3, 5, 8, 13, . . . . In Exercises 7–10, use induction to prove the identity. 7. F1 + F2 + · · · + Fn = Fn+2 − 1

n2 (n + 1)2 2. 13 + 23 + 33 + · · · + n3 = 4

8. F12 + F22 + · · · + Fn2 = Fn+1 Fn

1 1 n 1 + + ··· + = 3. 1·2 2·3 n(n + 1) n+1

9. Fn =

4. 1 + x + x 2 + · · · + x n =

1 − x n+1 for any x  = 1 1−x

5. Let P (n) be the statement 2n > n. (a) Show that P (1) is true. (b) Observe that if 2n > n, then 2n + 2n > 2n. Use this to show that if P (n) is true for n = k, then P (n) is true for n = k + 1. Conclude that P (n) is true for all n. 6. Use induction to prove that n! > 2n for n ≥ 4. Let {Fn } be the Fibonacci sequence, defined by the recursion formula Fn = Fn−1 + Fn−2 ,

F1 = F2 = 1

√ n − Rn R+ 1± 5 √ − , where R± = 2 5

10. Fn+1 Fn−1 = Fn2 + (−1)n . Hint: For the induction step, show that Fn+2 Fn = Fn+1 Fn + Fn2 2 Fn+1 = Fn+1 Fn + Fn+1 Fn−1

11. Use induction to prove that f (n) = 8n − 1 is divisible by 7 for all natural numbers n. Hint: For the induction step, show that 8k+1 − 1 = 7 · 8k + (8k − 1) 12. Use induction to prove that n3 − n is divisible by 3 for all natural numbers n.

APPENDIX C

13. Use induction to prove that 52n − 4n is divisible by 7 for all natural numbers n. 14. Use Pascal’s Triangle to write out the expansions of (a + b)6 and (a − b)4 . 15. Expand (x + x −1 )4 . 16. What is the coefficient of x 9 in (x 3 + x)5 ? n

 n 17. Let S(n) = . k k=0

INDUCTION AND THE BINOMIAL THEOREM

A17

(a) Use Pascal’s Triangle to compute S(n) for n = 1, 2, 3, 4. (b) Prove that S(n) = 2n for all n ≥ 1. Hint: Expand (a + b)n and evaluate at a = b = 1.

n  n k 18. Let T (n) = (−1) . k k=0

(a) Use Pascal’s Triangle to compute T (n) for n = 1, 2, 3, 4. (b) Prove that T (n) = 0 for all n ≥ 1. Hint: Expand (a + b)n and evaluate at a = 1, b = −1.

D ADDITIONAL PROOFS In this appendix, we provide proofs of several theorems that were stated or used in the text. Section 2.3

THEOREM 1 Basic Limit Laws Assume that lim f (x) and lim g(x) exist. Then: x→c x→c   (i) lim f (x) + g(x) = lim f (x) + lim g(x) x→c

x→c

x→c

(ii) For any number k, lim kf (x) = k lim f (x)  x→c   x→c  lim g(x) (iii) lim f (x)g(x) = lim f (x) x→c

x→c

x→c

(iv) If lim g(x)  = 0, then x→c

lim

x→c

lim f (x) f (x) x→c = g(x) lim g(x) x→c

Proof Let L = lim f (x) and M = lim g(x). The Sum Law (i) was proved in Section 2.6. x→c

x→c

Observe that (ii) is a special case of (iii), where g(x) = k is a constant function. Thus, it will suffice to prove the Product Law (iii). We write f (x)g(x) − LM = f (x)(g(x) − M) + M(f (x) − L) and apply the Triangle Inequality to obtain |f (x)g(x) − LM| ≤ |f (x)(g(x) − M)| + |M(f (x) − L)|

1

By the limit definition, we may choose δ > 0 so that |f (x) − L| < 1

if 0 < |x − c| < δ

If follows that |f (x)| < |L| + 1 for 0 < |x − c| < δ. Now choose any number  > 0. Applying the limit definition again, we see that by choosing a smaller δ if necessary, we may also ensure that if 0 < |x − c| < δ, then |f (x) − L| ≤

 2(|M| + 1)

and

|g(x) − M| ≤

 2(|L| + 1)

Using Eq. (1), we see that if 0 < |x − c| < δ, then |f (x)g(x) − LM| ≤ |f (x)| |g(x) − M| + |M| |f (x) − L|   + |M| ≤ (|L| + 1) 2(|L| + 1) 2(|M| + 1)   ≤ + = 2 2 Since  is arbitrary, this proves that lim f (x)g(x) = LM. To prove the Quotient Law x→c

(iv), it suffices to verify that if M  = 0, then lim

x→c

A18

1 1 = g(x) M

2

ADDITIONAL PROOFS

APPENDIX D

A19

For if Eq. (2) holds, then we may apply the Product Law to f (x) and g(x)−1 to obtain the Quotient Law: lim

x→c

  f (x) 1 1 lim = lim f (x) = lim f (x) x→c x→c x→c g(x) g(x) g(x)

1 L =L = M M

We now verify Eq. (2). Since g(x) approaches M and M  = 0, we may choose δ > 0 so that |g(x)| ≥ |M|/2 if 0 < |x − c| < δ. Now choose any number  > 0. By choosing a smaller δ if necessary, we may also ensure that

|M| |M − g(x)| < |M| 2

for 0 < |x − c| < δ

Then        1 1   M − g(x)   M − g(x)  |M|(|M|/2)   g(x) − M  =  Mg(x)  ≤  M(M/2)  ≤ |M|(|M|/2) =  Since  is arbitrary, the limit in Eq. (2) is proved. The following result was used in the text.

THEOREM 2 Limits Preserve Inequalities Let (a, b) be an open interval and let c ∈ (a, b). Suppose that f (x) and g(x) are defined on (a, b), except possibly at c. Assume that f (x) ≤ g(x)

for x ∈ (a, b),

x = c

and that the limits lim f (x) and lim g(x) exist. Then x→c

x→c

lim f (x) ≤ lim g(x)

x→c

x→c

Proof Let L = lim f (x) and M = lim g(x). To show that L ≤ M, we use proof by x→c

x→c

contradiction. If L > M, let  = 12 (L − M). By the formal definition of limits, we may choose δ > 0 so that the following two conditions are satisfied: |M − g(x)| < 

if |x − c| < δ

|L − f (x)| < 

if |x − c| < δ

But then f (x) > L −  = M +  > g(x) This is a contradiction since f (x) ≤ g(x). We conclude that L ≤ M.

A20

APPENDIX D

ADDITIONAL PROOFS

THEOREM 3 Limit of a Composite Function Assume that the following limits exist: L = lim g(x)

M = lim f (x)

and

x→c

x→L

Then lim f (g(x)) = M. x→c

Proof Let  > 0 be given. By the limit definition, there exists δ1 > 0 such that |f (x) − M| < 

if 0 < |x − L| < δ1

3

if 0 < |x − c| < δ

4

Similarly, there exists δ > 0 such that |g(x) − L| < δ1

We replace x by g(x) in Eq. (3) and apply Eq. (4) to obtain |f (g(x)) − M| < 

if 0 < |x − c| < δ

Since  is arbitrary, this proves that lim f (g(x)) = M. x→c

Section 2.4

THEOREM 4 Continuity of Composite Functions Let F (x) = f (g(x)) be a composite function. If g is continuous at x = c and f is continuous at x = g(c), then F (x) is continuous at x = c. Proof By definition of continuity, lim g(x) = g(c)

x→c

and

lim f (x) = f (g(c))

x→g(c)

Therefore, we may apply Theorem 3 to obtain lim f (g(x)) = f (g(c))

x→c

This proves that f (g(x)) is continuous at x = c. Section 2.6

THEOREM 5 Squeeze Theorem Assume that for x  = c (in some open interval containing c), l(x) ≤ f (x) ≤ u(x)

and

lim l(x) = lim u(x) = L

x→c

x→c

Then lim f (x) exists and x→c

lim f (x) = L

x→c

Proof Let  > 0 be given. We may choose δ > 0 such that |l(x) − L| < 

and

|u(x) − L| < 

if 0 < |x − c| < δ

In principle, a different δ may be required to obtain the two inequalities for l(x) and u(x), but we may choose the smaller of the two deltas. Thus, if 0 < |x − c| < δ, we have L −  < l(x) < L +  and L −  < u(x) < L + 

APPENDIX D

ADDITIONAL PROOFS

A21

Since f (x) lies between l(x) and u(x), it follows that L −  < l(x) ≤ f (x) ≤ u(x) < L +  and therefore |f (x) − L| <  if 0 < |x − c| < δ. Since  is arbitrary, this proves that lim f (x) = L as desired. x→c

Section 3.9

THEOREM 6 Derivative of the Inverse Assume that f (x) is differentiable and oneto-one on an open interval (r, s) with inverse g(x). If b belongs to the domain of g(x) and f  (g(b))  = 0, then g  (b) exists and g  (b) =

1 f  (g(b))

Proof The function f (x) is one-to-one and continuous (since it is differentiable). It follows that f (x) is monotonic increasing or decreasing on (r, s). For if not, then f (x) would have a local minimum or maximum at some point x = x0 . But then f (x) would not be one-to-one in a small interval around x0 by the IVT. Suppose that f (x) is increasing (the decreasing case is similar). We shall prove that g(x) is continuous at x = b. Let a = g(b), so that f (a) = b. Fix a small number  > 0. Since f (x) is an increasing function, it maps the open interval (a − , a + ) to the open interval (f (a − ), f (a + )) containing f (a) = b. We may choose a number δ > 0 so that (b − δ, b + δ) is contained in (f (a − ), f (a + )). Then g(x) maps (b − δ, b + δ) back into (a − , a + ). It follows that |g(y) − g(b)| < 

if 0 < |y − b| < δ

This proves that g is continuous at x = b. To complete the proof, we must show that the following limit exists and is equal to 1/f  (g(b)): g  (a) = lim

y→b

g(y) − g(b) y−b

By the inverse relationship, if y = f (x), then g(y) = x, and since g(y) is continuous, x approaches a as y approaches b. Thus, since f (x) is differentiable and f  (a) = 0, lim

y→b

Section 4.2

g(y) − g(b) x−a 1 1 = lim =  =  x→a f (x) − f (a) y−b f (a) f (g(b))

THEOREM 7 Existence of Extrema on a Closed Interval If f (x) is a continuous function on a closed (bounded) interval I = [a, b], then f (x) takes on a minimum and a maximum value on I . Proof We prove that f (x) takes on a maximum value in two steps (the case of a minimum is similar). Step 1. Prove that f (x) is bounded from above. We use proof by contradiction. If f (x) is not bounded from above, then there exist points an ∈ [a, b] such that f (an ) ≥ n for n = 1, 2, . . . . By Theorem 3 in Appendix B, we may choose a subsequence of elements an1 , an2 , . . . that converges to a limit in [a, b]—say, lim ank = L. Since f (x) is continuous, there exists δ > 0 such that k→∞

|f (x) − f (L)| < 1

if

x ∈ [a, b] and

|x − L| < δ

A22

ADDITIONAL PROOFS

APPENDIX D

Therefore, f (x) < f (L) + 1

x ∈ [a, b] and

if

x ∈ (L − δ, L + δ)

5

For k sufficiently large, ank lies in (L − δ, L + δ) because lim ank = L. By Eq. (5), k→∞

f (ank ) is bounded by f (L) + 1. However, f (ank ) = nk tends to infinity as k → ∞. This is a contradiction. Hence, our assumption that f (x) is not bounded from above is false. Step 2. Prove that f (x) takes on a maximum value. The range of f (x) on I = [a, b] is the set S = {f (x) : x ∈ [a, b]} By the previous step, S is bounded from above and therefore has a least upper bound M by the LUB Property. Thus f (x) ≤ M for all x ∈ [a, b]. To complete the proof, we show that f (c) = M for some c ∈ [a, b]. This will show that f (x) attains the maximum value M on [a, b]. By definition, M − 1/n is not an upper bound for n ≥ 1, and therefore, we may choose a point bn in [a, b] such that M−

1 ≤ f (bn ) ≤ M n

Again by Theorem 3 in Appendix B, there exists a subsequence of elements {bn1 , bn2 , . . . } in {b1 , b2 , . . . } that converges to a limit—say, lim bnk = c

k→∞

Let  > 0. Since f (x) is continuous, we may choose k so large that the following two conditions are satisfied: |f (c) − f (bnk )| < /2 and nk > 2/. Then |f (c) − M| ≤ |f (c) − f (bnk )| + |f (bnk ) − M| ≤

1    + ≤ + = 2 nk 2 2

Thus, |f (c) − M| is smaller than  for all positive numbers . But this is not possible unless |f (c) − M| = 0. Thus f (c) = M as desired. THEOREM 8 Continuous Functions Are Integrable If f (x) is continuous on [a, b], then f (x) is integrable over [a, b].

Section 5.2

Proof We shall make the simplifying assumption that f (x) is differentiable and that its derivative f  (x) is bounded. In other words, we assume that |f  (x)| ≤ K for some constant K. This assumption is used to show that f (x) cannot vary too much in a small interval. More precisely, let us prove that if [a0 , b0 ] is any closed interval contained in [a, b] and if m and M are the minimum and maximum values of f (x) on [a0 , b0 ], then

y

´

Slope f (c)

M

|M − m| ≤ K|b0 − a0 |

M−m m x a0

x1

c

x2

b0

FIGURE 1 Since M − m = f  (c)(x2 − x1 ), we conclude that M − m ≤ K(b0 − a0 ).

6

Figure 1 illustrates the idea behind this inequality. Suppose that f (x1 ) = m and f (x2 ) = M, where x1 and x2 lie in [a0 , b0 ]. If x1  = x2 , then by the Mean Value Theorem (MVT), there is a point c between x1 and x2 such that f (x2 ) − f (x1 ) M −m = = f  (c) x2 − x1 x2 − x 1

ADDITIONAL PROOFS

APPENDIX D

A23

Since x1 , x2 lie in [a0 , b0 ], we have |x2 − x1 | ≤ |b0 − a0 |, and thus, |M − m| = |f  (c)| |x2 − x1 | ≤ K|b0 − a0 | This proves Eq. (6). We divide the rest of the proof into two steps. Consider a partition P : P : y

x0 = a < x1
0. Since f (x) is continuous, there exists δ > 0 such that |f (x) − f (L)| < 

if 0 < |x − L| < δ

Since lim an = L, there exists N > 0 such that |an − L| < δ for n > N . Thus, n→∞

|f (an ) − f (L)| < 

for n > N

It follows that lim f (an ) = f (L). n→∞

Section 12.3

THEOREM 10 Clairaut’s Theorem If fxy and fyx are both continuous functions on a disk D, then fxy (a, b) = fyx (a, b) for all (a, b) ∈ D. Proof We prove that both fxy (a, b) and fyx (a, b) are equal to the limit L = lim

h→0

f (a + h, b + h) − f (a + h, b) − f (a, b + h) + f (a, b) h2

Let F (x) = f (x, b + h) − f (x, b). The numerator in the limit is equal to F (a + h) − F (a)

APPENDIX D

ADDITIONAL PROOFS

A25

and F  (x) = fx (x, b + h) − fx (x, b). By the MVT, there exists a1 between a and a + h such that F (a + h) − F (a) = hF  (a1 ) = h(fx (a1 , b + h) − fx (a1 , b)) By the MVT applied to fx , there exists b1 between b and b + h such that fx (a1 , b + h) − fx (a1 , b) = hfxy (a1 , b1 ) Thus, F (a + h) − F (a) = h2 fxy (a1 , b1 ) and h2 fxy (a1 , b1 ) = lim fxy (a1 , b1 ) = fxy (a, b) h→0 h→0 h2

L = lim

The last equality follows from the continuity of fxy since (a1 , b1 ) approaches (a, b) as h → 0. To prove that L = fyx (a, b), repeat the argument using the function F (y) = f (a + h, y) − f (a, y), with the roles of x and y reversed.

Section 12.4

THEOREM 11 Criterion for Differentiability If fx (x, y) and fy (x, y) exist and are continuous on an open disk D, then f (x, y) is differentiable on D. Proof Let (a, b) ∈ D and set L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) It is convenient to switch to the variables h and k, where x = a + h and y = b + k. Set f = f (a + h, b + k) − f (a, b) Then L(x, y) = f (a, b) + fx (a, b)h + fy (a, b)k and we may define the function e(h, k) = f (x, y) − L(x, y) = f − (fx (a, b)h + fy (a, b)k) To prove that f (x, y) is differentiable, we must show that lim

(h,k)→(0,0)

e(h, k) =0 √ h2 + k 2

To do this, we write f as a sum of two terms: f = (f (a + h, b + k) − f (a, b + k)) + (f (a, b + k) − f (a, b)) and apply the MVT to each term separately. We find that there exist a1 between a and a + h, and b1 between b and b + k, such that f (a + h, b + k) − f (a, b + k) = hfx (a1 , b + k) f (a, b + k) − f (a, b) = kfy (a, b1 )

A26

APPENDIX D

ADDITIONAL PROOFS

Therefore, e(h, k) = h(fx (a1 , b + k) − fx (a, b)) + k(fy (a, b1 ) − fy (a, b)) and for (h, k)  = (0, 0),      e(h, k)   h(fx (a1 , b + k) − fx (a, b)) + k(fy (a, b1 ) − fy (a, b))  √ =  √     h2 + k 2 h2 + k 2      h(fx (a1 , b + k) − fx (a, b))   k(fy (a, b1 ) − fy (a, b))  +  ≤  √ √    h2 + k 2 h2 + k 2   = |fx (a1 , b + k) − fx (a, b)| + fy (a, b1 ) − fy (a, b) In the second line, we use the Triangle Inequality (see Eq. (1) in Section 1.1), and we may  √   √  pass to the third line because h/ h2 + k 2  and k/ h2 + k 2  are both less than 1. Both terms in the last line tend to zero as (h, k) → (0, 0) because fx and fy are assumed to be continuous. This completes the proof that f (x, y) is differentiable.

ANSWERS TO ODDNUMBERED EXERCISES Chapter 1 Section 1.1 Preliminary Questions

53. Zeros: ±2; Increasing: x > 0; Decreasing: x < 0; Symmetry: f (−x) = f (x), so y-axis symmetry.

1. a = −3 and b = 1

y

2. The numbers a ≥ 0 satisfy |a| = a and | − a| = a. The numbers a ≤ 0 satisfy |a| = −a.

4 2

3. a = −3 and b = 1 4. (9, −4)

−2

−1

6. 3

7. (b)

8. Symmetry with respect to the origin

2

−4

5. (a) First quadrant. (b) Second quadrant. (c) Fourth quadrant. (d) Third quadrant.

x 1

−2

55. Zeros: 0, ±2; Symmetry: f (−x) = −f (x), so origin symmetry. y 10 5

Section 1.1 Exercises 1. r = 12337 3. |x| ≤ 2 5. |x − 2| < 2 7. |x − 3| ≤ 2 1250 9. −8 < x < 8 11. −3 < x < 2 13. (−4, 4) 15. √ (2, 6) √ 17. [− 74 , 94 ] 19. (−∞, 2) ∪ (6, ∞) 21. (−∞, − 3) ∪ ( 3, ∞) 23. (a) (i) (b) (iii) (c) (v) (d) (vi) (e) (ii) (f) (iv) 25. −3 < x < 1 29. |a + b − 13| = |(a − 5) + (b − 8)| ≤ |a − 5| + |b − 8| < 1 + 1 =1 2 2 31. (a) 11

−2

x

−1 −5

1

2

−10

57. This is an x-axis reflection of x 3 translated up 2 units. There is √ 3 one zero at x = 2. y

(b) 1

20 10

3 and r = 4 33. r1 = 11 2 15

−2

35. Let a = 1 and b = .9 (see the discussion before Example 1). The decimal expansions of a and b do not agree, but |1 − .9| < 10−k for all k.

x

−1 −10

1

2

−20

37. (a) (x − 2)2 + (y − 4)2 = 9

59. (B)

(b) (x − 2)2 + (y − 4)2 = 26

61. (a) Odd (b) Odd (c) Neither odd nor even

39. D = {r, s, t, u}; R = {A, B, E}

65. D : [0, 4]; R : [0, 4]

41. D : all reals; R : all reals

67.

y

y

y

4

4

8

3

3

6

45. D : all reals; R : {y : y ≥ 0}

2

2

4

47. D : {x : x = 0}; R : {y : y > 0}

1

1

43. D : all reals; R : all reals

49. On the interval (−1, ∞) 51. On the interval (0, ∞)

2

x 1

2 f (2x)

3

4

(d) Even

x

x 2

4 f (x/2)

6

8

1

2

3

4

2f (x)

A27

A28

ANSWERS TO ODD-NUMBERED EXERCISES

69.

25. (a) c = − 14

y 4

(c) No value for c that will make this slope equal to 0

3

1

29. b = 4

x

−4 −2

2

4

31. No, because the slopes between consecutive data points are not equal. √ 33. (a) 1 or − 14 (b) 1 ± 2

71. (a) D : [4, 8], R : [5, 9]. (b) D : [1, 5], R : [2, 6]. (c) D : [ 43 , 83 ], R : [2, 6]. (d) D : [4, 8], R : [6, 18]. 73. (a) h(x) = sin(2x − 10) (b) h(x) = sin(2x − 5) y y 75. 6

6

4

4

2 −2

−1

35. Minimum value is 0 37. Minimum value is −7 41. Maximum value is 13 39. Maximum value is 137 16 43.

y 10

2 x 1

2

−3

3

−2

8 x

−1

1

f (2x)

77.

2

6

3

f (x/2)

4

y

2 −4

2 1

−1

(d) c = 0

27. (a) 40.0248 cm (b) 64.9597 in (c) L = 65(1 + α(T − 100))

2

−3

(b) c = −2

x 1

2

3

D : all reals; R : {y | y ≥ 1}; f (x) = |x − 1| + 1 79. Even: even (f + g)(−x) = f (−x) + g(−x) = f (x) + g(x) = (f + g)(x) odd

Odd: (f + g)(−x) = f (−x) + g(−x) = −f (x) + −g(x) = −(f + g)(x) 85. (a) There are many possibilities, one of which is

−3

−2

x

−1

45. A double root occurs when c = ±2. There are no real roots when −2 < c < 2.  2 47. For all x ≥ 0, 0 ≤ x 1/2 − x −1/2 = x − 2 + x1 . √ √ 51. 4 + 2 2 and 4 − 2 2 x 2 −x 2 y 55. For x 2 , x = x22 −x11 = x2 + x1 . 59. (x − α)(x − β) = x 2 − αx − βx + αβ = x 2 + (−α − β)x + αβ

Section 1.3 Preliminary Questions

y

2

2 −2 1. One example is 3x 3

1

2. |x| is not a polynomial; |x 2 + 1| is a polynomial

7x +x−1

3. The domain of f (g(x)) is the empty set. −1

x 1

2

3

4

5

y = | x − 2|

(b) Let g(x) = f (x + a). Then g(−x) = f (−x + a) = f (a − x) = f (a + x) = g(x)

Section 1.2 Preliminary Questions 1. −4 2. No. 3. Parallel to the y-axis when b = 0; parallel to the x-axis when a=0 4. y = 9 5. −4 6. (x − 0)2 + 1

Section 1.2 Exercises 3. m = − 49 ; y = 13 ; x = 34 5. m = 3 7. m = − 34 9. y = 3x + 8 11. y = 3x − 12 13. y = −2 15. y = 3x − 2 17. y = 53 x − 13 19. y = 4 21. y = −2x + 9 23. 3x + 4y = 12 1. m = 3; y = 12; x = −4

4. Decreasing 5. One possibility is f (x) = ex − sin x

Section 1.3 Exercises 1. x ≥ 0 3. All reals 5. t = −2 7. u  = ±2 9. x  = 0, 1 11. y > 0 13. Polynomial 15. Algebraic 17. Transcendental 19. Rational 21. Transcendental 23. Rational 25. Yes √ √ 27. f (g(x)) = x + 1; D: x ≥ −1, g(f (x)) = x + 1; D: x ≥ 0 2

29. f (g(x)) = 2x ; D: R,

g(f (x)) = (2x )2 = 22x ; D: R

31. f (g(x)) = cos(x 3 + x 2 ); D: R, g(f (θ )) = cos3 θ + cos2 θ ; D: R 33. f (g(t)) = √ 1 2 ; D: Not valid for any t, −t  2 1 = − 1t ; D: t > 0 g(f (t)) = − √ t

35. 0.1(t+10) = 30 · 20.1t+1 = 2(30 · 20.1t ) = 2P (t); P (t + 10)  = 30 · 2 1 k(t+1/k) g t + k = a2 = a2kt+1 = 2a2kt = 2g(t)

ANSWERS TO ODD-NUMBERED EXERCISES

27. cos θ = − 45 29. Let’s start with the four points in Figure 23(A).

37. f (x) = x 2 : δf (x) = f (x + 1) − f (x) = (x + 1)2 − x 2 = 2x + 1 f (x) = x: δf (x) = x + 1 − x = 1 f (x) = x 3 : δf (x) = (x + 1)3 − x 3 = 3x 2 + 3x + 1 39.

•

The point in the first quadrant: sin θ = 0.918, cos θ = 0.3965, and tan θ =

δ(f + g) = (f (x + 1) + g(x + 1)) − (f (x) − g(x))

•

sin θ = 0.3965, cos θ = −0.918, and

δ(cf ) = cf (x + 1) − cf (x) = c(f (x + 1) − f (x)) = cδf (x).

tan θ =

Section 1.4 Preliminary Questions

•

1. Two rotations that differ by a whole number of full revolutions will have the same ending radius. 41π 3. − 5π 4. (a) 2. 9π 4 and 4 3 5. Let O denote the center of the unit circle, and let P be a point on the unit circle such that the radius OP makes an angle θ with the positive x-axis. Then, sin θ is the y-coordinate of the point P . 6. Let O denote the center of the unit circle, and let P be a point on the unit circle such that the radius OP makes an angle θ with the positive x-axis. The angle θ + 2π is obtained from the angle θ by making one full revolution around the circle. The angle θ + 2π will therefore have the radius OP as its terminal side.

The point in the third quadrant:

•

The point in the fourth quadrant: sin θ = −0.3965, cos θ = 0.918, and

−0.3965 = −0.4319. 0.918 Now consider the four points in Figure 23(B). tan θ =

The point in the first quadrant: sin θ = 0.918, cos θ = 0.3965, and

1. 5π/4 ◦ ◦ (b) 60◦ (c) 75◦ ≈ 23.87◦ 3. (a) 180 π ≈ 57.3 π 5. s = rθ = 3.6; s = rφ = 8 θ (cos θ, sin θ ) θ (cos θ, sin θ) 7. √   √ − 2, − 2 π 5π (0, 1) 4 2 2 2 √  √    2π −1 , 3 4π −1 , − 3 3 2 2 3 2 2  √ √  − 2, 2 3π 3π (0, −1) 4 2 2 2 √    √  − 3, 1 5π 1, − 3 5π 6 2 3 2 2 2 √  √ 2 − 2 7π π (−1, 0) 4 2 , 2  √  √  − 3 , −1 3 −1 11π 7π 6 2 2 6 2 , 2

(d) −135◦

7π 13. x = π , 2π 9. θ = π3 , 5π 11. θ = 3π 3 4 , 4 3 3 15. π π π π 2π 3π θ 6 4 3 2 3 4

sec θ

√2 3

√

2

√

3

2

und und

√ − 3

−1

−2

√ − 2

17. cos θ = sec1 θ = √ 1 2 = √ 1 2 1+c 1+tan θ 12 19. sin θ = 12 and tan θ = 5 13√ √ 2 53 21. sin θ = 53 , sec θ = 753 and cot θ = 72 23. 23/25

√

−0.918 = 2.3153. −0.3965

tan θ =

Section 1.4 Exercises

1

0.3965 = −0.4319. −0.918

sin θ = −0.918, cos θ = −0.3965, and

•

√1 3

0.918 = 2.3153. 0.3965

The point in the second quadrant:

= (f (x + 1) − f (x)) + (g(x + 1) − g(x)) = δf (x) + δg(x)

tan θ

A29

√

25. cos θ = − 521 and tan θ = − 2 2121

0.918 = 2.3153. 0.3965

tan θ = •

The point in the second quadrant: sin θ = 0.918, cos θ = −0.3965, and tan θ =

•

0.918 = −2.3153. 0.3965

The point in the third quadrant: sin θ = −0.918, cos θ = −0.3965, and tan θ =

•

−0.918 = 2.3153. −0.3965

The point in the fourth quadrant: sin θ = −0.918, cos θ = 0.3965, and tan θ =

5π 6

− √1

3

− √2

3

31. cos ψ = 0.3, sin ψ =

√ 0.91, cot ψ = √0.3 and 0.91

csc ψ = √ 1 0.91  √ √6 π 33. cos 3 + π4 = 2− 4 y 35. 2 1 x −1 −2

1

2

3

4

5

6

−0.918 = −2.3153. 0.3965

A30

ANSWERS TO ODD-NUMBERED EXERCISES

37.

1 + 3 11. f −1 (x) = 7x 7

y 1

4

x −0.5

y

y

0.5 1

2

3

4

5

4 y = f −1(x)

y = f (x)

6

2

−1

−4

39. If |c| > 1, no points of intersection; if |c| = 1, one point of intersection; if |c| < 1, two points of intersection. 4π 6π 8π 41. θ = 0, 2π 5 , 5 , π, 5 , 5 7π 3π 11π 43. θ = π6 , π2 , 5π 6 , 6 , 2 , 6

45. Starting from the double angle formula for cosine, cos2 θ = 12 (1 + cos 2θ), solve for cos 2θ.

2 x

−2

2

−4

4

x

−2

2

−2

−2

−4

−4

4

√

1−x 2 ; domain {x : x ≤ 0}: x

13. Domain {x : x ≥ 0}: f −1 (x) = √ 2 f −1 (x) = − 1−x x y

47. Substitute x = θ/2 into the double angle formula for sine, sin2 x = 12 (1 − cos 2x), then take the square root of both sides.

y = f −1(x)

1.5

49. cos(θ + π ) = cos θ cos π − sin θ sin π = cos θ(−1) = − cos θ

1

y = f (x)

sin(π−θ) − sin(−θ) sin θ 51. tan(π − θ ) = cos(π −θ) = − cos(−θ) = − cos θ = − tan θ.

0.5

sin 2x = 2 sin x cos x = 2 sin x cos x = sin x = tan x 53. 1+cos cos x 2x 1+2 cos2 x−1 2 cos2 x

−2

57. 16.928

x

−1

1

2

15. f −1 (x) = (x 2 − 9)1/3

Section 1.5 Preliminary Questions 1. (a), (b), (f)

y

2. No

8

3. Many different teenagers will have the same last name, so this function will not be one-to-one.

6

4. This function is one-to-one, and f −1 (6:27) = Hamilton Township.

2

4

−2

5. The graph of the inverse function is the reflection of the graph of y = f (x) through the line y = x. 6. (b) and (c)

y = f (x)

y = f −1(x) x 2

−2

4

6

8

17. Figures (B) and (C) y 19. (a)

7. Any angle θ < 0 or θ > π will work.

20 10

Section 1.5 Exercises 3. [−π/2, π/2]  3 • f (g(x)) = (x − 3)1/3 5. + 3 = x − 3 + 3 = x.  1/3  1/3 • g(f (x)) = x 3 + 3 − 3 = x3 = x. 7. v −1 (R) = 2GM 2 R

9. f −1 (x) = 4 − x. y 4

x

−1

1. f −1 (x) = x+4 7

1 −10 −20

(b) (−∞, ∞). (c) f −1 (3) = 1. √ 21. Domain x ≤ 1: f −1 (x) = 1 − x + 1; domain x ≥ 1: √ f −1 (x) = 1 + x + 1 23. 0 25. π4 27. π3 29. π3 31. π2 33. − π4 35. π √ √ 2 5 45. 4 1 √ 41. 43. 37. No inverse 39. 1−x x 3 3 x 2 −1 √ 1 47. 3 49. 20

f (x) = f −1(x) = 4 − x

3

Section 1.6 Preliminary Questions

2 1 x 1

2

3

4

1. (a) Correct (b) Correct (c) Incorrect (d) Correct 2. logb2 (b4 ) = 2 3. For 0 < x < 1 4. ln(−3) is not defined 5. This phrase is a verbal description of the general property of logarithms that states log(ab) = log a + log b.

A31

ANSWERS TO ODD-NUMBERED EXERCISES

7. Nothing. An appropriate viewing window: [50, 150] by [1000, 2000] y 9.

6. D: x > 0; R: real numbers 7. cosh x and sech x 8. sinh x and tanh x 9. Parity, identities and derivative formulas

2

Section 1.6 Exercises 1. 3. 15.

−8 −4

(a) 1 (b) 29 (c) 1 (d) 81 (e) 16 (f) 0 x = 1 5. x = −1/2 7. x = −1/3 9. k = 9 5 17. 1 19. 5 21. 1 23. 7 25. 29 3 3 6

27. (a) ln 1600 (b) ln(9x 7/2 )   31. x = −1 or x = 3 29. t = 15 ln 100 7

x 4

−2

11. 3

13. 0 11.

y

y 1

1

33. x = e

x 1

35. x

8 12 16

−3

0

5

sinh x =

ex − e−x 2

−10.0179

0

74.203

cosh x =

ex + e−x 2

10.0677

1

74.210

2

3

4

5

x

6

3.5 3.6 3.7 3.8 3.9

−1

4

−1

y 0.4 0.2 3.76 3.78 3.8 3.82 3.84 x −0.2

37. Let a = e2 and b = e3 39.

e−x − e−(−x) e−x − ex ex − e−x tanh(−x) = −x = = − = − tanh x −x x e +e ex + e−x e + e−(−x) 41. a = 8; 1000 earthquakes 47. (a) By Galileo’s law, w = 500 + 10 = 510 m/s. Using Einstein’s law, w = c · tanh(1.7 × 10−6 ) ≈ 510 m/s. (b) By Galileo’s law, u + v = 107 + 106 = 1.1 × 107 m/s. By Einstein’s law, w ≈ c · tanh(0.036679) ≈ 1.09988 × 107 m/s. 49. Let y = logb x. Then x = by and loga x = loga by = y loga b.

13. The table and graphs below suggest that as n gets large, n1/n approaches 1.

log x

Thus, y = loga b . a 51. 13 cosh x − 3 sinh x

n

n1/n

10 102 103 104 105 106

1.258925412 1.047128548 1.006931669 1.000921458 1.000115136 1.000013816

y

y

1

1

Section 1.7 Preliminary Questions 1. No 2. (a) The screen will display nothing. (b) The screen will display the portion of the parabola between the points (0, 3) and (1, 4). 3. No 4. Experiment with the viewing window to zoom in on the lowest point on the graph of the function. The y-coordinate of the lowest point on the graph is the minimum value of the function.

x 0

2

4

x

10

0

200 400 600 800 1000



n2 1 + n1

n 10 102 103 104 105 106

y 20 10 −4 −3 −2 −1 −10

8

15. The table and graphs below suggest that as n gets large, f (n) tends toward ∞.

Section 1.7 Exercises 1.

6

13780.61234 1.635828711 × 1043 1.195306603 × 10434 5.341783312 × 104342 1.702333054 × 1043429 1.839738749 × 10434294

y

y

x 1

2

3

1 × 10 43

10,000

−20

x = −3, x = −1.5, x = 1, and x = 2 3. Two positive solutions 5. There are no solutions

x 0

2

4

6

8

10

x 0

20

40

60

80 100

A32

ANSWERS TO ODD-NUMBERED EXERCISES

17. The table and graphs below suggest that as x gets large, f (x) approaches 1. x x tan x1

10 102 103 104 105 106

1. {x : |x − 7| < 3} 3. [−5, −1] ∪ [3, 7] 5. (x, 0) with x ≥ 0; (0, y) with y < 0



x

Chapter 1 Review

7.

1.033975759 1.003338973 1.000333389 1.000033334 1.000003333 1.000000333

y

y 5

5

4

4

3

3

2

2 1

1 x

−2 −1

1

2

3

x

−2 −1

4

1

f (x) + 2

y

y

1.5 1.4 1.3 1.2 1.1 1

9.

1.5 1.4 1.3 1.2 1.1 1

19.

10

15

20

20

40

60

1 6

−2

8

15. (a) Decreasing x 2

−1

4

6

8

3

4

(A, B) = (1, 2)

(b) Neither

(c) Neither

(d) Increasing

17. 2x − 3y = −14 19. 6x − y = 53 21. x + y = 5

−2

(A, B) = (1, 1)

2

13. D : {x : x  = 3}; R : {y : y  = 0}

2

4

x 1

11. D : {x : x ≥ −1}; R : {y : y ≥ 0}

1

2

y

−4 −3 −2 −1

80 100

y

−1

f (x + 2)

1

y

−2

4

2

x

x

3

3

x 5

2

23. Yes

25. Roots: x = −2, x = 0 and x = 2; decreasing: x < −1.4 and 0 < x < 1.4

y

y

4 2 −2

20 x

−2

2

4

6

10

8 −1

−4

1

−3 −2

x 2

3

(A, B) = (3, 4)

27. f (x) = 10x 2 + 2x + 5; minimum value is 49 10

21. x ∈ (−2, 0) ∪ (3, ∞)

29.

y 1

1





0.8

x 2 + 6x + 1 = 4(x + 1) ⎛ ⎞ 4 3 2 x 1 x 2 + 6x + 1 ⎠ = x + 28x + 70x + 28x + 1 + 2 f4 (x) = ⎝ x +6x+1 2 4(x + 1) 8(1 + x)(1 + 6x + x 2 ) 1 f3 (x) = 2

31.

y

23. 1 x (x + 1) + 1 2 (x + 1) 2

0.5

0.6 0.4

−1

−0.5

4(x+1)

x

−5

0.2

5

10

−0.5

x 0.5

1

−1

33.

y 2

and

1

1 + 120x + 1820x 2 + 8008x 3 + 12870x 4 + 8008x 5 + 1820x 6 + 120x 7 + x 8 f5 (x) = . 16(1 + x)(1 + 6x + x 2 )(1 + 28x + 70x 2 + 28x 3 + x 4 ) It appears as if the fn are asymptotic to

√ x.

−4 −3 −2 −1 −1 −2

x 1 2 3 4

ANSWERS TO ODD-NUMBERED EXERCISES

35. Let g(x) = f ( 13 x). Then     g(x − 3b) = f 13 (x − 3b) = f 13 x − b . The graph of y = | 13 x − 4|: y 4 3 2 1 x 0

5

10

15

20

f (t) = t 4 and g(t) = 12t + 9 39. 4π (a) a = b = π/2 (b) a = π x = π/2, x = 7π/6, x = 3π/2 and x = 11π/6 There are no solutions (a) No match. (b) No match. (c) (i) (d) (iii) 3 49. f −1 (x) = x 2 + 8; D : {x : x ≥ 0}; R : {y : y ≥ 2} √ √ 51. For {t : t ≤ 3}, h−1 (t) = 3 − t. For t ≥ 3, h−1 (t) = 3 + t. 53. (a) (iii) (b) (iv) (c) (ii) (d) (i)

37. 41. 43. 45. 47.

Chapter 2 Section 2.1 Preliminary Questions 1. The graph of position as a function of time 2. No. Instantaneous velocity is defined as the limit of average velocity as time elapsed shrinks to zero. 3. The slope of the line tangent to the graph of position as a function of time at t = t0 4. The slope of the secant line over the interval [x0 , x1 ] approaches the slope of the tangent line at x = x0 . 5. The graph of atmospheric temperature as a function of altitude. Possible units for this rate of change are ◦ F/ft or ◦ C/m.

Section 2.1 Exercises 1. (a) 11.025 m (b) 22.05 m/s (c)

7. (a) Dollars/year (b) [0, 0.5]: 7.8461; [0, 1]: 8 (c) Approximately $8/yr 9. (a) Approximately 0.283 million Internet users per year. (b) Decreases (c) Approximately 0.225 million Internet users per year. (d) Greater than 11. 12 13. −0.06 15. 1.00 17. 0.333 19. (a) [0, 0.1]: −144.721 cm/s; [3, 3.5]: 0 cm/s (b) 0 cm/s 21. (a) Seconds per meter; measures the sensitivity of the period of the pendulum to a change in the length of the pendulum. (b) B: average rate of change in T from L = 1 m to L = 3 m; A: instantaneous rate of change of T at L = 3 m. (c) 0.4330 s/m. 23. Sales decline more slowly as time increases. • In graph (A), the particle is (c) slowing down. 25. • In graph (B), the particle is (b) speeding up and then slowing down. • In graph (C), the particle is (d) slowing down and then speeding up. • In graph (D), the particle is (a) speeding up. 27. (a) Percent /day; measures how quickly the population of flax plants is becoming infected. (b) [40, 52], [0, 12], [20, 32] (c) The average rates of infection over the intervals [30, 40], [40, 50], [30, 50] are .9, .5, .7 %/d, respectively. (d) 0.55%/d 100 80 60 40 20 10

[2, 2.01]

[2, 2.005]

[2, 2.001]

[2, 2.00001]

average velocity

19.649

19.6245

19.6049

19.600049

The instantaneous velocity at t = 2 is 19.6 m/s. 3. 0.57735 m/(s · K) 5. 0.3 m/s

10 8 6 4 2 t 1

1.5

30

40

50

60

31. (B) 33. Interval [1, t]: average rate of change is t + 1; interval [2, t]: average rate of change is t + 2 35. x 2 + 2x + 4

1. 1 2. π 3. 20 4. Yes 5. limx→1− f (x) = ∞ and limx→1+ f (x) = 3 6. No 7. Yes

Section 2.2 Exercises 1.

h

0.5

20

Section 2.2 Preliminary Questions

time interval

2

2.5

A33

x

0.998

0.999

0.9995

0.99999

f (x)

1.498501

1.499250

1.499625

1.499993

x

1.00001

1.0005

1.001

1.002

f (x)

1.500008

1.500375

1.500750

1.501500

3

The limit as x → 1 is 32 .

A34

ANSWERS TO ODD-NUMBERED EXERCISES

3.

•

y

1.998

1.999

1.9999

f (y)

0.59984

0.59992

0.599992

y

2.0001

2.001

2.02

f (y)

0.600008

0.60008

0.601594

lim f (x) = lim f (x) = ∞

x→6− 55. 52

x→6+

y 2.50 2.48

The limit as y → 2 is 35 . 5. 1.5 7. 21 9. |3x − 12| = 3|x − 4| 11. |(5x + 2) − 17| = |5x − 15| = 5|x − 3| 13. Suppose |x| < 1, so that |x 2 − 0| = |x + 0||x − 0| = |x||x| < |x| 15. If |x| < 1, |4x + 2| can be no bigger than 6, so |4x 2 + 2x + 5 − 5| = |4x 2 + 2x| = |x||4x + 2| < 6|x| 17. 12 19. 53 21. 2 23. 0 25. As x → 4−, f (x) → −∞; similarly, as x → 4+, f (x) → ∞ 27. −∞ 29. 0 31. 1 33. 2.718 (The exact answer is e.) 35. ∞ y 37.

2.46 2.44 2.42

57. 0.693 (The exact answer is ln 2.) y 0.6940 0.6935 y=

x

2 − cos x x

0.6930

0.6925 0.6920

2

59. −12 1

y x

−1

1

2

x

−11.4

3

−11.6 −11.8 −12

(a) c − 1 (b) c 39. lim f (x) = −1, lim f (x) = 1 x→0−

61. For n even

x→0+

1 41. lim f (x) = ∞, lim f (x) = 6 x→0− x→0+

63. (a) No

(c) At x = 1, 13 , 15 , . . ., the value of f (x) is always −1. sin nθ 65. lim =n θ→0 θ n xn − 1 67. 12 , 2, 23 , 23 ; lim m = m x→1 x − 1 69. (a)

4x 2 + 7 4x 2 + 7 43. lim = −∞, lim =∞ 3 x→−2− x + 8 x→−2+ x 3 + 8 x5 + x − 2 =2 45. lim 2 x→1± x + x − 2 • lim f (x) = ∞ and lim f (x) = ∞. 47. •

x→2−

1 ) = 1 for all integers n. (b) f ( 2n

x→2+

lim f (x) = −∞ and lim f (x) = 10.

x→4−

x→4+

y

The vertical asymptotes are the vertical lines x = 2 and x = 4. y y 51. 49. 6

5.565 5.555 y=

3

5.545

2x − 8 x−3

2

4

5.535

1

2

5.525

x −1

1

2

3

4

5

x=3

x 1 •

53. • • • •

2

3

4

lim f (x) = lim f (x) = 3

x→1−

x→1+

lim f (x) = −∞

x→3−

lim f (x) = 4

x→3+

lim f (x) = 2

x→5−

lim f (x) = −3

x→5+

(b) L = 5.545.

Section 2.3 Preliminary Questions 1. Suppose limx→c f (x) and limx→c g(x) both exist. The Sum Law states that lim (f (x) + g(x)) = lim f (x) + lim g(x).

x→c

x→c

x→c

ANSWERS TO ODD-NUMBERED EXERCISES

Provided limx→c g(x) = 0, the Quotient Law states that lim

f (x)

x→c g(x)

2. (b)

=

limx→c f (x) . limx→c g(x)

A35

15. ex and cos 3x are continuous, so ex cos 3x is continuous by Continuity Law (iii). 17. Discontinuous at x = 0, at which there is an infinite discontinuity. The function is neither left- nor right-continuous at x = 0. 19. Discontinuous at x = 1, at which there is an infinite discontinuity. The function is neither left- nor right-continuous at x = 1.

3. (a)

Section 2.3 Exercises 1 1. 9 3. 16 5. 12 7. 4.6 9. 1 11. 9 13. − 25 15. 10 1 17. 15 19. 15 21. 25 23. 64 27. 3 29. 16 31. No 33. f (x) = 1/x and g(x) = −1/x 35. Write g(t) = tg(t) t 37. (b)

Section 2.4 Preliminary Questions 1. Continuity 2. f (3) = 12 3. No 4. No; Yes 5. (a) False. The correct statement is “f (x) is continuous at x = a if the left- and right-hand limits of f (x) as x → a exist and equal f (a).” (b) True. (c) False. The correct statement is “If the left- and right-hand limits of f (x) as x → a are equal but not equal to f (a), then f has a removable discontinuity at x = a."

21. Discontinuous at even integers, at which there are jump discontinuities. Function is right-continuous at the even integers but not left-continuous. 23. Discontinuous at x = 12 , at which there is an infinite discontinuity. The function is neither left- nor right-continuous at x = 12 . 25. Continuous for all x 27. Jump discontinuity at x = 2. Function is left-continuous at x = 2 but not right-continuous. 29. Discontinuous whenever t = (2n+1)π , where n is an integer. At 4 every such value of t there is an infinite discontinuity. The function is neither left- nor right-continuous at any of these points of discontinuity. 31. Continuous everywhere 33. Discontinuous at x = 0, at which there is an infinite discontinuity. The function is neither left- nor right-continuous at x = 0.

(e) False. The correct statement is “If f (x) and g(x) are continuous at x = a and g(a) = 0, then f (x)/g(x) is continuous at x = a."

35. The domain is all real numbers. Both sin x and cos x are continuous on this domain, so 2 sin x + 3 cos x is continuous by Continuity Laws (i) and (ii). √ 37. Domain is x ≥ 0. Since x and sin x are continuous, so is √ x sin x by Continuity Law (iii).

Section 2.4 Exercises

39. Domain is all real numbers. Both x 2/3 and 2x are continuous on this domain, so x 2/3 2x is continuous by Continuity Law (iii).

(d) True.

The function f is discontinuous at x = 1; it is right-continuous there. • The function f is discontinuous at x = 3; it is neither left-continuous nor right-continuous there. • The function f is discontinuous at x = 5; it is left-continuous there. None of these discontinuities is removable. 3. x = 3; redefine g(3) = 4 5. The function f is discontinuous at x = 0, at which lim f (x) = ∞ and lim f (x) = 2. The function f is also

1.

x→0−

•

x→0+

discontinuous at x = 2, at which lim f (x) = 6 and x→2−

lim f (x) = 6. The discontinuity at x = 2 is removable. Assigning

x→2+

f (2) = 6 makes f continuous at x = 2. 7. x and sin x are continuous, so is x + sin x by Continuity Law (i) 9. Since x and sin x are continuous, so are 3x and 4 sin x by Continuity Law (ii). Thus 3x + 4 sin x is continuous by Continuity Law (i). 11. Since x is continuous, so is x 2 by Continuity Law (iii). Recall that constant functions, such as 1, are continuous. Thus x 2 + 1 is 1 continuous by Continuity Law (i). Finally, 2 is continuous by x +1 2 Continuity Law (iv) because x + 1 is never 0. 13. The function f (x) is a composite of two continuous functions: cos x and x 2 , so f (x) is continuous by Theorem 5.

41. Domain is x  = 0. Because the function x 4/3 is continuous and not equal to zero for x  = 0, x −4/3 is continuous for x  = 0 by Continuity Law (iv). 43. Domain is all x  = ±(2n − 1)π/2 where n is a positive integer. Because tan x is continuous on this domain, it follows from Continuity Law (iii) that tan2 x is also continuous on this domain. 45. Domain of (x 4 + 1)3/2 is all real numbers. Because x 3/2 and the polynomial x 4 + 1 are both continuous, so is the composite function (x 4 + 1)3/2 . 47. Domain is all x  = ±1. Because the functions cos x and x 2 are continuous on this domain, so is the composite function cos(x 2 ). Finally, because the polynomial x 2 − 1 is continuous and not equal to 2) is continuous by Continuity zero for x  = ±1, the function cos(x x 2 −1 Law (iv).

49. f (x) is right-continuous at x = 1; f (x) is continuous at x = 2 51. The function f is continuous everywhere. y 1

x

−1

1 −1

2

3

A36

ANSWERS TO ODD-NUMBERED EXERCISES

53. The function f is neither left- nor right-continuous at x = 2. y

x→4

of 2 obtained in Exercise 23.

5 4 3 2 1 −2

√

23. 2 25. 14 27. 1 29. 9 31. 22 33. 12 35. lim f (x) ≈ 2.00; to two decimal places, this matches the value y 2.001 2.000 1.999 1.998 1.997 1.996

x 4

−1

6

−16 = lim (x + 4) = 8  = 10 = f (4) 55. lim xx−4 x→4 x→4 5 57. c = 3 59. a = 2 and b = 1 61. (a) No (b) g(1) = − π2 y y 63. 65. 2

4

4

3

3

x 3.6 3.8 4.0 4.2 4.4

37. 12 39. −1 41. 43 43. 14 45. 2a 47. −4 + 5a 49. 4a 1 51. √ 53. 3a 2 55. c = −1 and c = 6 57. c = 3 59. + 2 a

Section 2.6 Preliminary Questions 1. limx→0 f (x) = 0; No 2. Assume that for x  = c (in some open interval containing c),

2

1

1

x 1

2

3

4

l(x) ≤ f (x) ≤ u(x)

x

5

1

2

3

4

5

1 67. −6 69. 13 71. −1 73. 32 75. 27 77. 1000 79. π2 −1 81. No. Take f (x) = −x and g(x) = x −1 83. f (x) = |g(x)| is a composition of the continuous functions g(x) and |x| 85. No. y

and that lim l(x) = lim u(x) = L. Then lim f (x) exists and x→c

x→c

x→c

lim f (x) = L.

x→c

3. (a)

Section 2.6 Exercises 1. For all x  = 1 on the open interval (0, 2) containing x = 1,

(x) ≤ f (x) ≤ u(x). Moreover,

15,000 10,000

lim (x) = lim u(x) = 2.

5000 40,000 20,000

80,000

x→1 x

60,000

x→1

Therefore, by the Squeeze Theorem, lim f (x) = 2.

x→1

87. f (x) = 3 and g(x) = [x]

3. lim f (x) = 6 x→7

Section 2.5 Preliminary Questions 2 1. √x −1

x+3−2

−1 (b) f (x) = x −1 (c) f (x) = 1 2. (a) f (x) = xx−1 x x−1 3. The “simplify and plug-in” strategy is based on simplifying a function which is indeterminate to a continuous function. Once the simplification has been made, the limit of the remaining continuous function is obtained by evaluation. 2

2

Section 2.5 Exercises −36 = lim (x−6)(x+6) = lim (x + 6) = 12 1. lim xx−6 x−6 x→6 x→6 x→6 1 11 11. 2 13. 1 15. 2 17. 1 3. 0 5. 14 7. −1 9. 10 8 7 19. 17 21. Limit does not exist. √ h+2−2 • As h → 0+, → −∞. √ h h+2−2 • As h → 0−, → ∞. h 2

5. (a) not sufficient information (b) limx→1 f (x) = 1 (c) limx→1 f (x) = 3 π =0 7. lim x 2 cos x1 = 0 9. lim (x − 1) sin x−1 x→0

x→1

11. lim (2t − 1) cos 1t = 0 t→0

1 =0 13. lim (t 2 − 4) cos t−2 t→2

15. limπ cos θ cos(tan θ) = 0 θ→ 2

√

19. 3 21. 1 23. 0 25. 2 π 2 27. (b) L = 14 29. 9 1 33. 7 35. 1 3 1 6 5 3 25 37. 6 39. − 4 41. 2 43. 5 45. 0 9 0 49. −1 53. − 2 √ √ √ √ 1 − cos t 2 1 − cos t 2 = ; lim =− 55. lim t 2 t→0− t 2 t→0+ 59. (a) 17. 31. 47.

1

x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

.999983

.99999983

.99999983

.999983

Here c = 0 and cos c = 1.

ANSWERS TO ODD-NUMBERED EXERCISES

x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

.868511

.866275

.865775

.863511

Section 2.7 Exercises 1. y = 1 and y = 2 3.

y

√ Here c = π6 and cos c = 23 ≈ .866025.

3 −4

x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

.504322

.500433

.499567

.495662

−2

2

x

−1 −5 −9 −13

5. (a) From the table below, it appears that

Here c = π3 and cos c = 12 . x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

.710631

.707460

.706753

.703559

x→±∞ x 3 + x

x f (x)

x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

.005000

.000500

−.000500

−.005000

±50 .999600

±100 .999900

= 1.

±500 .999996

±1000 .999999

(b) From the graph below, it also appears that x3

lim

x→±∞ x 3 + x

Here c = π2 and cos c = 0. sin x − sin c = cos c. (b) lim x→c x−c (c)

= 1.

y 1.0

x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

−.411593

−.415692

−.416601

−.420686

0.8 0.6 0.4 0.2

Here c = 2 and cos c = cos 2 ≈ −.416147.

x

−5

x

c − .01

c − .001

c + .001

c + .01

sin x − sin c x−c

.863511

.865775

.866275

.868511

√

Here c = − π6 and cos c = 23 ≈ .866025.

Section 2.7 Preliminary Questions 1. (a) Correct (b) Not correct (c) Not correct (d) Correct 2. (a) limx→∞ x 3 = ∞ (b) limx→−∞ x 3 = −∞ (c) limx→−∞ x 4 = ∞ y 3.

x

7. 1 9. 0 11. 74 13. −∞ 15. ∞ 17. y = 14 19. y = 23 1 and y = − 23 21. y = 0 23. 0 25. 2 27. 16 29. 0 π −1 31. 2 ; the graph of y = tan x has a horizontal asymptote at y = π2 As A = lim 33. (a) lim R(s) = lim = A. s→∞ s→∞ K + s s→∞ 1 + K s

AK AK A (b) R(K) = = = half of the limiting value. K +K 2K 2 (c) 3.75 mM 35. 0 37. ∞ 39. ln 32 41. − π2 3 3x 2 − x 3−t 45. lim = = lim x→∞ 2x 2 + 5 2 t→0+ 2 + 5t 2 • b = 0.2: 47. x f (x)

1 = sin 0 = 0. x

x→∞ 1 On the other hand, x → ±∞ as x → 0, and as x1 → ±∞, sin x1

5

(c) The horizontal asymptote of f (x) is y = 1.

4. Negative 5. Negative 6. As x → ∞, x1 → 0, so lim sin

x3

lim

√ Here c = π4 and cos c = 22 ≈ 0.707107.

oscillates infinitely often.

A37

•

5 1.000064

10 1.000000

50 1.000000

100 1.000000

50 1.000000

100 1.000000

It appears that G(0.2) = 1. b = 0.8: x f (x)

5 1.058324

10 1.010251

A38

ANSWERS TO ODD-NUMBERED EXERCISES

•

x f (x) •

5 2.012347

50 2.000000

100 2.000000

10 3.000005

50 3.000000

100 3.000000

9. Let f (x) = x 2 . Observe that f is continuous with f (1) = 1 and f (2) = 4. Therefore, by the IVT there is a c ∈ [1, 2] such that f (c) = c2 = 2.

It appears that G(2) = 2. b = 3: x f (x)

•

10 2.000195

5. Let f (x) = x − cos x. Observe that f is continuous with f (0) = −1 and f (1) = 1 − cos 1 ≈ .46. Therefore, by the IVT there is a c ∈ [0, 1] such that f (c) = c − cos c = 0. √ √ that f is continuous on  f (x) = x + x + 2 − 3. Note

7. Let √ 1 , 2 with f ( 1 ) = −1 and f (2) = 2 − 1 ≈ .41. Therefore, by the 4 4  √ √ IVT there is a c ∈ 14 , 2 such that f (c) = c + c + 2 − 3 = 0.

It appears that G(0.8) = 1. b = 2:

5 3.002465

It appears that G(3) = 3. b = 5: x f (x)

5 5.000320

10 5.000000

50 5.000000

100 5.000000

It appears that G(5) = 5. Based on these observations we conjecture that G(b) = 1 if 0 ≤ b ≤ 1 and G(b) = b for b > 1. The graph of y = G(b) is shown below; the graph does appear to be continuous. y

11. For each positive integer k, let f (x) = x k − cos x. Observe that   k  f is continuous on 0, π2 with f (0) = −1 and f ( π2 ) = π2 > 0.  π Therefore, by the IVT there is a c ∈ 0, 2 such that f (c) = ck − cos(c) = 0. 13. Let f (x) = 2x + 3x − 4x . Observe that f is continuous on [0, 2] with f (0) = 1 > 0 and f (2) = −3 < 0. Therefore, by the IVT, there is a c ∈ (0, 2) such that f (c) = 2c + 3c − 4c = 0. 15. Let f (x) = ex + ln x. Observe that f is continuous on [e−2 , 1] −2 with f (e−2 ) = ee − 2 < 0 and f (1) = e > 0. Therefore, by the IVT, there is a c ∈ (e−2 , 1) ⊂ (0, 1) such that f (c) = ec + ln c = 0. 17. (a) f (1) = 1, f (1.5) = 21.5 − (1.5)3 < 3 − 3.375 < 0. Hence, f (x) = 0 for some x between 1 and 1.5.

4 3 2 1 x 0

1

2

3

4

Section 2.8 Preliminary Questions 1. Observe that f (x) = x 2 is continuous on [0, 1] with f (0) = 0

and f (1) = 1. Because f (0) < 0.5 < f (1), the Intermediate Value Theorem guarantees there is a c ∈ [0, 1] such that f (c) = 0.5. 2. We must assume that temperature is a continuous function of time. 3. If f is continuous on [a, b], then the horizontal line y = k for every k between f (a) and f (b) intersects the graph of y = f (x) at least once. y 4.

(b) f (1.25) ≈ 0.4253 > 0 and f (1.5) < 0. Hence, f (x) = 0 for some x between 1.25 and 1.5. (c) f (1.375) ≈ −0.0059. Hence, f (x) = 0 for some x between 1.25 and 1.375. 19. [0, .25] 21.

23.

y

y

4

6

3

5

2

4

1

3 x 1

2

3

4

2 1 x 1

2

3

4

−1

25. No; no f (a) f (b) a

x

b

5. (a) Sometimes true. (b) Always true. (c) Never true. (d) Sometimes true.

Section 2.8 Exercises 1. Observe that f (1) = 2 and f (2) = 10. Since f is a polynomial, it is continuous everywhere; in particular on [1, 2]. Therefore, by the IVT there is a c ∈ [1, 2] such that f (c) = 9. π2

3. g(0) = 0 and g( π4 ) = 16 . g(t) is continuous for all t between 0 2 and π4 , and 0 < 12 < π16 ; therefore, by the IVT, there is a c ∈ [0, π4 ] such that g(c) = 21 .

Section 2.9 Preliminary Questions 1. (c) 2. (b) and (d) are true

Section 2.9 Exercises 1. L = 4,  = .8, and δ = .1 3. (a) |f (x) − 35| = |8x + 3 − 35| = |8x − 32| = |8(x − 4)| = 8 |x − 4| (b) Let  > 0. Let δ = /8 and suppose |x − 4| < δ. By part (a), |f (x) − 35| = 8|x − 4| < 8δ. Substituting δ = /8, we see |f (x) − 35| < 8/8 = .

ANSWERS TO ODD-NUMBERED EXERCISES

 5. (a)  If 0 < |x − 2| < δ = .01, then |x| < 3 and  2  x − 4 = |x − 2||x + 2| ≤ |x − 2| (|x| + 2) < 5|x − 2| < .05. (b) If 0 < |x − 2| < δ = .0002, then |x| < 2.0002 and     2 x − 4 = |x − 2||x + 2| ≤ |x − 2| (|x| + 2) < 4.0002|x − 2| < .00080004 < .0009 (c) δ = 10−5 7. δ = 6 × 10−4 9. δ = 0.25

   . δ = min |c|, 3|c|

Then, for |x − c| < δ, we have |x 2 − c2 | = |x − c| |x + c| < 3|c|δ < 3|c|

 = . 3|c|

19. Let  > 0 be given. Let δ = min(1, 3). If |x − 4| < δ,    1  √  < |x − 4| 1 < δ 1 < 3 1 = . | x − 2| = |x − 4|  √ 3 3 3 x + 2 21. Let  > 0 be given. Let δ = min(1, 7 ), and assume |x − 1| < δ. Since δ < 1, 0 < x < 2. Since x 2 + x + 1 increases as x increases for x > 0, x 2 + x + 1 < 7 for 0 < x < 2, and so          3 x − 1 = |x − 1| x 2 + x + 1 < 7|x − 1| < 7 = . 7

y 3.1 3.0 2.9 x

2.8 4.6

4.8

5

5.2

5.4

11. δ = 0.02 y 1.02 1.01 1.00 0.99 0.98 0.97

17. Given  > 0, we let

A39

23. Let  > 0 be given. Let δ = min(1, 45 ), and suppose |x − 2| < δ. Since δ < 1, |x − 2| < 1, so 1 < x < 3. This means that 4x 2 > 4 and |2 + x| < 5, so that 2+x2 < 54 . We get: 4x     2 + x  5  −2 1  5 4  x −  = |2 − x|   4x 2  < 4 |x − 2| < 4 · 5  = .  4 25. Let L be any real number. Let δ > 0 be any small positive number. Let x = 2δ , which satisfies |x| < δ, and f (x) = 1. We consider two cases: •

−0.04 −0.02

x 0

•

0.02 0.04

13. (a) Since |x − 2| < 1, it follows that 1 < x < 3, in particular 1 that x > 1. Because x > 1, then < 1 and x     1     − 1  =  2 − x  = |x − 2| < 1 |x − 2|. x 2   2x  2x 2 (b) Let δ = min{1, 2} and suppose that |x − 2| < δ. Then by part (a) we have   1   − 1  < 1 |x − 2| < 1 δ < 1 · 2 = . x 2 2 2 2

(|f (x) − L| ≥ 12 ) : we are done. (|f (x) − L| < 12 ): This means 12 < L < 32 . In this case, let x = − 2δ . f (x) = −1, and so 32 < L − f (x).

In either case, there exists an x such that |x| < 2δ , but |f (x) − L| ≥ 12 . 27. Let  > 0 and let δ = min(1, 2 ). Then, whenever |x − 1| < δ, it follows that 0 < x < 2. If 1 < x < 2, then min(x, x 2 ) = x and  |f (x) − 1| = |x − 1| < δ < < . 2 On the other hand, if 0 < x < 1, then min(x, x 2 ) = x 2 , |x + 1| < 2 and |f (x) − 1| = |x 2 − 1| = |x − 1| |x + 1| < 2δ < . Thus, whenever |x − 1| < δ, |f (x) − 1| < . 31. Suppose that lim f (x) = L. Let  > 0 be given. Since x→c

lim f (x) = L, we know there is a δ > 0 such that |x − c| < δ forces

(c) Choose δ = .02.

x→c

(d) Let  > 0 be given. Then whenever 0 < |x − 2| < δ = min {1, 2}, we have    1  − 1  < 1 δ ≤ . x 2 2 15.

y 1.0 0.8

|f (x) − L| < /|a|. Suppose |x − c| < δ. Then |af (x) − aL| = |a||f (x) − aL| < |a|(/|a|) = .

Chapter 2 Review 1. average velocity approximately 0.954 m/s; instantaneous velocity approximately 0.894 m/s. 5. 1.50 7. 1.69 9. 2.00 3. 200 9 11. 5 13. − 12 15. 16 19. Does not exist;

0.6 0.4 0.2 x 0.25 0.5 0.75 1.00 1.25 1.50

17. 2

t −6 = −∞ lim √ t→9− t − 3 21. ∞

and

t −6 lim √ =∞ t→9+ t − 3

A40

ANSWERS TO ODD-NUMBERED EXERCISES

73. Let f (x) = e−x − x. Observe that f is continuous on [0, 1] with f (0) = e0 − 0 = 1 > 0 and f (1) = e−1 − 1 < 0. Therefore, the IVT 2 guarantees there exists a c ∈ (0, 1) such that f (c) = e−c − c = 0. 75. g(x) = [x]; On the interval   a a , ⊂ [−a, a], x∈ 2 + 2πa 2 2

23. Does not exist; x 3 − 2x =∞ x→1− x − 1 25. 2 27. 23 29. − 12 37. Does not exist;

x 3 − 2x = −∞ x→1+ x − 1

and

lim

31. 3b2

lim θ sec θ = ∞

θ→ π2 −

39. Does not exist; cos θ − 2 lim =∞ θ θ→0− 41. ∞ 43. ∞ 45. Does not exist; lim tan x = ∞

x→ π2 −

lim

33. 19

and

and

35. ∞

lim θ sec θ = −∞

θ→ π2 +

1 2 2 x runs from a to a + 2π, so the sine function covers one full period

cos θ − 2 lim = −∞ θ θ→0+

and clearly takes on every value from − sin a through sin a. 77. δ = 0.55; y 2.05

and

2.00

lim tan x = −∞

x→ π2 +

1.95

47. 0 49. 0 51. According to the graph of f (x),

7.0

lim f (x) = lim f (x) = 1

x→0−

7.5

8.0

8.5

79. Let  > 0 and take δ = /8. Then, whenever |x − (−1)| = |x + 1| < δ,

x→0+

lim f (x) = lim f (x) = ∞

x→2−

x

1.90

|f (x) − (−4)| = |4 + 8x + 4| = 8|x + 1| < 8δ = .

x→2+

lim f (x) = −∞

x→4−

lim f (x) = ∞.

Chapter 3

x→4+

The function is both left- and right-continuous at x = 0 and neither left- nor right-continuous at x = 2 and x = 4. 53. At x = 0, the function has an infinite discontinuity but is left-continuous. y 1

−4

x

−2

2

4

−1

Section 3.1 Preliminary Questions 1. B and D f (x) − f (a) f (a + h) − f (a) 2. and x−a h 3. a = 3 and h = 2 4. Derivative of the function f (x) = tan x at x = π4 5. (a) The difference in height between the points (0.9, sin 0.9) and (1.3, sin 1.3). (b) The slope of the secant line between the points (0.9, sin 0.9) and (1.3, sin 1.3). (c) The slope of the tangent line to the graph at x = 0.9.

55. g(x) has a jump discontinuity at x = −1; g(x) is left-continuous at x = −1. 57. b = 7; h(x) has a jump discontinuity at x = −2 59. Does not have any horizontal asymptotes 61. y = 2 63. y = 1 65. B =B ·1=B ·L= f (x)

f (x) = lim f (x) = A. x→a g(x) 1 1 67. f (x) = and g(x) = (x − a)5 (x − a)3 2 71. Let f (x) = x − cos x. Now, f (x) is continuous over the 2 interval [0, π2 ], f (0) = −1 < 0 and f ( π2 ) = π4 > 0. Therefore, by the Intermediate Value Theorem, there exists a c ∈ (0, π2 ) such that f (c) = 0; consequently, the curves y = x 2 and y = cos x intersect. lim g(x) · lim

x→a

x→a g(x)

= lim g(x) x→a

Section 3.1 Exercises 1. f (3) = 30 3. f (0) = 9 5. f (−1) = −2 7. Slope of the secant line = 1; the secant line through (2, f (2)) and (2.5, f (2.5)) has a larger slope than the tangent line at x = 2. 9. f (1) ≈ 0; f (2) ≈ 0.8 11. f (1) = f (2) = 0; f (4) = 12 ; f (7) = 0 13. f (5.5) 15. f (x) = 7 17. g (t) = −3 19. y = 2x − 1 21. The tangent line at any point is the line itself 1 1 23. f (−2 + h) = ;− −2 + h 3 25. f (5) = − 1√ 10 5

27. f (3) = 22; y = 22x − 18 29. f (3) = −11; y = −11t + 18 31. f (0) = 1; y = x

A41

ANSWERS TO ODD-NUMBERED EXERCISES

1 1 1 ;y =− x+ 64 64 4 f (−2) = −1; y = −x − 1 1 9 1 f (1) = √ ; y = √ x + √ 2 5 2 5 2 5 1 3 1

f (4) = − ; y = − x + 16 16 4 3 3 1

f (3) = √ ; y = √ t + √ 10 10 10 f (0) = 0; y = 1

33. f (8) = − 35. 37. 39. 41. 43.

y 80 60 40 20 x −20

1

2

y

y

2

2

1.5

1.5

1

1

0.5

0.5

y

x 0.2 0.4 0.6 0.8 1 1.2 1.4

3 2.5

x 0.2 0.4 0.6 0.8 1 1.2 1.4

2.4

65.

2.2

2 1.5 1 0.5

2.0

x 1

2

−0.2

−0.1

1

2

x 0.1

0.2

y 3 2.5 2 1.5 1 0.5 −2

−1

x

49. For 1 < x < 2.5 and for x > 3.5 51. f (x) = x 3 and a = 5 53. f (x) = sin x and a = π6 55. f (x) = 5x and a = 2 π  ≈ 0.7071 57. f 4 • On curve (A), f (1) is larger than 59. f (1 + h) − f (1) ; h the curve is bending downwards, so that the secant line to the right is at a lower angle than the tangent line. On curve (B), f (1) is smaller than f (1 + h) − f (1) ; h the curve is bending upwards, so that the secant line to the right is at a steeper angle than the tangent line. 61. (b) f (4) ≈ 20.0000

0.0808 − 0.0278 P (313) − P (293) = = 0.00265 atm/K; 20 20 0.1311 − 0.0482 P (323) − P (303) = = 0.004145 atm/K; P (313) ≈ 20 20 0.2067 − 0.0808 P (333) − P (313) = = 0.006295 atm/K; P (323) ≈ 20 20 0.3173 − 0.1311 P (343) − P (323) = = 0.00931 atm/K; P (333) ≈ 20 20 P (353) − P (333) 0.4754 − 0.2067 = = 0.013435 atm/K P (343) ≈ 20 20 67. −0.39375 kph·km/car 69. i(3) = 0.06 amperes 71. v (4) ≈ 160; C ≈ 0.2 farads 73. It is the slope of the secant line connecting the points (a − h, f (a − h)) and (a + h, f (a + h)) on the graph of f .

P (303) ≈

1.8

(b) y = −0.68x + 2

(c) y = 20x − 48

6

63. c ≈ 0.37.

y

•

5

−60

47. (a) f (0) ≈ −0.68

−1

4

−40

45. W (4) ≈ 0.9 kg/year; slope of the tangent is zero at t = 10 and at t = 11.6; slope of the tangent line is negative for 10 < t < 11.6.

−2

3

Section 3.2 Preliminary Questions 1. 2. 3. 4. 5.

8 (f − g) (1) = −2 and (3f + 2g) (1) = 19 (a), (b), (c) and (f) (b) The line tangent to f (x) = ex at x = 0 has slope equal to 1.

Section 3.2 Exercises 1 1. f (x) = 3 3. f (x) = 3x 2 5. f (x) = 1 − √ 2 x  d 4  3 7. x = 4(−2) = −32 dx x=−2  2 1 d 2/3  t  = (8)−1/3 = 9. dt 3 3 t=8 √ √ 11. 0.35x −0.65 13. 17t 17−1 15. f (x) = 4x 3 ; y = 32x − 48 17. f (x) = 5 − 16x −1/2 ; y = −3x − 32

A42

ANSWERS TO ODD-NUMBERED EXERCISES

19. (a)

d 12ex = 12ex . dx

(c)

(b)

d (25t − 8et ) = 25 − 8et . dt

d t−3 e = et−3 . dt

21. f (x) = 6x 2 − 6x

23. f (x) =

T 2 dP is roughly constant, suggesting that the Clausius–Clapeyron P dT law is valid, and that k ≈ 5000 y 67.

20 2/3 x + 6x −3 3

5 25. g (z) = − z−19/14 − 5z−6 2 1 1 27. f (s) = s −3/4 + s −2/3 4 3 29. g (x) = 0 31. h (t) = 5et−3 33. P (s) = 32s − 24 35. g (x) = −6x −5/2 37. 1 39. −60 41. 1 − e4 • The graph in (A) matches the derivative in (III). 43. • The graph in (B) matches the derivative in (I). • The graph in (C) matches the derivative in (II). • The graph in (D) matches the derivative in (III). (A) and (D) have the same derivative because the graph in (D) is just a vertical translation of the graph in (A). 45. Label the graph in (A) as f (x), the graph in (B) as h(x), and the graph in (C) as g(x) 47. (B) might be the graph of the derivative of f (x) d 3 49. (a) ct = 3ct 2 . dt d (b) (5z + 4cz2 ) = 5 + 8cz. dz d (c) (9c2 y 3 − 24c) = 27c2 y 2 . dy 51. x = 12 53. a = 2 and b = −3 • f (x) = 3x 2 − 3 ≥ −3 since 3x 2 is nonnegative. 55. • The two parallel tangent lines with slope 2 are shown with the graph of f (x) here.

x 1

y

2 −1

1

2

−2 −4

71. c = 1 73. c = 0 75. c = ±1 77. It appears that f is not differentiable at a = 0. Moreover, the tangent line does not exist at this point. 79. It appears that f is not differentiable at a = 3. Moreover, the tangent line appears to be vertical. 81. It appears that f is not differentiable at a = 0. Moreover, the tangent line does not exist at this point. 83. The graph of f (x) is shown in the figure below at the left and it is clear that f (x) > 0 for all x > 0. The positivity of f (x) tells us that the graph of f (x) is increasing for x > 0. y

x

y

400

800

300

600

200

400

100

2

200

−2

x 2

3 1/2 x 2

59. f (0) = 1; y = x 61. Decreasing; y = −0.63216(m − 33) + 83.445; y = −0.25606(m − 68) + 69.647 63. 0.0808 − 0.0278 P (313) − P (293) P (303) ≈ = = 0.00265 atm/K; 20 20 0.1311 − 0.0482 P (323) − P (303) = = 0.004145 atm/K; P (313) ≈ 20 20 P (333) − P (313) 0.2067 − 0.0808 = = 0.006295 atm/K; P (323) ≈ 20 20 0.3173 − 0.1311 P (343) − P (323) = = 0.00931 atm/K; P (333) ≈ 20 20 0.4754 − 0.2067 P (353) − P (333) = = 0.013435 atm/K P (343) ≈ 20 20

x

−2

2 −1

4

4

4 1

3

69. For x < 0, f (x) = −x 2 , and f (x) = −2x. For x > 0, f (x) = x 2 , and f (x) = 2x. Thus, f (0) = 0.

y

−2

2

4

6

8

x −200

2

4

6

8

57. f (x) =

10 7 87. The normal line intersects the x-axis at the point T with coordinates (x + f (x)f (x), 0). The point R has coordinates (a, 0), so the subnormal is |x + f (x)f (x) − x| = |f (x)f (x)|. 89. The tangent line to f at x = a is y = 2ax − a 2 . The x-intercept of this line is a2 so the subtangent is a − a/2 = a/2. 1 1 91. The subtangent is a. 93. r ≤ n 2 85.

Section 3.3 Preliminary Questions 1. (a) False. The notation fg denotes the function whose value at x is f (x)g(x). (b) True. (c) False. The derivative of a product fg is f (x)g(x) + f (x)g (x).

ANSWERS TO ODD-NUMBERED EXERCISES

(d) False.

  d (fg) = f (4)g (4) + g(4)f (4). dx x=4

A43

49. a = 1 51. (a) Given R(t) = N (t)S(t), it follows that dR = N (t)S (t) + S(t)N (t). dt

(e) True. 2. −1 3. 5

 dR  = 1, 250, 000 dt t=0 (c) The term 5S(0) is larger than the term 10, 000N(0). Thus, if only one leg of the campaign can be implemented, it should be part A: increase the number of stores by 5 per month. 1 • At x = −1, the tangent line is y = x+1 53. 2 1 • At x = 1, the tangent line is y = − x + 1 2 2 55. Let  g = f = ff . Then (b)

Section 3.3 Exercises 1. f (x) = 10x 4 + 3x 2 3. f (x) = ex (x 2 + 2x)  7 −3/2 3 −5/2 dh  871 dh =− s + s + 14; = 5. ds 2 2 ds s=4 64 −2 7. f (x) = (x − 2)2  4t dg dg  8 =− 2 9. ; =  2 dt 9 (t − 1) dt t=−2 x e 11. g (x) = − 13. f (t) = 6t 2 + 2t − 4 (1 + ex )2 15. h (t) = 1 for t = 1 17. f (x) = 6x 5 + 4x 3 + 18x 2 + 5  1 dy  1 dy =− ; =− 19.  2 dx 169 (x + 10) dx x=3 21. f (x) = 1  dy 2x 5 − 20x 3 + 8x dy  23. ; = −80 =   2 dx dx x=2 x2 − 5  dz 3x 2 dz  3 25. =− 3 ; =− dx 4 (x + 1)2 dx 

g = f 2

= (ff ) = ff + ff = 2ff .

57. Let p = fgh.Then  p = (fgh) = f gh + hg + ghf = f gh + fg h + fgh . 61. (x + h)f (x + h) − f (x) d (xf (x)) = lim dx h h→0   f (x + h) − f (x) + f (x + h) = lim x h h→0 = x lim

h→0

= xf (x) + f (x).

x=1

−2t 3 − t 2 + 1 27. h (t) =  2 t3 + t2 + t + 1

65. (a) Is a multiple root (b) Not a multiple root 67. d d  x x m(ab)(ab)x = (ab)x = a b dx dx d x d x b + bx a = ax dx dx

29. f (t) = 0 31. f (x) = 3x 2 − 6x − 13 xex 33. f (x) = (x + 1)2 35. For z  = −2 and z = 1, g (z) = 2z − 1 −xt 2 + 8t − x 2 37. f (t) = (t 2 − x)2

39. (f g) (4) = −20 and (f/g) (4) = 0

= m(b)a x bx + m(a)a x bx = (m(a) + m(b))(ab)x .

Section 3.4 Preliminary Questions

d 2x e = 2e2x dx 47. From the plot of f (x) shown below, we see that f (x) is decreasing on its domain {x : x = ±1}. Consequently, f (x) must be negative. Using the quotient rule, we find 41. G (4) = −10

43. F (0) = −7

f (x) =

45.

(x 2 − 1)(1) − x(2x) x2 + 1 = − , (x 2 − 1)2 (x 2 − 1)2

1. 10 square units per unit increase 3.

y

5 x 1 −5

1. (a) atmospheres/meter. (b) moles/(liter·hour). 2. 90 mph 3. f (26) ≈ 43.75 4. (a) P (2009) measures the rate of change of the population of Freedonia in the year 2009. (b) P (2010) ≈ 5.2 million.

Section 3.4 Exercises

which is negative for all x = ±1.

−4 −3 −2

f (x + h) − f (x) + lim f (x + h) h h→0

2

3

4

c

ROC of f (x) with respect to x at x = c.

1

f (1) = 13

8 27

1 f (8) = 12

1 f (27) = 27

A44

ANSWERS TO ODD-NUMBERED EXERCISES

5. d = 2

7. dV /dr = 3πr 2

9. (a) 100 km/hour (d) −50 km/hour

(b) 100 km/hour (c) 0 km/hour

11. (a) (i) (b) (ii) (c) (iii) dT ≈ −1.5625◦ C/hour 13. dt 15. −8 × 10−6 1/s    dT dT  ◦ C/km; dT  =0 ≈ 2.94 ≈ −3.33◦ C/km; 17.  dh h=30 dh h=70 dh over the interval [13, 23], and near the points h = 50 and h = 90.

(r) = −1.41 × 107 r −3/2 19. vesc

21. t = 52 s 23. The particle √ passes through the origin when t = 0 seconds and when t = 3 2 ≈ 4.24 seconds. The particle is instantaneously motionless when t = 0 seconds and when t = 3 seconds. 25. Maximum velocity: 200 m/s; maximum height: 2040.82 m 27. Initial velocity: v0 = 19.6 m/s; maximum height: 19.6 m dV = −1 (b) −4 31. (a) dv √ 5 35. Rate of change of BSA with respect to mass: √ ; m = 70 kg, 20 m 2

rate of change is ≈ 0.0133631 m kg ; m = 80 kg, rate of change is m2

1 80 kg ; BSA increases more rapidly at lower body mass.

37. 2 √ √ 39. 2 − 1 ≈ 12 ; the actual value, to six decimal places, is √ √ 0.414214. 101 − 100 ≈ .05; the actual value, to six decimal places, is 0.0498756. 41. • •

• F (65) = 282.75 ft Increasing speed from 65 to 66 therefore increases stopping distance by approximately 7.6 ft. The actual increase in stopping distance when speed increases from 65 mph to 66 mph is F (66) − F (65) = 290.4 − 282.75 = 7.65 feet, which differs by less than one percent from the estimate found using the derivative.

43. The cost of producing 2000 bagels is $796. The cost of the 2001st bagel is approximately $0.244, which is indistinguishable from the estimated cost. 45. An increase in oil prices of a dollar leads to a decrease in demand of 0.5625 barrels a year, and a decrease of a dollar leads to an increase in demand of 0.5625 barrels a year. dB 2k 3k 1/2 dH 47. = 1/3 ; = W dI dW 2 3I dB (a) As I increases, dI shrinks, so that the rate of change of perceived intensity decreases as the actual intensity increases. dH increases as well, so that the rate of change (b) As W increases, dW of perceived weight increases as weight increases. 49. (a) The average income among households in the bottom rth part is

F (r)T F (r) T F (r) = · = A. rN r N r

(b) The average income of households belonging to an interval [r, r + r] is equal to F (r + r)T − F (r)T F (r + r) − F (r) T = · rN r N F (r + r) − F (r) A = r (c) Take the result from part (b) and let r → 0. Because F (r + r) − F (r) lim = F (r), r r→0 we find that a household in the 100rth percentile has income F (r)A. (d) The point P in Figure 14(B) has an r-coordinate of 0.6, while the point Q has an r-coordinate of roughly 0.75. Thus, on curve L1 , 40% of households have F (r) > 1 and therefore have above-average income. On curve L2 , roughly 25% of households have above-average income. 53. By definition, the slope of the line through (0, 0) and (x, C(x)) is C(x) − 0 C(x) = Cavg (x). = x x−0 • • • •

At point A, average cost is greater than marginal cost. At point B, average cost is greater than marginal cost. At point C, average cost and marginal cost are nearly the same. At point D, average cost is less than marginal cost.

Section 3.5 Preliminary Questions 1. The first derivative of stock prices must be positive, while the second derivative must be negative. 2. True 3. All quadratic polynomials 4. ex

Section 3.5 Exercises 1. y

= 28 and y

= 0 3. y

= 12x 2 − 50 and y

= 24x 5. y

= 8πr and y

= 8π 16 4 96 −11/15 16 −7/3 t − t 7. y

= − t −6/5 + t −4/3 and y

= 5 3 25 9 9. y

= −8z−3 and y

= 24z−4 11. y

= 12θ + 14 and y

= 12 13. y

= −8x −3 and y

= 24x −4 15. y

= (x 5 + 10x 4 + 20x 3 )ex and 2 )ex y

= (x 5 + 15x 4 + 60x 3 + 60x   2 d y 17. f (4) (1) = 24 19. = 54  dt 2  t=1  3465 d 4 x  23. f

(−3) = 4e−3 − 6 = 21.  4  134217728 dt t=16

25. h

(1) = 74 e

27. y (0) (0) = d, y (1) (0) = c, y (2) (0) = 2b, y (3) (0) = 6a, y (4) (0) = 24, and y (5) (0) = 0 d 6 −1 29. x = 720x −7 dx 6

ANSWERS TO ODD-NUMBERED EXERCISES

31. f (n) (x) = (−1)n (n + 1)!x −(n+2)

51.

33. f (n) (x) = (−1)n (2n−1)×(2n−3)×...×1 x −(2n+1)/2 2n (n) n −x 35. f (x) = (−1) (x − n)e 37. (a) a(5) = −120 m/min2

f (x) = − f

(x) =

(b) The acceleration of the helicopter for 0 ≤ t ≤ 6 is shown in the figure below. As the acceleration of the helicopter is negative, the velocity of the helicopter must be decreasing. Because the velocity is positive for 0 ≤ t ≤ 6, the helicopter is slowing down.

f (4) (x) = x

39. 41. 43. 45. 47.

2

1

3

4

5

3 3·1 = (−1)1 ; (x − 1)2 (x − 1)1+1

6 3·2·1 = (−1)2 ; (x − 1)3 (x − 1)2+1

f

(x) = −

y −20 −40 −60 −80 −100 −120 −140

A45

3 · 3! 18 = (−1)3 ; and (x − 1)4 (x − 1)3+1

72 3 · 4! = (−1)4 . (x − 1)5 (x − 1)4+1

From the pattern observed above, we conjecture

6

f (k) (x) = (−1)k

(a) f

(b) f (c) f Roughly from time 10 to time 20 and from time 30 to time 40 n = −3 (a) v(t) = −5.12 m/s (b) v(t) = −7.25 m/s A possible plot of the drill bit’s vertical velocity follows: y 4 Metal 2

3 · k! . (x − 1)k+1

53. 99! 55. (fg)

= f

g + 3f

g + 3f g

+ fg

; n    n (n−k) (k) g f (fg)(n) = k k=0

57.

f (x) = x 2 ex + 2xex = (x 2 + 2x)ex ; f

(x) = (x 2 + 2x)ex + (2x + 2)ex = (x 2 + 4x + 2)ex ; f

(x) = (x 2 + 4x + 2)ex + (2x + 4)ex = (x 2 + 6x + 6)ex ; f (4) (x) = (x 2 + 6x + 6)ex + (2x + 6)ex = (x 2 + 8x + 12)ex .

From this information, we conjecture that the general formula is f (n) (x) = (x 2 + 2nx + n(n − 1))ex .

x 0.5

1

1.5

2

−2

Section 3.6 Preliminary Questions

−4

A graph of the acceleration is extracted from this graph: y

1. (a)

d (sin x + cos x) = − sin x + cos x dx

d sec x = sec x tan x dx d cot x = − csc2 x (c) dx 2. (a) This function can be differentiated using the Product Rule.

(b)

40 20 0.5

1

x

1.5

2

(b) We have not yet discussed how to differentiate a function like this.

−20 −40

Metal

49. (a) Traffic speed must be reduced when the road gets more dS to be negative. crowded so we expect dQ (b) The decrease in speed due to a one-unit increase in density is dS (a negative number). Since d 2 S = 5764Q−3 > 0 approximately dQ 2

(c) This function can be differentiated using the Product Rule. 3. 0 4. The difference quotient for the function sin x involves the expression sin(x + h). The addition formula for the sine function is used to expand this expression as sin(x + h) = sin x cos h + sin h cos x.

dQ

dS gets larger as Q increases. is positive, this tells us that dQ (c) dS/dQ is plotted below. The fact that this graph is increasing shows that d 2 S/dQ2 > 0. y 100 −0.2 −0.4 −0.6 −0.8 −1.0 −1.2

200

300

400

x

Section 3.6 Exercises 1. 5. 9. 11.

√  √ 2 2 π π x+ 1− 3. y = 2x + 1 − y= 2 2 4 2 f (x) = − sin2 x + cos2 x 7. f (x) = 2 sin x cos x H (t) = 2 sin t sec2 t tan t + sec t   f (θ) = tan2 θ + sec2 θ sec θ

13. f (x) = (2x 4 − 4x −1 ) sec x tan x + sec x(8x 3 + 4x −2 ) θ sec θ tan θ − sec θ 4 cos y − 3 17. R (y) = 15. y = 2 θ sin2 y

A46

ANSWERS TO ODD-NUMBERED EXERCISES

19. f (x) =

2 sec2 x

21. f (x) = ex (cos x + sin x) (1 − tan x)2 23. f (θ ) = eθ (5 sin θ + 5 cos θ − 4 tan θ − 4 sec2 θ)

25. y = 1

27. y = x + 3 √  √ π 29. y = (1 − 3) x − +1+ 3 3  π + eπ/2 31. y = x + 1 33. y = 2eπ/2 t − 2 x 35. cot x = cos sin x ; use the quotient rule 37. csc x = sin1 x ; use the quotient rule 39. f

(θ ) = −θ sin θ + 2 cos θ 41. y

= 2 sec2 x tan x y

= 2 sec4 x + 4 sec2 x tan2 x. • Then f (x) = − sin x, f

(x) = − cos x, f

(x) = sin x, (4) f (x) = cos x, and f (5) (x) = − sin x.

43. •

Accordingly, the successive derivatives of f cycle among {− sin x, − cos x, sin x, cos x}

•

in that order. Since 8 is a multiple of 4, we have f (8) (x) = cos x. Since 36 is a multiple of 4, we have f (36) (x) = cos x. Therefore, f (37) (x) = − sin x.

5π 7π 45. x = π4 , 3π 4 , 4 , 4

47. (a)

53. cos(x + h) − cos x cos x cos h − sin x sin h − cos x f (x) = lim = lim h h h→0 h→0 

= lim

h→0

(− sin x)

cos h − 1 sin h + (cos x) h h



= (− sin x) · 1 + (cos x) · 0 = − sin x.

Section 3.7 Preliminary Questions

√ 1. (a) The outer function is x, and the inner function is 4x + 9x 2 . (b) The outer function is tan x, and the inner function is x 2 + 1. (c) The outer function is x 5 , and the inner function is sec x. (d) The outer function is x 4 , and the inner function is 1 + ex . x can be differentiated using the Quotient Rule, 2. The function x+1 √ and the functions x · sec x and xex can be√differentiated using the Product Rule. The functions tan(7x 2 + 2), x cos x and esin x require the Chain Rule 3. (b) 4. We do not have enough information to compute F (4). We are missing the value of f (1).

Section 3.7 Exercises 1. f (g(x))

f (u)

f (g(x))

g (x)

(f ◦ g)

(x 4 + 1)3/2

3 u1/2 2

3 (x 4 + 1)1/2 2

4x 3

6x 3 (x 4 + 1)1/2

3.

y 12

f (g(x))

f (u)

f (g(x))

g (x)

(f ◦ g)

10

tan(x 4 )

sec2 u

sec2 (x 4 )

4x 3

4x 3 sec2 (x 4 )

8

5. 4(x + sin x)3 (1 + cos x)

6

7. (a) 2x sin(9 − x 2 )

4 2 x 2

4

6

8

10 12

(b) Since g (t) = 1 − cos t ≥ 0 for all t, the slope of the tangent line to g is always nonnegative. (c) t = 0, 2π, 4π

49. f (x) = sec2 x = 12 . Note that f (x) = 12 has numerator cos x cos x 1; the equation f (x) = 0 therefore has no solution. The least slope for a tangent line to tan x is 1. Here is a graph of f . y 14 12 10 8 6 4

−4

−2

x 2

4

(b)

sin(x −1 ) x2

(c) − sec2 x sin(tan x)

7 √ 2 7x − 3 15. −2(x 2 + 9x)−3 (2x + 9) 17. −4 cos3 θ sin θ 19. 9(2 cos θ + 5 sin θ)8 (5 cos θ − 2 sin θ) 21. ex−12 23. 2 cos(2x  + 1) 9. 12

11. 12x 3 (x 4 + 5)2

13.

−1 25. ex+x 1 − x −2

27. d f (g(x)) = − sin(x 2 + 1)(2x) = −2x sin(x 2 + 1) dx d g(f (x)) = −2 sin x cos x dx   t 29. 2x cos x 2 31. 2 t +9 −1/3   2 4 3 x −x −1 4x 3 − 3x 2 33. 3 sec (1/x) tan (1/x) 8(1 + x)3 37. − 35. (1 − x)5 x2 2

dR = (v02 /9.8)(− sin2 θ + cos2 θ ); if θ = 7π/24, increasing the 51. dθ angle will decrease the range.

39. (1 − sin θ) sec2 (θ + cos θ) 41. −18te2−9t 43. (2x + 4) sec2 (x 2 + 4x) 45. 3x sin (1 − 3x) + cos (1 − 3x) 47. 2(4t + 9)−1/2 49. 4(sin x − 3x 2 )(x 3 + cos x)−5

ANSWERS TO ODD-NUMBERED EXERCISES

x cos(x 2 ) − 3 sin 6x cos 2x 53. 51. √ 2 sin 2x cos 6x + sin(x 2 )  2 2 3  −1 55. 3 x sec (x ) + sec2 x tan2 x 57. √ z + 1 (z − 1)3/2     sin(−1) − sin(1 + x) 4 cot 6 x 5 csc2 x 5 59. 61. −35x (1 + cos x)2   8    63. −180x 3 cot 4 x 4 + 1 csc2 x 4 + 1 1 + cot 5 x 4 + 1 65. 24(2e3x + 3e−2x )3 (e3x − e−2x ) 67. 4(x + 1)(x 2 + 2x + 3)e(x +2x+3) 1 69.  √ √ √ 8 x 1+ x 1+ 1+ x     k 71. − (kx + b)−4/3 73. 2 cos x 2 − 4x 2 sin x 2 3 75. −336(9 − x)5 √  m dv  290 3 77. = dP P =1.5 3 s · atmospheres dV 79. (a) When r = 3, = 1.6π(3)2 ≈ 45.24 cm/s. dt dV (b) When t = 3, we have r = 1.2. Hence = 1.6π(1.2)2 ≈ 7.24 dt cm/s. π π 81. W (10) ≈ 0.3566 kg/yr 83. (a) (b) 1 + 360 90 √ 1 85. 5 3 87. 12 89. 16  dollars dP  = −0.727 91.  dt t=3 year dP = −4.03366 × 10−16 (288.14 − 0.000649 h)4.256 ; for 93. dh each additional meter of altitude, P ≈ −1.15 × 10−2 Pa 95. 0.0973 kelvins/yr  2 97. f (g(x))g

(x) + f

(g(x)) g (x) 99. Let u = h(x), v = g(u), and w = f (v). Then 2

2

df dv df dv du dw = = = f (g(h(x))g (h(x))h (x) dx dv dx dv du dx 103. For n = 1, we find  π d sin x = cos x = sin x + , dx 2 as required. Now, suppose that for some positive integer k,  dk kπ  . sin x = sin x + 2 dx k Then  d k+1 kπ  d sin x + sin x = dx 2 dx k+1     kπ (k + 1)π = cos x + = sin x + . 2 2

Section 3.8 Exercises

x x 2 − 9; g (x) = x2 − 9 1 1 3. g (x) = 5. g (x) = − x −6/5 7 5 1 1

11. g(1) = 0; 7. g (x) = 9. g(7) = 1; g (7) = 5 (1 − x)2 4 15. g(1/4) = 3; g (1) = 1 13. g(4) = 2; g (4) = 5 1 5 g (1/4) = −16 19. 21. √ 4 4 15 7 d −2x d sin−1 (7x) = cos−1 (x 2 ) = 25. 23. 2 dx dx 1 − (7x) 1 − x4   d 1 −1 −1 27. + tan x x tan x = x dx 1 + x2 d ex 29. sin−1 (ex ) = dx 1 − e2x  d  1−t −1 2 31. 1 − t + sin t = dt 1 − t2   −1 d 3(tan x)2 (tan−1 x)3 = 33. dx x2 + 1 d 35. (cos−1 t −1 − sec−1 t) = 0 dt 37. Let θ = cos−1 x. Then cos θ = x and 1. g(x) = f −1 (x) =

dθ 1 1 =− =− . dx sin θ sin(cos−1 x) Moreover, sin(cos−1 x) = sin θ = 1 − x 2 . 1 1 1 1 41. g (x) = = −1 = = f (g(x)) x f (f (x)) f (f −1 (x)) − sin θ

or

1 3. e2 4. e3 10 5. y (100) = cosh x and y (101) = sinh x 1. ln 4 2.

Section 3.9 Exercises 1. 5. 7. 9. 13.

Section 3.8 Preliminary Questions 1 3. g(x) = tan−1 x 3 4. Angles whose sine and cosine are x are complementary.

dθ =1 dx

Section 3.9 Preliminary Questions

15.

1. 2

A47

17.

2.

19.

d d 2 x ln x = ln x + 1 3. (ln x)2 = ln x dx dx x d 18x ln(9x 2 − 8) = 2 dx 9x − 8 d cos t ln(sin t + 1) = dt sin t + 1 d ln x 1 − ln x 1 d = ln(ln x) = 11. dx x dx x ln x x2 3(ln(ln x))2 d (ln(ln x))3 = dx x ln x d 4x + 11 ln((x + 1)(2x + 9)) = dx (x + 1)(2x + 9) d x 11 = ln 11 · 11x dx x(2x ln 2 + 3−x ln 3) − (2x − 3−x ) d 2x − 3−x = dx x x2

A48

ANSWERS TO ODD-NUMBERED EXERCISES

1 cot t 1 d · 23. log3 (sin t) = x ln 2 dt ln 3 25. y = 36 ln 6(x − 2) + 36 21. f (x) =

27. y = 320 ln 3(t − 2) + 318 29. y = 5−1 31. y = −1(t − 1) + ln 4   12 8 1 33. y = (z − 3) + 2 35. y = w− −3 25 ln 5 ln 2 8 37. y = 2x + 14 39. y = 3x 2 − 12x + 79   2x x(x 2 + 1) 1 1 + 2 41. y = √ − x x + 1 2(x + 1) x+1 43.    1 1 2 3 1 x(x + 2)

y = · + − − 2 (2x + 1)(3x + 2) x x + 2 2x + 1 3x + 2 45. 47. 49. 51. 53. 55. 57. 59. 61. 63. 65. 67.

d 3x x = x 3x (3 + 3 ln x) dx   x x d ex e x = xe + ex ln x dx x  x  x d 3x 3 x = x3 + (ln x)(ln 3)3x dx x d sinh(9x) = 9 cosh(9x) dx d cosh2 (9 − 3t) = −6 cosh(9 − 3t) sinh(9 − 3t) dt d √ 1 cosh x + 1 = (cosh x + 1)−1/2 sinh x dx 2 d coth t 1 = dt 1 + tanh t 1 + cosh t d cosh(ln x) sinh(ln x) = dx x d x x tanh(e ) = e sech2 (ex ) dx √ √ √ d 1 sech( x) = − x −1/2 sech x tanh x dx 2 d sech x coth x = − csch x coth x dx d 3 cosh−1 (3x) = dx 9x 2 − 1

d 2x (sinh−1 (x 2 ))3 = 3(sinh−1 (x 2 ))2 dx x4 + 1   −1 x d cosh−1 x 1 cosh 71. e =e dx x2 − 1 69.

73.

d 1 tanh−1 (ln t) = dt t (1 − (ln t)2 )

Section 3.10 Preliminary Questions 1. The chain rule 2. (a) This is correct (b) This is correct (c) This is incorrect. Because the differentiation is with respect to the variable x, the chain rule is needed to obtain dy d sin(y 2 ) = 2y cos(y 2 ) . dx dx 3. There are two mistakes in Jason’s answer. First, Jason should have applied the product rule to the second term to obtain d dy (2xy) = 2x + 2y. dx dx Second, he should have applied the general power rule to the third term to obtain d 3 dy y = 3y 2 . dx dx 4. (b)

Section 3.10 Exercises 2 dy =− 1. (2, 1), dx 3 d  2 3 x y = 3x 2 y 2 y + 2xy 3 3. dx  3/2    d  2 = 3 x + yy x 2 + y 2 x + y2 5. dx d y 2x y 7. 9. y = − 2 = dx y + 1 (y + 1)2 9y 1 − 2xy − 6x 2 y 3R 11. y = 13. R = − 2 3 5x x + 2x − 1 y(y 2 − x 2 ) 9

15. y = 17. y = x 1/2 y 5/3 4 x(y 2 − x 2 − 2xy 2 ) 2 (2x + 1)y 1 − cos(x + y) 19. y = 21. y = cos(x + y) + sin y y2 − 1 ey − 2y 23. y = 2x + 3y 2 − xey 1 xy −y 1 29. y = 31. y = − x + 2 25. y = xy + x 4 2 12 32 4 4 33. y = −2x + 2 35. y = − x + 37. y = x + 5 5 3 3 √ √ 39. The tangent is horizontal at the points (−1, 3) and (−1, − 3) 41. The tangent line is horizontal at  √  √ √  √  4 78 2 78 4 78 2 78 ,− and − , . 13 13 13 13 •

43.

y =

−1 d cosh x sinh2 x − cosh2 x d = = coth x = = dx dx sinh x sinh2 x sinh2 x 2 − csch x

75.

79. 1.22 cents per year dP 1 83. (a) =− (b) P ≈ −0.054 dT T ln 10 d d ln x 1 85. logb x = = dx dx ln b (ln b)x

When y = 21/4 , we have

√ √ −21/4 − 1 2+ 4 2   =− ≈ −0.3254. 8 4 23/4

When y = −21/4 , we have y =

√ √ 21/4 − 1 2− 4 2  =−  ≈ −0.02813. 8 −4 23/4

1 4 At the point (1, 1), the tangent line is y = x + . 5 5 √ 1 1/3 2/3 45. (2 , 2 ) 47. x = , 1 ± 2 2 •

ANSWERS TO ODD-NUMBERED EXERCISES

1 3 1 • At (1, −2), y = − 3 11 • At (1, 1 ), y = 2 12 11 • At (1, − 1 ), y = − 2 12 dx y(2y 2 − 1) 51. = ; The tangent line is vertical at: dy x •

49.

At (1, 2), y =

y 1

−4

−3

−2

x

−1 −1 −2

•

Upper part of lower left curve: y

√

√  3 2 (1, 0), (−1, 0), , , 2 2  √ √  √ √   √ √  3 2 3 2 3 2 , , ,− , − ,− . − 2 2 2 2 2 2

1

−4

−3

−2

x

−1 −1 −2

2y dx = 2 ; it follows that dx dy = 0 when y = 0, so the dy 3x − 4 tangent line to this curve is vertical at the points where the curve intersects the x-axis.

•

53.

Upper part of lower right curve: y 1 1

2

3

4

2

3

4

y 2

x

−1

1

−2

A49

−2

x

−1

1

2

•

3

Lower part of lower right curve:

−1

y 1

−2

y 3 − 2x 2 55. (b): y

= y5

1

10 27 dx dy y dx dy +y = 0, and =− 59. x dt dt dt x dt dy x 2 dx dy x + y dx 61. (a) = 2 (b) =− 3 dt dt y dt 2y + x dt 57. y

=

63. Let C1 be the curve described by x 2 − y 2 = c, and let C2 be the curve described by xy = d. Suppose that P = (x0 , y0 ) lies on the intersection of the two curves x 2 − y 2 = c and xy = d. Since x 2 − y 2 = c, y = xy . The slope to the tangent line to C1 is xy00 . On

the curve C2, since xy = d, y = − yx . Therefore the slope to the tangent line to C2 is − yx00 . The two slopes are negative reciprocals of one another, hence the tangents to the two curves are perpendicular. •

65.

−1 −2

Section 3.10 Preliminary Questions 1. Let s and V denote the length of the side and the corresponding ds volume of a cube, respectively. Determine dV dt if dt = 0.5 cm/s. dV dr 2. = 4πr 2 dt dt dV 3 3. Determine dh dt if dt = 2 cm /min dh 4. Determine dV dt if dt = 1 cm/min

Section 3.10 Exercises

Upper branch: y 2

−4

x

−2

2 −2

•

x

Lower part of lower left curve:

4

0.039 ft/min (a) 100π ≈ 314.16 m2 /min (b) 24π ≈ 75.40 m2 /min 27000π cm3 /min 7. 9600π cm2 /min −0.632 m/s 11. x = 4.737 m; dx dt ≈ 0.405 m/s 9 1000π 13. ≈ 0.36 m/min 15. ≈ 1047.20 cm3 /s 8π 3 17. 0.675 meters per second 19. (a) 799.91 km/h (b) 0 km/h 1. 3. 5. 9.

A50

ANSWERS TO ODD-NUMBERED EXERCISES

1200 21. 1.22 km/min 23. ≈ 4.98 rad/hr 241 √ 100 13 25. (a) ≈ 27.735 km/h (b) 112.962 km/h 13 √ 5 m/s 31. −1.92kPa/min 27. 16.2 ≈ 4.025 m 29. 3 1 33. − rad/s 8 16 35. (b): when x = 1, L (t) = 0; when x = 2, L (t) = 3 √ 37. −4 5 ≈ −8.94 ft/s 39. −0.79 m/min 41. Let the equation y = f (x) describe the shape of the roller coaster d of both sides of this equation yields dy = f (x) dx . track. Taking dt dt dt 43. (a) The distance formula gives  L = (x − r cos θ )2 + (−r sin θ )2 .

43.       √ −1/2 √ −1/2 1 1 1 x+ x+ x x+ x 1+ 1 + x −1/2 2 2 2 45. −3t −4 sec2 (t −3 ) 47. −6 sin2 x cos2 x + 2 cos4 x 49. 53. 55. 63. 67.

Thus, L2 = (x − r cos θ )2 + r 2 sin2 θ. (b) From (a) we have 0 = 2 (x − r cos θ)



dθ dx + r sin θ dt dt

71.

8 csc2 θ 1 + sec t − t sec t tan t 51. (1 + sec t)2 (1 + cot θ )2 √ 2 sec ( 1 + csc θ) csc θ cot θ − √ 2( 1 + csc θ) 8x 2 ln s 2 −36e−4x 57. (4 − 2t)e4t−t 59. 61. s 4x 2 + 1   1 cot θ 65. sec(z + ln z) tan(z + ln z) 1 + z 1 1 −2(ln 7)(7−2x ) 69. · 1 + (ln x)2 x 2 ln s ln s 1 s 73. − 2 −1 s |x| x − 1 csc x

75. 2(sin2 t)t (t cot t + ln sin t)

 + 2r 2 sin θ cos θ

dθ . dt

(c) −80π ≈ −251.33 cm/min √ 3 5 ≈ 0.0027 m/min 45. (c): 2500

Chapter 3 Review 1. 3; the slope of the secant line through the points (2, 7) and (0, 1) on the graph of f (x) 8 3. ; the value of the difference quotient should be larger than the 3 value of the derivative 5. f (1) = 1; y = x − 1 1 1 1 7. f (4) = − ; y = − x + 16 16 2 1

(1) where f (x) = √x 13. f 9. −2x 11. (2 − x)2

15. f (π ) where f (t) = sin t cos t 17. f (4) = −2; f (4) = 3 19. (C) is the graph of f (x) 21. (a) 8.05 cm/year (b) Larger over the first half (c) h (3) ≈ 7.8 cm/year; h (8) ≈ 6.0 cm/year 23. A (t) measures the rate of change in automobile production in the United States; A (1971) ≈ 0.25 million automobiles/year; A (1974) would be negative 25. (b) 1 1 1 27. g (x) = = 2 = x −2 = f (g(x)) f (g(x))2 x 1 − 2x − x 2 29. 15x 4 − 14x 31. −7.3t −8.3 33. (x 2 + 1)2 3 4 5 35. 6(4x − 9)(x − 9x) 37. 27x(2 + 9x 2 )1/2 2−z 3 39. 41. 2x − x −5/2 2 2(1 − z)3/2

77. 2t cosh(t 2 ) 79.

ex 1 − e2x

81. α = 0 and α > 1 83. Let f (x) = xe−x . Then f (x) = e−x (1 − x). On [1, ∞), f (x) < 0, so f (x) is decreasing and therefore one-to-one. The domain of g(x) is (0, e−1 ], and the range is [1, ∞). g (2e−2 ) = −e2 . 57 85. −27 87. − 89. −18 91. (−1, −1) and (3, 7) 16 1 93. a = 95. 72x − 10 97. −(2x + 3)−3/2 6 x2 dy = 2 99. 8x 2 sec2 (x 2 ) tan(x 2 ) + 2 sec2 (x 2 ) 101. dx y y 2 + 4x dy cos(x + y) dy = 105. = dx 1 − 2xy dx 1 − cos(x + y) 107. For the plot on the left, the red, green and blue curves, respectively, are the graphs of f , f and f

. For the plot on the right, the green, red and blue curves, respectively, are the graphs of f , f and f

.   4 3 (x + 1)3 − 109. (4x − 2)2 x + 1 2x − 1 103.

2 2 111. 4e(x−1) e(x−3) (x − 2)   2 2 e3x (x − 2)2 − 3 + 113. x−2 x+1 (x + 1)2

dq p dq dR =p +q =q + q = q(E + 1) dp dp q dp 117. E(150) = −3; number of passengers increases 3% when the ticket price is lowered 1% −11π 119. ≈ −0.407 cm/min 360 640 121. ≈ 0.00567 cm/s (336)2 123. 0.284 m/s 115.

ANSWERS TO ODD-NUMBERED EXERCISES

Chapter 4

y

Section 4.1 Preliminary Questions 1. True 2. g(1.2) − g(1) ≈ 0.8 3. f (2.1) ≈ 1.3 4. The Linear Approximation tells us that up to a small error, the change in output f is directly proportional to the change in input x when x is small.

Section 4.1 Exercises 1. f ≈ 0.12 3. f ≈ −0.00222 5. f ≈ 0.003333 7. f ≈ 0.0074074 9. f ≈ 0.049390; error is 0.000610; percentage error is 1.24% 11. f ≈ −0.0245283; error is 0.0054717; percentage error is 22.31% 13. y ≈ −0.007 15. y ≈ −0.026667 17. f ≈ 0.1; error is 0.000980486 19. f ≈ −0.0005; error is 3.71902 × 10−6 21. f ≈ 0.083333; error is 3.25 × 10−3 23. f ≈ −0.1; error is 4.84 × 10−3 25. f (4.03) ≈ 2.01 √ √ √ √ 27. 2.1 − 2 is larger than 9.1 − 9 29. R(9) = 25110 euros; if p is raised by 0.5 euros, then R ≈ 585 euros; on the other hand, if p is lowered by 0.5 euros, then R ≈ −585 euros. 2wR 2 2wh 31. (a) W ≈ W (R)x = − ≈ −0.0005wh h=− 3 R R (b) W ≈ −0.7 pounds 33. L ≈ −0.00171 cm 35. (a) P ≈ −0.434906 kilopascals

(b) If θ = 2◦ , this gives s ≈ 0.51 ft, in which case the shot would not have been successful, having been off half a foot. 41. V ≈ 4π(25)2 (0.5) ≈ 3927 cm3 ; S ≈ 8π(25)(0.5) ≈ 314.2 cm2 43. P = 6 atmospheres; P ≈ ±0.45 atmospheres 1 π 45. L(x) = 4x − 3 47. L(θ) = θ − + 4 2 1 1 49. L(x) = − x + 1 51. L(x) = 1 53. L(x) = e(x + 1) 2 2 55. f (2) = 8 √ 57. 16.2 ≈ L(16.2) = 4.025. Graphs of f (x) and L(x) are shown below. Because the graph of L(x) lies above the graph of f (x), we expect that the estimate from the Linear Approximation is too large.

5 4 3 2 1

L(x)

0

5

f (x) x 10

15

20

25

1 59. √ ≈ L(17) ≈ 0.24219; the percentage error is 0.14% 17 1 ≈ L(10.03) = 0.00994; the percentage error is 61. (10.03)2 0.0027% 63. (64.1)1/3 ≈ L(64.1) ≈ 4.002083; the percentage error is 0.000019% 65. cos−1 (0.52) ≈ L(0.02) = 1.024104; the percentage error is 0.015% 67. e−0.012 ≈ L(−0.012) = 0.988; the percentage error is 0.0073% √ 69. Let f (x) = x. Then f (9) = 3, f (x) = 12 x −1/2 and f (9) = 16 . Therefore, by the Linear Approximation, f (9 + h) − f (9) =

√

9+h−3≈

1 h. 6

Moreover, f

(x) = − 14 x −3/2 , so |f

(x)| = 14 x −3/2 . Because this is a decreasing function, it follows that for x ≥ 9, K = max |f

(x)| ≤ |f

(9)| =

1 < 0.01. 108

From the following table, we see that for h = 10−n , 1 ≤ n ≤ 4, E ≤ 12 Kh2 . h 10−1 10−2 10−3 10−4

(b) The actual change in pressure is −0.418274 kilopascals; the percentage error is 3.98% 37. (a) h ≈ 0.71 cm (b) h ≈ 1.02 cm. (c) There is a bigger effect at higher velocities. 39. (a) If θ = 34◦ (i.e., t = 17 90 π), then   17 625 cos π t s ≈ s (t)t = 16 45   17 π 625 cos π θ · ≈ 0.255θ. = 16 45 180

A51

√ E = | 9 + h − 3 − 16 h| 4.604 × 10−5 4.627 × 10−7 4.629 × 10−9 4.627 × 10−11

1 Kh2 2

5.00 × 10−5 5.00 × 10−7 5.00 × 10−9 5.00 × 10−11

 dy  1 = − ; y ≈ L(2.1) = 0.967 dx (2,1) 3 14 36 73. L(x) = − x + ; y ≈ L(−1.1) = 2.056 25 25 75. Let f (x) = x 2 . Then

71.

f = f (5 + h) − f (5) = (5 + h)2 − 52 = h2 + 10h and E = |f − f (5)h| = |h2 + 10h − 10h| = h2 =

1 1 (2)h2 = Kh2 . 2 2

Section 4.2 Preliminary Questions 1. A critical point is a value of the independent variable x in the domain of a function f at which either f (x) = 0 or f (x) does not exist. 2. (b) 3. (b) 4. Fermat’s Theorem claims: If f (c) is a local extreme value, then either f (c) = 0 or f (c) does not exist.

A52

ANSWERS TO ODD-NUMBERED EXERCISES

Section 4.2 Exercises 1. (a) 3 (b) 6 (c) Local maximum of 5 at x = 5 (d) Answers may vary. One example is the interval [4, 8]. Another is [2, 6]. (e) Answers may vary. One example is [0, 2]. 3. x = 1 5. x = −3 and x = 6 7. x = 0 9. x = ±1 11. t = 3 and t = −1 √ 13. x = 0, x = ± 2/3, x = ±1 √ 1 3 nπ 17. x = 19. x = ± 15. θ = 2 e 2 21. (a) c = 2 (b) f (0) = f (4) = 1 (c) Maximum value: 1; minimum value: −3. (d) Maximum value: 1; minimum value: −2. √ π 23. x = ; Maximum value: 2; minimum value: 1 4 25. Maximum value: 1 y 1 0.8 0.6 0.4 0.2 x 0

1

2

3

4

(e) We can see that there are six flat points on the graph between 0 and 2π, as predicted. There are 4 local extrema, and two points at ( π2 , 0) and ( 3π 2 , 0) where the graph has neither a local maximum nor a local minimum. y 2 1

2

3

5

1

−1

x

4

6

−2

61. Critical point: x = 2; minimum value: f (2) = 0, maximum: f (0) = f (4) = 2 63. Critical point: x = 2; minimum value: f (2) = 0, maximum: f (4) = 20 15 65. c = 1 67. c = 4 69. f (0) < 0 and f (2) > 0 so there is at least one root by the Intermediate Value Theorem; there cannot be another root because f (x) ≥ 4 for all x. 71. There cannot be a root c > 0 because f (x) > 4 for all x > 0. 75. b ≈ 2.86    v22 1 v 77. (a) F = 1− 2 1+ 2 2 v1 v

27. Critical point: x ≈ 0.652185; maximum value: approximately 0.561096 29. Minimum: f (−1) = 3, maximum: f (2) = 21 31. Minimum: f (0) = 0, maximum: f (3) = 9 33. Minimum: f (4) = −24, maximum: f (6) = 8 35. Minimum: f (1) = 5, maximum: f (2) = 28 37. Minimum: f (2) = −128, maximum: f (−2) = 128 39. Minimum: f (6) = 18.5, maximum: f (5) = 26 41. Minimum: f (1) = −1, maximum: f (0) = f (3) = 0 √ √ 43. Minimum: f (0) = 2 6 ≈ 4.9, maximum:f (2) = 4 2 ≈ 5.66 √  3 ≈ −0.589980, maximum: 45. Minimum: f 2 f (4) ≈ 0.472136 π  π  1 47. Minimum: f (0) = f = 0, maximum:f = 2 4 2 49. Minimum: f (0) =−1, maximum:   √ π π f = 2 − 1 ≈ −0.303493 4 4  π √ π = − 3 ≈ −0.685, maximum: 51. Minimum: g 3 3   √ 5 5 g π = π + 3 ≈ 6.968 3 3 π  π = 1 − ≈ −0.570796, maximum: f (0) = 0 53. Minimum: f 4 2 55. Minimum: f (1) = 0, maximum is f (e) = e−1 ≈ 0.367879 57. Minimum: f (5) = 5 tan−1 5 − 5 ≈ 1.867004. maximum: f (2) = 5 tan−1 2 − 2 ≈ 3.535744 7π 3π 11π 59. (d) π6 , π2 , 5π 6 , 6 , 2 , and 6 ; the maximum value is

There are also critical points where the derivative does not exist: √ 4 (0, 0), (± 27, 0).

3 3 the minimum value is f ( π6 ) = f ( 7π 6 ) = 2 and √ 3 3 5π 11π f( 6 ) = f( 6 ) = − 2

(b) The curve 27x 2 = (x 2 + y 2 )3 and its horizontal tangents are plotted below.

√

1

(b) F (r) achieves its maximum value when r = 1/3 (c) If v2 were 0, then no air would be passing through the turbine, which is not realistic. • The maximum value of f on [0, 1] is 81.      a 1/(b−a) a a/(b−a)  a b/(b−a) f − . = b b b 1 4 83. Critical points: x = 1, x = 4 and x = 52 ; maximum value: f (1) = f (4) = 54 , minimum value: f (−5) = 17 70 •

y 1.2 1 0.8 0.6 0.4 0.2 −5 −4 −3 −2 −1

x 1

2

3

4

5

85. (a) There are therefore four points at which the derivative is zero: √ √ √ √ (−1, − 2), (−1, 2), (1, − 2), (1, 2).

A53

ANSWERS TO ODD-NUMBERED EXERCISES y

•

The function graphed here is discontinuous at x = 0.

1 −2

y x

−1

1

8

2

6 4

−1

87.

−8 −6 −4 −2

x 2

4

6 8

y

Section 4.3 Preliminary Questions

10 x 1

2

3

−10

1. m = 3 2. (c) 3. Yes. The figure below displays a function that takes on only negative values but has a positive derivative.

89.

y x

y 4 3 2

4. (a) f (c) must be a local maximum.

(b) No.

1 x 0

1

2

3

4

91. If f (x) = a sin x + b cos x, then f (x) = a cos x − b sin x, so that f (x) = 0 implies a cos x − b sin x = 0. This implies tan x = ab . Then, ±a ±b sin x = and cos x = . 2 2 2 a +b a + b2 Therefore ±b ±a + b f (x) = a sin x + b cos x = a 2 2 2 a +b a + b2

Section 4.3 Exercises



1 − e−6 6

f (1) − f (0) 2−0 = = 2. 1−0 1 It appears that the x-coordinate of the point of tangency is approximately 0.62. y

y = x5 + x 2

y

93. Let f (x) = x 2 + rx + s and suppose that f (x) takes on both

Next, f (x) = 2x + r = 0 when x = − 2r and, because the graph of f (x) is an upward opening parabola, it follows that f (− 2r ) is a minimum. 95. b > 14 a 2 • Let f (x) be a continuous function with f (a) and f (b) local 97. minima on the interval [a, b]. By Theorem 1, f (x) must take on both a minimum and a maximum on [a, b]. Since local minima occur at f (a) and f (b), the maximum must occur at some other point in the interval, call it c, where f (c) is a local maximum.

1 7. c = − ln 2

9. The slope of the secant line between x = 0 and x = 1 is

a 2 + b2 = ± = ± a 2 + b2 . a 2 + b2 positive and negative values. This will guarantee that f has two real roots. By the quadratic formula, the roots of f are −r ± r 2 − 4s . x= 2 Observe that the midpoint between these roots is   −r − r 2 − 4s r 1 −r + r 2 − 4s + =− . 2 2 2 2

√ 5. c = ± 7

7π 1. c = 4 3. c = 4

0.6 0.5

4 y = 2x − 0.764

2

x 1

0.4 x

0.3 0.52

0.56

0.6

0.64

11. The derivative is positive on the intervals (0, 1) and (3, 5) and negative on the intervals (1, 3) and (5, 6). 13. f (2) is a local maximum; f (4) is a local minimum y 15. y 17. 10

8 6 4 2

8 6 4

x −2

2 x 0

1

2

3

4

5

1

2

3

4



A54

ANSWERS TO ODD-NUMBERED EXERCISES

19. critical point: x = 3 - local maximum

39. c = 0

21. critical point: x = −2 - local maximum; critical point: x = 0 local minimum

x

23. c = 72 

−∞, 72

x f f



 7/2



7,∞ 2

+

0

−



M



(−∞, 0) +

0

+

f



¬



41. c = π2 and c = π  π 0, 2 x

x

(−∞, 0)

0

(0, 8)

8

(8, ∞)

f

+

0

−

0

+

f



M



m



2 ,π

+

0

f



M

3π 43. c = π2 , 7π 6 , 2 , and  π x 0, 2

27. c = −2, −1, 1

π

π 2

f

25. c = 0, 8

(0, ∞)

0

f

11π 6



π 2



π

(π, 2π )

−

0

+



m



π , 7π 2 6





7π 6

7π , 3π 6 2

f

+

0

−

0

+

f



M



m





3π 2

x

3π , 11π 2 6





11π 6

11π , 2π 6

x

(−∞, −2)

−2

(−2, −1)

−1

(−1, 1)

1

(1, ∞)

f

−

0

+

0

−

0

+

f

0

−

0

+



f

M



m





f



m



M

m

(−∞, −2)

−2

(−2, −1)

−1

(−1, ∞)

x

(−∞, 0)

0

(0, ∞)

−

0

+



m



f

+

0

−

0

+

f

f



M



m



f 47. c = − π4

31. c = 0 x

(−∞, 0)

0

(0, ∞)

f

+

0

+

f



=



f

x f

 π  − 2 , − π4

− π4

+

0

 π π −4, 2 −



M



f 49. c = ±1

33. c = ( 32 )2/5 x



45. c = 0

29. c = −2, −1 x



x (0, ( 32 )2/5 )

3 2/5 2

(( 32 )2/5 , ∞)

−

0

+



m



f

(0, 1)

(1, ∞)

1

f

−

0

+

f



m



37. c = 0 x

(−∞, 0)

0

(0, ∞)

f

+

0

−

f



M



−1

(−1, 1)

1

(1, ∞)

−

0

+

0

−

f



m



M



x

(0, 1)

51. c = 1

35. c = 1 x

(−∞, −1)

f

1

(1, ∞)

f

−

0

+

f



m



 1/e 1 ≈ 0.692201 55. f (x) > 0 for all x e 57. The graph of h(x) is shown below at the left. Because h(x) is negative for x < −1 and for 0 < x < 1, it follows that f (x) is decreasing for x < −1 and for 0 < x < 1. Similarly, f (x) is increasing for −1 < x < 0 and for x > 1 because h(x) is positive on these intervals. Moreover, f (x) has local minima at x = −1 and x = 1 and a local maximum at x = 0. A plausible graph for f (x) is shown below at the right.

53.

A55

ANSWERS TO ODD-NUMBERED EXERCISES h(x)

Section 4.4 Exercises

f (x)

1.0

0.3 0.2

0.5 0.1 −2

x

−1

1

2

−2

−0.5

x

−1

1

2

−0.2

−1.0

59. f (x) < 0 as long as x < 500; so, 8002 + 2002 = f (200) > f (400) = 6002 + 4002 . 61. every point c ∈ (a, b) 69. (a) Let g(x) = cos x and f (x) = 1 − 12 x 2 . Then f (0) = g(0) = 1 and g (x) = − sin x ≥ −x = f (x) for x ≥ 0 by Exercise 67. Now apply Exercise 67 to conclude that cos x ≥ 1 − 12 x 2 for x ≥ 0. (b) Let g(x) = sin x and f (x) = x − 16 x 3 . Then f (0) = g(0) = 0 and g (x) = cos x ≥ 1 − 12 x 2 = f (x) for x ≥ 0 by part (a). Now apply Exercise 67 to conclude that sin x ≥ x − 16 x 3 for x ≥ 0. 1 x 4 and f (x) = cos x. Then (c) Let g(x) = 1 − 12 x 2 + 24 f (0) = g(0) = 1 and g (x) = −x + 16 x 3 ≥ − sin x = f (x) for x ≥ 0 by part (b). Now apply Exercise 67 to conclude that 1 x 4 for x ≥ 0. cos x ≤ 1 − 12 x 2 + 24 1 x5, (d) The next inequality in the series is sin x ≤ x − 16 x 3 + 120 valid for x ≥ 0.

71. •

• Let f

(x) = 0 for all x. Then f (x) = constant for all x. Since f (0) = m, we conclude that f (x) = m for all x. Let g(x) = f (x) − mx. Then g (x) = f (x) − m = m − m = 0 which implies that g(x) = constant for all x and consequently f (x) − mx = constant for all x. Rearranging the statement, f (x) = mx + constant. Since f (0) = b, we conclude that f (x) = mx + b for all x.

1. (a) In C, we have f

(x) < 0 for all x. (b) In A, f

(x) goes from + to −. (c) In B, we have f

(x) > 0 for all x. (d) In D, f

(x) goes from − to +. 3. concave up everywhere; no points of inflection √ √ 5.√ concave up for x < − √ 3 and for 0 < x < 3; concave down for − 3< √x < 0 and for x > 3; point of inflection at x = 0 and at x=± 3 7. concave up for 0 < θ < π; concave down for π < θ < 2π ; point of inflection at θ = π 9. concave down for 0 < x < 9; concave up for x > 9; point of inflection at x = 9 11. concave up on (0, 1); concave down on (−∞, 0) ∪ (1, ∞); point of inflection at both x = 0 and x = 1 13. concave up for |x| > 1; concave down for |x| < 1; point of inflection at both x = −1 and x = 1 15. concave down for x < 23 ; concave up for x > 23 ; point of inflection at x = 23 17. concave down for x < 12 ; concave up for x > 12 ; point of inflection at x = 12 19. The point of inflection in Figure 15 appears to occur at t = 40 days. The growth rate at the point of inflection is approximately 5.5 cm/day. Because the logistic curve changes from concave up to concave down at t = 40, the growth rate at this point is the maximum growth rate for the sunflower plant. Sketches of the first and second derivative of h(t) are shown below at the left and at the right, respectively. h 6

(b) Let f (x) = sin x. Then f (x) = cos x and f

(x) = − sin x, so f

(x) = −f (x). Next, let f (x) = cos x. Then f (x) = − sin x, f

(x) = − cos x, and we again have f

(x) = −f (x). Finally, if we take f (x) = sin x, the result from part (a) guarantees that sin2 x + cos2 x = sin2 0 + cos2 0 = 0 + 1 = 1.

Section 4.4 Preliminary Questions 1. (a) increasing 2. f (c) is a local maximum 3. False

4. False

´´

0.1

4

t

3

g (x) = 2f (x)f (x) + 2f (x)f

(x) = 2f (x)f (x) + 2f (x)(−f (x)) = 0,

f (x)2 + f (x)2 = f (0)2 + f (0)2 .

h

5

73. (a) Let g(x) = f (x)2 + f (x)2 . Then

Because g (0) = 0 for all x, g(x) = f (x)2 + f (x)2 must be a constant function. To determine the value of C, we can substitute any number for x. In particular, for this problem, we want to substitute x = 0 and find C = f (0)2 + f (0)2 . Hence,

´

20

2

40

60

80

100

−0.1

1 t 20

40

60

80

100

21. f (x) has an inflection point at x = b and another at x = e; f (x) is concave down for b < x < e. 23. (a) f is increasing on (0, 0.4). (b) f is decreasing on (0.4, 1) ∪ (1, 1.2). (c) f is concave up on (0, 0.17) ∪ (0.64, 1). (d) f is concave down on (0.17, 0.64) ∪ (1, 1.2). 25. critical points are x = 3 and x = 5; f (3) = 54 is a local maximum, and f (5) = 50 is a local minimum 27. critical points are x = 0 and x = 1; f (0) = 0 is a local minimum, Second derivative test is inconclusive at x = 1 29. critical points are x = −4 and x = 2; f (−4) = −16 is a local maximum and f (2) = −4 is a local minimum   31. critical points are x = 0 and x = 29 ; f 29 is a local minimum; f

(x) is undefined at x = 0, so the Second Derivative Test cannot be applied there

A56

ANSWERS TO ODD-NUMBERED EXERCISES

33. critical points are x = 0, x = π3 and x = π; f (0) is a local minimum, f ( π3 ) is a local maximum and f (π ) is a local minimum √ √  35. critical points are x = ± 22 ; f 22 is a local maximum and  √  f − 22 is a local minimum   37. critical point is x = e−1/3 ; f e−1/3 is a local minimum   1  1 39. − ∞, 13 x 1 (1, ∞) 3 3,1 f

+

0

−

0

+

f



M



m



x



f

 − ∞, 23

f 41.



0

+



I





m



M



f



0

−



I



√

−

f



M



x

(0, 4)

4

(4, ∞)

f

−

0

+

f



I





3, ∞

0

y

y 6

x

55.

x

4

2

4

y 10 5

3,∞

+

1+

+

2

3,∞

1

1 3

 3

f



−

 − ∞, 13

√

2

2

0

 f

2

+

f

1+

2

0

−

53.

√  3

4

2 3

0





0, 1 +

1

0, 3

− ∞, 0

t





x

3,∞

−

f

t

2

2 3

51.

−2

x

−1

1

2

−5 −10

43. f

(x) > 0 for all x ≥ 0, which means there are no inflection points     x 0 0, (2)2/3 (2)2/3 , ∞ (2)2/3 f

U

−

0

+

f

M



m



57. (a) Near the beginning of the epidemic, the graph of R is concave up. Near the epidemic’s end, R is concave down. (b) “Epidemic subsiding: number of new cases declining.” 59. The point of inflection should occur when the water level is equal to the radius of the sphere. A possible graph of V (t) is shown below. V

45.

√  −∞, −3 3

√ −3 3

f

−

f





x

√ √  −3 3, 3 3

√ 3 3

  √ 3 3, ∞

0

+

0

−

m



M





t

x

(−∞, −9)

−9

(−9, 0)

0

(0, 9)

9

(9, ∞)

f

−

0

+

0

−

0

+

f



I



I



I



47.

49.

θ

(0, π )

π

(π, 2π)

f

+

0

+

f



¬



θ

0

(0, π )

π

(π, 2π)

2π

f

0

−

0

+

0

f

¬



I



¬

f

 π π −2, 2 +

f

f



f

x

x

61. (a) f (u) =

beb(a−u) (1 + eb(a−u) )2

>0

1 ln 2 b 63. (a) From the definition of the derivative, we have

(b) u = a +

f (c + h) − f (c) f (c + h) = lim . h h h→0 h→0

f

(c) = lim

(b) We are given that f

(c) > 0. By part (a), it follows that f (c + h) > 0; h h→0 lim

 π  − 2 ,0

0



−

0

+



I



0, π2



in other words, for sufficiently small h, f (c + h) > 0. h Now, if h is sufficiently small but negative, then f (c + h) must also be negative (so that the ratio f (c + h)/ h will be positive) and

A57

ANSWERS TO ODD-NUMBERED EXERCISES

c + h < c. On the other hand, if h is sufficiently small but positive, then f (c + h) must also be positive and c + h > c. Thus, there exists an open interval (a, b) containing c such that f (x) < 0 for a < x < c and f (c) > 0 for c < x < b. Finally, because f (x) changes from negative to positive at x = c, f (c) must be a local minimum. 65. (b) f (x) has a point of inflection at x = 0 and at x = ±1. The figure below shows the graph of y = f (x) and its tangent lines at each of the points of inflection. It is clear that each tangent line crosses the graph of f (x) at the inflection point. y

x

67. Let f (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 be a polynomial of degree n. Then f (x) = nan x n−1 + (n − 1)an−1 x n−2 + · · · + 2a2 x + a1 and f

(x) = n(n − 1)an x n−2 + (n − 1)(n − 2)an−1 x n−3 + · · · + 6a3 x + 2a2 . If n ≥ 3 and is odd, then n − 2 is also odd and f

(x) is a polynomial of odd degree. Therefore f

(x) must take on both positive and negative values. It follows that f

(x) has at least one root c such that f

(x) changes sign at c. The function f (x) will then have a point of inflection at x = c. On the other hand, the functions f (x) = x 2 , x 4 and x 8 are polynomials of even degree that do not have any points of inflection.

57.

  1 ln(1 + x) lim ln (1 + x)1/x = lim ln(1 + x) = lim = 1, x x→0 x→0 x x→0

so lim (1 + x)1/x = e1 = e; x = 0.0005 x→0

59. (a) limx→0+ f (x) = 0; limx→∞ f (x) = e0 = 1. (b) f is increasing for 0 < x < e, is decreasing for x > e and has a maximum at x = e. The maximum value is f (e) = e1/e ≈ 1.444668. 61. Neither ln x x −1 1 = lim = lim x −a = 0 63. lim a x→∞ x x→∞ ax a−1 x→∞ a 67. (a) 1 ≤ 2 + sin x ≤ 3, so x x(2 + sin x) 3x ≤ ≤ 2 ; x2 + 1 x2 + 1 x +1 it follows by the Squeeze Theorem that lim

x→∞

x(2 + sin x) = 0. x2 + 1

(b) lim f (x) = lim x(2 + sin x) ≥ lim x = ∞ and x→∞

x→∞

x→∞

lim g(x) = lim (x 2 + 1) = ∞, but

x→∞

x→∞

lim

f (x)

x→∞ g (x)

= lim

x→∞

does not exist since cos x oscillates. This does not violate L’Hôpital’s Rule since the theorem clearly states lim

f (x)

x→∞ g(x)

1. Not of the form 00 or ∞ ∞ 2. No

Section 4.5 Exercises

11. 13. 15. 17. 31. 43. 55.

b = 0.25

4

L’Hôpital’s Rule does not apply. L’Hôpital’s Rule does not apply. L’Hôpital’s Rule does not apply. L’Hôpital’s Rule does not apply. 0 9 Quotient is of the form ∞ ∞; −2 Quotient is of the form ∞ ∞; 0 Quotient is of the form ∞ ∞; 0 5 3 7 9 2 19. − 21. − 23. 25. 27. 1 29. 2 6 5 3 7 7 1 2 −1 33. 35. 0 37. − 39. 1 41. Does not exist 2 π 1 0 45. ln a 47. e 49. e−3/2 51. 1 53. π ⎧ (−1)(m−n)/2 , ⎪ ⎪ ⎨ cos mx does not exist, = lim 0 ⎪ x→π/2 cos nx ⎪ ⎩ (−1)(m−n)/2 m n,

= lim

f (x)

x→∞ g (x)

“provided the limit on the right exists.” 69. (a) Using Exercise 68, we see that G(b) = eH (b) . Thus, G(b) = 1 if 0 ≤ b ≤ 1 and G(b) = b if b > 1. y (b) y

Section 4.5 Preliminary Questions

1. 3. 5. 7. 9.

x(cos x) + (2 + sin x) 2x

m, n even m even, n odd m odd, n even m, n odd

b = 0.5

4

3

3

2

2

1

1 x

x 5

10

15

5

10

15

y

y 6 5 4 3 2 1

b = 2.0

4 3 2 1

b = 3.0

x 5

10

x

15

5

10

71. lim f (x) k = lim x→0 x

1 2 . Let t = 1/x. As x → 0, t → ∞. x→0 x k e1/x

Thus, lim

1

x→0 x k e1/x 2

by Exercise 70.

15

= lim

tk

t→∞ et 2

=0

A58

ANSWERS TO ODD-NUMBERED EXERCISES

73. For x = 0, f (x) = e−1/x

2





2 . Here P (x) = 2 and r = 3. x3

•

2

−1/x . Then Assume f (k) (x) = P (x)ex r   2 x 3 P (x) + (2 − rx 2 )P (x) (k+1) −1/x f (x) = e x r+3

In D, f is decreasing and concave down, so f < 0 and f

< 0. In E, f is decreasing and concave up, so f < 0 and f

> 0. In F, f is increasing and concave up, so f > 0 and f

> 0. In G, f is increasing and concave down, so f > 0 and f

< 0.

• • •

3. This function changes from concave up to concave down at x = −1 and from increasing to decreasing at x = 0.

which is of the form desired. Moreover, from Exercise 71, f (0) = 0. Suppose f (k) (0) = 0. Then f (k) (x) − f (k) (0) P (x)e−1/x = lim x−0 x→0 x→0 x r+1

−1

0

1

x

2

f (k+1) (0) = lim

−1

f (x) = P (0) lim r+1 = 0. x→0 x

y

77. lim sinx x = lim cos1 x = 1. To use L’Hôpital’s Rule to evaluate x→0

x→0

limx→0 sinx x , we must know that the derivative of sin x is cos x, but to determine the derivative of sin x, we must be able to evaluate limx→0 sinx x .

5. The function is decreasing everywhere and changes from concave up to concave down at x = −1 and from concave down to concave up at x = − 12 .

79. (a) e−1/6 ≈ 0.846481724 x  sin x 1/x 2 x

y

1

0.1

0.01

0.841471

0.846435

0.846481

0.05

x

(b) 1/3 ±1

x 1 − 2 x sin2 x 1

±0.1

0.412283

0.334001

±0.01

7.

−1

0

9.

y

0.333340

2

10

1

5

x

−1

Section 4.6 Preliminary Questions

2

11.

4

1

−1

x

1. An arc with the sign combination ++ (increasing, concave up) is shown below at the left. An arc with the sign combination −+ (decreasing, concave up) is shown below at the right. y

y

15

6

2

3

−2

y 6

y

4 2 x 0

x

x

2. (c)

2

4

6

8 10 12 14

13. Local maximum at x = −16, a local minimum at x = 0, and an inflection point at x = −8.

3. x = 4 is not in the domain of f

y 3000

Section 4.6 Exercises •

1.

In A, f is decreasing and concave up, so f < 0 and

f

> 0.

• •

In B, f is increasing and concave up, so f > 0 and f

> 0. In C, f is increasing and concave down, so f > 0 and f

< 0.

2000 1000 −20 −15 −10 −5

x 5

ANSWERS TO ODD-NUMBERED EXERCISES

15. f (0) is a local minimum, f ( 16 ) is a local maximum, and there is a 1 . point of inflection at x = 12

A59

27. f has a local maximum at x = 6 and inflection points at x = 8 and x = 12.

y 0.04

y x

−0.2

0.2

−5

−0.04

x

−30

√

y

29. f has a local minimum at x = − 22 , a local maximum at √

10

√

x = 22 , inflection points at x = 0 and at x = ± 23 , and a horizontal asymptote at y = 0.

5 −2

2

x

19. Graph has no critical points and is always increasing, inflection point at (0, 0).

0.4

y

0.2

40

y

−3

−2

x

−1

20 −1

15

−20

√ 17. f has local minima at√x = ± 6, a local maximum at x = 0, and inflection points at x = ± 2.

−2

10

5 −10

1

2

3

–0.2 x 1

−20

2

−40

√ √ 21. f ( 1−8 33 ) and f (2) are local minima, and f ( 1+8 33 ) is a local maximum; points of inflection both at x = 0 and x = 32 .

31. f (2) is a local minimum and the graph is always concave up.

y y

6 8

4 2 −1

6 x 1

−2

4

2

2 x

23. f (0) is a local maximum, f (12) is a local minimum, and there is a point of inflection at x = 10.

2

4

6

8

y 1 × 107

33. f has a local maximum at x = 1 and a point of inflection at x = e−3/2 .

5 × 106 x

−5

5

10

−5 × 10

6

y x

25. f (4) is a local minimum, and the graph is always concave up. −2

y

−4

4

−6

2 x −2

0.5

5

10

15

20

1.0

1.5

2.0

2.5

A60

ANSWERS TO ODD-NUMBERED EXERCISES

35. Graph has an inflection point at x = 35 , a local maximum at x = 1 (at which the graph has a cusp), and a local minimum at x = 95 .

45. Local maximum at x = π6 and a point of inflection at x = 2π 3 . y 2

y

1

40 −1 20

−2

1

2

x

3

x

1

−1

−20 −40 −60 −80

2

3

−2

37. f has a local maximum at x = 0, local minima at x = ±3 and √ points of inflection at x = ± −6 + 3 5.

47. In both cases, there is a point where f is not differentiable at the transition from increasing to decreasing or decreasing to increasing. y

y

y 5 x

−6 −4 −2

2

4

6

x

−10

39. f has local minima at x = −1.473 and x = 1.347, a  local maximum at x = 0.126 and points of inflection at x = ± 23 . y 20 15 10 5

x

49. Graph (B) cannot be the graph of a polynomial. 3x 2 51. (B) is the graph of f (x) = 2 ; (A) is the graph of x −1 3x . f (x) = 2 x −1 53. f is decreasing for all x  = 13 , concave up for x > 13 , concave down for x < 13 , has a horizontal asymptote at y = 0 and a vertical asymptote at x = 13 . y

x −2

−1

1

−5

2

5 x

−2

41. Graph has an inflection point at x = π, and no local maxima or minima.

2 −5

y

55. f is decreasing for all x  = 2, concave up for x > 2, concave down for x < 2, has a horizontal asymptote at y = 1 and a vertical asymptote at x = 2.

6 5 4 3

y

2

10

1

5 x

0

1

2

3

4

5

6

43. Local maximum at x = π2 , a local minimum at x = 3π 2 , and π 5π inflection points at x = 6 and x = 6 . y

−10

x

−5

5

−5

10

−10

57. f is decreasing for all x  = 0, 1, concave up for 0 < x < 12 and x > 1, concave down for x < 0 and 12 < x < 1, has a horizontal asymptote at y = 0 and vertical asymptotes at x = 0 and x = 1.

2

y

1 3 −1 −2

1

2

x 4

5

5

6 −1

x −5

1

2

ANSWERS TO ODD-NUMBERED EXERCISES

59. f is increasing for x < 0 and 0 < x < 1 and decreasing for 1 < x < 2 and x > 2; f is concave up for x < 0 and x > 2 and concave down for 0 < x < 2; f has a horizontal asymptote at y = 0 and vertical asymptotes at x = 0 and x = 2.

A61

67. f is increasing for x < 0, decreasing for x >√0 and has a local maximum at x = 0; √f is concave up for |x| > 1/ 5, is concave √ down for |x| < 1/ 5, and has points of inflection at x = ±1/ 5; f has a horizontal asymptote at y = 0 and no vertical asymptotes.

y y 1 5 0.8 x

−2

4 −5

61. f is increasing for x < 2 and for 2 < x < 3, is decreasing for 3 < x < 4 and for x > 4, and has a local maximum at x = 3; f is concave up for x < 2 and for x > 4 and is concave down for 2 < x < 4; f has a horizontal asymptote at y = 0 and vertical asymptotes at x = 2 and x = 4. y

−4

x

−2

2

4

69. f is increasing √ for x < 0 and decreasing for √ x > 0; f is concave

down for |x| < 22 and concave up for |x| > 22 ; f has a horizontal asymptote at y = 0 and no vertical asymptotes.

5

y 3 1

1

x

2

4

5

6

0.8

−5

63. f is increasing for |x| > 2 and decreasing √ for −2 < x < 0 and for 0 < √x < 2; f is concave down for√−2 2 < x < 0 and for √ x > 2 2 and concave up for x < −2 2 and for 0 < x < 2 2; f has a horizontal asymptote at y = 1 and a vertical asymptote at x = 0. y 6 4 2 x

−6 −4 −2

2

−2

4

0.2 −10

73. f is increasing for x < −2 and for x > 0, is decreasing for −2 < x < −1 and for −1 < x < 0, has a local minimum at x = 0, has a local maximum at x = −2, is concave down on (−∞, −1) and concave up on (−1, ∞); f has a vertical asymptote at x = −1; by 1 and polynomial division, f (x) = x − 1 + x+1  1 lim x−1+ − (x − 1) = 0, x→±∞ x+1

65. f is increasing for x < 0 and for x > 2 and decreasing for 0 < x < 2; f is concave up for x < 0 and for 0 < x < 1, is concave down for 1 < x < 2 and for x > 2, and has a point of inflection at x = 1; f has a horizontal asymptote at y = 0 and vertical asymptotes at x = 0 and x = 2.

which implies that the slant asymptote is y = x − 1. y

y

4 2

4 2

−4

1

−4

10



−6

−2

5

6

−4

−2 −1

x

−5

2

3

4

x

−2

x −2 −4 −6

2

4

A62

ANSWERS TO ODD-NUMBERED EXERCISES

75. y = √ x + 2 is the slant asymptote of f (x); √ local minimum at x = 2 + 3, a local maximum at x = 2 − 3 and f is concave down on (−∞, 2) and concave up on (2, ∞); vertical asymptote at x = 2. y 10 5 −10

−5

5

10

x

−5 −10

Section 4.7 Preliminary Questions

1. b + h + b2 + h2 = 10 2. If the function tends to infinity at the endpoints of the interval, then the function must take on a minimum value at a critical point. 3. No

Section 4.7 Exercises 1. (a) y = 32 − x

(b) A = x( 32 − x) = 32 x − x 2

(c) Closed interval [0, 23 ] (d) 3. 5. 7.

51. There are N shipments per year, so the time interval between shipments is T = 1/N years. Hence, the total storage costs per year are sQ/N . The yearly delivery costs are dN and the total costs is C(N) = dN + sQ/N . Solving, sQ C (N) = d − 2 = 0 N √ for N yields N = sQ/d. N = 9. √ √ 53. (a) If b < 3a, then d = a − b/ 3 > 0 and the √ minimum occurs at this value of d. On the other hand, if b ≥ 3a, then the minimum occurs at the endpoint d = 0. √ (b) Plots of S(d) for b = 0.5, b = 3 and b = 3 are shown below. For b = 0.5, the √ results of (a) indicate the minimum should occur for d =√ 1 − 0.5/ 3 ≈ 0.711, and this is confirmed in the plot. For both b = 3 and b = 3, the results of (a) indicate that the minimum should occur at d = 0, and both of these conclusions are confirmed in the plots. y

y b = 0.5

2.1 2 1.9 1.8 1.7 1.6 1.5

x 0

The maximum area 0.5625 m2 is achieved with x = y = 34 m. Allot approximately 5.28 m of the wire to the circle. The middle of the wire The corral of maximum area has dimensions 300 150 x= m and y= m, 1 + π/4 1 + π/4

where x is the width of the corral and therefore the diameter of the semicircle and y is the height of the rectangular section   √ 1 1 9. Square of side length 4 2 11. , 2 2 √ π 3 3 2 13. (0.632784, −1.090410) 15. θ = 17. r 2 4 19. 60 cm wide by 100 cm high for the full poster (48 cm by 80 cm for the printed matter)  21. Radius: 23 R; half-height: √R 3 √ √ 23. x = 10 5 ≈ 22.36 m and y = 20 5 ≈ 44.72 m where x is the length of the brick wall and y is the length of an adjacent side 25. 1.0718 27. LH + 12 (L2 + H 2 ) 29. y = −3x + 24 √ √ 33. s = 3 3 4 m and h = 2 3 4 m, where s is the length of the side of the square bottom of the box and h is the height of the box 35. (a) Each compartment has length of 600 m and width of 400 m. (b) 240000 square meters. 37. N ≈ 58.14 pounds and P ≈ 77.33 pounds 39. $990 41. 1.2 million euros in equipment and 600000 euros in labor 43. Brandon swims diagonally to a point located 20.2 m downstream and then runs the rest of the way. 45. h = 3; dimensions are 9 × 18 × 3 47. A = B = 30 cm 49. x = bh + h2

0.2 0.4 0.6 0.8

b = 3

4.4 4.3 4.2 4.1 4

1

x 0

0.2 0.4 0.6 0.8

1

y b=3 6.8 6.6 6.4 x 0

0.2 0.4 0.6 0.8

1

f mg 55. minimum value of F (θ) is . 1 + f2 √ 57. s ≈ 30.07 59. 15 5 61. = (b2/3 + h2/3 )3/2 ft 63. (a) α = 0 corresponds to shooting the ball directly at the basket while α = π/2 corresponds to shooting the ball directly upward. In neither case is it possible for the ball to go into the basket. If the angle α is extremely close to 0, the ball is shot almost directly at the basket; on the other hand, if the angle α is extremely close to π/2, the ball is launched almost vertically. In either one of these cases, the ball has to travel at an enormous speed. (b) The minimum clearly occurs where θ = π/3.

π 6

π 4

π 3

5π 12

π 2

16d ; hence v 2 is smallest whenever F (θ ) is greatest. F (θ) (d) A critical point of F (θ) occurs where cos(α − 2θ ) = 0, so that α − 2θ = − π2 (negative because 2θ > θ > α), and this gives us θ = α/2 + π/4. The minimum value F (θ0 ) takes place at θ0 = α/2 + π/4. (c) v 2 =

ANSWERS TO ODD-NUMBERED EXERCISES

(e) Plug in θ0 = α/2 + π/4. From Figure 34 we see that d h cos α = and sin α = . 2 2 2 d +h d + h2 (f) This shows that the minimum velocity required to launch the ball to the basket drops as shooter height increases. This shows one of the ways height is an advantage in free throws; a taller shooter need not shoot the ball as hard to reach the basket.

A63

Section 4.8 Preliminary Questions 1. One 2. Every term in the Newton’s Method sequence will remain x0 . 3. Newton’s Method will fail. 4. Yes, that is a reasonable description. The iteration formula for Newton’s Method was derived by solving the equation of the tangent line to y = f (x) at x0 for its x-intercept.

y 600

Section 4.8 Exercises

500

1.

400 300 200 100 x 0

1

2

3

4

n

1

2

3

xn

2.5

2.45

2.44948980

5

65. (a) From the figure, we see that c − f (x) b − f (x) θ (x) = tan−1 − tan−1 . x x Then b − (f (x) − xf (x)) c − (f (x) − xf (x)) θ (x) = − x 2 + (b − f (x))2 x 2 + (c − f (x))2

3. n

1

2

3

xn

2.16666667

2.15450362

2.15443469

5. n

1

2

3

xn

0.28540361

0.24288009

0.24267469

= (b − c)

x 2 − bc + (b + c)(f (x) − xf (x)) − (f (x))2 + 2xf (x)f (x) (x 2 + (b − f (x))2 )(x 2 + (c − f (x))2 )

= (b − c)

(x 2 + (xf (x))2 − (bc − (b + c)(f (x) − xf (x)) + (f (x) − xf (x))2 ) (x 2 + (b − f (x))2 )(x 2 + (c − f (x))2 )

= (b − c)

(x 2 + (xf (x))2 − (b − (f (x) − xf (x)))(c − (f (x) − xf (x))) . (x 2 + (b − f (x))2 )(x 2 + (c − f (x))2 )

(b) The point Q is the y-intercept of the line tangent to the graph of f (x) at point P . The equation of this tangent line is

7. We take x0 = −1.4, based on the figure, and then calculate

Y − f (x) = f (x)(X − x). The y-coordinate of Q is then f (x) − xf (x). (c) From the figure, we see that BQ = b − (f (x) − xf (x)), CQ = c − (f (x) − xf (x)) and

  P Q = x 2 + (f (x) − (f (x) − xf (x)))2 = x 2 + (xf (x))2 .

Comparing these expressions with the numerator of dθ/dx, it follows dθ = 0 is equivalent to that dx P Q2 = BQ · CQ. (d) The equation P Q2 = BQ · CQ is equivalent to PQ CQ = . BQ PQ In other words, the sides CQ and P Q from the triangle QCP are proportional in length to the sides P Q and BQ from the triangle QP B. As  P QB =  CQP , it follows that triangles QCP and QP B are similar.

n

1

2

3

xn

−1.330964467

−1.328272820

−1.328268856

9. r1 ≈ 0.25917 and r2 ≈ 2.54264 √ 11 ≈ 3.317; a calculator yields 3.31662479

11.

13. 27/3 ≈ 5.040; a calculator yields 5.0396842 15. 2.093064358 17. −2.225 19. 1.749 21. x = 4.49341, which is approximately 1.4303π 23. (2.7984, −0.941684) 25. (a) P ≈ $156.69 (b) b ≈ 1.02121; the interest rate is around 25.45% 27. (a) The sector SAB is the slice OAB with the triangle OP S removed. OAB is a central sector with arc θ and radius OA = a, and 2 therefore has area a2θ . OP S is a triangle with height a sin θ and base length OS = ea. Hence, the area of the sector is a2 a2 1 θ − ea 2 sin θ = (θ − e sin θ ). 2 2 2

A64

ANSWERS TO ODD-NUMBERED EXERCISES

(b) Since Kepler’s second law indicates that the area of the sector is proportional to the time t since the planet passed point A, we get πa 2 (t/T ) = a 2 /2 (θ − e sin θ ) 2π

t = θ − e sin θ. T

(c) From the point of view of the Sun, Mercury has traversed an angle of approximately 1.76696 radians = 101.24◦ . Mercury has therefore traveled more than one fourth of the way around (from the point of view of central angle) during this time. 29. The sequence of iterates diverges spectacularly, since xn = (−2)n x0 . 31. (a) Let f (x) = x1 − c. Then 1 f (x) x −c = 2x − cx 2 . = x − f (x) −x −2

x−

(b) For c = 10.3, we have f (x) = x1 − 10.3 and thus xn+1 = 2xn − 10.3xn2 . •

Take x0 = 0.1. n xn

•

1 0.097

2 0.0970873

3 0.09708738

Take x0 = 0.5. n xn

1 −1.575

2 −28.7004375

3 −8541.66654

(c) The graph is disconnected. If x0 = .5, (x1 , f (x1 )) is on the other portion of the graph, which will never converge to any point under Newton’s Method. 1 − cos θ 33. θ ≈ 1.2757; hence, h = L ≈ 1.11181 2 sin θ 35. (a) a = 46.95 (b) s = 29.24 37. (a) a ≈ 28.46 (b) L = 1 foot yields s ≈ 0.61; L = 5 yields s ≈ 3.05 (c) s(161) − s(160) = 0.62, very close to the approximation obtained from the Linear Approximation; s(165) − s(160) = 3.02, again very close to the approximation obtained from the Linear Approximation.

Section 4.9 Preliminary Questions 1. Any constant function is an antiderivative for the function f (x) = 0. 2. No difference 3. No 4. (a) False. Even if f (x) = g(x), the antiderivatives F and G may differ by an additive constant. (b) True. This follows from the fact that the derivative of any constant is 0. (c) False. If the functions f and g are different, then the antiderivatives F and G differ by a linear function: F (x) − G(x) = ax + b for some constants a and b. 5. No

Section 4.9 Exercises 2 5 x − 8x 3 + 12 ln |x| + C 5 5. 2 sin x + 9 cos x + C 7. 12ex + 5x −1 + C 9. (a) (ii) (b) (iii) (c) (i) (d) (iv) 11 5/11 t +C 11. 4x − 9x 2 + C 13. 5 15. 3t 6 − 2t 5 − 14t 2 + C 3 4 17. 5z1/5 − z5/3 + z9/4 + C 5 9 18 3 2/3 + C 21. − 2 + C 19. x 2 t 2 5/2 1 2 2 3/2 23. t + t + t +t +C 5 2 3 1 25. x 2 + 3 ln |x| + 4x −1 + C 2 27. 12 sec x + C 29. − csc t + C 25 1 tan(3z + 1) + C 31. − tan(7 − 3x) + C 33. 3 3   θ 1 +C 35. sin(3θ) − 2 tan 3 4 3 37. e5x + C 39. 4x 2 + 2e5−2x + C 5 41. Graph (B) does not have the same local extrema as indicated by f (x) and therefore is not an antiderivative of f (x).   d 1 (x + 13)7 + C = (x + 13)6 43. dx 7   1 d 1 3 (4x + 13) + C = (4x + 13)2 (4) = (4x + 13)2 45. dx 12 4 1 4 2 1 47. y = x + 4 49. y = t 2 + 3t 3 − 2 51. y = t 3/2 + 4 3 3 1 1 4 53. y = (3x + 2) − 55. y = 1 − cos x 12 3 1 57. y = 3 + sin 5x 59. y = ex − e2 61. y = −3e12−3t + 10 5 63. f (x) = 6x 2 + 1; f (x) = 2x 3 + x + 2 1 5 1 3 1 2 x − x + x +x 65. f (x) = 14 x 4 − x 2 + x + 1; f (x) = 20 3 2 67. f (t) = −2t −1/2 + 2; f (t) = −4t 1/2 + 2t + 4 1 1 69. f (t) = t 2 − sin t + 2; f (t) = t 3 + cos t + 2t − 3 2 6 71. The differential equation satisfied by s(t) is 1. 6x 3 + C

3.

ds = v(t) = 6t 2 − t, dt and the associated initial condition is s(1) = 0; 1 3 s(t) = 2t 3 − t 2 − . 2 2 73. The differential equation satisfied by s(t) is ds = v(t) = sin(πt/2), dt and the associated initial condition is s(0) = 0; 2 s(t) = (1 − cos(πt/2)) π 75. 6.25 seconds; 78.125 meters 77. 300 m/s 81. c1 = 1 and c2 = −1

ANSWERS TO ODD-NUMBERED EXERCISES

83. (a) By the Chain Rule, we have   d 1 1 F (2x) = F (2x) · 2 = F (2x) = f (2x). dx 2 2

A65

49. No horizontal asymptotes; no vertical asymptotes y 10

Thus 12 F (2x) is an antiderivative of f (2x). 1 (b) F (kx) + C k

5 x

−1

1

−5

2

−10

Chapter 4 Review 1. 8.11/3 − 2 ≈ 0.00833333; error is 3.445 × 10−5 3. 6251/4 − 6241/4 ≈ 0.002; error is 1.201 × 10−6 1 ≈ 0.98; error is 3.922 × 10−4 5. 1.02 1 7. L(x) = 5 + (x − 25) 9. L(r) = 36π(r − 2) 10 1 11. L(x) = √ (2 − x) 13. s ≈ 0.632 e 15. (a) An increase of $1500 in revenue. (b) A small increase in price would result in a decrease in revenue. 3 ≈ 2.164 ∈ (1, 4) 17. 9% 21. c = ln 4 23. Let x > 0. Because f is continuous on [0, x] and differentiable on (0, x), the Mean Value Theorem guarantees there exists a c ∈ (0, x) such that f (c) =

f (x) − f (0) x−0

51. y = 0 is a horizontal asymptote; x = −1 is a vertical asymptote y 4 2 −3 −2 −1

f (x) ≤ 4 + x(2) = 2x + 4.

y 1 0.8 0.6 0.4 0.2

10 5 −5

−10

1

2

3

4

5

4

y 1

−1

4 1

2

3

x 5

6

57. y

−4

63. x

2

55.

59. b =

y

x

−8 −6 −4 −2

25. x = 23 and x = 2 are critical points; f ( 23 ) is a local maximum

while f (2) is a local minimum. 27. x = 0, x = −2 and x = − 45 are critical points; f (−2) is neither a local maximum nor a local minimum, f (− 45 ) is a local maximum and f (0) is a local minimum.   3π 3π + nπ is a critical point for all integers n; g + nπ 29. θ = 4 4 is neither a local maximum nor a local minimum for any integer n. 31. Maximum value is 21; minimum value is −11. 5 33. Minimum value is −1; maximum value is . 4 35. Minimum value is −1; maximum value is 3. 37. Minimum value is 12 − 12 ln 12 ≈ −17.818880; maximum value is 40 − 12 ln 40 ≈ −4.266553. 39. Minimum value is 2; maximum value is 17. 2 4 43. x = ± √ 45. x = 1 and x = 4 41. x = 3 3 47. No horizontal asymptotes; no vertical asymptotes

3

53. horizontal asymptote of y = 0; no vertical asymptotes

f (x) = f (0) + xf (c).

or

2

−4

Now, we are given that f (0) = 4 and that f (x) ≤ 2 for x > 0. Therefore, for all x ≥ 0,

−1

x 1

−2

16 π 9

√ 3

x 4

8

√ 12 meters and h = 13 3 12 meters 69.

√ 3

25 = 2.9240

2 71. x 4 − x 3 + C 73. − cos(θ − 8) + C 3 75. −2t −2 + 4t −3 + C

A66

ANSWERS TO ODD-NUMBERED EXERCISES

1 1 (y + 2)5 + C 81. ex − x 2 + C 5 2 83. 4 ln |x| + C 85. y(x) = x 4 + 3 87. y(x) = 2x 1/2 − 1 1 1 89. y(x) = 4 − e−x 91. f (t) = t 2 − t 3 − t + 2 2 3 93. (0, 2e ) is a local minimum 77. tan x + C

79.

Section 5.1 Exercises 1. Over the interval [0, 3]: 0.96 km; over the interval [1, 2.5]: 0.5 km 3. 28.5 cm; The figure below is a graph of the rainfall as a function of time. The area of the shaded region represents the total rainfall.

95. Local minimum at x = e−1 ; no points of inflection; limx→0+ x ln x = 0; limx→∞ x ln x = ∞

y 2.5 2.0 1.5

y 6

1.0 0.5

4

x 5

2 x 1

2

3

10

15

20

25

5. L5 = 46; R5 = 44

4

97. Local maximum at x = e−2 and a local minimum at x = 1; point of inflection at x = e−1 ; limx→0+ x(ln x)2 = 0; limx→∞ x(ln x)2 = ∞ y

7. (a) L6 = 16.5; R6 = 19.5 (b) Via geometry (see figure below), the exact area is A = 18. Thus, L6 underestimates the true area (L6 − A = −1.5), while R6 overestimates the true area (R6 − A = +1.5). y

0.8 0.6

9

0.4

6

0.2

3 x 0.5

1

x

1.5

0.5

99. As x → ∞, both 2x − sin x and 3x + cos 2x tend toward infinity, 2x − sin x so L’Hôpital’s Rule applies to lim ; however, the x→∞ 3x + cos 2x 2 − cos x resulting limit, lim , does not exist due to the x→∞ 3 − 2 sin 2x oscillation of sin x and cos x. To evaluate the limit, we note

103. 0

105. 3

107. ln 2

109.

1 6

1.5

2

2.5

3

9. R3 = 32; L3 = 20; the area under the graph is larger than L3 but smaller than R3 y 14 12 10 8 6 4

2 − sinx x 2x − sin x 2 lim = lim = . x→∞ 3x + cos 2x x→∞ 3 + cos 2x 3 x 101. 4

1

L3

x

111. 2

1.0

1.5

2.0

2.5

3.0

3.5

2.5

3.0

3.5

y 14 12 10 8 6 4

Chapter 5 Section 5.1 Preliminary Questions 1. The right endpoints of the subintervals are then 52 , 3, 72 , 4, 92 , 5, while the left endpoints are 2, 52 , 3, 72 , 4, 92 . 3 9 (b) and 2 2 2 3. (a) Are the same (b) Not the same

2. (a)

(d) Are the same  4. The first term in the sum 100 j =0 j is equal to zero, so it may be  dropped; on the other hand, the first term in 100 j =0 1 is not zero.

R3

x 1.0

1.5

2.0

11. R3 = 2.5; M3 = 2.875; L6 = 3.4375 13. R3 = 15. M6 = 87

17. L6 = 12.125 19. L4 ≈ 0.410236 21.

8  k=4

(c) Are the same

5. On [3, 7], the function f (x) = x −2 is a decreasing function.

16 3

23.

5 

(2k + 2)

k=2

25.

n  i=1

i (i + 1)(i + 2)

27. (a) 45 (b) 24 (c) 99

k7

ANSWERS TO ODD-NUMBERED EXERCISES

29. (a) −1 (b) 13 (c) 12 31. 15050 33. 352800 35. 1093350 37. 41650 1 1 43. 39. −123165 41. 2 3 45. 18; the region under the graph is a triangle with base 2 and height 18 47. 12; the region under the curve is a trapezoid with base width 4 and heights 2 and 4 49. 2; the region under the curve over [0, 2] is a triangle with base and height 2 51. limN→∞ RN = 16 1 1 1 1 53. RN = + + ; 3 2N 6N 2 3 27 189 55. RN = 222 + + 2 ; 222 N N 8 6 57. RN = 2 + + 2; 2 N N (b − a)2 ; 59. RN = (b − a)(2a + 1) + (b − a)2 + N 2 2 (b + b) − (a + a) 61. The area between the graph of f (x) = x 4 and the x-axis over the interval [0, 1] 63. The area between the graph of y = ex and the x-axis over the interval [−2, 3]   N kπ π  sin 65. lim RN = lim N N→∞ N→∞ N 67.

lim LN = lim

N→∞

4

N→∞ N

k=1 N−1 

15 +

j =0

8j N

   N 1 1  1 1 tan + j− 69. lim MN = lim 2 2N 2 N→∞ N→∞ 2N j =1 71. Represents the area between the graph of y = f (x) = 1 − x 2 and the x-axis over the interval [0, 1]. This is the portion of the circular disk x 2 + y 2 ≤ 1 that lies in the first quadrant. Accordingly, its area is π4 . 73. Of the three approximations, RN is the least accurate, then LN and finally MN is the most accurate. 75. The area A under the curve is somewhere between L4 ≈ 0.518 and R4 ≈ 0.768. 77. f (x) is increasing over the interval [0, π/2], so 0.79 ≈ L4 ≤ A ≤ R4 ≈ 1.18. 79. L100 = 0.793988; R100 = 0.80399; L200 = 0.797074; R200 = 0.802075; thus, A = 0.80 to two decimal places. 81. (a) Let f (x) = ex on [0, 1]. With n = N , x = (1 − 0)/N = 1/N and xj = a + j x =

j N

for j = 0, 1, 2, . . . , N. Therefore, LN = x

N−1  j =0

f (xj ) =

N−1 1  j/N e . N j =0

A67

(b) Applying Eq. (8) with r = e1/N , we have LN =

1 (e1/N )N − 1 e−1 = . N e1/N − 1 N(e1/N − 1)

(c) A = e − 1 83. 1

Graph of f (x)

1

Right endpt approx, n = 1

1

Right endpt approx, n = 2

0.8 0.6 0.5

0.5

0.4 0.2 0

0

0 0.5

0.2 0.4 0.6 0.8 1

1

0.5

1

x

85. When f is large, the graph of f is steeper and hence there is more gap between f and LN or RN . 89. N > 30000

Section 5.2 Preliminary Questions 1. 2 ! 2. (a) False. ab f (x) dx is the signed area between the graph and the x-axis. (b) True. (c) True. 3. Because cos(π − x) = − cos x, the “negative” area between the graph of y = cos x and the x-axis over [ π2 , π ] exactly cancels the “positive” area between the graph and the x-axis over [0, π2 ]. " −5 4. 8 dx −1

Section 5.2 Exercises 1. The region bounded by the graph of y = 2x and the x-axis over the interval [−3, 3] consists of two right triangles. One has area 1 (3)(6) = 9 below the axis, and the other has area 1 (3)(6) = 9 above 2 2 the axis. Hence, " 3 2x dx = 9 − 9 = 0. −3

y 6 4 2 −3

−2

−1 − 2

x 1

2

3

−4 −6

3. The region bounded by the graph of y = 3x + 4 and the x-axis over the interval [−2, 1] consists of two right triangles. One has area 1 ( 2 )(2) = 2 below the axis, and the other has area 1 ( 7 )(7) = 49 2 3 3 2 3 6 above the axis. Hence, " 1 15 49 2 − = . (3x + 4) dx = 6 3 2 −2

A68

ANSWERS TO ODD-NUMBERED EXERCISES

 50 = 30 N N→∞ N→∞ (b) The region bounded by the graph of y = 8 − x and the x-axis over the interval [0, 10] consists of two right triangles. One triangle has area 12 (8)(8) = 32 above the axis, and the other has area 1 (2)(2) = 2 below the axis. Hence, 2 " 10 (8 − x) dx = 32 − 2 = 30. 

y

11. (a)

8 6 4 2 −2

x

−1

1

−2

lim RN = lim

0

5. The region bounded by the graph of y = 7 − x and the x-axis over the interval [6, 8] consists of two right triangles. One triangle has area 12 (1)(1) = 12 above the axis, and the other has area 12 (1)(1) = 12 below the axis. Hence,

y 8 6

" 8

4 2

1 1 (7 − x) dx = − = 0. 2 2 6

y 1 0.5 x 2

−0.5

4

6

8

x 2

7. The region bounded by the graph of y = 25 − x 2 and the x-axis over the interval [0, 5] is one-quarter of a circle of radius 5. Hence, " 5 1 25π . 25 − x 2 dx = π(5)2 = 4 4 0

4

6

8

10

3π π (b) 13. (a) − 2 2 " 5 " 3 3 g(t) dt = ; g(t) dt = 0 15. 2 3 0 17. The partition P is defined by x0 = 0

−1

30 −


b be real numbers, and let f (x) be such that |f (x)| ≤ K for x ∈ [a, b]. By FTC, " x a

f (t) dt = f (x) − f (a).

Since f (x) ≥ −K for all x ∈ [a, b], we get: " x f (t) dt ≥ −K(x − a). f (x) − f (a) = a

Since f (x) ≤ K for all x ∈ [a, b], we get: " x f (t) dt ≤ K(x − a). f (x) − f (a) = a

Combining these two inequalities yields −K(x − a) ≤ f (x) − f (a) ≤ K(x − a), so that, by definition, |f (x) − f (a)| ≤ K|x − a|.

Section 5.4 Preliminary Questions 1. (a) No (b) Yes 2. (c) "3.x Yes. All continuous functions have an antiderivative, namely f (t) dt. a

2

3

4

2x 3 x2 + 1

31. − cos4 s sin s √ tan( x) 33. 2x tan(x 2 ) − √ 2 x 35. The minimum value of A(x) is A(1.5) = −1.25; the maximum value of A(x) is A(4.5) = 1.25. 37. A(x) = (x − 2) − 1 and B(x) = (x − 2) 39. (a) A(x) does not have a local maximum at P . (b) A(x) has a local minimum at R. (c) A(x) has a local maximum at S. (d) True. 41. g(x) = 2x + 1; c = 2 or c = −3 43. (a) If x = c is an inflection point of A(x), then A

(c) = f (c) = 0. (b) If A(x) is concave up, then A

(x) > 0. Since A(x) is the area function associated with f (x), A (x) = f (x) by FTC II, so A

(x) = f (x). Therefore f (x) > 0, so f (x) is increasing. (c) If A(x) is concave down, then A

(x) < 0. Since A(x) is the area function associated with f (x), A (x) = f (x) by FTC II, so A

(x) = f (x). Therefore, f (x) < 0 and so f (x) is decreasing. 45. (a) A(x) is increasing on the intervals (0, 4) and (8, 12) and is decreasing on the intervals (4, 8) and (12, ∞). (b) Local minimum: x = 8; local maximum: x = 4 and x = 12. (c) A(x) has inflection points at x = 2, x = 6, and x = 10. (d) A(x) is concave up on the intervals (0, 2) and (6, 10) and is concave down on the intervals (2, 6) and (10, ∞). 47. The graph of one such function is: 29.

y

4. (b), (e), and (f)

x

Section 5.4 Exercises 1. A(x) =

" x

−2

(2t + 4) dt = (x + 2)2 .

3. G(1) = 0; G (1) = −1 and G (2) = 2; G(x) = 5. G(1) = 0; G (0) = 0 and G ( π4 ) = 1 1 32 1 1 9. 1 − cos x 11. e3x − e12 7. x 5 − 5 5 3" x 3

15. −e−9x−2 + e−3x 17. F (x) = 5 " x sec t dt 21. x 5 − 9x 3 19. F (x) = 0

1 3 5 x − 2x + 3 3 13.

1 4 1 x − 2 2

t 3 + 1 dt

23. sec(5t − 9)

25. (a) A(2) = 4; A(3) = 6.5; A (2) = 2 and A (3) = 3. (b)  2x, 0≤x 0, while

when x < 0. Multiplying both sets of inequalities by n and passing to the limit as n → ∞, the squeeze theorem guarantees that   x n lim ln 1 + = x. n→∞ n

3.5

x

x

Following the proof in the text, we note that  x x x ≤ ln 1 + ≤ n+x n n

3.0

y

59. For m-fold growth, P (t) = mP0 for some t. Solving mP0 = P0 ekt for t, we find t = lnkm .

0.15t

2.5

M6 = 1127 16 35 30 25 20 15

57. P (t) = 204eae

2.0

37. 43. 49. 55.

" 9 N √ 38 5  4 + 5j/N = x dx = lim 3 N→∞ N 4 j =1  √  1 9 3 1 5 ln 19. 1− 4 3 5 32 9 4 4x 5 − x 4 − x 2 + C 23. x 5 − 3x 4 + 3x 3 + C 4 5 46 1 4 x + x 3 + C 27. 29. 3 4 3 1 1 15 (10t − 7) + C 33. − (3x 4 + 9x 2 )−4 + C 35. 506 150√ 24 3 3 1 1 − 39. tan(9t 3 + 1) + C 41. cot(9 − 2θ ) + C 2π √ 27 2 1 9−2x 1 x3 334 45. − e + C 47. e + C 3− 2 2 3 x x 10 e 1 1 + C 53. ln 2 + C 51. ln 10 + 1 2 2(e−x + 2)2   1 1 −1 2x −1 tan (ln t) + C 57. 59. tan +C 2 6 3

ANSWERS TO ODD-NUMBERED EXERCISES

π 1 65. sin−1 (x 2 ) + C 12 2 " 6 4 √ π 1 71. f (x) dx 67. √ tan−1 (4 2) 69. 1024 2 −2 73. Local minimum at x = 0, no local maxima, inflection points at x = ±1 75. Daily consumption: 9.312 million gallons; From 6 PM to midnight: 1.68 million gallons 77. $208,245 79. 0 83. The function f (x) = 2x is increasing, so 1 ≤ x ≤ 2 implies that 2 = 21 ≤ 2x ≤ 22 = 4. Consequently, " 2 " 2 " 2 2 dx ≤ 2x dx ≤ 4 dx = 4. 2= 61. sec−1 12 − sec−1 4

63.

1

1

1 −x On the other hand, the function f (x) = 3 is decreasing, so

1 ≤ x ≤ 2 implies that

1 1 = 3−2 ≤ 3−x ≤ 3−1 = . 9 3 It then follows that " 2 " 2 " 2 1 1 1 1 = dx ≤ dx = . 3−x dx ≤ 9 9 3 3 1 1 1 " 1 1 5 4 ≤ 87. − f (x) dx ≤ 85. 3 3 1+π 0 89. sin3 x cos x 91. −2 93. Consider the figure below, which displays a portion of the graph of a linear function.

d (cosh−1 x) = √ 1 Also, dx 2

x −1

; therefore, F (x) and cosh−1 x have the

same derivative. We conclude that F (x) and cosh−1 x differ by a constant: F (x) = cosh−1 x + C. Now, let x = 1. Because F (1) = 0 and cosh−1 1 = 0, it follows that C = 0. Therefore, F (x) = cosh−1 x. 99. Approximately 6065.9 years 101. 5.03% 103. $17,979.10

Chapter 6 Section 6.1 Preliminary Questions 1. Area of the region between the graphs of y = f (x) and y = g(x), bounded on the left by the vertical line x = a and on the right by the vertical line x = b. 2. Yes ! ! 3. 03 (f (x) − g(x)) dx − 35 (g(x) − f (x)) dx 4. Negative

Section 6.1 Exercises 1. 102 3. 32 3 √ 5. 2 − 1 y y = sin x

1

y

y = cos x π 4

x

The shaded rectangles represent the differences between the right-endpoint approximation RN and the left-endpoint approximation LN . Because the graph of y = f (x) is a line, the lower portion of each shaded rectangle is exactly the same size as the upper portion. Therefore, if we average LN and RN , the error in the two approximations will exactly cancel, leaving " b 1 (RN + LN ) = f (x) dx. 2 a 95. Let

x

y f (x) 10 g(x) x

−2

5

−5

9. 12 e2 − e + 12 11. π − 2 y

1

Then x2 dF − 2 x2 − 1 = x2 − 1 + dx x2 − 1 1 = − x2 − 1 = . 2 2 x −1 x −1

π 2

7. 343 3

" x F (x) = x x 2 − 1 − 2 t 2 − 1dt.

x2

A75

−

13. 160 3 21. 256

2

y=2

1

y = sec2 x

π 4

√  √ 12 3−12+ 3−2 π 15. 24 23. 32 25. 64 3 3

π 4

17. 2 − π2

x

19. 1,331 6

A76

ANSWERS TO ODD-NUMBERED EXERCISES

√

27. 64 3

39. 3 4 3 y

y 1

y = 4 − x2

4

y = cos x

2 −2

0.5 x

−1

1

−2

x

2

0.5

1.5

2

−0.5

y = x2 − 4

−4

1

y = cos 2x −1

√

29. 2

41. 2−2 2

y x+y=4

4

y

3 2 1

y = csc2 x

y + 3x = 4

y=x

1

y = sin x

x 0.5

1

1.5

2

π 4

31. 128 3

π 2

x

43. 4 ln 2 − 2 ≈ 0.77259 y

y

8

y = 8 − x

y=2

2 y = e −x

y = ex

y = x 0

x 0

16

x −ln 2

33. 12

ln 2

45. ≈ 0.7567130951 y

y 1.0

0.4

y=

0.8

0.2 x 0.2

−0.2

0.4

0.6

0.8

0.6 0.4

x = 1 − | y|

x = | y|

−0.4

1

0.2

y = (x − 1)2 x 0.5

35. 1,225 8 y + 2x = 0 2 x 10

−2

1.0

1.5

2.0

47. (a) (ii) (b) No (c) At 10 seconds, athlete 1; at 25 seconds, athlete 2.

y

−20 −10

x

x 2 + 1

x

20 = y3 −

1/3 49. 83 c3/2 ; c = 9 4 ≈ 0.520021.  √  ! (−1+ 5)/2  (1 + x 2 )−1 − x 2 dx 51. √

− (−1+ 5)/2

18y

37. 32 3

53. 0.8009772242 55. 214.75 in2 57. (b) 13 (c) 0 (d) 1  1/3 ≈ 0.206299 59. m = 1 − 12

y 4

Section 6.2 Preliminary Questions

x + 1 = ( y − 1) 2

3 x = 2y

2 1

x 2

4

6

8

1. 3 2. 15 3. Flow rate is the volume of fluid that passes through a cross-sectional area at a given point per unit time. 4. The fluid velocity depended only on the radial distance from the center of the tube. 5. 15

ANSWERS TO ODD-NUMBERED EXERCISES

Section 6.2 Exercises

√ x+1

(b) Disk with radius

4 (20 − y)2 1. (a) 25 (b) 1,280 3   2h 3 π R 3. 3 5. π Rh2 − h3

7. 16 abc

15. π3

9. 83

11. 36 13. 18

17. 96π 3 21. (a) 2 r 2 − y 2 (b) 4(r 2 − y 2 ) (c) 16 3 r 23. 160π 25. 5 kg 27. 0.36 g 29. P ≈ 4, 423.59 thousand 31. L10 = 442.24, R10 = 484.71 3 33. P ≈ 61 deer 35. Q = 128π cm3 /s 37. Q = 8π 3 cm /s 1 1 39. 16 41. π3 43. 10 45. −4 47. n+1 49. Over [0,24], the average temperature is 20; over [2,6] the average 15 ≈ 22.387325. temperature is 20 + 2π 51. 17 2 m/s

√ 53. Average acceleration = −80 m/s2 ; average speed = 20 5 + 104 m/s ≈ 148.7213596 m/s 3 ≈ 2.006221 55. 1/4

(c) V = 21π 2 24,573π 5. V = 81π 10 7. V = 13 π 2 13. (iv) 11. V = 2 e − 1 15. (a)

9. V = π

y y = 10 − x 2

10

y = x2 + 2

2 x

−2

2

(b) A washer with outer radius R = 10 − x 2 and inner radius r = x 2 + 2. (c) V = 256π 17. (a)

y

y = 16 − x y = 3x + 12

10

5

A 57. Mean Value Theorem for Integrals; c = √ 3

4

59. Over [0, 1], f (x); over [1, 2], g(x). 61. Many solutions exist. One could be:

0.5

1.0

(b) A washer with outer radius R = 16 − x and inner radius r = 3x + 12.

y 1 x 1

x

−1.0 −0.5

(c) V = 656π 3 19. (a)

2

y

y = sec x

1.2

−1

0.8 −2

0.4

63. v0 /2

x

−0.4

Section 6.3 Preliminary Questions 1. (a), (c) 2. True 3. False, the cross sections will be washers. 4. (b)

Section 6.3 Exercises 1. (a)

0.4

(b) A circular disk with radius R = sec x. (c) V = 2π 704π 21. V = 15π 23. V = 3π 2 10 25. V = 32π 27. V = 15 376π 128π 29. V = 5 31. V = 40π 33. V = 15 35. V = 824π 15 1,872π 1,400π 37. V = 32π 39. V = 41. V = 5 3 3 √  43. V = π 7π 45. V = 96π 47. V = 32π 5 9 − 3 35 49. V = 1184π 51. V = 7π (1 − ln 2) 15 1 2 55. V = 3 πr h

y 2

57. V = 32π 105

x 1

2

53. V ≈ 43, 000 cm3

3

−2

y 1

(b) Disk with radius x + 1 (c) V = 21π y 3. (a)

x

−1

2

1

1 x −1 −2

1

2

3

4

−1

√ 59. V = 4π 3 61. V = 43 πa 2 b

A77

A78

ANSWERS TO ODD-NUMBERED EXERCISES

19. V = 13 πa 3 + πa 2

Section 6.4 Preliminary Questions 1. (a) Radius h and height r. (b) Radius r and height h. 2. (a) With respect to x. (b) With respect to y.

a

Section 6.4 Exercises

−2 −a

1. V = 25 π

−2

−1

a

21. V = π3

y 1

y 1 x

−1

1

x=y

3. V = 4π

x=1 y x

y=0

0.8

1

0.6

23. V = 128π 3

0.2 −3

−2

x

−1

1

2

3

y

 √  5. V = 18π 2 2 − 1

4

x = y(4 − y)

y 4

2

x 4

1 −3

−2

−1

x 1

2

3

25. V = 8π

7. V = 32π 9. V = 16π 11. V = 32π 5 3 13. The point of intersection is x = 1.376769504; V = 1.321975576 15. V = 3π 5

y 4 y = 4 − x2

y 1

x 2 x 0

4

17. V = 280π 81 y

0.8

0.4

−2

x 10

27. (a) V = 576π 7

(b) V = 96π 5 29. (a) AB generates a disk with radius R = h(y); CB generates a shell with radius x and height f (x). ! ! (b) Shell, V = 2π 02 xf (x) dx ; Disk, V = π 01.3 (h(y))2 dy. 33. V = 8π 35. V = 40π 37. V = 1,024π = 602π 5 15 3 776π 45. V = 625π = 16π 41. V = 32π 43. V = 3 15 6 49. V = 563π 51. V = 43 π r 3 = 121π 525 30 = 2π 2 ab2    2 N R kR 55. (b) V ≈ 4π N (c) V = 43 π R 3 k=1 N

31. 39. 47. 53.

V V V V

ANSWERS TO ODD-NUMBERED EXERCISES

Section 6.5 Preliminary Questions 1. Because the required force is not constant through the stretching process. 2. The force involved in lifting the tank is the weight of the tank, which is constant. 3. 12 kx 2

Section 6.5 Exercises 1. W = 627.2 J 3. W = 5.76 J 5. W = 8 J 7. W = 11.25 J 9. W = 3.800 J 11. W = 105, 840 J 13. W = 56,448π J ≈ 3.547 × 104 J 15. W ≈ 1.842 × 1012 J 5

= 3.92 × 10−6 J 19. W ≈ 1.18 × 108 J = 9800π r 3 J 23. W = 2.94 × 106 J ≈ 1.222 × 106 J 27. W = 3920 J 29. W = 529.2 J = 1, 470 J 33. W = 374.85 J   1 37. W ≈ 5.16 × 109 J 41. 2GMe R1 − r+R m/s e e  e m/s 43. vesc = 2GM R

17. 21. 25. 31.

W W W W

e

√ 1. 32 3. 12 5. 24 7. 12 9. 3 2 − 1 11. e − 32 3 13. Intersection points x = 0, x = 0.7145563847; Area = 0.8235024596 15. V = 4π 17. 2.7552 kg 19. 94 21. 12 sinh 1 23. 3π 4 5 27. 2πm 15

1 e−5x (cos(x) + 5 sin(x)) + C 17. − 26   3 19. 14 x 2 (2 ln x − 1) + C 21. x3 ln x − 13 + C 

23. x (ln x)2 − 2 ln x + 2 + C

25. x tan x − ln | sec x| + C 27. x cos−1 x − 1 − x 2 + C 29. x sec−1 x − ln |x + x 2 − 1| + C 3x (sin x + ln 3 cos x) +C 31. 1 + (ln 3)2 33. (x 2 + 2) sinh x − 2x cosh x + C 35. x tanh−1 4x + 81 ln |1 − 16x 2 | + C √ √ 37. 2e x ( x − 1) + C 1 cos 4x + C 39. 14 x sin 4x + 16

41. 23 (x + 1)3/2 − 2(x + 1)1/2 + C 43. sin x ln(sin x) − sin x + C √ √ √ √ 45. 2xe x − 4 xe x + 4e x + C 47. 14 (ln x)2 [2 ln(ln x) − 1] + C

Chapter 6 Review

25. 27

A79

1 (11e12 + 1) 49. 16 y 30 20 10

29. V = 162π 5

31. V = 64π 33. V = 8π   39. V = 4π 1 − √1

37. V = 128π 35. V = 56π 15 15   3 c 43. V = cπ 41. V = 2π c + 3  ! 1  45. (a) 0 1 − (x − 1)2 − (1 − 1 − x 2 ) dx  ! (b) π 01 (1 − (x − 1)2 ) − (1 − 1 − x 2 )2 dx 47. W = 1.08 J 49. 0.75 ft 51. W = 117600π J ≈ 3.695 × 105 J

e

53. W = 98, 000 J

Section 7.1 Preliminary Questions 1. The Integration by Parts formula is derived from the Product Rule. 3. Transforming v = x into v = 12 x 2 increases the power of x and makes the new integral harder than the original.

Section 7.1 Exercises 4 5. x16 (4 ln x − 1) + C

0.2

0.4

0.6

0.8

1

π 53. e 2+1 55. ex (x 4 − 4x 3 + 12x 2 − 24x + 24) + C. ! ! 57. x n e−x dx = −x n e−x + n x n−1 e−x dx

51. 2 ln 2 − 34

59. Use Integration by Parts, with u = ln x and v =

√ x.

61. Use substitution, followed by algebraic manipulation, with u = 4 − x 2 and du = −2x dx. 63. Use substitution with u = x 2 + 4x + 3, du 2 = x + 2 dx.

Chapter 7

1. −x cos x + sin x + C

x −10

3. ex (2x + 7) + C 7. −e−x (4x + 1) + C

1 (5x − 1)e5x+2 + C 11. 1 x sin 2x + 1 cos 2x + C 9. 25 2 4 13. −x 2 cos x + 2x sin x + 2 cos x + C

15. − 12 e−x (sin x + cos x) + C

65. Use Integration by Parts, with u = x and v = sin(3x + 4). 67. x(sin−1 x)2 + 2 1 − x 2 sin−1 x − 2x + C 69. 14 x 4 sin(x 4 ) + 14 cos(x 4 ) + C 71. 2π(e2 + 1) 73. $42, 995 75. For k = 2: x(ln x)2 − 2x ln x + 2x + C; for k = 3: x(ln x)3 − 3x(ln x)2 + 6x ln x − 6x + C. 77. Use Integration by Parts with u = x and v = bx . 79. (b) V (x) = 12 x 2 + 12 is simpler, and yields 1 (x 2 tan−1 x − x + tan −1 x) + C. 2 81. An example of a function satisfying these properties for some λ is f (x) = sin πx. 83. (a) In = 12 x n−1 sin(x 2 ) − n−1 2 Jn−2 ; (c) 12 x 2 sin(x 2 ) + 21 cos(x 2 ) + C

A80

ANSWERS TO ODD-NUMBERED EXERCISES

Section 7.2 Preliminary Questions 1. Rewrite sin5 x = sin x sin4 x = sin x(1 − cos2 x)2 and then

substitute u = cos x. 3. No, a reduction formula is not needed because the sine function is raised to an odd power. 5. The second integral requires the use of reduction formulas, and therefore more work.

Section 7.2 Exercises 1. sin x − 13 sin3 x + C 3. − 13 cos3 θ + 15 cos5 θ + C 5. − 14 cos4 t + 16 cos6 t + C 7. 2 9. 11. 13. 15. 17. 19. 23. 25. 27. 29. 31. 35. 37. 39. 41. 43. 45. 47. 55.

1 4 1 6 1 5 1 3 1 5

cos3 y sin y + 38 cos y sin y + 38 y + C 1 sin3 x cos x − 1 sin x cos x + 1 x + C sin5 x cos x − 24 16 16 1 sin2 x cos x − 2 cos x + C sin4 x cos x − 15 15 sec3 x − sec x + C 1 tan(x) sec2 x − 2 tan x + C tan x sec4 x − 15 15

− 12 cot 2 x + ln | csc x| + C 21. − 16 cot 6 x + C − 16 cos6 x + C 1 3 3 12 (cos x sin x + 2 (x + sin x cos x)) + C 1 1 5 7 5π sin (πθ ) − 7π sin (πθ ) + C 1 sin3 (3x) cos(3x) − 1 sin(3x) cos(3x) + 9 x + C − 12 8 8 1 cot(3 − 2x) + C 33. 1 tan2 x + C 2 2 1 sec8 x − 1 sec6 x + 1 sec4 x + C 8 3 4 1 tan9 x + 1 tan7 x + C 7 9 − 19 csc9 x + 27 csc7 x − 15 csc5 x + C 1 sin2 2x + C 4 1 cos2 (t 2 ) sin(t 2 ) + 1 sin(t 2 ) + C 6 3 1 cos(sin t) sin(sin t) + 1 sin t + C 2 √ 2  8 π 49. 15 51. ln 2+1 53. ln 2 8 57. − 6 59. 8 7 3 15

61. First, observe sin 4x = 2 sin 2x cos 2x = 2 sin 2x(1 − 2 sin2 x) = 2 sin 2x − 4 sin 2x sin2 x = 2 sin 2x − 8 sin3 x cos x. Then 1 3 3 32 (12x − 8 sin 2x + sin 4x) + C = 8 x − 16 sin 2x − 1 sin3 x cos x + C = 3 x − 3 sin x cos x − 1 sin3 x cos x + C. 4 8 8 4 2 63. π2

1 sin 2x cos 2x + C 65. 18 x − 16

1 x − 1 sin 2x − 1 sin 2x cos 2x + 1 cos2 2x sin 2x + C 67. 16 48 32 48

69. Use the identity tan2 x = sec2 x − 1 and the substitution u = tan x, du = sec2 x dx. ! π/2 ! π/2 71. (a) I0 = 0 sin0 x dx = π2 ; I1 = 0 sin x dx = 1 ! π/2 m−2 (b) m−1 sin x dx m 0 8 (c) I2 = π4 ; I3 = 23 ; I4 = 3π 16 ; I5 = 15 73. cos(x) − cos(x) ln(sin(x)) + ln | csc(x) − cot(x)| + C 77. Use Integration by Parts with u = secm−2 x and v = sec2 x.

Section 7.3 Preliminary Questions 1. (a) x√= 3 sin θ (d) x = 5 sec θ 3. 2x 1 − x 2

(b) x = 4 sec θ

(c) x = 4 tan θ

Section 7.3 Exercises

  1. (a) θ + C (b) sin−1 x3 + C ! ! 3. (a) √ dx2 = 12 sec θ dθ

4x +9 1 (b) 2 ln | sec θ + tan θ| + C (c) ln | 4x 2 + 9 + 2x| + C

√ √ 2 2 ) + x 16−5x +C 5. √8 arccos( 16−5x 4 2 5 x  1 −x −1 √ 7. 3 sec 3 + C 9. 4 x 2 −4 + C 11. x 2 − 4 + C 13. (a) − 1 − x 2 (b) 18 (arcsin x − x 1 − x 2 (1 − 2x 2 )) 3

5

(c) − 13 (1 − x 2 ) 2 + 15 (1 − x 2 ) 2 3 3 (d) 1 − x 2 (− x4 − 3x 8 ) + 8 arcsin(x)   15. 92 sin−1 x3 − 12 x 9 − x 2 + C √   2   | + C 19. ln x + x 2 − 9  + C 17. 14 ln | x +16−4 x √ 5−y 2 21. − 5y + C 23. 15 ln 25x 2 + 25x + C √ 1 sec−1  z  + z2 −4 + C 25. 16 2 8z2 1 2 27. 12 x 6x − 49 + 12 ln x + x 2 − 1 + C 1 tan−1  t  + t +C 29. 54 3 18(t 2 +9) 31. √ x2 + ln x + x 2 − 1 + C x −1 √ 33. Use the substitution x = a u. 35. (a) x 2 − 4x + 8 = x 2 − 4x + 4 + 4 = (x − 2)2 + 4     (b) ln  u2 + 4 + u + C     (c) ln  (x − 2)2 + 4 + x − 2 + C     37. ln  x 2 + 4x + 13 + x + 2 + C √ 39. √1 ln 12x + 1 + 2 6 x + 6x 2 + C 6     1 41. 2 (x − 2) x 2 − 4x + 3 + 72 ln x − 2 + x 2 − 4x + 3 + C 43. Begin by multiplying by −1, then completing the square, and then follow up with u-substitution (u = (x + 3); du = dx) and then trigonometric substitution. 45. Use one of the following trigonometric methods: rewrite sin3 x = (1 − cos2 x) sin x and let u = cos x, or rewrite cos3 x = (1 − sin2 x) cos x and let u = sin x. 47. Use trigonometric substitution, with x = 3 sin θ or substitution with x = 3u and dx = 3 du. 49. The techniques learned thus far are insufficient to solve this integral.

ANSWERS TO ODD-NUMBERED EXERCISES

51. The techniques we have covered thus far are not sufficient to treat this integral. This integral requires a technique known as partial fractions.     53. x sec−1 x − ln x + x 2 − 1  + C 55. x(ln(x 2 + 1) − 2) + 2 tan−1 x + C 

√ √   57. π4 59. 4π 3 − ln 2 + 3 61. 12 ln |x − 1| − 12 ln |x + 1| + C V 63. (a) 1.789 × 106 m

V (b) 3.526 × 106 m

A81

Section 7.5 Preliminary Questions 1. No, f (x) cannot be a rational function because the integral of a rational √ function cannot contain a term with a non-integer exponent such as x + 1 3. (a) Square is already completed; irreducible. √ √ (b) Square is already completed; factors as (x − 5)(x + 5). (c) x 2 + 4x + 6 = (x + 2)2 + 2; irreducible. 2 (d) x 2 + √ 4x + 2 = (x + √2) − 2; factors as (x + 2 − 2)(x + 2 + 2).

Section 7.5 Exercises Section 7.4 Preliminary Questions 1. (a) x = sinh t     3. 12 ln  1+x 1−x 

(b) x = 3 sinh t

(c) 3x = sinh t

(b) (c)

Section 7.4 Exercises 1. 13 sinh 3x + C. 5. 11. 15. 19. 23. 27. 29.

3. 12 cosh(x 2 + 1) + C

(d)

tanh2 x − 12 tanh(1 − 2x) + C 7. + C 9. ln cosh x + C 2 1 ln | sinh x| + C 13. 16 sinh(8x − 18) − 12 x + C 1 1 −1 x + C 32 sinh 4x− 8x + C 17. cosh 1 sinh−1 5x + C 21. 1 x x 2 − 1 − 1 cosh−1 x + C 5 2 2  4 2 tanh−1 12 25. sinh−1 1      1 csch−1 − 1 − csch−1 − 3 4 4 4 √ x 2 −1 −1 cosh x − +C x

31. Let x = sinh t for the first formula and x = cosh t for the second.     x 2 + 1 + C 33. 12 x x 2 + 16 + 8 ln  x4 +  4 35. Using Integration by Parts with u = coshn−1 x and v = cosh x to begin proof. 2  37. − 12 tanh−1 x + C 39. x tanh−1 x + 12 ln |1 − x 2 | + C  x−1 1+u2 , 41. u = cosh 2 cosh x+1 . From this it follows that cosh x = sinh x = 2u 2 and dx = 2du2 . 1−u 1−u ! 43. du = u + C = tanh x2 + C

4 x 2 + 4x + 12 1 + = . x + 2 x2 + 4 (x + 2)(x 2 + 4) 2 −x + 2 2x 2 + 8x + 24 1 + + 2 = . x + 2 (x + 2)2 (x + 2)2 (x 2 + 4) x +4 4 5 −8 8 x 2 − 4x + 8 + + = + . x − 2 (x − 2)2 x − 1 (x − 1)2 (x − 1)2 (x − 2)2 x 4 − 4x + 8 4x − 4 4 − 2 =x−2+ . x+2 (x + 2)(x 2 + 4) x +4 −2 5. 19 (3x + 4 ln(3x − 4)) + C

1. (a)

1−u

45. Let gd(y) = tan−1 (sinh y). Then 1 d 1 cosh y = gd(y) = = sech y, 2 dy cosh y 1 + sinh y where we have used the identity 1 + sinh2 y = cosh2 y. 47. Let x = gd(y) = tan−1 (sinh y). Solving for y yields y = sinh−1 (tan x). Therefore, gd −1 (y) = sinh−1 (tan y). 49. Let x = it. Then cosh2 x = (cosh(it))2 = cos2 t and sinh2 x = (sinh(it))2 = i 2 sin2 t = − sin2 t. Thus, 1 = cosh2 (it) − sinh2 (it) = cos2 t − (− sin2 t) = cos2 t + sin2 t, as desired.

3. 7. 11. 13.

x 3 + ln(x + 2) + C 3

9. − 12 ln |x − 2| + 12 ln |x − 4| + C ln |x| − ln |2x + 1| + C x − 3 arctan x3 + C

15. 2 ln |x + 3| − ln |x + 5| − 23 ln |3x − 2| + C 5 +C 17. 3 ln |x − 1| − 2 ln |x + 1| − x+1 1 − 2 ln |x − 2| − 1 + C 19. 2 ln |x − 1| − x−1 x−2 2 + 2 +C 21. ln(x) − ln(x + 2) + x+2 (x+2)2 √   √ √ √  1 ln  2x − 3 − √ 1 ln  2x + 3 + C 23. √     2 6

2 6

5 − 5 25. 2x+5 + 12 ln(2x + 5) + C 4(2x+5)2 1 − 1 27. − ln |x| + ln |x − 1| + x−1 2 +C 2(x−1)

29. x + ln |x| − 3 ln |x + 1| + C 31. 2 ln |x − 1| + 12 ln |x 2 + 1| − 3 tan−1 x + C 33. 35. 37.

1 1 2 25 ln |x| − 50 ln |x + 25| + C

6x − 14 ln x + 3 + 2 ln x − 1 + C 1 + 1 ln |x 2 + 9| − 4 tan−1  x  + C − 15 ln |x − 1| − x−1 10 15 3

1 ln |x| − 1 ln |x 2 + 8| + 39. 64 128

1 +C 16(x 2 +8)

1 ln |x 2 + 4x + 10| + C 41. 16 ln |x + 2| − 12 5 43. ln |x| − 12 ln |x 2 + 2x + 5| − 5 − 2 + 2x + 5) 2(x   +C 3 tan−1 x+1 2

47. ln(ex − 1) − x + C √ 2 x + ln | x − 1| − ln | x + 1| + C         +C 51. ln  √ x2 − √ 12  + C = ln  √x−1 2 x −1 x −1 x −1  √  √ 4−x 2 4−x 2 +C + C = − 53. − 14 x 4x

45. 49.

1 arctan(x 2 ) + C 2√ √

55. 12 x + 18 sin 4x cos 4x + C

A82

ANSWERS TO ODD-NUMBERED EXERCISES

1 tan−1  x  + 57. 54 3

x +C 18(x 2 +9) 59. 15 sec5 x − 23 sec3 x + sec x + C 61. x ln(x 2 + 1) + (x + 1) ln(x + 1) + (x − 1) ln(x − 1) − 4x −

2 arctan x + C    63. ln x + x 2 − 1 − √ x2 65. 23 tan−1 (x 3/2 ) + C

x −1

9. 11. 13. 15. 17. 19.

+C

2 67. If θ = 2 tan−1 t, then dθ = 2 dt/(1 + t ). We also have that θ θ 2 cos( 2 ) = 1/ 1 + t and sin( 2 ) = t/ 1 + t 2 . To find cos θ, we use the double angle identity cos θ = 1 − 2 sin2 ( θ2 ). This gives us

2 cos θ = 1−t 2 . To find sin θ , we use the double angle identity

1+t

sin θ = 2 sin( θ2 ) cos( θ2 ). This gives us sin θ = 2t 2 . It follows then 1+t " dθ that = cos θ + 34 sin θ       θ  4  θ  4  + ln 1 + 2 tan + C. − ln 2 − tan   5 2 5 2  1

1

1 a−b b−a 69. Partial fraction decomposition shows (x−a)(x−b) = x−a + x−b .   !  x−a  dx 1 This can be used to show (x−a)(x−b) = a−b ln  x−b  + C. 2 + 1 71. x−6 x+2

Section 7.6 Preliminary Questions 1. (a) The integral is converges. (b) The integral is diverges. (c) The integral is diverges. (d) The integral is converges. 3. Any value of b satisfying |b| ≥ 2 will make this an improper integral. 5. Knowing that an integral is smaller than a divergent integral does not allow us to draw any conclusions using the comparison test.

Section 7.6 Exercises 1. (a) Improper. The function x −1/3 is infinite at 0. (b) Improper. Infinite interval of integration. (c) Improper. Infinite interval of integration. (d) Proper. The function e−x is continuous on the finite interval [0, 1]. (e) Improper. The function sec x is infinite at π2 . (f) Improper. Infinite interval of integration. (g) Proper. The function sin x is continuous on the finite interval [0, 1]. (h) Proper. The function 1/ 3 − x 2 is continuous on the finite interval [0, 1]. (i) Improper. Infinite interval of integration. (j) Improper. The function ln x is infinite at 0. ! ! 3. 1∞ x −2/3 dx = limR→∞ 1R x −2/3 dx =   limR→∞ 3 R 1/3 − 1 = ∞ 5. The integral does not converge. 7. The integral converges; I = 10,000e0.0004 .

The integral does not converge. The integral converges; I = 4. The integral converges; I = 18 . The integral converges; I = 2. The integral converges; I = 1.25. The integral converges; I = 112 . 3e

21. The integral converges; I = 13 . √ 23. The integral converges; I = 2 2. 25. The integral does not converge. 27. The integral converges; I = 12 . The integral converges; I = 12 . The integral converges; I = π2 . The integral does not converge. The integral does not converge. The integral converges; I = −1. The integral does not converge. dx dx − dx . This = x−3 (a) Partial fractions yields (x−2)(x−3) x−2   !   1 dx yields 4R (x−2)(x−3) = ln  R−3 − ln R−2  2      R−3  1 (b) I = limR→∞ ln  R−2  − ln 2 = ln 1 − ln 12 = ln 2 29. 31. 33. 35. 37. 39. 41.

43. The integral does not converge. 45. The integral does not converge. 47. The integral converges; I = 0. " 1 " 0 " 1 dx dx dx 49. = + =0 −1 x 1/3 −1 x 1/3 0 x 1/3 51. The integral converges for a < 0. ! ∞ dx 53. −∞ 2 = π.

1+x 1 55. 3 ≤ 13 . Therefore, by the comparison test, the integral x +4 x

converges.

57. For x ≥ 1, x 2 ≥ x, so −x 2 ≤ −x and e−x ≤ e−x . Now ! ∞ −x ! ∞ −x 2 dx converges, so dx converges by the 1 e 1 e comparison test. We conclude that our integral converges by writing it ! ! ! 2 2 2 as a sum: 0∞ e−x dx = 01 e−x dx + 1∞ e−x dx. " ∞ 2 1 − sin x . Since f (x) ≤ and 2x −2 dx = 2, 59. Let f (x) = 2 x2 1 " ∞ x 1 − sin x dx converges by the comparison test. it follows that x2 1 61. The integral converges. 63. The integral does not converge. 65. The integral converges. 67. The integral does not converge. 69. The integral converges. 71. The integral converges. 73. The integral does note converge. " 1 " ∞ dx dx 75. and both converge, therefore 0 x 1/2 (x + 1) 1 x 1/2 (x + 1) J converges. 250 79. $2,000,000 77. 0.07  ! 81. (a) π (b) 1∞ x1 1 + 14 dx diverges. 2

x

ANSWERS TO ODD-NUMBERED EXERCISES

  83. W = limT →∞ CV 2 12 − e−T /RC + 12 e−2T /RC =   CV 2 12 − 0 + 0 = 21 CV 2 85. The √ integrand is infinite at the upper limit of integration, x = 2E/k, so the integral  is improper.  T =

lim √

R→ 2E/k

−1 T (R) = 4 m k sin (1) = 2π

m. k

−1 87. Lf (s) = 2 lim e−st (s sin(αt) + αcos(αt)) − α. s + α 2 t→∞ n! 91. Jn = αn Jn−1 = αn · (n−1)! α n = α n+1 ! ∞ ν3 dν. Because α > 0 and 8πh/c3 is a 93. E = 8πh c3 0 eαν −1 constant, we know E is finite by Exercise 92. " x " x dt dt > > ln x. 95. Because t > ln t for t > 2, F (x) = 2 ln t 2 t Thus, F (x) → ∞ as x → ∞. Moreover, 1 F (x) lim G(x) = lim = lim x = ∞. Thus, lim is of the x→∞ x→∞ 1/x x→∞ x→∞ G(x) form ∞/∞, and L’Hôpital’s Rule applies. Finally, 1 F (x) ln x = lim ln x = lim = 1. L = lim x→∞ G(x) x→∞ ln x−1 x→∞ ln x − 1 2

89.

s s 2 +α 2

(ln x)

97. The integral is absolutely convergent. Use the comparison test with 12 . x

Section 7.7 Preliminary Questions 1. No, p(x) ≥ 0 fails.

3. p(x) = 4e−4x

Section 7.7 Exercises 1. C = 2; P (0 ≤ X ≤ 1) = 34   3. C = π1 ; P − 12 ≤ X ≤ 12 = 13 √   5. C = π2 ; P − 12 ≤ X ≤ 1 = 23 + 4π3 ! 7. 1∞ 3x −4 = 1; μ = 32 ! 1 e−t/50 = 1 9. Integration confirms 0∞ 50   3 11. e− 2 ≈ 0.2231 13. 12 2 − 10e−2 ≈ 0.32 15. F (− 23 ) − F (− 13 6 ) ≈ 0.2374

Section 7.8 Preliminary Questions 1. T1 = 6; T2 = 7 3. The Trapezoidal Rule integrates linear functions exactly, so the error will be zero. 5. The two graphical interpretations of the Midpoint Rule are the sum of the areas of the midpoint rectangles and the sum of the areas of the tangential trapezoids.

Section 7.8 Exercises 1. T4 = 2.75; M4 = 2.625 3. T6 = 64.6875; M6 ≈ 63.2813 5. T6 ≈ 1.4054; M6 ≈ 1.3769 7. T6 = 1.1703; M6 = 1.2063 9. T4 ≈ 0.3846; M5 ≈ 0.3871 11. T5 = 1.4807; M5 = 1.4537 13. S4 ≈ 5.2522 15. S6 ≈ 1.1090 17. S4 ≈ 0.7469 19. S8 ≈ 2.5450 21. S1 0 ≈ 0.3466 23. ≈ 2.4674 25. ≈ 1.8769 27. ≈ 608.611 29. (a) Assuming the speed of the tsunami √ is a continuous function, at x miles from the shore, the speed is 15f (x). Covering an infinitesimally small distance, dx, the time T required for the tsunami dx to cover that distance becomes √ . It follows from this that 15f (x) !M dx T = 0 √ . 15f (x)

(b) ≈ 3.347 hours. 31. (a) Since x 3 is concave up on [0, 2], T6 is too large. (b) We have f (x) = 3x 2 and f

(x) = 6x. Since |f

(x)| = |6x| is increasing on [0, 2], its maximum value occurs at x = 2 and we may take K2 = |f

(2)| = 12. Thus, Error(T6 ) ≤ 29 . (c) Error(T6 ) ≈ 0.1111 < 29

33. T1 0 will overestimate the integral. Error(T10 ) ≤ 0.045. 35. M1 0 will overestimate the integral. Error(M10 ) ≤ 0.0113

37. N ≥ 103 ; Error ≈ 3.333 × 10−7

39. N ≥ 750; Error ≈ 2.805 × 10−7 41. Error(T10 ) ≤ 0.0225; Error(M10 ) ≤ 0.01125 43. S8 ≈ 4.0467; N ≥ 23

17. (a) ≈ 0.8849 (b) ≈ 0.6554

45. Error(S40 ) ≤ 1.017 × 10−4 .

19. 1 − F (z) and F (−z) are the same area on opposite tails of the distribution function. Simple algebra with the standard normal cumulative distribution function shows P (μ − rσ ≤ X ≤ μ + rσ ) = 2F (r) − 1 √ 21. ≈ 0.0062 23. μ = 5/3; σ = 10/3 25. μ = 3; σ = 3

47. N ≥ 305 49. N ≥ 186

27. (a) f (t) is the fraction of initial atoms present at time t. Therefore, the fraction of atoms that decay is going to be the rate of change of the total number of atoms. Over a small interval, this is simply −f (t)t. (b) The fraction of atoms that decay over an arbitrarily small interval is equivalent to the probability that an individual atom will decay over that same interval. Thus, the probability density function becomes ! −f (t). (c) 0∞ −tf (t) dt = k1

A83

51. (a) The maximum value of |f (4) (x)| on the interval [0, 1] is 24. (b) N ≥ 20; S20 ≈ 0.785398; |0.785398 − π4 | ≈ 1.55 × 10−10 .

53. (a) Notice |f

(x)| = |2 cos(x 2 ) − 4x 2 sin(x 2 )|; proof follows. (b) When K2 = 2, Error(MN ) ≤ 1 2 . (c) N ≥ 16

4N

55. Error(T4 ) ≈ 0.1039; Error(T8 ) ≈ 0.0258; Error(T16 ) ≈ 0.0064; Error(T32 ) ≈ 0.0016; Error(T64 ) ≈ 0.0004. Thes are about twice as large as the error in MN . 57. S2 = 14 . This is the exact value of the integral. " b r(b2 − a 2 ) + s(b − a) = f (x) dx 59. TN = 2 a

A84

ANSWERS TO ODD-NUMBERED EXERCISES

61. (a) This result follows because the even-numbered interior endpoints overlap: (N−2)/2 

2j

S2 =

i=0

b−a [(y0 + 4y1 + y2 ) + (y2 + 4y3 + y4 ) + · · · ] 6

 b−a  y0 + 4y1 + 2y2 + 4y3 + 2y4 + · · · + 4yN−1 + yN = SN . = 6 (b) If f (x) is a quadratic polynomial, then by part (a) we have " b f (x) dx. SN = S20 + S22 + · · · + S2N−2 = a

63. Let f (x) = ax 3 + bx 2 + cx + d, with a = 0, be any cubic polynomial. Then, f (4) (x) = 0, so we can take K4 = 0. This yields Error(SN ) ≤ 0 4 = 0. In other words, SN is exact for all cubic 180N polynomials for all N .

Chapter 7 Review 1. (a) (v)

(b) (iv)

(c) (iii)

(d) (i)

(e) (ii)

9 11 3. sin9 θ − sin11 θ + C.

5.

tan θsec5 θ − 7 tan θ sec3 θ + tan θ sec θ + 1 ln | sec θ + tan θ| + C 6 24 16 16 √ 7. − √ 12 − sec−1 x + C 9. 2tan−1 x + C x −1   −1 11. − tanx x + ln |x| − 12 ln 1 + x 2 + C. 5 e4 − 1 ≈ 8.50 15. cos12 6θ − cos10 6θ + C 13. 32 32 72 60 17. 5 ln |x − 1| + ln |x + 1| + C 3 19. tan3 θ + tan θ + C

23. 27. 29. 33. 37.

39.

41. 43.

21. ≈ 1.0794

5 3θ − cos5 θ + 2cos − cos θ + C 25. − 14 3 2 (tan x)3/2 + C 3 sin6 θ − sin8 θ + C 31. − 1 u3 + C = − 1 cot 3 x + C 6 8 3  3  t+4  1 1 1 ≈ 0.4202 35. 49 ln  t−3  − 7 · t−3 + C 1 sec−1 x + C 2 2  ⎧ ⎪ √2 tan−1 x + C a>0 ⎪ a ⎪ a ⎪ " ⎨   √ √ dx = √1 ln  √x−√−a  + C a < 0 −a x+ −a ⎪ x 3/2 + ax 1/2 ⎪ ⎪ ⎪ ⎩− √2 + C a=0 x 5 − 3 +C ln |x + 2| + x+2 (x+2)2   1 + 1 ln x 2 + 4 + C − ln |x − 2| − 2 x−2 2

  5 − 3 45. 13 tan−1 x+4 +C + C 47. ln |x + 2| + x+2 3 (x+2)2  2 3/2 √ x +4 x 2 +4 + 16x + C 51. − 19 e4−3x (3x + 4) + C 49. − 3 48x

53. 12 x 2 sin x 2 + 12 cos x 2 + C   2   55. x2 tanh−1 x + x2 − 14 ln  1+x 1−x  + C     57. x ln x 2 + 9 − 2x + 6tan−1 x3 + C

59. 12 sinh 2 61. t + 14 coth(1 − 4t) + C 63. π3 65. tan−1 (tanh x) + C " " n−2 2 xn (x + 1 − 1) x 67. (a) In = dx = dx = 2+1 x x2 + 1 " " x n−2 x n−1 − In−2 dx = x n−2 dx − 2 n − 1 x +1 

(b) I0 = tan−1 x + C; I1 = 12 ln x 2 + 1 + C;   2 I2 = x − tan−1 x + C; I3 = x2 − 12 ln x 2 + 1 + C;   3 4 2 I4 = x3 − x + tan−1 x + C; I5 = x4 − x2 + 12 ln x 2 + 1 + C (c) Prove by induction; show it works for n = 1, then assume it works for n = k and use that to show it works for n = k + 1. 69. 34 71. C = 2; p(0 ≤ X ≤ 1) = 1 − 2e 73. (a) 0.1587 (b) 0.49997 75. Integral converges; I = 12 . √ 77. Integral converges; I = 3 3 4. 79. Integral converges; I = π2 . 81. The integral does not converge. 83. The integral does not converge. 85. The integral converges. 87. The integral converges. 2 89. The integral converges. 91. π 95. 3 (s−α)

97. (a) TN is smaller and MN is larger than the integral. (b) MN is smaller and TN is larger than the integral. (c) MN is smaller and TN is larger than the integral. (d) TN is smaller and MN is larger than the integral. 99. M5 ≈ 0.7481 101. M4 ≈ 0.7450 103. S6 ≈ 0.7469 3 . 105. V ≈ T9 ≈ 20 hectare-ft = 871,200 ft3 107. Error ≤ 128 109. N ≥ 29

Chapter 8 Section 8.1 Preliminary Questions

! 1. 0π 1 + sin2 x dx 2. The graph of y = f (x) + C is a vertical translation of the graph of y = f (x); hence, the two graphs should have the same arc length. We can explicitly establish this as follows:   2 " b d Length of y = f (x) + C = 1+ (f (x) + C) dx dx a " b = 1 + [f (x)]2 dx a

= length of y = f (x). 1 + f (x)2 ≥ 1 for any function f , we have " 4 Length of graph of f (x) over [1, 4] = 1 + f (x)2 dx

3. Since

1

≥

" 4 1

1 dx = 3

A85

ANSWERS TO ODD-NUMBERED EXERCISES

Section 8.1 Exercises 1. 7. 11. 13.

√ ! L = 26 1 + 16x 6 dx 3. 13 5. 3 10 12 √ √ 1 22 − 13 13) 9. e2 + ln22 + 14 27 (22 !2 6 1 1 + x dx ≈ 3.957736 !2 1 1 1 + 4 dx ≈ 1.132123 x

15. 6 √ 19. a = sinh−1 (5) = ln(5 + 26) 23. Let Then s denote the arc length. s = a2 1 + 4a 2 + 14 ln | 1 + 4a 2 + 2a|. Thus, when a = 1, √ √ s = 12 5 + 14 ln( 5 + 2) ≈ 1.478943. √ √ √ 2a √ 2 25. 1 + e2a + 12 ln √1+e2a −1 − 2 + 12 ln 1+ 2−1 1+e +1 √ √ 27. ln(1 + 2) 31. 1.552248 33. 16π 2 π (1453/2 − 1) 37. 384π π (e4 − 9) 35. 27 39. 16 5 ! 3 −1 1 + x −4 dx ≈ 7.60306 41. 2π 1 x ! 2 −x 2 /2 2 1 + x 2 e−x dx ≈ 8.222696 43. 2π 0 e 45. 2π ln 2 + 15π 8

47. 4π 2 br

√   b2 −a 2  2 b  ln + 49. 2π b2 + √2πba   a a 2 2 b −a

9. 11. 17. 19. 21.

F F F F F

≈ 321, 250, 000 lb N 13. F ≈ 5593.804 N 15. F ≈ 5652.4 N = 815360 3 = 940, 800 N 9 = 4, 532, 500, 000 sec( 7π 36 ) ≈ 5.53316 × 10 N 2 = (15b + 30a)h lb √

Section 8.3 Preliminary Questions 1. Mx = My = 0 2. Mx = 21 3. Mx = 5; My = 10 4. Because a rectangle is symmetric with respect to both the vertical line and the horizontal line through the center of the rectangle, the Symmetry Principle guarantees that the centroid of the rectangle must lie along both these lines. The only point in common to both lines of symmetry is the center of the rectangle, so the centroid of the rectangle must be the center of the rectangle.

Section 8.3 Exercises

  1. (a) Mx = 4m; My = 9m; center of mass: 94 , 1   14 (b) 46 17 , 17 5. A sketch of the lamina is shown here. y

Section 8.2 Preliminary Questions 1. Pressure is defined as force per unit area. 2. The factor of proportionality is the weight density of the fluid, w = ρg. 3. Fluid force acts in the direction perpendicular to the side of the submerged object. 4. Pressure depends only on depth and does not change horizontally at a given depth. 5. When a plate is submerged vertically, the pressure is not constant along the plate, so the fluid force is not equal to the pressure times the area.

Section 8.2 Exercises 1. (a) Top: F = 176, 500 N; bottom: F = 705, 600 N N !  (b) F ≈ ρg3yj y (c) F = 28 ρg3y dy j =1

(d) F = 882, 000 N 3. (a) The width of the triangle varies linearly from 0 at a depth of y = 3 m to 1 at a depth of y = 5 m. Thus, f (y) = 12 (y − 3). (b) The area of the strip at depth y is 12 (y − 3)y, and the pressure at depth y is ρgy, where ρ = 103 kg/m3 and g = 9.8. Thus, the fluid force acting on the strip at depth y is approximately equal to ρg 12 y(y − 3)y. N 

!5

ρg 12 yj (yj − 3) y → 3 ρg 12 y(y − 3) dy j =1 N (d) F = 127,400 3 19,600 3 5. (b) F = 3 r N 3 2 7. F = 19,600 3 r + 4,900πmr N (c) F ≈

√

23. Front and back: F = 62.59 3 H 3 ; slanted sides: F = 62.53 3 H 2 .

8 6 4 2 x 0

0.5 1 1.5 2 2.5 3

243 (a) Mx = 729 10 ; My = 4

  (b) Area = 9 cm2 ; center of mass: 94 , 27 10

  32ρ 8 16 7. Mx = 64ρ 7 ; My = 5 ; center of mass : 5 , 7 9. (a) Mx = 24 (b) M = 12, so ycm = 2; center of mass: (0, 2)     45 11. 93 13. 98 , 18 5 35 , 56     −4 −8 17. π2 , π8 15. 1−5e−4 , 1−e −4 1−e

)

4(1−e

19. A sketch of the region is shown here. y 5 4 3 2 1 x 0

0.5

1

1.5

2

The region is clearly symmetric about the line y = 3, so we expect the centroid of the region to lie along this line. We find Mx = 24, 7 My = 28 3 , centroid: 6 , 3 .     9 , 9 1 , e2 −3 23. 21. 20 20 2(e−2) 4(e−2)

A86

ANSWERS TO ODD-NUMBERED EXERCISES

 25.

 √ π√ 2−4 , √1 4 2−1 4( 2−1)

  27. A sketch of the region is shown here. Centroid: 0, 27

23.   1 x − π 2 + √ 1 x − π 3 · · · Tn (x) = √1 − √1 x − π4 − √ 4 4 2

2

2 2

In general, the coefficient of (x − π/4)n is

6 2

1 ± √ ( 2)n!

y 1

with the pattern of signs +, −, −, +, +, −, −, . . . .  2  25. T2 (x) = 1 + x + x ; T2 (−0.5) − f (−0.5) ≈ 0.018469 2

x

−1

1



   4b 4 , 4 0, 3π 31. 3π 3π   2 2 (r −h2 )3/2 √ ; with r = 1 and h = 12 : 33. 0, 2 −1 √3 2 /r 2 −h r 2 −h2 r sin 1−h √   0, 3 3√ ≈ (0, 0.71) 4π −3 3     4 , 4 35. 0, 49 37. − 9π 24 9π

29.

39.  For the square on the left: (4, 4); for the square on the right: 4, 25 7 .

27. T2 (x) = 1 − 23 (x − 1) + 59 (x − 1)2 ; |f (1.2) − T2 (1.2)| ≈ 0.00334008 1 (x − 1)3 29. T3 (x) = 1 + 12 (x − 1) − 18 (x − 1)2 + 16 1.1 |1.1|4 31. e 4! 6 2 4 33. T5 (x) = 1 − x2 + x24 ; maximum error = (0.25) 6! 1 (x − 4) + 3 (x − 4)2 − 5 (x − 4)3 ; 35. T3 (x) = 12 − 16 256 2048 4 maximum error = 35(0.3) 65,536   3 11 . With K = 5, 37. T3 (x) = x − x3 ; T3 12 = 24

    T3 12 − tan−1

Section 8.4 Preliminary Questions 1. 2. 3. 4.

T3 (x) = 9 + 8(x − 3) + 2(x − 3)2 + 2(x − 3)3 The polynomial graphed on the right is a Maclaurin polynomial. A Maclaurin polynomial gives the value of f (0) exactly. The correct statement is (b): |T3 (2) − f (2)| ≤ 23

Section 8.4 Exercises

 4

5

1 2

4!

5 . = 384

39. T3 (x) = cos(0.25) − sin(0.25)(x − 0.5) − cos(0.25)+2 sin(0.25) cos(0.25) (x − 0.5)2 + sin(0.25)−6 (x − 0.5)3 ; 2 6 ; K = 10 is acceptable. |T3 (0.6) − f (0.6)| ≤ K(0.0001) 24 41. n = 4 43. n = 6 47. n = 4 4 8 x 4n 51. T4n (x) = 1 − x2 + x4! + · · · + (−1)n (2n)!

53. At a = 0,

3 1. T2 (x) = x; T3 (x) = x − x6

T1 (x) = −4 − x

1 (x − 2)2 ; 3. T2 (x) = 13 − 19 (x − 2) + 27 1 1 1 1 (x − 2)3 T3 (x) = 3 − 9 (x − 2) + 27 (x − 2)2 − 81

T2 (x) = −4 − x + 2x 2 T3 (x) = −4 − x + 2x 2 + 3x 3 = f (x)

5. T2 (x) = 75 + 106(x − 3) + 54(x − 3)2 ; T3 (x) = 75 + 106(x − 3) + 54(x − 3)2 + 12(x − 3)3

T4 (x) = T3 (x)

3 7. T2 (x) = x; T3 (x) = x + x3

T5 (x) = T3 (x)

2 2 3 9. T2 (x) = 2 − 3x + 5x2 ; T3 (x) = 2 − 3x + 5x2 − 3x2 1 (x − 1)2 ; 11. T2 (x) = 1e + 1e (x − 1) − 2e 1 1 1 1 (x − 1)3 T3 (x) = e + e (x − 1) − 2e (x − 1)2 − 6e 2 13. T2 (x) = (x − 1) − 3(x−1) ; 2 3(x−1)2 11(x−1)3 T3 (x) = (x − 1) − + 2 6 x 15. Let f (x) = e . Then, for all n,

f (n) (x) = ex



1 ≤ 2

and f (n) (0) = 1.

It follows that x2

xn

x + Tn (x) = 1 + 1! 2! + · · · + n! .

19. Tn (x) = 1 + x + x 2 + x 3 + · · · + x n 2 n 21. Tn (x) = e + e(x − 1) + e(x−1) + · · · + e(x−1) 2! n!

At a = 1, T1 (x) = 12(x − 1) T2 (x) = 12(x − 1) + 11(x − 1)2 T3 (x) = 12(x − 1) + 11(x − 1)2 + 3(x − 1)3 = −4 − x + 2x 2 + 3x 3 = f (x) T4 (x) = T3 (x) T5 (x) = T3 (x) 55. T2 (t) = 60 + 24t − 32 t 2 ; truck’s distance from intersection after 4 s is ≈ 132 m 57. (a) T3 (x) = − k3 x + 3k5 x 3 R

2R

65. T4 (x) = 1 − x 2 + 12 x 4 ; the error is approximately |0.461458 − 0.461281| = 0.000177

A87

ANSWERS TO ODD-NUMBERED EXERCISES

! 1/2 67. (b) 0 T4 (x) dx = 1841 3840 ; error bound:  " " 1/2   1/2   cos x dx − T4 (x) dx  <    0 0

( 12 )7 6!

69. (a) T6 (x) = x 2 − 16 x 6

Chapter 8 Review 1. 779 240

√ 3. 4 17

√ 7. 24π 2

9. 67π 36

11. 12π + 4π 2 13. 176,400 N √ 15. Fluid force on a triangular face: 183, 750√ 3 + 306, 250 N; fluid force on a slanted rectangular edge: 122, 500 3 + 294, 000 N   17. Mx = 20480; My = 25600; center of mass: 2, 85   19. 0, π2 21. T3 (x) = 1 + 3(x − 1) + 3(x − 2)2 + (x − 1)3 1 (x − 1)4 23. T4 (x) = (x − 1) + 12 (x − 1)2 − 16 (x − 1)3 + 12 25. T4 (x) = x − x 3 1 (3x)2 + 1 (3x)3 + · · · + 1 (3x)n 27. Tn (x) = 1 + 3x + 2! 3!   n!   −1 29. T3 (1.1) = 0.832981496; T3 (1.1) − tan 1.1 = 2.301 × 10−7 31. n = 11 is sufficient. 1 is 33. The nth Maclaurin polynomial for g(x) = 1+x

Tn (x) = 1 − x + x 2 − x 3 + · · · + (−x)n .

Chapter 9 Section 9.1 Preliminary Questions 1. (a) First order (b) First order (c) Order 3 (d) Order 2 2. Yes 3. Example: y = y 2 4. Example: y = y 2 5. Example: y + y = x

Section 9.1 Exercises 1. (a) First order (b) Not first order (c) First order (d) First order (e) Not first order (f) First order 3. Let y = 4x 2 . Then y = 8x and y − 8x = 8x − 8x = 0. 5. Let y = 25e−2x . Then y = −100xe−2x and 2 2 y + 4xy = −100xe−2x + 4x(25e−2x ) = 0 7. Let y = 4x 4 − 12x 2 + 3. Then 2

2

y

− 2xy + 8y = (48x 2 − 24) − 2x(16x 3 − 24x) + 8(4x 4 − 12x 2 + 3)

9. (c) 11. 13. 15.

= 48x 2 − 24 − 32x 4 + 48x 2 + 32x 4 − 96x 2 + 24 = 0 sin x 3y (a) Separable: y = x9 y 2 (b) Separable: y = e 4 − x2 Not separable (d) Separable: y = (1)(9 − y 2 ) C=4 −1  , where C is an arbitrary constant. y = 2x 2 + C   y = ln 4t 5 + C , where C is an arbitrary constant.

17. y = Ce−(5/2)x + 54 , where C is an arbitrary constant.

√ 2 19. y = Ce− 1−x , where C is an arbitrary constant. 21. y = ± x 2 + C, where C is an arbitrary constant. 23. x = tan( 12 t 2 + t + C), where C is an arbitrary constant.   25. y = sin−1 12 x 2 + C , where C is an arbitrary constant. 27. y = C sec t, where C is an arbitrary constant.    29. y = 75e−2 x 31. y = − ln x 2 + e4   −t 37. y = e1−e 33. y = 2 + 2e x(x−2)/2 35. y = tan x 2 /2   39. y = et − 1 41. y = sin−1 12 ex 43. a = −3, 4 e1/t √ 45. t = ± π + 4 47. (a) ≈ 1145 s or 19.1 min (b) ≈ 3910 s or 65.2 min 49. y = 8 − (8 + 0.0002215t)2/3 ; te ≈ 66000 s or 18 hr, 20 min   53. (a) q(t) = CV 1 − e−t/RC (b)   lim q(t) = lim CV 1 − e−t/RC = lim CV (1 − 0) = CV t→∞ t→∞ t→∞   −1 (c) q(RC) = CV 1 − e ≈ (0.63) CV 55. V = (kt/3 + C)3 , V increases roughly with the cube of time. C , 57. g(x) = Ce(3/2)x , where C is an arbitrary constant; g(x) = x−1 where C is an arbitrary constant.  2 59. y = Cx 3 and y = ± A − x3 61. (b) v(t) = −9.8t + 100(ln(50) − ln(50 − 4.75t)); v(10) = −98 + 100(ln(50) − ln(2.5)) ≈ 201.573 m/s 7π R 5/2 67. (c) C = 60B

Section 9.2 Preliminary Questions 1. y(t) = 5 − ce4t for any positive constant c 2. No 3. True 4. The difference in temperature between a cooling object and the ambient temperature is decreasing. Hence the rate of cooling, which is proportional to this difference, is also decreasing in magnitude.

Section 9.2 Exercises 1. General solution: y(t) = 10 + ce2t ; solution satisfying y(0) = 25 : y(t) = 10 + 15e2t ; solution satisfying y(0) = 5: y(t) = 10 − 5e2t y

y

0.5

800

1

−50

600

−100

y(0) = 25

400

−150

y(0) = 5

−200

200 x 0.5

1

−250

1.5

3. y = −6 + 11e4x 1

5. (a) y = −0.02(y − 10) (b) y = 10 + 90e− 50 t (c) 100 ln 3 s ≈ 109.8 s 7. ≈ 5:50 AM 9. ≈ 0.77 min = 46.6 s

1.5

x

A88

ANSWERS TO ODD-NUMBERED EXERCISES

11. 500 ln 32 s ≈ 203 s = 3 min 23 s

11. (a)

y 2 1

y 40

yA yB

20

−1

x 200

400

600

800

−2 −2

1000

−20

13. −58.8 m/s 15. −11.8 m/s 17. (a) $17, 563.94 (b) 13.86 yr 19. $120, 000 21. 8%   1 ln 13,333.33 ≈ 15.4 yr 23. (b) t = 0.09 3,333.33

1

2

(b) y2 = 3.231

(d) y(2.2) ≈ 3.231, y(2.5) ≈ 3.799539 (c) No

15. y(0.5) ≈ 1.7210 17. y(3.3) ≈ 3.3364 19. y(2) ≈ 2.8838 23. y(0.5) ≈ 1.794894 25. y(0.25) ≈ 1.094871

Section 9.3 Preliminary Questions 3. (b)

0

(c) y3 = 3.3919, y4 = 3.58171, y5 = 3.799539, y6 = 4.0445851

(b) N(t) = 1 − e−kt (c) ≈ 64.63% −g  g  −kt 29. (a) v(t) = + v0 + e k k √ 2. y = ± 1 + t

−1

13. (a) y1 = 3.1

25. (a) N (t) = k(1 − N(t)) = −k(N(t) − 1)

1. 7

t

0

4. 20

Section 9.4 Preliminary Questions 1. (a) No (b) Yes 2. No

(c) No

(d) Yes

3. Yes

Section 9.3 Exercises 1.

Section 9.4 Exercises

3.

5 5 and y = 1 − e−3t /C 1 + (3/2)e−3t 3. lim y(t) = 2 1. y =

t→∞

5. (a) P (t) =

5. (a)

1 ln 3 ≈ 1.83 yrs (b) t = 0.6

ln 9 −1 7. k = ln 81 31 ≈ 0.96 yrs ; t = 2 ln 9−ln 31 ≈ 2.29 yrs 9. After t = 8 hours, or at 4:00 PM

y

10 1 11. (a) y1 (t) = 10−9e −t and y2 (t) = 1−2e−t

3 2

(b) t = ln 98

1

0

1

2

(c) t = ln 2

13. (a) A(t) = 16(1 − 53 et/40 )2 /(1 + 53 et/40 )2

t

0 −1 −1

2000 1 + 3e−0.6t

(b) A(10) ≈ 2.1 (c)

3

7. For y = t, y only depends on t. The isoclines of any slope c will be the vertical lines t = c.

y 16 A(t)

1

y

x 0

2

100

200

15. ≈ 943 million

1 t

0

17. (d) t = − k1 (ln y0 − ln (A − y0 ))

−1 −2 −2

Section 9.5 Preliminary Questions −1

0

1

9. (i) C (ii) B (iii) F (iv) D (v) A (vi) E

2

1. (a) Yes 2. (b)

(b) No

(c) Yes

(d) No

ANSWERS TO ODD-NUMBERED EXERCISES

Section 9.5 Exercises

15. y(0.1) ≈ 1.1; y(0.2) ≈ 1.209890; y(0.3) ≈ 1.329919

4 x4 1 1. (c) y = x5 + C x (d) y = 5 − 5x 5. y = 12 x + C x 7. y = − 14 x −1 + Cx 1/3 9. y = 15 x 2 + 13 + Cx −3 11. y = −x ln x + Cx 13. y = 12 ex + Ce−x 15. y = x cos x + C cos x 17. y = x x + Cx x e−x

−2x 19. y = 12 + e−x − 11 2e 21. y = 21 sin 2x − 2 cos x 23. y = 1 − t 2 + 15   25. w = tan k ln x + π4

|x| 1 5 19. y = 15 e2x − 65 e−3x 21. y = ln x+1 − x(x+1) + x+1 23. y = − cos x + sin x 25. y = tanh x + 3sech x 1 emx + Ce−nx ; for m = −n: 27. For m = −n: y = m+n −nx y = (x + C)e

29. (b) 31. 33. (b) (c)

17. y = x 2 + 2x

27. y = − cos x + sinx x + C x , where C is an arbitrary constant 29. Solution satisfying y(0) = 3: y(t) = 4 − e−2t ; solution satisfying y(0) = 4: y(t) = 4 y 4

40y (a) y = 4000 − 500+40t ; y = 1000 4t +100t+125 2t+25 40 g/L 50 g/L 20 (a) dV dt = 1+t − 5 and V (t) = 20 ln(1 + t) − 5t + 100 The maximum value is V (3) = 20 ln 4 − 15 + 100 ≈ 112.726 2

y(0) = 3 3 2 1 x

V

0

100 80 60 40 20

0.5

1

1.5

31. (a) 12 (b) ∞, if y(0) > 12; 12, if y(0) = 12; −∞, if y(0) < 12 (c) −3

t −20

y(0) = 4

10

20

30

40

33. 400, 000 − 200, 000e0.25 ≈ $143, 194.91

  1 1 − e−20t 35. I (t) = 10

35. $400, 000

37. (a) I (t) = VR − VR e−(R/L)t 39. (b) c1 (t) = 10e−t/6

√ −1.77 y

(c) Approximately 0.0184 s

39. 240y+64800 ; t = 9198 s about 2.56 hours. 41. 2 43. t = 5 ln 441 ≈ 30.45 days 2 1 47. (a) dc dt = − 5 c1

(b) c1 (t) = 8e(−2/5)t g/L

Chapter 9 Review 1. (a) No, first order (b) Yes, first order (c) No, order 3 (d) Yes, second order  1/4 3. y = ± 43 t 3 + C , where C is an arbitrary constant 5. y = Cx − 1, where C is an arbitrary constant

7. y = 12 x + 12 sin 2x + π4 y 11.

9. y =

2 2−x 2

Chapter 10 Section 10.1 Preliminary Questions 1. a4 = 12

3. lim an = n→∞

√ 2

4. (b)

5. (a) False. Counterexample: an = cos π n (b) True

2

2. (c)

(c) False. Counterexample: an = (−1)n

1

Section 10.1 Exercises

t

0 −1

1. (a) (iv) (b) (i) (c) (iii)

−2 −2 −1

0

1

(d) (ii)

3. c1 = 3, c2 = 92 , c3 = 92 , c4 = 27 8

2

5. a1 = 2, a2 = 5, a3 = 47, a4 = 4415

13. y(t) = tan t

7. b1 = 4, b2 = 6, b3 = 4, b4 = 6 25 9. c1 = 1, c2 = 32 , c3 = 11 6 , c4 = 12 11. b1 = 2, b2 = 3, b3 = 8, b4 = 19

y 2

n+1 13. (a) an = (−1)3

1 t

0 −1 −2 − 2 −1

0

1

2

n

(b) an = n+1 n+5

5n−1 = 5 15. lim 12 = 12 17. lim 12n+9 12 n→∞  n→∞  19. lim −2−n = 0 21. The sequence diverges. n→∞   12n+2 = ln 3 = 1 25. lim ln −9+4n 23. lim √ n2 n→∞

n +1

n→∞

A89

A90

ANSWERS TO ODD-NUMBERED EXERCISES

27. lim

n→∞

 4 + n1 = 2

29. lim cos−1



n→∞

n3 2n3 +1



  19. lim (−1)n n−1 does not exist. n

= π3

n→∞

31. (a) M = 999 (b) M = 99999   n  35. lim 10 + − 91 = 10 37. The sequence diverges.

1 = 1 = 0 21. lim an = cos n+1

39. lim 21/n = 1

1 7 29. S = e−1 31. S = 35 33. S = 4 35. S = 15 3 37. (b) and (c) ∞    1 n = 1. 41. (a) Counterexample: 2

n→∞

41. lim

9n

n→∞ n!

n→∞

=0

2 = 32 43. lim 3n +n+2 2

45. lim cosn n = 0 n→∞  1/3 47. The sequence diverges. 49. lim 2 + 42 = 21/3 n n→∞   2 51. lim ln 2n+1 3n+4 = ln 3 53. The sequence diverges. n→∞

2n −3

n→∞

n n =0 55. lim e +(−3) 5n

57. lim n sin πn = π n→∞  n n 3−4 1 59. lim 2+7·4n = − 7 61. lim 1 + n1 = e n→∞ n→∞   2 63. lim (lnnn) = 0 65. lim n n2 + 1 − n = 12 n→∞

n→∞ n→∞

n +n8

=0

69. lim (2n + 3n )1/n = 3 n→∞

71. (b)

73. Any number greater than or equal to 3 is an upper bound. 75. Example: an = (−1)n 79. Example: f (x) = sin πx  √  87. (e) AGM 1, 2 ≈ 1.198

Section 10.2 Preliminary Questions 1. The sum of an infinite series is defined as the limit of the sequence of partial sums. If the limit of this sequence does not exist, the series is said to diverge. 2. S = 12 3. The result is negative, so the result is not valid: a series with all positive terms cannot have a negative sum. The formula is not valid because a geometric series with |r| ≥ 1 diverges. 4. No 5. No 6. N = 13 7. No, SN is increasing and converges to 1, so SN ≤ 1 for all N. ∞  1 8. Example: 9/10 n=1

n

Section 10.2 Exercises 1. (a) an = 31n

 n−1 (b) an = 52 n

(c) an = (−1)n+1 nn!

(d) an =

n+1 +1

1+ (−1) 2 n2 +1

5369 3. S2 = 54 , S4 = 205 144 , S6 = 3600 5. S2 = 23 , S4 = 45 , S6 = 67 7. S6 = 1.24992 9. S10 = 0.03535167962, S100 = 0.03539810274, S500 = 0.03539816290, S1000 = 0.03539816334. Yes. ∞    3 ,S = 1,S = 5 , 1 1 1 11. S3 = 10 4 3 5 14 n+1 − n+2 = 2 n=1 5, 13. S3 = 37 , S4 = 49 , S5 = 11

∞  n=1

15. S = 12

1 = 12 4n2 −1

n 1 = 0 17. lim 10n+12 = 10 n→∞

27. S = 59049 3328

25. The series diverges.

n=1

(b) Counterexample: If an = 1, then SN = N . ∞  1 (c) Counterexample: n diverges. (d) Counterexample:

n→∞

67. lim √ 41

n→∞

23. S = 87

n=1 ∞ 

cos 2πn  = 1.

n=1 43. The total area is 14 .

45. The total length of the path is 2 +

√

2.

Section 10.3 Preliminary Questions 1. (b) 2. A function f (x) such that an = f (n) must be positive, decreasing, and continuous for x ≥ 1. 3. Convergence of p-series or integral test 4. Comparison Test ∞  −n 1 diverges, but since e n < n1 for n ≥ 1, the 5. No; n n=1

Comparison Test tells us nothing about the convergence of

∞  n=1

Section 10.3 Exercises

! 1. 1∞ dx4 dx converges, so the series converges. ! x 3. 1∞ x −1/3 dx = ∞, so the series diverges. !∞ x2 dx converges, so the series converges. 5. 25 (x 3 +9)5/2 ! ∞ dx 7. 1 2 converges, so the series converges. x +1 ! ∞ dx 9. 1 x(x+1) converges, so the series converges. ! 1 11. 2∞ dx converges, so the series converges. x(ln x)2 ! ∞ dx 13. 1 ln x = ∞, so the series diverges. 2

≤ 13 , so the series converges. n  n 19. n21n ≤ 12 , so the series converges.  n 21. 1/31 n ≤ 12 , so the series converges. n +2  m 4 1 , so the series converges. 23. m!+4 m ≤ 4 4 15.

1

n3 +8n

2 25. 0 ≤ sin2 k ≤ 12 , so the series converges. k k n 2 27. 3n +3−n ≤ 2 13 , so the series converges.

1 29. (n+1)! ≤ 12 , so the series converges. n

e−n . n

ANSWERS TO ODD-NUMBERED EXERCISES

31. ln3n ≤ 12 for n ≥ 1, so the series converges.

n n (ln n)100 1 for n sufficiently large, so the series converges. ≤ 1.09 33. n1.1  n n 35. 3nn ≤ 23 for n ≥ 1, so the series converges.

The series converges. 41. The series diverges. The series converges. 45. The series diverges. The series converges. 49. The series converges. The series diverges. 53. The series converges. The series diverges. 57. The series converges. The series diverges. 61. The series diverges. The series diverges. 65. The series converges. The series diverges. 69. The series diverges. The series converges. 73. The series converges. The series diverges. 77. The series converges. The series converges for a > 1 and diverges for a ≤ 1. ∞  n−5 ≈ 1.0369540120. 87.

39. 43. 47. 51. 55. 59. 63. 67. 71. 75. 79.

91.

n=1 1000  n=1

1 = 1.6439345667 and 1 + n2

100  n=1

1 = 1.6448848903. n2 (n+1)

The second sum is a better approximation to

π2 ≈ 1.6449340668. 6

Section 10.4 Preliminary Questions 1. Example:

 (−1)n √ 3

2. (b)

3. No.

n −3 4. |S − S100 | ≤ 10 , and S is larger than S100 .

A91

Section 10.5 Preliminary Questions   a  1. ρ = lim  n+1 an  n→∞

2. The Ratio Test is conclusive for for

∞ 

∞ 

1 2n and inconclusive

n=1 1 n.

n=1

3. No.

Section 10.5 Exercises 1. Converges absolutely 3. Converges absolutely 5. The ratio test is inconclusive. 7. Diverges 9. Converges absolutely 11. Converges absolutely 13. Diverges 15. The ratio test is inconclusive. 17. Converges absolutely 19. Converges absolutely 21. ρ = 13 < 1 23. ρ = 2|x| 25. ρ = |r| 29. Converges absolutely 31. The ratio test is inconclusive, so the series may converge or diverge. 33. Converges absolutely 35. The ratio test is inconclusive. 37. Converges absolutely 39. Converges absolutely 41. Converges absolutely 43. Converges (by geometric series and linearity) 45. Converges (by the Ratio Test) 47. Converges (by the Limit Comparison Test) 49. Diverges (by p-series) 51. Converges (by geometric series) 53. Converges (by Limit Comparison Test) 55. Diverges (by Divergence Test)

Section 10.4 Exercises 3. 5. 7. 9. 11.

13. 17. 19. 21. 23. 25. 27. 29. 31. 33.

Converges conditionally Converges absolutely Converges conditionally Converges conditionally (a) n Sn n Sn 1 1 6 0.899782407 2 0.875 7 0.902697859 3 0.912037037 8 0.900744734 4 0.896412037 9 0.902116476 5 0.904412037 10 0.901116476 S5 = 0.947 15. S44 = 0.06567457397 Converges (by geometric series) Converges (by Comparison Test) Converges (by Limit Comparison Test) Diverges (by Limit Comparison Test) Converges (by geometric series and linearity) Converges absolutely (by Integral Test) Converges conditionally (by Leibniz Test) Converges (by Integral Test) Converges conditionally

Section 10.6 Preliminary Questions 1. Yes. The series must converge for both x = 4 and x = −3. 2. (a), (c) 3. R = 4 ∞  n2 x n−1 ; R = 1 4. F (x) = n=1

Section 10.6 Exercises 1. 3. 9. 17. 27. 35.

37.

39.

R = 2. It does not converge at the endpoints. R = 3 for all three series. √ √ (−1, 1) 11. [− 2, 2] 13. [−1, 1] 15. (−∞, ∞) 1 1 [− 4 , 4 ) 19. (−1, 1] 21. (−1, 1) 23. [−1, 1) 25. (2, 4)     (6, 8) 29. − 72 , − 52 31. (−∞, ∞) 33. 2 − 1e , 2 + 1e ∞ 

n=0 ∞  n=0 ∞  n=0

  3n x n on the interval − 13 , 13 x n on the interval (−3, 3) 3n+1

(−1)n x 2n on the interval (−1, 1)

A92

43.

ANSWERS TO ODD-NUMBERED EXERCISES

∞ 

(−1)n+1 (x − 5)n on the interval (4, 6)

1 = 31. 1−x

n=0

n

(−1)n+1 (x−5) n+1 on the interval (1, 9) 4

n=0

69 and |S − S | ≈ 0.000386 < a = 1 47. (c) S4 = 640 5 4 1920 ∞  1−x−x 2 n 49. R = 1 51. 2n = 2 53. F (x) = 1−x 3 n=1 ∞  n 55. −1 ≤ x ≤ 1 57. P (x) = (−1)n xn! n=0

59. N must be at least 5; S5 = 0.3680555556 ∞  1·3·5···(2n−3) 2n 61. P (x) = 1 − 12 x 2 − x ;R=∞ (2n)! n=2

1. f (0) = 3 and f

(0) = 30 2. f (−2) = 0 and f (4) (−2) = 48 3. Substitute x 2 for x in the Maclaurin series for sin x. ∞  (x−3)n+1 4. f (x) = 4 + 5. (c) n(n+1) n=1

1. f (x) = 2 + 3x + 2x 2 + 2x 3 + · · · ∞    1 = 3. 1−2x 2n x n on the interval − 12 , 12 5. cos 3x =

n x 2n (−1)n 9(2n)! on the interval (−∞, ∞)

n=0 ∞ 

7. sin(x 2 ) =

4n+2

x (−1)n (2n+1)! on the interval (−∞, ∞)

n=0

9. ln(1 − x 2 ) = −

∞ 

x 2n on the interval (−1, 1) n

n=1

11. tan−1 (x 2 ) = ∞  n=0

∞ 

n=0 x n on the interval (−∞, ∞) e2 n!

15. ln(1 − 5x) = − 17. sinh x =

∞ 

1 = 1−x 2

5n x n n

n=0

(−1)n+1 (2n+1 −1) (x − 3)n on the interval (1, 5) 22n+3

39. cos2 x = 12 + 12

∞ 

n 2n

x (−1)n (4) (2n)!

45. S4 = 0.1822666667 47. (a) 4 (b) S4 = 0.7474867725 ∞  ! (−1)n 49. 01 cos(x 2 ) dx = (2n)!(4n+1) ; S3 = 0.9045227920 " 1 51. 0

n=0 ∞ 

e−x dx = 3

(−1)n n!(3n+1) ; S5 = 0.8074461996

n=0

n=1 ∞ 

!x

55. 0 ln(1 + t 2 ) dt =

 on the interval − 15 , 15 

n=1 x 2k+1 (2k+1)! on the interval (−∞, ∞)

k=0 3 5 x 19. e sin x = x + x 2 + x3 − x30 + · · · x 5x 3 5x 4 2 21. sin 1−x = x + x + 6 + 6 + · · · 3 x2 + 7 x3 + · · · 23. (1 + x)1/4 = 1 + 14 x − 32 128 25. ex tan−1 x = x + x 2 + 16 x 3 − 16 x 4 + · · · 27. esin x = 1 + x + 12 x 2 − 18 x 4 + · · · ∞  (−1)n (x − 1)n on the interval (0, 2) 29. x1 = n=0

(−1)n−1

n=1 1 57. 1+2x

3 63. ex

1 = 67. (1−2x)(1−x)

x 2n+1 n(2n + 1)

65. 1 − 5x + sin 5x ∞  

 2n+1 − 1 x n

n=0 ∞    (−1)n+1 Rt n 69. I (t) = VR L n! n=1 ∞  (−1)n x 6n (6) (0) = −360. 71. f (x) = (2n)! and f n=0

4n+2

(−1)n x2n+1 on the interval [−1, 1]

∞ 

37.

n=0 ∞ 

∞  ! x 2n 53. 0x 1−cos(t) dt = (−1)n+1 t (2n)!2n

Section 10.7 Exercises

n=0 ∞ 

33. 21 + 35(x − 2) + 24(x − 2)2 + 8(x − 2)3 + (x − 2)4 on the interval (−∞, ∞) ∞  (x − 4)n 35. 12 = (−1)n (n + 1) n+2 on the interval (0, 8) x 4

n=0

Section 10.7 Preliminary Questions

13. ex−2 =

∞ 

40 73. e20x = 1 + x 20 + x2 + · · · 3 sin x − x + x6 1 81. lim = 120 x→0 x5



sin(x 2 ) cos x − 2 83. lim x→0 x4 x 85. S = π4 − 12 ln 2



=

75. No.

1 2

89. L ≈ 28.369

Chapter 10 Review 1. (a) a12 = 4, a22 = 14 , a32 = 0 1 ,b = 1 ,b = 1 (b) b1 = 24 2 60 3 240 1 ,a b = − 1 ,a b = 0 (c) a1 b1 = − 12 2 2 120 3 3 1 (d) 2a2 − 3a1 = 5, 2a3 − 3a2 = 32 , 2a4 − 3a3 = 12 3. lim (5an − 2an2 ) = 2 5. lim ean = e2 n→∞

n→∞

7. lim (−1)n an does not exist. n→∞

A93

ANSWERS TO ODD-NUMBERED EXERCISES

9. lim

√

n→∞

n+5−

√

 n+2 =0

2

11. lim 21/n = 1 n→∞

13. The sequence diverges.   π 15. lim tan−1 n+2 n+5 = 4 n→∞   17. lim n2 + n − n2 + 1 = 12 n→∞   1 3m 19. lim 1 + = e3 21. lim (n ln(n + 1) − ln n) = 1 m→∞ n→∞ m an+1 11 41 25. lim = 3 27. S4 = − 60 , S7 = 630 n→∞ an ∞ ∞   2n+3 29. ( 23 )n = 43 31. = 36 3n n=2 n=−1  n 33. Example: an = 12 + 1, bn = −1 47 35. S = 180 !∞ 39. 1

41. 43. 45. 47.

37. The series diverges.

1 1 dx = , so the series converges. (x+2)(ln(x+2))3 2(ln(3))2 1 < 12 , so the series converges. (n+1)2 n ∞  1 converges, so the series converges. n1.5 n=0 1 , so the series converges. √ n < 3/2 n n5 +2 ∞    10 n converges, so the series converges. 11 n=0

49. Converges 53. (b) 0.3971162690 ≤ S ≤ 0.3971172688, so the maximum size of the error is 10−6 . 55. Converges absolutely 57. Diverges 59. (a) 500

(b) K ≈

499  (−1)k 2 = 0.9159650942 n=0

(2k+1)

61. (a) Converges (b) Converges (c) Diverges (d) Converges 63. Converges 65. Converges 67. Diverges 69. Diverges 71. Converges 73. Converges 75. Converges (by geometric series) 77. Converges (by geometric series) 79. Converges (by the Leibniz Test) 81. Converges (by the Leibniz Test) 83. Converges (by the Comparison Test) 85. Converges using partial sums (the series is telescoping) 87. Diverges (by the Comparison Test) 89. Converges (by the Comparison Test) 91. Converges (by the Comparison Test) 93. Converges on the interval (−∞, ∞) 95. Converges on the interval [2, 4] 97. Converges at x = 0 ∞    2 = 1 3 n x n . The series converges on the interval 99. 4−3x 2 4 4 ( −4 3 , 3)

n=0

101. (c)

y 7 6 5 4 3 2 1 −2

103. e4x =

x

−1

∞ 

1

2

4n x n n!

n=0

105. x 4 = 16 + 32(x − 2) + 24(x − 2)2 + 8(x − 2)3 + (x − 2)4 ∞  (−1)n+1 (x−π)2n+1 107. sin x = (2n+1)! n=0 ∞ 

1 = 109. 1−2x

n=0 2

2n (x 5n+1

113. (x 2 − x)ex =

∞ 

111. ln x2 =

+ 2)n

(x

∞  (−1)n+1 (x−2)n n2n

n=1 2n+2 −x 2n+1

n!

) so f (3) (0) = −6

n=0 1 4 3 2 4 2 (3) (0) = −8 115. 1+tan x = −x + x − 3 x + 3 x + · · · so f 3 5 7 117. π2 − π3 + π5 − π7 + · · · = sin π2 = 1 2 5!

2 3!

2 7!

Chapter 11 Section 11.1 Preliminary Questions 1. A circle of radius 3 centered at the origin. 2. The center is at (4, 5) 3. Maximum height: 4 4. Yes; no 5. (a) ↔ (iii), (b) ↔ (ii), (c) ↔ (i)

Section 11.1 Exercises 1. (t = 0)(1, 9); (t = 2)(9, −3); (t = 4)(65, −39) y (b) 5. (a)

y t=

t=0

x

t=

(c)

y t = 1(1, 1)

x

x

t = −1 (−1, −1)

7. y = 4x − 12



9. y = tan−1 x 3 + ex



11. y = 62 (where x > 0) 13. y = 2 − ex x

(1,1)

x

t = 2π

3π (−1,−1) 2

(d)

y

π 2

A94

ANSWERS TO ODD-NUMBERED EXERCISES

15.

17.

y

y

y

4 (4π 2

,0)

(−2π 2,0)

x

3 2

x

t=0

1

1

19. (a) ↔ (iv), (b) ↔ (ii), (c) ↔ (iii), (d) ↔ (i)

  2 π ≤ t ≤ 2π 23. c(t) = (t, 9 − 4t) 25. c(t) = 5+t 4 ,t c(t) = (−9 + 7 cos t, 4 + 7 sin t) 29. c(t) = (−4 + t, 9 + 8t) c(t) = (3 − 8t, 1 + 3t) 33. c(t) = (1 + t, 1+ 2t) (0 ≤ t ≤ 1) c(t) = (3 + 4 cos t, 9 + 4 sin t) 37. c(t) = −4 + t, −8 + t 2   39. c(t) = (2 + t, 2 + 3t) 41. c(t) = 3 + t, (3 + t)2 43. y = x 2 − 1 (1 ≤ x 0

Section 11.2 Preliminary Questions

! 1. S = ab x (t)2 + y (t)2 dt 2. The speed at time t 3. Displacement: 5; no 4. L = 180 cm

y

Section 11.2 Exercises

60

40 t = −3 (0,33) 20 t=0 (−9,0) −20 t=3

(0,−15)

−20

t=8 (55,0) 20

40

60

x

√ 1. S = 10 3. S = 16 13 5. S = 12 (653/2 − 53/2 ) ≈ 256.43  √ 7. S = 3π 9. S = −8 22 − 1 ≈ 2.34   √ ds 13. S = ln(cosh(A)) 15. dt  = 4 10 ≈ 12.65 m/s t=2     √ √  17. ds = 41 ≈ 6.4 m/s 19. ds dt  dt min ≈ 4.89 ≈ 2.21 t=9

t = 4 (7,−16)

The graph is in: quadrant (i) for t < −3 or t > 8, quadrant (ii) for −3 < t < 0, quadrant (iii) for 0 < t < 3, quadrant (iv) for 3 < t < 8. 61. (55, 0) 63. The coordinates of P , (R cos θ, r sin θ ), describe an ellipse for 0 ≤ θ ≤ 2π . 67. c(t) = (3 − 9t + 24t 2 − 16t 3 , 2 + 6t 2 − 4t 3 ), 0 ≤ t ≤ 1

21. ds dt = 8 23.

y t=

π 2

t = π, (−1, 1)

(0, e)

t = 0, t = 2π, (1, 1) x t=

3π 1 (0, ) 2 e

M10 = 6.903734, M20 = 6.915035, M30 = 6.914949, M50 = 6.914951

ANSWERS TO ODD-NUMBERED EXERCISES

25.

27.

y

A95

π 2 3π 4

t=0 x t = 2π

π 4

C π

M10 = 25.528309, M20 = 25.526999, M30 = 25.526999, M50 = 25.526999 27. S = 2π 2 R 29. S = m 1 + m2 πA2 31. S = 64π 3 33. (a)

H 0 2π

A O

E 5π 4

y

G 7π

F

4 3π 2

20

y

B

D

15

29. 10

t=0

3π 4

t = 10π x

π 2 y

π 4

5 π

0

x 10

20

1

0 2 x

30

(b) L ≈ 212.09

5π 4

7π 4 3π 2

Section 11.3 Preliminary Questions 1. (b)

    2. Positive: (r,θ) = 1, π2 ; Negative: (r, θ ) = −1, 3π 2 3. (a) Equation of the circle of radius 2 centered at the origin. √ (b) Equation of the circle of radius 2 centered at the origin. (c) Equation of the vertical line through the point (2, 0). 4. (a)

Section 11.3 Exercises 

√ 1. (A): 3 2, 3π ; (B): (3, π ); (C): √  4 √  √  √  5, π + 0.46 ≈ 5, 3.60 ; (D): 2, 5π 2, π4 ; (F): ; (E): 4    π 4, 6 ; (G): 4, 11π 6 √  √    3. (a) (1, 0) (b) 12, π6 8, 3π (c) (d) 2, 2π 4 3     √ (b) − √6 , √6 (c) (0, 0) (d) (0, −5) 5. (a) 3 2 3 , 32 2

2

7. (A): 0 ≤ r ≤ 3, π ≤ θ ≤ 2π , (B): 0 ≤ r ≤ 3, π4 ≤ θ ≤ π2 , (C): 3 ≤ r ≤ 5, 3π 4 ≤θ ≤π 2 2 2 9. m = tan 3π 5 ≈ −3.1 11. x + y = 7 √ 2 2 13. x + (y − 1) = 1 15. y = x − 1 17. r = 5 19. r = tan θ sec θ

21. (a)↔(iii), (b)↔(iv), (c)↔(i), (d)↔(ii) 23. (a) (r, 2π − θ ) (b) (r, θ + π) (d) r, π2 − θ   25. r cos θ − π3 = d

(c) (r, π − θ)

π 31. (a) A, θ = 0, r = 0; B, θ = π4 , r = sin 2π 4 = 1; C, θ = 2 , 2·3π 5π r = 0; D, θ = 3π 4 , r = sin 4 = −1; E, θ = π , r = 0; F, θ = 4 , 3π 7π r = 1; G, θ = 2 , r = 0; H, θ = 4 , r = −1; I, θ = 2π , r = 0

(b) 0 ≤ θ ≤ π2 is in the first quadrant. π2 ≤ θ ≤ π is in the fourth 3π quadrant. π ≤ θ ≤ 3π 2 is in the third quadrant. 2 ≤ θ ≤ 2π is in the second quadrant. 33. 3π 4

π

π 2 y

0

5π 4

π 4

1

0 2 x

7π 4 3π 2

2 2   2 2 35. x − a2 + y − b2 = a +b , r = a 2 + b2 , centered at the 4   point a2 , b2 2 2 3 2 37. r 2 = sec 2θ  39.  (x + y ) =√x − 3y x π 41. r = 2 sec θ − 9 43. r = 2 10 sec (θ − 4.39)

A96

ANSWERS TO ODD-NUMBERED EXERCISES

! √ 31. L = 02π 5 − 4 cos θ (2 − cos θ)−2 dθ

47. r 2 = 2a 2 cos 2θ

33. L ≈ 6.682

35. L ≈ 79.564

π 2

37. θ = π2 , m = − π2 ; θ = π, m = π  √  √  √  √ 2, π , 2 , 5π , 2 , 7π , 2 , 11π 39. 2 6 2 6 2 6 2 6 41. A: m = 1, B: m = −1, C: m = 1

π

0

Section 11.5 Preliminary Questions

r 2 = 8 cos 2θ

1. (a) True

(b) False

(c) True

(d) True

2. −3a = 15

3π 2

3. The components are not changed.

51. θ = π2 , m = − π2 ; θ = π, m = π √  √  √  √  2 π 2 5π 2 7π 2 11π 53. 2 , 6 , 2 , 6 , 2 , 6 , 2 , 6

4. (0, 0) 5. (a) True

(b) False

55. A: m = 1, B: m = −1, C: m = 1

Section 11.5 Exercises Section 11.4 Preliminary Questions 1. (b)

2. Yes

1. v1 = 2, 0 , v1  = 2

3. (c)

v2 = 2, 0 , v2  = 2

y

Section 11.4 Exercises

y

P

1 !π

v1

Q

1. A = 2 π/2 r 2 dθ = 25π 4

P

v2

Q x

x π θ= 2 y

v3 = 3, 1 , v3  = θ=π

√ 10

√ v4 = 2, 2 , v4  = 2 2

y

x

y

Q P

Q

v3 P

! 3. A = 12 0π r 2 dθ = 4π 5. A = 16 9. A = π8 ≈ 0.39 7. A = 3π 2 11.

3 A = π48

y

θ = π, r=π

x

x

Vectors v1 and v2 are equivalent. 3. (3, 5)

θ = π/2, r = π/2 θ = 2π, r = 2π x θ = 0, r=0

y

Q a

13. A = 15. A = 21. A = 25. L =

√

a0 P

 

15 −1 1 ≈ 11.163 4 2 + 7 cos √ 3 3 π − 2 ≈ 0.54 17. A = π8 − 14 ≈ 0.14 √ 9π − 4 2 23. A = 4π 2  3/2  1 2+4 − 8 ≈ 14.55 π 3

 √  27. L = 2 e2π − 1 ≈ 755.9 29. L = 8

v4

0

x

19. A = 4π −→ −→ 5. P Q = −1, 5# 7. P $ Q = −2, − 9 11. 30, 10 13. 52 , 5

9. 5, 5

A97

ANSWERS TO ODD-NUMBERED EXERCISES

%−→% √ % % 33. %OR % = 53 35. P = (0, 0) 37. ev = 15 3, 4 # √ √ $ 39. 4eu = −2 2, − 2 2 $ # 4π = −0.22, 0.97 , sin 41. e = cos 4π 7 7

15. Vector (B) −w

v−w

v

−w

w

17. 2v = 4, 6

43. λ = ± √1

45. P = (4, 6)

13

−w = −4, − 1

47. (a) → (ii), (b) → (iv), (c) → (iii), (d) → (i) 49. 9i + 7j 51. −5i − j

y

53.

y

y

y

y A

5 4 3 2 1

w sw

x

2v

−w

w

w

w B v

x 1

2

3

4

5

v

6

sw

rv

2v − w = 0, 5

rv

x

v + w = 6, 4

y

C

sw

v x

x rv

y v+w

2v − w

55. u = 2v − w

v

y w x

x

w v x u

19. 3v + w = −2, 10, 2v − 2w = 4, − 4 y 3v + w

57. The force on cable 1 is ≈ 45 lb, and force on cable 2 is ≈ 21 lb. 59. 230 km/hr 61. r = 6.45, 0.38

w v x 2v − 2w

21. 3

v1

1 2

−4

2. Obtuse

4. (a) v

(b) v

3. Distributive Law

6. (c)

Section 11.6 Exercises

1

−3

1. Scalar 5. (b); (c)

y

v2

Section 11.6 Preliminary Questions

x

v

23. (b) and (c) −→ −→ 25. AB = 2, 6 and P Q = 2, 6; equivalent −→ −→ 27. AB = 3, −2 and P Q = 3, −2; equivalent −→ −→ 29. AB = 2, 3 and P Q = 6, 9; parallel and point in the same direction −→ −→ 31. AB = −8, 1 and P Q = 8, −1; parallel and point in opposite directions

2 1. 5 3. 0 5. Acute 7. π/4 9. v2 11. v#2 − w $ 7 7 ◦ 13. 8 15. 2 17. π 19. (b) 7 23. 51.91 25. 2 , 2 # $ # $ √ 27. 17 29. a = 12 , 12 + 12 , − 12 35. ≈ 68.07 N

Section 11.7 Preliminary 1.

d

dt (f (t)r(t)) = f (t)r (t) + f (t)r(t) d

dt (r1 (t) · r2 (t)) = r1 (t) · r 2 (t) + r 1 (t) · r2 (t)

2. True

3. False

6. (a) Vector

4. True

(b) Scalar

5. False

A98

ANSWERS TO ODD-NUMBERED EXERCISES

Section# 11.7 Exercises $

25.

y

1. lim t 2 , 4t = 9, 12 t→3

2

3. lim (e2t i + ln(t + 1)j) = i

1

t→0

# $ = − 12 , cos t 5. lim r(t+h)−r(t) h

−2

t

h→0

7. dr dt = 1, 2t #

3s −s 9. dr ds = 3e , −e

x

−1

1

2

−1 −2

$

! s = 2 0π cos2 2t + sin2 t dt ≈ 6.0972   √    3 1 27. 1, π6 and 3, 5π 4 have rectangular coordinates 2 , 2 and √   √ −322,−322 . 2x 31. r = 3 + 2 sin θ 29. x 2 + y 2 = x−y

11. c (t) = −t −2 i 13. r (t) = 1, 2t, r

(t) = 0, 2 15.

5 4

´

r 1(1) r1(t)

´

r 2(1) r2(t)

2

# $ d r(g(t)) = 2e2t , −et 17. dt

1 0 −1

19. (t) = 4 − 4t, 16 − 32t # $ 21. 212 23. 0, 0 3 , 124

−2 −4

25. 1, 2 27. (ln 4)i + 56 3 j $ # 29. r(t) = −t 2 + t + 3, 2t 2 + 1 # $ 31. r(t) = 0, t 2 + c1 t + c2 ; with initial conditions, # $ r(t) = 1, t 2 − 6t + 10 # $ 33. r(3) = 45 4 , 5 35. r(t) = (t − 1)v + w

r = 3 + 2sin θ

3

37. r(t) = e2t c

−3

−2

−1

0

1

2

3

π 33. A = 16 35. e − 1e Note: One needs to double the integral from − π2 to π2 in order to account for both sides of the graph. 2 37. A = 3πa 2

# $ 41. √−2 , √5 29 29 %−→ % √ −→ % % 45. P Q = −4, 1 ; %P Q% = 17

39. 21, − 25 and −19, 31 2 v+ 5 w 43. i = 11 11 # $ 3 3 √ √ 47. , − 49. β = 32 2 2 y 51. v1 + v2 + v3

Chapter 11 Review 1. (a), (c)

5. c (θ) = (cos (θ + π) , sin (θ + π)) 9. y = − x4 + 37 4

 dy  13. dx 

t=3

3 = 14

7. c(t) = (1 + 2t, 3 + 4t)

11. y = 8 2 + 3−x 2 (3−x)

 dy  15. dx 

t=0

= cos2020 e

v2 v3

4

v1 + v2

3. c(t) =(1 + 2 cos t, 1 + 2 sin t). The intersection points with the √ y-axis are 0, 1 ± 3 . The intersection points with the x-axis are  √  1 ± 3, 0 .

4

v1 −2

x 3

√ 2  ; F1  = 980 N 55. F1  = 2F 3 √ 57. e − 4f  = 13 # $ 59. r (t) = −1, −2t −3 61. r (0) = 2, 0

17. (0, 1), (π, 2), (0.13, 0.40), and (1.41, 1.60)

d et 1, t = et 1, 1 + t 63. dt

19. x(t) = −2t 3 + 4t 2 − 1, y(t) = 2t 3 − 8t 2 + 6t − 1 √ √ 21. ds dt = 3 + 2(cos t − sin t); maximal speed: 3 + 2 2 √ 23. s = 2

d (6r (t) − 4r (t))| 65. dt 1 2 t=3 = 0, −8 $ !3# 2 dt = 27, 9 67. 0 4t + 3, t $ # 69. (3, 3) 71. r(t) = 2t 2 − 83 t 3 + t, t 4 − 16 t 3 + 1

ANSWERS TO ODD-NUMBERED EXERCISES

Vertical trace: z = (12 − 3a) − 4y and z = −3x + (12 − 4a) in the planes x = a, andy = a, respectively

73. v0 ≈ 67.279 m/s # $ −1 , √1 75. T(π ) = √ 2

A99

2

23.

z

Chapter 12 Section 12.1 Preliminary Questions 1. Same shape, but located in parallel planes 2. The parabola z = x 2 in the xz-plane 3. Not possible 4. The vertical lines x = c with distance of 1 unit between adjacent lines 5. In the contour map of g(x, y) = 2x, the distance between two adjacent vertical lines is 12 .

y x

The horizontal traces are ellipses for c > 0. The vertical trace in the plane x = a is the parabola z = a 2 + 4y 2 . The vertical trace in the plane y = a is the parabola z = x 2 + 4a 2 .

Section 12.1 Exercises 1. f (2, 2) = 18, f (−1, 4) = −5 3. h(3, 8, 2) = 6;h(3, − 2, − 6) = − 16 5. The domain is the entire xy-plane. & ' 9. D = (y, z) : z  = −y 2 7. y y = 4x 2

25.

z

z y x

x

y z = −y 2

z + y 2 = 0

11.

R

The horizontal traces in the plane z = c, |c| ≤ 1, are the lines x − y = sin−1 c + 2kπ and x − y = π − sin−1 c + 2kπ, for integer k The vertical trace in the plane x = a is z = sin (a − y). The vertical trace in the plane y = a is z = sin (x − a). 27. m = 1 : m = 2 :

m=1

I

IR≥0

13. 15. 17. 19. 21.

m=2

4

4

2

2

0

0

−2

−2

−4

Domain: entire (x, y, z)−space; range: entire real line Domain: {(r, s, t) : |rst| ≤ 4} ; range: {w : 0 ≤ w ≤ 4} f ↔ (B), g ↔ (A) (a) D (b) C (c) E (d) B (e) A (f) F z

−4 −4 −2

29.

0

2

2

3

6 5

12

4 3 2 3 4

1

y

x

Horizontal trace: 3x + 4y = 12 − c in the plane z = c

−3 −2 −1

0

1

4

−4 −2

0

2

4

A100

ANSWERS TO ODD-NUMBERED EXERCISES

31.

33.

y 1

4

3.

2

5.

0 0.5

7.

−4 −4

x 0.2 0.4 0.6 0.8

−2

0

2

4

9.

1

11. −0.5

lim

tan x cos y = 1

(x, y)→( π 4 , 0)

ex −e−y x+y (x, y)→(1, 1)

2

lim

= 12 (e − e−1 )

lim

(g(x, y) − 2f (x, y)) = 1

lim

2 ef (x, y) −g(x, y) = e2

(x, y)→(2, 5) (x, y)→(2, 5)

13. No; the limit along the x-axis and the limit along the y-axis are different.   1 17. lim (x 2 − 16) cos =0 (x, y)→(4, 0) (x − 4)2 + y 2

−1

35.

(xy − 3x 2 y 3 ) = 10

2

−2

0

lim

(x, y)→(2, −1)

19.

4 2

21.

0 −2 −4 −4 −2

0

2

4

37. m = 6 : f (x , y) = 2x + 6y + 6 m = 3 : f (x , y) = x + 3y + 3 39. (a) Only at (A) (b) Only at (C) (c) West 41. Average ROC from B to C = 0.000625 kg/m3 · ppt 43. At point A 45. Average ROC from A to B ≈ 0.0737 , average ROC from A to C ≈ 0.0457 47. C 540

B

lim

(z, w)→(−2, 1)

z4 cos(πw) = −16e ez+w

lim

√y−2

lim

tan(x 2 + y 2 )tan−1

lim

2 2 √ x +y

=0

x 2 −4 √ 1 = 15 23. lim (x, y)→(3, 4) x 2 +y 2 25. lim ex−y ln(x − y) = e4 ln(4) (x, y)→(1, −3) (x 2 y 3 + 4xy) = −48 27. lim (x, y)→(−3, −2)

29. 31. 35.

(x, y)→(4, 2)

(x, y)→(0, 0) (x, y)→(0, 0)

lim

(x, y)→Q

x 2 +y 2 +1−1



1



x 2 +y 2

=0

=2

g(x, y) = 4

37. Yes     41. (b) f 10−1 , 10−2 = 12 , f 10−5 , 10−10 = 12 ,  −20  f 10 , 10−40 = 12

500

Section 12.3 Preliminary Questions

i ii

400

D

iii A Contour interval = 20 m

0

1

2 km

49. f (r, θ ) = cos θ ; the level curves are θ = ±cos−1 (c)for |c| < 1, c = 0 ; the y−axis for c = 0; the positive x−axis for c = 1; the negative x−axis for c = −1.

Section 12.2 Preliminary Questions 1. D ∗ (p, r) consists of all points in D(p , r) other than p itself. 2. f (2, 3) = 27 3. All three statements are true 4. lim f (x, y) does not exist. (x, y)→(0, 0)

Section 12.2 Exercises 1.

lim

(x, y)→(1, 2)

(x 2 + y) = 3

∂ (x 2 y 2 ) = 2xy 2 1. ∂x 2. In this case, the Constant Multiple Rule can be used. In the second part, since y appears in both the numerator and the denominator, the Quotient Rule is preferred. 3. (a), (c) 4. fx = 0 5. (a), (d)

Section 12.3 Exercises ∂ y = x 3. ∂y 5. fz (2, 3, 1) = 6 z+y (x+y)2 7. m = 10 9. fx (A) ≈ 8, fy (A) ≈ −16.7

11. NW

∂ (x 2 + y 2 ) = 2x, ∂ (x 2 + y 2 ) = 2y 13. ∂x ∂y ∂ 4 −2 15. ∂x (x y + xy ) = 4x 3 y + y −2 , ∂ 4 −2 4 −3 ∂y (x y + xy ) = x − 2xy

 

 

x = −x y y2   ∂ 19. ∂x 9 − x 2 − y 2 = √ −x2 ∂ 17. ∂x

√ −y

x y

∂ = y1 , ∂y

9−x −y

, ∂ 2 ∂y

  9 − x2 − y2 =

9−x 2 −y 2 ∂ (sin x sin y) = sin y cos x, ∂ (sin x sin y) = sin x cos y 21. ∂x ∂y

ANSWERS TO ODD-NUMBERED EXERCISES

    ∂ tan x = 1  , ∂ tan x = −x  23. ∂x y y ∂y x 2 y cos y y 2 cos2 yx ∂ ln(x 2 + y 2 ) = 25. ∂x

27. 29. 31. 33. 35. 37. 39. ∂Q ∂t

41. 45. 47. 49. 51. 55. 59. 63. 67. 71. 77.

2x , ∂ ln(x 2 + y 2 ) = 2y x 2 +y 2 ∂y x 2 +y 2 ∂ er+s = er+s , ∂ er+s = er+s ∂r ∂s ∂ exy = yexy , ∂ exy = xexy ∂x ∂y ∂z = −2xe−x 2 −y 2 , ∂z = −2ye−x 2 −y 2 ∂y ∂y ∂U = −e−rt , ∂U = −e−rt (rt+1) ∂t ∂r r2 ∂ sinh(x 2 y) = 2xy cosh(x 2 y), ∂ sinh(x 2 y) = x 2 cosh(x 2 y) ∂x ∂y ∂w = y 2 z3 , ∂w = 2xz3 y, ∂w = 3xy 2 z2 ∂x ∂y ∂z ∂Q ∂Q M − L t −Lt/M e , ∂M = L(Lt −3 M) e−Lt/M , ∂L = M M2 2 = − L 2 e−Lt/M M fx (1, 2) = −164 43. gu (1, 2) = ln 3 + 13

N = 2865.058, N ≈ −217.74 ∂I ; 1.66 (a) I (95, 50) ≈ 73.1913 (b) ∂T A 1-cm increase in r ∂W = − 1 e−E/kT , ∂W = E e−E/kT 2 ∂E kT ∂T

kT ∂2f ∂2f (a), (b) 57. = 6y, 2 = −72xy 2 ∂x 2 ∂y hvv = 32u 3 61. fyy (2, 3) = − 49 (u + 4v) fxyxzy = 0 65. fuuv = 2v sin(u + v 2 ) Frst = 0 69. Fuuθ = cosh(uv + θ 2 ) · 2θv 2 gxyz = 2 3xyz 73. f (x, y) = x 2 y 5/2 (x +y 2 +z2 ) B = A2

Section 12.4 Preliminary Questions 1. L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)  2. f (x, y) − L(x, y) = ∈ (x, y) (x − a)2 + (y − b)2 3. (b) 4. f (2, 3, 1) ≈ 8.7 5. f ≈ −0.1 6. Criterion for Differentiability

Section 12.4 Exercises

Section 12.5 Preliminary Questions 1. (b) 3, 4 2. False 3. ∇f points in the direction of maximum rate of increase of f and is normal to the level curve of f . 4. (b) NW and (c) SE √ 5. 3 2

Section 12.5 Exercises # $

# $ 1. (a) ∇f = y 2 , 2xy , c (t) = t, 3t 2   d (f (c(t))) d (f (c(t))) (b) dt = 4; dt = −4   t=1 t=−1 3. A: zero, B: negative, C: positive, D: zero 5. ∇f = − sin(x 2 + y) 2x, 1 $ # 7. ∇h = yz−3 , xz−3 , − 3xyz−4   d f (c(t)) d f (c(t)) 9. dt = −7 11. dt = −3   t=0 t=0   d f (c(t))  13. dt = 5 cos 1 ≈ 2.702  t=0   d f (c(t))  15. dt = −56  t=4   d f (c(t))  17. dt = −1 + π8 ≈ 1.546  t=π/4  d g(c(t)) =0 19. dt  t=1   √ 21. Du f (1, 2) = 8.8 23. Du f 16 , 3 = 39 4 2 √ 27. Du f (1, 0) = 6 13 −50 29. Du f (1, 2, 0) = − √1 31. Du f (3, 2) = √ 3 13 5 33. Du f (P ) = − e3 ≈ −49.47 √ 25. Du f (3, 4) = 72902

35. f is increasing at P in the direction of v. √

37. Du f (P ) = 26 39. 6, 2, − 4     41. √4 , √9 , − √2 and − √4 , − √9 , √2 17

17

(a) f (x, y) = −16 + 4x + 12y f (2.01, 1.02) ≈ 4.28; f (1.97, 1.01) ≈ 4 f ≈ 3.56 17. f (0.01, − 0.02) ≈ 0.98 L(x, y, z) = −8.66025 + 0.721688x + 0.721688y + 3.4641z 5.07 23. 8.44 25. 4.998 27. 3.945 z = 3x − 3y + 13 31. I ≈ 0.5644 (b) H ≈ 0.022m (b) 6% (c) 1% error in r (a) $7.10 (b) $28.85, $57.69 (c) −$74.24 Maximum error in V is about 8.948 m.

17

17

17

17

43. 9x + 10y + 5z = 33 45. 0.5217x + 0.7826y − 1.2375z = −5.309 47. 49. f (x, y, z) = x 2 + y + 2z y

1. z = −34 − 20x + 16y 3. z = 5x + 10y − 14 5. z = 8x − 2y − 13 7. z = 4r − 5s + 2     12 12 1 1 1 9. z = 45 + 12 25 ln 2 − 25 x + 25 y 11. − 4 , 8 , 8 13. (b) 15. 19. 21. 29. 33. 35. 37. 39.

A101

4 2 x −4

−2

2

4

−2 −4

51. f (x, y, z) = xz + y 2 55. f ≈ 0.08 57. (a) 34, 18, 0 # $ (b) 2 + √32 t, 2 + √16 t, 8 − √8 t ; ≈ 4.58 s 21

21

21

61. x = 1 − 4t, y = 2 + 26t, z = 1 − 25t 73. y = 1 − ln(cos2 x)

A102

ANSWERS TO ODD-NUMBERED EXERCISES

Section 12.6 Preliminary Questions ∂f 1. (a) ∂f ∂x and ∂y (b) u and v 2. (a) 3. f (u, v)| (r,s)=(1,1) = e2

4. (b)

5. (c)

6. No

P

Q

−10

Section 12.6 Exercises 1. (b) (c) 3. 5. 7. 11. 13.

3 ∂f 2 2 ∂f 3 (a) ∂f ∂x = 2xy , ∂y = 3x y , ∂z = 4z ∂x = 2s, ∂y = 2t 2 , ∂z = 2st ∂s ∂s ∂s ∂f 6 t 6 + 8s 7 t 4 = 7s ∂s ∂f 2 ∂f 3 3 ∂s = 6rs , ∂r = 2s + 4r ∂g ∂g ∂u = −10 sin(10u − 20v), ∂v = 20 sin(10u − 20v) ∂F = xex 2 +xy 9. ∂h = 0 ∂y ∂t2   ∂f  ∂f  ∂u (u,v)=(−1,−1) = 1, ∂v (u,v)=(−1,−1) = −2   ∂g  ∂f  1   ∂θ (r,θ)= 2√2, π/4 = 6 15. ∂v (u,v)=(0,1) = 2 cos 2

19 √ 17. (b) ∂f ∂t = 2 7 23. (a) Fx = z2 + y, Fy = 2yz + x, Fz = 2xz + y 2 ∂z = − z2 + y , ∂z = − 2yz + x (b) ∂x 2 2 ∂y 2xz + y

2xz + y

∂z = − 2xy + z2 25. ∂x 2xz + y 2

29. ∂w ∂y =

w

∂z = − xexy + 1 27. ∂y x cos(xz)

−y(w2 + x 2 )



(w2

+

  33. ∇ 1r = − 13 r r

2 y2)

+

(w2

2

+ x2)

2

 ; at (1, 1, 1), ∂w = − 1 ∂y 2

∂z = x − 6 35. (c) ∂x z+4

nR ∂V 37. ∂P ∂T = − V − nb , ∂P =

nbV 3 − V 4 P V 3 + 2an3 b − an2 V

Section 12.7 Preliminary Questions 1. f has a local (and global) min at (0, 0); g has a saddle point at (0, 0). 2. 3 −3

1 −1

0

R

−1

−3

1 3 1

Point R is a saddle point.

−3 − 1

S

0

1

10

−6

3

Point S is neither a local extremum nor a saddle point.

6 −2

2 0

Point P is a local minimum and point Q is a local maximum. 3. Statement (a)

Section 12.7 Exercises

 √ √  1. (b) P1 = (0, 0) is a saddle point, P2 = 2 2, 2 and  √ √  P3 = −2 2, − 2 are local minima; absolute minimum value of f is −4.     13 and − 1 , 1 local minima 3. (0, 0) saddle point, 13 , − 64 32 4 2   5. (c) (0, 0), (1, 0), and (0, −1) saddle points, 13 , − 31 local minimum.   7. − 23 , − 13 local minimum   9. (−2, −1) local maximum, 53 , 56 saddle point    √  11. 0, ± 2 saddle points, 23 , 0 local maximum,   − 23 , 0 local minimum 13. (0, 0) saddle point, (1, 1) and (−1, −1) local minima     15. (0, 0) saddle point, √1 , √1 and − √1 , − √1 local 2   2  2  2 maximum, √1 , − √1 and − √1 , √1 local minimum 2 2 2 2   17. Critical points are j π, kπ + π2 , for j ,k even: saddle points j ,k odd: local maxima j even, k odd: local minima j odd, k even: saddle points     19. 1, 12 local maximum 21. 32 , − 12 saddle point   23. − 16 , − 17 18 local minimum 27. x = y = 0.27788 local minimum 29. Global maximum 2, global minimum 0 1 31. Global maximum 1, global minimum 35 35. Maximum value 13 37. Global minimum f (0, 1) = −2 , global maximum f (1, 0) = 1 39. Global maximum 3, global minimum 0 41. Global minimum f (1, 1) = −1 , global maximum f (1, 0) = f (0, 1) = 1 43. Global minimum f (1, 0) = f (−1, 0) = −0.368 , global maximum f (0, − 1) = f (0, 1) = 1.472 45. Maximum volume 34 49. (a) No. In the box B with minimal surface area, z is smaller than √ 3 V , which is the side of a cube with volume V .

ANSWERS TO ODD-NUMBERED EXERCISES

√

(b) Width: x = (2V )1/3 ; length: y = (2V )1/3 ;  1/3 height: z = V4

(b) f (3, 1) = 32 , f (−5, − 3) = −2 z 3.

(c)

A103

  − 53 , 1

51. f (x) = 1.9629x − 1.5519

Section 12.8 Preliminary Questions 1. Statement (b) 2. f had a local maximum 2, under the constraint, at A; f (B) is neither a local minimum nor a local maximum of f . −6 −2 2 6 3. (a) fA, gA

g (x, y) = 0

A

x

Vertical and horizontal traces: the line z = (c2 + 1) − y in the plane x = c, the parabola z = x 2 − c + 1 in the plane y = c. 5. (a) Graph (B)

B

(b) Graph (C)

(c) Graph (D) (d) Graph (A)

7. (a) Parallel lines 4x − y = ln c, c > 0, in the xy-plane

C D

E

y

(b) Parallel lines 4x − y = ec in the xy-plane (c) Hyperbolas 3x 2 − 4y 2 = c in the xy-plane (d) Parabolas x = c − y 2 in the xy-plane

2 −2 −6

6

Contour plot of f (x, y) (contour interval 2)

9.

(b) Global minimum −4, global maximum 6

17. (a) h =

√2 ≈ 0.6, r = 3π

√1 ≈ 0.43 3π

21. fxxyz = − cos(x + z) 23. z = 33x + 8y − 42 25. Estimate, 12.146; calculator value to three places, 11.996.

√ (b) hr = 2

(a+b)

31. r = 3, h = 6 33. x+ y + z = 3 39. √−6 , √−3 , √30 41. (−1, 0, 2) 105

105

43. Minimum 138 11 ≈ 12.545 , no maximum value 47. (b) λ = 2pcp 1 2

Chapter 12 Review 1. (a)

(2x + y)e−x+y = −e−4

19. fx = e−x−y (y cos(xy) − sin(xy)) fy = e−x−y (x cos(yx) − sin(yx))

(c) There is no cone of volume 1 and maximal surface area.   108 a a bb a a bb , 25. 23. 19. (8, − 2) 21. 48 a+b a+b 97 97

105

lim

(x,y)→(1,−3)

17. fx = 2, fy = 2y

(c) Critical points (−1, −2) and (1, 2) Maximum 10, √ minimum −10 √ Maximum 4 2 , minimum −4 2 Minimum 36 13 , no maximum value

7. Maximum 83 , minimum − 83 √ 9. Maximum 2 , minimum 1 11. Maximum 3.7, minimum −3.7 13. No maximum and minimum values 15. (−1, e−1) 

(xy + y 2 ) = 6

11. The limit does not exist. 13.

Section 12.8 Exercises 1. (d) 3. 5.

lim

(x,y)→(1,−3)

(a+b)

27. Statements (ii) and (iv) are true.  d f (c(t)) = 3 + 4e4 ≈ 221.4 29. dt  t=2  d f (c(t)) 31. dt = 4e − e3e ≈ −3469.3  t=1

54 33. Du f (3, − 1) = − √ √

35. Du f (P ) = − 52e

5

# $ 37. √1 , √1 , 0

2 2 ∂f 2 2 3 3 41. ∂s = 3s t + 4st + t − 2st + 6s 2 t 2 ∂f 2 2 3 3 2 2 ∂t = 4s t + 3st + s + 4s t − 3s t ∂z = − ez − 1 45. ∂x xez + ey

47. (0, 0) saddle point, (1, 1) and (−1, −1) local minima   49. 12 , 12 saddle point 53. Global maximum f (2, 4) = 10 , global minimum f (−2, 4) = −18

y

55. Maximum √26 , minimum − √26 13

−3

x

13

12 , minimum − √ 12 57. Maximum √ 3

3

59. f (0.8, 0.52, − 0.32) = 0.88 and f (−0.13, 0.15, 0.99) = 3.14  1/3  1/3 V V 61. r = 2π , h = 2 2π

ANSWERS TO PREPARING FOR THE AP EXAM Chapter 2 Solutions Multiple Choice Questions 1. B

3. C

5. E

7. E

9. C

11. C

13. D

15. B

17. D

19. E

π f ( 3π 2 )−f ( 2 ) 3π π 2 −2

1. (a)

=

π

x→∞

x→∞

lim sin x = 0. This means the line y = 0 is a horizontal asympx→∞ x tote. 3. (a) Since −5 ≤ f (x) ≤ 10, if x > 0 then −5x ≤ xf (x) ≤ 10x. Thus by the Squeeze Theorem lim xf (x) = 0. Next, if

Free Response Questions −1 − ( π1 ) ( 3π 2 2 )

sin x 1 (d) We know −1 ≤ sin x ≤ 1, so if x > 0, then −1 x ≤ x ≤ x, −1 1 and since lim x = 0 = lim x , the Squeeze Theorem implies

x→0+

=

−1 2 π ( 3π

+ π2 ) =

−8 3π 2

(b) lim f (x) = 1 x→0

(c) No, lim f (x) = 1, so neither the left-hand limit nor the right x→0

hand limit is infinite, which is needed for the graph to have a vertical asymptote.

x < 0, then −5x ≥ xf (x) ≥ 10x. Applying the Squeeze Theorem again, lim xf (x) = 0. Thus lim xf (x) = lim g(x) = 0. x→0−

x→0

x→0

Checking the functional value, we have g(0) = 0 · 3 = 0. Thus lim g(x) = g(0), so g is continuous at x = 0.

x→0

(b) No. lim

x→0

g(x)−0 x−0

= lim

x→0

xf (x) x

= lim f (x), which does not x→0

exist.

Chapter 3 Solutions Multiple Choice Questions 1. A

3. B

5. D

7. D

9. E

11. E

13. C

15. A

17. A

19. D

Free Response Questions 1. (a) The line through (3, −7) with slope −2 has equation y = −7 − 2(x − 3) = −2x − 1. To see where this line meets y = x 2 , set x 2 = −2x − 1; we get x = −1. The point (−1, 1) is on the graph of y = x 2 , and the derivative is 2x, so the slope of the tangent line is 2(−1) = −2. Thus y = −7 − 2(x − 3) is tangent to y = x 2 at (−1, 1). (b) Let the slope of the line be m. Then we have two equations to solve: first, as we did in (a), set x 2 = −7 + m(x − 3). Next, at the solution to that equation, we will have m = 2x. Thus we need to solve x 2 = −7 + 2x(x − 3), or x 2 − 6x − 7 = 0. That is, (x + 1)(x − 7) = 0, so x = −1 or 7. The x = −1 confirms A104

our solution to (a). The slope we want is m = 14, so the line is y = −7 + 14(x − 3). (c) No. The x-coordinates of the points on the graph of y = x 2 must satisfy the quadratic x 2 − 6x − 7 = 0, which has only two solutions. 3. (a) The volume of sand in the box is V = (20)(40)(y), where y is the depth of the sand in the box. Thus −300 = dV dt = dy dy 3 800 dt , so dt = − 8 . The depth of the sand is decreasing at the rate of 38 inch per minute. (b) (i) The area of the circular base is A = π r 2 , so dA dt = dr 2π r dt = 2π(8)(.75) = 12π . The area is increasing at the rate of 12π square inches per minute. Note that the diameter is twice the radius. π dr 2 dh (ii) dV dt = 3 [(2r dt )h + r dt ], and the sand is coming in at 300 cubic inches per minute, so 300 = π3 [(2(8)(.75))23 + (8)2 dh dt ], dh 1 900 or dt = 64 ( π − 276) inches per minute.

ANSWERS TO PREPARING FOR THE AP EXAM

A105

Chapter 4 Solutions Multiple Choice Questions 1. B

3. B

5. C

7. C

9. D

11. C

13. E

15. E

17. C

19. C

Intermediate Value Theorem says there is a rectangle with area exactly 5000 square yards.

Free Response Questions 1. (a) No. There are various justifications. For example, dx dt < > 0 when y < 0 since the runner is going 0 when y > 0 and dx dt counterclockwise. Or, dx dt = 0 when y = 0, and since the runner dx is moving, dt cannot be constantly zero. (b) Let P = (x, y) be a point in the first quadrant on the ellipse. Then construct the rectangle R with vertices (x, y), (−x, y), (x, −y), and√ (−x, −y). The area of R is A = 4xy. Since √ 2 y > 0, y = 50000−10x , so A = 2x 50000 − 10x 2 . A (x) = 2 2 √100000−4x = 0 when x = 50 (remember x > 0) and changes 50000−10x 2

sign from plus to minus. Thus A has a maximum at x = 50. √ A(50) = A = 5000 10. Next, A(1) < 5000 < A(50), so the

3. (a) If k = 30, then f (0) = 30. Since f (x) is a cubic and the coefficient of x 3 > 0, we know f (x) will be negative for some negative values of x. Experimenting, we find f (−3) = −15. Thus there is a c in (−3, 0) with f (c) = 0. (b) We have f  (x) = 6x 2 − 6x − 12 = 6(x + 1)(x − 2). Thus f (x) has a local maximum x = −1 and a local minimum at x = 2. With k = 30, f (x) is increasing on (−∞, −1] and f (−1) = 37. So f (x) = 0 has exactly one solution in (−∞, −1]. f (x) is decreasing on [−1, 2], and f (2) = 10, so there is no solution to f (x) = 0 in [−1, 2]. f (x) now is increasing for x > 2, so there is no solution to f (x) = 0 in [2, ∞). (c) We want the graph to intersect the x-axis exactly once, so we want either (i) the local maximum to be less than 0 or (ii) the local minimum to be greater than 0. For (i), f (−1) = 7 + k so k < −7. For (ii), f (2) = −20 + k, so k > 20. (Note, for k = −7 or 20, there are exactly two solutions.)

Chapter 5 Solutions Multiple Choice Questions

(d) When t =

1. B

3. E

5. C

7. A

9. C

11. C

13. C

15. E

17. C

19. E

√ − 2 2

v(t) =

1 2

−

√ 2 2

< 0 and a(t) = − cos t =

< 0. v(t) is negative and decreasing, so |v(t)|, or the speed, is increasing.

Free Response Questions 1. (a) If v(t) > 0, then x(t) will be increasing, so set sin t > 0. Solution is 0 ≤ t < π6 and 5π 6 < t ≤ 2π .  2π 1 (b) 3 + 0 ( 2 − sin t)dt = 3 + π π  5π  2π (c) 06 ( 12 − sin t)dt + π 6 −( 12 − sin t)dt + 5π ( 12 − 6 6 √ sin t)dt = 2 3 + π3

π 4,

1 2

−

3. (a) g has a local maximum when g  (x) = f (x) changes from positive to negative; this happens when x = 4. (b) The maximum occurs either at a local maximum, or at an end point. g(4) = 12 · 2 · 4 = 4, the area of the triangle; g decreases from 4 to 5, so we only  −3 2 need to check g(−3) = 2 f (x)dx = − −3 f (x)dx = 2 0 −( −3 f (x)dx + 0 f (x)dx) = −(−9 + 4) = 5. The maximum value of g(x) is 5. (c) The graph of g is concave up when g  = f is increasing, that is on (−3, 2).

Chapter 6 Solutions Multiple Choice Questions

Free Response Questions

1. E

3. C

5. B

7. E

9. C

11. D

13. C

15. A

17. D

19. E

1 1. (a) average acceleration = 10 10 1 2  10 (6t − t ) = −4 (ft/sec)/sec 0

 10 0

(6 − 2t) dt =

A106

ANSWERS TO PREPARING FOR THE AP EXAM

 1 10 2 (b) average velocity = 10 0 (6t − t + 7) dt =  10 1 t3 11 2 10 (3t − 3 + 7t)0 = 3 ft/sec (c) Note that v(t) ≥ 0 for 0 ≤ t ≤ 7, v(t) ≤ 0 for 7 ≤ t ≤ 10.   1 10  average speed = 10 0 6t − t 2 + 7 dt =    7 1 2 + 7t)dt − 10 (6t − t 2 + 7t)dt = (6t − t 0 7 10  7 10  3  1 t 2− 2 − t 3 + 7t) ( 3t = 38 + 7t) − (3t   10 3 3 3 ft/sec 0

7

3. (a) Let h be the depth of water in the bowl. Then the  −6+h amount of water is given by V (h) = −6 π(36 − y 2 )dy. Thus dV dV dh 2 dh dt = dh dt = π(36 − (−6 + h) ) dt . When h = 2, we have dh dh 4 4 = π 20 dt , so dt = 20π ft/min.  −1 cubic (b) The volume of water is −6 π(36 − y 2 )dy = 325π 3 feet. Water came in at 4 cubic feet per minute, so the time is 325π 12 minutes.

Chapter 7 Solutions Multiple Choice Questions 1. B

3. C

5. C

7. B

9. C

11. A

13. B

15. D

17. D

19. B

Free Response Questions 1. (a) u = sin−1 x ⇒ du = √ 1 2 dx and dv = dx ⇒ 1−x   v=x so sin−1 x dx = xsin−1 x − √ x 2 dx = 1−x √ xsin−1 x + 1 − x 2 + C 1 √  (b) ( xsin−1 x + 1 − x 2 ) = π2 − 1 0

(c) The area under the curve y = sin−1 x in the first quadrant plus the area to the left of this curve in the first quadrant forms a rectangle of height π2 and base 1, so total area is π2 . The area to

the left of the curve, when viewed from the y-axis, is under the π graph x = sin y, and so this area is 02 sin y dy. Thus total area is  1 −1  π2 π 2 = area to left + area under = 0 sin x dx + 0 sin y dy. 3. (a) Let g(x) = x1 . Then for x ≥ 2, x1 = √1 2 < √ 12 x x −1 w ∞ and 2 x1 dx = lim 2 x1 dx = lim (ln(w) − ln(2)) = ∞. w→∞  ∞ w→∞ Since f (x) > g(x) > 0, and 2 g(x)dx diverges, so does ∞ 2 f (x)dx. w ∞ 3 (b) 2 π f (x)2 dx = lim 2 π x 21−1 dx. Let g(x) = x 2 . w→∞

3

3 2 = 0, so for large values of x, x 2 < (x 2 − 1), Then lim xx2 −1 x→∞ ∞ or 13 > x 21−1 > 0. The integral 2 13 dx converges by the x2 x2 ∞ ∞ p-test, hence so does 2 x 21−1 dx and then so does 2 x 2π−1 dx.

Chapter 8 Solutions Multiple Choice Questions 1. D

3. E

5. D

7. E

9. E

11. E

13. C

15. A

17. D

19. E

Free Response Questions 1. (a) P (x) = 3 + 6x + 2x 2 + 2x 3 (b) First, g(0) = f (0) = 3. Next, g  (x) = f  (3x) · 3, so Next, g  (x) = f  (3x) · 9, so g  (0) = f  (0) · 3 = 18.   g (0) = f (0) · 9 = 36. Finally, g  (x) = f  (3x) · 27, so g  (0) = f  (0) · 27 = 324. The Taylor polynomial for 324 3 2 2 3 g is 3 + 18x + 36 2! x + 3! x = 3 + 18x + 18x + 54x = 2 3 3 + 6(3x) + 2(3x) + 2(3x) = P (3x). (c) First, h(0) = 0. Next, h (x) = f (x) + xf  (x), so h (0) = f (0) = 3. Next, h (x) = f  (x) + f  (x) + xf  (x), so h (0) = 2f  (0) = 12. Finally, h (x) = 2f  (x) + f  (x) + xf  (x), so h (0) = 3f  (0) = 12. Thus the third Maclaurin polynomial for

h is 3x + 6x 2 + 2x 3 = x(3 + 6x + 2x 2 ), which is x times the Maclaurin polynomial for f of degree two.  20 3. (a) 0 50(20 − y)10dy = 100000 pounds D (b) The force on the plate below L is 0 50(20 − y)10dy D which is half the force, so set 0 50(20 − y)10dy = 50000, or D D y2  D2 (20 − y)dy = 100. Thus 20y − 0 2  = 20D − 2 = 100, √

0

. Must select the or D 2 − 40D + 200 = 0. D = 40± 1600−800 √2 root between 0 and 20. D = 20 − 10 2   20 2 20 (c) A(x) = D 50(x − y)10dy = 500(xy − y2 ) = 2

D

500[(20x − 200) − (Dx − D2 ). Thus A (x) = 500(20 − D).  D 2 D B(x) = 0 50(x − y)10dy = 500(xy − y2 ) = 2

0

500(Dx − D2 ). √ Thus B  (x) = 500D. Since 2 > 1, D < 10, so 20–D > 10 > D. Thus A (x) > B  (x).

ANSWERS TO PREPARING FOR THE AP EXAM

A107

Chapter 9 Solutions Multiple Choice Questions 1. C

3. B

5. D

7. A

9. C

11. C

13. D

15. C

17. B

19. C

−6t − 2) = K e−6t + 1 for K = 0. FiNext, w = −1 3 3 6 (K2 e 3 nally, note that w = 13 is a constant solution to dw = 2 − 6w, dt so the general solution is w = Ce−6t + 13 for all real numbers C. 1 3 (c) y = w1 = −6t 1 = Ce−6t +1 +3

Ce

Free Response Questions 1. (a) w =

1 y

means

dw dt

1 = − y12 dy dt = − y 2 (y)(6 − 2y) =

−( y6 − 2) = −(6w − 2) = 2 − 6w. dw = dt, so − 16 ln |2 − 6w| = t + C1 ; ln |2 − 6w| = (b) 2−6w −6t + C2 |2 − 6w| = e−6t+C2 = eC2 e−6t = K1 e−6t for K1 > 0 2 − 6w = K2 e−6t for K2 = 0

3. (a) Write

dy dx

= 2x(y 2 + 1), so

dy y 2 +1

= 2xdx. Integrating

we get arctan(y) = + C. So y = + C). π (b) Using x = 0, y = 1 we have C = 4 , so the solution is y = tan(x 2 + π4 ). Since the domain includes x = 0, we must π π 2 2 have − π2 < x 2 + π4 < π2 , or − 3π 4 < x < 4 ; we need x < 4 , x2

so the domain is −

√

π 2

tan(x 2

<x
1. 3. (a) We have for F that x  (t) = 2x(t), so x(t) = Ce2t and 3 = Ce−2 , thus C = 3e2 and x(t) = 3e2t+2 . Similarly, we have y(t) = 4e2t+2 . F (t) = 3e2t+2 , 4e2t+2 . In like manner, G(t) = 9e−3t , 12e−3t . (b) F (t) = G(t) means 3e2t+2 = 9e−3t , or e5t+2 = 3; 5t + 2 = ln 3; t = ln(3)−2 . We must also have 4e2t+2 = 12e−3t , or 5 . The particles are at the same point when t = ln(3)−2 . t = ln(3)−2 5 5

(c) F (t) = e2t+2 3, 4 , since the range of e2t+2 is all positive numbers, F visits all points of the form y = 43 x, x > 0. Similarly, G(t) = e−3t 9, 12 = 3e−3t 3, 4 the same set of points.  ∞ 2 2 (d) 0 (−27e−3t ) + (−36e−3t ) dt = ∞ ∞ (272 + 362 )e−6t dt = 0 272 + 362 e−3t dt = 0   √ B  2 2 272 +362 −3t  lim e  = − 27 +36 lim (e−3B − 1) = −3 9 B→∞ B→∞ 0 √ √ 27 · 3 + 36 · 4 = 3 9 + 16 = 15. Alternatively, G goes in a straight line from 9, 12 to 0, 0 in the limit, so the distance is 92 + 122 = 15.

REFERENCES

The online source MacTutor History of Mathematics Archive www-history.mcs. st-and.ac.uk has been a valuable source of historical information. Section 1.1 (EX 77) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 9.

Section 3.1 (EX 75) Problem suggested by Dennis DeTurck, University of Pennsylvania. Section 3.2 (EX 92) Problem suggested by Chris Bishop, SUNY Stony Brook. (EX 93) Problem suggested by Chris Bishop, SUNY Stony Brook.

Section 1.2

Section 3.4

(EX 25) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 9.

(PQ 2) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 25.

Section 1.7 (EXMP 4) Adapted from B. Waits and F. Demana, “The Calculator and Computer Pre-Calculus Project, “ in The Impact of Calculators on Mathematics Instruction, University of Houston, 1994. (EX 12) Adapted from B. Waits and F. Demana, “The Calculator and Computer Pre-Calculus Project, “ in The Impact of Calculators on Mathematics Instruction, University of Houston, 1994. Section 2.2 (EX 61) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, Note 28. Section 2.3 (EX 38) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, Note 28. Chapter 2 Review (EX 68) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, Note 28.

(EX 48) Karl J. Niklas and Brian J. Enquist, “Invariant Scaling Relationships for Interspecific Plant Biomass Production Rates and Body Size,” Proc. Natl. Acad. Sci. 98, no. 5:2922-2927 (February 27, 2001) Section 3.5 (EX 47) Adapted from a contribution by Jo Hoffacker, University of Georgia. (EX 48-49) Adapted from a contribution by Thomas M. Smith, University of Illinois at Chicago, and Cindy S. Smith, Plainfield High School. (EX 45) Adapted from Walter Meyer, Falling Raindrops, in Applications of Calculus, P. Straffin, ed., Mathematical Association of America, Washington, DC, 1993 (EX 52, 56) Problems suggested by Chris Bishop, SUNY Stony Brook. Section 3.11 (EX 32) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. (EX 34) Problem suggested by Kay Dundas. (EX 38, 44) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. A109

A110

R EF E RE NCE S

Chapter 3 Review (EX 81, 94, 119) Problems suggested by Chris Bishop, SUNY Stony Brook. Section 4.2 (MN p. 216) Adapted from “Stories about Maxima and Minima,” V. M. Tikhomirov, AMS, (1990). (MN p. 221) From Pierre Fermat, On Maxima and Minima and on Tangents, translated by D.J. Struik (ed.), A Source Book in Mathematics, 1200-1800, Princeton University Press, Princeton, NJ, 1986. Section 4.5 (EX 48, 77) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. Section 4.6 (EX 26-27) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. (EX 32) From Michael Helfgott, Thomas Simpson and Maxima and Minima, Convergence Magazine, published online by the Mathematical Association of America.

Mathematical Association of America, Washington, DC, 1993, p. 52. (EX 32-33) Adapted from E. Packel and S. Wagon, Animating Calculus, Springer-Verlag, New York, 1997, p. 79. Chapter 4 Review (EX 68) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. Section 5.1 (EX 3) Problem suggested by John Polhill, Bloomsburg University. Section 5.2 (FI&C 84) Problem suggested by Chris Bishop, SUNY Stony Brook. Section 5.4 (EX 40-41) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 102. (EX 42) Problem suggested by Dennis DeTurck, University of Pennsylvania. Section 5.5.

(EX 40) Problem suggested by John Haverhals, Bradley University. Source: Illinois Agrinews.

(EX 25-26) M. Newman and G. Eble, “Decline in Extinction Rates and Scale Invariance in the Fossil Record.” Paleobiology 25:434-439 (1999).

(EX 42) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993.

(EX 28) From H. Flanders, R. Korfhage, and J. Price, Calculus, Academic Press, New York, 1970.

(EX 66-68) Adapted from B. Noble, Applications of Undergraduate Mathematics in Engineering, Macmillan, New York, 1967. (EX 70) Adapted from Roger Johnson, “A Problem in Maxima and Minima,” American Mathematical Monthly, 35:187-188 (1928). Section 4.7 (EX 67) Adapted from Robert J. Bumcrot, “Some Subtleties in L’ Hôpital’s Rule,” in A Century of Calculus, Part II, Mathematical Association of America, Washington, DC, 1992.

Section 5.6 (EX 74) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 121. Section 6.1 (EX 48) Adapted from Tom Farmer and Fred Gass, “Miami University: An Alternative Calculus” in Priming the Calculus Pump, Thomas Tucker, ed., Mathematical Association of America, Washington, DC, 1990, Note 17.

(EX 28) Adapted from “Calculus for a Real and Complex World” by Frank Wattenberg, PWS Publishing, Boston, 1995.

(EX 61) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993.

Section 4.8

Section 6.3

(EX 20) Adapted from Calculus Problems for a New Century, Robert Fraga, ed.,

(EX 60, 62) Adapted from G. Alexanderson and L. Klosinski, “Some Surprising Volumes

RE FE RE N CE S

of Revolution, “ Two-Year College Mathematics Journal 6, 3:13-15 (1975). Section 7.1 (EX 56-58, 59, 60-62, 65) Problems suggested by Brian Bradie, Christopher Newport University. (EX 70) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. (FI&C 79) Adapted from J. L. Borman, “A Remark on Integration by Parts,” American Mathematical Monthly 51:32-33 (1944). Section 7.3 (EX 43-47, 51) Problems suggested by Brian Bradie, Christopher Newport University. (EX 62) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 118. Section 7.6

A111

Section 10.1 (EX 68) Adapted from G. Klambauer, Aspects of Calculus, Springer-Verlag, New York, 1986, p. 393. (EX 92) Adapted from Apostol and Mnatsakanian, “New Insights into Cycloidal Areas,” American Math Monthly, August-September 2009. Section 10.2 (EX 42) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 137. (EX 43) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 138. (FI&C 51) Adapted from George Andrews, “The Geometric Series in Calculus,” American Mathematical Monthly 105, 1:36-40 (1998)

(EX 81) Problem suggested by Chris Bishop, SUNY Stony Brook.

(FI&C 54) Adapted from Larry E. Knop, “Cantor’s Disappearing Table,” The College Mathematics Journal 16, 5:398-399 (1985).

Section 7.8

Section 10.4

See R. Courant and F. John, Introduction to Calculus and Analysis, Vol. 1, Springer-Verlag, New York, 1989.

(EX 33) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993, p. 145.

Section 8.1 (FI&C 52) Adapted from G. Klambauer, Aspects of Calculus, Springer-Verlag, New York, 1986, Ch 6. Section 9.1 (EX 55) Adapted from E. Batschelet, Introduction to Mathematics for Life Scientists, Springer-Verlag, New York, 1979. (EX 57) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. (EX 58, 63) Adapted from M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover, New York, 1985.

Section 11.2 (FI&C 35) Adapted from Richard Courant and Fritz John, Differential and Integral Calculus, Wiley-Interscience, New York, 1965. Section 11.3 (EX 56) Adapted from Calculus Problems for a New Century, Robert Fraga, ed., Mathematical Association of America, Washington, DC, 1993. Section 12.8 (EX 42) Adapted from C. Henry Edwards, “Ladders, Moats, and Lagrange Multipliers,” Mathematica Journal 4, Issue 1 (Winter 1994).

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A113

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INDEX

abscissa, see x-coordinate absolute convergence, 569 absolute (global) extreme values, 735 absolute maximum and absolute minimum, 215 absolute value function: integral of, 314 nondifferentiability of, 136 absolute value of a real number, 2 absolutely convergent improper integral, 447 absorption spectra, 537 absorption wavelength of hydrogen atom, 537 acceleration, 161 addition formulas, 30 additivity: for adjacent intervals, 304 of moments, 481, 484, 485 adjacent intervals, 304 Agnesi, Maria, 149 air resistance, 525, 526, 530 algebra: vector, 641–646 algebraic functions, 22 alternating harmonic series, 572–573 alternating series, 570 amplitude (of a graph), 9 Andrews, George, 558 angle measurement: radians and degrees, 25, 26 angle of incidence, 261 angle of inclination, 719 angle of reflection, 261 angles: complementary, 30, 199 in radians, 25 obtuseness testing, 656 angles between vectors: and dot product, 653–655 angular coordinate, 627, 628, 631 angular momentum, 668 angular velocity, 623 annuity, 513 annuity, perpetual, 439 antiderivatives, 275–278, 328, 666 antiderivatives: computing definite integrals with, 316, 318

definition of, 275 and Fundamental Theorem of Calculus (FTC), 309, 310 general, 275 as integrals, 318, 322 antidifferentiation, see integration AP-style questions Applications of the Derivative, AP4-1–AP4-4 Applications of the Integral, AP6-1–AP6-4 Differentiation, AP3-1–AP3-4 Further Applications of the Integral and Taylor Polynomials, AP8-1–AP8-4 Infinite Series, AP10-1–AP10-4 Integrals, AP5-1–AP5-4 Introduction to Differential Equations, AP9-1–AP9-4 Limits, AP2-1–AP2-4 Parametric Equations, Polar Coordinates, and Vector Functions, AP11-1–AP11-4 Techniques of Integration, AP7-1–AP7-4 Apollonius of Perga, 142 Apollonius’s Theorem, 142 approximately equal to (≈), 207 approximations: to the derivative, 123 endpoint, 299, 300, 323 first-order, 488 of infinite sums, 548 left-endpoint, 301–302, 416 linear, 207–210, 488, 706–708 by linearization, 210 midpoint, 290, 299, 300 numerical, 269–270 parabolic, 460 polygonal, 467–468 by Riemann sum, 300 right-endpoint, 288, 291, 400 by Trapezoidal Rule, 454 arc length, 467–469, 620–621 Archimedes, 315, 555 Archimedes’s Law of the Lever, 485 arcs: circular, length of, 621 graph shapes of, 249–251 arcsine function, 38

derivative of, 179 arctangent, of linear or quadratic functions, 433 area, 286 approximating and computing of, 287–295 approximating of by rectangles, 287–289 approximating of under the graph, 287–295 between graphs, 357–359 between two curves, 357–360, 377, 486 calculating of as a limit, 292 calculating of by dividing the region, 292 computing of as the limit of approximations, 292–295 and polar coordinates, 635–637 signed, 302, 360 surface, 623 of a trapezoid, 454 under the graph approximation by trapezoids, 454 area function (cumulative area function), 316 and concavity, 321 derivative of, 316 arithmetic-geometric mean, 548 associative law, 645 asymptote, horizontal, 54, 252–253 asymptote, vertical, 53, 252–254 asymptotes: functions with, 54 asymptotic behavior, 248, 251, 253 atmospheric pressure, 350 average (mean) time of atom decay, 353 average cost function, 159 average rate of change, 63–64, 150, 694–696 average value of a function (mean value), 370–371 average velocity, 60–62 and slope of a secant line, 64 axes, 3 horizontal, 395–397, 404 Babylonians, ancient, and completing the square technique, 18 ball grid array (BGA), 694 Balmer series, 537, 540 I1

I2

IND EX

Balmer wavelengths, 539–540 Banker’s Rule of, 352 Barrow, Isaac, 309 base (of exponential function), 43 basepoint (in the plane), 641–642 Basic Limit Laws, 77–79 basic trigonometric integrals, 278 Beer–Lambert Law, 350 Bernoulli, Jacob, 295, 555, 612 Bernoulli, Johann, 555 Bernoulli numbers, 295 Bernoulli’s formula, 295 Bernstein, Sergei, 614 Bernstein polynomials, 614 Bessel functions, 580, 586 Bézier, Pierre, 614 Bézier curves, 614 binomial coefficient, 596, 597 recursion relation for, A15 binomial series, 596, 597 Binomial Theorem, 596, 597, A15 bird flight, 54 bird migration, 257, 266 Bisection Method, 107–108 Body Mass Index (BMI), 707 Boltzmann distribution, 754 Bolzano-Weierstrass Theorem, A10 boundary of the square, 735 boundary point of a domain, 735 bounded constraint curve, 745 bounded domains, 736 bounded monotonic sequences, 544 bounded sequences, 543–545 brachistochrone property, 612 branches (of a graph), 190 Bubble Sort, 244 cable position transducer, 209 calculators: and exponential functions, 43 graphing, 52–55 calculus: differential, 59, 120 infinite series, 537 integral, 59 inventors of, 60, 130, 221 and theory of infinite series, 537 see also Fundamental Theorem of Calculus (FTC) calorie, 391 Cantor’s disappearing table, 599 carbon dating, 344–345 carrying capacity, 524 Cartesian coordinates, see rectangular coordinates Cauchy, Augustin Louis, 69, 111, 575

center of mass (COM), 480–485 center of the linearization, 210 centripetal force, 510 centroid, 482–484 Chain Rule, 130, 143, 169–174, 318, 662, 696, 723–726 combining of with Fundamental Theorem of Calculus (FTC), 318 for gradients, 712 and implicit differentiation, 726–727 in partial derivatives, 693 proof of, 173 Chain Rule for Paths, 712–714 chambered nautilus, 607 change of base formula, 46 Change of Variables Formula, 329, 331–333 and Fundamental Theorem of Calculus (FTC), 336 linearity of, 303 properties of, 303–305 Chauvet Caves, 351 Chebyshev polynomials, 56 circle: area of, 636 equation of, 4 involute of, 629 moment of, 484 parametrization of, 610, 726–727 and polar equations, 631 circuits: current in, 516 circular arc: length of, 25 cissoid, 633 Clairaut, Alexis, 697 Clairaut’s Theorem, 697, A24 closed domains, 770 closed intervals, 3, 259–261 optimizing on, 218–220 versus optimizing on open intervals, 260–261 Cobb, Charles, 747 Cobb-Douglas Production Function, 747 coefficients, 21 binomial, 596 pattern of, 458 undetermined and partial fractions, 430 common ratio, 540, 551 Commutative Law, 645 commutativity: of dot product, 654 Comparison Test, 438, 441–442 for convergence of positive series, 562 for limits, 564 Comparison Theorem (for Integrals), 305 complementary angles, 30, 179 completeness property of real numbers, 108, 216

completing the square technique, 18, 738 complex numbers, 423 imaginary, 423 component of u, 657 components: and vector operations, 645 of vectors, 642 composite functions, 169, 318, 328, 687 and Chain Rule, 724 continuity of, 85 composition: and construction of new functions, 22 compound interest, 345–347 computational fluid dynamics, 467 computer algebra systems, 406, 433, 458-459, 674 and Euler’s Method, 520–521 computer technology, 51 concave up and concave down curves, 234–235 concavity: and area functions, 321 definition of, 234 and second derivative test, 237 test for, 235 conchoid, 194 conditional convergence, 570 Conductivity-Temperature-Depth instrument, 672 cone, 681 constant mass density, 480, 686 Constant Multiple Law, 77, 78 Constant Multiple Rule, 132, 662 constant: integral of a, 303 constant of integration, 276 constant vector, 666 constraint curve, 745–748 constraint equation, 257 constraints, 749 and Lagrange multipliers, 749 continuity, 81 of composite functions, 85 and differentiability, 136 for functions, 686, 687 of inverse function, 85 left-continuous, 82 and limits, 81–87 one-sided, 82 at a point, 81 of polynomial and rational functions, 84 of power series, 591 right-continuous, 82 and Substitution Method, 85–86 Continuity, Laws of, 83–85 Continuity Law for Quotients, 84–85 continuous functions, 81–82, 83–85, 86–87, 136, 215–216, 736, 762

I N DE X

integrability of, 301, A22 continuously compounded interest, 346 contour intervals, 676, 678, 679, 701 contour maps, 676–680 and critical points, 734 and directional derivative, 718 and estimating partial derivatives, 695 of a linear function, 678–679 convergence: absolute, 569 conditional, 569 of improper integral, 439 infinite radius of, 588 of an infinite series, 549 of positive series, 574–580 radius of, 594–596 convergent sequence, 542, 543–544 cooling, rate of, 511 cooling constant, 511 coordinates, 3 angular, 626–627 polar, 626–630, 634–637, 726 radial, 626 rectangular, 626–627, 629–630 x and y, 3 cosecant, 28 hyperbolic, 47, 451 cosine function: basic properties of, 31 derivative of, 165 Maclaurin expansions of, 596–597 period of, 27 unit circle definition of, 26 Cosines, Law of, 30 cost function, 325 cotangent, 28 hyperbolic, 47 Couette flow, see shear flow critical points, 217–218 analyzing, 229–230 first derivative test for, 229–230 and optimization problems, 261 outside the interval, 218–219 second derivative for, 236–237 testing of, 229–230 without a sign transition, 231 cross product: Product Rules for, 662–663 cross sections: horizontal, 365 see also washers cumulative area function, see area function cuneiform texts: completing the square, 18

current: in a circuit, 516 transient, 534 curve integrals, see line integrals curve length, see arc length curves: area between two, 357–361, 636 Bézier, 614 concave down and up, 235–236 and conic sections, 609 integral, 517 lemniscate, 633 orthogonal family of, 510 parametric (parametrized) 607–608, 611, 614, 618 resonance, 224 trident, 193 cycloid, 612–613, 618, 621 horizontal tangent vector on, 664–665 cylindrical shells, method of, see Shell Method decimal expansion, 1–2 decimals: finite, 1 repeating (or periodic), 1 decreasing function, 6 decreasing sequence, 544 definite integral, 300–301 degree of a polynomial, 21 degrees, 25, 26 delta () notation, 61 delta (δ), 110–111 density, see mass density dependent variable, 5 derivatives: acceleration, 161 of bx , 134–135, 182, 183 of constant function, 123 definition of, 120–121 directional, 715–717 discontinuous, 177 estimating, 123 first, 159, 235–236 as a function, 129–132 higher, 159–162 of hyperbolic functions, 185–186, 422 of inverse function, 178 of inverse hyperbolic functions, 185–186, 422 of inverse trigonometric functions, 179–180, 337–338 of logarithmic functions, 182–184 mixed partials, 696 nth-order, 159 partial, 672, 692–699, 706, 732 in polar coordinates, 626 of power series, 584

I3

primary, 724–725 scalar-valued, 621, 631, 633 second, 159 sign of the, 227 and tangent line, 121 as a tangent vector, 663–665 trigonometric, in degrees, 173 of trigonometric functions, 165–167 vector-valued, 660–661, 665 see also antiderivatives; First Derivative Test; Second Derivative Test Descartes, René, 3, 221 Descartes, folium of, 193, 618 Dichotomy Theorem for Positive Series, 560, 562 difference, see first difference difference quotient, 120 and approximations to the derivative, 123 difference quotient points, 735 Difference Rule, 132 differentiability, 703–707 and continuity, 136 criterion for, 704, A25 and local linearity, 137 and tangent plane, 704 differentiable functions, 121, 136 differential calculus, 59, 120 differential equations, 279, 502–507 and exponential functions, 341–342, 351 first-order linear, 528 first-order, 519 general solution, 502 Gompertz, 352 homogeneous, 534 linear, 503 logistic, 524–526 order of, 503 particular solution, 279 power series solutions of, 585–588 second-order, 423 for vector-valued functions, 666 differentials, 130, 208, 276, 329, 707 substitution using, 319–320 differentiation, 122, 160 and integration, 318 basic rules of, 131–133 implicit, 143, 188–192, 726–727 logarithmic, 184 of a power series, 584 differentiation rules, 661–662 directional derivatives, 715–720 Dirichlet, Lejeune, 575 discontinuity, 81–83 of a function, 215 infinite, 83, 440 jump, 82–83 removable, 82 discriminants, 17, 733–735, 738–739

I4

IND EX

disk method (for computing volume), 375–377 displacement, 622 and change in position, 61 Distance Formula, 3–4, 633 and vectors, 641 distance traveled: and displacement, 622 and velocity and time, 286 distinct linear factors, 427 Distributive Law, 655 and cross products, 663–664 and dot product, 653 distributive law for scalars, 645 divergence: of an improper integral, 437 of an infinite series, 549 of harmonic series, 555 of a sequence, 538 Divergence Test, 553 divergent sequences, 543 divergent series, 549, 553–554 domains, 4, 5, 34, 35, 735 bounded, 735 closed, 735 and differentiability, 704 and n variables, 672–673 open, 735 and sequence, 537 dot product, 653–654 and angle between vectors, 653–654 properties of, 653 Product Rules for, 665 and testing for orthogonality, 655 double-angle formulas, 30 double integration, 368 double roots, 18 doubling time, 343–344 doubly infinite improper integral, 437 Douglas, Paul, 747 dummy variable, 289, 301 Dürer, Albrecht, 630 e, 45, 134 irrationality of, 603 eccentricity: of Mercury’s orbit, 273 effect of a small change, 207 Einstein, Albert, 49, 155, 301, 392 Einstein’s Law of Velocity Addition, 49 elementary functions, 23 ellipse area of, 336 parametrization of, 610–611 elliptic function of the first kind, 598 elliptic function of the second kind, 603 elliptic integral of the second kind, 603 endpoint approximations, 295, 520

endpoints, 219–220 energy: conservation of, 439 kinetic, 439 and work, 391–394 epsilon (), 111 equation of a line, 13, 16 intercept-intercept form of, 19 point-point form of, 16 slope-intercept form of, 13, 16, 17 equations: constraint, 257 graphing of, 6 logistic differential, 524–526 parametric, 607–615 polar, 628 reverse logistic, 528 of tangent line, 121 equiangular spiral, 607 equilibrium solution, 524 equivalent vectors, 642 error, 111 in Linear Approximation, 207, 211–212 in linearization, 210–211 error, percentage, 211 Error Bound, 211 for Simpson’s Rule, 458–459 for Taylor polynomials, 492-495, 593 for Trapezoidal Rule and Midpoint Rule, 454–455 escape velocity, 397, 439 Euler, Leonhard, 45, 423, 555 Euler’s Constant, 548 Euler’s Formula, 424 Euler’s Method, 516, 519–521 Euler’s Midpoint Method, 523 even functions, 7 exponential functions, 22, 43, 134 with base b, 22, 45 continuity of, 84 derivatives of, 134, 182–183 differential equations of, 352 and financial calculations, 345 power series of, 579 properties of, 43–44 exponential growth and decay, 341–344 Exponents, Laws of, 43–44 exponents, negative, 44 extreme values (extrema), 215–218 existence on a closed interval, A21 Faraday’s Law of Induction, 157 Fermat, Pierre de, 221 Fermat’s Theorem on Local Extrema, 218, 834 Feynmann, Richard, 153, 391 Fibonacci sequence, A16

financial calculations: and exponential functions, 345–346 finite decimal expansion, 1 Fior, Antonio, 273 first derivative, 160 and points of inflection, 235 First Derivative Test, 229 for critical points, 229–231 first difference, 24 first octant, 668 first-order approximation, 488 first-order differential equations, 516, 519 first-order linear differential equations, 528–532 general solutions of, 529 flow rate, 368–369 fluid force, 474–477 fluid pressure, 474–477 folium of Descartes, 193, 618 foot-pounds (ft-lb), 391 force, 391, 474–477 calculating, 475 on an inclined surface, 476 force diagram, 648 force vectors, 647 forces: as vector quantity, 647 Fourier Series, 409 fractions: derivatives as, 171 Fractions, Method of Partial, 426–433 Franklin, Benjamin, 527 Fraunhofer diffraction pattern, 76 free fall, 511–512 Fresnel zones, 498 functions: algebraic, 22 and antiderivatives, 275 arccosine, 39 arcsine, 39 area, 316, 321 with asymptotes, 54 average value of, 370–371, 379–381 basic classes of, 21–23 Bessel, 22, 598 composite, 85, 87, 169, 666 constructing new, 22–23 continuity of, 81–82 continuous, 81–87, 136, 215–216, 686 decreasing, 6, 226 definition of, 4 derivative of, 121–123 derivatives as, 129–134 and differentiability, 703 differentiable, 120, 129, 703 discontinuous, 81–83, 119 elementary, 23 even, 6

I N DE X

exponential, 21, 22, 43, 54, 84, 131 extreme values of, 215–220 gamma, 22, 447 Gaussian, 454 gradient of, 711 graph of, 5 and graphs of two variables, 673–675 greatest integer function, 86 harmonic, 548, 561 higher-order partial derivatives of, 696–699 hyperbolic, 47–48, 184, 420 implicitly defined, 22 increasing, 6, 227 indeterminate forms of, 91 with infinite discontinuity, 83 integrable, 301 inverse, 34–38 inverse hyperbolic, 47–78, 184, 420 inverse trigonometric, 33–38 invertible, 34 level curves of, 674–676 linear, 674, 678–679 linear combination, 21 linear and nonlinear, 13, 16, 123 linearization of, 703–704 local extrema of, 216–218 local linearity of, 212 locally linear, 137 logarithmic, 22 monotonic, 226, 291 nondecreasing, 227 nondifferentiable, 217 numerical, 5 odd, 6 one-to-one, 35, 36 parity of, 6–7 periodic, 27, 28 piecewise-defined, 82 polynomials, 21 power, 21, 686 probability density, 448 product of, 687 product, 766 quadratic, 17, 19, 458 radial, 731 radial density, 368, 369 range of, 4, 5 rational, 21, 84, 102, 426, 686 real-valued of n variables, 672 real-world modeling of, 86–87 represented as a power series, 580 root of, 107 sequences defined by, 539 of several variables, 672 and Squeeze Theorem, 95–98 transcendental, 22

trigonometric, 22, 25–31, 165–167, 185, 336–337, 421, 423 of two or more variables, 672–680 value of, 4, 5 vector-valued, 660–666 vector-valued integral, 665 velocity, 666 zero (or root) of, 5, 107–108, 269 with zero derivative, 227 Fundamental Theorem of Calculus (FTC), 309–313, 316 proof of, 322 Galilei, Galileo, 60, 155 Galileo’s Law, 49 Galois Theory, 269 gamma function, 447 Gauss, C. F., 301 Gauss’s Theorem, see Divergence Theorem Gaussian function, 454 Gauss-Ostrogradsky Theorem, see Divergence Theorem general antiderivative, 275 General Exponential Rule, 171 General Power Rule, 171 general solution (of a differential equation), 502 general term (of a sequence), 537 in summation notation, 289–291 General Theory of Relativity, 155, 301 geometric sequence, 540 geometric series, 550, 551, 552, 555, 556 sum of, 552 global (absolute) extreme values, 731, 735–739 Gompertz, Benjamin, 352 Gompertz differential equation, 352 gradient vectors, 711–712, 714, 718 Product Rule and Chain Rule for, 712 properties of, 712 graphing, 3–9 of equations, 6 of functions of two variables, 673–674 graphing calculator, 52, 55 graphs: amplitude of, 8, 9 approximating area under, 286–288 branches of, 190 of a function, 5 of a linear function, 13, 16 of a nonlinear function, 16 of one-to-one function, 35 of a quadratic function, 16 and scales, 14 of trigonometric functions, 28 polar, 636 scaling (dilation) of, 8–9 shape of, 234–237

I5

sketching, 7, 248–254 translating (shifting of), 7–8 and viewing rectangle, 52 gravity: and acceleration, 154, 161 inverse square law of, 392 and motion, 153 and work, 39 Greek, ancient, mathematicians and philosophers, 485 Gregory, James, 408, 593 Gregory–Leibniz series, 537 grid lines: in polar coordinates, 626 gudermannian, 426 Guldin’s Rule, see Pappus’s Theorem half-life, 344 half-open interval, 2 Ham Sandwich Theorem, 110 harmonic series, 555, 561 alternating, 572–573 divergence of, 575 head (in the plane), 641 heat capacity, 327 heat equation, 698 height: maximum, 154–155 and velocity, 154 Heron of Alexandria, 261 homogeneous differential equation, 534 Hooke, Robert, 392 Hooke’s Law, 392 horizontal asymptote, 54, 100–101 horizontal axis, revolving about a, 387–379, 404 horizontal cross sections: and volume, 366 horizontal line, 14 Horizontal Line Test, 36 horizontal scaling, 8, 9 horizontal traces, 676–677 horizontal translation, 8 Huxley, Julian, 5 Huygens, Christiaan, 171, 498, 612 hyperbolas, 48 horizontal traces, 676 hyperbolic functions, 48–49, 420–425 derivatives of, 184–185, 422, 423 inverse, 185–186, 423 hyperbolic substitution, 421 hyperboloid, 383, 680 i and j components, 647 identities, 47 trigonometric, 29–31 implicit differentiation, 143, 188–192, 726–727

I6

IND EX

Implicit Function Theorem, 727, 746 implicitly defined function, 22 improper integrals, 436–449, 442 absolutely convergent, 447 Comparison Test for, 442 convergence of, 438 of x p , 441 income stream, 347 increasing function, 6 increasing sequence, 543 indefinite integral, 276, 277 and Fundamental Theorem of Calculus (FTC), 311 linearity of, 277 independent variables, 5, 723 indeterminate forms, 91, 243 index (of a sequence), 537 index (of an infinite series), 549 Indonesian tsunami (1996), 400 induction, principle of, A13 inequalities and intervals, 3 infinite discontinuity, 83, 441 infinite integrals, 444 infinite interval, 2, 440 infinite limit, 72–73 infinite radius of convergence, 588 infinite series, 537 convergence of, 549 linearity of, 551 summing of, 548–556 inflection points, 218, 235–238 initial condition, 504–505 and antiderivatives, 279–280 initial guess, 269, 271 initial value problem, 279 solution of, 279 inner product, see dot product instantaneous rate of change, 63–64, 150 instantaneous velocity, 60, 61–62 integrable functions, 301 integral calculus, 59, 286 integral curves, 517 integral formulas, 422 Integral Test, 560, 561 integrals: of an absolute value, 314 antiderivatives as, 318, 322 applications of, 357–388 arc length, 467–469, 621, 622 basic trigonometric, 278 and Change of Variables Formula, 329, 331 comparison of, 442 for computing net or total change, 322, 323 of a constant, 303 definite, 301–305, 309, 316–317, 331, 401 differentiating, 318 improper, 436–444 indefinite, 276–280, 301

infinite, 437, 438 reduction formulas for, 402–403 of velocity, 323–324 and volume, 365 integrands, 276, 301, 329 and improper integrals, 440 with infinite discontinuities, 440 and Integration by Parts formula, 400 integrating factor, 528 integration: and area of an irregular region, 286 for computing volume, 365–366 constant of, 276 to calculate work, 391 and differentiation, 318 and finding an antiderivative, 286 limits of, 301, 347 numerical, 400–424 of power series, 584 reversing limits of, 304 term-by-term, 584 using partial fractions, 426 using substitution, 328–329 using trigonometric substitution, 413 vector-valued, 665 Integration by Parts formula, 400 integration formulas, 278 of inverse trigonometric functions, 336 intercept-intercept form of an equation, 19 interest rate, 345 interior point of a domain, 735 Intermediate Value Theorem (IVT), 106–108, 371, A12 intervals: adjacent, 304 and test points, 230 closed, 2, 3, 216, 218, 259 critical points and endpoints of, 217 describing of via inequalities, 3 extreme values on, 215 half-open, 2 of increase and decrease, 230 infinite, 2 midpoint of, 3 open, 2, 3, 216, 259 radius of, 3 standard notation for, 2 test values within, 236 inverse functions, 33–38 continuity of, 85 defined, 34 derivative of, A24 existence of, 36 inverse hyperbolic functions, 49, 185–186, 422 inverse operations: integration and differentiation, 318

inverse trigonometric functions, 38–40, 336–338 derivatives of, 179–180, 336–338 integration formulas for, 338 invertible function, 34–35 invertible function: derivatives of, 178–179 involute, 625 irrational numbers, 1 irreducible quadratic factors, 430–431 isocline, 518 iteration: Newton’s Method, 269–271 joule, 391 jump discontinuity, 82 Kepler, Johannes, 60, 263, 273 Kepler’s Laws, 274, 392 Kepler’s Second Law, 274 Kepler’s Third Law, 626 Kepler’s Wine Barrel Problem, 263 kinetic energy, 397 Kleiber’s Law, 158 Koch snowflake, 559 Koch, Helge von, 559 Korteweg-deVries equation, 702 Kummer’s acceleration method, 568 Lagrange, Joseph Louis, 130 Lagrange condition, 746 Lagrange equations, 746–747 Lagrange multipliers: in three variables, 748–749 with multiple constraints, 749 laminar flow, 369 laminas, 481–484 Laplace, Pierre Simon, Marquis de, 423 Laplace operator (), 702, 731 Laplace transform, 446 Lascaux cave paintings, 345 latitude, 717 Law of Cosines, 30, 654 Laws of Continuity, 83–85 Laws of Exponents, 43–44 laws of logarithms, 46 leading coefficient: of a polynomial, 21 Least Distance, Principle of, 261 Least Time, Principle of, 261 Least Upper Bound (LUB) Property, 581 left-continuous function, 82 left-endpoint approximation, 290, 400 Leibniz, Gottfried Wilhelm von, 22, 122, 130, 145, 171, 221, 301, 329, 555, 594, 612 Leibniz notation, 130, 143, 150, 171 Chain Rule, 171 and definite integral, 301

I N DE X

differentials, 329 for higher derivatives, 159 for higher-order partial derivatives, 696 for partial derivatives, 692 vector-valued derivatives, 662 Leibniz Test for alternating series, 570–571 lemniscate curve, 194–195, 633 length: and dot product, 653 of a vector, 641 level curves, 674–680, 734 spacing of, 676–677 level surfaces of a function of three variables, 680 L’Hôpital, Guillaume François Antoine, Marquis de, 241 L’Hôpital’s Rule, 241–244 for limits, 244, 439, 539 proof of, 245 Libby, Willard, 344 light intensity, 350 limaçon, 629–630 limaçon of Pascal, 193 limit of approximations, 292–295 Limit Comparison Test, 564 Limit Laws, 70, 83, 686 for sequences, 541 Limit Laws, Basic, 77–79 Limit Laws of scalar functions, 661 limits, 59, 67–73, 685 calculating area as, 293 and continuity, 81–87 definition of, 68–69 discontinuous, 81 evaluating algebraically, 90–93 evaluating by substitution, 687 evaluation of with Substitution Method, 86–87 formal definition, 110–111 graphical and numerical investigation of, 69–71 indeterminate, 91 infinite, 72–73 at infinity, 251–252 and instantaneous velocity, 59 linearity rules for, 551 need for, 124 one-sided, 72, 440 of polygonal approximations, 620 of a sequence, 539–540 in several variables, 684–686 trigonometric, 95–98 of vector-valued functions, 660–663 verifying, 689 limits of integration, 301, 331 Linear Approximation, 207, 706–707 error in, 208, 211–212 and Taylor polynomials, 488

linear combination function, 22–23 linear combination of vectors, 645–646 linear differential equations, 503 linear equation, 16 first-order, 528–532 linear functions, 13 contour map of, 677–678 derivative of, 123 graph of, 16–17 traces of, 678 linear mass density, 367 linear motion, 153 linear regression, 16 linear relationship, 16 linearity: of indefinite integral, 277 local, 55, 137, 212 of summations, 289 linearization, 210–212 error in, 210–211 of a function, 703–704 lines: equation of a, 13–15 horizontal and vertical, 14 parallel, 14 perpendicular, 14 and slope of, 14 and traces of a linear function, 678 local extrema, 216–218, 731–732 local extreme values, 731 local linearity, 55, 212, 704 local linearity of differentiable functions, 212 local maximum, 216, 229, 731–733 local minimum, 216, 229, 731–735 locally linear functions, 137 logarithm functions, 22, 45 logarithmic differentiation, 183–184 logarithms, 45–46 with base b, 45 calculus of, 344 derivatives of, 182–184 laws of, 46 natural, 46, 183 logistic differential equation, 524–526 lower bound of a sequence, 543 Maclaurin, Colin, 491, 593 Maclaurin expansions, 593–594 Maclaurin polynomials, 489, 491–493 Maclaurin series, 592, 594–595, 597, 599 Madhava, 594 magnetic declination: of United States, 701 magnitude: of a vector, 641 Mandelbrot Set, 52 marginal cost, 152–153, 325 marginal cost of abatement, 327

I7

marginal utility, 753 Mars Climate Orbiter, 154 mass: center of, 480 computing of by mass density, 367–368 mass density constant, 481 linear, 367–368 and total mass, 367–368 maximum height, 154 maximum (max) value, 215–220 of unit square, 736 maximum volume, 737 Maxwell, James Clerk, 711 mean value, see average value Mean Value Theorem (MVT), 226–227, 231, 311, 371, 468, 555, 620 median: of a triangle, 487 Mengoli, Pietro, 555 Mercator map projection, 408 Method of Partial Fractions, 426–433 microchips: testing for reliability of, 694 midpoint approximations, 290, 299 Midpoint Rule, 456 midpoints, 456, 458 of intervals, 3 minimum (min) value, 215–220 mixed partial derivatives, 696 mixed partial derivatives: equality of, 697 mixing problem, 531 modeling: and differential equations, 505 moments, 480 additivity of, 481, 484 of the circle, 484 of a triangle, 484 monkey saddle, 837 monotonic functions, 291 monotonic sequences, bounded, 544 Moore, Gordon, 43, 351 Moore’s Law, 43, 351 motion: and gravity, 153 linear, 152 Newton’s laws of, 60, 155 nonuniform circular, 761 motion, laws of, for falling objects, 155 Mount Whitney Range, 678 mountains: and contour maps, 678–679 Multiples Rule, 277 multiplying by the conjugate, 92 multiplying Taylor series, 595

I8

IND EX

n variables, 672 nabla, 711 natural logarithm, 46, 183, 338 negative slope, 14 net change, 310, 322–323 newton, 391, 439 Newton, Isaac, 60, 122, 171, 309, 348, 392, 516, 596 Newton’s Law of Cooling, 511–512, 519 Newton’s laws of motion, 60, 155 Newton’s Method, 269–271 Newton’s Second Law of Motion, 397 Newton’s Universal Law of Gravitation, 485 nondifferentiable function, 217 nonlinear function, 15 nonzero vector, 644, 647 norm (of a vector), see length; magnitude norm (of partition), 301 normal, see perpendicular normal force, 657 nth-order derivative, 160 numbers: Bernoulli, 295 complex, 423 e, 45 imaginary complex, 423 irrational, 1 rational, 1 real, 1 sequences of, 537 whole, 1 numerical approximations, 269–271 numerical functions, 5 numerical integration, 454–460 obtuseness: testing for, 656 odd functions, 6–7 Ohm’s Law, 148, 692 one-sided continuity, 82 one-sided limits, 72–73 one-to-one functions, 35–37 graph of, 38 one-unit change, 151–152 open disk, 685 open domains, 704 open intervals, 2, 218 optimization problems, 257–258, 262 optimization, 215 with Lagrange multipliers, 745–749 on an open interval, 216 in several variables, 731–739 order: of a differential equation, 503 ordinate, see y-coordinate Oresme, Nicole d’, 555 origin, 1, 3 orthogonal families of curves, 510

orthogonal unit vectors, 657 orthogonal vectors, 656 orthogonality relations, 413 orthogonality: of vector-valued functions, 665 testing for, 655 p-series, 561 parabola, 17 graph of quadratic function, 14 vertical traces, 674 parabolic approximations: and Simpson’s Rule, 460 parallel lines, 14 parallel vectors, 642 parallelogram, 646 Parallelogram Law, 644, 647, 649 parameters, 607 and parametric equations, 607 parametric (parametrized) curve, 607–608, 610–611, 614 area under, 619 second derivative of, 618 parametric equations, 607–614 parametric line, 609 parametrization, 607 parity: of a function, 6–7, 47 partial derivatives, 672, 692–698 estimating with contour maps, 695 higher-order, 696–698 partial differential equation (PDE), 698 partial differentiation: and Clairaut’s Theorem, 697 partial fraction decomposition, 427–429 partial sums, 548, 549 even, 571 odd, 571 of positive series, 560 particular solution (of a differential equation), 279 partitions, 300 Pascal, Blaise, 183, 612 Pascal, limaçon of, 183 Pascal’s Triangle, A14–A15 path Chain Rule for, 712–714 path of steepest ascent, 680 path of steepest descent, 680 percentage error, 211 periodic function, 27 perpendicular lines, 14 perpetual annuity, 439 piecewise-defined function, 82 p-integral, 438, 440 Planck’s Radiation Law, 447

plane curve: arc length of, 620 planetary motion: Kepler’s laws of, 60 point masses, 481 point of inflection, 235–236 point-point equation of a line, 16 points (or real numbers), 1 point-slope equation of line, 16 Poiseuille, Jean, 369 Poiseuille’s Law of Laminar Flow, 369 polar coordinates, 626–630, 634–637, 726 and area, 634–637 arc length in, 634–637 derivative in, 634 polar equations, 631 polygonal approximations, 467–470, 620 polynomials, 21 Bernstein, 614 Chebyshev, 56 coefficients of, 21 continuity of, 84 continuous, 686 degree of a, 21 graphs of, 249–252 Maclaurin, 489, 491, 493 quadratic, 249–250, 430 Taylor, 488–495 population density, 368, 374 population growth, 524 position: and rates of change, 59 position vector, 642 positive series, 559–565 pound, 391 power consumption: and velocity, 36, 37 power functions, 21, 686 Power Rule, 131–133, 311 for derivatives, 276 for fractional exponents, 177 for integrals, 276 power series, 579–588, 591 adapting, 583 and integration, 584 differentiating, 584 finding radius of convergence, 581 interval of convergence of, 580 representing functions by, 556 solutions of differential equations, 585–588 term-by-term differentiating, 584 power series expansion, 583, 591–592 power sums, 292, 294 power to a power rule, 44 present value (PV), 347–348 of income stream, 348

I N DE X

pressure, 474 and depth, 474 atmospheric, 350 fluid, 474–477 primary derivatives, 724 prime meridian, 717 prime notation, 130 principal, 345 Principle of Equivalence, 155 Principle of Least Distance, 261 Principle of Least Time, 261 probability density function, 448 Product Formula: for cross products, 663 Product Law, 77–79, 686 Product Rule, 143, 161, 662–663 and computing derivatives, 143–145 for gradients, 712 production level, 152 products, 44, 46, 328 projection, 656 proper rational function, 426 psi (ψ), 719 punctured disk, 685 pyramid: volume of, 366 Pythagorean Theorem, 29 quadratic convergence to square roots, 274 quadratic factors, 430–433 quadratic forms, 738 quadratic formula, 17 quadratic functions, 17 finding minimum of, 18 graph of, 17 quadratic polynomials, 17–18, 249, 433 Quick Sort, 244 Quotient Law, 77, 686 Quotient Rule, 143, 145, 166 and computing derivatives, 143 quotients, 1, 21, 44, 46 continuity law for, 84–85 difference, 120 limits of, and L’Hôpital’s Rule, 244 radial coordinate, 626–628 radial density function, 368–369 radial functions, 731 radians, 25, 26 radius: of intervals, 3 radius of convergence, 580–588 infinite, 588 Radon-222, 344 range (of a function), 4, 5, 34 rate of change (ROC), 14, 59, 63, 195–199, 511 average, 63–64, 678–679

and exponential growth and decay, 342 of a function, 150–155 instantaneous, 63, 150 and Leibniz notation, 150 and partial derivatives, 794 Ratio Test, 575–578, 581–588 rational functions, 21, 426 continuity of, 87 continuous, 686 rational numbers, 1 real numbers, 1 absolute value of, 2 completeness property of, 108, 216, Appendix D distance between, 2 real roots, 17 real-valued functions of n variables, 672 real-world modeling by continuous functions, 86 reciprocals, 46 rectangle, viewing, 52 rectangles: and approximating area, 287–288 left-endpoint, 290–291 rectangular (or Cartesian) coordinates, 3, 607, 626–630 recursion relation, 586–587 recursive formulas, see reduction formulas, 402 recursive sequences, 538 recursively defined sequences, 537 reducible quadratic factors, 431 reduction formulas, 402–403 for integrals, 402–403 for sine and cosine, 406 reflection (of a function), 37 regions, see domains regression, linear, 16 regular parametrization, 750 related rate problems, 195–199 remainder term, 494 removable discontinuity, 82 repeated linear factors, 429 repeating decimal expansion, 1 resonance curve, 224 resultant force, 647 see fluid force reverse logistic equation, 528 Richter scale, 45 Riemann, Georg Friedrich, 301 Riemann hypothesis, 301 Riemann sum approximations, 300, 311 Riemann sums, 300–305, 348, 468, 470, 475–476, 620, 634 right-continuous function, 82–83 right cylinder: volume of, 365

I9

right-endpoint approximation, 288–290, 293, 385 right triangles, 26 Rolle’s Theorem, 220–221, 231 and Mean Value Theorem, 226 root (zero): as a function, 5, 107, 269 Root Test, 577 roots: double, 17 real, 17 saddle, 677 saddle point, 733, 734 scalar, 644 and dot product, 653 scalar functions: Limit Laws of, 661 scalar multiplication, 644 scalar product, see dot product scalar-valued derivatives, 664 scale, 14 scaling (dilation) of a graph, 8 seawater density, 672, 702 contour map of, 695 secant, 28 hyperbolic, 48 integral of, 408 secant line, 62, 120–121 and Mean Value Theorem, 226 slope of, 120 Second Derivative Test, 738 for critical points, 237, 733–735 proof of, 240 second derivatives, 159–162, 236–237 for a parametrized curve, 618 trapezoid, 455, 456 second-order differential equation, 423 seismic prospecting, 268 separable equations, 503 separation of variables, 503 sequences, 537 bounded, 542–545 bounded monotonic, 544 convergence of, 538–539, 543, 544 decreasing, 544 defined by a function, 539 difference from series, 550 divergence of, 538, 543 geometric, 540 increasing, 543 Limit Laws for, 541 limits of, 538–539, 540 recursive, 544 recursively defined, 537–538 Squeeze Theorem for, 541 term of, 537 unbounded, 543

I10

IND EX

series: absolutely convergent, 569–570 alternating, 570 alternating harmonic, 572–573 binomial, 597–598 conditionally convergent, 570 convergent, 549, 555 difference from a sequence, 550 divergent, 549, 553 geometric, 550, 551, 552, 553, 555, 583 Gregory–Leibniz, 537 harmonic, 555, 561, 572–573 infinite, 537, 548–555 Maclaurin, 592, 594, 595, 597–599 partial sums of, 548 positive, 559–565 power, 579–588, 591 p-series, 561 Taylor, 592–599 telescoping, 549–550 set (S), 3 of rational numbers, 1 set of intermediate points, 354 Shell Method, 384, 398 shift formulas, 30 Shifting and Scaling Rule, 172, 174 shifting: of a graph, 8 sigma (), 289 sign: on interval, 286 sign change, 228–229, 232, 248 sign combinations, 248 signed areas, 436 Simpson, Thomas, 264 Simpson’s Rule, 458, 460, 520 sine function: basic properties of, 31 derivative of, 165–166 Maclaurin expansion of, 594–595 period of, 27 unit circle definition of, 26 sine wave, 27 slope field, 516–522 slope-intercept form of an equation, 13 slope of a line, 14 and polar equation, 628 Snell’s Law, 261 solids: cross sections of, 375 volume of, 365 solids of revolution, 375 volume of, 375–380, 385–387 Solidum, Renato, 45 sound: speed of, 63 spanning with vectors, 646

Special Theory of Relativity, 49 speed, 60, 622 along parametrized path, 622 sphere: and gradient, 717 and level surfaces, 680 parametrization of, 746 volume of, 365 spring constant, 392 square root expressions, 413 square roots: quadratic convergence to, 274 Squeeze Theorem, 95–98 for sequences, 537–538 stable equilibrium, 524 standard basis vectors, 647, 649 steepness of the line, 13–14 stopping distance, 152 stream lines, see integral curves strictly decreasing function, 6 strictly increasing function, 6 subintervals, 287 substitution: and evaluating limits, 686–687 with hyperbolic functions, 420 and Maclaurin series, 592 and partial fractions, 426 trigonometic, 413–418 using differentials, 329–332 Substitution Method, 87–88, 329 Sum Law, 77–80, 114, 686 proof of, 114 Sum Law for Limits, 165 Sum Rule, 132–133, 277, 662 summation notation, 289–292 sums, partial, 548, 549, 560, 570–571 surface area, 623 surface of revolution, 469 symmetry, 483 and parametrization, 607 Symmetry Principle, 483, 487 tail (in the plane), 641 Tait, P. G., 711 tangent, 28 hyperbolic, 49, 50 tangent function: derivative of, 166 integral of, 416–417 tangent line approximation (Linear Approximation), 208 tangent lines, 59–60, 121, 734, 703 for a curve in parametric form, 612–613 defined, 121 limit of secant lines, 62, 120 and polar equation, 133

slopes of, 62, 63, 120, 613, 700, 717 vertical, 117 tangent plane, 704 and differentiability, 706 finding an equation of, 718 at a local extremum, 731 Tangent Rule, see Midpoint Rule, 455 tangent vectors, 665 derivatives, 663–665 horizontal, on the cycloid, 664–665 plotting of, 664 Tartaglia, Niccolo, 263, 273 Taylor, Brook, 488 Taylor expansions, 593, 594 Taylor polynomials, 488–499, 592, 593, 595 Taylor series, 591–603 integration of, 595 multiplying, 595 shortcuts to finding, 594–596 see also Maclaurin series Taylor’s Theorem: Version I, 494 Taylor’s Theorem: Version II, 495 temperature: directional derivative of, 717 term-by-term differentiation and integration, 584 terms (of the sequence), 537 test point, 229, 230 test values: within intervals, 236 tests for convergence and divergence of series: Comparison Test, 562 Dichotomy Theorem for Positive Series, 560 Divergence Test, 553–554 Integral Test, 560 Leibniz Test for alternating series, 570–571 Limit Comparison Test, 564 p-series, 561–562 Ratio Test, 575–576 Root Test, 577 theorems: analyzing, A6 uniqueness, 504 thermal expansion, 209 thin shell, 384 third derivative, 160 time interval: and average velocity, 60–61, 62 time step, 519, 520–521 Torricelli’s Law, 506 torus, 383 total cost, 325 total force, see fluid force total mass, 367–368

I N DE X

trace curves: and tangent lines, 703 traces, 674–676 tractrix, 383, 509, 619, 623 transcendental function, 22 transformations, see maps transient current, 534 transition points: of graphs, 248 translation (shifting) of a graph, 8 translation of a vector, 642 Trapezoidal Approximation, 323 Trapezoidal Rule, 454 trapezoidal sums, 455 trapezoids: area of, 455 area under the graph approximations, 454 Triangle Inequality, 2, 648, 649 triangles: and fluid force, 476 median of, 487 moment of, 484 trident curve, 193 trigonometric derivatives: in degrees, 173 trigonometric functions, 22, 25–30 derivatives of, 166, 185, 422 integrals of, 278 inverse, 336–337 trigonometric identities: 29–30 trigonometric integrals, 278, 405–410 table of, 405–409, 410 trigonometric limits, 95–98 trigonometric substitution, 413–417 two-dimensional vector, 641 unbounded sequences, 543 undetermined coefficients: and partial fractions, 430 uniform density, see constant mass density uniqueness theorem, 504 unit circle, 25, 26, 28, 48, 612 unit vectors, 657, 715 universal laws of motion and gravitation, 60 unknown quantity: estimating of, 119 unstable equilibrium, 324 upper bound of a sequence, 543 utility, 753 Valladas, Helene, 351 value: of a function, 4, 5

variables: Change of Variables Formula, 343 dummy, 284, 301 functions of two or more, 672–681 graphing functions of, 673–681 independent and dependent, 5, 734 and limits, 692–694 separation of, 514 vector addition, 643–645 vector algebra, 643–645 vector operations: using components, 645–651 vector parametrization, 654 of the cycloid, 664 vector product, see cross product vector quantity, 647 vector subtraction, 643 vector sum, 643 vectors: components of, 661 direction, 661 equivalent, 643 force, 641 gradient, 672 length of, 656 Lenz, 678–682 linear combination of, 645–646 nonzero, 647, 649 normal, 728 orthogonal unit, 657 position, 642, 649, 651 radial, 676 translation of, 642 two-dimensional, 641 unit, 726 unit tangent, 685 velocity, 664, 667 vector-valued functions, 660–667 calculus of, 666 differential equations of, 677 fundamental theory of calculus for, 666 limits of, 661 orthogonality of, 665 vector-valued integration, 665–666 velocity, 58–59, 631 and acceleration, 160 angular, 634 average, 60–62 Einstein’s Law of Velocity Addition, 51 escape, 397, 439 and flow rate, 368–369 graphical interpretation of, 62 instantaneous, 60–62

integral of, 346–347 and rates of change, 59 and speed, 60, 153 velocity vector, 664, 667, 724 Verhuls, Pierre-François, 524 vertical asymptote, 247–248 vertical axis: revolving about a, 396 rotating around a, 401 vertical cross sections: and volume, 380 vertical line, 14 Vertical Line Test, 6 vertical scaling, 8–9 vertical tangents, 137 vertical traces, 674 vertical translation, 7–8 Viète’s formula, 21 viewing rectangle, 51 volume: computing of by integration, 375–376 of a cylindrical shell, 384 as integral of cross-sectional area, 379 of a pyramid, 366–367 of a solid of revolution, 390–394 of a sphere, 377 washers (disks), 386–387 weighted average: and Simpson’s Rule, 458 whole numbers, 1 witch of Agnesi, 149 work: definition of, 391–392 and energy, 393–394 and gravity, 393, 967 using integration to calculate, 392 Work-Kinetic Energy Theorem, 439 Wright, Edward, 408 x-axis, 3 rotating about, 375 symmetry about, 629–630 x-coordinate, 3 x-moments, 484 y-axis, 3 integration along, 377–378 y-coordinate, 3 y-moments, 484 zeros of functions, 5, 103–104, 276

I11