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Calculus: Early Transcendentals 8th Edition
Calculus: Early Transcendentals 8th Edition
James Stewart
Success in your calculus course starts here! James Stewart's CALCULUS: EARLY TRANSCENDENTALS texts are worldwide bestsellers for a reason: they are clear, accurate, and filled with relevant, realworld examples. With CALCULUS: EARLY TRANSCENDENTALS, Eighth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic beauty of the subject. His patient examples and builtin learning aids will help you build your mathematical confidence and achieve your goals in the course.
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Mathematics
Volume:
8
Year:
2016
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english
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1404
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CALCULUS EARLY TRANSCENDENTALS EIGHTH EDITION JAMES STEWART M C MASTER UNIVERSITY AND UNIVERSITY OF TORONTO $XVWUDOLDä%UD]LOä0H[LFRä6LQJDSRUHä8QLWHG.LQJGRPä8QLWHG6WDWHV Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. te t portant otice e ia content reference a not e availa le in the e oo version ithin the pro ct escription or the pro ct Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Calculus: Early Transcendentals, Eighth Edition James Stewart © 2016, 2012 Cengage Learning Product Manager: Neha Taleja ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. Senior Content Developer: Stacy Green Associate Content Developer: Samantha Lugtu Product Assistant: Stephanie Kreuz Media Developer: Lynh Pham Marketing Manager: Ryan Ahern WCN: 02200203 Content Project Manager: Cheryll Linthicum Art Director: Vernon Boes Manufacturing Planner: Becky Cross Production Service: TECHarts Photo and Text Researcher: Lumina Datamatics Copy Editor: Kathi Townes, TECHarts Illustrator: TECHarts Text Designer: Diane Beasley Cover Designer: Irene Morris, Morris Design Compositor: Stephanie Kuhns, Kristina Elliott, and Kira Abdallah, TECHarts Cover Image: elisanth/123RF; tharrison/Getty Images For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 18003549706. FRUSHUPLVVLRQWRXVHPDWHULDOIURPWKLVWH[WRUSURGXFW sXEPLWDOOUHTXHVWVRQOLQHDWwww.cengage.com/permissions. FXUWKHUSHUPLVVLRQVTXHVWLRQVFDQEHHPDLOHGWR permissionrequest@cengage.com. Library of Congress Control Number: 2014951195 Student Edition: ISBN: 9781285741550 Looseleaf Edition: ISBN: 9781305272354 Cengage Learning 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with oﬃce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local oﬃce at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. Windows is a registered trademark of the Microsoft Corporation and used herein under license. Macintosh is a registered trademark of Apple Computer, Inc. Used herein under license. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. Tools for Enriching Calculus is a trademark used herein under license. Printed in the United States of America Print Number: 01 Print Year: 2014 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. K12T14 To learn more about Cengage Learning Solutions, visit www.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com. Contents PREFACE xi TO THE STUDENT xxiii CALCULATORS, COMPUTERS, AND OTHER GRAPHING DEVICES DIAGNOSTIC TESTS xxiv xxvi A Preview of Calculus 1 1 1.1 1.2 1.3 1.4 1.5 Four Ways to Represent a Function 10 Mathematical Models: A Catalog of Essential Functions 23 New Functions from Old Functions 36 Exponential Functions 45 Inverse Functions and Logarithms 55 Review 68 Principles of Problem Solving 71 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 The Tangent and Velocity Problems 78 The Limit of a Function 83 Calculating Limits Using the Limit Laws 95 The Precise Definition of a Limit 104 Continuity 114 Limits at Infinity; Horizontal Asymptotes 126 Derivatives and Rates of Change 140 8SJUJOH1SPKFDU t Early Methods for Finding Tangents 152 The Derivative as a Function 152 Review 165 Problems Plus 169 iii Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. iv Contents 3 3.1 Derivatives of Polynomials and Exponential Functions 172 "QQMJFE1SPKFDU t Building a Better Roller Coaster 182 3.2 The Product and Quotient Rules 183 3.3 Derivatives of Trigonometric Functions 190 3.4 The Chain Rule 197 "QQMJFE1SPKFDU t Where Should a Pilot Start Descent? 208 3.5 Implicit Diﬀerentiation 208 BCPSBUPSZ1SPKFDU t Families of Implicit Curves 217 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 "QQMJFE1SPKFDU t Controlling Red Blood Cell Loss During Surgery 244 3.9 Related Rates 245 3.10 Linear Approximations and Diﬀerentials 251 BCPSBUPSZ1SPKFDU t Taylor Polynomials 258 3.11 Hyperbolic Functions 259 Review 266 Problems Plus 270 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Maximum and Minimum Values 276 "QQMJFE1SPKFDU t The Calculus of Rainbows 285 The Mean Value Theorem 287 How Derivatives Aﬀect the Shape of a Graph 293 Indeterminate Forms and l’Hospital’s Rule 304 8SJUJOH1SPKFDU t The Origins of l’Hospital’s Rule 314 Summary of Curve Sketching 315 Graphing with Calculus and Calculators 323 Optimization Problems 330 "QQMJFE1SPKFDU t The Shape of a Can 343 "QQMJFE1SPKFDU t Planes and Birds: Minimizing Energy 344 Newton’s Method 345 Antiderivatives 350 Review 358 Problems Plus 363 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents 5 5.1 5.2 5.3 5.4 5.5 Areas and Distances 366 The Definite Integral 378 %JTDPWFSZ1SPKFDU t Area Functions 391 The Fundamental Theorem of Calculus 392 Indefinite Integrals and the Net Change Theorem 402 8SJUJOH1SPKFDU t Newton, Leibniz, and the Invention of Calculus 411 The Substitution Rule 412 Review 421 Problems Plus 425 6 6.1 6.2 6.3 6.4 6.5 Areas Between Curves 428 "QQMJFE1SPKFDU t The Gini Index 436 Volumes 438 Volumes by Cylindrical Shells 449 Work 455 Average Value of a Function 461 "QQMJFE1SPKFDU t Calculus and Baseball 464 "QQMJFE1SPKFDU t Where to Sit at the Movies 465 Review 466 Problems Plus 468 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Integration by Parts 472 Trigonometric Integrals 479 Trigonometric Substitution 486 Integration of Rational Functions by Partial Fractions 493 Strategy for Integration 503 Integration Using Tables and Computer Algebra Systems 508 %JTDPWFSZ1SPKFDU t Patterns in Integrals 513 Approximate Integration 514 Improper Integrals 527 Review 537 Problems Plus 540 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. v vi Contents 8 8.1 8.2 8.3 8.4 8.5 Arc Length 544 %JTDPWFSZ1SPKFDU t Arc Length Contest 550 Area of a Surface of Revolution 551 %JTDPWFSZ1SPKFDU t Rotating on a Slant 557 Applications to Physics and Engineering 558 %JTDPWFSZ1SPKFDU t Complementary Coﬀee Cups 568 Applications to Economics and Biology 569 Probability 573 Review 581 Problems Plus 583 9 9.1 9.2 9.3 9.4 9.5 9.6 Modeling with Diﬀerential Equations 586 Direction Fields and Euler’s Method 591 Separable Equations 599 "QQMJFE1SPKFDU t How Fast Does a Tank Drain? 608 "QQMJFE1SPKFDU t Which Is Faster, Going Up or Coming Down? 609 Models for Population Growth 610 Linear Equations 620 PredatorPrey Systems 627 Review 634 Problems Plus 637 10 10.1 10.2 10.3 10.4 Curves Defined by Parametric Equations 640 BCPSBUPSZ1SPKFDU t Running Circles Around Circles 648 Calculus with Parametric Curves 649 BCPSBUPSZ1SPKFDU t Bézier Curves 657 Polar Coordinates 658 BCPSBUPSZ1SPKFDU t Families of Polar Curves 668 Areas and Lengths in Polar Coordinates 669 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents 10.5 10.6 Conic Sections 674 Conic Sections in Polar Coordinates 682 Review 689 Problems Plus 692 11 11.1 Sequences 694 BCPSBUPSZ1SPKFDU t Logistic Sequences 707 11.2 Series 707 11.3 The Integral Test and Estimates of Sums 719 11.4 The Comparison Tests 727 11.5 Alternating Series 732 11.6 Absolute Convergence and the Ratio and Root Tests 737 11.7 Strategy for Testing Series 744 11.8 Power Series 746 11.9 Representations of Functions as Power Series 752 11.10 Taylor and Maclaurin Series 759 BCPSBUPSZ1SPKFDU t An Elusive Limit 773 8SJUJOH1SPKFDU t How Newton Discovered the Binomial Series 773 11.11 Applications of Taylor Polynomials 774 "QQMJFE1SPKFDU t Radiation from the Stars 783 Review 784 Problems Plus 787 12 12.1 12.2 12.3 12.4 12.5 12.6 ThreeDimensional Coordinate Systems 792 Vectors 798 The Dot Product 807 The Cross Product 814 %JTDPWFSZ1SPKFDU t The Geometry of a Tetrahedron 823 Equations of Lines and Planes 823 BCPSBUPSZ1SPKFDU t Putting 3D in Perspective 833 Cylinders and Quadric Surfaces 834 Review 841 Problems Plus 844 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. vii viii Contents 13 13.1 13.2 13.3 13.4 Vector Functions and Space Curves 848 Derivatives and Integrals of Vector Functions 855 Arc Length and Curvature 861 Motion in Space: Velocity and Acceleration 870 "QQMJFE1SPKFDU t Kepler’s Laws 880 Review 881 Problems Plus 884 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 Functions of Several Variables 888 Limits and Continuity 903 Partial Derivatives 911 Tangent Planes and Linear Approximations 927 "QQMJFE1SPKFDU t The Speedo LZR Racer 936 The Chain Rule 937 Directional Derivatives and the Gradient Vector 946 Maximum and Minimum Values 959 "QQMJFE1SPKFDU t Designing a Dumpster 970 %JTDPWFSZ1SPKFDU t Quadratic Approximations and Critical Points 970 Lagrange Multipliers 971 "QQMJFE1SPKFDU t Rocket Science 979 "QQMJFE1SPKFDU t HydroTurbine Optimization 980 Review 981 Problems Plus 985 15 15.1 15.2 15.3 15.4 15.5 Double Integrals over Rectangles 988 Double Integrals over General Regions 1001 Double Integrals in Polar Coordinates 1010 Applications of Double Integrals 1016 Surface Area 1026 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Contents 15.6 Triple Integrals 1029 %JTDPWFSZ1SPKFDU t Volumes of Hyperspheres 1040 15.7 Triple Integrals in Cylindrical Coordinates 1040 %JTDPWFSZ1SPKFDU t The Intersection of Three Cylinders 1044 15.8 Triple Integrals in Spherical Coordinates 1045 "QQMJFE1SPKFDU t Roller Derby 1052 15.9 Change of Variables in Multiple Integrals 1052 Review 1061 Problems Plus 1065 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 Vector Fields 1068 Line Integrals 1075 The Fundamental Theorem for Line Integrals 1087 Green’s Theorem 1096 Curl and Divergence 1103 Parametric Surfaces and Their Areas 1111 Surface Integrals 1122 Stokes’ Theorem 1134 8SJUJOH1SPKFDU t Three Men and Two Theorems 1140 16.9 The Divergence Theorem 1141 16.10 Summary 1147 Review 1148 Problems Plus 1151 17 17.1 17.2 17.3 17.4 SecondOrder Linear Equations 1154 Nonhomogeneous Linear Equations 1160 Applications of SecondOrder Diﬀerential Equations 1168 Series Solutions 1176 Review 1181 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. ix x Contents A B C D E F G H I Numbers, Inequalities, and Absolute Values A2 Coordinate Geometry and Lines A10 Graphs of SecondDegree Equations A16 Trigonometry A24 Sigma Notation A34 Proofs of Theorems A39 The Logarithm Defined as an Integral A50 Complex Numbers A57 Answers to OddNumbered Exercises A65 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Preface A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. G E O R G E P O LYA The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition, as in the first seven editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement. The emphasis is on understanding concepts. I think that nearly everybody agrees that this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation: Focus on conceptual understanding. I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently, the Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well. In writing the eighth edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform, but within the context of a traditional curriculum. I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions. ● ● ● Calculus, Eighth Edition, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the second semester. Essential Calculus, Second Edition, is a much briefer book (840 pages), though it contains almost all of the topics in Calculus, Eighth Edition. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website. Essential Calculus: Early Transcendentals, Second Edition, resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3. xi Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xii Preface ● ● ● ● ● Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters. Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking engineering and physics courses concurrently with calculus. Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences. Biocalculus: Calculus for the Life Sciences is intended to show students in the life sciences how calculus relates to biology. Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all the content of Biocalculus: Calculus for the Life Sciences as well as three additional chapters covering probability and statistics. The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I’ve incorporated into this edition: ● ● ● ● ● The data in examples and exercises have been updated to be more timely. New examples have been added (see Examples 6.1.5, 11.2.5, and 14.3.3, for instance). And the solutions to some of the existing examples have been amplified. Three new projects have been added: The project Controlling Red Blood Cell Loss During Surgery (page 244) describes the ANH procedure, in which blood is extracted from the patient before an operation and is replaced by saline solution. This dilutes the patient’s blood so that fewer red blood cells are lost during bleeding and the extracted blood is returned to the patient after surgery. The project Planes and Birds: Minimizing Energy (page 344) asks how birds can minimize power and energy by flapping their wings versus gliding. In the project The Speedo LZR Racer (page 936) it is explained that this suit reduces drag in the water and, as a result, many swimming records were broken. Students are asked why a small decrease in drag can have a big effect on performance. I have streamlined Chapter 15 (Multiple Integrals) by combining the first two sections so that iterated integrals are treated earlier. More than 20% of the exercises in each chapter are new. Here are some of my favorites: 2.7.61, 2.8.36–38, 3.1.79–80, 3.11.54, 4.1.69, 4.3.34, 4.3.66, 4.4.80, 4.7.39, 4.7.67, 5.1.19–20, 5.2.67–68, 5.4.70, 6.1.51, 8.1.39, 12.5.81, 12.6.29–30, 14.6.65–66. In addition, there are some good new Problems Plus. (See Problems 12–14 on page 272, Problem 13 on page 363, Problems 16–17 on page 426, and Problem 8 on page 986.) Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Preface xiii Conceptual Exercises The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 2.2, 2.5, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a TrueFalse Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.35–38, 2.8.47–52, 9.1.11–13, 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–38, 14.1.41–44, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.6–8, 16.1.11–18, 16.2.17–18, and 16.3.1–2). Another type of exercise uses verbal description to test conceptual understanding (see Exercises 2.5.10, 2.8.66, 4.3.69–70, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.45–46, 3.7.27, and 9.4.4). Graded Exercise Sets Each exercise set is carefully graded, progressing from basic conceptual exercises and skilldevelopment problems to more challenging problems involving applications and proofs. RealWorld Data My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting realworld data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.35 (unemployment rates), Exercise 5.1.16 (velocity of the space shuttle Endeavour), and Figure 4 in Section 5.4 (San Francisco power consumption). Functions of two variables are illustrated by a table of values of the windchill index as a function of air temperature and wind speed (Example 14.1.2). Partial derivatives are introduced in Section 14.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 14.4.3). Directional derivatives are introduced in Section 14.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December 20–21, 2006 (Example 15.1.9). Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns. Projects One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) The project after Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xiv Preface velocity. Laboratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projects ask students to compare presentday methods with those of the founders of calculus—Fermat’s method for finding tangents, for instance. Suggested references are supplied. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 7.6). Others explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6), and intersections of three cylinders (after Section 15.7). Additional projects can be found in the Instructor’s Guide (see, for instance, Group Exercise 5.1: Position from Samples). Problem Solving Students usually have difficulties with problems for which there is no single welldefined procedure for obtaining the answer. I think nobody has improved very much on George Polya’s fourstage problemsolving strategy and, accordingly, I have included a version of his problemsolving principles following Chapter 1. They are applied, both explicitly and implicitly, throughout the book. After the other chapters I have placed sections called Problems Plus, which feature examples of how to tackle challenging calculus problems. In selecting the varied problems for these sections I kept in mind the following advice from David Hilbert: “A mathematical problem should be difficult in order to entice us, yet not inaccessible lest it mock our efforts.” When I put these challenging problems on assignments and tests I grade them in a different way. Here I reward a student significantly for ideas toward a solution and for recognizing which problemsolving principles are relevant. Technology The availability of technology makes it not less important but more important to clearly understand the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology and I use two special symbols to indicate clearly when a particular type of machine is required. The icon ; indicates an exercise that definitely requires the use of such technology, but that is not to say that it can’t be used on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Maple, Mathematica, or the TI89) are required. But technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate. Tools for Enriching Calculus TEC is a companion to the text and is intended to enrich and complement its contents. (It is now accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC Modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Preface xv TEC also includes Homework Hints for representative exercises (usually oddnumbered) in every section of the text, indicated by printing the exercise number in red. These hints are usually presented in the form of questions and try to imitate an effective teaching assistant by functioning as a silent tutor. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress. Enhanced WebAssign Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the Eighth Edition we have been working with the calculus community and WebAssign to develop an online homework system. Up to 70% of the exercises in each section are assignable as online homework, including free response, multiple choice, and multipart formats. The system also includes Active Examples, in which students are guided in stepbystep tutorials through text examples, with links to the textbook and to video solutions. Website Visit CengageBrain.com or stewartcalculus.com for these additional materials: ● Homework Hints ● Algebra Review ● Lies My Calculator and Computer Told Me ● History of Mathematics, with links to the better historical websites ● ● Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes Archived Problems (Drill exercises that appeared in previous editions, together with their solutions) ● Challenge Problems (some from the Problems Plus sections from prior editions) ● Links, for particular topics, to outside Web resources ● Selected Visuals and Modules from Tools for Enriching Calculus (TEC) Diagnostic Tests The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry. A Preview of Calculus This is an overview of the subject and includes a list of questions to motivate the study of calculus. 1 Functions and Models From the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view. 2 Limits and Derivatives The material on limits is motivated by a prior discussion of the tangent and velocity problems. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Section 2.4, on the precise definition of a limit, is an optional section. Sections Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xvi Preface 2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3. Here the examples and exercises explore the meanings of derivatives in various contexts. Higher derivatives are introduced in Section 2.8. 3 Differentiation Rules All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here. When derivatives are computed in applied situations, students are asked to explain their meanings. Exponential growth and decay are now covered in this chapter. 4 Applications of Differentiation The basic facts concerning extreme values and shapes of curves are deduced from the Mean Value Theorem. Graphing with technology emphasizes the interaction between calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to raise your head 42° to see the top of a rainbow. 5 Integrals The area problem and the distance problem serve to motivate the definite integral, with sigma notation introduced as needed. (Full coverage of sigma notation is provided in Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables. 6 Applications of Integration Here I present the applications of integration—area, volume, work, average value—that can reasonably be done without specialized techniques of integration. General methods are emphasized. The goal is for students to be able to divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit as an integral. 7 Techniques of Integration All the standard methods are covered but, of course, the real challenge is to be able to recognize which technique is best used in a given situation. Accordingly, in Section 7.5, I present a strategy for integration. The use of computer algebra systems is discussed in Section 7.6. 8 Further Applications of Integration Here are the applications of integration—arc length and surface area—for which it is useful to have available all the techniques of integration, as well as applications to biology, economics, and physics (hydrostatic force and centers of mass). I have also included a section on probability. There are more applications here than can realistically be covered in a given course. Instructors should select applications suitable for their students and for which they themselves have enthusiasm. 9 Differential Equations Modeling is the theme that unifies this introductory treatment of differential equations. Direction fields and Euler’s method are studied before separable and linear equations are solved explicitly, so that qualitative, numerical, and analytic approaches are given equal consideration. These methods are applied to the exponential, logistic, and other models for population growth. The first four or five sections of this chapter serve as a good introduction to firstorder differential equations. An optional final section uses predatorprey models to illustrate systems of differential equations. 10 Parametric Equations and Polar Coordinates This chapter introduces parametric and polar curves and applies the methods of calculus to them. Parametric curves are well suited to laboratory projects; the two presented here involve families of curves and Bézier curves. A brief treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in Chapter 13. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Preface xvii 11 Infinite Sequences and Series The convergence tests have intuitive justifications (see page 719) as well as formal proofs. Numerical estimates of sums of series are based on which test was used to prove convergence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices. 12 Vectors and the Geometry of Space The material on threedimensional analytic geometry and vectors is divided into two chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces. 13 Vector Functions This chapter covers vectorvalued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler’s laws. 14 Partial Derivatives Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. 15 Multiple Integrals Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders. Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals. 16 Vector Calculus Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized. 17 SecondOrder Differential Equations Since firstorder differential equations are covered in Chapter 9, this final chapter deals with secondorder linear differential equations, their application to vibrating springs and electric circuits, and series solutions. Calculus, Early Transcendentals, Eighth Edition, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to enhance student understanding and to facilitate creative instruction. The tables on pages xxi–xxii describe each of these ancillaries. The preparation of this and previous editions has involved much time spent reading the reasoned (but sometimes contradictory) advice from a large number of astute reviewers. I greatly appreciate the time they spent to understand my motivation for the approach taken. I have learned something from each of them. Eighth Edition Reviewers Jay Abramson, Arizona State University Adam Bowers, University of California San Diego Neena Chopra, The Pennsylvania State University Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xviii Preface Edward Dobson, Mississippi State University Isaac Goldbring, University of Illinois at Chicago Lea Jenkins, Clemson University Rebecca Wahl, Butler University Technology Reviewers Maria Andersen, Muskegon Community College Eric Aurand, Eastfield College Joy Becker, University of Wisconsin–Stout Przemyslaw Bogacki, Old Dominion University Amy Elizabeth Bowman, University of Alabama in Huntsville Monica Brown, University of Missouri–St. Louis Roxanne Byrne, University of Colorado at Denver and Health Sciences Center Teri Christiansen, University of Missouri–Columbia Bobby Dale Daniel, Lamar University Jennifer Daniel, Lamar University Andras Domokos, California State University, Sacramento Timothy Flaherty, Carnegie Mellon University Lee Gibson, University of Louisville Jane Golden, Hillsborough Community College Semion Gutman, University of Oklahoma Diane Hoffoss, University of San Diego Lorraine Hughes, Mississippi State University Jay Jahangiri, Kent State University John Jernigan, Community College of Philadelphia Brian Karasek, South Mountain Community College Jason Kozinski, University of Florida Carole Krueger, The University of Texas at Arlington Ken Kubota, University of Kentucky John Mitchell, Clark College Donald Paul, Tulsa Community College Chad Pierson, University of Minnesota, Duluth Lanita Presson, University of Alabama in Huntsville Karin Reinhold, State University of New York at Albany Thomas Riedel, University of Louisville Christopher Schroeder, Morehead State University Angela Sharp, University of Minnesota, Duluth Patricia Shaw, Mississippi State University Carl Spitznagel, John Carroll University Mohammad Tabanjeh, Virginia State University Capt. Koichi Takagi, United States Naval Academy Lorna TenEyck, Chemeketa Community College Roger Werbylo, Pima Community College David Williams, Clayton State University Zhuan Ye, Northern Illinois University Previous Edition Reviewers B. D. Aggarwala, University of Calgary John Alberghini, Manchester Community College Michael Albert, CarnegieMellon University Daniel Anderson, University of Iowa Amy Austin, Texas A&M University Donna J. Bailey, Northeast Missouri State University Wayne Barber, Chemeketa Community College Marilyn Belkin, Villanova University Neil Berger, University of Illinois, Chicago David Berman, University of New Orleans Anthony J. Bevelacqua, University of North Dakota Richard Biggs, University of Western Ontario Robert Blumenthal, Oglethorpe University Martina Bode, Northwestern University Barbara Bohannon, Hofstra University Jay Bourland, Colorado State University Philip L. Bowers, Florida State University Amy Elizabeth Bowman, University of Alabama in Huntsville Stephen W. Brady, Wichita State University Michael Breen, Tennessee Technological University Robert N. Bryan, University of Western Ontario David Buchthal, University of Akron Jenna Carpenter, Louisiana Tech University Jorge Cassio, MiamiDade Community College Jack Ceder, University of California, Santa Barbara Scott Chapman, Trinity University ZhenQing Chen, University of Washington—Seattle James Choike, Oklahoma State University Barbara Cortzen, DePaul University Carl Cowen, Purdue University Philip S. Crooke, Vanderbilt University Charles N. Curtis, Missouri Southern State College Daniel Cyphert, Armstrong State College Robert Dahlin M. Hilary Davies, University of Alaska Anchorage Gregory J. Davis, University of Wisconsin–Green Bay Elias Deeba, University of Houston–Downtown Daniel DiMaria, Suffolk Community College Seymour Ditor, University of Western Ontario Greg Dresden, Washington and Lee University Daniel Drucker, Wayne State University Kenn Dunn, Dalhousie University Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Preface Dennis Dunninger, Michigan State University Bruce Edwards, University of Florida David Ellis, San Francisco State University John Ellison, Grove City College Martin Erickson, Truman State University Garret Etgen, University of Houston Theodore G. Faticoni, Fordham University Laurene V. Fausett, Georgia Southern University Norman Feldman, Sonoma State University Le Baron O. Ferguson, University of California—Riverside Newman Fisher, San Francisco State University José D. Flores, The University of South Dakota William Francis, Michigan Technological University James T. Franklin, Valencia Community College, East Stanley Friedlander, Bronx Community College Patrick Gallagher, Columbia University–New York Paul Garrett, University of Minnesota–Minneapolis Frederick Gass, Miami University of Ohio Bruce Gilligan, University of Regina Matthias K. Gobbert, University of Maryland, Baltimore County Gerald Goff, Oklahoma State University Stuart Goldenberg, California Polytechnic State University John A. Graham, Buckingham Browne & Nichols School Richard Grassl, University of New Mexico Michael Gregory, University of North Dakota Charles Groetsch, University of Cincinnati Paul Triantafilos Hadavas, Armstrong Atlantic State University Salim M. Haïdar, Grand Valley State University D. W. Hall, Michigan State University Robert L. Hall, University of Wisconsin–Milwaukee Howard B. Hamilton, California State University, Sacramento Darel Hardy, Colorado State University Shari Harris, John Wood Community College Gary W. Harrison, College of Charleston Melvin Hausner, New York University/Courant Institute Curtis Herink, Mercer University Russell Herman, University of North Carolina at Wilmington Allen Hesse, Rochester Community College Randall R. Holmes, Auburn University James F. Hurley, University of Connecticut Amer Iqbal, University of Washington—Seattle Matthew A. Isom, Arizona State University Gerald Janusz, University of Illinois at UrbanaChampaign John H. Jenkins, EmbryRiddle Aeronautical University, Prescott Campus Clement Jeske, University of Wisconsin, Platteville Carl Jockusch, University of Illinois at UrbanaChampaign Jan E. H. Johansson, University of Vermont Jerry Johnson, Oklahoma State University Zsuzsanna M. Kadas, St. Michael’s College Nets Katz, Indiana University Bloomington Matt Kaufman Matthias Kawski, Arizona State University Frederick W. Keene, Pasadena City College Robert L. Kelley, University of Miami Akhtar Khan, Rochester Institute of Technology xix Marianne Korten, Kansas State University Virgil Kowalik, Texas A&I University Kevin Kreider, University of Akron Leonard Krop, DePaul University Mark Krusemeyer, Carleton College John C. Lawlor, University of Vermont Christopher C. Leary, State University of New York at Geneseo David Leeming, University of Victoria Sam Lesseig, Northeast Missouri State University Phil Locke, University of Maine Joyce Longman, Villanova University Joan McCarter, Arizona State University Phil McCartney, Northern Kentucky University Igor Malyshev, San Jose State University Larry Mansfield, Queens College Mary Martin, Colgate University Nathaniel F. G. Martin, University of Virginia Gerald Y. Matsumoto, American River College James McKinney, California State Polytechnic University, Pomona Tom Metzger, University of Pittsburgh Richard Millspaugh, University of North Dakota Lon H. Mitchell, Virginia Commonwealth University Michael Montaño, Riverside Community College Teri Jo Murphy, University of Oklahoma Martin Nakashima, California State Polytechnic University, Pomona Ho Kuen Ng, San Jose State University Richard Nowakowski, Dalhousie University Hussain S. Nur, California State University, Fresno Norma OrtizRobinson, Virginia Commonwealth University Wayne N. Palmer, Utica College Vincent Panico, University of the Pacific F. J. Papp, University of Michigan–Dearborn Mike Penna, Indiana University–Purdue University Indianapolis Mark Pinsky, Northwestern University Lothar Redlin, The Pennsylvania State University Joel W. Robbin, University of Wisconsin–Madison Lila Roberts, Georgia College and State University E. Arthur Robinson, Jr., The George Washington University Richard Rockwell, Pacific Union College Rob Root, Lafayette College Richard Ruedemann, Arizona State University David Ryeburn, Simon Fraser University Richard St. Andre, Central Michigan University Ricardo Salinas, San Antonio College Robert Schmidt, South Dakota State University Eric Schreiner, Western Michigan University Mihr J. Shah, Kent State University–Trumbull Qin Sheng, Baylor University Theodore Shifrin, University of Georgia Wayne Skrapek, University of Saskatchewan Larry Small, Los Angeles Pierce College Teresa Morgan Smith, Blinn College William Smith, University of North Carolina Donald W. Solomon, University of Wisconsin–Milwaukee Edward Spitznagel, Washington University Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xx Preface Joseph Stampfli, Indiana University Kristin Stoley, Blinn College M. B. Tavakoli, Chaffey College Magdalena Toda, Texas Tech University Ruth Trygstad, Salt Lake Community College Paul Xavier Uhlig, St. Mary’s University, San Antonio Stan Ver Nooy, University of Oregon Andrei Verona, California State University–Los Angeles Klaus Volpert, Villanova University Russell C. Walker, Carnegie Mellon University William L. Walton, McCallie School Peiyong Wang, Wayne State University Jack Weiner, University of Guelph Alan Weinstein, University of California, Berkeley Theodore W. Wilcox, Rochester Institute of Technology Steven Willard, University of Alberta Robert Wilson, University of Wisconsin–Madison Jerome Wolbert, University of Michigan–Ann Arbor Dennis H. Wortman, University of Massachusetts, Boston Mary Wright, Southern Illinois University–Carbondale Paul M. Wright, Austin Community College Xian Wu, University of South Carolina In addition, I would like to thank R. B. Burckel, Bruce Colletti, David Behrman, John Dersch, Gove Effinger, Bill Emerson, Dan Kalman, Quyan Khan, Alfonso GraciaSaz, Allan MacIsaac, Tami Martin, Monica Nitsche, Lamia Raffo, Norton Starr, and Jim Trefzger for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; George Bergman, David Bleecker, Dan Clegg, Victor Kaftal, Anthony Lam, Jamie Lawson, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, Philip Straffin, and Klaus Volpert for ideas for projects; Dan Anderson, Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proofreading; Andy BulmanFleming, Lothar Redlin, Gina Sanders, and Saleem Watson for additional proofreading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the answer manuscript. In addition, I thank those who have contributed to past editions: Ed Barbeau, George Bergman, Fred Brauer, Andy BulmanFleming, Bob Burton, David Cusick, Tom DiCiccio, Garret Etgen, Chris Fisher, Leon Gerber, Stuart Goldenberg, Arnold Good, Gene Hecht, Harvey Keynes, E. L. Koh, Zdislav Kovarik, Kevin Kreider, Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Mary Pugh, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Dusty Sabo, Doug Shaw, Dan Silver, Simon Smith, Saleem Watson, Alan Weinstein, and Gail Wolkowicz. I also thank Kathi Townes, Stephanie Kuhns, Kristina Elliott, and Kira Abdallah of TECHarts for their production services and the following Cengage Learning staff: Cheryll Linthicum, content project manager; Stacy Green, senior content developer; Samantha Lugtu, associate content developer; Stephanie Kreuz, product assistant; Lynh Pham, media developer; Ryan Ahern, marketing manager; and Vernon Boes, art director. They have all done an outstanding job. I have been very fortunate to have worked with some of the best mathematics editors in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, Richard Stratton, Liz Covello, and now Neha Taleja. All of them have contributed greatly to the success of this book. james stewart Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Instructor’s Guide by Douglas Shaw ISBN 9781305393714 Each section of the text is discussed from several viewpoints. The Instructor’s Guide contains suggested time to allot, points to stress, text discussion topics, core materials for lecture, workshop/discussion suggestions, group work exercises in a form suitable for handout, and suggested homework assignments. Complete Solutions Manual Single Variable Early Transcendentals By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker ISBN 9781305272392 Multivariable By Dan Clegg and Barbara Frank ISBN 9781305276116 Includes workedout solutions to all exercises in the text. Printed Test Bank By William Steven Harmon ISBN 9781305387225 Contains textspecific multiplechoice and free response test items. Cengage Learning Testing Powered by Cognero (login.cengage.com) This flexible online system allows you to author, edit, and manage test bank content from multiple Cengage Learning solutions; create multiple test versions in an instant; and deliver tests from your LMS, your classroom, or wherever you want. TEC TOOLS FOR ENRICHING™ CALCULUS By James Stewart, Harvey Keynes, Dan Clegg, and developer Hubert Hohn Tools for Enriching Calculus (TEC) functions as both a powerful tool for instructors and as a tutorial environment in which students can explore and review selected topics. The Flash simulation modules in TEC include instructions, written and audio explanations of the concepts, and exercises. TEC is accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com. Enhanced WebAssign® www.webassign.net Printed Access Code: ISBN 9781285858265 Instant Access Code ISBN: 9781285858258 Exclusively from Cengage Learning, Enhanced WebAssign offers an extensive online program for Stewart’s Calculus to encourage the practice that is so critical for concept mastery. The meticulously crafted pedagogy and exercises in our proven texts become even more effective in Enhanced WebAssign, supplemented by multimedia tutorial support and immediate feedback as students complete their assignments. Key features include: ■ ■ ■ ■ ■ ■ Stewart Website www.stewartcalculus.com Contents: Homework Hints ■ Algebra Review ■ Additional Topics ■ Drill exercises ■ Challenge Problems ■ Web Links ■ History of Mathematics ■ Tools for Enriching Calculus (TEC) ■ Electronic items ■ Printed items ■ ■ Thousands of homework problems that match your textbook’s endofsection exercises Opportunities for students to review prerequisite skills and content both at the start of the course and at the beginning of each section Read It eBook pages, Watch It videos, Master It tutorials, and Chat About It links A customizable Cengage YouBook with highlighting, notetaking, and search features, as well as links to multimedia resources Personal Study Plans (based on diagnostic quizzing) that identify chapter topics that students will need to master A WebAssign Answer Evaluator that recognizes and accepts equivalent mathematical responses in the same way an instructor grades A Show My Work feature that gives instructors the option of seeing students’ detailed solutions Visualizing Calculus Animations, Lecture Videos, and more (Table continues on page xxii) xxi Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Cengage Customizable YouBook YouBook is an eBook that is both interactive and customizable. Containing all the content from Stewart’s Calculus, YouBook features a text edit tool that allows instructors to modify the textbook narrative as needed. With YouBook, instructors can quickly reorder entire sections and chapters or hide any content they don’t teach to create an eBook that perfectly matches their syllabus. Instructors can further customize the text by adding instructorcreated or YouTube video links. Additional media assets include animated figures, video clips, highlighting and notetaking features, and more. YouBook is available within Enhanced WebAssign. CourseMate CourseMate is a perfect selfstudy tool for students, and requires no set up from instructors. CourseMate brings course concepts to life with interactive learning, study, and exam preparation tools that support the printed textbook. CourseMate for Stewart’s Calculus includes an interactive eBook, Tools for Enriching Calculus, videos, quizzes, flashcards, and more. For instructors, CourseMate includes Engagement Tracker, a firstofitskind tool that monitors student engagement. CengageBrain.com To access additional course materials, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found. Student Solutions Manual Single Variable Early Transcendentals By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker ISBN 9781305272422 Multivariable By Dan Clegg and Barbara Frank ISBN 9781305271821 Provides completely workedout solutions to all oddnumbered exercises in the text, giving students a chance to ■ Electronic items check their answer and ensure they took the correct steps to arrive at the answer. The Student Solutions Manual can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN. Study Guide Single Variable Early Transcendentals By Richard St. Andre ISBN 9781305279148 Multivariable By Richard St. Andre ISBN 9781305271845 For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, and summary and focus questions with explained answers. The Study Guide also contains “Technology Plus” questions and multiplechoice “On Your Own” examstyle questions. The Study Guide can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN. A Companion to Calculus By Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and Kay Somers ISBN 9780495011248 Written to improve algebra and problemsolving skills of students taking a calculus course, every chapter in this companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to understand and solve calculus problems related to that topic. It is designed for calculus courses that integrate the review of precalculus concepts or for individual use. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN. Linear Algebra for Calculus by Konrad J. Heuvers, William P. Francis, John H. Kuisti, Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner ISBN 9780534252489 This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic ideas of linear algebra. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN. ■ Printed items xxii Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. To the Student Reading a calculus textbook is different from reading a newspaper or a novel, or even a physics book. Don’t be discouraged if you have to read a passage more than once in order to understand it. You should have pencil and paper and calculator at hand to sketch a diagram or make a calculation. Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises. In particular, you should look at the definitions to see the exact meanings of the terms. And before you read each example, I suggest that you cover up the solution and try solving the problem yourself. You’ll get a lot more from looking at the solution if you do so. Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected, stepbystep fashion with explanatory sentences— not just a string of disconnected equations or formulas. The answers to the oddnumbered exercises appear at the back of the book, in Appendix I. Some exercises ask for a verbal explanation or interpretation or description. In such cases there is no single correct way of expressing the answer, so don’t worry that you haven’t found the definitive answer. In addition, there are often several different forms in which to express a numerical or algebraic answer, so if your answer differs from mine, don’t immediately assume you’re wrong. For example, if the answer given in the back of the book is s2 2 1 and you obtain 1y (1 1 s2 ), then you’re right and rationalizing the denominator will show that the answers are equivalent. The icon ; indicates an exercise that definitely requires the use of either a graphing calculator or a computer with graphing software. But that doesn’t mean that graphing devices can’t be used to check your work on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Maple, Mathematica, or the TI89) are required. You will also encounter the symbol , which warns you against committing an error. I have placed this symbol in the margin in situations where I have observed that a large proportion of my students tend to make the same mistake. Tools for Enriching Calculus, which is a companion to this text, is referred to by means of the symbol TEC and can be accessed in the eBook via Enhanced WebAssign and CourseMate (selected Visuals and Modules are available at www.stewartcalculus.com). It directs you to modules in which you can explore aspects of calculus for which the computer is particularly useful. You will notice that some exercise numbers are printed in red: 5. This indicates that Homework Hints are available for the exercise. These hints can be found on stewartcalculus.com as well as Enhanced WebAssign and CourseMate. The homework hints ask you questions that allow you to make progress toward a solution without actually giving you the answer. You need to pursue each hint in an active manner with pencil and paper to work out the details. If a particular hint doesn’t enable you to solve the problem, you can click to reveal the next hint. I recommend that you keep this book for reference purposes after you finish the course. Because you will likely forget some of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses. And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer. Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. I hope you will discover that it is not only useful but also intrinsically beautiful. JAMES STEWART xxiii Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Calculators, Computers, and Other Graphing Devices xxiv © Dan Clegg You can also use computer software such as Graphing Calculator by Pacific Tech (www.pacifict.com) to perform many of these functions, as well as apps for phones and tablets, like Quick Graph (Colombiamug) or MathStudio (Pomegranate Apps). Similar functionality is available using a web interface at WolframAlpha.com. © Dan Clegg © Dan Clegg Advances in technology continue to bring a wider variety of tools for doing mathematics. Handheld calculators are becoming more powerful, as are software programs and Internet resources. In addition, many mathematical applications have been released for smartphones and tablets such as the iPad. Some exercises in this text are marked with a graphing icon ; , which indicates that the use of some technology is required. Often this means that we intend for a graphing device to be used in drawing the graph of a function or equation. You might also need technology to find the zeros of a graph or the points of intersection of two graphs. In some cases we will use a calculating device to solve an equation or evaluate a definite integral numerically. Many scientific and graphing calculators have these features built in, such as the Texas Instruments TI84 or TINspire CX. Similar calculators are made by Hewlett Packard, Casio, and Sharp. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. The CAS icon is reserved for problems in which the full resources of a computer algebra system (CAS) are required. A CAS is capable of doing mathematics (like solving equations, computing derivatives or integrals) symbolically rather than just numerically. Examples of wellestablished computer algebra systems are the computer software packages Maple and Mathematica. The WolframAlpha website uses the Mathematica engine to provide CAS functionality via the Web. Many handheld graphing calculators have CAS capabilities, such as the TI89 and TINspire CX CAS from Texas Instruments. Some tablet and smartphone apps also provide these capabilities, such as the previously mentioned MathStudio. © Dan Clegg © Dan Clegg © Dan Clegg In general, when we use the term “calculator” in this book, we mean the use of any of the resources we have mentioned. xxv Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Diagnostic Tests Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas. After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided. A . Evaluate each expression without using a calculator. (a) s23d4 (b) 234 (c) 324 (d) 5 23 5 21 (e) SD 2 3 22 (f) 16 23y4 . Simplify each expression. Write your answer without negative exponents. (a) s200 2 s32 (b) s3a 3b 3 ds4ab 2 d 2 (c) S 3x 3y2 y 3 x 2 y21y2 D 22 . Expand and simplify. (a) 3sx 1 6d 1 4s2x 2 5d (b) sx 1 3ds4x 2 5d (c) ssa 1 sb dssa 2 sb d (d) s2x 1 3d2 (e) sx 1 2d3 . Factor each expression. (a) 4x 2 2 25 (c) x 3 2 3x 2 2 4x 1 12 (e) 3x 3y2 2 9x 1y2 1 6x 21y2 (b) 2x 2 1 5x 2 12 (d) x 4 1 27x (f) x 3 y 2 4xy . Simplify the rational expression. (a) x 2 1 3x 1 2 x2 2 x 2 2 (c) x2 x11 2 x2 2 4 x12 2x 2 2 x 2 1 x13 ? x2 2 9 2x 1 1 y x 2 x y (d) 1 1 2 y x (b) xxvi Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Diagnostic Tests . Rationalize the expression and simplify. s10 s5 2 2 (a) (b) . Rewrite by completing the square. (a) x 2 1 x 1 1 s4 1 h 2 2 h (b) 2x 2 2 12x 1 11 . Solve the equation. (Find only the real solutions.) (c) x 2 2 x 2 12 − 0 2x 2x 2 1 − x11 x (d) 2x 2 1 4x 1 1 − 0 (e) x 4 2 3x 2 1 2 − 0 (f) 3 x 2 4 − 10 (a) x 1 5 − 14 2 12 x (g) 2xs4 2 xd 21y2 (b)  2 3 s4 2 x − 0  . Solve each inequality. Write your answer using interval notation. (a) 24 , 5 2 3x < 17 (b) x 2 , 2x 1 8 (c) xsx 2 1dsx 1 2d . 0 (d) x 2 4 , 3  2x 2 3 (e) <1 x11 . State whether each equation is true or false. (a) s p 1 qd2 − p 2 1 q 2 (c) sa 2 1 b 2 − a 1 b 1 1 1 − 2 x2y x y (e)  (b) sab − sa sb 1 1 TC (d) −11T C 1yx 1 (f) − ayx 2 byx a2b ANSWERS TO DIAGNOSTIC TEST A: ALGEBRA . (a) 81 (d) 25 . (a) 6s2 (b) 281 (c) 9 4 (f) (e) (b) 48a 5b7 (c) 1 81 1 8 x 9y7 . (a) 11x 2 2 (b) 4x 2 1 7x 2 15 (c) a 2 b (d) 4x 2 1 12x 1 9 3 2 (e) x 1 6x 1 12x 1 8 . (a) s2x 2 5ds2x 1 5d (c) sx 2 3dsx 2 2dsx 1 2d (e) 3x21y2sx 2 1dsx 2 2d x12 x22 1 (c) x22 . (a) (b) s2x 2 3dsx 1 4d (d) xsx 1 3dsx 2 2 3x 1 9d (f) xysx 2 2dsx 1 2d (b) x21 x23 (d) 2sx 1 yd 1 . (a) 5s2 1 2s10 (b) . (a) s x 1 12 d 1 34 (b) 2sx 2 3d2 2 7 2 . (a) 6 (d) 21 6 (g) 12 5 1 2 s2 (b) 1 (c) 23, 4 (e) 61, 6s2 (f) 23 , 22 3 . (a) f24, 3d (c) s22, 0d ø s1, `d (e) s21, 4g . (a) False (d) False s4 1 h 1 2 (b) s22, 4d (d) s1, 7d (b) True (e) False (c) False (f) True If you had difficulty with these problems, you may wish to consult the Review of Algebra on the website www.stewartcalculus.com. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xxvii xxviii Diagnostic Tests B . Find an equation for the line that passes through the point s2, 25d and (a) has slope 23 (b) is parallel to the xaxis (c) is parallel to the yaxis (d) is parallel to the line 2x 2 4y − 3 . Find an equation for the circle that has center s21, 4d and passes through the point s3, 22d. . Find the center and radius of the circle with equation x 2 1 y 2 2 6x 1 10y 1 9 − 0. . Let As27, 4d and Bs5, 212d be points in the plane. (a) Find the slope of the line that contains A and B. (b) Find an equation of the line that passes through A and B. What are the intercepts? (c) Find the midpoint of the segment AB. (d) Find the length of the segment AB. (e) Find an equation of the perpendicular bisector of AB. (f) Find an equation of the circle for which AB is a diameter. . Sketch the region in the xyplane defined by the equation or inequalities. (a) 21 < y < 3 (c) y , 1 2 (b) 1 2x  x  , 4 and  y  , 2 (d) y > x 2 2 1 (e) x 2 1 y 2 , 4 (f) 9x 2 1 16y 2 − 144 ANSWERS TO DIAGNOSTIC TEST B: ANALYTIC GEOMETRY . (a) y − 23x 1 1 (c) x − 2 (b) y − 25 (d) y − 1 2x 26 . (a) (b) 3 . sx 1 1d2 1 s y 2 4d2 − 52 x _1 234 4x 1 3y 1 16 − 0; xintercept 24, yintercept 2 16 3 s21, 24d 20 3x 2 4y − 13 sx 1 1d2 1 s y 1 4d2 − 100 (d) y _4 1 1 4x 0 (e) y 2 0 y 0 y=1 2 x 2 x _2 y _1 (c) 2 0 . Center s3, 25d, radius 5 . (a) (b) (c) (d) (e) (f) y 1 x (f ) ≈+¥=4 0 y=≈1 2 x y 3 0 4 x If you had difficulty with these problems, you may wish to consult the review of analytic geometry in Appendixes B and C. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Diagnostic Tests xxix C y . The graph of a function f is given at the left. (a) State the value of f s21d. (b) Estimate the value of f s2d. (c) For what values of x is f sxd − 2? (d) Estimate the values of x such that f sxd − 0. (e) State the domain and range of f. 1 0 1 x . If f sxd − x 3, evaluate the difference quotient f s2 1 hd 2 f s2d and simplify your answer. h . Find the domain of the function. FIGURE FOR PROBLEM 1 (a) f sxd − 2x 1 1 x 1x22 (b) tsxd − 2 3 x s x 11 (c) hsxd − s4 2 x 1 sx 2 2 1 2 . How are graphs of the functions obtained from the graph of f ? (a) y − 2f sxd (b) y − 2 f sxd 2 1 (c) y − f sx 2 3d 1 2 . Without using a calculator, make a rough sketch of the graph. (a) y − x 3 (b) y − sx 1 1d3 (c) y − sx 2 2d3 1 3 (d) y − 4 2 x 2 (e) y − sx (f) y − 2 sx (g) y − 22 x (h) y − 1 1 x21 . Let f sxd − H 1 2 x 2 if x < 0 2x 1 1 if x . 0 (a) Evaluate f s22d and f s1d. (b) Sketch the graph of f. . If f sxd − x 2 1 2x 2 1 and tsxd − 2x 2 3, find each of the following functions. (a) f 8 t (b) t 8 f (c) t 8 t 8 t ANSWERS TO DIAGNOSTIC TEST C: FUNCTIONS . (a) 22 (c) 23, 1 (e) f23, 3g, f22, 3g (b) 2.8 (d) 22.5, 0.3 . (a) 0 . (a) Reflect about the xaxis (b) Stretch vertically by a factor of 2, then shift 1 unit downward (c) Shift 3 units to the right and 2 units upward (d) (g) 1 x _1 (e) 2 x (2, 3) x 0 1 x 1 x x 0 (f) y 0 (h) y y 0 1 y 1 0 _1 y 1 y 4 0 (c) y 1 . 12 1 6h 1 h 2 . (a) s2`, 22d ø s22, 1d ø s1, `d (b) s2`, `d (c) s2`, 21g ø f1, 4g (b) y 1 x 0 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. x xxx Diagnostic Tests . (a) 23, 3 (b) _1 y . (a) s f 8 tdsxd − 4x 2 2 8x 1 2 1 (b) s t 8 f dsxd − 2x 2 1 4x 2 5 (c) s t 8 t 8 tdsxd − 8x 2 21 x 0 If you had difficulty with these problems, you should look at sections 1.1–1.3 of this book. D . Convert from degrees to radians. (a) 3008 (b) 2188 . Convert from radians to degrees. (a) 5!y6 (b) 2 . Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of 308. . Find the exact values. (a) tans!y3d (b) sins7!y6d 24 ¨ (c) secs5!y3d . Express the lengths a and b in the figure in terms of ". a . If sin x − 13 and sec y − 54, where x and y lie between 0 and !y2, evaluate sinsx 1 yd. . Prove the identities. b (a) tan " sin " 1 cos " − sec " FIGURE FOR PROBLEM 5 (b) 2 tan x − sin 2x 1 1 tan 2x . Find all values of x such that sin 2x − sin x and 0 < x < 2!. . Sketch the graph of the function y − 1 1 sin 2x without using a calculator. ANSWERS TO DIAGNOSTIC TEST D: TRIGONOMETRY s4 1 6 s2 d . (a) 5!y3 (b) 2!y10 . . (a) 1508 (b) 3608y! < 114.68 . 0, !y3, !, 5!y3, 2! . 2! cm 1 15 y 2 . 221 . (a) s3 (b) . (a) 24 sin " (b) 24 cos " (c) 2 _π 0 π x If you had difficulty with these problems, you should look at Appendix D of this book. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. A Preview of Calculus By the time you finish this course, you will be able to calculate the length of the curve used to design the Gateway Arch in St. Louis, determine where a pilot should start descent for a smooth landing, compute the force on a baseball bat when it strikes the ball, and measure the amount of light sensed by the human eye as the pupil changes size. CALCULUS IS FUNDAMENTALLY DIFFERENT FROM the mathematics that you have studied previously: calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems. 1 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2 A PREVIEW OF CALCULUS The Area Problem A¡ The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles. It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons. A∞ A™ A¢ A£ A=A¡+A™+A£+A¢+A∞ FIGURE 1 A£ A¢ A∞ Aß A¶ !!! A¡™ !!! FIGURE 2 Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write TEC In the Preview Visual, you can see how areas of inscribed and circumscribed polygons approximate the area of a circle. y A − lim An nl` The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century bc) used exhaustion to prove the familiar formula for the area of a circle: A − !r 2. We will use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles. y y (1, 1) y (1, 1) (1, 1) (1, 1) y=≈ A 0 1 FIGURE 3 x 0 1 4 1 2 3 4 1 x 0 1 x 0 1 n 1 x FIGURE 4 The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank. The Tangent Problem Consider the problem of trying to find an equation of the tangent line t to a curve with equation y − f sxd at a given point P. (We will give a precise definition of a tangent line in Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. A PREVIEW OF CALCULUS y Chapter 2. For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can find the equation of t if we know its slope m. The problem is that we need two points to compute the slope and we know only one point, P, on t. To get around the problem we first find an approximation to m by taking a nearby point Q on the curve and computing the slope mPQ of the secant line PQ. From Figure 6 we see that t y=ƒ P 0 x FIGURE 5 1 mPQ − t m − lim mPQ Q lP Q { x, ƒ} ƒf(a) P { a, f(a)} and we say that m is the limit of mPQ as Q approaches P along the curve. Because x approaches a as Q approaches P, we could also use Equation 1 to write xa a 0 x x FIGURE 6 The secant line at PQ y f sxd 2 f sad x2a Now imagine that Q moves along the curve toward P as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope mPQ of the secant line becomes closer and closer to the slope m of the tangent line. We write The tangent line at P y 3 t Q P 0 FIGURE 7 Secant lines approaching the tangent line x 2 f sxd 2 f sad x2a m − lim xla Specific examples of this procedure will be given in Chapter 2. The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main ideas behind differential calculus are due to the French mathematician Pierre Fermat (1601–1665) and were developed by the English mathematicians John Wallis (1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the German mathematician Gottfried Leibniz (1646–1716). The two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems in a sense that will be described in Chapter 5. Velocity When we look at the speedometer of a car and read that the car is traveling at 48 miyh, what does that information indicate to us? We know that if the velocity remains constant, then after an hour we will have traveled 48 mi. But if the velocity of the car varies, what does it mean to say that the velocity at a given instant is 48 miyh? In order to analyze this question, let’s examine the motion of a car that travels along a straight road and assume that we can measure the distance traveled by the car (in feet) at lsecond intervals as in the following chart: t − Time elapsed ssd 0 1 2 3 4 5 d − Distance sftd 0 2 9 24 42 71 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4 A PREVIEW OF CALCULUS As a first step toward finding the velocity after 2 seconds have elapsed, we find the average velocity during the time interval 2 < t < 4: change in position time elapsed average velocity − 42 2 9 422 − − 16.5 ftys Similarly, the average velocity in the time interval 2 < t < 3 is average velocity − 24 2 9 − 15 ftys 322 We have the feeling that the velocity at the instant t − 2 can’t be much different from the average velocity during a short time interval starting at t − 2. So let’s imagine that the distance traveled has been measured at 0.lsecond time intervals as in the following chart: t 2.0 2.1 2.2 2.3 2.4 2.5 d 9.00 10.02 11.16 12.45 13.96 15.80 Then we can compute, for instance, the average velocity over the time interval f2, 2.5g: average velocity − 15.80 2 9.00 − 13.6 ftys 2.5 2 2 The results of such calculations are shown in the following chart: Time interval f2, 3g f2, 2.5g f2, 2.4g f2, 2.3g f2, 2.2g f2, 2.1g Average velocity sftysd 15.0 13.6 12.4 11.5 10.8 10.2 The average velocities over successively smaller intervals appear to be getting closer to a number near 10, and so we expect that the velocity at exactly t − 2 is about 10 ftys. In Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value of the average velocities over smaller and smaller time intervals. In Figure 8 we show a graphical representation of the motion of the car by plotting the distance traveled as a function of time. If we write d − f std, then f std is the number of feet traveled after t seconds. The average velocity in the time interval f2, tg is d Q { t, f(t)} average velocity − which is the same as the slope of the secant line PQ in Figure 8. The velocity v when t − 2 is the limiting value of this average velocity as t approaches 2; that is, 20 10 0 change in position f std 2 f s2d − time elapsed t22 P { 2, f(2)} 1 2 FIGURE 8 3 4 5 t v − lim tl2 f std 2 f s2d t22 and we recognize from Equation 2 that this is the same as the slope of the tangent line to the curve at P. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5 A PREVIEW OF CALCULUS Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities. The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences. The Limit of a Sequence In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning space and time that were held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as follows, that Achilles could never pass the tortoise: Suppose that Achilles starts at position a 1 and the tortoise starts at position t1. (See Figure 9.) When Achilles reaches the point a 2 − t1, the tortoise is farther ahead at position t2. When Achilles reaches a 3 − t2, the tortoise is at t3. This process continues indefinitely and so it appears that the tortoise will always be ahead! But this defies common sense. Achilles FIGURE 9 a¡ tortoise a™ a£ a¢ a∞ ... t¡ t™ t£ t¢ ... One way of explaining this paradox is with the idea of a sequence. The successive positions of Achilles sa 1, a 2 , a 3 , . . .d or the successive positions of the tortoise st1, t2 , t3 , . . .d form what is known as a sequence. In general, a sequence ha nj is a set of numbers written in a definite order. For instance, the sequence h1, 12 , 13 , 14 , 15 , . . . j can be described by giving the following formula for the nth term: an − a¢ a £ a™ 0 We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the terms of the sequence a n − 1yn are becoming closer and closer to 0 as n increases. In fact, we can find terms as small as we please by making n large enough. We say that the limit of the sequence is 0, and we indicate this by writing a¡ 1 (a) 1 lim nl` 1 2 3 4 5 6 7 8 (b) FIGURE 10 1 n n 1 −0 n In general, the notation lim a n − L nl` is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 6 A PREVIEW OF CALCULUS The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number. For instance, if a 1 − 3.1 a 2 − 3.14 a 3 − 3.141 a 4 − 3.1415 a 5 − 3.14159 a 6 − 3.141592 a 7 − 3.1415926 f then lim a n − ! nl` The terms in this sequence are rational approximations to !. Let’s return to Zeno’s paradox. The successive positions of Achilles and the tortoise form sequences ha nj and htn j, where a n , tn for all n. It can be shown that both sequences have the same limit: lim a n − p − lim tn nl` nl` It is precisely at this point p that Achilles overtakes the tortoise. The Sum of a Series Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man standing in a room cannot walk to the wall. In order to do so, he would first have to go half the distance, then half the remaining distance, and then again half of what still remains. This process can always be continued and can never be ended.” (See Figure 11.) 1 2 FIGURE 11 1 4 1 8 1 16 Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances as follows: 3 1− 1 1 1 1 1 1 1 1 1 ∙∙∙ 1 n 1 ∙∙∙ 2 4 8 16 2 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. A PREVIEW OF CALCULUS 7 Zeno was arguing that it doesn’t make sense to add infinitely many numbers together. But there are other situations in which we implicitly use infinite sums. For instance, in decimal notation, the symbol 0.3 − 0.3333 . . . means 3 3 3 3 1 1 1 1 ∙∙∙ 10 100 1000 10,000 and so, in some sense, it must be true that 3 3 3 3 1 1 1 1 1 ∙∙∙ − 10 100 1000 10,000 3 More generally, if dn denotes the nth digit in the decimal representation of a number, then 0.d1 d2 d3 d4 . . . − d1 d2 d3 dn 1 2 1 3 1 ∙∙∙ 1 n 1 ∙∙∙ 10 10 10 10 Therefore some infinite sums, or infinite series as they are called, have a meaning. But we must define carefully what the sum of an infinite series is. Returning to the series in Equation 3, we denote by sn the sum of the first n terms of the series. Thus s1 − 12 − 0.5 s2 − 12 1 14 − 0.75 s3 − 12 1 14 1 18 − 0.875 1 s4 − 12 1 14 1 18 1 16 − 0.9375 1 1 s5 − 12 1 14 1 18 1 16 1 32 − 0.96875 1 1 1 s6 − 12 1 14 1 18 1 16 1 32 1 64 − 0.984375 1 1 1 1 s7 − 12 1 14 1 18 1 16 1 32 1 64 1 128 − 0.9921875 f 1 s10 − 12 1 14 1 ∙ ∙ ∙ 1 1024 < 0.99902344 f s16 − 1 1 1 1 1 ∙ ∙ ∙ 1 16 < 0.99998474 2 4 2 Observe that as we add more and more terms, the partial sums become closer and closer to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently many terms of the series), we can make the partial sum sn as close as we please to the number 1. It therefore seems reasonable to say that the sum of the infinite series is 1 and to write 1 1 1 1 1 1 1 ∙∙∙ 1 n 1 ∙∙∙ − 1 2 4 8 2 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 8 A PREVIEW OF CALCULUS In other words, the reason the sum of the series is 1 is that lim sn − 1 nl` In Chapter 11 we will discuss these ideas further. We will then use Newton’s idea of combining infinite series with differential and integral calculus. Summary We have seen that the concept of a limit arises in trying to find the area of a region, the slope of a tangent to a curve, the velocity of a car, or the sum of an infinite series. In each case the common theme is the calculation of a quantity as the limit of other, easily calculated quantities. It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits. After Sir Isaac Newton invented his version of calculus, he used it to explain the motion of the planets around the sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast oil prices rise or fall, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas. We will explore some of these uses of calculus in this book. In order to convey a sense of the power of the subject, we end this preview with a list of some of the questions that you will be able to answer using calculus: 1. How can we explain the fact, illustrated in Figure 12, that the angle of elevation from an observer up to the highest point in a rainbow is 42°? (See page 285.) rays from sun 138° rays from sun 42° 2. How can we explain the shapes of cans on supermarket shelves? (See page 343.) 3. Where is the best place to sit in a movie theater? (See page 465.) 4. How can we design a roller coaster for a smooth ride? (See page 182.) 5. How far away from an airport should a pilot start descent? (See page 208.) observer FIGURE 12 6. How can we fit curves together to design shapes to represent letters on a laser printer? (See page 657.) 7. How can we estimate the number of workers that were needed to build the Great Pyramid of Khufu in ancient Egypt? (See page 460.) 8. Where should an infielder position himself to catch a baseball thrown by an outfielder and relay it to home plate? (See page 465.) 9. Does a ball thrown upward take longer to reach its maximum height or to fall back to its original height? (See page 609.) 10. How can we explain the fact that planets and satellites move in elliptical orbits? (See page 868.) 11. How can we distribute water flow among turbines at a hydroelectric station so as to maximize the total energy production? (See page 980.) 12. If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which of them reaches the bottom first? (See page 1052.) Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1 Often a graph is the best way to represent a function because it conveys so much information at a glance. Shown is a graph of the vertical ground acceleration created by the 2011 earthquake near Tohoku, Japan. The earthquake had a magnitude of 9.0 on the Richter scale and was so powerful that it moved northern Japan 8 feet closer to North America. Functions and Models Pictura Collectus/Alamy (cm/s@) 2000 1000 0 time _1000 _2000 0 50 100 150 200 Seismological Society of America THE FUNDAMENTAL OBJECTS THAT WE deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of realworld phenomena. 9 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 10 CHAPTER 1 Functions and Models Year Population (millions) 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870 Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A − !r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. B. The human population of the world P depends on the time t. The table gives estimates of the world population Pstd at time t, for certain years. For instance, Ps1950d < 2,560,000,000 But for each value of the time t there is a corresponding value of P, and we say that P is a function of t. C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of