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97909_FrontEP_FrontEP_pF2_RefPage1-2_97909_FrontEP_FrontEP_pF2_RefPage1-2 9/24/10 5:36 PM Page R E F E R E N C E PA G E Cut here and keep for reference ALGEBRA GEOMETRY Arithmetic Operations Geometric Formulas a c ad bc 苷 b d bd a d ad b a 苷 苷 c b c bc d a共b c兲 苷 ab ac a c ac 苷 b b b Formulas for area A, circumference C, and volume V: Triangle Circle Sector of Circle A 苷 12 bh A 苷 r A 苷 12 r 2 C 苷 2 r s 苷 r 共 in radians兲 苷 12 ab sin a Exponents and Radicals xm 苷 x mn xn xn 苷 n x x m x n 苷 x mn 共x m兲n 苷 x m n 冉冊 x y 共xy兲n 苷 x n y n n 苷 xn yn n n x m兾n 苷 s x m 苷 (s x )m n x 1兾n 苷 s x 冑 n n n xy 苷 s xs y s n r h ă r s ă b r Sphere V 苷 43 r Cylinder V 苷 r 2h Cone V 苷 13 r 2h A 苷 4 r A 苷 rsr h n x x s 苷 n y sy r r h h Factoring Special Polynomials r x y 苷 共x y兲共x y兲 x y 苷 共x y兲共x xy y 2兲 x y 苷 共x y兲共x xy y 2兲 Distance and Midpoint Formulas Binomial Theorem 共x y兲2 苷 x 2xy y 共x y兲2 苷 x 2xy y Distance between P1共x1, y1兲 and P2共x 2, y2兲: d 苷 s共x x1兲2 共 y2 y1兲2 共x y兲3 苷 x 3x y 3xy y 共x y兲3 苷 x 3x y 3xy y 共x y兲n 苷 x n nx n1y where 冉冊 n共n 1兲 n2 x y 冉冊 n nk k x y nxy n1 y n k n共n 1兲 共n k 1兲 n 苷 k ⴢ ⴢ ⴢ ⴢ k Midpoint of P1 P2 : 冉 x1 x y1 y2 , 2 Lines Slope of line through P1共x1, y1兲 and P2共x 2, y2兲: m苷 Quadratic Formula If ax bx c 苷 0, then x 苷 冊 b sb 4ac 2a y2 y1 x x1 Point-slope equation of line through P1共x1, y1兲 with slope m: Inequalities and Absolute Value y y1 苷 m共x x1兲 If a b and b c, then a c Slope-intercept equation of line with slope m and y-intercept b: If a b, then a c b c If a b and c 0, then ca cb y 苷 mx b If a b and c 0, then ca cb If a 0, then ⱍxⱍ 苷 a ⱍxⱍ a ⱍxⱍ a means x 苷 a or x 苷 a means a x a means xa or x a Circles Equation of the circle with center 共h, k兲 and radius r: 共x h兲2 共 y k兲2 苷 r Copyright 2010 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 97909_FrontEP_FrontEP_pF2_RefPage1-2_97909_FrontEP_FrontEP_pF2_RefPage1-2 9/24/10 5:36 PM Page R E F E R E N C E PA G E TRIGONOMETRY Angle Measurement Fundamental Identities radians 苷 180⬚ 1⬚ 苷 rad 180 rad 苷 s r 180⬚ r 共 in radians兲 Right Angle Trigonometry hyp csc 苷 opp cos 苷 adj hyp sec 苷 hyp adj tan 苷 opp adj cot 苷 adj opp hyp y r csc ă adj x r sec 苷 r x tan 苷 y x cot 苷 x y cot 苷 cos sin cot 苷 tan sin 2 ⫹ cos 2 苷 1 ⫹ tan 2 苷 sec 2 ⫹ cot 2 苷 csc 2 sin共⫺兲 苷 ⫺sin cos共⫺兲 苷 cos tan共⫺兲 苷 ⫺tan sin ⫺ 苷 cos tan ⫺ 苷 cot 冉 冊 冉 冊 ⫺ 苷 sin B sin A sin B sin C 苷 苷 a b c (x, y) a r C c ă The Law of Cosines x b a 苷 b ⫹ c ⫺ 2bc cos A b 苷 a ⫹ c ⫺ 2ac cos B y A c 苷 a ⫹ b ⫺ 2ab cos C y=tan x y=cos x 1 π sin cos The Law of Sines y y y=sin x tan 苷 冉 冊 Graphs of Trigonometric Functions y cos cos r y cos 苷 sec 苷 opp Trigonometric Functions sin sin ă s r opp sin 苷 hyp csc 苷 2π Addition and Subtraction Formulas 2π x _1 π 2π x π x sin共x ⫹ y兲 苷 sin x cos y ⫹ cos x sin y sin共x ⫺ y兲 苷 sin x cos y ⫺ cos x sin y _1 cos共x ⫹ y兲 苷 cos x cos y ⫺ sin x sin y y y y=csc x y y=sec x cos共x ⫺ y兲 苷 cos x cos y ⫹ sin x sin y y=cot x 1 π 2π x π 2π x π 2π x tan共x ⫹ y兲 苷 tan x ⫹ tan y ⫺ tan x tan y tan共x ⫺ y兲 苷 tan x ⫺ tan y ⫹ tan x tan y _1 _1 Double-Angle Formulas sin 2x 苷 sin x cos x Trigonometric Functions of Important Angles cos 2x 苷 cos 2x ⫺ sin 2x 苷 cos 2x ⫺ 苷 ⫺ sin 2x radians sin cos tan 0⬚ 30⬚ 45⬚ 60⬚ 90⬚ 兾6 兾4 兾3 兾2 1兾2 s2兾2 s3兾2 1 s3兾2 s2兾2 1兾2 0 s3兾3 s3 — tan 2x 苷 tan x ⫺ tan2x Half-Angle Formulas sin 2x 苷 ⫺ cos 2x cos 2x 苷 ⫹ cos 2x Copyright 2010 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 Copyright 2010 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 97909_FM_FM_pi-xxviii.qk_97909_FM_FM_pi-xxviii 10/15/10 10:53 AM Page i CA L C U L U S EARLY TRANSCENDENTALS SEVENTH EDITION JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States Copyright 2010 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 97909_FM_FM_pi-xxviii.qk_97909_FM_FM_pi-xxviii 10/15/10 10:53 AM Page ii Calculus: Early Transcendentals, Seventh Edition James Stewart Executive Editor: Liz Covello Assistant Editor: Liza Neustaetter Editorial Assistant: Jennifer Staller Media Editor : Maureen Ross Marketing Manager: Jennifer Jones Marketing Coordinator: Michael Ledesma Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Cheryll Linthicum Art Director: Vernon T Boes Print Buyer: Becky Cross Rights Acquisitions Specialist: Don Schlotman Production Service: TECH· arts Text Designer: TECH· arts Photo Researcher: Terri Wright, www.terriwright.com Copy Editor: Kathi Townes Cover Designer: Irene Morris Cover Illustration: Irene Morris Compositor: Stephanie Kuhns, TECH· arts © 2012, 2008 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com Library of Congress Control Number: 2010936599 Student Edition: ISBN-13: 978-0-538-49790-9 ISBN-10: 0-538-49790-4 Loose-leaf Edition: ISBN-13: 978-0-8400-5885-0 ISBN-10: 0-8400-5885-3 Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 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 To learn more about Brooks/Cole, visit www.cengage.com/brookscole Printed in the United States of America 11 Trademarks ExamView ® and ExamViewPro ® are registered trademarks of FSCreations, Inc Windows is a registered trademark of the Microsoft Corporation and used herein under license Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc Used herein under license Derive is a registered trademark of Soft Warehouse, Inc Maple is a registered trademark of Waterloo Maple, Inc Mathematica is a registered trademark of Wolfram Research, Inc Tools for Enriching is a trademark used herein under license Copyright 2010 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 K09T10 Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com 97909_FM_FM_pi-xxviii.qk_97909_FM_FM_pi-xxviii 10/15/10 10:53 AM Page iii Contents Preface xi To the Student xxiii Diagnostic Tests xxiv A PREVIEW OF CALCULUS Functions and Models 9 1.1 Four Ways to Represent a Function 1.2 Mathematical Models: A Catalog of Essential Functions 1.3 New Functions from Old Functions 1.4 Graphing Calculators and Computers 1.5 Exponential Functions 1.6 Inverse Functions and Logarithms Review 10 23 36 44 51 58 72 Principles of Problem Solving 75 Limits and Derivatives 81 2.1 The Tangent and Velocity Problems 2.2 The Limit of a Function 2.3 Calculating Limits Using the Limit Laws 2.4 The Precise Definition of a Limit 2.5 Continuity 2.6 Limits at Infinity; Horizontal Asymptotes 2.7 Derivatives and Rates of Change 87 N Problems Plus 108 130 143 Early Methods for Finding Tangents The Derivative as a Function Review 99 118 Writing Project 2.8 82 153 154 165 170 iii Copyright 2010 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 97909_FM_FM_pi-xxviii.qk_97909_FM_FM_pi-xxviii 10/15/10 10:53 AM Page iv iv CONTENTS Differentiation Rules 173 3.1 Derivatives of Polynomials and Exponential Functions Applied Project N Building a Better Roller Coaster 3.2 The Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 The Chain Rule Applied Project 3.5 184 191 Where Should a Pilot Start Descent? Implicit Differentiation N Families of Implicit Curves 217 Derivatives of Logarithmic Functions 3.7 Rates of Change in the Natural and Social Sciences 3.8 Exponential Growth and Decay 3.9 Related Rates 3.10 Linear Approximations and Differentials Problems Plus 218 224 237 244 N Taylor Polynomials Hyperbolic Functions Review 208 209 3.6 Laboratory Project 184 198 N Laboratory Project 3.11 174 250 256 257 264 268 Applications of Differentiation 273 4.1 Maximum and Minimum Values Applied Project N 274 The Calculus of Rainbows 282 4.2 The Mean Value Theorem 4.3 How Derivatives Affect the Shape of a Graph 4.4 Indeterminate Forms and l’Hospital’s Rule Writing Project N 284 Summary of Curve Sketching 4.6 Graphing with Calculus and Calculators 4.7 Optimization Problems Applied Project N 4.8 Newton’s Method 4.9 Antiderivatives Review Problems Plus 301 The Origins of l’Hospital’s Rule 4.5 290 310 310 318 325 The Shape of a Can 337 338 344 351 355 Copyright 2010 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 97909_FM_FM_pi-xxviii.qk_97909_FM_FM_pi-xxviii 10/15/10 10:53 AM Page v CONTENTS Integrals 359 5.1 Areas and Distances 360 5.2 The Definite Integral 371 Discovery Project 385 The Fundamental Theorem of Calculus 5.4 Indefinite Integrals and the Net Change Theorem 5.5 N Problems Plus 386 397 Newton, Leibniz, and the Invention of Calculus The Substitution Rule Review 406 407 415 419 Applications of Integration 421 6.1 Areas Between Curves Applied Project N 422 The Gini Index 6.2 Volumes 6.3 Volumes by Cylindrical Shells 6.4 Work 6.5 Average Value of a Function 429 430 441 446 451 Applied Project N Calculus and Baseball Applied Project N Where to Sit at the Movies Review Problems Plus Area Functions 5.3 Writing Project N 455 456 457 459 Techniques of Integration 463 7.1 Integration by Parts 7.2 Trigonometric Integrals 7.3 Trigonometric Substitution 7.4 Integration of Rational Functions by Partial Fractions 7.5 Strategy for Integration 7.6 Integration Using Tables and Computer Algebra Systems Discovery Project N 464 471 478 484 494 Patterns in Integrals 500 505 Copyright 2010 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 97909_FM_FM_pi-xxviii.qk_97909_FM_FM_pi-xxviii 10/15/10 10:53 AM Page vi vi CONTENTS 7.7 Approximate Integration 7.8 Improper Integrals Review Problems Plus 519 529 533 Further Applications of Integration 537 8.1 Arc Length 538 Discovery Project 8.2 8.3 N Arc Length Contest Area of a Surface of Revolution Discovery Project N 545 545 Rotating on a Slant 551 Applications to Physics and Engineering Discovery Project N Applications to Economics and Biology 8.5 Probability Problems Plus 552 Complementary Coffee Cups 8.4 Review 506 562 563 568 575 577 Differential Equations 579 9.1 Modeling with Differential Equations 9.2 Direction Fields and Euler’s Method 9.3 Separable Equations 580 585 594 Applied Project N How Fast Does a Tank Drain? Applied Project N Which Is Faster, Going Up or Coming Down? 9.4 Models for Population Growth 9.5 Linear Equations 9.6 Predator-Prey Systems Review Problems Plus 603 604 605 616 622 629 633 Copyright 2010 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