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stewart Calculus 6E Single Variable Calculus Early Transcendentals Visit Thomson Brooks/Cole at www.thomsonedu.com Single Variable Calculus Early Transcendentals j a m e s st ewa rt E SE/Stewart, Single Variable Calculus: Early Transcendentals, 6 ISBN-13: 978-0-495-01169-9 ©2007 Designer: Irene Morris Text printer: RR Donnelley Cover printer: Phoenix Binding: Case Trim: 8.5" x 10" CMYK S I N G L E VA R I A B L E CA L C U L U S E A R LY T R A N S C E N D E N TA L S SIXTH EDITION J A M E S S T E WA RT McMASTER UNIVERSITY AUSTRALIA N BRAZIL N C A N A DA N MEXICO N SINGAPORE N S PA I N N UNITED KINGDOM N U N I T E D S TAT E S Single Variable Calculus: Early Transcendentals, Sixth Edition James Stewart Publisher Bob Pirtle Assistant Editor Stacy Green Editorial Assistant Elizabeth Rodio Technology Project Manager Sam Subity Marketing Manager Mark Santee Marketing Assistant Melissa Wong Marketing Communications Manager Bryan Vann Project Manager, Editorial Production Cheryll Linthicum Creative Director Rob Hugel Art Director Vernon T Boes Print Buyer Becky Cross Permissions Editor Bob Kauser Production Service TECH·arts Text Designer Kathi Townes Photo Researcher Stephanie Kuhns Copy Editor Kathi Townes Illustrator Brian Betsill Cover Designer Irene Morris Cover Image Amelie Fear, Folkmusician.com Cover Printer R R Donnelley /Willard Compositor Stephanie Kuhns, TECH·arts Printer R R Donnelley /Willard COPYRIGHT © 2008, 2003 Thomson Brooks/Cole, a part of The Thomson Corporation Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license Trademarks Exam View ® and Exam View Pro ® 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 N N N N N N N N N N N ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher Printed in the United States of America N N N N N N N N N N N 11 10 09 08 07 For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 Thomson Higher Education 10 Davis Drive Belmont, CA 94002 USA For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com Any additional questions about permissions can be submitted by email to thomsonrights@thomson.com © 2008 Thomson Learning, Inc All Rights Reserved Thomson Learning WebTutor ™ is a trademark of Thomson Learning, Inc Library of Congress Control Number: 2006939531 K12T06 ISBN-13: 978-0-495-01169-9 ISBN-10: 0-495-01169-X FOR SALLY AND DON FOR ALAN AND SHARON FOR KELLY, KIM, AND C ALLUM FOR JACKIE AND NINO CONTENTS Preface xi To the Student xxii Diagnostic Tests xxiv A PREVIEW OF C ALCULUS FUNCTIONS AND MODELS 10 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 11 37 46 52 59 76 LIMITS AND DERIVATIVES 82 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 83 88 109 The Derivative as a Function Problems Plus 130 143 Writing Project Early Methods for Finding Tangents Review 99 119 N 2.8 24 73 Principles of Problem Solving 2 153 154 165 170 v vi |||| CONTENTS DIFFERENTIATION RULES 3.1 172 Derivatives of Polynomials and Exponential Functions Applied Project Building a Better Roller Coaster N m=0 m=1 m=_1 π π The Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 The Chain Rule π 182 183 189 197 Applied Project Where Should a Pilot Start Descent? 206 N y 3.2 π 3.5 Implicit Differentiation 3.6 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 207 241 Laboratory Project Taylor Polynomials Hyperbolic Functions Review Problems Plus 247 253 254 261 265 APPLIC ATIONS OF DIFFERENTIATION 4.1 215 233 N 3.11 Maximum and Minimum Values 270 271 Applied Project The Calculus of Rainbows N 279 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 280 Writing Project The Origins of L’Hospital’s Rule N 4.5 Summary of Curve Sketching 4.6 Graphing with Calculus and Calculators 4.7 Optimization Problems 4.8 Newton’s Method 4.9 Antiderivatives Problems Plus 347 351 334 340 287 298 307 307 322 Applied Project The Shape of a Can N Review 173 333 315 221 CONTENTS INTEGRALS 354 5.1 Areas and Distances 355 5.2 The Definite Integral 366 Discovery Project Area Functions N 379 5.3 The Fundamental Theorem of Calculus 379 5.4 Indefinite Integrals and the Net Change Theorem Writing Project Newton, Leibniz, and the Invention of Calculus N 5.5 The Substitution Rule Review Problems Plus 408 412 6.1 Areas between Curves 6.2 Volumes 6.3 Volumes by Cylindrical Shells 6.4 Work 6.5 Average Value of a Function 414 415 422 433 438 442 Applied Project Where to Sit at the Movies N Problems Plus 399 400 APPLIC ATIONS OF INTEGRATION Review 391 446 446 448 TECHNIQUES OF INTEGRATION 452 7.1 Integration by Parts 453 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 460 467 483 Discovery Project Patterns in Integrals N 473 494 489 |||| VII viii |||| CONTENTS 7.7 Approximate Integration 7.8 Improper Integrals Review Problems Plus 495 508 518 521 FURTHER APPLIC ATIONS OF INTEGRATION 8.1 Arc Length 525 Discovery Project Arc Length Contest N 8.2 532 Area of a Surface of Revolution Discovery Project Rotating on a Slant N 8.3 532 538 Applications to Physics and Engineering Discovery Project Complementary Coffee Cups N Applications to Economics and Biology 8.5 Probability Problems Plus 539 550 8.4 Review 524 550 555 562 564 DIFFERENTIAL EQUATIONS 566 9.1 Modeling with Differential Equations 9.2 Direction Fields and Euler’s Method 9.3 Separable Equations 567 572 580 Applied Project How Fast Does a Tank Drain? N 588 Applied Project Which Is Faster, Going Up or Coming Down? N 9.4 Models for Population Growth Applied Project Calculus and Baseball N 9.5 Linear Equations 9.6 Predator-Prey Systems Review Problems Plus 614 618 602 608 591 601 590 CONTENTS 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 10.1 Curves Defined by Parametric Equations 621 Laboratory Project Running Circles around Circles N 10.2 Calculus with Parametric Curves Laboratory Project Bézier Curves N 639 Polar Coordinates 10.4 Areas and Lengths in Polar Coordinates 10.5 Conic Sections 10.6 Conic Sections in Polar Coordinates Problems Plus 11 630 10.3 Review 629 639 650 654 662 669 672 INFINITE SEQUENCES AND SERIES 11.1 Sequences 674 675 Laboratory Project Logistic Sequences 687 N 11.2 Series 687 11.3 The Integral Test and Estimates of Sums 11.4 The Comparison Tests 11.5 Alternating Series 11.6 Absolute Convergence and the Ratio and Root Tests 11.7 Strategy for Testing Series 11.8 Power Series 11.9 Representations of Functions as Power Series 11.10 Taylor and Maclaurin Series 697 705 710 721 723 Laboratory Project An Elusive Limit N 734 748 Writing Project How Newton Discovered the Binomial Series N 11.11 Applications of Taylor Polynomials Applied Project Radiation from the Stars N Review Problems Plus 758 761 728 749 757 748 714 620 |||| ix A106 |||| (a) C APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES (b) D 11 19 D 21 D 31 D 33 e͑͞e Ϫ 1͒ 41 23 D 25 35 43 1138͞333 17 27 D 29 D 11 39 e Ϫ 37 1 1 Ϫ 4x (b) 59 EXERCISES 11.6 (a) D AC 15 AC 25 AC 661 35 (a) 960 2 73 (a) 0, , , , , , , ͑n ϩ 1͒! Ϫ ͑n ϩ 1͒! N C N (c) 1 C 15 C 27 C PAGE 703 y y= (b) C x 1.3 N D 17 D 29 C a£ a¢ a∞ EXERCISES 11.4 1, ͓Ϫ1, 1͒ x 41 b Ͻ 1͞e N PAGE 727 and a and the cn’s are constants 1, ͓Ϫ1, 1͔ D C C D 11 C 13 D 15 C 17 C 19 C 21 D 23 C 25 C 27 p Ͼ 29 p Ͻ Ϫ1 31 ͑1, ϱ͒ 33 (a) 1.54977, error ഛ 0.1 (b) 1.64522, error ഛ 0.005 (c) n Ͼ 1000 35 0.00145 N A series of the form ϱn0 cn͑x Ϫ a͒n, where x is a variable a™ PAGE 722 C D C 11 C 13 C 19 C 21 C 23 D 25 C 31 D 33 C 35 C 37 C EXERCISES 11.8 PAGE 719 CC 17 CC 27 D EXERCISES 11.7 EXERCISES 11.3 10 (c) May converge or diverge AC D 11 AC 13 AC 19 AC 21 AC 23 D 29 D 31 (a) and (d) Ϸ 0.68854, error Ͻ 0.00521 (b) n ജ 11, 0.693109 71 ͕sn ͖ is bounded and increasing 23 119 Abbreviations: AC, absolutely convergent; CC, conditionally convergent (s3 Ϫ 1) 65 The series is divergent 75 (a) , , 24 , 120 ; ssn d _1 53 D͑1 Ϫ c n ͒ 57 (a) Sn 1Ϫc n͑n ϩ 1͒ 23 25 27 0.9721 29 0.0676 31 An underestimate 33 p is not a negative integer 35 ͕bn ͖ is not decreasing for n Ͼ 1, sum n͑n ϩ 1͒ 55 a1 0, an 21 1.0000, 0.6464, 0.8389, 0.7139, 0.8033, 0.7353, 0.7893, 0.7451, 0.7821, 0.7505; error Ͻ 0.0275 san d 49 Ϫ Ͻ x Ͻ ; 2 Ϫ cos x 51 All x ; 15 60 45 5063͞3300 x 3Ϫx 47 Ϫ3 Ͻ x Ͻ 3; 63 13 D 2, ͑Ϫ2, 2͒ 11 15 1, ͓1, 3͔ [ 13 11 13 4, ͑Ϫ4, 4͔ ) 19 ϱ, ͑Ϫϱ, ϱ͒ 23 0, { } 21 b, ͑a Ϫ b, a ϩ b͒ 27 ϱ, ͑Ϫϱ, ϱ͒ 35 (a) ͑Ϫϱ, ϱ͒ ] , (Ϫ , 17 , Ϫ , Ϫ ϱ, ͑Ϫϱ, ϱ͒ 1 2 29 (a) Yes (b), (c) (b) No N 17 D 29 C D C 19 D 31 D C 21 C 0.76352, error Ͻ 0.001 PAGE 713 negative (b) Ͻ bnϩ1 ഛ bn and lim n l ϱ bn 0, where bn an (c) Rn ഛ bnϩ1 C C D C 11 C 13 D 15 C 17 C 19 D Խ Խ 33 No _8 _2 EXERCISES 11.9 (a) A series whose terms are alternately positive and Խ Խ 31 k k J¡ C s¡ s£ s∞ 37 ͑Ϫ1, 1͒, f ͑x͒ ͑1 ϩ 2x͒͑͞1 Ϫ x ͒ EXERCISES 11.5 ] sá s s PAGE 709 (a) Nothing (b) C 11 C 13 C 15 C 23 C 25 D 27 C 33 1.249, error Ͻ 0.1 35 45 Yes [ 25 4, Ϫ2, N 41 PAGE 733 ϱ 10 ͚ ͑Ϫ1͒ x , ͑Ϫ1, 1͒ n n n0 ϱ ͚ ͑Ϫ1͒ n0 n nϩ1 x 2nϩ1, ͑Ϫ3, 3͒ ϱ ͚ n0 ϩ x n, ͑Ϫ3, 3͒ nϩ1 ϱ ͚ x , ͑Ϫ1, 1͒ n n1 APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES ͫ ϱ 11 ͚ ͑Ϫ1͒nϩ1 Ϫ n0 nϩ1 ϱ 13 (a) (b) (c) ϱ x n, ͑Ϫ1, 1͒ 15 n 17 19 n0 ϱ n 25 ϩ n2 ϱ ͚ n1 ͚ ͚ ͑Ϫ1͒ ͑Ϫ1͒n n0 ϱ xn ,R5 n5 n 17 ͚ n3 nϪ2 n x ,R2 nϪ1 ͚ ͑Ϫ1͒ n n0 ϱ 29 31 s∞ ͚ n0 f s™ _4 n0 33 ͚ ͑Ϫ1͒ n n0 35 xϩ x 4nϩ1 , R ϱ 2n ͑2n͒! ϱ ͚ ͑Ϫ1͒ n n1 s¡ ϱ 37 _0.25 s£ ͚ ϱ s¢ s™ s¢ f s∞ ϱ 2nϩ1 x 2nϩ1, R ϱ ͑2n ϩ 1͒! 2n ϩ n x ,Rϱ n! n ϱ s¡ ͑n ϩ 1͒͑n ϩ 2͒ n x ,R2 nϩ4 ͚ ͑Ϫ1͒ n0 s£ ؒ ؒ ؒ и и и ؒ ͑2n Ϫ 1͒ ͑x Ϫ 9͒ n, R n ؒ 2nϩ1 ؒ n! ϱ x ؒ ؒ ؒ и и и ؒ ͑2n Ϫ 3͒ n ϩ ͚ ͑Ϫ1͒nϪ1 x ,R1 2 nn! n2 ϱ 27 x 2nϩ1, R 16nϩ1 0.25 21 n n0 ͚ ͑Ϫ1͒ n͑n Ϫ 1͒x , R ϱ nϩ1 ϱ ͚ ͑Ϫ1͒ ͑n ϩ 2͒͑n ϩ 1͒x , R n n ͑x Ϫ ͒2n, R ϱ ͑2n͒! ͚ ͑Ϫ1͒ n0 n0 ϱ ͚ ͑Ϫ1͒ nϩ1 n1 2x 2nϩ1 ,R1 2n ϩ s£ ϱ s™ 39 s¡ Ϫ2 ͚ ͑Ϫ1͒ n n0 f ؒ ؒ ؒ и и и ؒ ͑2n Ϫ 1͒ 2nϩ1 x ,R2 n! 3nϩ1 2nϪ1 2n x ,Rϱ ͑2n͒! x 4n, R ϱ ͑2n͒! 1.5 T¸=T¡=T™=T£ _1.5 1.5 Tˆ=T˜=T¡¸=T¡¡ Ϫ3 23 C ϩ ϱ ͚ n0 f 8nϩ2 t ,R1 8n TÂ=T=Tò=Tả _1.5 x 2n1 ,R1 25 C ϩ ͚ ͑Ϫ1͒ nϩ1 4n Ϫ n1 29 0.000983 31 0.09531 27 0.199989 33 (b) 0.920 37 ͓Ϫ1, 1͔, ͓Ϫ1, 1͒, ͑Ϫ1, 1͒ ϱ 41 ͚ n1 ͑Ϫ1͒nϪ1 n x ,Rϱ ͑n Ϫ 1͒! T£ T∞ T¡ EXERCISES 11.10 N b f ͑8͒͑5͒͞8! PAGE 746 ϱ ͚ f _3 ͑n ϩ 1͒x n, R T™ n0 T™ ͚ ͑n ϩ 1͒x , R n 2nϩ1 ͚ ͑Ϫ1͒ n x 2nϩ1, R ϱ ͑2n ϩ 1͒! n0 ϱ ͚ n0 5n n x ,Rϱ n! ϱ 11 ͚ n0 x 2nϩ1 ,Rϱ ͑2n ϩ 1͒! 13 Ϫ1 Ϫ 2͑x Ϫ 1͒ ϩ 3͑x Ϫ 1͒2 ϩ 4x 13 x 14, R T Tò f T£ n0 ϱ T¡ ϱ A107 e3 ͑x Ϫ 3͒n, R ϱ n! ϱ n n ͚ n0 ͚ ͑Ϫ1͒ ͑n ϩ 1͒x , R 15 ln Ϫ 19 ͬ |||| _6 T¢ T∞ Tß 43 0.81873 45 (a) ϩ ϱ ͚ n1 (b) x ϩ ϱ ͚ n1 ؒ ؒ ؒ и и и ؒ ͑2n Ϫ 1͒ 2n x n n! ؒ ؒ ؒ и и и ؒ ͑2n Ϫ 1͒ 2nϩ1 x ͑2n ϩ 1͒2 n n! A108 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES ϱ x ϩ x x 6nϩ2 ,Rϱ ͑6n ϩ 2͒͑2n͒! n0 ϱ x 2n, R ϱ 49 C ϩ ͚ ͑Ϫ1͒ n 2n ͑2n͒! n1 ͚ ͑Ϫ1͒ 47 C ϩ 53 0.40102 55 61 ϩ x ϩ n 360 57 x 63 e 51 0.440 59 Ϫ x ϩ 120 Ϫx 1.6 25 24 x4 _1 T£ 67 e Ϫ 65 1͞s2 EXERCISES 11.11 f PAGE 755 N _1.6 x Ϫ 2x ϩ 2x (a) T0 ͑x͒ T1͑x͒, T2͑x͒ Ϫ x T3͑x͒, T4͑x͒ Ϫ 12 x ϩ 241 x T5͑x͒, T6͑x͒ Ϫ x ϩ 24 x Ϫ 720 x 1 _1 TÂ=T 1.5 Tá=TĂ f _2 T£ f 2π ͩ ͪ ͩ ͪ ͩ ͪ ͩ ͪ ͩ ͪ 11 T5 ͑x͒ Ϫ x Ϫ _2 Tß (b) x T0 T1 f T™=T£ T2 T3 T4 T5 ϩ2 xϪ T6 xϪ Ϫ 10 xϪ 4 Ϫ 64 xϪ 15 T¢ T∞ 0.7071 0.6916 0.7074 0.7071 Ϫ0.2337 0.0200 Ϫ0.0009 Ϫ1 Ϫ3.9348 T£ T™ T¢ π ͩ ͪ ͩ ͪ xϪ (b) 1.5625 ϫ 10Ϫ5 N PAGE 759 True-False Quiz False False 17 True f T£ π π f True 11 True 19 True False 13 True False 15 False Exercises T£ _1.1 ͑x Ϫ 4͒2 (a) ϩ 23 ͑x Ϫ 1͒ Ϫ ͑x Ϫ 1͒ ϩ 814 ͑x Ϫ 1͒ (b) 0.000097 (a) ϩ 12 x (b) 0.0015 19 (a) ϩ x (b) 0.00006 (a) x Ϫ 16 x (b) 0.042 23 0.17365 25 Four 29 Ϫ0.86 Ͻ x Ͻ 0.86 Ϫ1.037 Ͻ x Ͻ 1.037 21 m, no 37 (c) They differ by about ϫ 10Ϫ9 km CHAPTER 11 REVIEW 1.1 64 ϩ T£ _2 15 17 21 27 31 T£ f T∞ f T™ π 13 (a) ϩ ͑x Ϫ 4͒ Ϫ f Ϫ1.2114 0.1239 Ϫ 14 ͑x Ϫ 2͒ ϩ 18 ͑x Ϫ 2͒ Ϫ 161 ͑x Ϫ 2͒3 Ϫ x Ϫ ϩ (c) As n increases, Tn͑x͒ is a good approximation to f ͑x͒ on a larger and larger interval _4 e 12 11 C D 13 C 15 D 17 C 19 C 21 C 23 CC 25 AC 27 11 29 ͞4 31 e Ϫe 35 0.9721 APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES 37 0.18976224, error Ͻ 6.4 ϫ 10Ϫ7 41 4, ͓Ϫ6, 2͒ 43 0.5, [2.5, 3.5) 45 ͫ ͩ ͪ n ϱ ͚ ͑Ϫ1͒ x n0 ͩ ͪ ͬ ϱ n1 _ œ„ 2nϩ1 s3 ϩ xϪ ͑2n ϩ 1͒! xn ,R1 n 33 ͑Ϫϱ, 1͔ 37 ͑Ϫϱ, 0͒ ʜ x 8nϩ4 ,Rϱ 51 ͚ ͑Ϫ1͒n ͑2n ϩ 1͒! n0 ϱ ؒ ؒ ؒ и и и ؒ ͑4n Ϫ 3͒ n ϩ ͚ x , R 16 n! 6nϩ1 n1 ϱ n x 55 C ϩ ln x ϩ ͚ n1 n ؒ n! 1 57 (a) ϩ ͑x Ϫ 1͒ Ϫ ͑x Ϫ 1͒2 ϩ 16 ͑x Ϫ 1͒3 (b) 1.5 (c) 0.000006 53 Խ Խ _1 ( 14 , ϱ) N PAGE A15 s74 17 y s37 Ϫ 19 y xy=0 x=3 N PAGE 762 15!͞5! 10,897,286,400 (b) if x 0, ͑1͞x͒ Ϫ cot x if x k, k an integer (a) sn ؒ n, ln 1͞3 n, pn n͞3 nϪ1 (c) 52 s3 x ϩ 4x ϩ x ͑Ϫ1, 1͒, 11 ln ͑1 Ϫ x͒4 250 (b) 250 13 (a) 101 ͑eϪ͑nϪ1͒͞5 Ϫ eϪn͞5 ͒ 101 x 59 Ϫ PROBLEMS PLUS 37 m Ϫ , 39 m 0, b0 41 m , b Ϫ2 b Ϫ3 y y y x _2 _3 PAGE A9 N 5 Ϫ s5 ͭ xϩ1 x ϩ Ϫx Ϫ 13 ͑Ϫ2, ϱ͒ Ϫ x for x ജ Ϫ1 for x Ͻ Ϫ1 43 y 11 x ϩ _1 17 ͑3, ϱ͒ 45 y 15 ͓Ϫ1, ϱ͒ _2 x y=_2 x APPENDIXES Խ x 21 y 6x Ϫ 15 23 2x Ϫ 3y ϩ 19 25 5x ϩ y 11 27 y 3x Ϫ 29 y 3x Ϫ 31 y 33 x ϩ 2y ϩ 11 35 5x Ϫ 2y ϩ 0 EXERCISES A 41 (a) T 20 Ϫ 10h, ഛ h ഛ 12 (b) Ϫ30ЊC ഛ T ഛ 20ЊC 43 Ϯ 45 2, Ϫ 47 ͑Ϫ3, 3͒ 49 ͑3, 5͒ 51 ͑Ϫϱ, Ϫ7͔ ʜ ͓Ϫ3, ϱ͒ 53 ͓1.3, 1.7͔ 55 ͓Ϫ4, Ϫ1͔ ʜ ͓1, 4͔ 57 x ജ ͑a ϩ b͒c͑͞ab͒ 59 x Ͼ ͑c Ϫ b͒͞a Խ 39 10 ഛ C ഛ 35 f 18 œ„ EXERCISES B T£ 35 ͑Ϫ1, 0͒ ʜ ͑1, ϱ͒ ϱ A109 31 (Ϫs3, s3 ) 29 ͑Ϫϱ, ϱ͒ 49 Ϫ ͚ ,R1 n nϩ2 47 2n ͑Ϫ1͒ xϪ ͚ ͑2n͒! n0 ϱ |||| x x 0 19 ͑2, 6͒ 47 49 y y 23 [Ϫ1, ) 21 ͑0, 1͔ 1 25 ͑Ϫϱ, 1͒ ʜ ͑2, ϱ͒ _1 27 [Ϫ1, 2 _2 yϭ4 xϭ2 x ] _1 x A110 |||| 51 APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES 27 Parabola y y=1+x 29 Parabola y y ͑0, 1͒ x 0 (3, 4) y=1-2x x Ϫ2 (b) ͑3.5, Ϫ3͒ 53 ͑0, Ϫ4͒ 55 (a) ͑4, 9͒ 59 y x Ϫ 61 (b) 4x Ϫ 3y Ϫ 24 57 ͑1, Ϫ2͒ x 31 Ellipse 33 y y (3, 9) EXERCISES C PAGE A23 N ͑x Ϫ 3͒2 ϩ ͑ y ϩ 1͒2 25 (Ϫ , 0), ͑2, Ϫ5͒, 11 Parabola 1 x x ϩ y 65 ( 14 , Ϫ 14 ), s10͞4 35 y x Ϫ 2x 37 13 Ellipse y y 39 y x y x _4 x x _2 x Ϫ1 15 Hyperbola y 17 Ellipse EXERCISES D y y= x x _2 PAGE A32 7͞6 ͞20 5 720° 75° 11 Ϫ67.5Њ 13 3 cm 15 rad ͑120͒͞Њ 17 _5 N 19 y y x _1 y=_ x 0 x x 315° 19 Parabola 3π _ 21 Hyperbola y y x 21 x y=_ 1 y= y rad _1 _1 x x _1 23 Hyperbola 23 sin͑3͞4͒ 1͞s2, cos͑3͞4͒ Ϫ1͞s2, tan͑3͞4͒ Ϫ1, 25 Ellipse y csc͑3͞4͒ s2, sec͑3͞4͒ Ϫs2, cot͑3͞4͒ Ϫ1 25 sin͑9͞2͒ 1, cos͑9͞2͒ 0, csc͑9͞2͒ 1, cot͑9͞2͒ 0, tan͑9͞2͒ and sec͑9͞2͒ undefined y (1, 2) x 27 sin͑5͞6͒ , cos͑5͞6͒ Ϫs3͞2, tan͑5͞6͒ Ϫ1͞s3, x csc͑5͞6͒ 2, sec͑5͞6͒ Ϫ2͞s3, cot͑5͞6͒ Ϫs3 29 cos , tan , csc , sec , cot x 5 APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES 31 sin s5͞3, cos Ϫ , tan Ϫs5͞2, csc 3͞s5, 19 cot Ϫ2͞s5 i1 33 sin  Ϫ1͞s10, cos  Ϫ3͞s10, tan  , 35 5.73576 cm 61 (3 ϩ s2 ) 63 71 0, , 2 59 15 (4 ϩ s2 ) 65 ͞3, 5͞3 67 ͞4, 3͞4, 5͞4, 7͞4 69 ͞6, ͞2, 5͞6, 3͞2 73 ഛ x ഛ ͞6 and 5͞6 ഛ x ഛ 2 43 (b) 0.405 EXERCISES H 79 π i1 x 5π Ϫ 4i n i ͚x 19 i0 25 A111 i i1 29 n͑n ϩ 1͒ 27 61 33 n͑n ϩ 6n ϩ 11͒͞3 97 300 (d) an Ϫ a0 49 nϩ1 ϩ n ϩ n Ϫ 45 14 77 ͚2 17 35 n͑n ϩ 2n Ϫ n Ϫ 10͒͞4 (b) 100 Ϫ (c) 41 (a) n EXERCISES G y ͚ 2i 23 3276 75 ഛ x Ͻ ͞4, 3͞4 Ͻ x Ͻ 5͞4, 7͞4 Ͻ x ഛ 2 15 31 n͑n ϩ 6n ϩ 17͒͞3 37 24.62147 cm 24 25 n i iϩ1 21 80 csc  Ϫs10, sec  Ϫs10͞3 15 ͚ 13 |||| N PAGE A54 N PAGE A62 13 ϩ 18i 12 Ϫ 7i 11 13 ϩ 10 13 i 11 Ϫi 13 5i 15 12 ϩ 5i, 13 Ϫ i 17 4i, 19 Ϯ i 21 Ϫ1 Ϯ 2i 23 Ϫ Ϯ (s7͞2)i 25 s2 ͓cos͑3͞4͒ ϩ i sin͑3͞4͔͒ y 2 [ ] [ ]} 27 5{cos tanϪ1( ) ϩ i sin tanϪ1( 3) 4 29 4͓cos͑͞2͒ ϩ i sin͑͞2͔͒, cos͑Ϫ͞6͒ ϩ i sin͑Ϫ͞6͒, π 3π 2π π 5π 3π 2 x ͓cos͑Ϫ͞6͒ ϩ i sin͑Ϫ͞6͔͒ 31 s2 ͓cos͑7͞12͒ ϩ i sin͑7͞12͔͒, (2 s2 )͓cos͑13͞12͒ ϩ i sin͑13͞12͔͒, 14 ͓cos͑͞6͒ ϩ i sin͑͞6͔͒ 81 33 Ϫ1024 37 Ϯ1, Ϯi, (1͞s2 )͑Ϯ1 Ϯ i ͒ _π π N 39 Ϯ(s3͞2) ϩ i, Ϫi Im i x 2π 89 14.34457 cm EXERCISES E 35 Ϫ512 s3 ϩ 512i y Im Re _i PAGE A38 s1 ϩ s2 ϩ s3 ϩ s4 ϩ s5 3 ϩ ϩ 10 Ϫ1 ϩ ϩ ϩ ϩ ϩ 210 ϩ 310 ϩ и и и ϩ n10 10 Ϫ ϩ Ϫ ϩ и и и ϩ ͑Ϫ1͒nϪ1 11 ͚ i1 i 41 i 43 ϩ (s3͞2) i 45 Ϫe 47 cos 3 cos3 Ϫ cos sin2, sin 3 cos2 sin Ϫ sin3 Re INDEX RP denotes Reference Page numbers Abel, Niels, 210 absolutely convergent series, 740 absolute maximum and minimum, 271 absolute value, 17, A6, A56 absolute value function, 17 acceleration, 160, 221 Achilles and the tortoise, adaptive numerical integration, 504 addition formulas for sine and cosine, A28, A29 algebraic function, 31 alternating harmonic series, 711, 714 alternating series, 710 Alternating Series Estimation Theorem, 712 Alternating Series Test, 711 analytic geometry, A10 angle, A24 between curves, 268 of deviation, 279 negative, A25 positive, A25 standard position, A25 antiderivative, 340 antidifferentiation formulas, 351 ? aphelion, 667 apolune, 661 approach path of an aircraft, 206 approximate integration, 495 approximating cylinder, 424 approximating surface, 532 approximation by differentials, 250 to e, 179 linear, 247 by the Midpoint Rule, 496 by Newton’s method, 335 by an nth-degree Taylor polynomial, 253 quadratic, 253 by Riemann sums, 367 by Simpson’s Rule, 500, 501 tangent line, 247 by Taylor’s Inequality, 737 by the Trapezoidal Rule, 497 Archimedes’ Principle, 449 arc length, 525, 633, 634, 652 arc length contest, 532 arc length formula, 526 arc length function, 528 area, 3, 355 of a circle, 469 under a curve, 355, 360, 365 between curves, 415, 418 of an ellipse, 469 by exhaustion, under a parametric curve, 632 in polar coordinates, 639 of a sector of a circle, 650 surface, 635 of a surface of a revolution, 532, 538 area function, 379 Area Problem, 3, 355 argument of a complex number, A57 arithmetic-geometric mean, 686 arrow diagram, 12 astroid, 213, 629 asymptote, 308 horizontal, 132, 308 of a hyperbola, 658, A20 slant, 312 vertical, 95, 308 asymptotic curve, 315 autonomous differential equation, 575 average cost function, 330 average rate of change, 148, 221 average speed of molecules, 516 average value of a function, 443, 557 average velocity, 5, 85, 145, 221 axes, coordinate, A11 axes of ellipse, A19 bacterial growth, 591, 598 Barrow, Isaac, 4, 153, 380 baseball and calculus, 601 base of a cylinder, 422 base of logarithm, 63, 428, A53 change of, 66 Bernoulli, James, 580, 607 Bernoulli, John, 307, 580 Bernoulli differential equation, 607 Bessel, Friedrich, 724 Bessel function, 674, 724 Bézier, Pierre, 639 Bézier curves, 624, 639 binomial series, 742, 748 Binomial Theorem, 772 discovery by Newton, 748 blackbody radiation, 757 blood flow, 227, 332, 551 bounded sequence, 682 Boyle’s Law, 221, 246 brachistochrone problem, 625 branches of hyperbola, 629, A20 Buffon’s needle problem, 565 bullet-nose curve, 51, 204 A113 A114 |||| INDEX cable (hanging), 255 calculator, graphing, 46, 315, 624, 646 calculus, invention of, 399 cancellation equations, 62 for inverse trigonometric functions, 67 for logarithms, 64 cans, manufacturing, 333 Cantor, Georg, 696 Cantor set, 696 capital formation, 554 cardiac output, 552 cardioid, 213, 643 carrying capacity, 233, 568 Cartesian coordinate system, A11 Cartesian plane, A11 Cassini, Giovanni, 649 catenary, 255 Cauchy, Augustin-Louis, 113, A45 Cauchy’s Mean Value Theorem, A45 Cavalieri’s Principle, 432 center of gravity, 542 center of mass, 542 centroid of a plane region, 543 Chain Rule, 197 change of base, 66 change of variables in integration, 401 charge, 224 chemical reaction, 224 circle, A16 area of, 469 circular cylinder, 422 cissoid, 629, 648 closed interval, A3 Closed Interval Method, 275 cochleoid, 670 coefficient of friction, 196, 278 of inequality, 399 of a polynomial, 28 of a power series, 723 combinations of functions, 41 comets, orbits of, 668 comparison properties of the integral, 375 Comparison Test, 514, 706 comparison tests for series, 705 Comparison Theorem for integrals, 514 Completeness Axiom, 682 complex conjugate, A55 complex exponentials, A57 complex number(s), A55 argument of, A57 division of, A55, A58 equality of, A55 imaginary part of, A55 modulus of, A56 multiplication of, A55, A58 polar form, A57 powers of, A59 principal square root of, A56 real part of, A46 roots of, A60 composition of functions, 41, 197 continuity of, 125 derivative of, 199 compound interest, 238, 306, 618 compressibility, 225 computer algebra system, 91, 624 for graphing sequences, 680 for integration, 491 computer, graphing with, 46, 315, 646 concavity, 291 Concavity Test, 291, A44 concentration, 224 conchoid, 626, 648 conditionally convergent series, 715 cone, 431 conic section, 654, 662 directrix, 662 eccentricity, 662 focus, 662 polar equation, 663 shifted, 659, A21 conjugates, properties of, A56 constant function, 173 Constant Multiple Law of limits, 100 Constant Multiple Rule, 176 consumer surplus, 550 continued fraction expansion, 686 continuity of a function, 119 on an interval, 121 from the left, 121 from the right, 121 continuous compounding of interest, 306, 618 continuous random variable, 555 convergence absolute, 714 conditional, 715 of an improper integral, 509, 512 interval of, 725 radius of, 725 of a sequence, 677 of a series, 688 convergent improper integral, 509, 512 convergent sequence, 677 convergent series, 688 properties of, 693 coordinate(s), A2 Cartesian, A11 polar, 639 rectangular, A11 coordinate axes, A11 Cornu’s spiral, 637 cosine function, A26 derivative, 192 graph, 33, A31 power series, 740 cost function, 228, 327 critical number, 274 cross-section, 422 cubic function, 28 current, 201 curvature, 638 curve(s) asymptotic, 315 Bézier, 624, 639 bullet-nose, 51, 204 Devil’s, 213 length of, 525 orthogonal, 214 parametric, 621 polar, 641 smooth, 525 swallowtail catastrophe, 629 curve fitting, 25 curve-sketching procedure, 308 cusp, 626 cycloid, 624 cylinder, 422 cylindrical shell, 433 decay, law of natural, 236 decay, radioactive, 236 decreasing function, 20 decreasing sequence, 681 definite integral, 366 properties of, 373 Substitution Rule for, 404 definite integration by parts, 456 by substitution, 404 degree of a polynomial, 28 delta (⌬) notation, 148 demand curve, 327, 550 demand function, 327, 550 De Moivre, Abraham, A59 De Moivre’s Theorem, A59 density linear, 223 liquid, 540 INDEX dependent variable, 12 derivative(s), 143,154 of a composite function, 199 of a constant function, 173 domain of, 154 of exponential functions, 180, 201, A51, A53 as a function, 154 higher, 160 of hyperbolic functions, 254 of an integral, 380 of an inverse function, 212 of inverse trigonometric functions, 211, 213 left-hand, 165 of logarithmic functions, 215, A49, A54 notation, 157 of a power function, 174 of a power series, 729 of a product, 183, 184 of a quotient, 185 as a rate of change, 148 right-hand, 165 second, 160 as the slope of a tangent, 146 third, 161 of trigonometric functions, 189, 193 Descartes, René, A11 descent of aircraft, determining start of, 206 Devil’s curve, 213 Difference Law of limits, 100 Difference Rule, 177 differentiable function, 157 differential, 250 differential equation, 234, 342, 566, 569 autonomous, 575 Bernoulli, 607 first-order, 580 general solution of, 569 linear, 602 logistic, 592, 687 order of, 569 second-order, 569 separable, 580 solution of, 569 differentiation, 157 formulas for, 187, RP5 implicit, 207, 208 logarithmic, 217 of a power series, 729 term by term, 729 differentiation operator, 157 Direct Substitution Property, 102 direction field, 572, 573, 600 directrix, 655, 662 discontinuity, 119, 120 discontinuous function, 119 discontinuous integrand, 511 disk method, 424 dispersion, 280 displacement, 145, 395 distance between points in a plane, A11 between real numbers, A7 distance formula, A12 distance problem, 362 divergence of an improper integral, 509, 512 of an infinite series, 688 of a sequence, 677 Test for, 692 divergent improper integral, 509, 512 divergent sequence, 677 divergent series, 688 division of power series, 745 domain of a function, 11 double-angle formulas, A29 drumhead, vibration of, 724 dye dilution method, 552 e (the number), 56, 179, A50 as a limit, 219 eccentricity, 662 electric circuit, 605 to a flash bulb, 84, 205 elementary function, 487 ellipse, 213, 656, 662, A19 area, 491 directrix, 662 eccentricity, 662 foci, 656, 662 major axis, 667, 657 polar equation, 663 reflection property, 658 rotated, 214 vertices, 666, 657 empirical model, 25 end behavior of a function, 142 endpoint extreme values, 272 epicycloid, 630 equation(s) of a circle, A17 differential (see Differential equation) of an ellipse, A19 of a graph, A10, A16 of a hyperbola, A20 of a line, A12, A13, A14, A16 linear, A14 logistic, 568 |||| A115 logistic differential, 568, 600 Lotka-Volterra, 609 nth-degree, 210 of a parabola, A18 parametric, 631 point-slope, 19, A12 polar, 641 predator-prey, 609 second-degree, A16 slope-intercept, A13 two-intercept form, A16 equilateral hyperbola, A21 equilibrium point, 610 equilibrium position, 196 equilibrium solution, 568, 609 error in approximate integration, 521 percentage, 251 relative, 251 in Taylor approximation, 749 error bounds, 499, 503 error estimate for alternating series, 712 for the Midpoint Rule, 521 for Simpson’s Rule, 525 for the Trapezoidal Rule, 521 escape velocity, 539 estimate of the sum of a series, 700, 708, 712, 717 Eudoxus, Euler’s constant, 704 Euler’s formula, A61 Euler’s Method, 575 even function, 19, 308 exponential decay, 591 exponential function(s), 33, 92, 180 with base a, A53 derivatives of, 180, 201, A51, A53 graphs of, 53, 180 integration of, 370, 385, 400 limits of, 137, A51 power series for, 736 properties of, A51 exponential graph, 53 exponential growth, 591 exponents, laws of, 54, A51, A53 extrapolation, 27 extreme value, 271 Extreme Value Theorem, 272 family of functions, 50, 320 family of hypocycloids, 629 family of solutions, 568 fat circles, 211, 531 Fermat, Pierre, 4, 153, 273 A116 |||| INDEX Fermat’s Principle, 331 Fermat’s Theorem, 273 Fibonacci, 686 Fibonacci sequence, 676 First Derivative Test, 288 for Absolute Extreme Values, 324 first-order linear differential equation, 602 fixed point of a function, 171, 286 flash bulb, current to, 84 flux, 551, 552 FM synthesis, 318 focus of a conic section, 662 of an ellipse, 656, 662 of a hyperbola, 658 of a parabola, 655 folium of Descartes, 208, 672 force, 438 exerted by fluid, 540 Fourier, Joseph, 230 Fourier sequence, 467 four-leaved rose, 643 fractions (partial), 473, 474 Fresnel, Augustin, 383 Fresnel function, 383 frustum of a cone, 431 of a pyramid, 432 function(s), 12 absolute value, 17 algebraic, 31 arc length, 528 arrow diagram of, 12 average cost, 330 average value of, 433, 557 Bessel, 674, 724 combinations of, 41 composite, 41, 197 constant, 173 continuous, 119 cost, 228, 327 cubic, 28 decreasing, 20 demand, 327 derivative of, 146 differentiable, 157 discontinuous, 119 domain of, 11 elementary, 487 even, 19, 308 exponential, 33, 52, 180 extreme values of, 271 family of, 50, 320 fixed point of, 171, 286 Fresnel, 397 Gompertz, 600 graph of, 12 greatest integer, 105 Heaviside, 45, 92 hyperbolic, 254 implicit, 207 increasing, 20 inverse, 61 inverse hyperbolic, 254 inverse trigonometric, 67, 68 limit of, 88, 110 linear, 24 logarithmic, 34, 63, A48, A53 machine diagram of, 12 marginal cost, 229, 327 marginal profit, 327 marginal revenue, 327 natural logarithmic, 64 nondifferentiable, 60 odd, 19, 308 one-to-one, 60 periodic, 308 piecewise defined, 17 polynomial, 28 position, 145 power, 29, 173 profit, 327 quadratic, 28 ramp, 45 range of, 11 rational, 31, 473 reciprocal, 31 reflected, 38 representation as a power series, 728 representations of, 11, 13 revenue, 327 root, 30 shifted, 37 sine integral, 389 step, 18 stretched, 38 transcendental, 34 transformation of, 37, 38 translation of, 38 trigonometric, 32, A26 value of, 12 Fundamental Theorem of Calculus, 381, 384, 387 G (gravitational constant), 231, 442 Gabriel’s horn, 537 Galileo, 625, 633 Galois, Evariste, 210 Gause, G F., 596 Gauss, Karl Friedrich, A35 geometric series, 688 Gompertz function, 600 Gourdon, Xavier, 739 graph(s) of an equation, A10, A16 of exponential functions, 53 of a function, 12 of logarithmic functions, 66 of a parametric curve, 632 polar, 641, 646 of power functions, 30, RP3 of trignometric functions, A30, RP2 graphing calculator, 46, 315, 624, 646 gravitational acceleration, 438 gravitation law, 442 greatest integer function, 105 Gregory, James, 732 Gregory’s series, 732 growth, law of natural, 234, 591 growth rate, 226 relative, 234 half-angle formulas, A29 half-life, 236 hare-lynx system, 612 harmonic series, 791 Heaviside, Oliver, 92 Heaviside function, 45, 92 Hecht, Eugene, 250, 754 height of a rocket 459 higher derivatives, 160 Hooke’s Law, 439 horizontal asymptote, 132 horizontal line, A13 Horizontal Line Test, 60 Hubble Space Telescope, 276 Huygens, 625 hydrostatic pressure and force, 539 hyperbola, 213, 658, 662, A20 asymptotes, 629, A20 branches, 629, A20 directrix, 662 eccentricity, 662 equation, 658, A20 equilateral, A21 foci, 658, 662 polar equation, 663 reflection property, 662 vertices, 658 hyperbolic function(s), 254 derivatives, 254 inverse, 257, 258 hyperbolic identities, 255 hyperbolic substitution, 471 hypocycloid, 629 INDEX i (imaginary number), A46 I/D Test, 287 ideal gas law, 233 implicit differentiation, 207, 208 implicit function, 207 improper integral, 508 impulse of a force, 601 Increasing/Decreasing Test, 287 increasing function, 20 increasing sequence, 681 increment, 147 indefinite integrals, 392 table of, 392 independent variable, 11 indeterminate difference, 298 indeterminate forms of limits, 298 indeterminate product, 302 indeterminate power, 303 index of summation, A34 inequalities, rules for, A4 infinite discontinuity, 120 infinite interval, 530, 531 infinite limit, 94, 116, 136 infinite sequence (see Sequence) infinite series (see Series) inflection point, 219 initial condition, 570 initial point of a parametric curve, 662 initial-value problem, 570 instantaneous rate of change, 85, 148, 148, 221 instantaneous rate of growth, 221 instantaneous rate of reaction, 225 instantaneous velocity, 85, 145, 221 integer, A2 integral(s) approximations to, 372 change of variables in, 400 comparison properties of, 375 definite, 366 derivative of, 401 evaluating, 369 improper, 508 indefinite, 391 patterns in, 494 properties of, 373 of symmetric functions, 405 table of, 484, RP6–10 units for, 396 Integral Test, 697 integrand, 366 discontinuous, 511 integration, 366 approximate, 495 by computer algebra system, 491 of exponential functions, 370, 385, 400 formulas, 452, 484, RP6–10 indefinite, 391 limits of, 366 numerical, 496 by partial fractions, 473 left-hand derivative, 165 left-hand limit, 192, 113 Leibniz, Gottfried Wilhelm, 4, 157, 399, 580, 748 Leibniz notation, 157 lemniscate, 213 length of a curve, 325 of a line segment, A7, A12 of a parametric curve, 635 of a polar curve, 652 l’Hospital, Marquis de, 299, 307 l’Hospital’s Rule, 299, 307 origins of, 307 libration point, 340 limaỗon, 647 Limit Comparison Test, 707 Limit Laws, 99, A39 Limit Laws for sequences, 778 limit(s), calculating, 99 of exponential functions, 136, 137 of a function, 88, 110 infinite, 94, 116, 136 at infinity, 130, 131, 137 of integration, 366 left-hand, 93, 113 one-sided, 93, 113 precise definitions, 109, 113, 116, 138, 140 properties of, 99 right-hand, 93, 163 of a sequence, 6, 359, 677 of a trigonometric function, 190 linear approximation, 246 linear density, 223 linear differential equation, 602 linear equation, A14 linear function, 24 linearization, 48 linear model, 24 linear regression, 27 line(s) in the plane, A12 equations of, A12, A13, A14 horizontal, A13 normal, 181 parallel, A14 perpendicular, A14 secant, 4, 83 |||| slope of, A12 tangent, 4, 83, 144 liquid force, 539, 540 Lissajous figure, 673 lithotripsy, 658 local maximum and minimum, 271 logarithm(s), 34, 63 laws of, 64, A49 natural, 64, A48 notation for, 64 logarithmic differentiation, 217 logarithmic function(s), 34, 63 with base a, A53 derivatives of, 215, A49, A53 graphs of, 64, 66 integration of, 218 limits of, 101, A50 properties of, 68, A49 logistic difference equation, 687 logistic differential equation, 568, 592 logistic model, 600 logistic sequence, 687 LORAN system, 661 Lotka-Volterra equations, 609 machine diagram of a function, 12 Maclaurin, Colin, 736 Maclaurin series, 734, 736 table of, 741 major axis of ellipse, 657 marginal cost function, 228, 327 marginal profit function, 327 marginal propensity to consume or save, 695 marginal revenue function, 327 mass, center of, 542 mathematical induction, 695 principle of, 77 mathematical model, 14, 24 maximum and minimum values, 271 mean life of an atom, 517 mean of a probability density function, 557 Mean Value Theorem, 282 for integrals, 443 mean waiting time, 557 median of a probability density function, 558 method of cylindrical shells, 433 method of exhaustion, 3, 102 method of least squares, 27 midpoint formula, A16 Midpoint Rule, 372, 496 error in using, 497 mixing problems, 584 A117 A118 |||| INDEX modeling with differential equations, 567 motion of a spring, 568 population growth, 55, 567, 591, 600, 597, 616 vibration of membrane, 724 model(s), mathematical, 24 comparison of natural growth vs logistic, 596 empirical, 25 exponential, 33 Gompertz function, 600 linear, 24 logarithmic, 34 polynomial, 28 power function, 29 predator-prey, 233, 609 rational function, 31 seasonal growth, 600 trigonometric, 32, 33 von Bertalanffy, 616 modulus, A56 moment about an axis, 543 of a lamina, 543 of a mass, 543 of a system of particles, 543 momentum of an object, 601 monotonic sequence, 681 Monotonic Sequence Theorem, 683 movie theater seating, 446 multiplication of power series, 745 multiplier effect, 695 natural exponential function, 56, A50 derivatives of, 180, A51 graph of, 180 properties of, A51 natural growth law, 234, 591 natural logarithm function, 64, A48 derivatives of, 215, A49 limits of, 424 properties of, A49 negative angle, A25 net area, 367 Net Change Theorem, 394 net investment flow, 554 Newton, Sir Isaac, 4, 9, 102, 153, 157, 380, 399, 748 newton (unit of force), 438 Newton’s Law of Cooling, 237 Newton’s Law of Gravitation, 231, 442 Newton’s method, 334, 335 Newton’s Second Law, 438 Nicomedes, 626 nondifferentiable function, 160 normal distribution, 559 normal line, 181 nth-degree equation, finding roots of, 210 nth-degree Taylor polynomial, 254, 737 number complex, A55 integer, A2 irrational, A2 rational, A2 real, A2 numerical integration, 496 odd function, 19, 308 one-sided limits, 92, 113 one-to-one function, 60 open interval, A3 optics first-order, 754 Gaussian, 7554 third-order, 755 optimization problems, 271, 322 order of a differential equation, 569 ordered pair, A10 Oresme, Nicole, 692 origin, A2, A10 orthogonal curves, 214 orthogonal trajectory, 214, 583 ovals of Cassini, 649 Pappus, Theorem of, 546 Pappus of Alexandria, 546 parabola, 655, 663, A18 axis, 655 directrix, 655 equation, 656 focus, 655, 662 polar equation, 663 reflection property, 268, 269 vertex, 655 paradoxes of Zeno, 6, parallelepiped, 422 parallel lines, A14 parameter, 621 parametric curve, 621 parametric equations, 621 paraxial rays, 249 partial fractions, 473, 474 partial integration, 453, 454 partial sum of a series, 688 parts, integration by, 453, 454 patterns in integrals, 494 pendulum, approximating the period of, 249, 253 percentage error, 251 perihelion, 667 perilune, 661 period, 308 periodic function, 308 perpendicular lines, A14 phase plane, 610 phase portrait, 610 phase trajectory, 610 piecewise defined function, 17 Planck’s Law, 757 point of inflection, 291 point-slope equation of a line, 18, A12 Poiseuille, Jean-Louis-Marie, 227 Poiseuille’s Laws, 253, 332, 552 polar axis, 639 polar coordinates, 639 polar equation(s), 639 of conics, 663 graph of, 641 polar form of a complex number, A57 pole, 639 polynomial, 28 population growth, 591 of bacteria, 227, 591, 598 of fish, 599 of insects, 483 models, 567 world, 55, 235 position function, 145 positive angle, A25 potential, 520 pound, 438 power consumption, approximation of, 396 power function, 29, 173 Power Law of limits, 101 Power Rule, 174, 218 power series, 723 coefficient of, 723 differentiation of, 729 division of, 745 integration of, 729 interval of convergence, 725 multiplication of, 745 radius of convergence, 725 representations of functions as, 728 predator, 608 predator-prey model, 233, 609 pressure exerted by a fluid, 540 prey, 609 prime notation, 146, 177 principal square root of a complex number, A56 principle of mathematical induction, 77, 80, A36 INDEX probability density function, 555 for customer waiting time, 561 problem-solving principles, 76 producer surplus, 553 product formulas, A29 Product Law of limits, 100 Product Rule, 183 profit function, 327 projectile, 629 p-series, 699 quadrant, A11 quadratic approximation, 254 quadratic function, 28 Quotient Law of limits, 100 Quotient Rule, 185 radian measure, 189, A24 radiation from stars, 757 radioactive decay, 235 radiocarbon dating, 240 radius of convergence, 725 rainbow, formation and location of, 279 rainbow angle, 279 ramp function, 45 range of a function, 11 rate of change average, 148, 221 derivative as, 148 instantaneous, 85, 148, 221 rate of growth, 226 rate of reaction, 225 rates, related, 241 rational function, 31 integration of, 473 rationalizing substitution, 481 rational number, A2 Ratio Test, 716 Rayleigh-Jeans Law, 757 real line, A3 real number, A2 rearrangement of a series, 719 reciprocal function, 31 Reciprocal Rule, 189 rectangular coordinate system, A11 rectilinear motion, 343 reduction formula, 457 reflecting a function, 38 reflection property of an ellipse, 658 of a hyperbola, 662 of a parabola, 268, 269 region under a graph, 355, 360 between two graphs, 415 related rates, 241 relative error, 251 relative growth rate, 234, 592 relative maximum and minimum, 271 remainder estimates for the Alternating Series, 712 for the Comparison Test, 707 for the Integral Test, 701 for the Ratio Test, 716 remainder of the Taylor series, 737 removable discontinuity, 120 representations of functions, 11, 13 revenue function, 327 revolution, solid of, 427 revolution, surface of, 532 Riemann, Georg Bernhard, 367 Riemann sum(s), 367 right circular cylinder, 422 right-hand derivative, 165 right-hand limit, 92, 113 RMS voltage, 466 Roberval, Gilles de, 386, 633 Rolle, Michel, 280 roller coaster, design of, 182 Rolle’s Theorem, 280 root function, 30 roots of a complex number, A60 roots of an nth-degree equation, 210 Root Test, 714 rumors, rate of spread of, 230 sample point, 360 scatter plot, 14 seasonal growth model, 600 secant function, A26 derivative, 193 graph, A31 secant line, 4, 83 second derivative, 160 Second Derivative Test, 292 second-order differential equation, 569 sector of a circle, 650 sensitivity, 233 separable differential equation, 580 sequence, 6, 675 bounded, 682 convergent, 677 decreasing, 681 divergent, 677 Fibonacci, 676 graph of, 681 increasing, 681 limit of, 6, 357, 677 monotonic, 681 |||| of partial sums, 688 term of, 675 series, 7, 687 absolutely convergent, 714 alternating, 710 alternating harmonic, 711, 715 binomial, 742, 748 coefficient of, 723 conditionally convergent, 715 convergent, 688 divergent, 688 geometric, 688 Gregory’s, 732 harmonic, 691 infinite, 687 Maclaurin, 734, 736 p-, 699 partial sum of, 688 power, 723 rearrangement of, 719 strategy for testing, 721 sum of, 7, 688 Taylor, 734, 736 term of, 687 trigonometric, 723 set, A3 serpentine, 188 shell method, 433 shifted conics, 659, A21 shift of a function, 37 Sierpinski carpet, 696 sigma notation, 360, A34 simple harmonic motion, 205 Simpson, Thomas, 501, 502 Simpson’s Rule, 500, 502 error bounds for, 503 sine function, A26 derivative, 193 graph, 32, A31 power series, 740 sine integral function, 389 slant asymptote, 312 slope, A12 slope-intercept equation of a line, A13 slope field, 573 smooth curve, 525 smooth function, 525 Snell’s Law, 331 snowflake curve, 761 solid, 422 volume of, 423 solid of revolution, 427 rotated on a slant, 538 volume of, 430, 434, 538 A119 A120 |||| INDEX solution of predator-prey equations, 609 solution curve, 572 speed, 148 speedometer readings, interpretation of, 85 spherical zones, 564 spring constant, 439, 568 Squeeze Theorem, 105, A42 for sequences, 679 standard deviation, 559 stellar stereography, 539 step function, 18 strategy for integration, 483, 484 for optimization problems, 322 for problem solving, 76 for related rates, 243 for testing series, 721 for trigonometric integrals, 462, 463 stretching a function, 38 strophoid, 653, 671 Substitution Rule, 400, 401, 404 subtraction formulas for sine and cosine, A29 sum of a geometric series, 689 of an infinite series, 688 of partial fractions, 474 Riemann, 367 telescoping, 791 Sum Law of limits, 100 summation notation, A34 Sum Rule, 177 supply function, 553 surface area, 534, 538 of a parametric surface, 635 surface of revolution, 532 area of, 534, 538 swallowtail catastrophe curve, 629 symmetric functions, integrals of, 405 symmetry, 19, 308, 405 in polar graphs, 644 symmetry principle, 543 table of differentiation formulas, 187, RP5 tables of integrals, 484, RP6–10 use of, 489 tangent function, A26 derivative, 192 graph, 33, A31 tangent line(s), 143 to a curve, 4, 88, 144 early methods of finding, 153 to a parametric curve, 630, 631 to a polar curve, 644 tangent line approximation, 247 tangent problem, 4, 83, 144 tautochrone problem, 625 Taylor, Brook, 736 Taylor polynomial, 254, 737, 749 Taylor series, 734, 736 Taylor’s Inequality, 737 techniques of integration, summary, 484 telescoping sum, 691 term of a sequence, 675 term of a series, 687 terminal point of a curve, 622 terminal velocity, 587 term-by-term differentiation and integration, 729 tests for convergence and divergence of series Alternating Series Test, 710 Comparison Test, 705 Integral Test, 697 Limit Comparison Test, 707 Ratio Test, 716 Root Test, 714 summary of tests, 721 Test for Divergence, 692 third derivative, 161 Torricelli, Evangelista, 633 Torricelli’s Law, 231 torus, 432, 473 total fertility rate, 169 transcendental function, 34 transformation of a function, 37 translation of a function, 38 Trapezoidal Rule, 497 error in, 497 Triangle Inequality, A8 trigonometric functions, 32, A26 derivatives of, 189, 193 graphs of, A30 integrals of, 460 inverse, 67 limits of, 190 trigonometric identities, A28 trigonometric integrals, 460 strategy for evaluating, 462, 463 trigonometric series, 723 trigonometric substitutions, 467 table of, 467 trochoid, 628 Tschirnhausen cubic, 214, 421 ultraviolet catastrophe, 757 union of sets, A3 value of a function, 11 variable dependent, 11 independent, 11 variables, change of, 401 vascular branching, 332 velocity, 4, 83, 145, 221 average, 5, 86, 145, 221 instantaneous, 86, 145, 221 velocity gradient, 228 velocity problem, 85, 145 Verhulst, 568 vertex of a parabola, 655 vertical asymptote, 95, 308 Vertical Line Test, 16 vertical tangent line, 159 vertical translation of a graph, 37 vertices of an ellipse, 656, 657 of a hyperbola, 658 vibration of a rubber membrane, model of, 724 viewing rectangle, 46 volume, 423 by cross-sections, 422 by cylindrical shells, 433 by disks, 424, 425 of a solid, 422 of a solid of revolution, 427, 538 of a solid on a slant, 538 by washers, 426, 427 Volterra, Vito, 609 Von Bertalanffy model, 616 Wallis, John, Wallis product, 459 washer method, 426 Weierstrass, Karl, 482 weight, 438 witch of Maria Agnesi, 188 work, 438 Wren, Sir Christopher, 635 x-axis, A10 x-coordinate, A10 x-intercept, A19 y-axis, A10 y-coordinate, A10 y-intercept, A19 Zeno, Zeno’s paradoxes, 6, zone of a sphere, 538 ... that might be preferable for some instructors Most of them also come in single variable and multivariable versions N N N Calculus, Sixth Edition, is similar to the present textbook except that... Engineering and Physics courses concurrently with calculus WHAT’S NEW IN THE SIXTH EDITION? Here are some of the changes for the sixth edition of Single Variable Calculus: Early Transcendentals: N N N N... Essential Calculus: Early Transcendentals resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter xi xii |||| PREFACE N N Calculus:
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