- STUDENT SOLUTIONS MANUAL for STEWART'S DAN CLEGG • BARBARA FRANK 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page i Student Solutions Manual for MULTIVARIABLE CALCULUS SEVENTH EDITION DAN CLEGG Palomar College BARBARA FRANK Cape Fear Community College Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . UnitedKingdom . United States 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page ii © 2012 Brooks/Cole, Cengage Learning ISBN-13: 987-0-8400-4945-2 ISBN-10: 0-8400-4945-5 ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, Brooks/Cole stored, or used in any form or by any means graphic, elec- 20 Davis Drive tronic, or mechanical, including but not limited to photocopy- Belmont, CA 94002-3098 ing, recording, scanning, digitizing, taping, Web distribution, USA information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of Cengage Learning is a leading provider of customized learn- the 1976 United States Copyright Act, without the prior ing solutions with office locations around the globe, including written permission of the publisher. Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: For product information and technology assistance, www.cengage.com/global contact us at Cengage Learning Customer & Sales Support, Cengage Learning products are represented in Canada by 1-800-354-9706 Nelson Education, Ltd. For permission to use material from this text or product, To learn more about Brooks/Cole, visit submit all requests online at www.cengage.com/brookscole www.cengage.com/permissions Further permissions questions can be emailed to Purchase any of our products at your local college store or [email protected] at our preferred online store www.cengagebrain.com Printed in the United States of America 1 2 3 4 5 6 7 15 14 13 12 11 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page iii ■ PREFACE This Student Solutions Manualcontains detailed solutions to selected exercises in the text Multivariable Calculus, Seventh Edition (Chapters 10–17 of Calculus, Seventh Edition, and Calculus: Early Transcendentals, Seventh Edition) by James Stewart. Specifically, it includes solu- tions to the odd-numbered exercises in each chapter section, review section, True-False Quiz, and Problems Plus section. Also included are all solutions to the Concept Check questions. Because of differences between the regular version and the Early Transcendentalsversion of the text, some references are given in a dual format. In these cases, readers of the Early Transcendentals text should use the references denoted by “ET.” Each solution is presented in the context of the corresponding section of the text. In general, solutions to the initial exercises involving a new concept illustrate that concept in more detail; this knowledge is then utilized in subsequent solutions. Thus, while the intermediate steps of a solution are given, you may need to refer back to earlier exercises in the section or prior sections for addition- al explanation of the concepts involved. Note that, in many cases, different routes to an answer may exist which are equally valid; also, answers can be expressed in different but equivalent forms. Thus, the goal of this manual is not to give the definitive solution to each exercise, but rather to assist you as a student in understanding the concepts of the text and learning how to apply them to the chal- lenge of solving a problem. We would like to thank James Stewart for entrusting us with the writing of this manual and offer- ing suggestions and Kathi Townes of TECH-arts for typesetting and producing this manual as well as creating the illustrations. We also thank Richard Stratton, Liz Covello, and Elizabeth Neustaetter of Brooks/Cole, Cengage Learning, for their trust, assistance, and patience. DAN CLEGG Palomar College BARBARA FRANK Cape Fear Community College © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. iii 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page iv 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page v ■ ABBREVIATIONS AND SYMBOLS CD concavedownward CU concaveupward D thedomainoff FDT FirstDerivativeTest HA horizontalasymptote(s) I intervalofconvergence I/D Increasing/DecreasingTest IP inflectionpoint(s) R radiusofconvergence VA verticalasymptote(s) C=AS indicatestheuseofacomputeralgebrasystem. =H indicatestheuseofl’Hospital’sRule. j = indicatestheuseofFormulajintheTableofIntegralsinthebackendpapers. =s indicatestheuseofthesubstitution u=sinx,du=cosxdx . { } =c indicatestheuseofthesubstitution u=cosx,du= sinxdx . { − } © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. v 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page vi 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page vii ■ CONTENTS ■ 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 1 10.1 Curves Defined by Parametric Equations 1 10.2 Calculus with Parametric Curves 7 10.3 Polar Coordinates 13 10.4 Areas and Lengths in Polar Coordinates 20 10.5 Conic Sections 26 10.6 Conic Sections in Polar Coordinates 32 Review 35 Problems Plus 43 ■ 11 INFINITE SEQUENCES AND SERIES 45 11.1 Sequences 45 11.2 Series 51 11.3 The Integral Test and Estimates of Sums 59 11.4 The Comparison Tests 62 11.5 Alternating Series 65 11.6 Absolute Convergence and the Ratio and Root Tests 68 11.7 Strategy for Testing Series 72 11.8 Power Series 74 11.9 Representations of Functions as Power Series 78 11.10 Taylor and Maclaurin Series 83 11.11 Applications of Taylor Polynomials 90 Review 97 Problems Plus 105 ■ 12 VECTORS AND THE GEOMETRY OF SPACE 111 12.1 Three-Dimensional Coordinate Systems 111 12.2 Vectors 114 12.3 The Dot Product 119 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. vii 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page viii viii ■ CONTENTS 12.4 The Cross Product 123 12.5 Equations of Lines and Planes 128 12.6 Cylinders and Quadric Surfaces 135 Review 140 Problems Plus 147 ■ 13 VECTOR FUNCTIONS 151 13.1 Vector Functions and Space Curves 151 13.2 Derivatives and Integrals of Vector Functions 157 13.3 Arc Length and Curvature 161 13.4 Motion in Space: Velocity and Acceleration 168 Review 173 Problems Plus 179 ■ 14 PARTIAL DERIVATIVES 183 14.1 Functions of Several Variables 183 14.2 Limits and Continuity 192 14.3 Partial Derivatives 195 14.4 Tangent Planes and Linear Approximations 203 14.5 The Chain Rule 207 14.6 Directional Derivatives and the Gradient Vector 213 14.7 Maximum and Minimum Values 220 14.8 Lagrange Multipliers 229 Review 234 Problems Plus 245 ■ 15 MULTIPLE INTEGRALS 247 15.1 Double Integrals over Rectangles 247 15.2 Iterated Integrals 249 15.3 Double Integrals over General Regions 251 15.4 Double Integrals in Polar Coordinates 258 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part. 49455_SSMFM_7eMV_pi-ix.qk_12289_SSMFM_6eMV_pi-xii.qk 2/24/11 5:19 PM Page ix CONTENTS ■ ix 15.5 Applications of Double Integrals 261 15.6 Surface Area 267 15.7 Triple Integrals 269 15.8 Triple Integrals in Cylindrical Coordinates 276 15.9 Triple Integrals in Spherical Coordinates 280 15.10 Change of Variables in Multiple Integrals 285 Review 289 Problems Plus 297 ■ 16 VECTOR CALCULUS 303 16.1 Vector Fields 303 16.2 Line Integrals 305 16.3 The Fundamental Theorem for Line Integrals 310 16.4 Green’s Theorem 313 16.5 Curl and Divergence 316 16.6 Parametric Surfaces and Their Areas 321 16.7 Surface Integrals 328 16.8 Stokes’ Theorem 333 16.9 The Divergence Theorem 335 Review 337 Problems Plus 343 ■ 17 SECOND-ORDER DIFFERENTIAL EQUATIONS 345 17.1 Second-Order Linear Equations 345 17.2 Nonhomogeneous Linear Equations 347 17.3 Applications of Second-Order Differential Equations 350 17.4 Series Solutions 352 Review 354 ■ APPENDIX 359 H Complex Numbers 359 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied, or duplicated, or posted to a publicly accessible website, in whole or in part.