INSTRUCTOR SOLUTIONS MANUAL Complete Solutions Manual for MULTIVARIABLE CALCULUS SEVENTH EDITION DAN CLEGG Palomar College BARBARA FRANK Cape Fear Community College Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States © Cengage Learning g All A Rights Reserved © 2012 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 except as may be permitted by the license terms below 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 ISBN-13: 978-0-8400-4947-6 ISBN-10: 0-8400-4947-1 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 Purchase any of our products at our preferred online store www.cengagebrain.com NOTE: UNDER NO CIRCUMSTANCES MAY THIS MATERIAL OR ANY PORTION THEREOF BE SOLD, LICENSED, AUCTIONED, OR OTHERWISE REDISTRIBUTED EXCEPT AS MAY BE PERMITTED BY THE LICENSE TERMS HEREIN READ IMPORTANT LICENSE INFORMATION Dear Professor or 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Learning and/or its licensors The Supplement is furnished by Cengage Learning on an “as is” basis without any warranties, express or implied This Agreement will be governed by and construed pursuant to the laws of the State of New York, without regard to such State’s conflict of law rules Thank you for your assistance in helping to safeguard the integrity of the content contained in this Supplement We trust you find the Supplement a useful teaching tool Printed in the United States of America 15 14 11 © Cengage Learning g All A Rights Reserved ■ PREFACE This Complete Solutions Manual contains detailed solutions to all exercises in the text Multivariable Calculus, Seventh Edition (Chapters 10–17 of Calculus, Seventh Edition, and Calculus: Early Transcendentals, Seventh Edition) by James Stewart A Student Solutions Manual is also available, which contains solutions to the odd-numbered exercises in each chapter section, review section, True-False Quiz, and Problems Plus section as well as all solutions to the Concept Check questions (It does not, however, include solutions to any of the projects.) Because of differences between the regular version and the Early Transcendentals version of the text, some references are given in a dual format In these cases, users of the Early Transcendentals text should use the references denoted by “ET.” While we have extended every effort to ensure the accuracy of the solutions presented, we would appreciate correspondence regarding any errors that may exist Other suggestions or comments are also welcome, and can be sent to dan clegg at dclegg@palomar.edu or in care of the publisher: Brooks/Cole, Cengage Learning, 20 Davis Drive, Belmont CA 94002-3098 We would like to thank James Stewart for entrusting us with the writing of this manual and offering 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 © Cengage Learning g All A Rights Reserved © Cengage Learning g All A Rights Reserved ■ ABBREVIATIONS AND SYMBOLS CD concave downward CU concave upward D the domain of i FDT First Derivative Test HA horizontal asymptote(s) I interval of convergence I/D Increasing/Decreasing Test IP inÀection point(s) R radius of convergence VA vertical asymptote(s) CAS = indicates the use of a computer algebra system H indicates the use of l’Hospital’s Rule m indicates the use of Formula m in the Table of Integrals in the back endpapers s indicates the use of the substitution {x = sin {> gx = cos { g{} = = = c = indicates the use of the substitution {x = cos {> gx = sin { g{} © Cengage Learning g All A Rights Reserved © Cengage Learning g All A Rights Reserved ■ CONTENTS ■ 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 10.1 Curves Defined by Parametric Equations Laboratory Project 10.2 Polar Coordinates ■ 15 18 Bézier Curves 32 33 Laboratory Project ■ Families of Polar Curves 10.4 Areas and Lengths in Polar Coordinates 10.5 Conic Sections 10.6 Conic Sections in Polar Coordinates Review Running Circles Around Circles Calculus with Parametric Curves Laboratory Project 10.3 ■ 48 51 63 74 80 Problems Plus 93 ■ 11 INFINITE SEQUENCES AND SERIES 11.1 Sequences 97 97 Laboratory Project ■ Logistic Sequences 110 11.2 Series 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 114 138 143 156 160 11.10 Taylor and Maclaurin Series Laboratory Project ■ Applied Project Problems Plus ■ 169 179 An Elusive Limit 11.11 Applications of Taylor Polynomials Review 129 194 195 Radiation from the Stars 210 223 © Cengage Learning g All A Rights Reserved 209 149 viii ■ CONTENTS ■ 12 VECTORS AND THE GEOMETRY OF SPACE 12.1 Three-Dimensional Coordinate Systems 12.2 12.3 12.4 Vectors 242 The Dot Product 251 The Cross Product 260 Discovery Project 12.5 Equations of Lines and Planes ■ 13 VECTOR FUNCTIONS 273 Putting 3D in Perspective 285 287 313 13.1 Vector Functions and Space Curves 13.2 13.3 13.4 Derivatives and Integrals of Vector Functions 324 Arc Length and Curvature 333 Motion in Space: Velocity and Acceleration 348 Review Problems Plus 271 307 Applied Project ■ ■ Cylinders and Quadric Surfaces Review 297 Problems Plus 235 The Geometry of a Tetrahedron ■ Laboratory Project 12.6 235 ■ Kepler’s Laws 313 359 360 367 14 PARTIAL DERIVATIVES 373 14.1 Functions of Several Variables 14.2 14.3 14.4 14.5 14.6 14.7 Limits and Continuity 391 Partial Derivatives 398 Tangent Planes and Linear Approximations 416 The Chain Rule 425 Directional Derivatives and the Gradient Vector 437 Maximum and Minimum Values 449 Applied Project ■ Discovery Project 373 Designing a Dumpster ■ 469 Quadratic Approximations and Critical Points © Cengage Learning g All A Rights Reserved 471 CONTENTS 14.8 Lagrange Multipliers ■ Rocket Science Applied Project ■ Hydro-Turbine Optimization Review Problems Plus ■ 474 Applied Project 485 490 505 15 MULTIPLE INTEGRALS 511 15.1 Double Integrals over Rectangles 15.2 15.3 15.4 15.5 15.6 15.7 Iterated Integrals 516 Double Integrals over General Regions 521 Double Integrals in Polar Coordinates 534 Applications of Double Integrals 542 Surface Area 553 Triple Integrals 557 Discovery Project 15.8 ■ 15.9 511 Volumes of Hyperspheres Triple Integrals in Cylindrical Coordinates Discovery Project ■ ■ Roller Derby 575 584 594 15.10 Change of Variables in Multiple Integrals Review 601 Problems Plus 595 615 16 VECTOR CALCULUS 623 16.1 Vector Fields 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 Line Integrals 628 The Fundamental Theorem for Line Integrals Green’s Theorem 643 Curl and Divergence 650 Parametric Surfaces and Their Areas 659 Surface Integrals 673 Stokes’ Theorem 684 The Divergence Theorem 689 Review 694 Problems Plus 574 The Intersection of Three Cylinders Triple Integrals in Spherical Coordinates Applied Project ■ 488 623 705 © Cengage Learning g All A Rights Reserved 637 582 ■ ix CHAPTER 15 PROBLEMS PLUS But by Definition 15.1.5 this is equal to ]] Ô i ({> |) gD, so U lim q32 q