www.elsolucionario.org www.elsolucionario.org Student Solutions Manual for MULTIVARIABLE CALCULUS EIGHTH EDITION DAN CLEGG Palomar College BARBARA FRANK Cape Fear Community College Australia Brazil Mexico Singapore United Kingdom United States 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 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02-200-203 Cengage Learning 20 Channel Center Street Boston, MA 02210 USA 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 emailed to permissionrequest@cengage.com Cengage Learning is a leading provider of customized learning solutions with employees residing in nearly 40 different countries and sales in more than 125 countries around the world Find your local representative at: www.cengage.com Cengage Learning products are represented in Canada by Nelson Education, Ltd 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 Printed in the United States of America Print Number: 01 Print Year: 2015 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 This Student Solutions Manual contains detailed solutions to selected exercises in the text Multivariable Calculus, Eighth Edition (Chapters 10–17 of Calculus, Eighth Edition, and Calculus: Early Transcendentals, Eighth Edition) by James Stewart Specifically, it includes solutions to the odd-numbered exercises in each chapter section, review section, True-False Quiz, and Problems Plus section 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, 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 additional 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 challenge of solving a problem We would like to thank James Stewart for entrusting us with the writing of this manual and offering suggestions, Gina Sanders for reviewing solutions for accuracy and style, and the staff of TECH-arts for typesetting and producing this manual as well as creating the illustrations We also thank Richard Stratton, Neha Taleja, Samantha Lugtu, Stacy Green, and Terry Boyle of Cengage Learning, for their trust, assistance, and patience www.elsolucionario.org DAN CLEGG Palomar College BARBARA FRANK Cape Fear Community College 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 www.elsolucionario.org 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 ■ ABBREVIATIONS AND SYMBOLS CD concave downward CU concave upward D the domain of f FDT First Derivative Test HA horizontal asymptote(s) I interval of convergence IP inflection point(s) R radius of convergence VA vertical asymptote(s) CAS = PR QR CR H = j = indicates the use of a computer algebra system indicates the use of the Product Rule indicates the use of the Quotient Rule indicates the use of the Chain Rule indicates the use of l’Hospital’s Rule indicates the use of Formula j in the Table of Integrals in the back endpapers s indicates the use of the substitution {u sin x, du cos x dx} c indicates the use of the substitution {u cos x, du Ϫsin x dx} www.elsolucionario.org = = v 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 www.elsolucionario.org 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 PARAMETRIC EQUATIONS AND POLAR COORDINATES 10.1 Curves Defined by Parametric Equations 10.2 10.3 10.4 10.5 10.6 Calculus with Parametric Curves Polar Coordinates 14 Areas and Lengths in Polar Coordinates 20 Conic Sections 26 Conic Sections in Polar Coordinates 31 Review 35 Problems Plus ■ 1 41 11 INFINITE SEQUENCES AND SERIES 43 11.1 Sequences 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 Series 50 The Integral Test and Estimates of Sums 58 The Comparison Tests 62 Alternating Series 65 Absolute Convergence and the Ratio and Root Tests Strategy for Testing Series 71 Power Series 73 Representations of Functions as Power Series 77 Taylor and Maclaurin Series 82 Applications of Taylor Polynomials 91 Review 97 43 www.elsolucionario.org Problems Plus ■ 67 105 12 VECTORS AND THE GEOMETRY OF SPACE 12.1 Three-Dimensional Coordinate Systems 12.2 12.3 Vectors 114 The Dot Product 111 111 119 vii 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 www.elsolucionario.org 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 www.elsolucionario.org 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 www.elsolucionario.org 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 www.elsolucionario.org 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 www.elsolucionario.org 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 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remove additional content at any time if subsequent rights restrictions require it ... PREFACE This Student Solutions Manual contains detailed solutions to selected exercises in the text Multivariable Calculus, Eighth Edition (Chapters 10–17 of Calculus, Eighth Edition, and Calculus: ... www .elsolucionario. org Student Solutions Manual for MULTIVARIABLE CALCULUS EIGHTH EDITION DAN CLEGG Palomar College BARBARA FRANK Cape Fear... 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 www .elsolucionario. org