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INSTRUCTOR’S SOLUTIONS MANUAL MULTIVARIABLE ELKA BLOCK

FRANK PURCELL

T HOMAS ’ C ALCULUS THIRTEENTH EDITION AND

T HOMAS ’ C ALCULUS E ARLY T RANSCENDENTALS THIRTEENTH EDITION Based on the original work by

George B. Thomas, Jr Massachusetts Institute of Technology

as revised by Maurice

D. Weir Naval

Postgraduate School

Joel

Hass University of California, Davis

with the assistance of

Christopher Heil Georgia Institute of Technology

Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Reproduced by Pearson from electronic files supplied by the author. Copyright © 2014, 2010, 2008 Pearson Education, Inc. Publishing as Pearson, 75 Arlington Street, Boston, MA 02116. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. ISBN-13: 978-0-321-87901-1 ISBN-10: 0-321-87901-5 1 2 3 4 5 6 ?? 17 16 15 14 13

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TABLE OF CONTENTS 10 Infinite Sequences and Series 701 10.1 Sequences 701 10.2 Infinite Series 712 10.3 The Integral Test 720 10.4 Comparison Tests 728 10.5 Absolute Convergence; The Ratio and Root Tests 738 10.6 Alternating Series and Conditional Convergence 744 10.7 Power Series 752 10.8 Taylor and Maclaurin Series 764 10.9 Convergence of Taylor Series 769 10.10 The Binomial Series and Applications of Taylor Series 777 Practice Exercises 786 Additional and Advanced Exercises 795

11 Parametric Equations and Polar Coordinates 801 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Parametrizations of Plane Curves 801 Calculus with Parametric Curves 809 Polar Coordinates 819 Graphing Polar Coordinate Equations 825 Areas and Lengths in Polar Coordinates 832 Conic Sections 838 Conics in Polar Coordinates 849 Practice Exercises 860 Additional and Advanced Exercises 871

12 Vectors and the Geometry of Space 877 12.1 12.2 12.3 12.4 12.5 12.6

Three-Dimensional Coordinate Systems 877 Vectors 881 The Dot Product 886 The Cross Product 892 Lines and Planes in Space 898 Cylinders and Quadric Surfaces 906 Practice Exercises 912 Additional and Advanced Exercises 920

13 Vector-Valued Functions and Motion in Space 927 13.1 13.2 13.3 13.4 13.5

Curves in Space and Their Tangents 927 Integrals of Vector Functions; Projectile Motion 933 Arc Length in Space 941 Curvature and Normal Vectors of a Curve 945 Tangential and Normal Components of Acceleration 952

13.6 Velocity and Acceleration in Polar Coordinates 959 Practice Exercises 962 Additional and Advanced Exercises 969

14 Partial Derivatives 973 14.1 Functions of Several Variables 973 14.2 Limits and Continuity in Higher Dimensions 984 14.3 Partial Derivatives 990 14.4 The Chain Rule 999 14.5 Directional Derivatives and Gradient Vectors 1008 14.6 Tangent Planes and Differentials 1014 14.7 Extreme Values and Saddle Points 1024 14.8 Lagrange Multipliers 1040 14.9 Taylor's Formula for Two Variables 1052 14.10 Partial Derivatives with Constrained Variables 1055 Practice Exercises 1059 Additional and Advanced Exercises 1076

15 Multiple Integrals 1083 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8

Double and Iterated Integrals over Rectangles 1083 Double Integrals over General Regions 1086 Area by Double Integration 1100 Double Integrals in Polar Form 1105 Triple Integrals in Rectangular Coordinates 1112 Moments and Centers of Mass 1118 Triple Integrals in Cylindrical and Spherical Coordinates 1124 Substitutions in Multiple Integrals 1134 Practice Exercises 1142 Additional and Advanced Exercises 1149

16 Integrals and Vector Fields 1155 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8

Line Integrals 1155 Vector Fields and Line Integrals: Work, Circulation, and Flux 1161 Path Independence, Conservative Fields, and Potential Functions 1172 Green's Theorem in the Plane 1178 Surfaces and Area 1185 Surface Integrals 1196 Stokes' Theorem 1206 The Divergence Theorem and a Unified Theory 1213 Practice Exercises 1219 Additional and Advanced Exercises 1230

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