Hass weir thomas university calculus early transcendentals 2nd txtbk

www.elsolucionario.net UNIVERSITY CALCULUS EARLY TRANSCENDENTALS Second Edition Joel Hass University of California, Davis Maurice D Weir Naval Postgraduate School George B Thomas, Jr Massachusetts Institute of Technology www.elsolucionario.net Editor in Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Sponsoring Editor: Caroline Celano Senior Content Editor: Elizabeth Bernardi Editorial Assistant: Brandon Rawnsley Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Sheila Spinney Digital Assets Manager: Marianne Groth Supplements Production Coordinator: Kerri McQueen Associate Media Producer: Stephanie Green Software Development: Kristina Evans and Marty Wright Executive Marketing Manager: Jeff Weidenaar Marketing Coordinator: Kendra Bassi Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Advisor: Michael Joyce Image Manager: Rachel Youdelman Manufacturing Manager: Evelyn Beaton Senior Manufacturing Buyer: Carol Melville Senior Media Buyer: Ginny Michaud Design Manager: Andrea Nix Production Coordination, Composition, and Illustrations: Nesbitt Graphics, Inc Cover Design: Andrea Nix Cover Image: Black Shore III—Iceland, 2007 All content copyright © 2009 Josef Hoflehner For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page C-1, which is hereby made part of this copyright page Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Hass, Joel University calculus: early transcendentals/Joel Hass, Maurice D Weir, George B Thomas, Jr.—2nd ed p cm Rev ed of: University calculus c2007 ISBN 978-0-321-71739-9 (alk paper) Calculus—Textbooks I Weir, Maurice D II Thomas, George B (George Brinton), 1914–2006 III Title QA303.2.H373 2011 515—dc22 2010035141 Copyright © 2012, 2007, Pearson Education, Inc 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 For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm 6—CRK—14 13 12 11 www.pearsonhighered.com ISBN 13: 978-0-321-71739-9 ISBN 10: 0-321-71739-2 www.elsolucionario.net Student resources: Maple and Mathematica modules, expanded biographies, flashcards, http://wps.aw.com/aw_thomas_calculus_series/ CONTENTS Preface ix Functions 1.1 1.2 1.3 1.4 1.5 1.6 14 Limits and Continuity 2.1 2.2 2.3 2.4 2.5 2.6 Functions and Their Graphs Combining Functions; Shifting and Scaling Graphs Trigonometric Functions 21 Graphing with Calculators and Computers 29 Exponential Functions 33 Inverse Functions and Logarithms 39 Rates of Change and Tangents to Curves 52 Limit of a Function and Limit Laws 59 The Precise Definition of a Limit 70 One-Sided Limits 79 Continuity 86 Limits Involving Infinity; Asymptotes of Graphs QUESTIONS TO GUIDE YOUR REVIEW 110 PRACTICE EXERCISES 111 ADDITIONAL AND ADVANCED EXERCISES 113 52 97 Differentiation 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 116 Tangents and the Derivative at a Point 116 The Derivative as a Function 120 Differentiation Rules 129 The Derivative as a Rate of Change 139 Derivatives of Trigonometric Functions 149 The Chain Rule 156 Implicit Differentiation 164 Derivatives of Inverse Functions and Logarithms Inverse Trigonometric Functions 180 Related Rates 186 Linearization and Differentials 195 QUESTIONS TO GUIDE YOUR REVIEW 206 PRACTICE EXERCISES 206 ADDITIONAL AND ADVANCED EXERCISES 211 www.elsolucionario.net 170 iii iv Contents Applications of Derivatives 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 289 Area and Estimating with Finite Sums 289 Sigma Notation and Limits of Finite Sums 299 The Definite Integral 305 The Fundamental Theorem of Calculus 317 Indefinite Integrals and the Substitution Method 328 Substitution and Area Between Curves 335 QUESTIONS TO GUIDE YOUR REVIEW 345 PRACTICE EXERCISES 345 ADDITIONAL AND ADVANCED EXERCISES 349 Applications of Definite Integrals 6.1 6.2 6.3 6.4 6.5 6.6 230 Integration 5.1 5.2 5.3 5.4 5.5 5.6 Extreme Values of Functions 214 The Mean Value Theorem 222 Monotonic Functions and the First Derivative Test Concavity and Curve Sketching 235 Indeterminate Forms and L’Hôpital’s Rule 246 Applied Optimization 255 Newton’s Method 266 Antiderivatives 271 QUESTIONS TO GUIDE YOUR REVIEW 281 PRACTICE EXERCISES 281 ADDITIONAL AND ADVANCED EXERCISES 285 214 353 Volumes Using Cross-Sections 353 Volumes Using Cylindrical Shells 364 Arc Length 372 Areas of Surfaces of Revolution 378 Work 383 Moments and Centers of Mass 389 QUESTIONS TO GUIDE YOUR REVIEW 397 PRACTICE EXERCISES 397 ADDITIONAL AND ADVANCED EXERCISES 399 Integrals and Transcendental Functions 7.1 7.2 7.3 The Logarithm Defined as an Integral 401 Exponential Change and Separable Differential Equations Hyperbolic Functions 420 QUESTIONS TO GUIDE YOUR REVIEW 428 PRACTICE EXERCISES 428 ADDITIONAL AND ADVANCED EXERCISES 429 www.elsolucionario.net 401 411 Contents Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 8.7 448 563 Parametrizations of Plane Curves 563 Calculus with Parametric Curves 570 Polar Coordinates 579 Graphing in Polar Coordinates 583 Areas and Lengths in Polar Coordinates 587 Conics in Polar Coordinates 591 QUESTIONS TO GUIDE YOUR REVIEW 598 PRACTICE EXERCISES 599 ADDITIONAL AND ADVANCED EXERCISES 600 Vectors and the Geometry of Space 11.1 11.2 11.3 11.4 11.5 11.6 486 Sequences 486 Infinite Series 498 The Integral Test 507 Comparison Tests 512 The Ratio and Root Tests 517 Alternating Series, Absolute and Conditional Convergence 522 Power Series 529 Taylor and Maclaurin Series 538 Convergence of Taylor Series 543 The Binomial Series and Applications of Taylor Series 550 QUESTIONS TO GUIDE YOUR REVIEW 559 PRACTICE EXERCISES 559 ADDITIONAL AND ADVANCED EXERCISES 561 Parametric Equations and Polar Coordinates 10.1 10.2 10.3 10.4 10.5 10.6 11 431 Infinite Sequences and Series 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 10 Integration by Parts 432 Trigonometric Integrals 439 Trigonometric Substitutions 444 Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems 456 Numerical Integration 461 Improper Integrals 471 QUESTIONS TO GUIDE YOUR REVIEW 481 PRACTICE EXERCISES 481 ADDITIONAL AND ADVANCED EXERCISES 483 v Three-Dimensional Coordinate Systems 602 Vectors 607 The Dot Product 616 The Cross Product 624 Lines and Planes in Space 630 Cylinders and Quadric Surfaces 638 QUESTIONS TO GUIDE YOUR REVIEW 643 PRACTICE EXERCISES 644 ADDITIONAL AND ADVANCED EXERCISES 646 www.elsolucionario.net 602 vi Contents 12 Vector-Valued Functions and Motion in Space 12.1 12.2 12.3 12.4 12.5 12.6 13 Curves in Space and Their Tangents 649 Integrals of Vector Functions; Projectile Motion 657 Arc Length in Space 664 Curvature and Normal Vectors of a Curve 668 Tangential and Normal Components of Acceleration 674 Velocity and Acceleration in Polar Coordinates 679 QUESTIONS TO GUIDE YOUR REVIEW 682 PRACTICE EXERCISES 683 ADDITIONAL AND ADVANCED EXERCISES 685 Partial Derivatives 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 14 649 686 Functions of Several Variables 686 Limits and Continuity in Higher Dimensions 694 Partial Derivatives 703 The Chain Rule 714 Directional Derivatives and Gradient Vectors 723 Tangent Planes and Differentials 730 Extreme Values and Saddle Points 740 Lagrange Multipliers 748 QUESTIONS TO GUIDE YOUR REVIEW 757 PRACTICE EXERCISES 758 ADDITIONAL AND ADVANCED EXERCISES 761 Multiple Integrals 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 763 Double and Iterated Integrals over Rectangles 763 Double Integrals over General Regions 768 Area by Double Integration 777 Double Integrals in Polar Form 780 Triple Integrals in Rectangular Coordinates 786 Moments and Centers of Mass 795 Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 814 QUESTIONS TO GUIDE YOUR REVIEW 823 PRACTICE EXERCISES 823 ADDITIONAL AND ADVANCED EXERCISES 825 www.elsolucionario.net 802 Contents 15 Integration in Vector Fields 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 16 17 828 Line Integrals 828 Vector Fields and Line Integrals: Work, Circulation, and Flux 834 Path Independence, Conservative Fields, and Potential Functions 847 Green’s Theorem in the Plane 858 Surfaces and Area 870 Surface Integrals 880 Stokes’ Theorem 889 The Divergence Theorem and a Unified Theory 900 QUESTIONS TO GUIDE YOUR REVIEW 911 PRACTICE EXERCISES 911 ADDITIONAL AND ADVANCED EXERCISES 914 First-Order Differential Equations 16.1 16.2 16.3 16.4 16.5 Online Solutions, Slope Fields, and Euler’s Method 16-2 First-Order Linear Equations 16-10 Applications 16-16 Graphical Solutions of Autonomous Equations 16-22 Systems of Equations and Phase Planes 16-29 Second-Order Differential Equations 17.1 17.2 17.3 17.4 17.5 vii Online Second-Order Linear Equations 17-1 Nonhomogeneous Linear Equations 17-8 Applications 17-17 Euler Equations 17-23 Power Series Solutions 17-26 Appendices AP-1 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.11 Real Numbers and the Real Line AP-1 Mathematical Induction AP-6 Lines, Circles, and Parabolas AP-10 Conic Sections AP-18 Proofs of Limit Theorems AP-26 Commonly Occurring Limits AP-29 Theory of the Real Numbers AP-31 Complex Numbers AP-33 The Distributive Law for Vector Cross Products AP-43 The Mixed Derivative Theorem and the Increment Theorem Taylor’s Formula for Two Variables AP-48 AP-44 Answers to Odd-Numbered Exercises A-1 Index I-1 Credits C-1 A Brief Table of Integrals www.elsolucionario.net T-1 This page intentionally left blank www.elsolucionario.net ... Data Hass, Joel University calculus: early transcendentals/ Joel Hass, Maurice D Weir, George B Thomas, Jr.? ?2nd ed p cm Rev ed of: University calculus c2007 ISBN 978-0-321-71739-9 (alk paper) Calculus? ??Textbooks... through the University Calculus, Early Transcendentals Web site, www.pearsonhighered.com /thomas, and MyMathLab WEB SITE www.pearsonhighered.com /thomas The University Calculus, Early Transcendentals. .. UNIVERSITY CALCULUS EARLY TRANSCENDENTALS Second Edition Joel Hass University of California, Davis Maurice D Weir Naval Postgraduate School George B Thomas, Jr Massachusetts