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 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 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 211 170 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 429 401 411 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 646 602 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 825 802 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 T-1 This page intentionally left blank PREFACE We have significantly revised this edition of University Calculus, Early Transcendentals to meet the changing needs of today’s instructors and students The result is a book with more examples, more mid-level exercises, more figures, better conceptual flow, and increased clarity and precision As with the previous edition, this new edition provides a briefer, modern introduction to calculus that supports conceptual understanding but retains the essential elements of a traditional course These enhancements are closely tied to an expanded version of MyMathLab® for this text (discussed further on), providing additional support for students and flexibility for instructors In this second edition, we introduce the basic transcendental functions in Chapter After reviewing the basic trigonometric functions, we present the family of exponential functions using an algebraic and graphical approach, with the natural exponential described as a particular member of this family Logarithms are then defined as the inverse functions of the exponentials, and we also discuss briefly the inverse trigonometric functions We fully incorporate these functions throughout our developments of limits, derivatives, and integrals in the next five chapters of the book, including the examples and exercises This approach gives students the opportunity to work early with exponential and logarithmic functions in combinations with polynomials, rational and algebraic functions, and trigonometric functions as they learn the concepts, operations, and applications of single-variable calculus Later, in Chapter 7, we revisit the definition of transcendental functions, now giving a more rigorous presentation Here we define the natural logarithm function as an integral with the natural exponential as its inverse Today, an increasing number of students become familiar with the terminology and operational methods of calculus in high school However, their conceptual understanding of calculus is often quite limited when they enter college We have acknowledged this reality by concentrating on concepts and their applications throughout We encourage students to think beyond memorizing formulas and to generalize concepts as they are introduced Our hope is that after taking calculus, students will be confident in their problem-solving and reasoning abilities Mastering a beautiful subject with practical applications to the world is its own reward, but the real gift is the ability to think and generalize We intend this book to provide support and encouragement for both Changes for the Second Edition CONTENT In preparing this edition we have maintained the basic structure of the Table of Contents from the first edition, yet we have paid attention to requests by current users and reviewers to postpone the introduction of parametric equations until we present polar coordinates We have made numerous revisions to most of the chapters, detailed as follows: • Functions We condensed this chapter to focus on reviewing function concepts and introducing the transcendental functions Prerequisite material covering real numbers, intervals, increments, straight lines, distances, circles, and the conic sections is presented in Appendices 1–4 ix This page intentionally left blank A BRIEF TABLE OF INTEGRALS Basic Forms L L k dx = kx + C sany number kd dx x = ln ƒ x ƒ + C ax a x dx = + C ln a L 11 13 15 L L L L L sa 0, a Z 1d cos x dx = sin x + C csc2 x dx = -cot x + C 10 csc x cot x dx = -csc x + C 12 cot x dx = ln ƒ sin x ƒ + C 14 cosh x dx = sinh x + C 16 x dx = a tan-1 a + C 2 La + x x dx = sinh-1 a + C 19 2 L 2a + x L L L L L L L x n dx = xn+1 + C n + sn Z -1d e x dx = e x + C sin x dx = -cos x + C sec2 x dx = tan x + C sec x tan x dx = sec x + C tan x dx = ln ƒ sec x ƒ + C sinh x dx = cosh x + C x dx = sin-1 a + C L 2a - x x dx 18 = a sec-1 ` a ` + C L x2x - a dx x = cosh-1 a + C sx a 0d 20 2 L 2x - a 17 sa 0d Forms Involving ax ؉ b 21 22 23 25 27 L L L L L sax + bdn dx = sax + bdn + + C, asn + 1d xsax + bdn dx = n Z -1 sax + bdn + ax + b b c d + C, n + n + a2 sax + bd-1 dx = a ln ƒ ax + b ƒ + C xsax + bd-2 dx = A 2ax + b B n b c ln ƒ ax + b ƒ + d + C ax + b a2 A 2ax + b B dx = a n + n Z -1, -2 24 x b xsax + bd-1 dx = a - ln ƒ ax + b ƒ + C a L 26 dx x ` + C = ln ` b ax + b L xsax + bd 28 2ax + b dx = 22ax + b + b x n+2 + C, n Z -2 L dx L x2ax + b T-1 T-2 A Brief Table of Integrals 29 (a) 30 L dx L x2ax + b 2ax + b - 2b ln ` ` + C 2b 2ax + b + 2b = 2ax + b 2ax + b a dx + dx = + C x L x2ax + b x2 (b) 31 dx L x2ax - b = ax - b tan-1 + C b A 2b 2ax + b dx a dx = + C bx 2b L x2ax + b L x 2ax + b Forms Involving a2 ؉ x2 x dx x x dx 1 + = a tan-1 a + C 33 = tan-1 a + C 2 2 2 2 a + x sa + x d 2a sa + x d 2a L L dx -1 x 2 = sinh a + C = ln A x + 2a + x B + C 34 L 2a + x a2 x 35 2a + x dx = 2a + x + ln A x + 2a + x B + C 2 L x a4 ln A x + 2a + x B + C 36 x 2a + x dx = sa + 2x d2a + x 8 L 32 37 38 39 40 L 2a + x a + 2a + x dx = 2a + x - a ln ` ` +C x x 2a + x 2a + x 2 dx = ln x + 2a + x + C A B x x2 L x2 L 2a + x dx = - x2a + x a2 ln A x + 2a + x B + + C 2 a + 2a + x = - a ln ` ` + C x dx L x2a + x 41 dx L x 2a + x = - 2a + x + C a 2x Forms Involving a2 ؊ x2 42 44 46 47 49 51 dx x + a = ln ` x - a ` + C 2 2a La - x dx L 2a - x L L x = sin-1 a + C x 2a - x dx = 43 45 x2 L 2a - x dx = dx L x 2a - x 50 2a - x + C a 2x Forms Involving x2 ؊ a2 52 53 dx L 2x - a 2 L = ln ƒ x + 2x - a ƒ + C 2x - a dx = x a -1 x 2a - x + sin a + C 2 a x sin-1 a - x2a - x sa - 2x d + C 8 a -1 x sin a - x2a - x + C 2 = - L 2a - x dx = 2a - x a + 2a - x dx = 2a - x - a ln ` ` + C 48 x x dx x x + a = ln ` x - a ` + C + 2 2a 2sa - x d 4a L sa - x d x a2 2x - a ln ƒ x + 2x - a ƒ + C 2 2a - x 2a - x x dx = -sin-1 a + C x x L dx L x2a - x 2 a + 2a - x = - a ln ` ` + C x A Brief Table of Integrals 54 55 56 57 58 A 2x - a B dx = L dx L A 2x - a L B n x A 2x - a L x A 2x - a B n = n + x A 2x - a B s2 - nda B dx = - n-2 na 2x - a B dx, A n + 1L - n - dx , 2 sn - 2da L A 2x - a B n - 2-n A 2x - a B n x 2x - a dx = n n Z -1 n Z n+2 n + + C, n Z -2 x a4 s2x - a d2x - a ln ƒ x + 2x - a ƒ + C 8 2x - a x dx = 2x - a - a sec-1 ` a ` + C x L 2x - a 2x - a 2 dx = ln x + 2x a + C ƒ ƒ x x2 L x2 a2 x dx = 60 ln ƒ x + 2x - a ƒ + 2x - a + C 2 2 L 2x - a 59 61 dx L x2x - a x a 1 = a sec-1 ` a ` + C = a cos-1 ` x ` + C 62 dx L x 2x - a = 2x - a + C a 2x Trigonometric Forms 63 65 67 L sin ax dx = - a cos ax + C sin2 ax dx = L sin 2ax x + C 4a sinn ax dx = - L 64 sinn - ax cos ax n - + n na 66 L cosn ax dx = L (b) (c) 70 72 74 75 76 L L L cos2 ax dx = sin 2ax x + + C 4a cosn - ax sin ax n - + n na a2 Z b2 sinsa - bdx sinsa + bdx + C, 2sa - bd 2sa + bd a2 Z b2 cos ax cos bx dx = sinsa - bdx sinsa + bdx + + C, 2sa - bd 2sa + bd a2 Z b2 sin ax cos ax dx = - cos 2ax + C 4a cos ax dx = a ln ƒ sin ax ƒ + C sin ax L L L sin ax sin bx dx = L L cos ax dx = a sin ax + C sinn - ax dx cosn - ax dx L cossa + bdx cossa - bdx 69 (a) sin ax cos bx dx = + C, 2sa + bd 2sa - bd L 68 L 71 73 L L sinn ax cos ax dx = sinn + ax + C, sn + 1da cosn ax sin ax dx = - n Z -1 cosn + ax + C, sn + 1da sin ax cos ax dx = - a ln ƒ cos ax ƒ + C sinn ax cosm ax dx = sinn ax cosm ax dx = sinn - ax cosm + ax n - + sinn - ax cosm ax dx, m + nL asm + nd sinn + ax cosm - ax m - sinn ax cosm - ax dx, + m + nL asm + nd n Z -m m Z -n sreduces sinn axd sreduces cosm axd n Z -1 T-3 T-4 A Brief Table of Integrals 77 dx ax b - c p -2 b d + C, = tan-1 c tan a Ab + c L b + c sin ax a2b - c 78 c + b sin ax + 2c - b cos ax dx -1 ln ` ` + C, = 2 b + c sin ax b + c sin ax L a2c - b 79 dx ax p = - a tan a b + C L + sin ax 81 dx b - c ax = tan-1 c tan d + C, Ab + c L b + c cos ax a2b - c 82 c + b cos ax + 2c - b sin ax dx = ln ` ` + C, 2 b + c cos ax b + c cos ax L a2c - b 83 ax dx = a tan + C + cos ax L 85 L 87 89 91 93 97 84 x sin ax - a cos ax + C a2 86 L L tan ax dx = a ln ƒ sec ax ƒ + C 90 L tan2 ax dx = a tan ax - x + C 92 tann ax dx = L x n - cos ax dx n-1 ax tan tann - ax dx, asn - 1d L 88 n Z 94 L sec ax dx = a ln ƒ sec ax + tan ax ƒ + C 96 L sec2 ax dx = a tan ax + C 98 secn - ax tan ax n - secn - ax dx, secn ax dx = + 99 n - 1L asn 1d L 100 L 101 L cscn ax dx = - secn ax na + C, n Z ax dx p = a tan a + b + C L - sin ax b2 c2 dx ax = - a cot + C cos ax L L x cos ax dx = x cos ax + a sin ax + C a2 L n xn x n cos ax dx = a sin ax - a L cot ax dx = a ln ƒ sin ax ƒ + C L cot2 ax dx = - a cot ax - x + C L cotn ax dx = - L x n - sin ax dx cotn - ax cotn - ax dx, asn - 1d L L csc ax dx = - a ln ƒ csc ax + cot ax ƒ + C L csc2 ax dx = - a cot ax + C n Z n Z cscn - ax cot ax n - + cscn - ax dx, n - 1L asn - 1d secn ax tan ax dx = b2 c2 b2 c2 n xn x n sin ax dx = - a cos ax + a L 95 x sin ax dx = 80 b2 c2 n Z 102 L cscn ax cot ax dx = - cscn ax na + C, n Z Inverse Trigonometric Forms 103 105 L L sin-1 ax dx = x sin-1 ax + a 21 - a 2x + C tan-1 ax dx = x tan-1 ax - 104 L cos-1 ax dx = x cos-1 ax - a 21 - a 2x + C ln s1 + a 2x d + C 2a xn+1 a x n + dx sin-1 ax , n Z -1 n + n + L 21 - a 2x L xn+1 a x n + dx 107 cos-1 ax + , n Z -1 x n cos-1 ax dx = n + n + L 21 - a 2x L 106 108 L x n sin-1 ax dx = x n tan-1 ax dx = x n + dx xn+1 a tan-1 ax , n + n + L + a 2x n Z -1 A Brief Table of Integrals Exponential and Logarithmic Forms 109 111 113 114 115 L L L L L e ax dx = a e ax + C xe ax dx = 110 e ax sax - 1d + C a2 x nb ax dx = 112 x nb ax n x n - 1b ax dx, a ln b a ln b L b ax b ax dx = a + C, ln b L L n x ne ax dx = a x ne ax - a b 0, b Z L x n - 1e ax dx b 0, b Z e ax sin bx dx = e ax sa sin bx - b cos bxd + C a + b2 e ax cos bx dx = e ax sa cos bx + b sin bxd + C a + b2 2 116 L ln ax dx = x ln ax - x + C x n + 1sln axdm m x nsln axdm - dx, n Z -1 n + n + 1L L sln axdm + dx + C, m Z -1 x -1sln axdm dx = = ln ƒ ln ax ƒ + C 118 119 m + x ln ax L L x nsln axdm dx = 117 Forms Involving 22ax ؊ x2, a>0 120 121 122 123 124 125 dx x - a = sin-1 a a b + C L 22ax - x L x - a a -1 x - a 22ax - x + sin a a b + C 2 22ax - x dx = A 22ax - x B dx = n L dx L A 22ax - x L L B n sx - ad A 22ax - x B n + sx - ad A 22ax - x B = sn - 2da x22ax - x dx = n + n-2 na 22ax - x B dx A n + 1L + n - dx sn - 2da L A 22ax - x B n - 2-n sx + ads2x - 3ad22ax - x a -1 x - a + sin a a b + C 22ax - x x - a dx = 22ax - x + a sin-1 a a b + C x 126 22ax - x 2a - x x - a dx = -2 - sin-1 a a b + C A x x2 L 127 x dx x - a = a sin-1 a a b - 22ax - x + C L 22ax - x 128 dx 2a - x = -a + C A x L x22ax - x Hyperbolic Forms 129 131 133 L L L sinh ax dx = a cosh ax + C sinh2 ax dx = sinhn ax dx = 130 sinh 2ax x - + C 4a sinh n-1 ax cosh ax n - - n na 132 L sinhn - ax dx, L L cosh ax dx = a sinh ax + C cosh2 ax dx = n Z sinh 2ax x + + C 4a T-5 T-6 134 135 137 139 141 143 144 145 147 149 150 151 153 154 A Brief Table of Integrals L coshn ax dx = coshn - ax sinh ax n - + n na L coshn - ax dx, x x sinh ax dx = a cosh ax - sinh ax + C a L xn n x n sinh ax dx = a cosh ax - a L L ax dx = a ln scosh axd + C L tanh2 ax dx = x - a ax + C L L L n Z 136 x n - cosh ax dx 138 140 142 tanhn ax dx = - tanhn - ax + tanhn - ax dx, sn - 1da L n Z cothn ax dx = - cothn - ax + cothn - ax dx, sn - 1da L n Z L sech ax dx = a sin-1 stanh axd + C 146 L sech2 ax dx = a ax + C 148 L L L L L sechn ax dx = sechn - ax ax n - + sechn - ax dx, n - 1L sn - 1da cschn ax dx = - sechn ax + C, na n Z 152 e ax sinh bx dx = e ax e bx e -bx c d + C, a + b a - b a2 Z b2 e ax cosh bx dx = e ax e bx e -bx c + d + C, a + b a - b a2 Z b2 xn n x n cosh ax dx = a sinh ax - a L L coth ax dx = a ln ƒ sinh ax ƒ + C L coth2 ax dx = x - a coth ax + C L ax csch ax dx = a ln ` ` + C L csch2 ax dx = - a coth ax + C L n Z L cschn ax coth ax dx = - cschn ax + C, na Some Definite Integrals q 155 L0 x n - 1e -x dx = ≠snd = sn - 1d!, p>2 157 p>2 sin x dx = n L0 L0 q n 156 # # # Á # sn - 1d # p , 2#4#6# Á #n n cos x dx = d # # # Á # sn - 1d , 3#5#7# Á #n x n - sinh ax dx n Z cschn - ax coth ax n - cschn - ax dx, n - 1L sn - 1da sechn ax ax dx = - x x cosh ax dx = a sinh ax - cosh ax + C a L L0 e -ax dx = p , 2A a a if n is an even integer Ú2 if n is an odd integer Ú3 n Z BASIC ALGEBRA FORMULAS Arithmetic Operations asb + cd = ab + ac, a#c ac = b d bd c ad + bc a + = , b d bd a>b a d = #c b c>d Laws of Signs -a a a = - = b b -b -s -ad = a, Zero Division by zero is not defined If a Z 0: a = 0, a = 1, 0a = For any number a: a # = # a = Laws of Exponents a ma n = a m + n, sabdm = a mb m, sa m dn = a mn, n a m>n = 2a m = n aBm A2 If a Z 0, am = a m - n, an a = 1, a -m = am The Binomial Theorem For any positive integer n, sa + bdn = a n + na n - 1b + + nsn - 1d n - 2 a b 1#2 nsn - 1dsn - 2d n - 3 a b + Á + nab n - + b n 1#2#3 For instance, sa + bd2 = a + 2ab + b 2, sa - bd2 = a - 2ab + b sa + bd3 = a + 3a 2b + 3ab + b 3, sa - bd3 = a - 3a 2b + 3ab - b Factoring the Difference of Like Integer Powers, n>1 a n - b n = sa - bdsa n - + a n - 2b + a n - 3b + Á + ab n - + b n - d For instance, a - b = sa - bdsa + bd, a - b = sa - bdsa + ab + b d, a - b = sa - bdsa + a 2b + ab + b d Completing the Square If a Z 0, ax + bx + c = au + C au = x + sb>2ad, C = c - The Quadratic Formula If a Z and ax + bx + c = 0, then x = -b ; 2b - 4ac 2a b2 b 4a GEOMETRY FORMULAS A = area, B = area of base, C = circumference, S = lateral area or surface area, V = volume Triangle Similar Triangles c' c h Pythagorean Theorem a' a c b b' b b a a' b' c' a5b5c A bh Parallelogram a2 Trapezoid c2 Circle a h h r b A pr 2, C 2pr b A bh A ( )h Any Cylinder or Prism with Parallel Bases Right Circular Cylinder r h h h V Bh B B V pr2h S 2prh Area of side Any Cone or Pyramid Right Circular Cone h h B Sphere V5 Bh B V pr2h S prs Area of side V 43 pr3, S 4pr2 LIMITS General Laws Specific Formulas If L, M, c, and k are real numbers and If Psxd = an x n + an - x n - + Á + a0 , then lim ƒsxd = L and x:c lim gsxd = M, x:c then lim Psxd = Pscd = an c n + an - c n - + Á + a0 x:c lim sƒsxd + gsxdd = L + M Sum Rule: x:c lim sƒsxd - gsxdd = L - M Difference Rule: x:c If P(x) and Q(x) are polynomials and Qscd Z 0, then lim sƒsxd # gsxdd = L # M Product Rule: Psxd Pscd = Qscd x:c Qsxd lim x:c Constant Multiple Rule: lim sk # ƒsxdd = k # L x:c ƒsxd L = , M x:c gsxd Quotient Rule: lim M Z If ƒ(x) is continuous at x = c, then The Sandwich Theorem lim ƒsxd = ƒscd If gsxd … ƒsxd … hsxd in an open interval containing c, except possibly at x = c, and if x:c lim gsxd = lim hsxd = L, x:c x:c then limx:c ƒsxd = L lim x:0 sin x x = and lim x:0 - cos x = x Inequalities L’Hôpital’s Rule If ƒsxd … gsxd in an open interval containing c, except possibly at x = c, and both limits exist, then If ƒsad = gsad = 0, both ƒ¿ and g¿ exist in an open interval I containing a, and g¿sxd Z on I if x Z a, then lim ƒsxd … lim gsxd x:c x:c lim x:a Continuity ƒsxd ƒ¿sxd = lim , gsxd x:a g¿sxd assuming the limit on the right side exists If g is continuous at L and limx:c ƒsxd = L, then lim g(ƒsxdd = gsLd x:c DIFFERENTIATION RULES General Formulas Inverse Trigonometric Functions Assume u and y are differentiable functions of x d Constant: scd = dx du d dy su + yd = Sum: + dx dx dx d du dy su - yd = Difference: dx dx dx d du scud = c Constant Multiple: dx dx d dy du suyd = u Product: + y dx dx dx d ssin-1 xd = dx 21 - x d scos-1 xd = dx 21 - x d stan-1 xd = dx + x2 d ssec-1 xd = dx - x 2x ƒ ƒ d scot-1 xd = dx + x2 d scsc-1 xd = dx x 2x - ƒ ƒ du dy y - u d u dx dx a b = dx y y2 d n x = nx n - dx d sƒsgsxdd = ƒ¿sgsxdd # g¿sxd dx Quotient: Power: Chain Rule: Hyperbolic Functions d ssinh xd = cosh x dx d stanh xd = sech2 x dx d scoth xd = -csch2 x dx d scosh xd = sinh x dx d ssech xd = -sech x x dx d scsch xd = -csch x coth x dx Inverse Hyperbolic Functions Trigonometric Functions d ssin xd = cos x dx d stan xd = sec2 x dx d scot xd = -csc2 x dx d scos xd = -sin x dx d ssec xd = sec x tan x dx d scsc xd = -csc x cot x dx Exponential and Logarithmic Functions d x e = ex dx d x a = a x ln a dx d ln x = x dx d sloga xd = dx x ln a d ssinh-1 xd = dx 21 + x d scosh-1 xd = dx 2x - d stanh-1 xd = dx - x2 d ssech-1 xd = dx x21 - x d scoth-1 xd = dx - x2 d scsch-1 xd = dx ƒ x ƒ 21 + x Parametric Equations If x = ƒstd and y = gstd are differentiable, then y¿ = dy>dt dy = dx dx>dt and d 2y dx = dy¿>dt dx>dt INTEGRATION RULES General Formulas a Zero: ƒsxd dx = La a Order of Integration: b ƒsxd dx = - Lb ƒsxd dx La b b kƒsxd dx = k ƒsxd dx La La Constant Multiples: b b -ƒsxd dx = - La La b sƒsxd ; gsxdd dx = La b La b ƒsxd dx ; c ƒsxd dx + Additivity: sk = -1d ƒsxd dx b Sums and Differences: sAny number kd La gsxd dx c ƒsxd dx = ƒsxd dx La Lb La Max-Min Inequality: If max ƒ and ƒ are the maximum and minimum values of ƒ on [a, b], then ƒ # sb - ad … ƒsxd dx … max ƒ # sb - ad b La b ƒsxd Ú gsxd Domination: on [a, b] implies La b ƒsxd dx Ú La gsxd dx b ƒsxd Ú on [a, b] implies La ƒsxd dx Ú The Fundamental Theorem of Calculus x Part If ƒ is continuous on [a, b], then Fsxd = 1a ƒstd dt is continuous on [a, b] and differentiable on (a, b) and its derivative is ƒ(x); x F¿(x) = d ƒstd dt = ƒsxd dx La Part If ƒ is continuous at every point of [a, b] and F is any antiderivative of ƒ on [a, b], then b La ƒsxd dx = Fsbd - Fsad Integration by Parts Substitution in Definite Integrals ƒsgsxdd # g¿sxd dx = b La gsbd Lgsad ƒsud du ƒsxdg¿sxd dx = ƒsxdgsxd D a - b La b b La ƒ¿sxdgsxd dx tan A + tan B - tan A tan B tan A - tan B tan sA - Bd = + tan A tan B Trigonometry Formulas tan sA + Bd = y Definitions and Fundamental Identities y Sine: sin u = r = csc u Cosine: x cos u = r = sec u Tangent: y tan u = x = cot u P(x, y) r y x x cos s -ud = cos u sin2 u + cos2 u = 1, sec2 u = + tan2 u, sin 2u = sin u cos u, cos2 u = + cos 2u , p b = -cos A, sin aA + p b = cos A, sin A sin B = Identities sin s -ud = -sin u, sin aA - csc2 u = + cot2 u cos 2u = cos2 u - sin2 u sin2 u = - cos 2u 1 cos sA - Bd + cos sA + Bd 2 sin A cos B = 1 sin sA - Bd + sin sA + Bd 2 sin A + sin B = sin 1 sA + Bd cos sA - Bd 2 sin A - sin B = cos 1 sA + Bd sin sA - Bd 2 1 sA + Bd cos sA - Bd 2 cos A - cos B = -2 sin cos sA - Bd = cos A cos B + sin A sin B 1 sA + Bd sin sA - Bd 2 y y y ϭ sin x y ϭ cos x Trigonometric Functions Radian Measure Degrees ͙2 45 C ir ͙2 90 3 2 x – – 3 2 x Domain: (–ϱ, ϱ) Range: [–1, 1] y y y ϭ tan x y ϭ sec x e it cir cl cle of r Un θ y ϭ sinx Domain: (–ϱ, ϱ) Range: [–1, 1] 45 s – – Radians p b = -sin A cos A cos B = cos A + cos B = cos cos sA + Bd = cos A cos B - sin A sin B cos aA + p b = sin A 1 cos sA - Bd - cos sA + Bd 2 sin sA + Bd = sin A cos B + cos A sin B sin sA - Bd = sin A cos B - cos A sin B cos aA - s diu r 30 ͙3 60 u s s or u = r , r = = u 180° = p radians 90 – 3 – – 2 ͙3 3 2 x Domain: All real numbers except odd integer multiples of /2 Range: (–ϱ, ϱ) 3 2 – 3 – – 2 x Domain: All real numbers except odd integer multiples of /2 Range: (–ϱ, –1] h [1, ϱ) y y y ϭ csc x The angles of two common triangles, in degrees and radians – – y ϭ cot x 3 2 Domain: x 0, Ϯ, Ϯ2, Range: (–ϱ, –1] h [1, ϱ) x – – 3 2 Domain: x 0, Ϯ, Ϯ2, Range: (–ϱ, ϱ) x SERIES Tests for Convergence of Infinite Series The nth-Term Test: Unless an : 0, the series diverges Geometric series: gar n converges if ƒ r ƒ 1; otherwise it diverges p-series: g1>n p converges if p 1; otherwise it diverges Series with nonnegative terms: Try the Integral Test, Ratio Test, or Root Test Try comparing to a known series with the Comparison Test or the Limit Comparison Test Series with some negative terms: Does g ƒ an ƒ converge? If yes, so does gan since absolute convergence implies convergence Alternating series: gan converges if the series satisfies the conditions of the Alternating Series Test Taylor Series q = + x + x + Á + x n + Á = a x n, - x n=0 ƒxƒ q = - x + x - Á + s -xdn + Á = a s -1dnx n, + x n=0 q ex = + x + sin x = x cos x = - xn x2 xn + Á + + Á = a , 2! n! n = n! ƒxƒ ƒxƒ q q s -1dnx 2n + x5 x3 x 2n + + - Á + s -1dn + Á = a , 3! 5! s2n + 1d! n = s2n + 1d! q s -1dnx 2n x4 x2 x 2n + - Á + s -1dn + Á = a , 2! 4! s2nd! s2nd! n=0 ln s1 + xd = x - ƒxƒ q ƒxƒ q q s -1dn - 1x n x2 x3 xn + - Á + s -1dn - n + Á = a , n n=1 -1 x … q ln + x x5 x3 x 2n + x 2n + = tanh-1 x = ax + + + Á b = 2a , + Á + - x 2n + 2n + n=0 tan-1 x = x - q s -1dnx 2n + x5 x3 x 2n + + + Á = a , - Á + s -1dn 2n + 2n + n=0 ƒxƒ ƒxƒ … Binomial Series s1 + xdm = + mx + msm - 1dx msm - 1dsm - 2dx msm - 1dsm - 2d Á sm - k + 1dx k + + Á + + Á 2! 3! k! q m = + a a bx k, k=1 k ƒ x ƒ 1, where m a b = m, msm - 1d m , a b = 2! msm - 1d Á sm - k + 1d m a b = k! k for k Ú VECTOR OPERATOR FORMULAS (CARTESIAN FORM) Formulas for Grad, Div, Curl, and the Laplacian Cartesian (x, y, z) i, j, and k are unit vectors in the directions of increasing x, y, and z M, N, and P are the scalar components of F(x, y, z) in these directions Gradient Divergence §ƒ = 0ƒ 0ƒ 0ƒ i + j + k 0x 0y 0z §#F = The Fundamental Theorem of Line Integrals Let F = Mi + Nj + Pk be a vector field whose components are continuous throughout an open connected region D in space Then there exists a differentiable function ƒ such that 0ƒ 0ƒ 0ƒ F = §ƒ = i + j + k 0x 0y 0z if and only if for all points A and B in D the value of 1A F # dr is independent of the path joining A to B in D If the integral is independent of the path from A to B, its value is B F # dr = ƒsBd - ƒsAd B LA 0N 0P 0M + + 0x 0y 0z Green’s Theorem and Its Generalization to Three Dimensions Curl i j k § * F = 0x 0y 0z M N P ƒ Laplacian §2ƒ = 0x 0ƒ + 0y Normal form of Green’s Theorem: C Divergence Theorem: 0ƒ + 0z F # n ds = § # F dA F F # n ds = § # F dV S Tangential form of Green’s Theorem: u * sv * wd = su # wdv - su # vdw Stokes’ Theorem: D F # dr = § * F # k dA F C Vector Triple Products su * vd # w = sv * wd # u = sw * ud # v R R F # dr = § * F # n ds F C S Vector Identities In the identities here, ƒ and g are differentiable scalar functions, F, F1 , and F2 are differentiable vector fields, and a and b are real constants § * s§ƒd = §sƒgd = ƒ§g + g§ƒ § # sgFd = g§ # F + §g # F § * sgFd = g§ * F + §g * F § # saF1 + bF2 d = a§ # F1 + b§ # F2 § * saF1 + bF2 d = a§ * F1 + b§ * F2 §sF1 # F2 d = sF1 # §dF2 + sF2 # §dF1 + F1 * s§ * F2 d + F2 * s§ * F1 d § # sF1 * F2 d = F2 # § * F1 - F1 # § * F2 § * sF1 * F2 d = sF2 # §dF1 - sF1 # §dF2 + s§ # F2 dF1 - s§ # F1 dF2 § * s§ * Fd = §s§ # Fd - s§ # §dF = §s§ # Fd - §2F s§ * Fd * F = sF # §dF - §sF # Fd ... 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... solutions to all of the oddnumbered exercises in University Calculus, Early Transcendentals Preface xiii JUST-IN-TIME ALGEBRA AND TRIGONOMETRY FOR EARLY TRANSCENDENTALS CALCULUS, Third Edition ISBN... 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