Thomas calculus early transcendentals 12th

Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th Thomas calculus early transcendentals 12th

7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page i THOMAS’ CALCULUS EARLY TRANSCENDENTALS Twelfth 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 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page ii Editor-in-Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Senior Project Editor: Rachel S Reeve Associate Editor: Caroline Celano Associate Project Editor: Leah Goldberg Senior Managing Editor: Karen Wernholm Senior Production Project Manager: Sheila Spinney Senior Design Supervisor: Andrea Nix Digital Assets Manager: Marianne Groth Media Producer: Lin Mahoney Software Development: Mary Durnwald and Bob Carroll Executive Marketing Manager: Jeff Weidenaar Marketing Assistant: Kendra Bassi Senior Author Support/Technology Specialist: Joe Vetere Senior Prepress Supervisor: Caroline Fell Manufacturing Manager: Evelyn Beaton Production Coordinator: Kathy Diamond Composition: Nesbitt Graphics, Inc Illustrations: Karen Heyt, IllustraTech Cover Design: Rokusek Design Cover image: Forest Edge, Hokuto, Hokkaido, Japan 2004 © Michael Kenna About the cover: The cover image of a tree line on a snow-swept landscape, by the photographer Michael Kenna, was taken in Hokkaido, Japan The artist was not thinking of calculus when he composed the image, but rather, of a visual haiku consisting of a few elements that would spark the viewer’s imagination Similarly, the minimal design of this text allows the central ideas of calculus developed in this book to unfold to ignite the learner’s imagination 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 Addison-Wesley was aware of a trademark claim, the designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Weir, Maurice D Thomas’ calculus : early transcendentals / Maurice D Weir, Joel Hass, George B Thomas.—12th ed p cm Includes index ISBN 978-0-321-58876-0 Calculus—Textbooks Geometry, Analytic—Textbooks I Hass, Joel II Thomas, George B (George Brinton), 1914–2006 III Title IV Title: Calculus QA303.2.W45 2009 515—dc22 2009023070 Copyright © 2010, 2006, 2001 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 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm 10—CRK—12 11 10 09 www.pearsoned.com ISBN-10: 0-321-58876-2 ISBN-13: 978-0-321-58876-0 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page iii 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 22 Graphing with Calculators and Computers 30 Exponential Functions 34 Inverse Functions and Logarithms 40 QUESTIONS TO GUIDE YOUR REVIEW 52 PRACTICE EXERCISES 53 ADDITIONAL AND ADVANCED EXERCISES 55 58 Rates of Change and Tangents to Curves 58 Limit of a Function and Limit Laws 65 The Precise Definition of a Limit 76 One-Sided Limits 85 Continuity 92 Limits Involving Infinity; Asymptotes of Graphs QUESTIONS TO GUIDE YOUR REVIEW 116 PRACTICE EXERCISES 117 ADDITIONAL AND ADVANCED EXERCISES 119 Differentiation 3.1 3.2 103 122 Tangents and the Derivative at a Point The Derivative as a Function 126 122 iii 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page iv iv Contents 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 Extreme Values of Functions 222 The Mean Value Theorem 230 Monotonic Functions and the First Derivative Test Concavity and Curve Sketching 243 Indeterminate Forms and L’Hôpital’s Rule 254 Applied Optimization 263 Newton’s Method 274 Antiderivatives 279 QUESTIONS TO GUIDE YOUR REVIEW 289 PRACTICE EXERCISES 289 ADDITIONAL AND ADVANCED EXERCISES 293 222 238 Integration 297 5.1 5.2 5.3 5.4 5.5 5.6 176 Applications of Derivatives 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Differentiation Rules 135 The Derivative as a Rate of Change 145 Derivatives of Trigonometric Functions 155 The Chain Rule 162 Implicit Differentiation 170 Derivatives of Inverse Functions and Logarithms Inverse Trigonometric Functions 186 Related Rates 192 Linearization and Differentials 201 QUESTIONS TO GUIDE YOUR REVIEW 212 PRACTICE EXERCISES 213 ADDITIONAL AND ADVANCED EXERCISES 218 Area and Estimating with Finite Sums 297 Sigma Notation and Limits of Finite Sums 307 The Definite Integral 313 The Fundamental Theorem of Calculus 325 Indefinite Integrals and the Substitution Method 336 Substitution and Area Between Curves 344 QUESTIONS TO GUIDE YOUR REVIEW 354 PRACTICE EXERCISES 354 ADDITIONAL AND ADVANCED EXERCISES 358 Applications of Definite Integrals 6.1 6.2 6.3 Volumes Using Cross-Sections 363 Volumes Using Cylindrical Shells 374 Arc Length 382 363 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page v Contents 6.4 6.5 6.6 The Logarithm Defined as an Integral 417 Exponential Change and Separable Differential Equations Hyperbolic Functions 436 Relative Rates of Growth 444 QUESTIONS TO GUIDE YOUR REVIEW 450 PRACTICE EXERCISES 450 ADDITIONAL AND ADVANCED EXERCISES 451 Integration by Parts 454 Trigonometric Integrals 462 Trigonometric Substitutions 467 Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems 481 Numerical Integration 486 Improper Integrals 496 QUESTIONS TO GUIDE YOUR REVIEW 507 PRACTICE EXERCISES 507 ADDITIONAL AND ADVANCED EXERCISES 509 First-Order Differential Equations 9.1 9.2 9.3 9.4 9.5 10 417 Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Areas of Surfaces of Revolution 388 Work and Fluid Forces 393 Moments and Centers of Mass 402 QUESTIONS TO GUIDE YOUR REVIEW 413 PRACTICE EXERCISES 413 ADDITIONAL AND ADVANCED EXERCISES 415 Integrals and Transcendental Functions 7.1 7.2 7.3 7.4 427 453 471 514 Solutions, Slope Fields, and Euler’s Method 514 First-Order Linear Equations 522 Applications 528 Graphical Solutions of Autonomous Equations 534 Systems of Equations and Phase Planes 541 QUESTIONS TO GUIDE YOUR REVIEW 547 PRACTICE EXERCISES 547 ADDITIONAL AND ADVANCED EXERCISES 548 Infinite Sequences and Series 10.1 10.2 v Sequences 550 Infinite Series 562 550 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page vi vi Contents 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 11 Parametric Equations and Polar Coordinates 11.1 11.2 11.3 11.4 11.5 11.6 11.7 12 678 Three-Dimensional Coordinate Systems 678 Vectors 683 The Dot Product 692 The Cross Product 700 Lines and Planes in Space 706 Cylinders and Quadric Surfaces 714 QUESTIONS TO GUIDE YOUR REVIEW 719 PRACTICE EXERCISES 720 ADDITIONAL AND ADVANCED EXERCISES 722 Vector-Valued Functions and Motion in Space 13.1 13.2 13.3 13.4 13.5 13.6 628 Parametrizations of Plane Curves 628 Calculus with Parametric Curves 636 Polar Coordinates 645 Graphing in Polar Coordinates 649 Areas and Lengths in Polar Coordinates 653 Conic Sections 657 Conics in Polar Coordinates 666 QUESTIONS TO GUIDE YOUR REVIEW 672 PRACTICE EXERCISES 673 ADDITIONAL AND ADVANCED EXERCISES 675 Vectors and the Geometry of Space 12.1 12.2 12.3 12.4 12.5 12.6 13 The Integral Test 571 Comparison Tests 576 The Ratio and Root Tests 581 Alternating Series, Absolute and Conditional Convergence 586 Power Series 593 Taylor and Maclaurin Series 602 Convergence of Taylor Series 607 The Binomial Series and Applications of Taylor Series 614 QUESTIONS TO GUIDE YOUR REVIEW 623 PRACTICE EXERCISES 623 ADDITIONAL AND ADVANCED EXERCISES 625 Curves in Space and Their Tangents 725 Integrals of Vector Functions; Projectile Motion 733 Arc Length in Space 742 Curvature and Normal Vectors of a Curve 746 Tangential and Normal Components of Acceleration 752 Velocity and Acceleration in Polar Coordinates 757 725 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page vii Contents vii QUESTIONS TO GUIDE YOUR REVIEW 760 PRACTICE EXERCISES 761 ADDITIONAL AND ADVANCED EXERCISES 763 14 Partial Derivatives 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 15 Functions of Several Variables 765 Limits and Continuity in Higher Dimensions 773 Partial Derivatives 782 The Chain Rule 793 Directional Derivatives and Gradient Vectors 802 Tangent Planes and Differentials 809 Extreme Values and Saddle Points 820 Lagrange Multipliers 829 Taylor’s Formula for Two Variables 838 Partial Derivatives with Constrained Variables 842 QUESTIONS TO GUIDE YOUR REVIEW 847 PRACTICE EXERCISES 847 ADDITIONAL AND ADVANCED EXERCISES 851 Multiple Integrals 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 16 765 854 Double and Iterated Integrals over Rectangles 854 Double Integrals over General Regions 859 Area by Double Integration 868 Double Integrals in Polar Form 871 Triple Integrals in Rectangular Coordinates 877 Moments and Centers of Mass 886 Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 905 QUESTIONS TO GUIDE YOUR REVIEW 914 PRACTICE EXERCISES 914 ADDITIONAL AND ADVANCED EXERCISES 916 893 Integration in Vector Fields 16.1 16.2 16.3 16.4 16.5 16.6 Line Integrals 919 Vector Fields and Line Integrals: Work, Circulation, and Flux 925 Path Independence, Conservative Fields, and Potential Functions 938 Green’s Theorem in the Plane 949 Surfaces and Area 961 Surface Integrals 971 919 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page viii viii Contents 16.7 16.8 17 Stokes’ Theorem 980 The Divergence Theorem and a Unified Theory QUESTIONS TO GUIDE YOUR REVIEW 1001 PRACTICE EXERCISES 1001 ADDITIONAL AND ADVANCED EXERCISES 1004 990 Second-Order Differential Equations 17.1 17.2 17.3 17.4 17.5 online Second-Order Linear Equations Nonhomogeneous Linear Equations Applications Euler Equations Power Series Solutions Appendices AP-1 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 Real Numbers and the Real Line AP-1 Mathematical Induction AP-6 Lines, Circles, and Parabolas AP-10 Proofs of Limit Theorems AP-18 Commonly Occurring Limits AP-21 Theory of the Real Numbers AP-23 Complex Numbers AP-25 The Distributive Law for Vector Cross Products AP-35 The Mixed Derivative Theorem and the Increment Theorem AP-36 Answers to Odd-Numbered Exercises A-1 Index I-1 Credits C-1 A Brief Table of Integrals T-1 7001_ThomasET_FM_SE_pi-xvi.qxd 11/3/09 3:18 PM Page ix PREFACE We have significantly revised this edition of Thomas’ 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 previous editions, this new edition provides a 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 twelfth edition early transcendentals version, 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 Many of our students were exposed to the terminology and computational aspects of calculus during high school Despite this familiarity, students’ algebra and trigonometry skills often hinder their success in the college calculus sequence With this text, we have sought to balance the students’ prior experience with calculus with the algebraic skill development they may still need, all without undermining or derailing their confidence We have taken care to provide enough review material, fully stepped-out solutions, and exercises to support complete understanding for students of all levels 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 Twelfth Edition CONTENT In preparing this edition we have maintained the basic structure of the Table of Contents from the eleventh 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: ix 7001_ThomasET_Credits_ppC1-C2 11/3/09 2:48 PM Page 7001_ThomasET_BTIpT1-T6.qxd 11/3/09 12:17 PM Page A BRIEF TABLE OF INTEGRALS Basic Forms L k dx = kx + C L dx x = ln ƒ x ƒ + C L a x dx = L cos x dx = sin x + C L 11 L x n dx = L e x dx = e x + C L sin x dx = - cos x + C L sec2 x dx = tan x + C csc2 x dx = - cot x + C 10 L sec x tan x dx = sec x + C L csc x cot x dx = - csc x + C 12 L tan x dx = ln ƒ sec x ƒ + C 13 L cot x dx = ln ƒ sin x ƒ + C 14 L sinh x dx = cosh x + C 15 L cosh x dx = sinh x + C 16 sany number kd ax + C ln a sa 0, a Z 1d dx x = a tan-1 a + C 2 La + x dx x 19 = sinh-1 a + C L 2a + x xn+1 + C n + sn Z - 1d L 2a - x dx 18 = a sec-1 L x2x - a dx x 20 = cosh-1 a L 2x - a 17 sa 0d dx 2 x = sin-1 a + C x ` a ` + C + C sx a 0d Forms Involving ax ؉ b sax + bdn + + C, asn + 1d 21 L sax + bdn dx = 22 L xsax + bdn dx = 23 L sax + bd-1 dx = a ln ƒ ax + b ƒ + C 25 L xsax + bd-2 dx = 27 L A 2ax + b B n n Z -1 sax + bdn + ax + b b d + C, c n + n + a2 b d + C c ln ƒ ax + b ƒ + ax + b a A 2ax + b B dx = a n + n Z - 1, -2 24 26 n+2 + C, n Z -2 28 b x xsax + bd-1 dx = a - ln ƒ ax + b ƒ + C a L dx x ` + C = ln ` b ax + b xsax + bd L L dx L x2ax + b 2ax + b dx = 22ax + b + b x T-1 7001_ThomasET_BTIpT1-T6.qxd 11/3/09 12:17 PM Page T-2 A Brief Table of Integrals 29 (a) 30 2ax + b - 2b dx = ln ` ` + C L x2ax + b 2b 2ax + b + 2b 2ax + b 2ax + b dx a + dx = + C x 2 x L x2ax + b L (b) 31 Forms Involving a2 ؉ x2 dx ax - b = tan-1 + C A b L x2ax - b 2b L x 2ax + b dx = - 2ax + b a dx + C bx 2b L x2ax + b dx x dx x x 1 = a tan-1 a + C 33 = tan-1 a + C + 2 2 2a 2sa + x d 2a La + x L sa + x d dx x 34 = sinh-1 a + C = ln A x + 2a + x B + C L 2a + x x a2 35 2a + x dx = 2a + x + ln A x + 2a + x B + C 2 L x a4 36 x 2a + x dx = sa + 2x d2a + x ln A x + 2a + x B + C 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 + C dx = ln A x + 2a + x B x x L L 2a + x x2 2 dx = - L x 2a + x x2a + x a2 + C ln A x + 2a + x B + 2 a + 2a + x = - a ln ` ` + C x dx 41 L x 2a + x dx = - 2a + x + C a 2x Forms Involving a2 ؊ x2 42 44 46 47 49 51 dx x + a = ln ` x - a ` + C 2a La - x L 2a - x dx L L x = sin-1 a + C x 2a - x dx = 43 45 a x sin-1 a - x 2a - x sa - 2x d + C 8 2a - x a + 2a - x dx = 2a - x - a ln ` ` + C 48 x x L 2a - x x2 dx = L x 2a - x dx a -1 x sin a - x2a - x + C 2 = - 50 2a - x + C a 2x Forms Involving x2 ؊ a2 52 53 L 2x - a dx L = ln ƒ x + 2x - a ƒ + C 2x - a dx = x a2 2x - a ln ƒ x + 2x - a ƒ + C 2 dx x x + a = ln ` x - a ` + C + 2 2 2a sa - x d 4a L sa - x d L 2a - x dx = x a -1 x 2a - x + sin a + C 2 2a - x 2a - x x + C dx = - sin-1 a x x L L x2a - x dx a + 2a - x = - a ln ` ` + C x 7001_ThomasET_BTIpT1-T6.qxd 11/3/09 12:17 PM Page A Brief Table of Integrals 54 55 56 L A 2x - a B dx = L A 2x - a dx L 57 L 58 L x A 2x - a B n 2 n x A 2x - a B = - n + x A 2x - a B 2-n s2 - nda A 2x - a B n B dx = x 2x - a dx = n n+2 n + n-2 na A 2x - a B dx, n + 1L n - dx , sn - 2da L A 2x - a B n - + C, n Z -2 x a4 s2x - a d2x - a ln ƒ x + 2x - a ƒ + C 8 2x - a 2x - a + C dx = ln ƒ x + 2x - a ƒ x x L x2 a2 x dx = ln ƒ x + 2x - a ƒ + 2x - a + C 60 2 2 L 2x - a L x 2x - a dx x a 1 = a sec-1 ` a ` + C = a cos-1 ` x ` + C 62 Trigonometric Forms 63 L sin ax dx = - a cos ax + C 65 L sin2 ax dx = 67 L sinn ax dx = - sin 2ax x + C 4a sinn - ax cos ax n - + n na L L cosn ax dx = cosn - ax sin ax n - + n na L x 2x - a dx L cos ax dx = a sin ax + C 66 L cos2 ax dx = a2 Z b2 a2 Z b2 cos ax cos bx dx = sinsa + bdx sinsa - bdx + + C, 2sa - bd 2sa + bd a2 Z b2 (c) L 70 L 72 cos ax dx = a ln ƒ sin ax ƒ + C L sin ax 74 L sin ax cos ax dx = - a ln ƒ cos ax ƒ + C 75 L sinn ax cosm ax dx = - 76 L sinn ax cosm ax dx = sin ax cos ax dx = - cos 2ax + C 4a sin 2ax x + + C 4a cosn - ax dx sinsa + bdx sinsa - bdx + C, 2sa - bd 2sa + bd L 2x - a + C a 2x 64 sin ax sin bx dx = (b) = sinn - ax dx L cossa - bdx cossa + bdx + C, sin ax cos bx dx = 69 (a) 2sa + bd 2sa - bd L 68 n Z 2x - a x dx = 2x - a - a sec-1 ` a ` + C x 59 61 n Z -1 71 L sinn ax cos ax dx = 73 L cosn ax sin 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 sinn + ax + C, sn + 1da n Z -m m Z -n n Z -1 cosn + ax + C, sn + 1da sreduces sinn axd sreduces cosm axd n Z -1 T-3 7001_ThomasET_BTIpT1-T6.qxd 11/3/09 12:17 PM Page T-4 A Brief Table of Integrals ax dx b - c p -2 = b d + C, tan-1 c tan a 2 b + c sin ax b + c A L a2b - c 77 b2 c2 c + b sin ax + 2c - b cos ax dx -1 = ln ` ` + C, b + c sin ax L b + c sin ax a2c - b 78 ax dx p = - a tan a b + C + sin ax L 79 80 dx b - c ax = tan-1 c tan d + C, 2 b + c cos ax b + c A L a2b - c 81 ax dx p = a tan a + b + C sin ax L b2 c2 c + b cos ax + 2c - b sin ax dx = ln ` ` + C, b + c cos ax L b + c cos ax a2c - b 82 b2 c2 b2 c2 dx ax = a tan + C + cos ax L 84 dx ax = - a cot + C cos ax L 85 L x sin ax dx = 86 L x cos ax dx = 87 L xn n x n sin ax dx = - a cos ax + a 88 L xn n x n cos ax dx = a sin ax - a 89 L tan ax dx = a ln ƒ sec ax ƒ + C 90 L cot ax dx = a ln ƒ sin ax ƒ + C 91 L tan2 ax dx = a tan ax - x + C 92 L cot2 ax dx = - a cot ax - x + C 93 L tann ax dx = 94 L cotn ax dx = - 95 L sec ax dx = a ln ƒ sec ax + tan ax ƒ + C 96 L csc ax dx = - a ln ƒ csc ax + cot ax ƒ + C 97 L sec2 ax dx = a tan ax + C 98 L csc2 ax dx = - a cot ax + C 99 L secn ax dx = 100 L cscn ax dx = - 101 L secn ax tan ax dx = 83 x sin ax - a cos ax + C a2 L x n - cos ax dx tan ax tann - ax dx, asn - 1d L n-1 n Z secn - ax tan ax n - secn - ax dx, + n - 1L asn - 1d n Z L x n - sin ax dx cotn - ax cotn - ax dx, asn - 1d L n Z n Z cscn - ax cot ax n - cscn - ax dx, + n - 1L asn - 1d secn ax na + C, x cos ax + a sin ax + C a2 n Z 102 L cscn ax cot ax dx = - 104 L cos-1 ax dx = x cos-1 ax - a 21 - a 2x + C cscn ax na + C, n Z Inverse Trigonometric Forms 103 L 105 L sin-1 ax dx = x sin-1 ax + a 21 - a 2x + C tan-1 ax dx = x tan-1 ax - 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 7001_ThomasET_BTIpT1-T6.qxd 11/3/09 12:17 PM Page A Brief Table of Integrals Exponential and Logarithmic Forms 109 L e ax dx = a e ax + C 111 L xe ax dx = 113 L x nb ax dx = 114 L e ax sin bx dx = e ax sa sin bx - b cos bxd + C a + b2 115 L e ax cos bx dx = e ax sa cos bx + b sin bxd + C a2 + b2 e ax sax - 1d + C a2 n x b x n - 1b ax dx, a ln b a ln b L n ax L b ax + C, b ax dx = a ln b b 0, b Z 112 L n x ne ax dx = a x ne ax - a L 116 L ln ax dx = x ln ax - x + C 110 x n - 1e ax dx b 0, b Z x n + 1sln axdm m x nsln axdm - dx, n Z - n + n + L L m+1 sln axd dx + C, m Z - = ln ƒ ln ax ƒ + C 118 119 x -1sln axdm dx = m + L x ln ax L x nsln axdm dx = 117 Forms Involving 22ax ؊ x2, a>0 120 121 122 123 124 125 126 127 L 22ax - x dx L L n A 22ax - x B dx = L A 22ax - x L n B x - a a b + C x - a a -1 x - a 22ax - x + sin a a b + C 2 22ax - x dx = dx L = sin-1 a sx - ad A 22ax - x B n + sx - ad A 22ax - x B = n 2-n + sn - 2da x22ax - x dx = + n-2 na A 22ax - x B dx n + 1L n - dx sn - 2da L A 22ax - x B n - sx + ads2x - 3ad 22ax - 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 22ax - x 2a - x x - a dx = - - sin-1 a a b + C A x x2 L L 22ax - x x dx = a sin-1 a x - a a b - 22ax - x + C 128 L x22ax - x 130 L cosh ax dx = a sinh ax + C 132 L cosh2 ax dx = dx 2a - x = -a + C A x Hyperbolic Forms 129 L sinh ax dx = a cosh ax + C 131 L sinh2 ax dx = 133 L sinhn ax dx = sinh 2ax x - + C 4a n-1 sinh ax cosh ax n - - n na L sinhn - ax dx, n Z sinh 2ax x + + C 4a T-5 7001_ThomasET_BTIpT1-T6.qxd 11/3/09 12:17 PM Page T-6 134 A Brief Table of Integrals L coshn ax dx = coshn - ax sinh ax n - + n na L coshn - ax dx, n Z 136 x x cosh ax dx = a sinh ax - cosh ax + C a L 138 L xn n x n cosh ax dx = a sinh ax - a ax dx = a ln scosh axd + C 140 L coth ax dx = a ln ƒ sinh ax ƒ + C tanh2 ax dx = x - a ax + C 142 L coth2 ax dx = x - a coth ax + C ax csch ax dx = a ln ` ` + C 135 x x sinh ax dx = a cosh ax - sinh ax + C a L 137 L xn n x n sinh ax dx = a cosh ax - a 139 L 141 L L x n - cosh ax dx L tanhn ax dx = - ax + tanhn - ax dx, sn - 1da L n Z 144 L cothn ax dx = - cothn - ax + cothn - ax dx, sn - 1da L n Z 145 L sech ax dx = a sin-1 stanh axd + C 146 L 147 L sech2 ax dx = a ax + C 148 L 149 L sechn ax dx = 150 L cschn ax dx = - 151 L sechn ax ax dx = - 153 L e ax sinh bx dx = 154 L e ax cosh bx dx = 143 n-1 ax ax n - + sechn - ax dx, n - 1L sn - 1da n-2 sech n Z e -bx e ax e bx d + C, c a + b a - b e -bx e ax e bx + d + C, c a + b a - b csch2 ax dx = - a coth ax + C 152 n Z L cschn ax coth ax dx = - cschn ax + C, na a2 Z b2 a2 Z b2 Some Definite Integrals L0 q 155 157 L0 p>2 x n - 1e -x dx = ≠snd = sn - 1d!, n sin x dx = L0 p>2 n 156 # # # Á # sn - 1d # p , 2#4#6# Á #n n cos x dx = d # # # Á # sn - 1d # # # Á #n , x n - sinh ax dx n Z cschn - ax coth ax n - cschn - ax dx, n - 1L sn - 1da sechn ax + C, na L L0 q e -ax dx = p , 2A a a if n is an even integer Ú if n is an odd integer Ú3 n Z 7001_ThomasET_Last3Bkpgs 11/3/09 12:22 PM Page 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 sin u = r = Sine: csc u Cosine: x cos u = r = sec u Tangent: y tan u = x = cot u P(x, y) r y ␪ x x sin aA + 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, 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 sin sA + Bd = sin A cos B + cos A sin B cos sA + Bd = cos A cos B - sin A sin B 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 1 sA + Bd sin sA - Bd 2 y y y ϭ sin x y ϭ cos x Trigonometric Functions Radian Measure Degrees ␲ 45 s ͙2 C ir ͙2 ␲ 90 r ␲ ␲ 3␲ 2␲ x y ϭ sinx Domain: (–ϱ, ϱ) Range: [–1, 1] –␲ – ␲ ␲ ␲ 3␲ 2␲ x Domain: (–ϱ, ϱ) Range: [–1, 1] ␲ y y y ϭ tan x it cir cl y ϭ sec x e Un θ 45 –␲ – ␲ Radians p b = - sin A cos aA + cos A cos B = cos A - cos B = - sin cos sA - Bd = cos A cos B + sin A sin B p b = sin A 1 cos sA - Bd - cos sA + Bd 2 cos A + cos B = cos sin sA - Bd = sin A cos B - cos A sin B cos aA - s cle of 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) y ␲ ␲ 3␲ 2␲ Domain: x 0, Ϯ␲, Ϯ2␲, Range: (–ϱ, –1] h [1, ϱ) ␲ ␲ 3␲ 2 y x x Domain: All real numbers except odd integer multiples of ␲/2 Range: (–ϱ, –1] h [1, ϱ) y ϭ csc x The angles of two common triangles, in degrees and radians –␲ – ␲ – 3␲ –␲ – ␲ 2 y ϭ cot x –␲ – ␲ ␲ ␲ 3␲ 2␲ Domain: x 0, Ϯ␲, Ϯ2␲, Range: (–ϱ, ϱ) x 7001_ThomasET_Last3Bkpgs 11/3/09 12:22 PM Page SERIES Tests for Convergence of Infinite Series The nth-Term Test: Unless an : 0, the series diverges Geometric series: g ar 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 = + x + x + Á + x n + Á = a x n, - x n=0 q ƒxƒ 1 = - 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! q 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 … x 2n + x5 + x x3 x 2n + + Á + = tanh-1 x = ax + + + Á b = 2a , - x 2n + n = 2n + q ln tan-1 x = x - q s - 1dnx 2n + x5 x3 x 2n + - Á + s -1dn + + Á = a , 2n + 2n + n=0 ƒxƒ ƒxƒ … Binomial Series s1 + xdm = + mx + where msm - 1dsm - 2dx msm - 1dx msm - 1dsm - 2d Á sm - k + 1dx k + Á + + Á + 2! 3! k! q m = + a a bx k, k=1 k m a b = m, ƒ x ƒ 1, msm - 1d m a b = , 2! msm - 1d Á sm - k + 1d m a b = k! k for k Ú 7001_ThomasET_Last3Bkpgs 11/3/09 12:23 PM Page 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 0ƒ 0ƒ 0ƒ §ƒ = i + j + k 0x 0y 0z Divergence §#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 i + j + k F = §ƒ = 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 LA 0N 0P 0M + + 0x 0y 0z B F # dr = ƒsBd - ƒsAd Green’s Theorem and Its Generalization to Three Dimensions Curl Laplacian i j k § * F = 0x 0y 0z M N P §2ƒ = 2ƒ 0x + 2ƒ 0y + 0z su * = sv * u * sv * wd = Divergence Theorem: = sw * - su # vdw ud # v R F # n ds = S Tangential form of Green’s Theorem: Stokes’ Theorem: § # F dV D F # dr = § * F # k dA F C wd # u su # wdv F # n ds = § # F dA F C 2ƒ Vector Triple Products vd # w Normal form of Green’s Theorem: 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 7001_ThomasET_EPp01-07 11/3/09 12:18 PM Page BASIC ALGEBRA FORMULAS Arithmetic Operations asb + cd = ab + ac, ac a#c = b d bd a c ad + bc + = , 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, If a Z 0, am = a m - n, an a = 1, a -m = n a m>n = 2a m = am n aBm A2 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 The Quadratic Formula au = x + sb>2ad, C = c - If a Z and ax + bx + c = 0, then x = -b ; 2b - 4ac 2a b2 b 4a 7001_ThomasET_EPp01-07 11/3/09 12:18 PM Page GEOMETRY FORMULAS A = area, B = area of base, C = circumference, S = lateral area or surface area, V = volume Triangle Similar Triangles c' c h a' Pythagorean Theorem a c b b' b a a' b' c' aϭbϭc a2 ϩ b2 ϭ c2 b A ϭ bh Parallelogram Trapezoid Circle a h h b A ϭ ␲r 2, C ϭ 2␲r r b A ϭ bh A ϭ (a ϩ b)h Any Cylinder or Prism with Parallel Bases Right Circular Cylinder r h h h V ϭ Bh B B V ϭ ␲r2h S ϭ 2␲rh ϭ Area of side Any Cone or Pyramid Right Circular Cone h h Sphere h s r V ϭ Bh B B V ϭ ␲r2h S ϭ ␲rs ϭ Area of side V ϭ 43 ␲r3, S ϭ 4␲r2 7001_ThomasET_EPp01-07 11/3/09 12:18 PM Page 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 lim Quotient Rule: x:c ƒsxd L = , M gsxd M Z If ƒ(x) is continuous at x = c , then The Sandwich Theorem lim ƒsxd = ƒscd x: c If gsxd … ƒsxd … hsxd in an open interval containing c, except possibly at x = c, and if lim gsxd = lim hsxd = L, x: c x:c lim x:0 then limx:c ƒsxd = L sin x x = and lim x: - 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 Continuity lim x:a 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 7001_ThomasET_EPp01-07 11/3/09 12:18 PM Page DIFFERENTIATION RULES General Formulas Inverse Trigonometric Functions Assume u and y are differentiable functions of x d Constant: scd = dx dy d du Sum: + su + yd = dx dx dx dy d du Difference: su - yd = dx dx dx d du Constant Multiple: scud = c dx dx d dy du Product: + y suyd = u dx dx dx d ssin-1 xd = dx 21 - x d scos-1 xd = dx 21 - x d scot-1 xd = dx + x2 d scsc-1 xd = dx ƒ x ƒ 2x - 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: 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 d stan-1 xd = dx + x2 Hyperbolic Functions d ssinh xd = cosh x dx d stanh xd = sech2 x dx d scoth xd = - csch2 x dx d ssec-1 xd = dx ƒ x ƒ 2x - d scosh xd = sinh x dx d ssech xd = - sech x x dx d scsch xd = - csch x coth x dx Inverse Hyperbolic Functions d ssinh-1 xd = dx 21 + x d scosh-1 xd = dx 2x - d scoth-1 xd = dx - x2 d scsch-1 xd = dx ƒ x ƒ 21 + x d stanh-1 xd = dx - x2 d ssech-1 xd = dx x21 - x Exponential and Logarithmic Functions Parametric Equations d x e = ex dx d x a = a x ln a dx If x = ƒstd and y = gstd are differentiable, then d ln x = x dx d sloga xd = dx x ln a y¿ = dy>dt dy = dx dx>dt and d 2y dx = dy¿>dt dx>dt 7001_ThomasET_EPp01-07 11/3/09 12:18 PM Page INTEGRATION RULES General Formulas Zero: Order of Integration: Constant Multiples: Sums and Differences: La a Lb a La b La b La b ƒsxd dx = ƒsxd dx = - La b La b ƒsxd dx kƒsxd dx = k -ƒsxd dx = - La b sk = - 1d ƒsxd dx sƒsxd ; gsxdd dx = b sAny number kd ƒsxd dx La b ƒsxd dx ; c La b gsxd dx c 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 ƒsxd dx + Additivity: ƒsxd dx = ƒ # sb - ad … ƒsxd Ú gsxd Domination: ƒsxd Ú on on [a, b] [a, b] implies implies La La La b ƒsxd dx … max ƒ # sb - ad b ƒsxd dx Ú La b gsxd dx b ƒsxd dx Ú The Fundamental Theorem of Calculus 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 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 La b ƒsxd dx = Fsbd - Fsad Substitution in Definite Integrals La b ƒsgsxdd # g¿sxd dx = Integration by Parts gsbd Lgsad ƒsud du La b b ƒsxd g¿sxd dx = ƒsxd gsxd D a - La b ƒ¿sxdgsxd dx ... instructors through the Thomas Calculus: Early Transcendentals Web site, www.pearsonhighered.com /thomas, and MyMathLab WEB SITE www.pearsonhighered.com /thomas The Thomas Calculus: Early Transcendentals. .. 1–6 University Calculus (Early Transcendentals) University Calculus: Alternate Edition (Late Transcendentals) University Calculus: Elements with Early Transcendentals The University Calculus texts... of the early chapters alongside the algebraic functions The Multivariable book for Thomas Calculus: Early Transcendentals is the same text as Thomas Calculus, Multivariable THOMAS CALCULUS,