Thomas’ Thirteenth Edition CalCulus THoMaS’ CalCUlUS 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 Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_THOM8960_FM_ppi-xiv.indd 27/12/13 10:50 AM Editor-in-Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Senior Content Editor: Rachel S Reeve Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Sheila Spinney; Sherry Berg Associate Design Director, USHE EMSS, TED and HSC: Andrea Nix Art Director and Cover Design: Beth Paquin Digital Assets Manager: Marianne Groth Associate Producer Multimedia: Nicholas Sweeny Software Development: John Flanagan and Kristina Evans Executive Marketing Manager: Jeff Weidenaar Marketing Assistant: Caitlin Crain Senior Author Support/Technology Specialist: Joe Vetere Manufacturing Manager: Carol Melville Text Design, Production Coordination, Composition: Cenveo® Publisher Services Illustrations: Karen Hartpence, IlustraTech; Cenveo® Publisher Services Cover image: Art on File/Corbis 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 Weir, Maurice D Thomas’ calculus / 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 — Thirteenth edition pages cm Updated edition of: Thomas’ calculus : early transcendentals / as revised by Maurice D Weir, Joel Hass c2010 ISBN 0-321-87896-5 (hardcover) 1 Calculus—Textbooks. 2 Geometry, Analytic—Textbooks. I Hass, Joel. II Weir, Maurice D QA303.2.W45 2013 515–dc23 2013023097 Copyright © 2014, 2010, 2008 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—18 17 16 15 14 www.pearsonhighered.com A01_THOM8960_FM_ppi-xiv.indd ISBN-10: 0-321-87896-5 ISBN-13: 978-0-321-87896-0 10/01/14 9:57 AM Contents Preface ix Functions 1.1 1.2 1.3 1.4 Functions and Their Graphs Combining Functions; Shifting and Scaling Graphs Trigonometric Functions 21 Graphing with Software 29 Questions to Guide Your Review 36 Practice Exercises 36 Additional and Advanced Exercises 38 Limits and Continuity 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 41 Rates of Change and Tangents to Curves 41 Limit of a Function and Limit Laws 48 The Precise Definition of a Limit 59 One-Sided Limits 68 Continuity 75 Limits Involving Infinity; Asymptotes of Graphs Questions to Guide Your Review 99 Practice Exercises 100 Additional and Advanced Exercises 102 Derivatives 86 105 Tangents and the Derivative at a Point 105 The Derivative as a Function 110 Differentiation Rules 118 The Derivative as a Rate of Change 127 Derivatives of Trigonometric Functions 137 The Chain Rule 144 Implicit Differentiation 151 Related Rates 156 Linearization and Differentials 165 Questions to Guide Your Review 177 Practice Exercises 177 Additional and Advanced Exercises 182 Applications of Derivatives 4.1 4.2 4.3 4.4 14 185 Extreme Values of Functions 185 The Mean Value Theorem 193 Monotonic Functions and the First Derivative Test Concavity and Curve Sketching 204 199 iii A01_THOM8960_FM_ppi-xiv.indd 27/12/13 10:50 AM iv Contents 4.5 4.6 4.7 Applied Optimization 215 Newton’s Method 227 Antiderivatives 232 Questions to Guide Your Review 242 Practice Exercises 243 Additional and Advanced Exercises 245 Integrals 5.1 5.2 5.3 5.4 5.5 5.6 249 Area and Estimating with Finite Sums 249 Sigma Notation and Limits of Finite Sums 259 The Definite Integral 266 The Fundamental Theorem of Calculus 278 Indefinite Integrals and the Substitution Method 289 Definite Integral Substitutions and the Area Between Curves Questions to Guide Your Review 306 Practice Exercises 306 Additional and Advanced Exercises 309 Applications of Definite Integrals 6.1 6.2 6.3 6.4 6.5 6.6 A01_THOM8960_FM_ppi-xiv.indd 366 Inverse Functions and Their Derivatives 366 Natural Logarithms 374 Exponential Functions 382 Exponential Change and Separable Differential Equations Indeterminate Forms and L’Hôpital’s Rule 403 Inverse Trigonometric Functions 411 Hyperbolic Functions 424 Relative Rates of Growth 433 Questions to Guide Your Review 438 Practice Exercises 439 Additional and Advanced Exercises 442 Techniques of Integration 8.1 8.2 313 Volumes Using Cross-Sections 313 Volumes Using Cylindrical Shells 324 Arc Length 331 Areas of Surfaces of Revolution 337 Work and Fluid Forces 342 Moments and Centers of Mass 351 Questions to Guide Your Review 362 Practice Exercises 362 Additional and Advanced Exercises 364 Transcendental Functions 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 296 Using Basic Integration Formulas Integration by Parts 449 393 444 444 07/01/14 10:34 AM Contents 8.3 8.4 8.5 8.6 8.7 8.8 8.9 First-Order Differential Equations 9.1 9.2 9.3 9.4 9.5 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 A01_THOM8960_FM_ppi-xiv.indd Trigonometric Integrals 457 Trigonometric Substitutions 463 Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems 477 Numerical Integration 482 Improper Integrals 492 Probability 503 Questions to Guide Your Review 516 Practice Exercises 517 Additional and Advanced Exercises 519 v 468 524 Solutions, Slope Fields, and Euler’s Method 524 First-Order Linear Equations 532 Applications 538 Graphical Solutions of Autonomous Equations 544 Systems of Equations and Phase Planes 551 Questions to Guide Your Review 557 Practice Exercises 557 Additional and Advanced Exercises 558 Infinite Sequences and Series 560 Sequences 560 Infinite Series 572 The Integral Test 581 Comparison Tests 588 Absolute Convergence; The Ratio and Root Tests 592 Alternating Series and Conditional Convergence 598 Power Series 604 Taylor and Maclaurin Series 614 Convergence of Taylor Series 619 The Binomial Series and Applications of Taylor Series 626 Questions to Guide Your Review 635 Practice Exercises 636 Additional and Advanced Exercises 638 Parametric Equations and Polar Coordinates 641 Parametrizations of Plane Curves 641 Calculus with Parametric Curves 649 Polar Coordinates 659 Graphing Polar Coordinate Equations 663 Areas and Lengths in Polar Coordinates 667 Conic Sections 671 Conics in Polar Coordinates 680 Questions to Guide Your Review 687 Practice Exercises 687 Additional and Advanced Exercises 689 27/12/13 10:50 AM vi Contents 12 12.1 12.2 12.3 12.4 12.5 12.6 13 13.1 13.2 13.3 13.4 13.5 13.6 14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 A01_THOM8960_FM_ppi-xiv.indd Vectors and the Geometry of Space 692 Three-Dimensional Coordinate Systems 692 Vectors 697 The Dot Product 706 The Cross Product 714 Lines and Planes in Space 720 Cylinders and Quadric Surfaces 728 Questions to Guide Your Review 733 Practice Exercises 734 Additional and Advanced Exercises 736 Vector-Valued Functions and Motion in Space 739 Curves in Space and Their Tangents 739 Integrals of Vector Functions; Projectile Motion 747 Arc Length in Space 756 Curvature and Normal Vectors of a Curve 760 Tangential and Normal Components of Acceleration 766 Velocity and Acceleration in Polar Coordinates 772 Questions to Guide Your Review 776 Practice Exercises 776 Additional and Advanced Exercises 778 Partial Derivatives 781 Functions of Several Variables 781 Limits and Continuity in Higher Dimensions 789 Partial Derivatives 798 The Chain Rule 809 Directional Derivatives and Gradient Vectors 818 Tangent Planes and Differentials 827 Extreme Values and Saddle Points 836 Lagrange Multipliers 845 Taylor’s Formula for Two Variables 854 Partial Derivatives with Constrained Variables 858 Questions to Guide Your Review 863 Practice Exercises 864 Additional and Advanced Exercises 867 Multiple Integrals 870 Double and Iterated Integrals over Rectangles 870 Double Integrals over General Regions 875 Area by Double Integration 884 Double Integrals in Polar Form 888 Triple Integrals in Rectangular Coordinates 894 Moments and Centers of Mass 903 Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 922 910 27/12/13 10:50 AM Contents vii Questions to Guide Your Review 932 Practice Exercises 932 Additional and Advanced Exercises 935 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 17 17.1 17.2 17.3 17.4 17.5 Integrals and Vector Fields Line Integrals 938 Vector Fields and Line Integrals: Work, Circulation, and Flux 945 Path Independence, Conservative Fields, and Potential Functions 957 Green’s Theorem in the Plane 968 Surfaces and Area 980 Surface Integrals 991 Stokes’ Theorem 1002 The Divergence Theorem and a Unified Theory 1015 Questions to Guide Your Review 1027 Practice Exercises 1028 Additional and Advanced Exercises 1030 Second-Order Differential Equations AP-1 Real Numbers and the Real Line AP-1 Mathematical Induction AP-6 Lines, Circles, and Parabolas AP-10 Proofs of Limit Theorems AP-19 Commonly Occurring Limits AP-22 Theory of the Real Numbers AP-23 Complex Numbers AP-26 The Distributive Law for Vector Cross Products AP-35 The Mixed Derivative Theorem and the Increment Theorem AP-36 Answers to Odd-Numbered Exercises Credits Index A-1 C-1 I-1 A Brief Table of Integrals A01_THOM8960_FM_ppi-xiv.indd online Second-Order Linear Equations Nonhomogeneous Linear Equations Applications Euler Equations Power Series Solutions Appendices A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 938 T-1 10/01/14 9:57 AM This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM Preface Thomas’ Calculus, Thirteenth Edition, provides a modern introduction to calculus that focuses on conceptual understanding in developing the essential elements of a traditional course This material supports a three-semester or four-quarter calculus sequence typically taken by students in mathematics, engineering, and the natural sciences Precise explanations, thoughtfully chosen examples, superior figures, and time-tested exercise sets are the foundation of this text We continue to improve this text in keeping with shifts in both the preparation and the ambitions of today’s students, and the applications of calculus to a changing world Many of today’s students have been exposed to the terminology and computational methods of calculus in high school Despite this familiarity, their acquired algebra and trigonometry skills sometimes limit their ability to master calculus at the college level In this text, we seek to balance students’ prior experience in calculus with the algebraic skill development they may still need, without slowing their progress through calculus itself We have taken care to provide enough review material (in the text and appendices), detailed solutions, and variety of examples and exercises, to support a complete understanding of calculus for students at varying levels We present the material in a way to encourage student thinking, going beyond memorizing formulas and routine procedures, and we show students how to generalize key concepts once they are introduced References are made throughout which tie a new concept to a related one that was studied earlier, or to a generalization they will see later on After studying calculus from Thomas, students will have developed problem solving and reasoning abilities that will serve them well in many important aspects of their lives Mastering this beautiful and creative subject, with its many practical applications across so many fields of endeavor, is its own reward But the real gift of studying calculus is acquiring the ability to think logically and factually, and learning how to generalize conceptually We intend this book to encourage and support those goals New to this Edition In this new edition we further blend conceptual thinking with the overall logic and structure of single and multivariable calculus We continue to improve clarity and precision, taking into account helpful suggestions from readers and users of our previous texts While keeping a careful eye on length, we have created additional examples throughout the text Numerous new exercises have been added at all levels of difficulty, but the focus in this revision has been on the mid-level exercises A number of figures have been reworked and new ones added to improve visualization We have written a new section on probability, which provides an important application of integration to the life sciences We have maintained the basic structure of the Table of Contents, and retained improvements from the twelfth edition In keeping with this process, we have added more improvements throughout, which we detail here: ix A01_THOM8960_FM_ppi-xiv.indd 27/12/13 10:50 AM T-2 A Brief Table of Integrals 29 (a) 30 L 2ax + b - 2b dx = ln ` ` + C 2b 2ax + b + 2b L x 2ax + b 2ax + b x2 dx = - 2ax + b x + Forms Involving a + x 32 34 35 31 33 L 37 L 38 L 2a2 + x2 dx = 44 46 47 49 51 2 L (a + x ) dx = x x + tan-1 a + C 2a 2a2 ( a2 + x2 ) x a2 2a2 + x2 + ln x + 2a2 + x2 + C 2 x2 2a2 + x2 dx = 2a2 + x2 x 2a2 + x2 x x a4 ( ln x + 2a2 + x2 + C a + 2x2 ) 2a2 + x2 8 dx = 2a2 + x2 - a ln ` a + 2a2 + x2 ` + C x dx = ln x + 2a2 + x2 - 2a2 + x2 x + C x 2a2 + x2 x2 a2 dx = - ln x + 2a2 + x2 + + C 2 L 2a + x a + 2a2 + x2 dx = ln ` ` + C a x L x 2a + x 41 Forms Involving a − x 42 2ax + b dx a dx = + C bx 2bL x 2ax + b L x2 2ax + b dx x = sinh-1 a + C = ln x + 2a2 + x2 + C L 2a2 + x2 L 40 a dx + C 2L x 2ax + b dx x = a tan-1 a + C 2 a + x L 36 39 dx ax - b = tan-1 + C A b 2b L x 2ax - b (b) dx x + a = ln x - a + C 2 2a La - x dx x = sin-1 a + C 2 L 2a - x L L x2 2a2 - x2 dx = 2a2 - x2 x 2a + x dx = + C a2x L x2 2a2 + x2 43 dx x x + a = 2 + ln x - a + C 2 4a 2a ( a - x2 ) L (a - x ) 45 L 2a2 - x2 dx = x a2 -1 x 2a - x + sin a + C 2 a x sin-1 a - x 2a2 - x2 ( a2 - 2x2 ) + C 8 dx = 2a2 - x2 - a ln ` a + 2a2 - x2 ` + C 48 x x2 a2 -1 x dx = sin a - x 2a2 - x2 + C 2 L 2a - x 2a2 - x2 dx = + C a2x L x2 2a2 - x2 50 L 2a2 - x2 x 2a2 - x2 x dx = -sin-1 a + C x a + 2a2 - x2 dx = - a ln ` ` + C x 2 L x 2a - x Forms Involving x − a 52 53 dx = ln x + 2x2 - a2 + C L 2x2 - a2 L 2x2 - a2 dx = Z05_THOM4077_BTIpT1-T6.indd x a2 2x2 - a2 ln x + 2x2 - a2 + C 2 8/29/13 11:35 AM A Brief Table of Integrals x1 2x2 - a2 na2 dx = 2x2 - a2 2n - dx, n ≠ -1 n + n + 1L n L 2x 56 L 57 L x1 2x2 - a2 2n dx = 58 L 59 L 54 T-3 - a 2 n x1 2x2 - a2 22 - n dx dx n - 55 = n-2 , n ≠ 2 2 n (2 n)a (n 2)a L 2x - a L 2x - a2 60 61 x2 2x2 - a2 dx = 2x2 - a2 x 2x2 - a2 x 2x2 - a2 2n + + C, n ≠ -2 n + x ( 2x2 - a2 ) 2x2 - a2 - a8 ln x + 2x2 - a2 + C x dx = 2x2 - a2 - a sec-1 ` a ` + C dx = ln x + 2x2 - a2 - 2x2 - a2 x + C x2 a2 x dx = ln x + 2x2 - a2 + 2x2 - a2 + C 2 L 2x - a dx x a 1 = a sec-1 ` a ` + C = a cos-1 ` x ` + C 2 L x 2x - a 62 Trigonometric Forms 63 L sin ax dx = - a cos ax + C 65 L sin2 ax dx = 67 L sinn ax dx = - L cosn ax dx = x sin 2ax + C 4a sin n-1 ax cos ax n - + n na L 2x2 - a2 dx + C = a2x L x2 2x2 - a2 64 L cos ax dx = a sin ax + C 66 L cos2 ax dx = x sin 2ax + + C 4a sinn - ax dx L cos(a + b)x cos(a - b)x + C, a2 ≠ b2 69 (a) sin ax cos bx dx = 2(a + b) 2(a - b) L 68 cosn - ax sin ax n - + n na cosn - ax dx sin ax sin bx dx = sin(a - b)x sin(a + b)x + C, a2 ≠ b2 2(a - b) 2(a + b) cos ax cos bx dx = sin(a - b)x sin(a + b)x + + C, a2 ≠ b2 2(a - b) 2(a + b) (b) L (c) L 70 L 72 cos ax dx = a ln ͉ sin ax ͉ + C L sin ax 74 L sin ax cos ax dx = - a ln ͉ cos ax ͉ + C 75 L sinn ax cosm ax dx = - 76 L sinn ax cosm ax dx = sin ax cos ax dx = - Z05_THOM4077_BTIpT1-T6.indd cos 2ax + C 4a 71 L sinn ax cos ax dx = 73 L cosn ax sin ax dx = - sinn + ax + C, n ≠ -1 (n + 1)a cosn + ax + C, n ≠ -1 (n + 1)a sinn - ax cosm + ax n - + sinn - ax cosm ax dx, n ≠ -m (reduces sinn ax) m + nL a(m + n) sinn + ax cosm - ax m - + sinn ax cosm - ax dx, m ≠ -n (reduces cosm ax) m + nL a(m + n) 8/29/13 11:35 AM T-4 77 78 79 81 82 A Brief Table of Integrals dx b - c p ax -2 = tan-1 c tana - b d + C, b2 c2 Ab + c L b + c sin ax a 2b2 - c2 c + b sin ax + 2c2 - b2 cos ax dx -1 = ln ` ` + C, 2 b + c sin ax b + c sin ax a 2c - b L dx p ax = - a tan a - b + C + sin ax L 80 b2 c2 dx p ax = a tan a + b + C sin ax L dx b - c ax = tan-1 c tan d + C, b2 c2 2 b + c cos ax b + c A a 2b - c L c + b cos ax + 2c2 - b2 sin ax dx = ln ` ` + C, 2 b + c cos ax L b + c cos ax a 2c - b b2 c2 dx ax = a tan + C L + cos ax 84 dx ax = - a cot + C L - cos ax 85 L 86 L x cos ax dx = 87 L x n xn sin ax dx = - a cos ax + a 88 L xn n xn cos ax dx = a sin ax - a 89 L 90 L 91 L tan ax dx = a ln sec ax + C 92 L cot ax dx = a ln sin ax + C 93 L 94 L 95 L 96 L 97 L 98 L 99 L secn ax dx = 100 L cscn ax dx = - 101 L secn ax tan ax dx = 83 x sin ax dx = x sin ax - a cos ax + C a2 n L xn - cos ax dx tan2 ax dx = a tan ax - x + C tann ax dx = tann - ax tann - ax dx, n ≠ a(n - 1) L sec ax dx = a ln sec ax + tan ax + C sec2 ax dx = a tan ax + C x cos ax + a sin ax + C a2 L xn - sin ax dx cot2 ax dx = - a cot ax - x + C cotn ax dx = - cotn - ax cotn - ax dx, n ≠ a(n - 1) L csc ax dx = - a ln csc ax + cot ax + C csc2 ax dx = - a cot ax + C secn - ax tan ax n - secn - ax dx, n ≠ + a(n - 1) n - 1L cscn - ax cot ax n - cscn - ax dx, n ≠ + a(n - 1) n - 1L secn ax na + C, n ≠ 102 L cscn ax cot ax dx = - 104 L cos-1 ax dx = x cos-1 ax - a 21 - a2x2 + C cscn ax na + C, n ≠ Inverse Trigonometric Forms 103 L 105 L 106 L xn sin-1 ax dx = 107 L xn cos-1 ax dx = 108 L xn tan-1 ax dx = sin-1 ax dx = x sin-1 ax + a 21 - a2x2 + C tan-1 ax dx = x tan-1 ax - Z05_THOM4077_BTIpT1-T6.indd ln ( + a2x2 ) + C 2a xn + a xn + dx , n ≠ -1 sin-1 ax n + n + L 21 - a2x2 xn + a xn + dx , n ≠ -1 cos-1 ax + n + n + L 21 - a2x2 xn + a xn + dx , n ≠ -1 tan-1 ax n + n + L + a2x2 8/29/13 11:35 AM A Brief Table of Integrals T-5 Exponential and Logarithmic Forms 109 L eax dx = a eax + C 111 L xeax dx = 113 L xnbax dx = 114 L eax sin bx dx = eax (a sin bx - b cos bx) + C a + b2 115 L eax cos bx dx = eax (a cos bx + b sin bx) + C a + b2 eax (ax - 1) + C a2 110 L bax dx = 112 L n xneax dx = a xneax - a L ln ax dx = x ln ax - x + C bax + C, b 0, b ≠ a ln b L xn - 1eax dx xnbax n xn - 1bax dx, b 0, b ≠ a ln b a ln b L 2 116 xn + 1(ln ax)m m xn(ln ax)m - dx, n ≠ -1 n + n + 1L L (ln ax)m + dx 118 x-1(ln ax)m dx = 119 = ln ln ax + C + C, m ≠ -1 m + x ln ax L L xn(ln ax)m dx = 117 Forms Involving 22ax − x 2, a + 120 121 122 dx x - a = sin-1 a a b + C L 22ax - x2 L L 22ax - x2 dx = 22ax x - a a2 -1 x - a 22ax - x2 + sin a a b + C 2 - x2 dx = n (x - a)1 22ax - x2 na2 + 22ax - x2 2n - dx n + n + 1L n (x - a)1 22ax - x2 22 - n dx n - dx = + n (n - 2)a2 (n - 2)a2L 22ax - x2 2n - L 22ax - x2 (x + a)(2x - 3a) 22ax - x2 a3 -1 x - a 124 x 22ax - x2 dx = sin a a b + C + L 123 125 L 22ax - x2 126 L 22ax - x2 127 x x2 dx = 22ax - x2 + a sin-1 a dx = -2 A x - a a b + C 2a - x x - a - sin-1 a a b + C x x dx x - a = a sin-1 a a b - 22ax - x2 + C L 22ax - x2 128 Hyperbolic Forms 129 L sinh ax dx = a cosh ax + C 131 L sinh2 ax dx = sinh 2ax x - + C 4a 133 L sinhn ax dx = sinhn - ax cosh ax n - - n na Z05_THOM4077_BTIpT1-T6.indd L dx 2a - x = -a + C A x L x 22ax - x2 130 L cosh ax dx = a sinh ax + C 132 L cosh2 ax dx = sinh 2ax x + + C 4a sinhn - ax dx, n ≠ 8/29/13 11:35 AM T-6 A Brief Table of Integrals 134 L 135 x x sinh ax dx = a cosh ax - sinh ax + C a L 136 x x cosh ax dx = a sinh ax - cosh ax + C a L 137 L xn n xn sinh ax dx = a cosh ax - a 138 L xn n xn cosh ax dx = a sinh ax - a 139 L ax dx = a ln (cosh ax) + C 140 L coth ax dx = a ln ͉ sinh ax ͉ + C 141 L tanh2 ax dx = x - a ax + C 142 L coth2 ax dx = x - a coth ax + C 143 L tanhn ax dx = - tanhn - ax + tanhn - ax dx, n ≠ (n - 1)a L 144 L cothn ax dx = - cothn - ax + cothn - ax dx, n ≠ (n - 1)a L 145 L sech ax dx = a sin-1 (tanh ax) + C 146 L ax csch ax dx = a ln + C 147 L sech2 ax dx = a ax + C 148 L csch2 ax dx = - a coth ax + C 149 L sechn ax dx = 150 L cschn ax dx = - 151 L sechn ax ax dx = - 153 L eax sinh bx dx = 154 L eax cosh bx dx = coshn ax dx = Some Definite Integrals 155 157 L0 L0 coshn - ax sinh ax n - + n na L L coshn - ax dx, n ≠ xn - cosh ax dx cschn - ax coth ax n - cschn - ax dx, n ≠ (n - 1)a n - 1L ax sechn ax + C, n ≠ na bx 152 sin x dx = Z05_THOM4077_BTIpT1-T6.indd cschn ax + C, n ≠ na e e e c d + C, a2 ≠ b2 a + b a - b eax ebx e-bx c + d + C, a2 ≠ b2 a + b a - b e dx = Γ(n) = (n - 1)!, n n L cschn ax coth ax dx = - -bx n - -x p>2 xn - sinh ax dx sechn - ax ax n - + sechn - ax dx, n ≠ (n - 1)a n - 1L q x L L0 p>2 n cos x dx = d 156 L0 # # # g # (n - 1) # p , 2#4#6# g#n # # # g # (n - 1) , 3#5#7# g#n q e-ax dx = p , a 2A a if n is an even integer Ú if n is an odd integer Ú 8/29/13 11:35 AM Trigonometry Formulas y Definitions and Fundamental Identities y Sine: sin u = r = csc u Tangent: P(x, y) r x cos u = r = sec u y tan u = x = cot u Cosine: tan A + tan B - tan A tan B tan A - tan B tan (A - B) = + tan A tan B p p sin aA - b = -cos A, cos aA - b = sin A 2 tan (A + B) = y u x x p p b = cos A, cos aA + b = -sin A 2 1 sin A sin B = cos (A - B) - cos (A + B) 2 1 cos A cos B = cos (A - B) + cos (A + B) 2 1 sin A cos B = sin (A - B) + sin (A + B) 2 1 sin A + sin B = sin (A + B) cos (A - B) 2 1 sin A - sin B = cos (A + B) sin (A - B) 2 1 cos A + cos B = cos (A + B) cos (A - B) 2 1 cos A - cos B = -2 sin (A + B) sin (A - B) 2 sin aA + Identities sin (-u) = -sin u, cos (-u) = cos u sin2 u + cos2 u = 1, sec2 u = + tan2 u, csc2 u = + cot2 u sin 2u = sin u cos u, cos 2u = cos2 u - sin2 u + cos 2u - cos 2u cos2 u = , sin2 u = 2 sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B Trigonometric Functions y Degrees Radian Measure y = sin x Radians p 45 s "2 C ir "2 u p 90 r p –p – p p p y = cos x 3p 2p x Domain: (−∞, ∞) Range: [−1, 1] –p – p p p 3p 2p x Domain: (−∞, ∞) Range: [−1, 1] l it c irc e Un 45 y cle of rad y r ius u s s r = = u or u = r , 180° = p radians p 30 "3 60 90 "3 p – 3p –p – p 2 p p 3p 2 x y = sec x – 3p –p – p 2 p p 3p 2 x p The angles of two common triangles, in degrees and radians Domain: All real numbers except odd integer multiples of p͞2 Range: (−∞, ∞) y Domain: All real numbers except odd integer multiples of p͞2 Range: (−∞, −1] ´ [1, ∞) y y = csc x –p – p p p 3p 2p Domain: x ≠ 0, ±p, ±2p, Range: (−∞, −1] ´ [1, ∞) Z06_THOM4077_Last3Bkpgs.indd y y = tan x x y = cot x –p – p p p 3p 2p x Domain: x ≠ 0, ±p, ±2p, Range: (−∞, ∞) 8/29/13 4:59 PM SerieS Tests for Convergence of Infinite Series The nth-Term Test: Unless an S 0, the series diverges Geometric series: g ar converges if ͉ r ͉ 1; otherwise it diverges p-series: g 1>np 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 n Taylor Series q = + x + x2 + g + xn + g = a xn, - x n=0 0x0 1 = - x + x2 - g + (-x)n + g = a (-1)nxn, + x n=0 q ex = + x + sin x = x cos x = - x2 xn xn + g + + g = a , 2! n! n = n! q 0x0 q (-1)nx2n + x3 x2n + x5 + - g + (-1)n + g = a , 3! 5! (2n + 1)! n = (2n + 1)! q (-1)nx2n x2 x2n x4 + - g + (-1)n + g = a , 2! 4! (2n)! n = (2n)! ln (1 + x) = x ln 0x0 q Series with some negative terms: Does g ͉ an ͉ converge? If yes, so does g an since absolute convergence implies convergence Alternating series: g an converges if the series satisfies the conditions of the Alternating Series Test 0x0 q q (-1)n - 1xn x2 x3 xn + - g + (-1)n - n + g = a , n n=1 0x0 q -1 x … 1 + x x3 x5 x2n + x2n + = tanh-1 x = 2ax + + + g + + gb = a , - x 2n + 2n + n=0 q tan-1 x = x - q (-1)nx2n + x3 x5 x2n + + - g + (-1)n + g = a , 2n + n = 2n + Binomial Series (1 + x)m = + mx + where Z06_THOM4077_Last3Bkpgs.indd 0x0 … m(m - 1)x2 m(m - 1)(m - 2)x3 m(m - 1)(m - 2) g(m - k + 1)xk + + g + + g 2! 3! k! q m = + a a b xk, k=1 k m a b = m, 0x0 x 1, m(m - 1) m a b = , 2! m(m - 1) g(m - k + 1) m a b = k! k for k Ú 8/29/13 4:59 PM 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 0ƒ 0ƒ 0ƒ i + j + k 0x 0y 0z Gradient ∇ƒ = Divergence 0M 0N 0P + + ∇#F = 0x 0y 0z Curl Laplacian i j k ∇ * F = 0x 0y 04 0z M N P 2ƒ 2ƒ 2ƒ ∇ƒ = + + 0x 0y 0z The Fundamental Theorem of Line Integrals Part Let F = M i + N j + P k 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 F = ∇ƒ = 0ƒ 0ƒ 0ƒ 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 Part If the integral is independent of the path from A to B, its value is B LA B F # dr = ƒ(B) - ƒ(A) Green’s Theorem and Its Generalization to Three Dimensions Tangential form of Green’s Theorem: F F # T ds = (u * v) # w = (v * w) # u = (w * u) # v u * (v * w) = (u # w)v - (u # v)w ∇ * F # k dA O ∇ * F # n ds O ∇ # F dA C Stokes’ Theorem: R F F # T ds = F F # n ds = C Vector Triple Products O Normal form of Green’s Theorem: S C Divergence Theorem: O S R F # n ds = l ∇ # F dV D 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 ∇ * ( ∇ƒ ) = ∇ ( ƒg ) = ƒ∇g + g∇ƒ ∇ # ( gF ) = g∇ # F + ∇g # F ∇ * ( gF ) = g∇ * F + ∇g * F ∇ # ( aF1 + bF2 ) = a∇ # F1 + b∇ # F2 ∇ * ( aF1 + bF2 ) = a∇ * F1 + b∇ * F2 ∇ # ( F1 * F2 ) = F2 # ∇ * F1 - F1 # ∇ * F2 ∇ * ( F1 * F2 ) = ( F2 # ∇ ) F1 - ( F1 # ∇ ) F2 + ( ∇ # F2 ) F1 - ( ∇ # F1 ) F2 ∇ * ( ∇ * F ) = ∇ ( ∇ # F ) - ( ∇ # ∇ ) F = ∇ ( ∇ # F ) - ∇ 2F ( ∇ * F ) * F = ( F # ∇ ) F - 12 ∇ ( F # F ) ∇ ( F1 # F2 ) = ( F1 # ∇ ) F2 + ( F2 # ∇ ) F1 + F1 * ( ∇ * F2 ) + F2 * ( ∇ * F1 ) Z06_THOM4077_Last3Bkpgs.indd 8/29/13 4:59 PM This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM Basic algeBra Formulas Arithmetic Operations a#c ac = b d bd a>b a d = # b c c>d a(b + c) = ab + ac, a c ad + bc + = , b d bd Laws of Signs -a a a = - = b b -b -(-a) = a, Zero Division by zero is not defined 0 a a = 0, a = 1, = For any number a: a # = # a = If a ≠ 0: Laws of Exponents aman = am + n, (ab)m = ambm, (am)n = amn, If a ≠ 0, am = am - n, an The Binomial Theorem a0 = 1, am n m 12 a2 For any positive integer n, (a + b)n = an + nan - 1b + + For instance, a-m = n am>n = 2am = n(n - 1) n - 2 a b 1#2 n(n - 1)(n - 2) n - 3 a b + g + nabn - + bn 1#2#3 (a + b)2 = a2 + 2ab + b2, (a + b)3 = a3 + 3a2b + 3ab2 + b3, (a - b)2 = a2 - 2ab + b2 (a - b)3 = a3 - 3a2b + 3ab2 - b3 Factoring the Difference of Like Integer Powers, n + an - bn = (a - b)(an - + an - 2b + an - 3b2 + g + abn - + bn - 1) For instance, a2 - b2 = (a - b)(a + b), a3 - b3 = (a - b) ( a2 + ab + b2 ) , a4 - b4 = (a - b) ( a3 + a2b + ab2 + b3 ) Completing the Square If a ≠ 0, ax2 + bx + c = au + C The Quadratic Formula If a ≠ and ax2 + bx + c = 0, then x = Z07_THOM4077_Epp01-07.indd au = x + (b>2a), C = c - b2 b 4a -b { 2b2 - 4ac 2a 8/29/13 12:10 PM 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 c a b b′ b a b a′ = b′ = c′ a b c A = bh Parallelogram Trapezoid a2 + b2 = c2 Circle a h h b A = pr 2, C = 2pr r b A = bh A = (a + b)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 Z07_THOM4077_Epp01-07.indd Sphere V= Bh B V = pr2h S = prs = Area of side V = 43 pr3, S = 4pr2 8/29/13 12:10 PM limiTs General Laws Specific Formulas If L, M, c, and k are real numbers and If P(x) = an xn + an - xn - + g + a0, then lim ƒ(x) = L lim g(x) = M, then and xSc xSc lim (ƒ(x) + g(x)) = L + M Sum Rule: xSc lim P(x) = P(c) = an cn + an - cn - + g + a0 xSc If P(x) and Q(x) are polynomials and Q(c) ≠ 0, then lim (ƒ(x) - g(x)) = L - M Difference Rule: xSc lim (ƒ(x) # g(x)) = L # M Product Rule: lim xSc xSc Constant Multiple Rule: lim (k # ƒ(x)) = k # L xSc lim Quotient Rule: xSc ƒ(x) L = , M≠0 g(x) M P(x) P(c) = Q(x) Q(c) If ƒ(x) is continuous at x = c, then lim ƒ(x) = ƒ(c) xSc The Sandwich Theorem If g(x) … ƒ(x) … h(x) in an open interval containing c, except possibly at x = c, and if lim g(x) = lim h(x) = L, xSc sin x lim x = xS0 and lim xS0 - cos x = x xSc then limx S c ƒ(x) = L L’Hôpital’s Rule Inequalities If ƒ(a) = g(a) = 0, both ƒ′ and g′ exist in an open interval I containing a, and g′(x) ≠ on I if x ≠ a, then If ƒ(x) … g(x) in an open interval containing c, except possibly at x = c, and both limits exist, then lim ƒ(x) … lim g(x) xSc xSc lim xSa ƒ(x) ƒ′(x) = lim , g(x) x S a g′(x) assuming the limit on the right side exists Continuity If g is continuous at L and limx S c ƒ(x) = L, then lim g(ƒ(x)) = g(L) xSc Z07_THOM4077_Epp01-07.indd 8/29/13 12:10 PM DiFFerenTiaTion rules General Formulas Assume u and y are differentiable functions of x d Constant: (c) = dx d du dy Sum: (u + y) = + dx dx dx d du dy Difference: (u - y) = dx dx dx d du Constant Multiple: (cu) = c dx dx d dy du Product: (uy) = u + y dx dx dx du dy - u y dx dx d u Quotient: ayb = dx y2 d n x = nxn - dx d (ƒ(g(x)) = ƒ′(g(x)) # g′(x) dx Power: Chain Rule: Trigonometric Functions d (sin x) = cos x dx d (tan x) = sec2 x dx d (cot x) = -csc2 x dx d (cos x) = -sin x dx d (sec x) = sec x tan x dx d (csc x) = -csc x cot x dx Exponential and Logarithmic Functions d x e = ex dx d x a = ax ln a dx Z07_THOM4077_Epp01-07.indd d ln x = x dx d (loga x) = dx x ln a Inverse Trigonometric Functions d d 1 (sin-1 x) = (cos-1 x) = dx dx 21 - x 21 - x2 d (tan-1 x) = dx + x2 d (cot-1 x) = dx + x2 Hyperbolic Functions d (sinh x) = cosh x dx d (tanh x) = sech2 x dx d (coth x) = -csch2 x dx d (sec-1 x) = dx x 2x2 - d (csc-1 x) = dx x 2x2 - d (cosh x) = sinh x dx d (sech x) = -sech x x dx d (csch x) = -csch x coth x dx Inverse Hyperbolic Functions d d 1 (sinh-1 x) = (cosh-1 x) = dx 21 + x2 dx 2x2 - d d 1 (tanh-1 x) = (sech-1 x) = dx dx - x2 x 21 - x2 d d 1 (coth-1 x) = (csch-1 x) = dx dx - x2 x 21 + x2 Parametric Equations If x = ƒ(t) and y = g(t) are differentiable, then y′ = dy dy>dt = dx dx>dt and d 2y dy′>dt = dx>dt dx2 8/29/13 12:10 PM inTegraTion rules General Formulas Zero: Order of Integration: La a Lb a ƒ(x) dx = ƒ(x) dx = - La b Constant Multiples: Sums and Differences: b ƒ(x) dx b kƒ(x) dx = k ƒ(x) dx La La La b La b -ƒ(x) dx = - La b ƒ(x) dx (k = -1) b b La (ƒ(x) { g(x)) dx = b (Any number k) ƒ(x) dx { c La g(x) dx c ƒ(x) dx La Lb La Max-Min Inequality: If max ƒ and ƒ are the maximum and minimum values of ƒ on a, b4 , then ƒ(x) dx + Additivity: ƒ(x) dx = ƒ # (b - a) … a, b4 ƒ(x) Ú g(x) on Domination: a, b4 ƒ(x) Ú on implies implies La La La b ƒ(x) dx … max ƒ # (b - a) b ƒ(x) dx Ú La b g(x) dx b ƒ(x) dx Ú The Fundamental Theorem of Calculus Part If ƒ is continuous on a, b4 , then F(x) = 1a ƒ(t) dt is continuous on a, b4 and differentiable on (a, b) and its derivative is ƒ(x): x x F′(x) = d ƒ(t) dt = ƒ(x) dxLa Part If ƒ is continuous at every point of a, b4 and F is any antiderivative of ƒ on a, b4 , then La b ƒ(x) dx = F(b) - F(a) Substitution in Definite Integrals La Z07_THOM4077_Epp01-07.indd b ƒ(g(x)) # g′(x) dx = Integration by Parts g(b) Lg(a) ƒ(u) du La b b ƒ(x)g′(x) dx = ƒ(x)g(x) d a La b ƒ′(x)g(x) dx 8/29/13 12:10 PM ... 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... Sharp algebra and trigonometry skills are critical to mastering calculus, and Just-in-Time Algebra and Trigonometry for Calculus by Guntram Mueller and Ronald I Brent is designed to bolster these... www.pearsonhighered.com /thomas, and MyMathLab WEB SITE www.pearsonhighered.com /thomas The Thomas? ?? Calculus Web site contains the chapter on Second-Order Differential Equations, including odd-numbered answers, and provides