Thomas’ Calculus Early Transcendentals 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 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 : early transcendentals : based on the original work by George B Thomas, Jr., Massachusetts Institute of Technology.—Thirteenth edition / as revised by Maurice D Weir, Naval Postgraduate School, Joel Hass, University of California, Davis pages cm ISBN 978-0-321-88407-7 (hardcover) I Hass, Joel. II Thomas, George B (George Brinton), Jr., 1914–2006 Calculus Based on (Work): III Title IV Title: Calculus QA303.2.W45 2014 515–dc23 2013023096 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—17 16 15 14 13 www.pearsonhighered.com ISBN-10: 0-321-88407-8 ISBN-13: 978-0-321-88407-7 Contents 1 Functions 1 1.1 1.2 1.3 1.4 1.5 1.6 Preface ix Functions and Their Graphs Combining Functions; Shifting and Scaling Graphs 14 Trigonometric Functions 21 Graphing with Software 29 Exponential Functions 36 Inverse Functions and Logarithms 41 Questions to Guide Your Review 54 Practice Exercises 54 Additional and Advanced Exercises 57 Limits and Continuity 59 2.1 Rates of Change and Tangents to Curves 59 2.2 Limit of a Function and Limit Laws 66 2.3 The Precise Definition of a Limit 77 2.4 One-Sided Limits 86 2.5 Continuity 93 2.6 Limits Involving Infinity; Asymptotes of Graphs 104 Questions to Guide Your Review 118 Practice Exercises 118 Additional and Advanced Exercises 120 3 Derivatives 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 123 Tangents and the Derivative at a Point 123 The Derivative as a Function 128 Differentiation Rules 136 The Derivative as a Rate of Change 146 Derivatives of Trigonometric Functions 156 The Chain Rule 163 Implicit Differentiation 171 Derivatives of Inverse Functions and Logarithms 177 Inverse Trigonometric Functions 187 Related Rates 193 Linearization and Differentials 202 Questions to Guide Your Review 214 Practice Exercises 215 Additional and Advanced Exercises 219 iii iv Contents Applications of Derivatives 223 4.1 Extreme Values of Functions 223 4.2 The Mean Value Theorem 231 4.3 Monotonic Functions and the First Derivative Test 239 4.4 Concavity and Curve Sketching 244 4.5 Indeterminate Forms and L’Hôpital’s Rule 255 4.6 Applied Optimization 264 4.7 Newton’s Method 276 4.8 Antiderivatives 281 Questions to Guide Your Review 291 Practice Exercises 291 Additional and Advanced Exercises 295 5 Integrals 299 5.1 5.2 5.3 5.4 5.5 5.6 6.1 6.2 6.3 6.4 6.5 6.6 Area and Estimating with Finite Sums 299 Sigma Notation and Limits of Finite Sums 309 The Definite Integral 316 The Fundamental Theorem of Calculus 328 Indefinite Integrals and the Substitution Method 339 Definite Integral Substitutions and the Area Between Curves 347 Questions to Guide Your Review 357 Practice Exercises 357 Additional and Advanced Exercises 361 Applications of Definite Integrals 365 Volumes Using Cross-Sections 365 Volumes Using Cylindrical Shells 376 Arc Length 384 Areas of Surfaces of Revolution 390 Work and Fluid Forces 395 Moments and Centers of Mass 404 Questions to Guide Your Review 415 Practice Exercises 416 Additional and Advanced Exercises 417 Integrals and Transcendental Functions 420 7.1 7.2 7.3 7.4 The Logarithm Defined as an Integral 420 Exponential Change and Separable Differential Equations 430 Hyperbolic Functions 439 Relative Rates of Growth 448 Questions to Guide Your Review 453 Practice Exercises 453 Additional and Advanced Exercises 455 Contents Techniques of Integration 456 8.1 Using Basic Integration Formulas 456 8.2 Integration by Parts 461 8.3 Trigonometric Integrals 469 8.4 Trigonometric Substitutions 475 8.5 Integration of Rational Functions by Partial Fractions 480 8.6 Integral Tables and Computer Algebra Systems 489 8.7 Numerical Integration 494 8.8 Improper Integrals 504 8.9 Probability 515 Questions to Guide Your Review 528 Practice Exercises 529 Additional and Advanced Exercises 531 First-Order Differential Equations 536 9.1 Solutions, Slope Fields, and Euler’s Method 536 9.2 First-Order Linear Equations 544 9.3 Applications 550 9.4 Graphical Solutions of Autonomous Equations 556 9.5 Systems of Equations and Phase Planes 563 Questions to Guide Your Review 569 Practice Exercises 569 Additional and Advanced Exercises 570 10 Infinite Sequences and Series 572 10.1 Sequences 572 10.2 Infinite Series 584 10.3 The Integral Test 593 10.4 Comparison Tests 600 10.5 Absolute Convergence; The Ratio and Root Tests 604 10.6 Alternating Series and Conditional Convergence 610 10.7 Power Series 616 10.8 Taylor and Maclaurin Series 626 10.9 Convergence of Taylor Series 631 10.10 The Binomial Series and Applications of Taylor Series 638 Questions to Guide Your Review 647 Practice Exercises 648 Additional and Advanced Exercises 650 11 Parametric Equations and Polar Coordinates 653 11.1 Parametrizations of Plane Curves 653 11.2 Calculus with Parametric Curves 661 11.3 Polar Coordinates 671 v vi Contents 11.4 11.5 11.6 11.7 12 Graphing Polar Coordinate Equations 675 Areas and Lengths in Polar Coordinates 679 Conic Sections 683 Conics in Polar Coordinates 692 Questions to Guide Your Review 699 Practice Exercises 699 Additional and Advanced Exercises 701 Vectors and the Geometry of Space 704 12.1 Three-Dimensional Coordinate Systems 704 12.2 Vectors 709 12.3 The Dot Product 718 12.4 The Cross Product 726 12.5 Lines and Planes in Space 732 12.6 Cylinders and Quadric Surfaces 740 Questions to Guide Your Review 745 Practice Exercises 746 Additional and Advanced Exercises 748 13 13.1 13.2 13.3 13.4 13.5 13.6 14 Vector-Valued Functions and Motion in Space 751 Curves in Space and Their Tangents 751 Integrals of Vector Functions; Projectile Motion 759 Arc Length in Space 768 Curvature and Normal Vectors of a Curve 772 Tangential and Normal Components of Acceleration 778 Velocity and Acceleration in Polar Coordinates 784 Questions to Guide Your Review 788 Practice Exercises 788 Additional and Advanced Exercises 790 Partial Derivatives 793 14.1 Functions of Several Variables 793 14.2 Limits and Continuity in Higher Dimensions 801 14.3 Partial Derivatives 810 14.4 The Chain Rule 821 14.5 Directional Derivatives and Gradient Vectors 830 14.6 Tangent Planes and Differentials 839 14.7 Extreme Values and Saddle Points 848 14.8 Lagrange Multipliers 857 14.9 Taylor’s Formula for Two Variables 866 14.10 Partial Derivatives with Constrained Variables 870 Questions to Guide Your Review 875 Practice Exercises 876 Additional and Advanced Exercises 879 Contents 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 17 Multiple Integrals 882 Double and Iterated Integrals over Rectangles 882 Double Integrals over General Regions 887 Area by Double Integration 896 Double Integrals in Polar Form 900 Triple Integrals in Rectangular Coordinates 906 Moments and Centers of Mass 915 Triple Integrals in Cylindrical and Spherical Coordinates 922 Substitutions in Multiple Integrals 934 Questions to Guide Your Review 944 Practice Exercises 944 Additional and Advanced Exercises 947 Integrals and Vector Fields 950 Line Integrals 950 Vector Fields and Line Integrals: Work, Circulation, and Flux 957 Path Independence, Conservative Fields, and Potential Functions 969 Green’s Theorem in the Plane 980 Surfaces and Area 992 Surface Integrals 1003 Stokes’ Theorem 1014 The Divergence Theorem and a Unified Theory 1027 Questions to Guide Your Review 1039 Practice Exercises 1040 Additional and Advanced Exercises 1042 Second-Order Differential Equations online 17.1 Second-Order Linear Equations 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4 Euler Equations 17.5 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-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 A-1 Index I-1 Credits C-1 A Brief Table of Integrals T-1 vii This page intentionally left blank Preface Thomas’ Calculus: Early Transcendentals, 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 timetested 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 x Preface • Functions In discussing the use of software for graphing purposes, we added a brief subsection on least squares curve fitting, which allows students to take advantage of this widely used and available application Prerequisite material continues to be reviewed in Appendices 1–3 • Continuity We clarified the continuity definitions by confining the term “endpoints” to intervals instead of more general domains, and we moved the subsection on continuous extension of a function to the end of the continuity section • Derivatives We included a brief geometric insight justifying l’Hôpital’s Rule We also enhanced and clarified the meaning of differentiability for functions of several variables, and added a result on the Chain Rule for functions defined along a path • Integrals We wrote a new section reviewing basic integration formulas and the Substitution Rule, using them in combination with algebraic and trigonometric identities, before presenting other techniques of integration • Probability We created a new section applying improper integrals to some commonly used probability distributions, including the exponential and normal distributions Many examples and exercises apply to the life sciences • Series We now present the idea of absolute convergence before giving the Ratio and Root Tests, and then state these tests in their stronger form Conditional convergence is introduced later on with the Alternating Series Test • Multivariable and Vector Calculus We give more geometric insight into the idea of multiple integrals, and we enhance the meaning of the Jacobian in using substitutions to evaluate them The idea of surface integrals of vector fields now parallels the notion for line integrals of vector fields We have improved our discussion of the divergence and curl of a vector field • Exercises and Examples Strong exercise sets are traditional with Thomas’ Calculus, and we continue to strengthen them with each new edition Here, we have updated, changed, and added many new exercises and examples, with particular attention to including more applications to the life science areas and to contemporary problems For instance, we updated an exercise on the growth of the U.S GNP and added new exercises addressing drug concentrations and dosages, estimating the spill rate of a ruptured oil pipeline, and predicting rising costs for college tuition Continuing Features RIGOR The level of rigor is consistent with that of earlier editions We continue to distinguish between formal and informal discussions and to point out their differences We think starting with a more intuitive, less formal, approach helps students understand a new or difficult concept so they can then appreciate its full mathematical precision and outcomes We pay attention to defining ideas carefully and to proving theorems appropriate for calculus students, while mentioning deeper or subtler issues they would study in a more advanced course Our organization and distinctions between informal and formal discussions give the instructor a degree of flexibility in the amount and depth of coverage of the various topics For example, while we not prove the Intermediate Value Theorem or the Extreme Value Theorem for continuous functions on a # x # b, we state these theorems precisely, illustrate their meanings in numerous examples, and use them to prove other important results Furthermore, for those instructors who desire greater depth of coverage, in Appendix we discuss the reliance of the validity of these theorems on the completeness of the real numbers A Brief Table of Integrals Basic Forms 11 13 15 L L L k dx = kx + C (any number k) dx x = ln x + C ax dx = cos x dx = sin x + C L L L L L ax + C (a 0, a ≠ 1) ln a L L xn dx = xn + + C (n ≠ -1) n + ex dx = ex + C sin x dx = -cos x + C L sec2 x dx = tan x + C L csc2 x dx = -cot x + C 10 sec x tan x dx = sec x + C csc x cot x dx = -csc x + C 12 cot x dx = ln sin x + C 14 cosh x dx = sinh x + C 16 dx x = sin-1 a + C L 2a - x L L L tan x dx = ln sec x + C sinh x dx = cosh x + C 17 dx x = a tan-1 a + C 2 a + x L 18 dx x = a sec-1 a + C 2 L x 2x - a 19 dx x = sinh-1 a + C (a 0) L 2a + x 20 dx x = cosh-1 a + C (x a 0) L 2x - a 2 Forms Involving ax + b 21 22 23 25 27 L L L L L (ax + b)n dx = (ax + b)n + + C, n ≠ -1 a(n + 1) x(ax + b)n dx = (ax + b)n + ax + b b c d + C, n ≠ -1, -2 n + n + a2 (ax + b)-1 dx = a ln ax + b + C x(ax + b)-2 dx = 2ax b c ln ax + b + d + C ax + b a n+2 2ax + b n + b dx = a + C, n ≠ -2 n + 24 26 28 L x b x(ax + b)-1 dx = a - ln ͉ ax + b ͉ + C a dx x = ln + C x(ax + b) b ax + b L 2ax + b L x dx dx = 2ax + b + b L x 2ax + b T-1 T-2 A Brief Table of Integrals 29 (a) 30 2ax + b - 2b dx = ln ` ` + C 2b 2ax + b + 2b L x 2ax + b 2ax + b L x2 dx = - 2ax + b x + a dx + C 2L x 2ax + b (b) dx ax - b = tan-1 + C A b 2b L x 2ax - b 31 2ax + b a dx dx = + C bx 2bL x 2ax + b L x2 2ax + b Forms Involving a + x 32 dx x = a tan-1 a + C 2 a + x L 34 dx x = sinh-1 a + C = ln x + 2a2 + x2 + C L 2a2 + x2 35 36 37 38 L L 2a2 + x2 dx = dx = 2a2 + x2 - a ln ` 2a2 + x2 dx = ln x + 2a2 + x2 - x = x x + tan-1 a + C 2a 2a2 ( a2 + x2 ) x a4 ( ln x + 2a2 + x2 + C a + 2x2 ) 2a2 + x2 8 2a2 + x2 x dx 2 L (a + x ) x a2 2a2 + x2 + ln x + 2a2 + x2 + C 2 x2 2a2 + x2 dx = L L 33 a + 2a2 + x2 ` + C x 2a2 + x2 x + C 39 x 2a2 + x2 x2 a2 dx = - ln x + 2a2 + x2 + + C 2 L 2a + x 40 a + 2a2 + x2 dx = ` ` + C ln a x L x 2a2 + x2 41 2a2 + x2 dx + C = a2x L x2 2a2 + x2 x dx x + a = 2 + ln x - a + C 2 4a 2a ( a - x2 ) L (a - x ) Forms Involving a − x 42 dx x + a = ln x - a + C 2 2a La - x 43 44 x dx = sin-1 a + C 2 L 2a - x 45 46 47 L x2 2a2 - x2 dx = 2a2 - x2 L x a x sin-1 a - x 2a2 - x2 ( a2 - 2x2 ) + C 8 dx = 2a2 - x2 - a ln ` a + 2a2 - x2 ` + C 48 x a2 -1 x x2 dx = sin a - x 2a2 - x2 + C 2 L 2a - x 51 2a2 - x2 dx + C = a2x L x2 2a2 - x2 Forms Involving x − a 53 dx = ln x + 2x2 - a2 + C L 2x2 - a2 L 2x2 - a2 dx = x a2 -1 x 2a2 - x2 + sin a + C 2 49 52 L 2a2 - x2 dx = x a2 2x2 - a2 ln x + 2x2 - a2 + C 2 50 2a2 - x2 L x 2a2 - x2 x dx = -sin-1 a + C x a + 2a2 - x2 dx = - a ln ` ` + C x 2 L x 2a - x A Brief Table of Integrals 54 L 2x - a 2 n x1 2x2 - a2 na2 dx = 2x2 - a2 2n - dx, n ≠ -1 n + n + 1L n x1 2x2 - a2 22 - n n - dx dx 55 = n-2 , n ≠ 2 2 n (2 n)a (n 2)a L 2x - a L 2x - a2 56 57 58 59 L L x1 2x2 - a2 2n dx = 2x2 - a2 2n + + C, n ≠ -2 n + x ( 2x2 - a2 ) 2x2 - a2 - a8 ln x + 2x2 - a2 + C x2 2x2 - a2 dx = 2x2 - a2 x dx = 2x2 - a2 - a sec-1 ` a ` + C 2x2 - a2 dx = ln x + 2x2 - a2 - x L L x 2x2 - a2 x + C 60 a2 x x2 dx = ln x + 2x2 - a2 + 2x2 - a2 + C 2 L 2x - a 61 dx x a 1 = a sec-1 ` a ` + C = a cos-1 ` x ` + C 2 L x 2x - a 62 2x2 - a2 dx + C = a2x L x2 2x2 - a2 Trigonometric Forms 63 65 67 68 L sin ax dx = - a cos ax + C sin2 ax dx = L x sin 2ax + C 4a sinn ax dx = - L cosn ax dx = L 64 n-1 sin ax cos ax n - + n na cosn - ax sin ax n - + n na 66 L L L cos ax dx = a sin ax + C cos2 ax dx = x sin 2ax + + C 4a sinn - ax dx cosn - 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 (b) (c) 70 72 74 75 76 L 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) L L sin ax cos ax dx = - cos 2ax + C 4a cos ax dx = a ln ͉ sin ax ͉ + C L sin ax L L L 71 73 L L sinn ax cos ax dx = sinn + ax + C, n ≠ -1 (n + 1)a cosn ax sin ax dx = - cosn + ax + C, n ≠ -1 (n + 1)a sin ax cos ax dx = - a ln ͉ cos ax ͉ + C sinn ax cosm ax dx = sinn ax cosm ax dx = 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) T-3 T-4 A Brief Table of Integrals 77 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 78 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 79 p ax dx = - a tan a - b + C + sin ax L 81 ax dx b - c = tan-1 c tan d + C, b2 c2 2 b + c cos ax b + c A a 2b - c L 82 c + b cos ax + 2c2 - b2 sin ax dx = ln ` ` + C, 2 b + c cos ax L b + c cos ax a 2c - b 83 ax dx = a tan + C L + cos ax 85 L x sin ax dx = 84 x sin ax - a cos ax + C a2 n 87 89 91 93 95 97 99 100 101 80 86 L x n xn sin ax dx = - a cos ax + a L tan ax dx = a ln sec ax + C 90 L tan2 ax dx = a tan ax - x + C 92 L tann ax dx = L xn - cos ax dx tann - ax tann - ax dx, n ≠ a(n - 1) L 88 94 L sec ax dx = a ln sec ax + tan ax + C 96 L sec2 ax dx = a tan ax + C 98 L L L secn ax dx = b2 c2 dx p ax = a tan a + b + C sin ax L b2 c2 dx ax = - a cot + C L - cos ax L x cos ax dx = x cos ax + a sin ax + C a2 L xn n xn cos ax dx = a sin ax - a L cot ax dx = a ln sin ax + C L cot2 ax dx = - a cot ax - x + C cotn ax dx = - L L xn - sin ax dx cotn - ax cotn - ax dx, n ≠ a(n - 1) L L csc ax dx = - a ln csc ax + cot ax + C L csc2 ax dx = - a cot ax + C secn - ax tan ax n - secn - ax dx, n ≠ + a(n - 1) n - 1L cscn ax dx = - cscn - ax cot ax n - cscn - ax dx, n ≠ + a(n - 1) n - 1L secn ax tan ax dx = secn ax na + C, n ≠ 102 L cscn ax cot ax dx = - cscn ax na + C, n ≠ Inverse Trigonometric Forms 103 105 106 107 108 L L L L L sin-1 ax dx = x sin-1 ax + a 21 - a2x2 + C tan-1 ax dx = x tan-1 ax - 104 L cos-1 ax dx = x cos-1 ax - a 21 - a2x2 + C ln ( + a2x2 ) + C 2a xn sin-1 ax dx = xn + a xn + dx , n ≠ -1 sin-1 ax n + n + L 21 - a2x2 xn cos-1 ax dx = xn + a xn + dx , n ≠ -1 cos-1 ax + n + n + L 21 - a2x2 xn tan-1 ax dx = xn + a xn + dx , n ≠ -1 tan-1 ax n + n + L + a2x2 A Brief Table of Integrals Exponential and Logarithmic Forms 109 111 113 114 115 L L L L L eax dx = a eax + C xeax dx = xnbax dx = 110 eax (ax - 1) + C a2 112 L L bax dx = bax + C, b 0, b ≠ a ln b n xneax dx = a xneax - a L xn - 1eax dx xnbax n xn - 1bax dx, b 0, b ≠ a ln b a ln b L eax sin bx dx = eax (a sin bx - b cos bx) + C a + b2 eax cos bx dx = eax (a cos bx + b sin bx) + C a + b2 2 116 L ln ax dx = x ln ax - x + C xn + 1(ln ax)m m xn(ln ax)m - dx, n ≠ -1 n + n + 1L L (ln ax)m + dx = ln ln ax + C 118 x-1(ln ax)m dx = 119 + 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 n - dx 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 123 125 126 127 22ax - x2 L x 22ax - x2 L 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 - a x dx = a sin-1 a a b - 22ax - x2 + C L 22ax - x2 128 dx 2a - x = -a + C A x L x 22ax - x2 Hyperbolic Forms 129 131 133 L L L sinh ax dx = a cosh ax + C sinh2 ax dx = sinh 2ax x - + C 4a sinhn ax dx = sinhn - ax cosh ax n - - n na 130 132 L L L cosh ax dx = a sinh ax + C cosh2 ax dx = sinhn - ax dx, n ≠ 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, n ≠ x x sinh ax dx = a cosh ax - sinh ax + C a L 136 L xn n xn sinh ax dx = a cosh ax - a L ax dx = a ln (cosh ax) + C 140 L tanh2 ax dx = x - a ax + C 142 L L L xn - cosh ax dx 138 tanhn ax dx = - tanhn - ax tanhn - ax dx, n ≠ + (n - 1)a L cothn ax dx = - cothn - ax cothn - ax dx, n ≠ + (n - 1)a L L sech ax dx = a sin-1 (tanh ax) + C 146 L sech2 ax dx = a ax + C 148 L L L L L sechn ax dx = x x cosh ax dx = a sinh ax - cosh ax + C a L L xn n xn cosh ax dx = a sinh ax - a L coth ax dx = a ln ͉ sinh ax ͉ + 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 cschn ax dx = - cschn - ax coth ax n - cschn - ax dx, n ≠ (n - 1)a n - 1L sechn ax ax dx = - sechn ax + C, n ≠ na eax sinh bx dx = e e e d + C, a2 ≠ b2 c a + b a - b eax cosh bx dx = e-bx eax ebx + d + C, a2 ≠ b2 c a + b a - b ax bx 152 L cschn ax coth ax dx = - cschn ax + C, n ≠ na -bx q L0 x q n - -x e dx = Γ(n) = (n - 1)!, n p>2 157 p>2 sin x dx = n L0 xn - sinh ax dx sechn - ax ax n - sechn - ax dx, n ≠ + (n - 1)a n - 1L Some Definite Integrals 155 L cos x dx = d n L0 156 L0 # # # g # (n - 1) # p , 2#4#6# g#n # # # g # (n - 1) , 3#5#7# g#n e-ax dx = p , a 2A a if n is an even integer Ú if n is an odd integer Ú Trigonometry Formulas y Definitions and Fundamental Identities y Sine: sin u = r = csc u P(x, y) r x cos u = r = sec u y tan u = x = cot u Cosine: Tangent: tan A + tan B - tan A tan B tan A - tan B tan (A - B) = + tan A tan B p p cos aA - b = sin A sin aA - b = -cos A, 2 tan (A + B) = u x y sin aA + 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 x 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 , sin2 u = cos2 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 y y = sin x Degrees Radian Measure "2 C ir "2 u 45 p 90 r p p p 3p 2p x Domain: (−∞, ∞) Range: [−1, 1] –p – p p p 3p 2p x Domain: (−∞, ∞) Range: [−1, 1] l it circ e Un –p – p p 45 s y = cos x Radians cle of rad y r ius u s s r = = u or u = r , 180° = p radians p 30 "3 60 90 y = sec x "3 p y y = tan x – 3p –p – p 2 p p 3p 2 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 y y = csc x –p – p Domain: All real numbers except odd integer multiples of p͞2 Range: (−∞, −1] ´ [1, ∞) y = cot x p p 3p 2p Domain: x ≠ 0, ±p, ±2p, Range: (−∞, −1] ´ [1, ∞) x –p – p p p 3p 2p Domain: x ≠ 0, ±p, ±2p, Range: (−∞, ∞) x 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 q = - x + x2 - g + (-x)n + g = a (-1)nxn, + x n=0 ex = + x + sin x = x cos x = - q x2 xn xn + g + + g = a , 2! n! n = n! 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 0x0 q 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 - 0x0 q 0x0 q q (-1)n - 1xn x2 x3 xn , - g + (-1)n - n + g = a + n n=1 -1 x … q ln x3 x5 + x x2n + x2n + + + g + = tanh-1 x = 2ax + + gb = a , - x 2n + 2n + n=0 tan-1 x = x - q (-1)nx2n + x3 x5 x2n + + - g + (-1)n + g = a , 2n + n = 2n + 0x0 0x0 … Binomial Series (1 + x)m = + mx + 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 x 1, where m a b = m, m(m - 1) m a b = , 2! m(m - 1) g(m - k + 1) 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 0ƒ 0ƒ 0ƒ i + j + k 0x 0y 0z Gradient ∇ƒ = Divergence 0M 0N 0P + + ∇#F = 0x 0y 0z Curl Laplacian The Fundamental Theorem of Line Integrals Part 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 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 B LA F # dr = ƒ(B) - ƒ(A) i j k ∇ * F = 0x 0y 04 0z Green’s Theorem and Its Generalization to Three Dimensions M N P Tangential form of Green’s Theorem: 2ƒ 2ƒ 2ƒ ∇ƒ = + + 0x 0y 0z F F # T ds = C Stokes’ Theorem: F (u * v) # w = (v * w) # u = (w * u) # v u * (v * w) = (u # w)v - (u # v)w Normal form of Green’s Theorem: F F # T ds = F # n ds = O S O ∇ * F # n ds S C Divergence Theorem: ∇ * F # k dA R C Vector Triple Products O O ∇ # F dA 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 ) = ( F1 # ∇ ) F2 + ( F2 # ∇ ) F1 + F1 * ( ∇ * F2 ) + F2 * ( ∇ * F1 ) ∇ # ( 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 - 21 ∇ ( F # F ) This page intentionally left blank BASIC ALGEBRA FORMULAS Arithmetic Operations a#c ac = b d bd a>b a d = # b c c>d a(b + c) = ab + ac, c ad + bc a + = , 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, n am>n = 2am = n m 12 a2 If a ≠ 0, am = am - n, an a0 = 1, a-m = am The Binomial Theorem For any positive integer n, (a + b)n = an + nan - 1b + + n(n - 1) n - 2 a b 1#2 n(n - 1)(n - 2) n - 3 a b + g + nabn - + bn 1#2#3 For instance, (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 au = x + (b>2a), C = c - The Quadratic Formula If a ≠ and ax2 + bx + c = 0, then x = -b { 2b2 - 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 a′ Pythagorean Theorem c a b b′ b a b a′ = b′ = c′ b c a A = bh Parallelogram Trapezoid a2 + b2 = c2 Circle a h h A = pr 2, C = 2pr r b 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 Sphere V= 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 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: Quotient Rule: lim (k # ƒ(x)) = k # L P(x) P(c) = Q(x) Q(c) xSc lim xSc ƒ(x) L = , M≠0 g(x) M 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 Continuity If g is continuous at L and limx S c ƒ(x) = L, then lim g(ƒ(x)) = g(L) xSc lim xSa ƒ(x) ƒ′(x) = lim , g(x) x S a g′(x) assuming the limit on the right side exists 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 du d (cu) = c Constant Multiple: dx dx d dy du Product: (uy) = u + y dx dx dx dy du y - u d u dx dx 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 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 INTEGRATION RULES General Formulas a Zero: ƒ(x) dx = La a Order of Integration: b ƒ(x) dx = - Lb b Constant Multiples: ƒ(x) dx La b kƒ(x) dx = k ƒ(x) dx La La b b -ƒ(x) dx = - La La ƒ(x) dx (k = -1) b b b Sums and Differences: (ƒ(x) { g(x)) dx = La (Any number k) b ƒ(x) dx { La c ƒ(x) dx + La g(x) dx c ƒ(x) dx = ƒ(x) dx Lb La La Max-Min Inequality: If max ƒ and ƒ are the maximum and minimum values of ƒ on a, b4 , then Additivity: ƒ # (b - a) … a, b4 ƒ(x) Ú g(x) on Domination: La ƒ(x) dx … max ƒ # (b - a) b implies b ƒ(x) dx Ú La La g(x) dx b a, b4 ƒ(x) Ú on b implies La 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 b La ƒ(x) dx = F(b) - F(a) Substitution in Definite Integrals b La ƒ(g(x)) # g′(x) dx = Integration by Parts g(b) Lg(a) ƒ(u) du b La b b ƒ(x)g′(x) dx = ƒ(x)g(x) d a La ƒ′(x)g(x) 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... the exercises in Thomas? ?? Calculus: Early Transcendentals STUDENT’S SOLUTIONS MANUAL Single Variable Calculus (Chapters 1–11), ISBN 0-321-88410-8 | 978-0-321-88410-7 Multivariable Calculus (Chapters... Integrals T-1 vii This page intentionally left blank Preface Thomas? ?? Calculus: Early Transcendentals, Thirteenth Edition, provides a modern introduction to calculus that focuses on conceptual understanding