George b thomas, jr , maurice d weir, joel hass et al thomas calculus in SI units pearson (2016)

THOMAS’ CALCULUS Thirteenth Edition in SI Units 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 SI conversion by Antonio Behn Universidad de Chile A01_THOM9799_13_SE_FM.indd 08/04/16 2:46 PM www.elsolucionario.org Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2016 Authorized adaptation from the United States edition, entitled Thomas’ Calculus, Thirteenth Edition, ISBN 978-0-321-87896-0, by Maurice D Weir and Joel Hass published by Pearson Education © 2016 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 ISBN 10: 1-292-08979-2 ISBN 13: 978-1-292-08979-9 Typeset by S4Carlisle Printed and bound in Italy by L.E.G.O A01_THOM9799_13_SE_FM.indd 07/04/16 7:10 PM Contents 1 Functions   15 1.1 1.2 1.3 1.4 Preface  Functions and Their Graphs  15 Combining Functions; Shifting and Scaling Graphs 28 Trigonometric Functions 35 Graphing with Software 43 Questions to Guide Your Review 50 Practice Exercises 50 Additional and Advanced Exercises 52 Limits and Continuity  55 2.1 Rates of Change and Tangents to Curves 55 2.2 Limit of a Function and Limit Laws 62 2.3 The Precise Definition of a Limit 73 2.4 One-Sided Limits 82 2.5 Continuity 89 2.6 Limits Involving Infinity; Asymptotes of Graphs 100 Questions to Guide Your Review 113 Practice Exercises 114 Additional and Advanced Exercises 116 3 Derivatives   119 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Tangents and the Derivative at a Point 119 The Derivative as a Function 124 Differentiation Rules 132 The Derivative as a Rate of Change 141 Derivatives of Trigonometric Functions 151 The Chain Rule 158 Implicit Differentiation 165 Related Rates 170 Linearization and Differentials 179 Questions to Guide Your Review 191 Practice Exercises 191 Additional and Advanced Exercises 196 Applications of Derivatives  199 A01_THOM9799_13_SE_FM.indd 4.1 4.2 4.3 4.4 Extreme Values of Functions 199 The Mean Value Theorem 207 Monotonic Functions and the First Derivative Test 213 Concavity and Curve Sketching 218 07/04/16 7:10 PM www.elsolucionario.org 4.5 Applied Optimization 229 4.6 Newton’s Method 241 4.7 Antiderivatives 246 Questions to Guide Your Review 256 Practice Exercises 257 Additional and Advanced Exercises 259 5 Integrals   263 5.1 5.2 5.3 5.4 5.5 5.6 Applications of Definite Integrals  327 6.1 6.2 6.3 6.4 6.5 6.6 Volumes Using Cross-Sections 327 Volumes Using Cylindrical Shells 338 Arc Length 345 Areas of Surfaces of Revolution 351 Work and Fluid Forces 356 Moments and Centers of Mass 365 Questions to Guide Your Review 376 Practice Exercises 376 Additional and Advanced Exercises 378 Transcendental Functions  380 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Area and Estimating with Finite Sums 263 Sigma Notation and Limits of Finite Sums 273 The Definite Integral 280 The Fundamental Theorem of Calculus 292 Indefinite Integrals and the Substitution Method 303 Definite Integral Substitutions and the Area Between Curves 310 Questions to Guide Your Review 320 Practice Exercises 320 Additional and Advanced Exercises 323 Inverse Functions and Their Derivatives 380 Natural Logarithms  388 Exponential Functions  396 Exponential Change and Separable Differential Equations 407 Indeterminate Forms and L’Hôpital’s Rule  417 Inverse Trigonometric Functions  425 Hyperbolic Functions 438 Relative Rates of Growth 447 Questions to Guide Your Review 452 Practice Exercises 453 Additional and Advanced Exercises 456 Techniques of Integration  458 8.1 8.2 A01_THOM9799_13_SE_FM.indd Using Basic Integration Formulas 458 Integration by Parts 463 07/04/16 7:10 PM 8.3 Trigonometric Integrals 471 8.4 Trigonometric Substitutions 477 8.5 Integration of Rational Functions by Partial Fractions 482 8.6 Integral Tables and Computer Algebra Systems 491 8.7 Numerical Integration 496 8.8 Improper Integrals 506 8.9 Probability 517 Questions to Guide Your Review 530 Practice Exercises 531 Additional and Advanced Exercises 533 First-Order Differential Equations  538 9.1 Solutions, Slope Fields, and Euler’s Method 538 9.2 First-Order Linear Equations 546 9.3 Applications 552 9.4 Graphical Solutions of Autonomous Equations 558 9.5 Systems of Equations and Phase Planes 565 Questions to Guide Your Review 571 Practice Exercises 571 Additional and Advanced Exercises 572 10 Infinite Sequences and Series  574 10.1 Sequences 574 10.2 Infinite Series 586 10.3 The Integral Test 595 10.4 Comparison Tests 602 10.5 Absolute Convergence; The Ratio and Root Tests 606 10.6 Alternating Series and Conditional Convergence 612 10.7 Power Series 618 10.8 Taylor and Maclaurin Series 628 10.9 Convergence of Taylor Series 633 10.10 The Binomial Series and Applications of Taylor Series 640 Questions to Guide Your Review 649 Practice Exercises 650 Additional and Advanced Exercises 652 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 A01_THOM9799_13_SE_FM.indd Parametric Equations and Polar Coordinates  655 Parametrizations of Plane Curves 655 Calculus with Parametric Curves 663 Polar Coordinates 673 Graphing Polar Coordinate Equations 677 Areas and Lengths in Polar Coordinates 681 Conic Sections 685 Conics in Polar Coordinates 694 Questions to Guide Your Review 701 Practice Exercises 701 Additional and Advanced Exercises 703 07/04/16 7:10 PM www.elsolucionario.org 12 Vectors and the Geometry of Space  706 12.1 Three-Dimensional Coordinate Systems 706 12.2 Vectors 711 12.3 The Dot Product 720 12.4 The Cross Product 728 12.5 Lines and Planes in Space 734 12.6 Cylinders and Quadric Surfaces 742 Questions to Guide Your Review 747 Practice Exercises 748 Additional and Advanced Exercises 750 13 13.1 13.2 13.3 13.4 13.5 13.6 14 Vector-Valued Functions and Motion in Space  753 Curves in Space and Their Tangents 753 Integrals of Vector Functions; Projectile Motion 761 Arc Length in Space 770 Curvature and Normal Vectors of a Curve 774 Tangential and Normal Components of Acceleration 780 Velocity and Acceleration in Polar Coordinates 786 Questions to Guide Your Review 790 Practice Exercises 790 Additional and Advanced Exercises 792 Partial Derivatives  795 14.1 Functions of Several Variables 795 14.2 Limits and Continuity in Higher Dimensions 803 14.3 Partial Derivatives 812 14.4 The Chain Rule 823 14.5 Directional Derivatives and Gradient Vectors 832 14.6 Tangent Planes and Differentials 841 14.7 Extreme Values and Saddle Points 850 14.8 Lagrange Multipliers 859 14.9 Taylor’s Formula for Two Variables 868 14.10 Partial Derivatives with Constrained Variables 872 Questions to Guide Your Review 877 Practice Exercises 878 Additional and Advanced Exercises 881 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 A01_THOM9799_13_SE_FM.indd Multiple Integrals  884 Double and Iterated Integrals over Rectangles 884 Double Integrals over General Regions 889 Area by Double Integration 898 Double Integrals in Polar Form 902 Triple Integrals in Rectangular Coordinates 908 Moments and Centers of Mass 917 Triple Integrals in Cylindrical and Spherical Coordinates 924 Substitutions in Multiple Integrals 936 07/04/16 7:10 PM 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 17 Questions to Guide Your Review 946 Practice Exercises 946 Additional and Advanced Exercises 949 Integrals and Vector Fields  952 Line Integrals 952 Vector Fields and Line Integrals: Work, Circulation, and Flux 959 Path Independence, Conservative Fields, and Potential Functions 971 Green’s Theorem in the Plane 982 Surfaces and Area 994 Surface Integrals 1005 Stokes’ Theorem 1016 The Divergence Theorem and a Unified Theory 1029 Questions to Guide Your Review 1041 Practice Exercises 1042 Additional and Advanced Exercises 1044 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 Credits  C-1 Index  I-1 A Brief Table of Integrals  T-1 Basic Formulas and Rules  F-1 A01_THOM9799_13_SE_FM.indd 07/04/16 7:10 PM www.elsolucionario.org 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: A01_THOM9799_13_SE_FM.indd 07/04/16 7:10 PM 10 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 added new exercises addressing drug concentrations and dosages, ­estimating the spill rate of a ruptured oil pipeline, and predicting rising costs for college tuition • The Use of SI Units All the units in this edition have been converted to SI units, except where a non-SI unit is commonly used in scientific, technical, and commercial literature in most regions 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 A01_THOM9799_13_SE_FM.indd 10 07/04/16 7:10 PM I-16 Index Washer method, 333–334, 343 Wave equation, 823 Weierstrass, Karl, 513 Weierstrass function, 132 Whirlpool effect, 984 Windows, graphing, 43–46 Work by constant force, 356 by force over curve in space, 964–966 by force through displacement, 725 Hooke’s Law for springs, 357–358 and kinetic energy, 363 kinetic energy and, 363 pumping liquids from containers, 358–359 Z04_THOM9799_13_SE_IND.indd 16 by variable force along curve, 964 by variable force along line, 356–357 Work done by the heart, 189–190 x-coordinate, AP-10 x-intercept, AP-13 x-limits of integration, 911, 913 xy-plane, definition of, 706 xz-plane, 706 y = ƒ(x) graphing of, 223–225 length of, 345–347, 668–669 y, integration with respect to, 315–316 y-axis, revolution about, 353–354 y-coordinate, AP-10 y-intercept, AP-13 y-limits of integration, 910, 912 yz-plane, 706 Zero denominators, algebraic elimination of, 66–67 Zero vector, 712 Zero Width Interval Rule, 284, 389 z-limits of integration, 910, 911, 912, 927 04/04/16 4:33 PM www.elsolucionario.org Brief Table of Integrals Basic Forms L L L ax dx = L cos x dx = sin x + C 9 L 11 L 13 L 15 L 17 dx x = a tan-1 a + C 2 La + x 19 L xn dx = L ex dx = ex + 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 csc x cot x dx = -csc x + C 12 L cot x dx = ln sin x + C 14 L k dx = kx + C (any number k) dx x = ln x + C ax + C (a 0, a ≠ 1) ln a cosh x dx = sinh x + C dx x = sinh-1 a + C (a 0) L 2a + x 16 18 20 Forms Involving ax + b 21 L (ax + b)n dx = 22 L x(ax + b)n dx = 23 L 25 L 27 L x(ax + b)-2 dx = 2ax Z05_THOM9799_13_SE_BTI.indd + b2 tan x dx = ln sec x + C sinh x dx = cosh x + C dx x = sin-1 a + C L 2a - x dx x = a sec-1 a + C L x 2x2 - a2 dx x = cosh-1 a + C (x a 0) L 2x - a (ax + b)n + + C, n ≠ -1 a(n + 1) (ax + b)n + ax + b b c d + C, n ≠ -1, -2 n + n + a2 (ax + b)-1 dx = a ln ax + b + C n xn + + C (n ≠ -1) n + b c ln ax + b + d + C ax + b a2 n+2 2ax + b dx = a + C, n ≠ -2 n + 24 x b x(ax + b)-1 dx = a - ln ͉ ax + b ͉ + C a L 26 dx x = ln + C x(ax + b) b ax + b L 28 L 2ax + b x dx dx = 2ax + b + b L x 2ax + b 04/04/16 4:34 PM 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 L 38 L 2a2 + x dx = 46 47 49 51 dx x x = 2 + tan-1 a + C 2 2a 2a ( a + x2 ) L (a + x ) x a2 2a2 + x + ln x + 2a2 + x2 + C 2 x2 2a2 + x2 dx = 2a2 + x x 2a2 + x x2 x a4 ( a + 2x2 ) 2a2 + x2 ln x + 2a2 + x2 + C 8 dx = 2a2 + x2 - a ln ` a + 2a + x ` + C x dx = ln x + 2a2 + x2 - 2a2 + x2 x + C x 2a2 + x2 x2 a2 dx = x + 2a2 + x2 + ln + C 2 L 2a2 + x2 a + 2a2 + x2 dx = - a ln ` ` + C x 2 L x 2a + x 41 Forms Involving a − x 44 2ax + b a dx dx = + C bx 2b L x 2ax + b L x 2ax + b dx x = sinh-1 a + C = ln x + 2a2 + x2 + C L 2a + x 37 42 dx ax - b = tan-1 + C b A 2b L x 2ax - b L 40 dx a + C 2L x 2ax + b dx x = a tan-1 a + C 2 La + x 36 39 + (b) dx x + a = ln x - a + C 2 2a a x L dx x = sin-1 a + C L 2a - x2 L L x2 2a2 - x2 dx = 2a2 - x2 x 2a2 + x2 dx = + C 2 a2x L x 2a + x 43 2 L (a - x ) 45 L dx a4 -1 x sin 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 2 L 2a - x 2a - x dx = + C 2 a2x L x 2a - x 50 = x x + a + ln x - a + C 4a 2a2 ( a2 - x2 ) 2a2 - x dx = L 2a2 - x2 x2 x a2 -1 x 2a2 - x + sin a + C 2 2a2 - x2 x dx = -sin-1 a + C x a + 2a2 - x2 dx = ln ` ` + C a x L x 2a2 - x2 Forms Involving x − a 52 53 dx = ln x + 2x2 - a2 + C L x - a2 L 2x2 - a2 dx = Z05_THOM9799_13_SE_BTI.indd x a2 x - a2 ln x + 2x2 - a2 + C 2 04/04/16 4:34 PM www.elsolucionario.org A Brief Table of Integrals L 56 L 57 L x1 2x - a 58 L 59 L 55 60 61 x1 2x2 - a2 na2 2x2 - a2 2n - dx, n ≠ -1 n + n + 1L n 2x 54 T-3 - a2 dx = n x1 2x2 - a2 22 - n dx n - dx = , n≠2 n (2 - n)a2 (n - 2)a2L 2x2 - a2 2n - L 2x - a 2 2 n dx = x 2x2 - a2 x2 - a2 2n + + C, n ≠ -2 n + x a4 ( 2x2 - a2 ) 2x2 - a2 ln x + 2x2 - a2 + C 8 x2 2x2 - a2 dx = 2x2 - a2 2x 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 2 L 2x - a dx x a 1 = a sec-1 ` a ` + C = a cos-1 ` x ` + C 2 x x a L 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 L n-1 sin ax cos ax n - + n na 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 cosn - ax dx L cos(a + b)x cos(a - b)x 69 (a)  sin ax cos bx dx = + C, a2 ≠ b2 2(a + b) 2(a - b) L 68 cosn - ax sin ax n - + n na 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 sin ax L 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_THOM9799_13_SE_BTI.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) 04/04/16 4:34 PM T-4 77 78 79 81 82 A Brief Table of Integrals dx b - c p ax -2 = tan-1 c tan a - b d + C, b2 c2 2 b + c sin ax b + c A a 2b - c L c + b sin ax + 2c2 - b2 cos ax dx -1 = ln ` ` + C, b + c sin ax L b + c sin ax a 2c2 - b2 dx p ax = - a tan a - b + C L + sin ax 80 b2 c2 dx p ax = a tan a + b + C L - sin ax ax dx b - c = tan-1 c tan d + C, b2 c2 Ab + c L b + c cos ax a 2b2 - c2 c + b cos ax + 2c2 - b2 sin ax dx = ln ` ` + C, b + c cos ax L b + c cos ax a 2c2 - b2 b2 c2 ax dx = a tan + C + cos ax L 84 ax dx = - a cot + C cos ax L 85 L x sin ax dx = 86 L x cos ax dx = 87 L xn 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 94 L 96 L 98 L 83 x sin ax - a cos ax + C a2 L xn - cos ax dx tan2 ax dx = a tan ax - x + C 93 L 95 L 97 L 99 L secn ax dx = 100 L cscn ax dx = - 101 L secn ax tan ax dx = tann ax dx = tan ax tann - ax dx, n ≠ a(n - 1) L n-1 sec ax dx = a ln sec ax + tan ax + C sec2 ax dx = a tan ax + C sec 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 ax tan ax n - secn - ax dx, n ≠ + a(n - 1) n - 1L n-2 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_THOM9799_13_SE_BTI.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 - a 2x xn + a xn + dx , n ≠ -1 tan-1 ax n + n + L + a2x2 04/04/16 4:34 PM www.elsolucionario.org A Brief Table of Integrals T-5 Exponential and Logarithmic Forms 109 L 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 a2 + b2 eax dx = a eax + C ax e (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 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 + L L x ln ax 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 - x L L 22ax - x2 dx = 22ax - x 2 n x - a a2 -1 x - a 22ax - x2 + sin a a b + C 2 (x - a)1 22ax - x2 na2 dx = 22ax - x2 2n - dx + n + n + 1L n (x - a)1 22ax - x2 22 - n n - dx dx 123 = + n-2 2 n (n 2)a (n 2)a L 22ax - x L 22ax - x2 (x + a)(2x - 3a) 22ax - x2 a3 -1 x - a 124 x 22ax - x2 dx = + sin a a b + C L 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 dx 2a - x = a sin-1 a a b - 22ax - x2 + C 128 = -a + C 2 A x L 22ax - x L x 22ax - x Hyperbolic Forms 129 L sinh ax dx = a cosh ax + C 131 L sinh2 ax dx = 133 L sinhn ax dx = Z05_THOM9799_13_SE_BTI.indd sinh 2ax x - + C 4a n-1 sinh ax cosh ax n - - n na L 130 L cosh ax dx = a sinh ax + C 132 L cosh2 ax dx = sinh 2ax x + + C 4a sinhn - ax dx, n ≠ 04/04/16 4:34 PM 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 xn n xn sinh ax dx = a cosh ax - a xn - cosh ax dx L L 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 coshn ax dx = coshn - ax sinh ax n - + n na L coshn - ax dx, n ≠ L L tanhn ax dx = - 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 ax csch ax dx = a ln + C 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 eax sinh bx dx = 154 L eax cosh bx dx = 143 Some Definite Integrals 155 157 L0 L0 n-1 sech ax ax n - + sechn - ax dx, n ≠ (n - 1)a n - 1L cschn - ax coth ax n - cschn - ax dx, n ≠ (n - 1)a n - 1L ax sechn ax + C, n ≠ na bx 152 Z05_THOM9799_13_SE_BTI.indd L cschn ax coth ax dx = - cschn ax + C, n ≠ na -bx 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 xn - 1e-x dx = Γ(n) = (n - 1)!, n sinn x dx = csch2 ax dx = - a coth ax + C n-2 q p>2 xn - sinh ax dx L0 p>2 cosn x dx = 156 L0 # # # g # (n - 1) # p , 2#4#6# g#n d # # # 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 04/04/16 4:35 PM www.elsolucionario.org Basic Formulas and Rules 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: y u x 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 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 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) = 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 + Trigonometric Functions y y y = sin x Degrees Radian Measure p 45 s u "2 C ir 90 "2 p p –p – p p p 3p 2p x Domain: (−∞, ∞) Range: [−1, 1] –p – p p p 3p 2p x Domain: (−∞, ∞) Range: [−1, 1] cl e Un 45 r y = cos x Radians it cir cle of r a di u y sr u s s r = = u or u = r , 180° = p radians p 30 "3 60 90 p p "3 The angles of two common triangles, in degrees and radians – 3p –p – p 2 p p 3p 2 x Domain: All real numbers except odd integer multiples of p͞2 Range: (−∞, ∞) y –p – p – 3p –p – p 2 p p 3p 2p Domain: x ≠ 0, ±p, ±2p, Range: (−∞, −1] ´ [1, ∞) p p 3p 2 x Domain: All real numbers except odd integer multiples of p͞2 Range: (−∞, −1] ´ [1, ∞) y x y = sec x y = csc x Z06_THOM9799_13_SE_BFR.indd y y = tan x y = cot x –p – p p p 3p 2p x Domain: x ≠ 0, ±p, ±2p, Range: (−∞, ∞) 05/04/16 5:15 PM Basic Formulas and Rules F-2 SERIES Tests for Convergence of Infinite Series The nth-Term Test: Unless an S 0, the series diverges Geometric series: g ar n 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 Taylor Series q = + x + x + g + x n + g = a x n, - 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 0x0 q q (-1)n - 1xn x2 xn x3 ,  -1 x … - g + (-1)n - n + g = a + n n=1 + x x3 x5 x2n + x2n + + + g + = tanh-1 x = 2ax + + gb = a , - x 2n + n = 2n + q tan-1 x = x - q (-1)nx2n + x3 x2n + x5 + - g + (-1)n + g = a ,   2n + n = 2n + Binomial Series 0x0 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 x 1, = + a a b xk, k=1 k (1 + x)m = + mx + where m a b = m, Z06_THOM9799_13_SE_BFR.indd m(m - 1) m a b = , 2! m(m - 1) g(m - k + 1) m a b = k! k for k Ú 05/04/16 5:15 PM www.elsolucionario.org F-3 Basic Formulas and Rules 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 ∇#F = Curl Laplacian The Fundamental Theorem of Line Integrals Part 1 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 = 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 2 If the integral is independent of the path from A to B, its value is B LA 0M 0N 0P + + 0x 0y 0z i j k ∇ * F = 0x 0y 04 0z M N P 2ƒ 2ƒ 2ƒ ∇ 2ƒ = + + 0x 0y 0z 0ƒ 0ƒ 0ƒ i + j + k 0x 0y 0z 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 = F F # T ds = F F # n 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 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_THOM9799_13_SE_BFR.indd 05/04/16 5:15 PM Basic Formulas and Rules F-4 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 If a ≠ 0:  a = 0, a0 = 1, 0a = For any number a: a # = # a = Laws of Exponents aman = am + n, (ab)m = ambm, (am)n = amn, If a ≠ 0, am = am - n, an a0 = 1, a-m = n am>n = 2am = am n m 12 a2 The Binomial Theorem  For any positive integer n, 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)n = an + nan - 1b +     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 = Z06_THOM9799_13_SE_BFR.indd b2 b 4a -b { 2b2 - 4ac 2a 05/04/16 5:14 PM www.elsolucionario.org F-5 Basic Formulas and Rules 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 b a a′ = b′ = c′ a b c A = bh Parallelogram a2 + b2 = c2 Trapezoid 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 Sphere V= Z06_THOM9799_13_SE_BFR.indd Bh B V = pr2h S = prs = Area of side V = 43 pr3, S = 4pr2 05/04/16 5:14 PM Basic Formulas and Rules F-6 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: P(x) P(c) = Q(c) x S c Q(x) xSc Product Rule: Constant Multiple Rule: lim (ƒ(x) # g(x)) = L # M xSc lim (k # ƒ(x)) = k # L xSc ƒ(x) L = , M≠0 M x S c g(x) lim Quotient Rule: lim 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 lim x 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 lim x S c ƒ(x) = L, then lim g(ƒ(x)) = g(L) xSc Z06_THOM9799_13_SE_BFR.indd 05/04/16 5:14 PM www.elsolucionario.org F-7 Basic Formulas and Rules 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 y - u d u dx dx Quotient: a b = dx y 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 Z06_THOM9799_13_SE_BFR.indd Inverse Trigonometric Functions d d 1 (sin-1 x) = (cos-1 x) = dx dx 21 - x 21 - x 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 2x - 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 dx 21 + x 2x - d d 1 (tanh-1 x) = (sech-1 x) = dx dx - x2 x 21 - x d d 1 (coth-1 x) = (csch-1 x) = dx dx - x x 21 + x 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 05/04/16 5:14 PM Basic Formulas and Rules F-8 INTEGRATION RULES General Formulas Zero: Order of Integration: Constant Multiples: Sums and Differences: La a Lb a La ƒ(x) dx = ƒ(x) dx = - La b b ƒ(x) dx b La b La b kƒ(x) dx = k ƒ(x) dx La -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 c ƒ # (b - a) b La g(x) dx ƒ(x) dx + ƒ(x) dx = 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 Additivity: 3a, b ƒ(x) Ú g(x) on Domination: 3a, b ƒ(x) Ú on … implies implies La La La 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 1 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 d ƒ(t) dt = ƒ(x) dxLa Part 2 If ƒ is continuous at every point of a, b4 and F is any antiderivative of ƒ on a, b4 , then F′(x) = La b ƒ(x) dx = F(b) - F(a) Substitution in Definite Integrals La Z06_THOM9799_13_SE_BFR.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 05/04/16 5:14 PM ... a, b, and c are the sides opposite the angles A, B, and C in a triangle, then sin A sin B sin C a = b = c Use the accompanying figures and the identity sin(p - u) = sin u, if required, to derive... www.pearsonglobaleditions /thomas The Thomas? ?? Calculus Web site contains the chapter on Second-Order Differential Equations, including odd-numbered answers, and provides the expanded historical biographies... course, including syntax and commands These manuals are available to qualified instructors through the Thomas? ?? Calculus Web site, www.pearsonglobaleditions /thomas, and MyMathLab WEB SITE www.pearsonglobaleditions/thomas