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Thomas Calculus không chỉ giúp học sinh đạt được mức độ thành thạo trong toán học, mà còn hỗ trợ cho sinh viên, những người cần Giải tích. Thomas Calculus giải thích rõ ràng và trực quan các ứng dụng hiện tại và khái niệm tổng quát của Giải tích. Trong phiên bản thứ 14, đồng tác giả mới Christopher Heil (Viện Công nghệ Georgia) hợp tác với tác giả Joel Hass để lưu giữ những gì tốt nhất về văn phong được thử nghiệm theo thời gian của Thomas, đồng thời xem xét từng từ và các tác phẩm khác với các sinh viên ngày nay. Kết quả thu được là một cuốn sách vượt ra ngoài việc ghi nhớ các công thức và quy trình thường quy, giúp học sinh khái quát các khái niệm chính và phát triển tư duy sâu sắc.
THOMAS’ HASS • HEIL • WEIR FOURTEENTH EDITION CALCULUS THOMAS’ CALCULUS FOURTEENTH EDITION Based on the original work by GEORGE B THOMAS, JR Massachusetts Institute of Technology as revised by JOEL HASS University of California, Davis CHRISTOPHER HEIL Georgia Institute of Technology MAURICE D WEIR Naval Postgraduate School Director, Portfolio Management: Deirdre Lynch Executive Editor: Jef Weidenaar Editorial Assistant: Jennifer Snyder Content Producer: Rachel S Reeve Managing Producer: Scott Disanno Producer: Stephanie Green TestGen Content Manager: Mary Durnwald Manager: Content Development, Math: Kristina Evans Product Marketing Manager: Emily Ockay Field Marketing Manager: Evan St Cyr Marketing Assistants: Jennifer Myers, Erin Rush Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Project Manager: Gina M Cheselka Manufacturing Buyer: Carol Melville, LSC Communications, Inc Program Design Lead: Barbara T Atkinson Associate Director of Design: Blair Brown Text and Cover Design, Production Coordination, Composition: Cenveo® Publisher Services Illustrations: Network Graphics, Cenveo® Publisher Services Cover Image: Te Rewa Rewa Bridge, Getty Images/Kanwal Sandhu Copyright © 2018, 2014, 2010 by Pearson Education, Inc All Rights Reserved Printed in the United States of America This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/ Attributions of third party content appear on page C-1, which constitutes an extension of this copyright page PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks owned by Pearson Education, Inc or its ailiates in the U.S and/or other countries Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its ailiates, authors, licensees or distributors Library of Congress Cataloging-in-Publication Data Names: Hass, Joel | Heil, Christopher, 1960- | Weir, Maurice D Title: Thomas’ calculus / based on the original work by George B Thomas, Jr., Massachusetts Institute of Technology, as revised by Joel Hass, University of California, Davis, Christopher Heil, Georgia Institute of Technology, Maurice D Weir, Naval Postgraduate School Description: Fourteenth edition | Boston : Pearson, [2018] | Includes index Identiiers: LCCN 2016055262 | ISBN 9780134438986 | ISBN 0134438981 Subjects: LCSH: Calculus Textbooks | Geometry, Analytic Textbooks Classiication: LCC QA303.2.W45 2018 | DDC 515 dc23 LC record available at https://lccn.loc.gov/2016055262 17 Instructor’s Edition ISBN 13: 978-0-13-443909-9 ISBN 10: 0-13-443909-0 Student Edition ISBN 13: 978-0-13-443898-6 ISBN 10: 0-13-443898-1 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 33 Practice Exercises 34 Additional and Advanced Exercises 35 Technology Application Projects 37 Limits and Continuity 2.1 2.2 2.3 2.4 2.5 2.6 38 Rates of Change and Tangent Lines to Curves 38 Limit of a Function and Limit Laws 45 The Precise Definition of a Limit 56 One-Sided Limits 65 Continuity 72 Limits Involving Infinity; Asymptotes of Graphs 83 Questions to Guide Your Review 96 Practice Exercises 97 Additional and Advanced Exercises 98 Technology Application Projects 101 Derivatives 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 14 102 Tangent Lines and the Derivative at a Point 102 The Derivative as a Function 106 Differentiation Rules 115 The Derivative as a Rate of Change 124 Derivatives of Trigonometric Functions 134 The Chain Rule 140 Implicit Differentiation 148 Related Rates 153 Linearization and Differentials 162 Questions to Guide Your Review 174 Practice Exercises 174 Additional and Advanced Exercises 179 Technology Application Projects 182 iii iv Contents Applications of Derivatives 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Extreme Values of Functions on Closed Intervals 183 The Mean Value Theorem 191 Monotonic Functions and the First Derivative Test 197 Concavity and Curve Sketching 202 Applied Optimization 214 Newton’s Method 226 Antiderivatives 231 Questions to Guide Your Review 241 Practice Exercises 241 Additional and Advanced Exercises 244 Technology Application Projects 247 Integrals 5.1 5.2 5.3 5.4 5.5 5.6 183 248 Area and Estimating with Finite Sums 248 Sigma Notation and Limits of Finite Sums 258 The Definite Integral 265 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 307 Additional and Advanced Exercises 310 Technology Application Projects 313 Applications of Definite Integrals 6.1 6.2 6.3 6.4 6.5 6.6 314 Volumes Using Cross-Sections 314 Volumes Using Cylindrical Shells 325 Arc Length 333 Areas of Surfaces of Revolution 338 Work and Fluid Forces 344 Moments and Centers of Mass 353 Questions to Guide Your Review 365 Practice Exercises 366 Additional and Advanced Exercises 368 Technology Application Projects 369 Transcendental Functions 7.1 7.2 7.3 7.4 7.5 7.6 7.7 296 370 Inverse Functions and Their Derivatives 370 Natural Logarithms 378 Exponential Functions 386 Exponential Change and Separable Differential Equations Indeterminate Forms and L’Hôpital’s Rule 407 Inverse Trigonometric Functions 416 Hyperbolic Functions 428 397 Contents 7.8 Relative Rates of Growth 436 Questions to Guide Your Review 441 Practice Exercises 442 Additional and Advanced Exercises 445 Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 447 Using Basic Integration Formulas 447 Integration by Parts 452 Trigonometric Integrals 460 Trigonometric Substitutions 466 Integration of Rational Functions by Partial Fractions 471 Integral Tables and Computer Algebra Systems 479 Numerical Integration 485 Improper Integrals 494 Probability 505 Questions to Guide Your Review 518 Practice Exercises 519 Additional and Advanced Exercises 522 Technology Application Projects 525 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 10.10 526 Solutions, Slope Fields, and Euler’s Method 526 First-Order Linear Equations 534 Applications 540 Graphical Solutions of Autonomous Equations 546 Systems of Equations and Phase Planes 553 Questions to Guide Your Review 559 Practice Exercises 559 Additional and Advanced Exercises 561 Technology Application Projects 562 Infinite Sequences and Series 563 Sequences 563 Infinite Series 576 The Integral Test 586 Comparison Tests 592 Absolute Convergence; The Ratio and Root Tests 597 Alternating Series and Conditional Convergence 604 Power Series 611 Taylor and Maclaurin Series 622 Convergence of Taylor Series 627 Applications of Taylor Series 634 Questions to Guide Your Review 643 Practice Exercises 644 Additional and Advanced Exercises 646 Technology Application Projects 648 v vi Contents 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 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 Parametric Equations and Polar Coordinates 649 Parametrizations of Plane Curves 649 Calculus with Parametric Curves 658 Polar Coordinates 667 Graphing Polar Coordinate Equations 671 Areas and Lengths in Polar Coordinates 675 Conic Sections 680 Conics in Polar Coordinates 688 Questions to Guide Your Review 694 Practice Exercises 695 Additional and Advanced Exercises 697 Technology Application Projects 699 Vectors and the Geometry of Space 700 Three-Dimensional Coordinate Systems 700 Vectors 705 The Dot Product 714 The Cross Product 722 Lines and Planes in Space 728 Cylinders and Quadric Surfaces 737 Questions to Guide Your Review 743 Practice Exercises 743 Additional and Advanced Exercises 745 Technology Application Projects 748 Vector-Valued Functions and Motion in Space Curves in Space and Their Tangents 749 Integrals of Vector Functions; Projectile Motion 758 Arc Length in Space 767 Curvature and Normal Vectors of a Curve 771 Tangential and Normal Components of Acceleration 777 Velocity and Acceleration in Polar Coordinates 783 Questions to Guide Your Review 787 Practice Exercises 788 Additional and Advanced Exercises 790 Technology Application Projects 791 749 Contents 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 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 Partial Derivatives 792 Functions of Several Variables 792 Limits and Continuity in Higher Dimensions 800 Partial Derivatives 809 The Chain Rule 821 Directional Derivatives and Gradient Vectors 831 Tangent Planes and Differentials 839 Extreme Values and Saddle Points 849 Lagrange Multipliers 858 Taylor’s Formula for Two Variables 868 Partial Derivatives with Constrained Variables 872 Questions to Guide Your Review 876 Practice Exercises 877 Additional and Advanced Exercises 880 Technology Application Projects 882 Multiple Integrals 883 Double and Iterated Integrals over Rectangles 883 Double Integrals over General Regions 888 Area by Double Integration 897 Double Integrals in Polar Form 900 Triple Integrals in Rectangular Coordinates 907 Applications 917 Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 939 Questions to Guide Your Review 949 Practice Exercises 949 Additional and Advanced Exercises 952 Technology Application Projects 954 Integrals and Vector Fields 927 955 Line Integrals of Scalar Functions 955 Vector Fields and Line Integrals: Work, Circulation, and Flux 962 Path Independence, Conservative Fields, and Potential Functions 975 Green’s Theorem in the Plane 986 Surfaces and Area 998 Surface Integrals 1008 Stokes’ Theorem 1018 The Divergence Theorem and a Unified Theory 1031 Questions to Guide Your Review 1044 Practice Exercises 1044 Additional and Advanced Exercises 1047 Technology Application Projects 1048 vii viii Contents 17 17.1 17.2 17.3 17.4 17.5 Second-Order Differential Equations 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 (Online at www.goo.gl/MgDXPY) AP-1 Real Numbers and the Real Line AP-1 Mathematical Induction AP-6 Lines, Circles, and Parabolas AP-9 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-34 The Mixed Derivative Theorem and the Increment Theorem AP-35 Answers to Odd-Numbered Exercises Applications Index Subject Index AI-1 I-1 A Brief Table of Integrals Credits C-1 T-1 A-1 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 + (b) a dx + C L x 2ax + b 31 dx ax - b = tan-1 + C A b 2b L x 2ax - b 2ax + b a dx dx = + C bx 2b L x 2ax + b L x2 2ax + b Forms Involving a + x 32 34 35 36 37 38 39 x dx = tan-1 + C 2 a a a + x L dx x = sinh-1 a + C = ln x + 2a2 + x2 + C L 2a + x L L L L 2a2 + x2 dx = x 2a2 + x2 dx = x x + tan-1 + C a 2a 2a2 ( a2 + x2 ) x a4 ( a + 2x2 ) 2a2 + x2 ln x + 2a2 + x2 + C 8 dx = 2a2 + x2 - a ln ` a + 2a2 + x2 ` + C x dx = ln x + 2a2 + x2 - 2a2 + x2 + C x 2a2 + x2 x2 a2 dx = x + 2a2 + x2 + ln + C 2 L 2a2 + x2 x2 x a + 2a2 + x2 dx ln = ` ` + C x a L x 2a2 + x2 41 42 dx x + a = ln x - a + C 2 2a La - x 43 44 x dx = sin-1 + C a L 2a - x 45 40 2 L (a + x ) x a2 2a2 + x2 + ln x + 2a2 + x2 + C 2 x2 2a2 + x2 dx = 2a2 + x2 33 2a2 + x2 dx + C = a2x L x2 2a2 + x2 Forms Involving a − x 46 47 49 51 L L x2 2a2 - x2 dx = 2a2 - x2 x a4 -1 x - x 2a2 - x2 ( a2 - 2x2 ) + C sin a dx = 2a2 - x2 - a ln ` a + 2a2 - x2 ` + C x a2 -1 x x2 - x 2a2 - x2 + C dx = sin 2 a L 2a - x 2a2 - x2 dx + C = 2 a2x L x 2a - x dx = ln x + 2x2 - a2 + C L 2x - a2 Forms Involving x − a 52 53 L 2x2 - a2 dx = x a2 2x2 - a2 ln x + 2x2 - a2 + C 2 x dx x + a + ln x - a + C = 2 2 4a 2a ( a - x2 ) L (a - x ) L 2a2 - x2 dx = 48 50 L 2a2 - x2 x x a2 -1 x + C 2a2 - x2 + sin 2 a dx = -sin-1 x 2a2 - x2 + C x a a + 2a2 - x2 dx = - ln ` ` + C x 2 a L x 2a - x 54 55 56 57 58 59 60 L 2x2 - a2 dx = L x1 2x2 - a2 2n dx = A Brief Table of Integrals x1 2x2 - a2 2n na2 2x2 - a2 2n - dx, n ≠ -1 n + n + 1L x1 2x2 - a2 22 - n n - dx dx = n-2 , n ≠ 2 2 n (2 n)a (n 2)a x a 2 x - a2 L L L n x2 2x2 - a2 dx = 2x2 - a2 L x 2x2 - a2 L 2x2 - a2 2n + + C, n ≠ -2 n + x a4 ( 2x2 - a2 ) 2x2 - a2 ln x + 2x2 - a2 + C 8 x dx = 2x2 - a2 - a sec-1 ` a ` + C dx = ln x + 2x2 - a2 - 2x2 - a2 x + C a2 x x2 dx = ln x + 2x2 - a2 + 2x2 - a2 + C 2 2 L 2x - a x x a dx = sec-1 ` ` + C = cos-1 ` ` + C 2 a a a x L x 2x - a 62 63 L sin ax dx = - a cos ax + C 64 L cos ax dx = a sin ax + C 65 L sin2 ax dx = 66 L cos2 ax dx = 67 L sinn ax dx = - 68 L cosn ax dx = 61 2x2 - a2 dx = + C a2x L x2 2x2 - a2 Trigonometric Forms x sin 2ax + C 4a x sin 2ax + + C 4a sinn - ax cos ax n - + n sinn - ax dx na L cosn - ax sin ax n - + n cosn - ax dx na L 69 (a) L sin ax cos bx dx = - (b) L sin ax sin bx dx = (c) L cos ax cos bx dx = cos(a + b)x cos(a - b)x + C, a2 ≠ b2 2(a + b) 2(a - b) sin(a - b)x sin(a + b)x + C, a2 ≠ b2 2(a - b) 2(a + b) sin(a - b)x sin(a + b)x + + C, a2 ≠ b2 2(a - b) 2(a + b) 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 = - cos 2ax + C 4a sinn + ax + C, n ≠ -1 (n + 1)a 71 L sinn ax cos ax dx = 73 L cosn ax sin ax dx = - 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) T-3 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 L b + c sin ax a 2c - b dx p ax = - tan a - b + C a 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, 2 b + c cos ax L b + c cos ax a 2c - b b2 c2 83 ax dx = tan + C L + cos ax a 84 ax dx = - cot + C a L - cos ax 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 92 L 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 = x sin ax - cos ax + C a a tan ax dx = a ln sec ax + C 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 + sin ax + C a a L cot ax dx = a ln sin ax + C 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 Inverse Trigonometric Forms sin-1 ax dx = x sin-1 ax + a 21 - a2x2 + C 103 L 105 L tan-1 ax dx = x tan-1 ax - 106 L xn sin-1 ax dx = 104 L cscn ax cot ax dx = - cscn ax na + C, n ≠ cos-1 ax dx = x cos-1 ax - a 21 - a2x2 + C ln ( + a2x2 ) + C 2a a xn + xn + dx , n ≠ -1 sin-1 ax n + n + L 21 - a2x2 A Brief Table of Integrals 107 108 L xn cos-1 ax dx = xn + a xn + dx , n ≠ -1 cos-1 ax + n + n + L 21 - a2x2 L xn tan-1 ax dx = xn + a xn + dx , n ≠ -1 tan-1 ax n + n + L + a2x2 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 a + b2 117 L xn(ln ax)m dx = xn + 1(ln ax)m m xn(ln ax)m - dx, n ≠ -1 n + n + 1L 118 L x-1(ln ax)m dx = eax dx = a eax + C 110 L bax dx = 112 L n xneax dx = a xneax - a L ln ax dx = x ln ax - x + C ax e (ax - 1) + C a2 L 122 (ln ax)m + + C, m ≠ -1 m + L 123 124 125 126 127 22ax - x2 dx = 22ax xn - 1eax dx 116 119 dx x - a = sin-1 a a b + C L 22ax - x2 121 L xnbax n xn - 1bax dx, b 0, b ≠ a ln b a ln b L Forms Involving 22ax − x2, a + 120 bax + C, b 0, b ≠ a ln b dx = ln ln ax + C x ln ax L x - a a2 -1 x - a 22ax - x2 + sin a a b + C 2 - x2 dx = (x - a)1 22ax - x2 2n na2 22ax - x2 2n - dx + n + n + 1L (x - a)1 22ax - x2 22 - n n - dx dx = + n (n - 2)a2 (n - 2)a2 L 22ax - x2 2n - L 22ax - x2 L L L n x 22ax - x2 dx = 22ax - x2 x 22ax - x2 x2 (x + a)(2x - 3a) 22ax - x2 a3 -1 x - a + sin a a b + C 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 2a - x dx = + C aA x L x 22ax - x Hyperbolic Forms 129 L sinh ax dx = a cosh ax + C 131 L sinh2 ax dx = sinh 2ax x - + C 4a 130 L cosh ax dx = a sinh ax + C 132 L cosh2 ax dx = sinh 2ax x + + C 4a T-5 T-6 A Brief Table of Integrals L sinhn ax dx = sinhn - ax cosh ax n - - n na 134 L coshn ax dx = coshn - ax sinh ax n - + n na 135 x x sinh ax dx = a cosh ax - sinh ax + C a L 136 L x cosh ax dx = 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 = 133 157 L0 L0 L sinhn - ax dx, n ≠ coshn - ax dx, n ≠ xn - cosh ax dx sechn ax + C, n ≠ na 152 e-bx eax ebx c d + C, a2 ≠ b2 a + b a - b e dx = Γ(n) = (n - 1)!, n sin x dx = n L cschn ax coth ax dx = - xn - sinh ax dx cschn ax + C, n ≠ na e-bx eax ebx + c d + C, a2 ≠ b2 a + b a - b n - -x p>2 L cschn - ax coth ax n - cschn - ax dx, n ≠ (n - 1)a n - 1L q x x sinh ax - cosh ax + C a a sechn - ax ax n - sechn - ax dx, n ≠ + (n - 1)a n - 1L Some Definite Integrals 155 L L L0 p>2 cos x dx = d n 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 Basic Algebra Formulas Arithmetic Operations a#c ac = b d bd a(b + c) = ab + ac, a>b a d = # c>d b c c ad + bc a + = , b d bd Laws of Signs -a a a = - = b b -b -(-a) = a, Zero Division by zero is not deined If a ≠ 0: 0 a a = 0, a = 1, = For any number a: a # = # a = Laws of Exponents aman = am + n, (ab)m = ambm, (am)n = amn, am>n = 2am = n If a ≠ 0, then am = am - n, an a0 = 1, a-m = n m 12 a2 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 Diference of Like Integer Powers, n + an - bn = (a - b)(an - + an - 2b + an - 3b2 + g + ab n - + b n - 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, then ax2 + bx + c = au + C The Quadratic Formula If a ≠ and ax2 + bx + c = 0, then x = au = x + (b>2a), C = c - -b { 2b2 - 4ac 2a b2 b 4a Geometry Formulas A = area, B = area of base, C = circumference, S = 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 a2 + b2 = c2 Trapezoid 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 = 13 Bh B V = pr2h S = prs = Area of side V = 43 pr3, S = 4pr2 Trigonometry Formulas y Definitions and Fundamental Identities y sin u = r = csc u Sine: x cos u = r = sec u y tan u = x = cot u Cosine: Tangent: y u x tan A + tan B - tan A tan B tan (A - B) = tan A - tan B + tan A tan B sin aA - P(x, y) r tan (A + B) = sin aA + x 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 = 2 sin (A + B) = sin A cos B + cos A sin B cos2 u = 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 Degrees "2 "2 u 45 r p 90 p b = -sin A y = sin x p p p b = sin A y Radians 45 C ir cos aA + y Radian Measure –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 e Un cos aA - 1 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 (-u) = -sin u, cos (-u) = cos u p b = cos A, sin A sin B = Identities s p b = -cos A, it circ cle of radiu y sr u s s r = = u or u = r , 180° = p radians "3 60 90 y = sec x p 30 y y = tan x "3 p p 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: (−∞, ∞) – 3p –p – p 2 p p 3p 2 Domain: All real numbers except odd integer multiples of p͞2 Range: (−∞, −1] ´ [1, ∞) y y y = csc x y = cot x –p – p 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 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 0x0 Taylor Series = + x + x2 + g + xn + g = a xn, - x n=0 q = - x + x2 - g + (-x)n + g = a (-1)nxn, + x n=0 q x2 xn xn + g + + g = a , 2! n! n = n! q ex = + x + sin x = x cos x = - 0x0 q Series with some negative terms: Does g ͉ an ͉ converge? If yes, so does g an because absolute convergence implies convergence Alternating series: g an converges if the series satisies the conditions of the Alternating Series Test 0x0 0x0 q q (-1)nx2n + x2n + x3 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 q (-1)n - 1xn x2 x3 xn , + - g + (-1)n - n + g = a n n=1 -1 x … 1 + x x3 x5 x2n + x2n + + g + = tanh-1 x = 2ax + + + gb = a , - x 2n + n = 2n + q ln 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 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 = + a a b xk, k=1 k m a b = m, x 1, 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 ∇#F = i j k Curl ∇ * F = 0x 0y 04 0z M N P Laplacian ∇ 2ƒ = 0M 0N 0P + + 0x 0y 0z 2ƒ 2ƒ 2ƒ + + 0x2 0y 0z The Fundamental Theorem of Line Integrals Part Let F = M i + N j + P k be a vector ield whose components are continuous throughout an open connected region D in space Then there exists a diferentiable 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 B Part If the integral is independent of the path from A to B, its value is LA B F # dr = ƒ(B) - ƒ(A) Green’s Theorem and Its Generalization to Three Dimensions Tangential form of Green’s Theorem: F # T ds = C F # T ds = C F # n ds = C Stokes’ Theorem: Normal form of Green’s Theorem: (∇ * F) # k dA O (∇ * F) # n ds O (∇ # F) dA S C Divergence Theorem: O R C Vector Triple Products (u * v) # w = (v * w) # u = (w * u) # v u * (v * w) = (u # w)v - (u # v)w C O R F # n ds = S l ∇ # F dV D Vector Identities In the identities here, ƒ and g are diferentiable scalar functions; F, F1, and F2 are diferentiable vector ields; and a and b are real constants ∇ # (F1 * F2) = F2 # (∇ * F1) - F1 # (∇ * F2) ∇ * ( ∇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 + ( ∇ # F2 ) F1 - ( ∇ # F1 ) F2 ∇ * ( ∇ * F ) = ∇ ( ∇ # F ) - ( ∇ # ∇ ) F = ∇ ( ∇ # F ) - ∇ 2F ( ∇ * F ) * F = ( F # ∇ ) F - 12 ∇ ( F # F ) 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 and xSc lim g(x) = M, then 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 ƒ(x) L = , M≠0 M x S c g(x) lim Quotient Rule: 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 xSc sin x lim x = xS0 and lim xS0 - cos x = x then limxSc ƒ(x) = L L’Hôpital’s Rule Inequalities 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 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 ƒ(x) ƒ′(x) , = lim g(x) x S a g′(x) xSa lim assuming the limit on the right side exists Continuity If g is continuous at L and limxSc ƒ(x) = L, then lim g(ƒ(x)) = g(L) xSc Diferentiation Rules General Formulas Inverse Trigonometric Functions Assume u and y are differentiable functions of x Constant: d (c) = dx Sum: d du dy (u + y) = + dx dx dx Difference: d du dy (u - y) = dx dx dx Constant Multiple: d du (cu) = c dx dx Product: d dy du (uy) = u + y dx dx dx Quotient: d u a b = dx y y du dy - u dx dx y2 Power: d n x = nxn - dx Chain Rule: d (ƒ(g(x)) = ƒ′(g(x)) # g′(x) dx Trigonometric Functions d (sin x) = cos x dx d (cos x) = -sin x dx d (tan x) = sec2 x dx d (sec x) = sec x tan x dx d (cot x) = -csc2 x dx d (csc x) = -csc x cot x dx Exponential and Logarithmic Functions d x e = ex dx d ln x = dx x d x a = ax ln a dx d (loga x) = dx x ln a d (sin-1 x) = dx 21 - x2 d (tan-1 x) = dx + x2 d (cot-1 x) = dx + x2 Hyperbolic Functions d (cos-1 x) = dx 21 - x2 d (sec-1 x) = dx x 2x2 - d (csc-1 x) = dx x 2x2 - d (sinh x) = cosh x dx d (cosh x) = sinh x dx d (tanh x) = sech2 x dx d (sech x) = -sech x x dx d (coth x) = -csch2 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 (tanh-1 x) = dx - x2 d (coth-1 x) = dx - x2 Parametric Equations d (sech-1 x) = dx x 21 - x2 d (csch-1 x) = dx x 21 + x2 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 La Zero: a ƒ(x) dx = a Order of Integration: b ƒ(x) dx = - ƒ(x) dx Lb La b Constant Multiples: b kƒ(x) dx = k ƒ(x) dx, La La b k any number b -ƒ(x) dx = - ƒ(x) dx, La La Sums and Differences: Additivity: La b La b (ƒ(x) { g(x)) dx = ƒ(x) dx + Lb La k = -1 b ƒ(x) dx { c ƒ(x) dx = La La b g(x) dx c ƒ(x) dx Max-Min Inequality: If max ƒ and ƒ are the maximum and minimum values of ƒ on a, b4 , then ƒ # (b - a) … a, b4 ƒ(x) Ú g(x) on Domination: a, b4 ƒ(x) Ú on La b ƒ(x) dx … max ƒ # (b - a) implies implies La La b ƒ(x) dx Ú La b g(x) dx b ƒ(x) dx Ú The Fundamental Theorem of Calculus x Part If ƒ is continuous on a, b4 , then F(x) = 1a ƒ(t)dt is continuous on a, b4 and diferentiable on (a, b) and its derivative is ƒ(x): x F′(x) = d ƒ(t) dt = ƒ(x) dx La 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 b ƒ(g(x)) # g′(x) dx = Integration by Parts g(b) Lg(a) ƒ(u) du La b u(x) y′(x) dx = u(x) y(x) d - A Brief Table of Integrals follows the Index at the back of the text b a La b y(x) u′(x) dx Credits Frontmatter: Page ix, Pling/Shutterstock Chapter 1: Page 1, Chapter opening photo, Lebrecht Music and Arts Photo Library/ Alamy Stock Photo Chapter 2: Page 38, Chapter opening photo, Gui Jun Peng/Shutterstock Chapter 3: Page 102, Chapter opening photo, Yellowj/Shutterstock; Page 132, Exercise 19, PSSC Physics, 2nd ed., Reprinted by permission of Educational Development Center, Inc.; Page 177, Exercise 94, NCPMF “Differentiation” by W.U Walton et al., Project CALC Reprinted by permission of Educational Development Center, Inc Chapter 4: Page 183, Chapter opening photo, Carlos Castilla/Shutterstock Chapter 5: Page 248, Chapter opening photo, Patrick Pleul/Dpa picture alliance archive/Alamy Stock Photo Chapter 6: Page 314, Chapter opening photo, Pling/Shutterstock; Page 354, Figure 6.44, PSSC Physics, 2nd ed., Reprinted by permission of Education Development Center, Inc Chapter 7: Page 370, Chapter opening photo, Markus Gann/Shutterstock Chapter 8: Page 447, Chapter opening photo, Petr Petrovich/Shutterstock Chapter 9: Page 526, Chapter opening photo, Rich Carey/Shutterstock; Page 542, Table 9.3, U.S Bureau of the Census (Sept., 2007): www.census.gov/ipc/www/idb Chapter 10: Page 563, Chapter 10 opening photo, Fotomak/Shutterstock; Page 579, Figure 10.10, PSSC Physics, 2nd ed., Reprinted by permission of Educational Development Center, Inc Chapter 11: Page 649, Chapter 11 opening photo, Kjpargeter/Shutterstock Chapter 12: Page 700, Chapter 12 opening photo, Dudarev Mikhail/Shutterstock Chapter 13: Page 749, Chapter 13 opening photo, EPA European pressphoto agency b.v./Alamy Stock Photo; Page 766, Exercise 39, PSSC Physics, 2nd ed., Reprinted by permission of Educational Development Center, Inc Chapter 14: Page 792, Chapter 14 opening photo, Alberto Loyo/Shutterstock; Page 796, Figure 14.7, From Appalachian Mountain Club Copyright by Appalachian Mountain Club; Page 831, Figure 14.26, Yosemite National Park Map, U.S Geological Survey Chapter 15: Page 883, Chapter 15 opening photo, Viappy/Shutterstock Chapter 16: Page 955, Chapter 16 opening photo, Szefei/Shutterstock; Page 962, Figure 16.7, Reprinted by permission of Educational Development Center, Inc.; Page 962, Figure 16.8, Reprinted by permission of Educational Development Center, Inc Appendices: Page AP-1, Appendices opening photo, Lebrecht Music and Arts Photo Library/Alamy Stock Photo C-1 Get the Most Out of ® MyMathLab MyMathLab is the leading online homework, tutorial, and assessment program designed to help you learn and understand mathematics S Personalized and adaptive learning S Interactive practice with immediate feedback S Multimedia learning resources S Complete eText S Mobile-friendly design MyMathLab is available for this textbook To learn more, visit www.mymathlab.com www.pearsonhighered.com ISBN-13: 978-0-13-443898-6 ISBN-10: 0-13-443898-1 0 0 780134 438986 ... Cataloging-in-Publication Data Names: Hass, Joel | Heil, Christopher, 1960- | Weir, Maurice D Title: Thomas calculus / based on the original work by George B Thomas, Jr., Massachusetts Institute.. .THOMAS CALCULUS FOURTEENTH EDITION Based on the original work by GEORGE B THOMAS, JR Massachusetts Institute of Technology as revised... Massachusetts Institute of Technology, as revised by Joel Hass, University of California, Davis, Christopher Heil, Georgia Institute of Technology, Maurice D Weir, Naval Postgraduate School Description: Fourteenth