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Calculus A Complete Course NINTH EDITION 9780134154367_Calculus 1 05/12/16 3:09 pm

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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page vi October 14, 2016 ROBERT A. ADAMS University of British Columbia CHRISTOPHER ESSEX University of Western Ontario Calculus A Complete Course NINTH EDITION 9780134154367_Calculus 3 05/12/16 3:09 pm

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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page v October 14, 2016 Editorial dirEctor: Claudine O’Donnell MEdia dEvEloPEr: Kelli Cadet acquisitions Editor: Claudine O’Donnell coMPositor: Robert Adams MarkEting ManagEr: Euan White PrEflight sErvicEs: Cenveo® Publisher Services PrograM ManagEr: Kamilah Reid-Burrell PErMissions ProjEct ManagEr: Joanne Tang ProjEct ManagEr: Susan Johnson intErior dEsignEr: Anthony Leung Production Editor: Leanne Rancourt covEr dEsignEr: Anthony Leung ManagEr of contEnt dEvEloPMEnt: Suzanne Schaan covEr iMagE: © Hiroshi Watanabe / Getty Images dEvEloPMEntal Editor: Charlotte Morrison-Reed vicE-PrEsidEnt, Cross Media and Publishing Services: MEdia Editor: Charlotte Morrison-Reed Gary Bennett Pearson Canada Inc., 26 Prince Andrew Place, Don Mills, Ontario M3C 2T8. Copyright © 2018, 2013, 2010 Pearson Canada 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, please contact Pearson Canada’s Rights and Permissions Department by visiting www. pearsoncanada.ca/contact-information/permissions-requests. Attributions of third-party content appear on the appropriate page within the text. PEARSON is an exclusive trademark owned by Pearson Canada Inc. or its afliates in Canada and/or other countries. Unless otherwise indicated herein, any third party trademarks that may appear in this work are the property of their respec- tive owners and any references to third party trademarks, logos, or other trade dress are for demonstrative or descriptive pur- To Noreen and Sheran poses only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson Canada products by the owners of such marks, or any relationship between the owner and Pearson Canada or its afliates, authors, licensees, or distributors. ISBN 978-0-13-415436-7 10 9 8 7 6 5 4 3 2 1 Library and Archives Canada Cataloguing in Publication Adams, Robert A. (Robert Alexander), 1940-, author Calculus : a complete course / Robert A. Adams, Christopher Essex. -- Ninth edition. Includes index. ISBN 978-0-13-415436-7 (hardback) 1. Calculus--Textbooks. I. Essex, Christopher, author II. Title. QA303.2.A33 2017 515 C2016-904267-7 Co9m7p8le0t1e3 C41o5u4rs3e6_7t_eCxta_lcpu_lutes m 4plate_8-25x10-875.indd 1 16/162/126/ 1 62 : 2 21: 5p3m pm

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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page vi October 14, 2016 ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page v October 14, 2016 To Noreen and Sheran 9780134154367_Calculus 5 05/12/16 3:09 pm

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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page vi October 14, 2016 ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page vii October 14, 2016 vii Contents Preface xv Maple Calculations 54 To the Student xvii Trigonometry Review 55 To the Instructor xviii Acknowledgments xix 1 Limits and Continuity 59 What Is Calculus? 1 1.1 Examples of Velocity, Growth Rate, and 59 P Preliminaries 3 Area P.1 Real Numbers and the Real Line 3 Average Velocity and Instantaneous 59 Intervals 5 Velocity The Absolute Value 8 The Growth of an Algal Culture 61 Equations and Inequalities Involving 9 The Area of a Circle 62 Absolute Values 1.2 Limits of Functions 64 P.2 Cartesian Coordinates in the Plane 11 One-Sided Limits 68 Axis Scales 11 Rules for Calculating Limits 69 Increments and Distances 12 The Squeeze Theorem 69 Graphs 13 1.3 Limits at Inﬁnity and Inﬁnite Limits 73 Straight Lines 13 Limits at Inﬁnity 73 Equations of Lines 15 Limits at Inﬁnity for Rational Functions 74 P.3 Graphs of Quadratic Equations 17 Inﬁnite Limits 75 Circles and Disks 17 Using Maple to Calculate Limits 77 Equations of Parabolas 19 1.4 Continuity 79 Reﬂective Properties of Parabolas 20 Continuity at a Point 79 Scaling a Graph 20 Continuity on an Interval 81 Shifting a Graph 20 There Are Lots of Continuous Functions 81 Ellipses and Hyperbolas 21 Continuous Extensions and Removable 82 P.4 Functions and Their Graphs 23 Discontinuities The Domain Convention 25 Continuous Functions on Closed, Finite 83 Graphs of Functions 26 Intervals Finding Roots of Equations 85 Even and Odd Functions; Symmetry and 28 Reﬂections 1.5 The Formal Deﬁnition of Limit 88 Reﬂections in Straight Lines 29 Using the Deﬁnition of Limit to Prove 90 Deﬁning and Graphing Functions with 30 Theorems Maple Other Kinds of Limits 90 Chapter Review 93 P.5 Combining Functions to Make New 33 Functions Sums, Differences, Products, Quotients, 33 2 Differentiation 95 and Multiples 2.1 Tangent Lines and Their Slopes 95 Composite Functions 35 Normals 99 Piecewise Deﬁned Functions 36 2.2 The Derivative 100 P.6 Polynomials and Rational Functions 39 Some Important Derivatives 102 Roots, Zeros, and Factors 41 Leibniz Notation 104 Roots and Factors of Quadratic 42 Differentials 106 Polynomials Derivatives Have the Intermediate-Value 107 Miscellaneous Factorings 44 Property P.7 The Trigonometric Functions 46 2.3 Differentiation Rules 108 Some Useful Identities 48 Sums and Constant Multiples 109 Some Special Angles 49 The Product Rule 110 The Addition Formulas 51 The Reciprocal Rule 112 Other Trigonometric Functions 53 The Quotient Rule 113 9780134154367_Calculus 7 05/12/16 3:09 pm

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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page viii October 14, 2016 ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page ix October 14, 2016 viii ix 2.4 The Chain Rule 116 Interest on Investments 188 4.9 Linear Approximations 269 Other Inverse Substitutions 353 Finding Derivatives with Maple 119 Logistic Growth 190 Approximating Values of Functions 270 The tan(/2) Substitution 354 Building the Chain Rule into Differentiation 119 Error Analysis 271 3.5 The Inverse Trigonometric Functions 192 6.4 Other Methods for Evaluating Integrals 356 Formulas The Inverse Sine (or Arcsine) Function 192 4.10 Taylor Polynomials 275 The Method of Undetermined Coefﬁcients 357 Proof of the Chain Rule (Theorem 6) 120 The Inverse Tangent (or Arctangent) 195 Taylor’s Formula 277 Using Maple for Integration 359 2.5 Derivatives of Trigonometric Functions 121 Function Big-O Notation 280 Using Integral Tables 360 Some Special Limits 121 Other Inverse Trigonometric Functions 197 Evaluating Limits of Indeterminate Forms 282 Special Functions Arising from Integrals 361 The Derivatives of Sine and Cosine 123 The Derivatives of the Other Trigonometric 125 3.6 Hyperbolic Functions 200 4.11 Roundoff Error, Truncation Error, and 284 6.5 Improper Integrals 363 Functions Inverse Hyperbolic Functions 203 Computers Improper Integrals of Type I 363 3.7 Second-Order Linear DEs with Constant 206 Taylor Polynomials in Maple 284 Improper Integrals of Type II 365 2.6 Higher-Order Derivatives 127 Coefﬁcients Persistent Roundoff Error 285 Estimating Convergence and Divergence 368 Recipe for Solving ay” + by’ + cy = 0 206 Truncation, Roundoff, and Computer 286 6.6 The Trapezoid and Midpoint Rules 371 2.7 Using Differentials and Derivatives 131 Simple Harmonic Motion 209 Algebra The Trapezoid Rule 372 Approximating Small Changes 131 Damped Harmonic Motion 212 Chapter Review 287 The Midpoint Rule 374 Average and Instantaneous Rates of 133 Chapter Review 213 Error Estimates 375 Change Sensitivity to Change 134 5 Integration 291 6.7 Simpson’s Rule 378 4 More Applications of 216 Derivatives in Economics 135 5.1 Sums and Sigma Notation 291 2.8 The Mean-Value Theorem 138 Differentiation Evaluating Sums 293 6.8 Other Aspects of Approximate Integration 382 Increasing and Decreasing Functions 140 Approximating Improper Integrals 383 Proof of the Mean-Value Theorem 142 4.1 Related Rates 216 5.2 Areas as Limits of Sums 296 Using Taylor’s Formula 383 Procedures for Related-Rates Problems 217 The Basic Area Problem 297 Romberg Integration 384 2.9 Implicit Differentiation 145 4.2 Finding Roots of Equations 222 Some Area Calculations 298 The Importance of Higher-Order Methods 387 Higher-Order Derivatives 148 Discrete Maps and Fixed-Point Iteration 223 Other Methods 388 The General Power Rule 149 5.3 The Deﬁnite Integral 302 Newton’s Method 225 Chapter Review 389 Partitions and Riemann Sums 302 2.10 Antiderivatives and Initial-Value Problems 150 “Solve” Routines 229 The Deﬁnite Integral 303 Antiderivatives 150 4.3 Indeterminate Forms 230 General Riemann Sums 305 7 Applications of Integration 393 The Indeﬁnite Integral 151 l’Ho^pital’s Rules 231 Differential Equations and Initial-Value 153 5.4 Properties of the Deﬁnite Integral 307 Problems 4.4 Extreme Values 236 A Mean-Value Theorem for Integrals 310 7.1 Volumes by Slicing—Solids of Revolution 393 Maximum and Minimum Values 236 Deﬁnite Integrals of Piecewise Continuous 311 Volumes by Slicing 394 2.11 Velocity and Acceleration 156 Critical Points, Singular Points, and 237 Functions Solids of Revolution 395 Velocity and Speed 156 Endpoints Cylindrical Shells 398 Acceleration 157 5.5 The Fundamental Theorem of Calculus 313 Falling Under Gravity 160 Finding Absolute Extreme Values 238 7.2 More Volumes by Slicing 402 The First Derivative Test 238 Chapter Review 163 Functions Not Deﬁned on Closed, Finite 240 5.6 The Method of Substitution 319 Intervals Trigonometric Integrals 323 7.3 Arc Length and Surface Area 406 3 Transcendental Functions 166 Arc Length 406 4.5 Concavity and Inﬂections 242 5.7 Areas of Plane Regions 327 The Arc Length of the Graph of a 407 3.1 Inverse Functions 166 The Second Derivative Test 245 Areas Between Two Curves 328 Function Inverting Non–One-to-One Functions 170 4.6 Sketching the Graph of a Function 248 Chapter Review 331 Areas of Surfaces of Revolution 410 Derivatives of Inverse Functions 170 Asymptotes 247 7.4 Mass, Moments, and Centre of Mass 413 3.2 Exponential and Logarithmic Functions 172 Examples of Formal Curve Sketching 251 6 Techniques of Integration 334 Mass and Density 413 Exponentials 172 4.7 Graphing with Computers 256 Moments and Centres of Mass 416 Logarithms 173 6.1 Integration by Parts 334 Numerical Monsters and Computer 256 Two- and Three-Dimensional Examples 417 Reduction Formulas 338 3.3 The Natural Logarithm and Exponential 176 Graphing 7.5 Centroids 420 The Natural Logarithm 176 Floating-Point Representation of Numbers 257 6.2 Integrals of Rational Functions 340 Pappus’s Theorem 423 The Exponential Function 179 in Computers Linear and Quadratic Denominators 341 General Exponentials and Logarithms 181 Machine Epsilon and Its Effect on 259 Partial Fractions 343 7.6 Other Physical Applications 425 Logarithmic Differentiation 182 Figure 4.45 Completing the Square 345 Hydrostatic Pressure 426 3.4 Growth and Decay 185 Determining Machine Epsilon 260 Denominators with Repeated Factors 346 Work 427 Potential Energy and Kinetic Energy 430 The Growth of Exponentials and 185 4.8 Extreme-Value Problems 261 6.3 Inverse Substitutions 349 Logarithms Procedure for Solving Extreme-Value 263 The Inverse Trigonometric Substitutions 349 7.7 Applications in Business, Finance, and 432 Exponential Growth and Decay Models 186 Problems Inverse Hyperbolic Substitutions 352 Ecology 9780134154367_Calculus 8 05/12/16 3:09 pm

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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page viii October 14, 2016 ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page ix October 14, 2016 viii ix 2.4 The Chain Rule 116 Interest on Investments 188 4.9 Linear Approximations 269 Other Inverse Substitutions 353 Finding Derivatives with Maple 119 Logistic Growth 190 Approximating Values of Functions 270 The tan(/2) Substitution 354 Building the Chain Rule into Differentiation 119 Error Analysis 271 3.5 The Inverse Trigonometric Functions 192 6.4 Other Methods for Evaluating Integrals 356 Formulas The Inverse Sine (or Arcsine) Function 192 4.10 Taylor Polynomials 275 The Method of Undetermined Coefﬁcients 357 Proof of the Chain Rule (Theorem 6) 120 The Inverse Tangent (or Arctangent) 195 Taylor’s Formula 277 Using Maple for Integration 359 2.5 Derivatives of Trigonometric Functions 121 Function Big-O Notation 280 Using Integral Tables 360 Some Special Limits 121 Other Inverse Trigonometric Functions 197 Evaluating Limits of Indeterminate Forms 282 Special Functions Arising from Integrals 361 The Derivatives of Sine and Cosine 123 The Derivatives of the Other Trigonometric 125 3.6 Hyperbolic Functions 200 4.11 Roundoff Error, Truncation Error, and 284 6.5 Improper Integrals 363 Functions Inverse Hyperbolic Functions 203 Computers Improper Integrals of Type I 363 3.7 Second-Order Linear DEs with Constant 206 Taylor Polynomials in Maple 284 Improper Integrals of Type II 365 2.6 Higher-Order Derivatives 127 Coefﬁcients Persistent Roundoff Error 285 Estimating Convergence and Divergence 368 Recipe for Solving ay” + by’ + cy = 0 206 Truncation, Roundoff, and Computer 286 6.6 The Trapezoid and Midpoint Rules 371 2.7 Using Differentials and Derivatives 131 Simple Harmonic Motion 209 Algebra The Trapezoid Rule 372 Approximating Small Changes 131 Damped Harmonic Motion 212 Chapter Review 287 The Midpoint Rule 374 Average and Instantaneous Rates of 133 Chapter Review 213 Error Estimates 375 Change Sensitivity to Change 134 5 Integration 291 6.7 Simpson’s Rule 378 4 More Applications of 216 Derivatives in Economics 135 5.1 Sums and Sigma Notation 291 2.8 The Mean-Value Theorem 138 Differentiation Evaluating Sums 293 6.8 Other Aspects of Approximate Integration 382 Increasing and Decreasing Functions 140 Approximating Improper Integrals 383 Proof of the Mean-Value Theorem 142 4.1 Related Rates 216 5.2 Areas as Limits of Sums 296 Using Taylor’s Formula 383 Procedures for Related-Rates Problems 217 The Basic Area Problem 297 Romberg Integration 384 2.9 Implicit Differentiation 145 4.2 Finding Roots of Equations 222 Some Area Calculations 298 The Importance of Higher-Order Methods 387 Higher-Order Derivatives 148 Discrete Maps and Fixed-Point Iteration 223 Other Methods 388 The General Power Rule 149 5.3 The Deﬁnite Integral 302 Newton’s Method 225 Chapter Review 389 Partitions and Riemann Sums 302 2.10 Antiderivatives and Initial-Value Problems 150 “Solve” Routines 229 The Deﬁnite Integral 303 Antiderivatives 150 4.3 Indeterminate Forms 230 General Riemann Sums 305 7 Applications of Integration 393 The Indeﬁnite Integral 151 l’Ho^pital’s Rules 231 Differential Equations and Initial-Value 153 5.4 Properties of the Deﬁnite Integral 307 Problems 4.4 Extreme Values 236 A Mean-Value Theorem for Integrals 310 7.1 Volumes by Slicing—Solids of Revolution 393 Maximum and Minimum Values 236 Deﬁnite Integrals of Piecewise Continuous 311 Volumes by Slicing 394 2.11 Velocity and Acceleration 156 Critical Points, Singular Points, and 237 Functions Solids of Revolution 395 Velocity and Speed 156 Endpoints Cylindrical Shells 398 Acceleration 157 5.5 The Fundamental Theorem of Calculus 313 Falling Under Gravity 160 Finding Absolute Extreme Values 238 7.2 More Volumes by Slicing 402 The First Derivative Test 238 Chapter Review 163 Functions Not Deﬁned on Closed, Finite 240 5.6 The Method of Substitution 319 Intervals Trigonometric Integrals 323 7.3 Arc Length and Surface Area 406 3 Transcendental Functions 166 Arc Length 406 4.5 Concavity and Inﬂections 242 5.7 Areas of Plane Regions 327 The Arc Length of the Graph of a 407 3.1 Inverse Functions 166 The Second Derivative Test 245 Areas Between Two Curves 328 Function Inverting Non–One-to-One Functions 170 4.6 Sketching the Graph of a Function 248 Chapter Review 331 Areas of Surfaces of Revolution 410 Derivatives of Inverse Functions 170 Asymptotes 247 7.4 Mass, Moments, and Centre of Mass 413 3.2 Exponential and Logarithmic Functions 172 Examples of Formal Curve Sketching 251 6 Techniques of Integration 334 Mass and Density 413 Exponentials 172 4.7 Graphing with Computers 256 Moments and Centres of Mass 416 Logarithms 173 6.1 Integration by Parts 334 Numerical Monsters and Computer 256 Two- and Three-Dimensional Examples 417 Reduction Formulas 338 3.3 The Natural Logarithm and Exponential 176 Graphing 7.5 Centroids 420 The Natural Logarithm 176 Floating-Point Representation of Numbers 257 6.2 Integrals of Rational Functions 340 Pappus’s Theorem 423 The Exponential Function 179 in Computers Linear and Quadratic Denominators 341 General Exponentials and Logarithms 181 Machine Epsilon and Its Effect on 259 Partial Fractions 343 7.6 Other Physical Applications 425 Logarithmic Differentiation 182 Figure 4.45 Completing the Square 345 Hydrostatic Pressure 426 3.4 Growth and Decay 185 Determining Machine Epsilon 260 Denominators with Repeated Factors 346 Work 427 Potential Energy and Kinetic Energy 430 The Growth of Exponentials and 185 4.8 Extreme-Value Problems 261 6.3 Inverse Substitutions 349 Logarithms Procedure for Solving Extreme-Value 263 The Inverse Trigonometric Substitutions 349 7.7 Applications in Business, Finance, and 432 Exponential Growth and Decay Models 186 Problems Inverse Hyperbolic Substitutions 352 Ecology 9780134154367_Calculus 9 05/12/16 3:09 pm

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