Calculus: A Complete Course

1176 Pages·2018·54.78 MB·English

by  Robert A. Adams, Christopher Essex

Calculus
A Complete Course
NINTH EDITION
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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
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ADAMS & ESSEX: Calculus: a Complete Course, 8th Edition. Front – page v October 14, 2016
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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

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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
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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
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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
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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
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