,
, TABLE OF CONTENTS
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 8
1.3 Trigonometric Functions 18
1.4 Graphing with Software 26
1.5 Exponential Functions 31
1.6 Inverse Functions and Logarithms 34
Practice Exercises 43
Additional and Advanced Exercises 52
2 Limits and Continuity 59
2.1 Rates of Change and Tangents to Curves 59
2.2 Limit of a Function and Limit Laws 62
2.3 The Precise Definition of a Limit 73
2.4 One-Sided Limits 81
2.5 Continuity 86
2.6 Limits Involving Infinity; Asymptotes of Graphs 92
Practice Exercises 102
Additional and Advanced Exercises 108
3 Differentiation 115
3.1 Tangents and the Derivative at a Point 115
3.2 The Derivative as a Function 121
3.3 Differentiation Rules 131
3.4 The Derivative as a Rate of Change 138
3.5 Derivatives of Trigonometric Functions 144
3.6 The Chain Rule 152
3.7 Implicit Differentiation 162
3.8 Derivatives of Inverse Functions and Logarithms 170
3.9 Inverse Trigonometric Functions 180
3.10 Related Rates 186
3.11 Linearization and Differentials 192
Practice Exercises 199
Additional and Advanced Exercises 214
, 4 Applications of Derivatives 219
4.1 Extreme Values of Functions 219
4.2 The Mean Value Theorem 233
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 253
4.5 Indeterminate Forms and L Hôpital s Rule 280
4.6 Applied Optimization 290
4.7 Newton's Method 304
4.8 Antiderivatives 309
Practice Exercises 318
Additional and Advanced Exercises 336
5 Integration 343
5.1 Area and Estimating with Finite Sums 343
5.2 Sigma Notation and Limits of Finite Sums 348
5.3 The Definite Integral 354
5.4 The Fundamental Theorem of Calculus 369
5.5 Indefinite Integrals and the Substitution Method 379
5.6 Substitution and Area Between Curves 387
Practice Exercises 407
Additional and Advanced Exercises 422
6 Applications of Definite Integrals 431
6.1 Volumes Using Cross-Sections 431
6.2 Volumes Using Cylindrical Shells 443
6.3 Arc Length 454
6.4 Areas of Surfaces of Revolution 462
6.5 Work and Fluid Forces 468
6.6 Moments and Centers of Mass 479
Practice Exercises 492
Additional and Advanced Exercises 501
7 Integrals and Transcendental Functions 507
7.1 The Logarithm Defined as an Integral 507
7.2 Exponential Change and Separable Differential Equations 515
7.3 Hyperbolic Functions 521
7.4 Relative Rates of Growth 529
Practice Exercises 535
Additional and Advanced Exercises 540
8 Techniques of Integration 543
8.1 Using Basic Integration Formulas 543
8.2 Integration by Parts 555
, TABLE OF CONTENTS
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 8
1.3 Trigonometric Functions 18
1.4 Graphing with Software 26
1.5 Exponential Functions 31
1.6 Inverse Functions and Logarithms 34
Practice Exercises 43
Additional and Advanced Exercises 52
2 Limits and Continuity 59
2.1 Rates of Change and Tangents to Curves 59
2.2 Limit of a Function and Limit Laws 62
2.3 The Precise Definition of a Limit 73
2.4 One-Sided Limits 81
2.5 Continuity 86
2.6 Limits Involving Infinity; Asymptotes of Graphs 92
Practice Exercises 102
Additional and Advanced Exercises 108
3 Differentiation 115
3.1 Tangents and the Derivative at a Point 115
3.2 The Derivative as a Function 121
3.3 Differentiation Rules 131
3.4 The Derivative as a Rate of Change 138
3.5 Derivatives of Trigonometric Functions 144
3.6 The Chain Rule 152
3.7 Implicit Differentiation 162
3.8 Derivatives of Inverse Functions and Logarithms 170
3.9 Inverse Trigonometric Functions 180
3.10 Related Rates 186
3.11 Linearization and Differentials 192
Practice Exercises 199
Additional and Advanced Exercises 214
, 4 Applications of Derivatives 219
4.1 Extreme Values of Functions 219
4.2 The Mean Value Theorem 233
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 253
4.5 Indeterminate Forms and L Hôpital s Rule 280
4.6 Applied Optimization 290
4.7 Newton's Method 304
4.8 Antiderivatives 309
Practice Exercises 318
Additional and Advanced Exercises 336
5 Integration 343
5.1 Area and Estimating with Finite Sums 343
5.2 Sigma Notation and Limits of Finite Sums 348
5.3 The Definite Integral 354
5.4 The Fundamental Theorem of Calculus 369
5.5 Indefinite Integrals and the Substitution Method 379
5.6 Substitution and Area Between Curves 387
Practice Exercises 407
Additional and Advanced Exercises 422
6 Applications of Definite Integrals 431
6.1 Volumes Using Cross-Sections 431
6.2 Volumes Using Cylindrical Shells 443
6.3 Arc Length 454
6.4 Areas of Surfaces of Revolution 462
6.5 Work and Fluid Forces 468
6.6 Moments and Centers of Mass 479
Practice Exercises 492
Additional and Advanced Exercises 501
7 Integrals and Transcendental Functions 507
7.1 The Logarithm Defined as an Integral 507
7.2 Exponential Change and Separable Differential Equations 515
7.3 Hyperbolic Functions 521
7.4 Relative Rates of Growth 529
Practice Exercises 535
Additional and Advanced Exercises 540
8 Techniques of Integration 543
8.1 Using Basic Integration Formulas 543
8.2 Integration by Parts 555
