iv
CONTENTS
3.2 Using derivatives to describe families of functions . . . . . . . . . . . . . . 174
3.3 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.4 Applied Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.5 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4 The Definite Integral
207
4.1 Determining distance traveled from velocity . . . . . . . . . . . . . . . . . 207
4.2 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . 250
5 Finding Antiderivatives and Evaluating Integrals
265
5.1 Constructing Accurate Graphs of Antiderivatives . . . . . . . . . . . . . . 265
5.2 The Second Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 276
5.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.5 Other Options for Finding Algebraic Antiderivatives . . . . . . . . . . . . 310
5.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6 Using Definite Integrals
333
6.1 Using Definite Integrals to Find Area and Length . . . . . . . . . . . . . . 333
6.2 Using Definite Integrals to Find Volume . . . . . . . . . . . . . . . . . . . . 343
6.3 Density, Mass, and Center of Mass . . . . . . . . . . . . . . . . . . . . . . . 354
6.4 Physics Applications: Work, Force, and Pressure . . . . . . . . . . . . . . . 365
6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7 Differential Equations
387
7.1 An Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . 387
7.2 Qualitative behavior of solutions to DEs . . . . . . . . . . . . . . . . . . . 398
7.3 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
7.4 Separable differential equations . . . . . . . . . . . . . . . . . . . . . . . . 419
7.5 Modeling with differential equations . . . . . . . . . . . . . . . . . . . . . . 427
7.6 Population Growth and the Logistic Equation . . . . . . . . . . . . . . . . 435
CONTENTS
3.2 Using derivatives to describe families of functions . . . . . . . . . . . . . . 174
3.3 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.4 Applied Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.5 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4 The Definite Integral
207
4.1 Determining distance traveled from velocity . . . . . . . . . . . . . . . . . 207
4.2 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . 250
5 Finding Antiderivatives and Evaluating Integrals
265
5.1 Constructing Accurate Graphs of Antiderivatives . . . . . . . . . . . . . . 265
5.2 The Second Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 276
5.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.5 Other Options for Finding Algebraic Antiderivatives . . . . . . . . . . . . 310
5.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6 Using Definite Integrals
333
6.1 Using Definite Integrals to Find Area and Length . . . . . . . . . . . . . . 333
6.2 Using Definite Integrals to Find Volume . . . . . . . . . . . . . . . . . . . . 343
6.3 Density, Mass, and Center of Mass . . . . . . . . . . . . . . . . . . . . . . . 354
6.4 Physics Applications: Work, Force, and Pressure . . . . . . . . . . . . . . . 365
6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7 Differential Equations
387
7.1 An Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . 387
7.2 Qualitative behavior of solutions to DEs . . . . . . . . . . . . . . . . . . . 398
7.3 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
7.4 Separable differential equations . . . . . . . . . . . . . . . . . . . . . . . . 419
7.5 Modeling with differential equations . . . . . . . . . . . . . . . . . . . . . . 427
7.6 Population Growth and the Logistic Equation . . . . . . . . . . . . . . . . 435