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The Definite Integral

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The Definite Integral Table of Contents Preface . 1 Chapter 1: Functions and Graphs . 7 1.1 Review of Functions . 8 1.2 Basic Classes of Functions . 36 1.3 Trigonometric Functions . 62 1.4 Inverse Functions . 78 1.5 Exponential and Logarithmic Functions . 96 Chapter 2: Limits . 123 2.1 A Preview of Calculus . 124 2.2 The Limit of a Function . 135 2.3 The Limit Laws . 160 2.4 Continuity . 179 2.5 The Precise Definition of a Limit . 194 Chapter 3: Derivatives . 213 3.1 Defining the Derivative . 214 3.2 The Derivative as a Function . 232 3.3 Differentiation Rules . 247 3.4 Derivatives as Rates of Change . 266 3.5 Derivatives of Trigonometric Functions . 277 3.6 The Chain Rule . 287 3.7 Derivatives of Inverse Functions . 299 3.8 Implicit Differentiation . 309 3.9 Derivatives of Exponential and Logarithmic Functions . 319 Chapter 4: Applications of Derivatives . 341 4.1 Related Rates . 342 4.2 Linear Approximations and Differentials . 354 4.3 Maxima and Minima . 366 4.4 The Mean Value Theorem . 379 4.5 Derivatives and the Shape of a Graph . 390 4.6 Limits at Infinity and Asymptotes . 407 4.7 Applied Optimization Problems . 439 4.8 L’Hôpital’s Rule . 454 4.9 Newton’s Method . 472 4.10 Antiderivatives . 485 Chapter 5: Integration . 507 5.1 Approximating Areas . 508 5.2 The Definite Integral . 529 5.3 The Fundamental Theorem of Calculus . 549 5.4 Integration Formulas and the Net Change Theorem . 566 5.5 Substitution . 584 5.6 Integrals Involving Exponential and Logarithmic Functions . 595 5.7 Integrals Resulting in Inverse Trigonometric Functions . 608 Chapter 6: Applications of Integration . 623 6.1 Areas between Curves . 624 6.2 Determining Volumes by Slicing . 636 6.3 Volumes of Revolution: Cylindrical Shells . 656 6.4 Arc Length of a Curve and Surface Area . 671 6.5 Physical Applications . 685 6.6 Moments and Centers of Mass . 703 6.7 Integrals, Exponential Functions, and Logarithms . 721 6.8 Exponential Growth and Decay . 734 6.9 Calculus of the Hyperbolic Functions . 745 Appendix A: Table of Integrals . 763 Appendix B: Table of Derivatives . 769 Appendix C: Review of Pre-Calculus . 771 Index . 863 Table of Contents Preface . 1 Chapter 1: Integration . 5 1.1 Approximating Areas . 6 1.2 The Definite Integral . 27 1.3 The Fundamental Theorem of Calculus . 47 1.4 Integration Formulas and the Net Change Theorem . 64 1.5 Substitution . 82 1.6 Integrals Involving Exponential and Logarithmic Functions . 93 1.7 Integrals Resulting in Inverse Trigonometric Functions . 106 Chapter 2: Applications of Integration . 121 2.1 Areas between Curves . 122 2.2 Determining Volumes by Slicing . 134 2.3 Volumes of Revolution: Cylindrical Shells . 154 2.4 Arc Length of a Curve and Surface Area . 169 2.5 Physical Applications . 183 2.6 Moments and Centers of Mass . 201 2.7 Integrals, Exponential Functions, and Logarithms . 219 2.8 Exponential Growth and Decay . 232 2.9 Calculus of the Hyperbolic Functions . 243 Chapter 3: Techniques of Integration . 261 3.1 Integration by Parts . 262 3.2 Trigonometric Integrals . 273 3.3 Trigonometric Substitution . 285 3.4 Partial Fractions . 298 3.5 Other Strategies for Integration . 311 3.6 Numerical Integration . 316 3.7 Improper Integrals . 330 Chapter 4: Introduction to Differential Equations . 351 4.1 Basics of Differential Equations . 352 4.2 Direction Fields and Numerical Methods . 365 4.3 Separable Equations . 381 4.4 The Logistic Equation . 393 4.5 First-order Linear Equations . 408 Chapter 5: Sequences and Series . 427 5.1 Sequences . ..
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