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• Table of Contents

Table of Contents

1. FUNCTIONS AND LIMITS.
Functions and Their Representations. A Catalog of Essential Functions. The Limit of a Function. Calculating Limits. Continuity. Limits Involving Infinity. Review.
2. DERIVATIVES.
Derivatives and Rates of Change. The Derivative as a Function. Basic Differentiation Formulas. The Product and Quotient Rules. The Chain Rule. Implicit Differentiation. Related Rates. Linear Approximations and Differentials. Review.
3. INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS.
Exponential Functions. Inverse Functions and Logarithms. Derivatives of Logarithmic and Exponential Functions. Exponential Growth and Decay. Inverse Trigonometric Functions. Hyperbolic Functions. Indeterminate Forms and l'Hospital's Rule. Review.
4. APPLICATIONS OF DIFFERENTIATION.
Maximum and Minimum Values. The Mean Value Theorem. Derivatives and the Shapes of Graphs. Curve Sketching. Optimization Problems. Newton's Method. Antiderivatives. Review.
5. INTEGRALS.
Areas and Distances. The Definite Integral. Evaluating Definite Integrals. The Fundamental Theorem of Calculus. The Substitution Rule. Review.
6. TECHNIQUES OF INTEGRATION.
Integration by Parts. Trigonometric Integrals and Substitutions. Partial Fractions. Integration with Tables and Computer Algebra Systems. Approximate Integration. Improper Integrals. Review.
7. APPLICATIONS OF INTEGRATION.
Areas between Curves. Volumes. Volumes by Cylindrical Shells. Arc Length. Applications to Physics and Engineering. Differential Equations. Review.
8. SERIES.
Sequences. Series. The Integral and Comparison Tests. Other Convergence Tests. Power Series. Representing Functions as Power Series. Taylor and Maclaurin Series. Applications of Taylor Polynomials. Review.
9. PARAMETRIC EQUATIONS AND POLAR COORDINATES.
Parametric Curves. Calculus with Parametric Curves. Polar Coordinates. Areas and Lengths in Polar Coordinates. Conic Sections in Polar Coordinates. Review.
Appendix A: Trigonometry.
Appendix B: Proofs.
Appendix C: Sigma Notation.
Appendix D: The Logarithm Defined as an Integral.