| 0: PRELIMINARIES. Lines. Functions and Their Graphs. The Trigonometric Functions. Combining Functions. Graphing Calculators and Computers. Mathematical Models. Chapter Review. 1: LIMITS. An Intuitive Introduction to Limits. Techniques for Finding Limits. A Precise Definition of a Limit. Continuous Functions. Tangent Lines and Rates of Change. Chapter Review. Problem-Solving Techniques. Challenge Problems. 2: THE DERIVATIVE. The Derivative. Basic Rules of Differentiation. The Product and Quotient Rules. The Role of the Derivative in the Real World. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation. Related Rates. Differentials and Linear Approximations. Chapter Review. Problem-Solving Techniques. Challenge Problems. 3: APPLICATIONS OF THE DERIVATIVE. Extrema of Functions. The Mean Value Theorem. Increasing and Decreasing Functions and the First. Derivative Test. Concavity and Inflection Points. Limits Involving Infinity; Asymptotes. Curve Sketching. Optimization Problems. Newton's Method. Chapter Review. Problem-Solving Techniques. Challenge Problems. 4: INTEGRATION. Indefinite Integrals. Integration by Substitution. Area. The Definite Integral. The Fundamental Theorem of Calculus. Numerical Integration. Chapter Review. Problem-Solving Techniques. Challenge Problems. 5: APPLICATIONS OF THE DEFINITE INTEGRAL. Areas Between Curves. Volumes: Disks, Washers, and Cross Sections. Volumes Using Cylindrical Shells. Arc Length and Areas of Surfaces of Revolution. Work. Fluid Pressure and Force. Moments and Centers of Mass. Chapter Review. Problem-Solving Techniques. Challenge Problems. 6: THE TRANSCENDENTAL FUNCTIONS. The Natural Logarithmic Function. Inverse Functions. Exponential Functions. General Exponential and Logarithmic Functions. Inverse Trigonometric Functions. Hyperbolic Functions. Indeterminate Forms and L'Hopital's Rule. Chapter Review. Challenge Problems. 7: TECHNIQUES OF INTEGRATION. Integration by Parts. Trigonometric Integrals. Trigonometric Substitutions. The Method of Partial Fractions. Integration Using Tables of Integrals and CAS. Improper Integrals. Chapter Review. Problem-Solving Techniques. Challenge Problems. 8: DIFFERENTIAL EQUATIONS. Differential Equations: Separable Equations. Direction Fields and Euler's Method. The Logistic Equation. First-Order Linear Differential Equations. Chapter Review. Challenge Problems. 9: INFINITE SEQUENCES AND SERIES. Sequences. Series. The Integral Test. The Comparison Tests. Alternating Series. Absolute Convergence; The Ratio and Root Tests. Power Series. Taylor and Maclaurin Series. Approximation by Taylor Polynomials. Chapter Review. Problem-Solving Techniques. Challenge Problems. |
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