Topic Outline for Math 141, Calculus with Analytic Geometry II I. Inverse Functions a. Inverse Functions- One-to-one functions, horizontal line test, finding inverse functions, finding the derivative of an inverse function at a point. b. Exponential Functions and Their Derivatives- Graphing exponential functions, derivatives and integrals involving the natural exponential function. c. Logarithmic Functions- Graphing logarithmic functions, properties of logarithms. d. Derivatives of Logarithmic Functions- Derivatives and integrals involving logarithmic functions, logarithmic differentiation. e. Inverse Trigonometric Functions- Definition of the inverse trig functions, derivatives and integrals involving inverse trig functions. f. Indeterminate Forms and L’Hospital’s Rule- Discussion of limits that are in indeterminate forms and how to evaluate them using L’Hospital’s Rule or other methods. II. Techniques of Integration a. Integration by Parts b. Trigonometric Integrals- Use trigonometric identities to evaluate certain combinations of trigonometric integrals. c. Trigonometric Substitution- Evaluating certain types of algebraic integrals using trigonometric substitutions. d. Integration of Rational Functions by Partial Fractions- (Including repeated linear and/or irreducible quadratic factors.) e. Improper Integrals- Discussion of integrals involving infinite limits and/or discontinuities. III. Infinite Sequences and Series a. Sequences- Definition of a sequence, determining whether a sequence is convergent or divergent, increasing/decreasing sequences. b. Series- Definition of an infinite series, partial sums, geometric series, telescoping series, the Test for Divergence. c. The Integral Test- Using the integral test to determine whether a series is convergent/divergent, p-series. d. The Comparison Tests- Using the Direct Comparison Test and the Limit Comparison Test to determine whether a series is convergent/divergent. e. Alternating Series- Using the alternating series test to determine whether a series is convergent/divergent, Alternating Series Estimation Theorem. f. Absolute Convergence and the Ratio and Root Tests- Determining whether a series is absolutely or conditionally convergent, The Ratio and Root Tests. g. Power Series- Finding the radius and interval of convergence for power series. h. Representations of Functions as Power Series- Finding the power series representation of a function, differentiation and integration of power series, using power series to approximate definite integrals. i. Taylor and Maclaurin Series- Finding the Taylor Series/Maclaurin Series representation for a function, multiplication and division of power series. j. Applications of Taylor Polynomials- Approximating functions by Taylor polynomials, using Taylor’s Inequality to estimate the accuracy of a given approximation. IV. Parametric Equations and Polar Coordinates a. Curves defined by Parametric Equations- Graphing parametric equations, eliminating the parameter to find the Cartesian equation. b. Calculus with Parametric Curves- Derivatives of parametric functions. c. Polar Coordinates- Discussion of the polar coordinate system, graphing polar curves, Derivatives of polar functions. d. Areas in Polar Coordinates- Finding the area enclosed by polar curves. Note: Those items in italics are covered in Math 141 but may/may not have been covered in AP Calculus.
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