• Solving logistic differential equations and using them in modeling.

Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

*IV. Polynomial Approximations and Series

* Concept of series. A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.

* Series of constants.

+ Motivating examples, including decimal expansion.

+ Geometric series with applications.

+ The harmonic series.

+ Alternating series with error bound.

+ Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.

+ The ratio test for convergence and divergence.

+ Comparing series to test for convergence or divergence.

* Taylor series.

+ Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.)

+ Maclaurin series and the general Taylor series centered at x = a.

+ Maclaurin series for the functions ex, sin x, cos x, and

+ Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.

+ Functions defined by power series.

+ Radius and interval of convergence of power series.

+ Lagrange error bound for Taylor polynomials.

SOURCE: College Entrance Examination Board. 1999. Advanced Placement Program Course Description: Calculus – May 2000, May 2001 (pp. 9-14). New York: author.

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