Computation of derivatives.

  • Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

  • Basic rules for the derivative of sums, products, and quotients of functions.

  • Chain rule and implicit differentiation.

  • Derivatives of parametric, polar, and vector functions.

III. Integrals

Interpretations and properties of definite integrals.

  • Computation of Riemann sums using left, right, and midpoint evaluation points.

  • Definite integral as a limit of Riemann sums over equal subdivisions.

  • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

  • Basic properties of definite integrals. (Examples include additivity and linearity.)

* Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form).

Fundamental Theorem of Calculus.

  • Use of the Fundamental Theorem to evaluate definite integrals.

  • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

Techniques of antidifferentiation.

  • Antiderivatives following directly from derivatives of basic functions.

  • Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).

  • Improper integrals (as limits of definite integrals).

Applications of antidifferentiation.

  • Finding specific antiderivatives using initial conditions, including applications to motion along a line.

  • Solving separable differential equations and using them in modeling. In particular, studying the equation y’ = ky and exponential growth.



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OCR for page 45
Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics Computation of derivatives. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Basic rules for the derivative of sums, products, and quotients of functions. Chain rule and implicit differentiation. Derivatives of parametric, polar, and vector functions. III. Integrals Interpretations and properties of definite integrals. Computation of Riemann sums using left, right, and midpoint evaluation points. Definite integral as a limit of Riemann sums over equal subdivisions. Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: Basic properties of definite integrals. (Examples include additivity and linearity.) * Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curve given in parametric form). Fundamental Theorem of Calculus. Use of the Fundamental Theorem to evaluate definite integrals. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Techniques of antidifferentiation. Antiderivatives following directly from derivatives of basic functions. Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only). Improper integrals (as limits of definite integrals). Applications of antidifferentiation. Finding specific antiderivatives using initial conditions, including applications to motion along a line. Solving separable differential equations and using them in modeling. In particular, studying the equation y’ = ky and exponential growth.

OCR for page 45
Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools - Report of the Content Panel for Mathematics 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.