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# Calculus: A Complete Course ©2007

Ross L. Finney, Franklin D. Demana, Bert K. Waits, Daniel Kennedy

## Calculus BC

SE = Student Edition
TE = Teacher Edition

## I. FUNCTIONS, GRAPHS, AND LIMITS

 Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form SE/TE: 531, 541, 545, 550, 557

## II. DERIVATIVES

 Applications of derivatives —Analysis of planar curves given in paremetric form, polar form, and vector form, including velocity and acceleration vectors. SE/TE: 221, 222, 224–227 —Numerical solution of differential equations using Euler's method SE/TE: 326, 328, 326, 328, 329 —L'Hôpital's Rule, including its use in determining limits and convergence of improper integrals and series. SE/TE: 446–448, 450, 451, 461 Computation of derivatives —Derivatives of parametric, polar, and vector functions SE/TE: 532, 535, 543, 545, 552

## III. INTEGRALS

 Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, biological, 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 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.) SE/TE: 386, 387, 396, 397, 406, 407, 416, 417, 425–427 Techniques of antidifferentiation —Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating llinear factors only) SE/TE: 333–338 Applications of antidifferentiation —Solving logistic differential equations and using them in modeling SE/TE: 365–371

## 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 SE/TE: 474, 481, 482 Series of constants —Motivating examples, including decimal expansion SE/TE: 474, 481, 482 —Geometric series with applications SE/TE: 482 —The harmonic series SE/TE: 514, 523 —Alternating series with error bound SE/TE: 518, 523, 524 —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. SE/TE: 513, 523 —The ratio test for convergence and divergence SE/TE: 508, 509, 511, 512 —Comparing series to test for convergence or divergence SE/TE: 505, 506, 511, 515, 516, 523 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.) SE/TE: 495, 496, 500, 496, 500, 501 —Maclaurin series and the general Taylor series centered at x = a SE/TE: 488, 490, 492 —Maclaurin series for the functions ex, sin x, cos x, and 1/ 1 – x SE/TE: 491–493 —Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series SE/TE: 484–486, 490, 492 —Functions defined by power series SE/TE: 485, 486, 492, 493 —Radius and interval of convergence of power series SE/TE: 508, 511, 520, 523 —Lagrange error bound for Taylor polynomials SE/TE: 497–501

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