AP Calculus

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Special Focus: The Fundamental Theorem of Calculus (b) Consider the function H defined by H( x) = t − 4 e dt. Although this is ∫ x 0 2 ( ) −t 10 still an accumulation function, this is slightly different than the previous three examples because the upper bound is x 2 (a function of x), not simply x. Rewrite H as a composition function (similar to part (a)) and find H′ ( x). (c) Use your result in part (b) to find the absolute maximum value and the absolute minimum value of the function H (if they exist). (d) Use technology to sketch a graph of y = H( x) and confirm your results in part (c). Additional Problems 1. Suppose f is a continuous function such that x 2 x 2 ∫ 0 ∫ 0 f ( t) dt = x sin x − t f ( t) dt for all x. Find an explicit formula for f(x) and sketch a graph of y = f(x). 2. Consider the function F defined by F( x) = ∫ t sin( t) dt . 0 (a) Compute F′ ( x) and use this to find the intervals on which the graph of y = f(x) is increasing and the intervals on which the graph is decreasing. (b) Find the absolute maximum value and the absolute minimum value of y = f(x) (if they exist). (c) Sketch a graph of y = f(x). x 3. Consider the function F defined by F( x) = ∫ (sin t + ln t) dt for x > 0. 1 (a) Approximate the value(s) of x for which the graph of y = f(x) has a local minimum value or local maximum value. (b) Sketch a graph of y = f(x). x AP® Calculus: 2006–2007 Workshop Materials 115
Special Focus: The Fundamental Theorem of Calculus Solutions 1. (a) Using the FTC, the derivative of S(x) is the integrand evaluated at x: ⎛ π x S '( x) = sin⎜ ⎝ 2 2 ⎞ ⎟ . ⎠ (b) Technology produces the following graph of y = S"(x). Notice that the graph oscillates faster and faster as |x| increases. By using a calculator to find the zeros, students might recognize the numerical approximations for ± 2k . The zeros of S′ ( x) are 0,± 2,± 4, ± 6,.... By hand, students can set S′ ( x) = 0 and continue in the following manner: ⎛ sin π x2 ⎞ ⎜ ⎟ = 0 (Use the expression for S′ ( x).) ⎝ 2 ⎠ π x kπ 2 = (Set the argument equal to kπ.) x 2 = ± 2 k (Solve for x.) x = 0, ± 2, ± 4, ± 6, ... (Let k = 0, 1, 2, 3, ....) 116 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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