AP® Calculus BC · Free response

Graphical FTC with a lower limit inside the interval

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Graph of f′ or graphical FTC · Part B: no calculator · 9 points

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The question

Name: ______________________   Date: ____________   Graphical FTC with a lower limit inside the interval (9 points)

Graph

The graph of ff on [−2,6][-2, 6]

The continuous function ff is defined on [−2,6][-2, 6]. Its graph is a semicircle of radius 2 above the x-axis from x=−2x = -2 to x=2x = 2, a segment from (2,0)(2, 0) to (4,−2)(4, -2), and a segment from (4,−2)(4, -2) to (6,2)(6, 2). Let g(x)g(x)=∫2xf(t)dt= \int_2^x f(t)\,dt for −2≤x≤6-2 \le x \le 6.
−2−1123456−2−112xyf

  1. (a) Find g(0)g(0) and g(6)g(6). [2 points]

  2. (b) Find the absolute minimum value of gg on the closed interval [−2,6][-2, 6]. Justify your answer. [3 points]

  3. (c) Let k(x)k(x)=∫23xf(t)dt= \int_2^{3x} f(t)\,dt. Find k′(1)k'(1). [2 points]

  4. (d) Find the value of limn→∞∑i=1nf(−2+4in)4n\lim_{n \to \infty} \sum_{i=1}^{n} f\left(-2 + \frac{4i}{n}\right)\frac{4}{n}. [2 points]