Free response · Unit 6 Beta

Graphical FTC: lines and a semicircle, with a limit

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

The question

Name: ______________________   Date: ____________   Graphical FTC: lines and a semicircle, with a limit (9 points)

Graph

The graph of ff on [−4,6][-4, 6]

The continuous function ff is defined on [−4,6][-4, 6]. Its graph consists of a segment from (−4,2)(-4, 2) to (−2,0)(-2, 0), a semicircle of radius 1 below the x-axis from x=−2x = -2 to x=0x = 0, a segment from (0,0)(0, 0) to (2,2)(2, 2), and a segment from (2,2)(2, 2) to (6,−2)(6, -2). Let g(x)g(x)=∫0xf(t)dt= \int_0^x f(t)\,dt for −4≤x≤6-4 \le x \le 6.
−4−3−2−1123456−2−112xyf

  1. (a) Find g(4)g(4) and g(−4)g(-4). [2 points]

  2. (b) On the open interval −4<x<6-4 < x < 6, find all values of xx at which gg has a relative maximum. Justify your answer. [2 points]

  3. (c) Find the xx-coordinate of each point of inflection of the graph of gg on −4<x<6-4 < x < 6. Give a reason for your answer. [2 points]

  4. (d) Find limx→2g(x)−2x2−4\lim_{x \to 2} \frac{g(x) - 2}{x^2 - 4}. Show the work that leads to your answer. [3 points]