Free response · Unit 5 Beta

Graph of the derivative: extrema, concavity and a tangent line

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

The question

Name: ______________________   Date: ____________   Graph of the derivative: extrema, concavity and a tangent line (9 points)

Graph

The graph of f′f', the derivative of ff

The function ff is defined and differentiable on the closed interval [−4,5][-4, 5]. The graph of f′f', the derivative of ff, consists of a semicircle below the xx-axis from x=−4x = -4 to x=0x = 0 and two line segments, from (0,0)(0, 0) to (2,3)(2, 3) and from (2,3)(2, 3) to (5,−3)(5, -3). It is given that f(2)=5f(2) = 5.
−4−3−2−112345−3−2−1123xyf'(-2, -2)(2, 3)

  1. (a) Find the xx-coordinate of each point at which ff has a relative minimum and each point at which ff has a relative maximum on the open interval −4<x<5-4 < x < 5. Justify your answers. [2 points]

  2. (b) Find the open intervals on which the graph of ff is concave down. Give a reason for your answer. [2 points]

  3. (c) Find the xx-coordinate of each point of inflection of the graph of ff on −4<x<5-4 < x < 5. Justify your answer. [2 points]

  4. (d) Write an equation for the line tangent to the graph of ff at x=2x = 2. Use it to approximate f(2.2)f(2.2). Is this approximation an overestimate or an underestimate of f(2.2)f(2.2)? Explain your reasoning. [3 points]