MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
The graph of f′, the derivative of a function f, consists of three straight–line segments and one constant segment as described below. Assume f′ is continuous on −4≤x≤5.
• Segment S1: A line from (−4,3) to (−1,−3).
• Segment S2: A horizontal line at y=−3 on −1≤x≤2.
• Segment S3: A line from (2,−3) to (5,3).
Respond to the prompts about f′′.
(a) Compute the numerical value of the slope of each segment S1, S2, and S3. State these values explicitly.
(b) Using the slopes found in part (a), give the exact value of f′′(x) on each open interval (−4,−1), (−1,2), and (2,5).
(c) Identify all x–values in −4≤x≤5 at which either f′′(x)=0 or f′′ is not defined. Classify the type of point (zero, discontinuity, or endpoint) for each value and explain your reasoning.
(d) On the axes below, sketch the graph of f′′ on the interval −4≤x≤5, labeling key coordinates determined in parts (b) and (c). (An accurately labeled hand–drawn sketch is sufficient.)