The continuous function is described verbally as follows
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The continuous function f is described verbally as follows.
• On the interval [−3,0] the graph rises concavely from the point (−3,−2) to the point (0,1), where it has a horizontal tangent.
• On the interval (0,2] the graph is decreasing and concave down, reaching a relative minimum of f(2)=0 at x=2.
• On the interval (2,4] the graph is a straight line segment that rises from (2,0) to (4,3).
Use this information to analyze the first derivative of f.
(a) Estimate the slope of the tangent line to the graph of f at x=−2, at x=1, and at x=3. Explain how you obtained each estimate from the verbal description.
(b) Identify all x–values in the domain of f at which f′(x)=0 or f′(x) does not exist. Give a brief justification in each case.
(c) Determine the intervals on which f′(x)>0 and the intervals on which f′(x)<0. State your reasoning clearly.
(d) Sketch a qualitative (not necessarily to scale) graph of y=f′(x) on the domain [−3,4]. Label all critical x–values and indicate any discontinuities.
(e) Provide a concise explanation that connects each key feature of your graph of f′ in part (d) to the corresponding feature described for f. Your explanation should reference slopes, extrema, and concavity as appropriate.