A piecewise–smooth curve modeling the height (in meters) of a balloon above the…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A piecewise–smooth curve y=h(x) modeling the height (in meters) of a balloon above the ground is described for 0≤x≤10 minutes.
• For 0≤x≤3, the curve is increasing and concave down, passing through (0,2) and (3,8). The tangent at x=1 intersects (1,5) and (2,6).
• For 3≤x≤7, the balloon drifts downward. The tangent at x=5 passes through the points (5,6) and (7,0).
• For 7≤x≤10, the balloon rises again. At x=8, the tangent line through (8,1) also passes through (9,3).
Answer each part thoroughly.
(a) Estimate h′(1), h′(5), and h′(8), showing each slope calculation.
(b) Using your results from part (a), determine on which of the three subintervals the balloon’s vertical speed changes most rapidly in magnitude. Justify your answer with quantitative evidence.
(c) Identify all x–values in 0<x<10 at which h′(x)=0 must occur and classify each as a local maximum, local minimum, or neither for h. Provide reasoning based on the verbal description of the graph.