MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Two hiking trails, Trail A and Trail B, ascend a mountain from an elevation of 0 meters at the trailhead to 1200 meters at the summit. For each trail the slope (instantaneous rate of change of elevation with respect to horizontal distance) is measured at various horizontal distances x (in kilometers) from the trailhead, producing the smooth graphs of A′(x) and B′(x) described below.
• Along Trail A, the slope decreases steadily from 0.18mm at x=0 km to 0.05 at x=4 km, then increases steadily to 0.20 at x=8 km.
• Along Trail B, the slope increases steadily from 0.12 at x=0 km to 0.24 at x=4 km, then decreases steadily to 0.08 at x=8 km.
Assume each trail is 8 km long, twice differentiable on (0,8), and that the slope changes exactly as stated with no additional fluctuations.
(a) For each trail, identify the intervals of horizontal distance on which the ascent is concave upward and the intervals on which it is concave downward. Provide justification based on the stated behavior of the slope.
(b) Determine whether either trail must have a point of inflection at x=4 km. Support your conclusion with a clear explanation.
(c) Between x=2 km and x=6 km, which trail is steeper on average? Does this fact correlate with which trail is more concave upward on that same interval? Explain your reasoning.
(d) Suppose a hiker prefers a trail whose steepness increases slowly at first, then levels off before getting steeper again. Based on your concavity analysis, recommend which trail better matches the hiker’s preference and justify your recommendation.