MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The elevation profile of a mountain hiking trail, measured in meters relative to the trailhead, is modeled for 0≤x≤5 kilometers by
H(x)=∣x−2∣3/2−21x.
A group of hikers wants to know the highest and lowest elevations they will encounter along this segment of the trail.
(a) Explain why H is continuous on [0,5] even though the expression involves an absolute value.
(b) Determine all interior points of [0,5] at which H′(x)=0, showing all calculus steps.
(c) Identify any interior point(s) of [0,5] where H′(x) does not exist, and describe the geometric feature of the graph at each such point.
(d) Using the points found in parts (b) and (c), together with the interval endpoints, create a table of elevations for all candidates.
(e) From the data gathered in part (d), determine the maximum elevation and the minimum elevation encountered on the trail. Conclude with a brief interpretation for the hikers, including the corresponding distances from the trailhead.