A hiker’s elevation (in meters) above sea level is modeled by a continuous function…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A hiker’s elevation (in meters) above sea level is modeled by a continuous function h of horizontal distance x (in kilometers) along a mountain trail. A detailed topographic profile, drawn to scale so that one centimeter horizontally represents 0.2 km and one centimeter vertically represents 100 m, shows the following local features:
• Near x=1.0 km, the graph appears almost linear: moving 0.5 cm to the right on the profile corresponds to a drop of 0.4 cm vertically.
• Near x=2.2 km, the graph has a horizontal tangent.
• Near x=3.5 km, the graph is steeply uphill: moving 0.3 cm to the right corresponds to a rise of 0.9 cm vertically.
Assume the profile is smooth at all three locations described.
(a) Convert each of the three centimetre‐based measurements into real‐world units of kilometers and meters, and then compute an approximate value for h′(1.0), h′(2.2), and h′(3.5) in units of meters per kilometer.
(b) Interpret the sign and magnitude of each derivative in terms of the hiker’s instantaneous rate of ascent or descent.
(c) Based on your results, describe between which two of the given points the hiker experiences the greatest physical effort due solely to slope, and justify your answer quantitatively.