A mountain road climbs from an elevation of at its base to a pass away
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A mountain road climbs from an elevation of 700 m at its base to a pass 10 km away. Transportation planners recorded the road’s instantaneous grade r (percentage slope) at every kilometer, as shown in Table 5.
Table 5. Road grade r(s), where s is distance from the base (km)
s (km) 0 1 2 3 4 5 6 7 8 9 10
r(\%) 0 1.5 3.2 4.1 4.8 5.0 4.6 3.8 2.5 1.0 0
Assume r is differentiable on 0≤s≤10.
Respond to the following, providing full justification based on the numerical data.
(a) Determine the intervals of distance s on which the elevation profile of the road is concave up and the intervals on which it is concave down.
(b) Estimate, to the nearest 0.5 km, the location of every point of inflection of the elevation profile. Explain clearly how the data support your estimate.
(c) Interpret the meaning of the concave-up and concave-down intervals from part (a) for a bicyclist ascending the road. Address how the cyclist’s experience of the climb changes across the point(s) you identified in part (b).
(d) Suppose road safety regulations require that no single concave-up segment exceed 4 km in length. Determine whether the recorded data satisfy this requirement, and justify your conclusion with precise calculations.