Two hikers are ascending separate trails that start at the same parking lot
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Two hikers are ascending separate trails that start at the same parking lot. Their altitudes above the trailhead (in meters) are recorded every hour. The table below lists the instantaneous rate of change of altitude (in meters per hour) at selected times.
Time t (hours)
Trail P: p′(t)
Trail Q: q′(t)
1
140
125
2
135
150
3
120
155
4
110
140
Assume both p and q are differentiable for 0≤t≤4.
(a) Using the data, rank the four recorded times from fastest overall ascent (largest instantaneous rate of change regardless of hiker) to slowest overall ascent. Clearly indicate the rate that justifies each ranking step.
(b) Identify the time(s) during which Hiker Q is ascending faster than Hiker P. Provide evidence from the table, including units.
(c) Explain why a comparison based solely on p(t) and q(t) (altitude values) at a given time might lead to a different conclusion about “who is higher,” and discuss how the derivative information clarifies the notion of “changing faster.”