A cyclist’s velocity in meters per second is modeled for the first 10 seconds of a…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A cyclist’s velocity in meters per second is modeled for the first 10 seconds of a sprint by the differentiable function
v(t)=4+3sin(10πt),0≤t≤10.
Answer the following questions about the ride.
(a) Describe, in the context of the sprint, the meaning of the average value of v on the interval [0,10]. Include appropriate units in your description.
(b) Without performing any calculations, explain why there must be at least one instant c in 0<c<10 at which the cyclist’s instantaneous velocity equals the average velocity you described in part (a).
(c) Suppose a coach claims that the cyclist spends more time above the average velocity than below it during the interval [0,10]. Explain how you would determine whether the coach’s claim is valid, citing specific features of the function v or its graph in your reasoning.