CALCULUS 1 What result tells us that if a biker travels at 45 km in 3 hours, then her speedometer must have read 15 km/h at least once during the trip?

Question

What result tells us that if a biker travels at 45 km in 3 hours, then her speedometer must have read 15 km/h at least once during the trip? Can we guarantee it will read 12 km/h during the trip? How about 18 km/h

Answer 1

Let f(t) = distance traveled in t hours. So, f(3) - f(0) = 45.

Then, f'(t) = velocity at time t

Assuming that f(t) is continuous on [0,3] and is differentiable on (0,3), then, by the Mean Value Theorem, there is at least one value, c, in the interval (0,3) so that:

f'(c) = [f(3) - f(0)] / (3-0) = 45/3 = 15