Mean Value Theorem?

Question

Car X and Car Y are 5 miles apart. Car Z passes Car X at 55mph and passes car Y at 50mph. The time it takes is 4 minutes. Prove Car Z went over 55mph during those 4 minutes.

Answer 1

The mean value theorem tells us that for a continuous and differentiable function f on [a,b], there is a point c in (a,b) with f'(c) = (f(b) - f(a))/(b-a).

In this case, f(x) = velocity at time x, and f'(x) = velocity at time x.

Our interval is [0,4/60], and we know f(b) - f(a) = 5.

So we know there is a point c inside that interval with f'(c) = (f(b)-f(a))/(b-a) = 5/(4/60) = 75 > 55.

The starting and ending velocities seem to be irrelevant.

Answer 2

In order to calculate velocity, divide the distance by the time. 5 miles/ 4 minutes. Since you want this answer in miles/hour, multiply 5 miles/4 minutes by 60 minutes/1 hour = 75 miles / hr