Driving Process (Unit: mile)

Peter is driving on on a road where the speed limit is 65 mi/hr. By 4 p.m. he has drove 2000 miles and by 6 p.m. his total trip mileage goes up to 2240 miles. Was Peter speeding during in this period of time?

Let f(t) be the car's position at time t.

By 4 p.m. he traveled 2000 miles, so f(4) = 2000

By 6 p.m. he traveled 2240 miles, so f(6) = 2240

The mean value theorem gives that


In this case, the mean value on interval [4, 6] is

     df(c)/dx = (f(6) - f(4))/(6 - 4) = (2240 - 2000)/2 = 120

Since f(t) is a position function, the derivative of it is velocity. In other words, the above equation means that there is a time c in the interval of 4 and 6 such that the velocity is 120 mi/hr. Therefore, Peter exceeded the speed limit.