can you guys give me an example where this theorem would be used in the real world? Im trying to explain it to my friend but im having little luck with it since he's not a big math fan.
thanks a lot
Let $\displaystyle s(t)$ be your distance from your starting point at any time $\displaystyle t\geq 0$. You know that $\displaystyle s'(t_0)$ is the velocity at that point. And $\displaystyle \frac{s(t_2)-s(t_1)}{t_2-t_1}$ is the average (mean) velocity.
Therefore we must have the mean velocity equal to the instatensous velocity at some point during the travel.
To prove someone had been speeding even though the speed of their vehicle
when passing the check points was less than the speed limit. Since if the
average speed between the check points d/t (where d is the distance
between the check points and t the difference in the times of passing the
check points) is greater than the speed limit then by the mean value theorem
the vehicles instantaneous speed was greater than the speed limit at some
point between the check points.
RonL