Use the Mean Value Theorem to show that a car that travels $110$ mi in $2$ h must have had a speed of $55$ miles per hour (mph) at least once during the 2 h.

Solution Let $s=$ $f(t)$ represent the distance $s$ the car has traveled after $t$ hours. Its average velocity during the time period from $0$ to $2$ h is $$\hbox{Average velocity}=\frac{f(2)-f(0)}{2-0}=\frac{110-0}{2-0}=55\ {\rm mph}$$

Using the Mean Value Theorem, there is a time $t_{0}$, $0<t_{0}<2,$ at which $$f^\prime (t_{0}) =\dfrac{f(2) -f( 0) }{2-0}=55$$

That is, the car had a velocity of $55$ mph at least once during the $2$ hour period.