Rolle's Theorem is a special case of the Mean Value Theorem which
says that there has to be a point between $a$ and $b$ where the
instantaneous rate of change
is equal to the average rate of change between $a$ and
$b$. More precisely:

Mean Value Theorem: If $f$ is a
function that is continuous on the closed interval
$[a,b]$ and differentiable on the open
interval $(a,b)$, then there is a point $c$
between $a$ and $b$ such that $\displaystyle
f'(c)=\frac{f(b)-f(a)}{b-a}$.