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 $c$ in $(a,b)$
such that $\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$.
In the following animation (or screen shot if the plugin is not installed),
the red line goes from $(a,f(a))$ to $(b,f(b))$. If you adjust the height
of the parallel teal line until it is tangent to the curve $y=f(x)$ at a point
$(c,f(c))$, then
the derivative at the point of tangency will be
exactly $f'(c)=\displaystyle{\frac{f(b)-f(a)}{b-a}}$.