Concept preview

What you will be able to do

1. 01State the Mean Value Theorem — if $f$ is continuous on $[a,\,b]$ and differentiable on $(a,\,b)$, then there is a number $c$ in $(a,\,b)$ with $f'(c)=\dfrac{f(b)-f(a)}{b-a}$ — and read it geometrically as a tangent line parallel to the secant line.
2. 02Recognise Rolle's Theorem as the special case $f(a)=f(b)$, and always verify BOTH hypotheses before applying the theorem, since a corner such as $y=|x|$ can make the conclusion fail.
3. 03Find the guaranteed value (or values) of $c$ for a concrete function on an interval, and interpret the theorem in context: the average rate of change over $[a,\,b]$ is achieved as an instantaneous rate somewhere inside.

Here is the theorem in full. If $f$ is continuous on the closed interval $[a,\,b]$ AND differentiable on the open interval $(a,\,b)$, then there exists at least one number $c$ in $(a,\,b)$ such that $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Read the right-hand side first: it is the average rate of change of $f$ across the whole interval, which is the slope of the secant line joining the endpoints $\left(a,\,f(a)\right)$ and $\left(b,\,f(b)\right)$. Read the left-hand side next: $f'(c)$ is the instantaneous rate of change at the single interior point $c$, which is the slope of the tangent line there. The theorem guarantees that these two slopes are equal for some $c$.