The Mean Value Theorem

The average rate of change of $f$ on $[a, b]$ is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. The Mean Value Theorem says that when $f$ is smooth enough, some tangent line between $a$ and $b$ has that same slope.

The right side is the average rate of change of $f$ on $[a, b]$, and the left side is the instantaneous rate of change at $c$. The theorem promises at least one such $c$. It doesn't say where it is, and there can be more than one.

Continuity is needed on the whole closed interval, but a derivative is needed only inside it. Since differentiability implies continuity, a function that is differentiable at every point of $[a, b]$ meets both hypotheses. On a free-response answer, name the theorem and say in words why each hypothesis holds.

This is the Mean Value Theorem when the secant line is horizontal. The average rate of change is $\frac{f(b) - f(a)}{b - a} = 0$, so somewhere between $a$ and $b$ the tangent line is horizontal too.

The Mean Value Theorem also follows from Rolle's Theorem. Let $m$ be the slope of the secant line of $f$ on $[a, b]$, and subtract that line from $f$:

Then $g$ is continuous on $[a, b]$ and differentiable on $(a, b)$, with $g(a) = g(b) = 0$. By Rolle's Theorem, $g'(c) = f'(c) - m = 0$ for some $c$ in $(a, b)$, so $f'(c) = m$.

When $f$ is given by a formula, check the two hypotheses, compute the average rate of change, and solve $f'(c)$ equal to it. Keep only the solutions that lie in $(a, b)$.

When $f$ is known only at a few points, the theorem still guarantees values of $f'$. For any two points in the table, $f'$ equals the slope between them at some $c$ in between. The problem has to say that $f$ is differentiable. A table can't show that, and if $f$ is only continuous, the theorem can't be used.

If $f$ fails either hypothesis on $[a, b]$, the theorem says nothing about $f$ there. The conclusion can fail, or it can hold anyway.

For an object with position $s(t)$ on a line, the average rate of change $\frac{s(t_2) - s(t_1)}{t_2 - t_1}$ is its average velocity on $[t_1, t_2]$. When $s$ is differentiable, the Mean Value Theorem says the velocity $s'(t)$ equals that average at some instant in $(t_1, t_2)$.