• Understand and use Rolle’s Theorem.
  • Understand and use the Mean Value Theorem.

Assignment

  • Vocabulary and teal boxes
  • p224 1, 2, 9–21 odd, 24–26, 29, 31, 35, 40–48 even, 52, 53 64–66, 81–83

Additional Resources

Rolle’s Theorem

Rolle’s theorem is another theorem that applies a rigorous definition to something that, at least in insight, is fairly simple.

If a real-valued function $f$ is continuous on a proper closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the open interval $(a, b)$ such that $f’( c ) = 0$.

Drawing a curve between two points with the same vertical means at some point that curve is flat.

Mean Value Theorem

Rolle’s Theorem is really a jumping off point for the meat of the section, the Mean Value Theorem.

If $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists a number $c$ in $(a,b)$ such that

\[f'(c)=\frac{f(b)-f(a)}{b-a}\]

In other words, if you draw a secant line from the point on the curve at the beginning of the interval to the end, there is at least one point on the curve itself with a matching slope.

The proof for this is in the book, but a visual representation might help first. Here is a graph in Desmos that shows a function on a closed interval with a secant line connecting the two endpoints. The third function $h(x)$, which is just the difference between the first two, fits Rolle’s Theorem since the endpoints of the two are the same. That guarantees at least one $h’(c)=0$, meaning the derivatives of the other two functions at $c$ must be the same.

The Mean Value Theorem is an important underlying concept for both problem solving and proving other theorems. It’s also the entire title of AP topic 5.1, so don’t overlook it.