Mean Value Theorem

Definition

The Mean Value Theorem states that for a continuous function $f$ on the interval $[a, b]$ that is differentiable on $(a, b)$, there exists at least one point $c$ in $(a, b)$ where the instantaneous rate of change (the derivative) equals the average rate of change over $[a, b]$. Mathematically, this is expressed as $f'(c) = \frac{f(b) - f(a)}{b - a}$.

5 Must Know Facts For Your Next Test

1. The function must be continuous on the closed interval $[a, b]$.
2. The function must be differentiable on the open interval $(a, b)$.
3. There exists at least one point $c$ in the interval $(a, b)$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$.
4. The theorem can be used to prove other important results in calculus such as Rolle's Theorem.
5. It provides a formal way to find points where the tangent to the curve is parallel to the secant line joining $(a, f(a))$ and $(b, f(b))$.

Review Questions

• What are the two main conditions required for applying the Mean Value Theorem?
• How does the Mean Value Theorem formally express that there is a point where the instantaneous rate of change equals the average rate of change?
• Why is differentiability on an open interval $(a, b)$ critical for applying this theorem?

Related terms

Rolle's Theorem: A special case of the Mean Value Theorem where if $f(a) = f(b)$ for a continuous function on $[a,b]$, then there exists some $c$ in $(a,b)$ such that $f'(c) = 0$.

Continuity: A property of a function if it is defined and does not have any gaps or jumps over its domain.

Differentiability: A property of a function if it has a derivative at each point in its domain.