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 $f'(c) = \frac{f(b) - f(a)}{b - a}$.
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The function must be continuous on the closed interval $[a, b]$.
The function must be differentiable on the open interval $(a, b)$.
There exists at least one point $c$ in the interval $(a, b)$ where $f'(c) = \frac{f(b) - f(a)}{b - a}$.
The theorem can be used to prove other important results in calculus such as Rolle's Theorem.
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 $f'(c) = 0$.