The Mean Value Theorem

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The Fundamental Theorem of Calculus

Three Different Quantities
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The Mean Value Theorem

Rolle's Theorem is a special case of the Mean Value Theorem which says that there has to be a point between $a$ and $b$ where the instantaneous rate of change is equal to the average rate of change between $a$ and $b$ . More precisely:

Mean Value Theorem: If

$f$ is a function that is continuous on the closed interval

$[a,b]$ and differentiable on the open interval

$(a,b)$ , then there is a point

$c$ between

$a$ and

$b$ such that

$\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$ .