mean-value theoremMeanValueTheorem2002-02-15 21:44:172004-07-21 08:51:04Theorem\usepackage{amssymb}
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\newcommand{\set}[1]{\{#1\}}Let $f:\reals \to \reals$ be a function which is continuous on the interval $[a,b]$ and differentiable on $(a,b)$. Then there exists a number $c: a < c < b$ such that
\begin{equation}
f'(c) = \frac{f(b) - f(a)}{b - a}.
\end{equation}
The geometrical meaning of this theorem is illustrated in the picture:\\[10pt]
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The dashed line connects the points $(a,f(a))$ and $(b,f(b))$. There is $c$ between $a$ and $b$ at which the tangent to $f$ has the same slope as the dashed line.
The mean-value theorem is often used in the integral context: There is a $c \in [a,b]$ such that
\begin{equation}
(b-a)f(c) = \int_{a}^{b} f(x) dx.
\end{equation}