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Mean Value Theorem

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For a continuously differentiable function $ f$, there exists $ t \in (a,b)$ with

$\displaystyle f(b)-f(a) = f^\prime(t)(b-a) \,. $

\includegraphics[width=0.6\linewidth]{mittelwertsatz}

The geometric interpretation of this identity is that the tangent at $ t$ is parallel to the secant connecting the points $ (a,f(a))$ and $ (b,f(b))$. This suggests the alternative notation $ \Delta y = f^\prime(t)\Delta x$ for describing the mean value theorem.

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  automatically generated 6/14/2016