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Mathematics-Online lexicon: Annotation to

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.

The function

$\displaystyle g(x)= f(x)- f(a) - \frac{f(b)-f(a)}{b-a}(x-a) $

vanishes at $ x=a$ and $ x=b$. Since $ f$ is continuously differentiable, by Rolle's theorem, there exists $ t \in (a,b)$ with

$\displaystyle 0 = g^\prime(t)$ $\displaystyle = f^\prime(t) - \frac{f(b)-f(a)}{b-a}$    

i.e.


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

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  automatisch erstellt am 14.  6. 2016