# mean-value theorem

Let $f:\mathbb{R}\to\mathbb{R}$ be a function which is continuous on the interval $[a,b]$ and differentiable on $(a,b)$. Then there exists a number $c:a such that

 $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}.$ (1)

The geometrical meaning of this theorem is illustrated in the picture:

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

 $(b-a)f(c)=\int_{a}^{b}f(x)dx.$ (2)
Title mean-value theorem MeanvalueTheorem 2013-03-22 12:20:39 2013-03-22 12:20:39 mathwizard (128) mathwizard (128) 9 mathwizard (128) Theorem msc 26A06 RollesTheorem IntermediateValueTheorem ExtendedMeanValueTheorem ProofOfExtendedMeanValueTheorem DerivationOfWaveEquation