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Homemean-value theorem

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# 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 $c:a<c<b$ 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.

$(b-a)f(c)=\int_{{a}}^{{b}}f(x)dx.$ | (2) |

Related:

RollesTheorem, IntermediateValueTheorem, ExtendedMeanValueTheorem, ProofOfExtendedMeanValueTheorem, DerivationOfWaveEquation

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

26A06*no label found*

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