# 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 $$ 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

$$(b-a)f(c)={\int}_{a}^{b}f(x)\mathit{d}x.$$ | (2) |

Title | mean-value theorem |
---|---|

Canonical name | MeanvalueTheorem |

Date of creation | 2013-03-22 12:20:39 |

Last modified on | 2013-03-22 12:20:39 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 9 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 26A06 |

Related topic | RollesTheorem |

Related topic | IntermediateValueTheorem |

Related topic | ExtendedMeanValueTheorem |

Related topic | ProofOfExtendedMeanValueTheorem |

Related topic | DerivationOfWaveEquation |