mean-value theorem

Let f: be a function which is continuousMathworldPlanetmath on the interval [a,b] and differentiableMathworldPlanetmathPlanetmath on (a,b). Then there exists a number c:a<c<b such that

f(c)=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 tangentPlanetmathPlanetmath 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[a,b] such that

(b-a)f(c)=abf(x)𝑑x. (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