PlanetMath: mean-value theorem

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mean-value theorem

(Theorem)

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

$\displaystyle f'(c) = \frac{f(b) - f(a)}{b - a}.$

(1)

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

$\includegraphics[scale=0.4]{mittelwertsatz.ps}$

The dashed line connects the points

and

. There is

between

and

at which the tangent to

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

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

(2)

"mean-value theorem" is owned by mathwizard. [ full author list (2) | owner history (1) ]

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Attachments:: proof of mean value theorem (Proof) by Andrea Ambrosio
complex mean-value theorem (Theorem) by matte
fundamental theorem of integral calculus (Theorem) by pahio

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Cross-references: integral, slope, tangent, points, line, theorem, number, differentiable, interval, continuous, function
There are 17 references to this entry.

This is version 6 of mean-value theorem, born on 2002-02-15, modified 2004-07-21.
Object id is 1990, canonical name is MeanValueTheorem.
Accessed 21269 times total.

Classification:

26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda

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