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Mean Value 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)

1	Incidence Geometry
1	Point
1	Differntiable Function
1	Continuous
1	Interval
1	Function
1	Number
2	Integral
3	Definitions In Trigonometry
3	Slope