Mean value theorem

The mean value theorem is a key theorem of differential calculus, a section of the Analysis ( Mathematics ).

Demonstrates the mean value theorem can be geometrically interpreted that there is under the below mentioned conditions between two points of a function graph at least one point on the curve for which the tangent is parallel to the secant line through the two given points.

The statement of the theorem can be applied both transferred to the quotient of two functions as well as functions of several variables.

Statement of the Mean Value Theorem

It is a function, which is (partly) defined in the closed interval, and continuously. Also, the function is differentiable in the open interval. Under these conditions, there is at least one, so that

Applies.

Interpreted Geometrically this means that the Sekantensteigung occurs at least an intermediate point as the slope of the tangent to the graph of the function.

Evidence in the one-dimensional case

It should define a helper function

Is continuous in and differentiable. It is true.

After the set of role therefore exists with. because

The assertion follows.

Example of an application of the mean value theorem

As a typical application of the average rate can be shown that

Are all: Without limitation, we may assume. Since the sine function is differentiable in the interval, there exists by the mean value theorem, so that

Applies. Because for all, one obtains

In general it can be so demonstrated that continuously differentiable functions are locally Lipschitz continuous.

Extended mean value theorem of differential calculus

The mean value theorem can be generalized in the following way:

Let and two functions that are on the closed interval ( with ) defined and differentiable on the open interval and steadily. Under these conditions, there is at least one, so that

Applies.

If additionally provided on the interval, so is particularly well and you can write the extended mean value theorem in the usual form of a fraction,

Mean value theorem for real-valued functions of several variables

In the multivariate analysis the mean value theorem is as follows:

It is a picture with further is differentiable on an open set. Also, be with and their link. Then there exists at least one with and and we have:

For the record corresponds to the above-mentioned mean value theorem of one-dimensional differential calculus. Herein, the gradient at the site, which occurs in a scalar.

Geometrically interpreted, the Sekantensteigung between and at least one point from occurs as the slope in the direction of the vector.

Proof in the multidimensional case

Looking at the function with

So on and differentiable on steadily. Thus it follows from the mean value theorem of dimensional analysis, that a exists such that

From the chain rule now follows:

This can be summarized as follows:

Is substituted by now, it is clear

Which the statement of the theorem would be proved.

Mean value theorem for vector-valued functions of several variables

An extension of the theorem to functions is only possible under different geometric conditions or tightening. In particular, the pool of eligible linear maps is extended significantly beyond the discharges on the track out:

If the discharges are limited by the entire route (it is on to Jacobians, so limited relative to a standard, for example, the operator norm ), so there is a linear map from the closed convex hull of discharges on the link, so that

Applies.

The proof requires some preliminary work, including the Hahn- Banach'schen separation theorems, but ultimately follows the principle of returning to the real-valued case. Why the derivatives are not enough on the track, can be understood as follows: The individual components of the vector-valued function on the one hand the mean value theorem for real-valued functions of several variables are applied. On the other hand there is no assurance that the associated location on, at the appropriate derivative is found for all of the components have the same functions. One must, therefore, in a larger quantity around, just the convex hull of discharges on the track.

Descriptive meaning

Describes the function of, for example, a distance a function of a time, then the derivative of the speed. The mean value theorem then states: On the way from A to B you have to be at least at one time was as fast as his average speed.