Traces of the general mean value theorem are also found in his works.

The mean value theorem is still valid in a slightly more general setting.

He also refined the second mean value theorem of integration.

Indeed, it is needed to prove both the mean value theorem and the existence of Taylor series.

It can be generalized to more variables according by the mean value theorem for divided differences.

It can be obtained from the mean value theorem by choosing .

If we place and we get Lagrange's mean value theorem.

There are various slightly different theorems called the "'second mean value theorem for integration"'.

In practice, what the mean value theorem does is control a function in terms of its derivative.

More formally, P is found in the mean value theorem of calculus, which says:

There is no exact analog of the mean value theorem for vector-valued functions.

To begin, we use the Mean value theorem:

Next, we employ the mean value theorem.

From the mean value theorem, we know that the vehicle's speed must equal its average speed at some time between the measurements.

Since 'g' is a differentiable function in one variable, the mean value theorem gives:

The mean value theorem gives a relationship between values of the derivative and values of the original function.

This exposes Taylor's theorem as a generalization of the mean value theorem.

One can use the mean value theorem (for real-valued functions of many variables) to see that this does not rely on taking first order approximation.

The remainder term in the Lagrange form can be derived by the mean value theorem in the following way:

The proof is easier for twice continuously differentiable (mean value theorem), but may be proved in a distributional sense as well.

Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value.

The first derivative test depends on the "increasing-decreasing test", which is itself ultimately a consequence of the mean value theorem.

These refinements of Taylor's theorem are usually proved using the mean value theorem, whence the name.

He worked on functional analysis, harmonic analysis, ergodic theory, mean value theorems, and numerical integration.

That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.