Finite-increments formula

Lagrange finite-increments formula

A formula expressing the increment of a function in terms of the value of its derivative at an intermediate point. If a function is continuous on an interval on the real axis and is differentiable at the interior points of it, then

The finite-increments formula can also be written in the form

The geometric meaning of the finite-increments formula is: Given the chord of the graph of the function with end points , , then there exists a point , , such that the tangent to the graph of the function at the point is parallel to the chord (see Fig.).

Figure: f040300a

The finite-increments formula can be generalized to functions of several variables: If a function is differentiable at each point of a convex domain in an -dimensional Euclidean space, then there exists for each pair of points , a point lying on the segment joining and and such that

Comments

This formula is usually called the mean-value theorem (for derivatives). It is a statement for real-valued functions only; consider, e.g., .