Mean-Value Theorem

Mean-Value Theorem

First presented in a variant form by the French mathematician Michel Rolle (1652-1719) in an obscure book, the mean-value theorem is one of the fundamental principles of the discipline of calculus. Stated mathematically, the mean-value theorem asserts that if a function f(x) is both continuous and differentiable over the closed interval [a, b] (that is, the interval includes the endpoints a and b), then there exists at least one number c (and perhaps more) such that the first derivative of the function (symbolized by f^'(x)), evaluated at c, is equal to the difference of the function evaluated at b and a divided by the difference of b and a. Given the conditions of continuity and a closed interval, the mean-value theorem can be written (in symbolic form) f^'(c) = [f(b)--f(a)] / (b--a). If the quantity [f(b)--f(a)] / (b--a) is considered the average, or mean, rate of change of the function f over the given interval [a, b] and f^'(c) is considered an instantaneous change, there must exist at least one interior point c at which the instantaneous rate of change equals the average rate of change.

Relationship to Rolle's Theorem

Published in 1691, Rolle's theorem is a simplified version of the mean-value theorem. Rolle's theorem asserts that between any two "zeros" a and b of a continuous and differentiable function f(x) (that is, f(a) = f(b) = 0), there exists at least one point c (and perhaps more) such that f^'(c) = 0 in the interval [a, b]. The value c is called a critical point. If f(x) is not equal to a constant over this interval, Rolle's theorem implies that there exists at least one (and perhaps more) local maximum or local minimum in the interval at which the slope of the function changes sign, such as f^' > 0 changing to f^' < 0 or f^' < 0 changing to f^' > 0).

Integral Form of the Mean-Value Theorem

The form of the mean-value theorem given above is its differential case, and as one would expect, a form of the mean-value theorem also exists for the integral case. This case involves the definite integral over the interval [a, b], and states that at some point c, the continuous function f(x) is equal to 1 / (b--a) multiplied by the integral from a to b of f(x)dx. Written symbolically, if f is continuous on [a, b], then there exists some point c such that f(c) = 1 /(b--a) * int(f(x)dx).

Like the differential case, the integral case of the mean-value theorem over a closed interval [a, b] also has a relationship to an average value of the function. It turns out that if f is integrable over the interval [a, b], then the average, or mean, value of the function in this interval is exactly 1 / (b--a) * int(f(x)dx). Any points satisfying the condition that f(c) is equal to this quantity are mean values. It should be noted that if f is integrable, the quantity 1 /(b--a) * int(f(x)dx) is in itself an expression of the average change of the function F(x), where F^'(x) = f(x). This means that f(c) is actually a point on the interval [a, b] where the instantaneous rate of change of the function F(x) is equal to its average rate of change over the same interval.