Mean value theorem

In calculus, the Mean Value Theorem states: for some continually differentiable curve; for every secant, there is some parallel tangent. In addition, the tangent runs through a point located between the intersection points of said secant. This theorem was developed by Lagrange. Some mathematicians consider this theorem to be the most important theorem of calculus (see also: the fundamental theorem of calculus). The theorem is not often used to solve mathematical problems; rather, it is more commonly used to prove other theorems.

Let f : [a, b] → R be continuous on the closed interval [a, b], and differentiable on the open interval (a, b). Then there exists some x in (a, b) such that

<math>f ' (x) = \frac{f(b) - f(a)}{b - a}</math>

An understanding of this and the Point-Slope Formula will make it clear that the equation of a secant (which intersects (a, f(a)) and (b, f(b)) ) is: y = {[f(b) - f(a)] / [b - a]}(x - a) - f(a).

The formula ( f(b) - f(a) ) / (b - a) gives the slope of the line joining the points (a , f(a)) and (b , f(b)), which we call a chord of the curve, while f ' (x) gives the slope of the tangent to the curve at the point (x , f(x) ). Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the cord such that the tangent at that point is parallel to the chord.

The mean value theorem can be used to prove Taylor's theorem, of which it is a special case.

Proof of the theorem: Define g(x) = f(x) + rx , where r is a constant. Since f is continuous on [a , b] and differentiable on (a , b), the same is true of g. We choose r so that g satisfies the conditions of Rolle's theorem, which means

<math>

f(a) + ra = f(b) + rb </math>

<math>

\Rightarrow r = - \frac{ f(b) - f(a) }{ b - a} </math>

By Rolle's Theorem, there is some x in (a , b) for which g '(x) = 0, and it follows

<math>

f ' (x) = g ' (x) - r = 0 - r = \frac{ f(b) - f(a) }{ b - a} </math> as required.

Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : [a , b] → R is continuous on [a , b], and that for every x in (a , b) the limit lim_h→0 (f(x+h)-f(x))/h exists or is equal to ± infinity.

Mean value theorems for integration

The first mean value theorem for integration states:

If f : [a , b] → R is a continuous function and φ : [a , b] → R is an integrable positive function, then there exists a number x in (a , b) such that

<math>

\int_a^b f(t) \; \varphi (t) \; dt \quad = \quad f(x) \int_a^b \varphi (t) \; dt </math>

In particular (φ(t) = 1), there exists x in (a , b) with

<math>

\int_a^b f(t) \; dt \quad = \quad f(x) (b - a) </math>

The second mean value theorem for integration states:

If f : [a , b] → R is a positive and monotone decreasing function and φ : [a , b] → R is an integrable function, then there exists a number x in (a , b] such that

<math>

\int_a^b f(t) \; \varphi (t) \; dt \quad = \quad ( \lim_{t \to a} f(t) ) \cdot \int_a^x \varphi (t) \; dt </math>