Mean value theorem

The Mean Value Theorem is a theorem in calculus. Roughly, it states that given a curve and a section of that curve, there is a point on that section at which the gradient of the curve is equal to the "average" gradient of the section.

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

More precisely, the 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.

Sketch of mean value theorem

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 c in (a, b) such that

$f'(c)={\frac {f(b)-f(a)}{b-a}}$

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.

Proof

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 chord such that the tangent at that point is parallel to the chord. The following proof illustrates this idea.

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

$g(a)=g(b)\qquad \Rightarrow \qquad f(a)+ra=f(b)+rb$

$\Rightarrow \qquad r=-{\frac {f(b)-f(a)}{b-a}}$

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

$f'(c)=g'(c)-r=0-r={\frac {f(b)-f(a)}{b-a}}$

as required.

The mean value theorem in the following form is considered more useful.

$f(b)-f(a)=f'(c)\times (b-a)$

Cauchy's mean value theorem

Cauchy's mean value is the more generalised form of mean value theorem. It states: If functions y(t) and x(t) are both continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then exist some c such that

${\frac {y'(c)}{x'(c)}}={\frac {y(b)-x(a)}{y(b)-x(a)}}$

is true.

Cauchy's mean value theorem can be used to proof l'Hopital's rule.

Proof of Cauchy's mean value theorem

The proof of Cauchy's mean value theorem is based on the same idea as the proof of mean value theorem. We aim to transform the curve defined by y = y(t) and x = x(t), so that it satifies the conditions of Rolle's theorem.

We define a new function:

$F(t)=y(t)-m\times x(t)$

where m is a constant, so that

$F(a)=F(b)\qquad \Rightarrow \qquad m={\frac {y(b)-y(a)}{x(b)-x(a)}}$

Since F is continuous and F(a) = F(b), by Rolles theorem, there exist some c in (a,b) such that F'(c) = 0, i.e.

$F'(c)=0\ =\ y'(c)-{\frac {y(b)-y(a)}{x(b)-x(a)}}\times x'(c)$

$\Rightarrow \qquad {\frac {y'(c)}{x'(c)}}\ =\ {\frac {y(b)-y(a)}{x(b)-x(a)}}$

as required.

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

$\int _{a}^{b}f(t)\;\varphi (t)\;dt\quad =\quad f(x)\int _{a}^{b}\varphi (t)\;dt$

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

$\int _{a}^{b}f(t)\;dt\quad =\quad f(x)(b-a)$

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

$\int _{a}^{b}f(t)\;\varphi (t)\;dt\quad =\quad (\lim _{t\to a}f(t))\cdot \int _{a}^{x}\varphi (t)\;dt$