# The Mean-value Theorem

Consider the quantity Q defined by the equation

 (13.1)

or

 (13.2)

Let be a function formed by replacing by in the left-hand member of (13.2); that is,

 (13.3)

From (13.2), , and from (13.3), ; therefore, by Rolle's Theorem (see §13.1), must be zero for at least one value of between and , say for . But by differentiating (13.3) we get

Therefore, since , then also , and . Substituting this value of Q in (13.1), we get the Theorem of Mean Value13.1,

 (13.4)

where in general all we know about is that it lies between and .

The Theorem of Mean Value interpreted Geometrically.

Let the curve in the figure be the locus of .

Figure 13.3: Geometric illustration of the Mean value theorem.

Take and ; then and , giving and . Therefore the slope of the chord AB is

There is at least one point on the curve between A and B (as P) where the tangent (or curve) is parallel to the chord AB. If the abscissa of P is the slope at P is

Equating these last two equations, we get

which is the Theorem of Mean Value.

The student should draw curves (as the one in §13.1), to show that there may be more than one such point in the interval; and curves to illustrate, on the other hand, that the theorem may not be true if becomes discontinuous for any value of between and (see Figure 13.2 (a)), or if becomes discontinuous (see Figure 13.2 (b)).

Clearing (13.4) of fractions, we may also write the theorem in the form

 (13.5)

Let ; then , and since is a number lying between and , we may write

where is a positive proper fraction. Substituting in (13.4), we get another form of the Theorem of Mean Value.

 (13.6)

david joyner 2008-11-22