The Mean-value Theorem

Consider the quantity Q defined by the equation

or

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

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

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 .

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

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

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

david joyner 2008-11-22