Some Geometric Considerations Related to the Mean Value Theorem

Abstract

In the teaching of differential calculus, insufficient attention is given to the geometric interpretations of both the generating function for the mean value theorem (hereafter referred to as MVT) and the conclusion of the MVT. An article to this effect was written some 34 years ago by A. A. Bennett and published in the AMERICAN MATHEMATICAL MONTHLY (The Consequences of Rolle's Theorem,' Vol. 31, p. 40). The few words written here have little in common with those of Dr. Bennett except to emphasize the need for the understanding of the MVT which can come most easily from interpreting it geometrically. Geometric interpretation of the MVT can be divided into three phases: (a) interpreting a generating function to be used in obtaining the desired results, (b) interpreting the fashion in which the generating function satisfies Rolle's Theorem, and (c) interpreting the conclusion of the MVT relative to the generating function. Many generating functions (and, by generating function is meant the function which will be shown to possess the properties of Rolle's Theorem and which will be differentiated to yield the conclusion of the MVT) are available. Various generating functions lead to various forms of the conclusion of the MVT. Let us suppose that the functions with which we deal all satisfy the ordinary continuity conditions imposed by the MVT. To obtain the ordinary MVT, the generating functions may be interpreted primarily as either area functions or length functions. Consider the function f(x) and the chord joining [a, f(a)] to [b, f(b)]. The generating function which most texts ask the student to consider is