Some Functional Equations for the Calculus Student

Abstract

Among the errors frequently made by students in a calculus course are false assumptions such as the following: (a + b)2 = a2 + b2, sin(a + b) = sin a + sin b, log(a + b) = log a + log b, for all real a and b. These would seem to be examples of "the student's tendency to overgeneralize the properties of operations," in this case the distributive property, as discussed by Shumway [3]. Here the function operator is assumed to "distribute over" the real operation of addition. Typically, one points out the mistake of such assumptions by providing counterexamples or "negative instances." Shumway's review of the literature finds that this strategy is not always successful and, in fact, that the use of negative instances can have a detrimental effect on concept learning. One might then wish to try an alternative approach and address the question directly: What functions f, if any, have the property f(a + b) = f(a) + f(b) for all real a and b? The problem, identified as Cauchy's equation, has received a great deal of attention over the years. Young [4] makes reference to the problem as one frequently posed in advanced calculus. Aczel [1] provides an interesting historical account of the problem. In both of these works, the emphasis seems to be on looking for solutions to the problem under the conditions of continuity or boundedness of the function f on a closed interval. Arguments under these conditions require a degree of sophistication generally identified with upper level undergraduate mathematics students. If, however, one is willing to require the stronger condition of differentiability, then the solution to the problem is accessible to the calculus student. Since most of the functions in elementary calculus are differentiable at all but perhaps a few points, it seems reasonable to work under this condition. The problem then which will be considered here is: What differentiable