THE APPLICATIONS OF CALCULUS TO BUSINESS FIELD

The Applications of Calculus to Business Field

The Applications of Calculus to Business Field

Background

Calculus, the branch of mathematics designed to represent and study continually changing quantities (Allen, G. D., and Chui, Charles F., 1983). Though the word "calculus" is sometimes used to mean any procedure for calculating some quantity, usually when one speaks of "calculus" or "the calculus," one means the particular branch of mathematics created in the 17th century and expanded and perfected in the next two centuries. The basic laws of science and engineering are expressed in terms of concepts of the calculus, and the deduction of knowledge about the physical phenomena so represented makes intensive use of the techniques of the calculus.

The Completion of the Calculus

The two basic operational ideas of the calculus are differentiation (together with its inverse, antidifferentiation) and the definite integral. Underlying each of these is a limit concept. The derivative rests on the notion of a limit of a function, and the definite integral rests on the notion of a limit of a sequence. Let us reconsider the first of these limit notions, though much of what we shall say applies to both (Baum, Alan, et al., 1985).

The description given earlier of the limit of a function is admittedly vague. In particular the word "approach" is suspect. If for smaller and smaller values of h the ratio k/h should have the values 1/4, 3/8, 7/16, 15/32…, are these values approaching 1? They are indeed getting closer to 1, but it is also clear that they are always less than 1/2, and so the limit might very well be 1/2. How closely must the values of k/h approach a particular number before we can decide that that number is the limit of k/h? Though we had no difficulty in determining the limit in the cases of particular functions, we have used loose language to describe what we mean by a limit. That we were obliged to do so will be apparent in a moment (Edwards, C. H., Jr., and David E. Penney, 1988).

The history of the efforts of mathematicians to grasp the limit concept properly is extensive and instructive as to how mathematics develops. We have already mentioned that many mathematicians of the 17th century made contributions to the calculus even before Newton and Leibniz began to work on the subject. These forerunners realized that they were unable to give satisfactory expositions of their ideas and, in fact, hardly comprehended the significance of what they were creating. Despite the long tradition of rigorous proof in mathematics, the early workers in the calculus did not hesitate to defend their work in ways that are outlandish for mathematicians. Rigor, said Bonaventura Cavalieri, a pupil of Galileo and professor at the University of Bologna, is the concern of philosophy and not of geometry. Pascal argued that the heart intervenes to assure us of the correctness of mathematical steps. Proper finesse rather than logic is what is needed to do the correct thing (Ross, ...