This chapter brings together all the basic theoretical tools that enable the application of calculus to other areas. We then go on to work through some of those applications, optimization problems, and the qualitative analysis of functions (curve sketching).
The Mean Value Theorem, or MVT, is a cornerstone of the theory behind calculus. As a first application it establishes, theoretically, the connection between the sign of the derivative (positive or negative) and the behavior of tht the function itself. It also justifies the idea of an antiderivative of a function, almost totally reconstructing a function just from knowing the derivative. As preliminary results needed to prove the mean value theorem, we also establish the basis for our optimization problems to come (the Maximum Theorem) and a classic lemma (a preliminary result whose only purpose is to prove another theorem) called Rolle's Theorem.
The main purpose of the MVT is not the silly sorts of problems mentioned in most texts, but in proving other theoretical results. We show here how to derive a number of consequences from the MVT. Some of them will seem trivial, but there is a subtlety in that they are not so trivial to prove without the MVT or something equivalent.
In the process of proving the MVT, we showed that you could locate the maximum or minimum of a function by looking at where the derivative of the function is zero (or at endpoints, or at places where the derivative does not exist). The implication of that observation is that, given any situation, you can optimize any component, find where its value is the ``best'', by looking at only a few values rather than finding the value of that component at all points.
l'Hôpital's rule is a simple way to compute difficult limits. It is useful to help with curve sketching, but the reason it is in this chapter is that it as well is a consequence of the MVT.
The idea behind curve sketching is to translate the quantitative information you can determine from the formula describing a function into qualitative information, interpreting the quantitative information in terms of the dynamics of the function (expressed as the graph, or picture, of the function). Conversely, given that qualitative information, you can conclude a number of things about the function that have quantatitive meaning. The two ways of understanding a function complement each other.
Copyright (c) 2000 by David L. Johnson.