ABSTRACT
In this chapter, calculus techniques are applied to the study of a variety of problems, beginning with optimization in Section 6.2. In an optimization problem, given a function f of one variable x, the problem is to find the value of x, x* that maximizes f. Such an x* is characterized by the condition that f(x*) ≥ f(x) for all x. If the choice of x is restricted to some set X, then the requirement is that f(x*) ≥ f(x) for all x ϵ X. Calculus techniques are used to find x* by considering how f varies with x. These techniques are local in nature and so locate “local” maxima (or minima), where there is no alternative choice close to the candidate optimum that yields a higher value of the function f (or lower value if the task is to minimize the value of f). In contrast, x* is a global maximizer if f(x*) is as large as f(x) for any possible x. Identifying whether a value x* is a local or global maximizer is considered in Section 6.3. This discussion focuses on the shape of the function f, and the use of calculus techniques to identify the shape.
