chapter  6
44 Pages

Functions of one variable: applications

In this chapter, calculus techniques are applied to the study of a variety of prob-

lems, 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 ∈ 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.