Derivative:
                   Calculus is the mathematics of change, and the primary tool for studying change is a
procedure called differentiation. In this section, we shall introduce this procedure and
examine some of its uses, especially in computing rates of change. Here, and later in
this chapter, we shall encounter rates such as velocity, acceleration, production rates
with respect to labor level or capital expenditure, the rate of growth of a population,
the infection rate of a susceptible population during an epidemic, and many others.
Calculus was developed in the seventeenth century by Isaac Newton (1642–1727)
and G. W. Leibniz (1646–1716) and others at least in part in an attempt to deal with
two geometric problems:
Tangent problem: Find a tangent line at a particular point on a given curve.
Area problem: Find the area of the region under a given curve.
The area problem involves a procedure called integration in which quantities such as
area, average value, present value of an income stream, and blood flow rate are
computed as a special kind of limit of a sum. We shall study this procedure in Chapters
5 and 6. The tangent problem is closely related to the study of rates of change, and
we will begin our study of calculus by examining this connection.
Recall from Section 1.3 that a linear function L(x) mx b changes at the constant rate
m with respect to the independent variable x. That is, the rate of change of L(x) is given
by the slope or steepness of its graph, the line y mx b (Figure 2.1a). For a function
f(x) that is not linear, the rate of change is not constant but varies with x. In particular,
when x c the rate is given by the steepness of the graph of f(x) at the point P(c, f(c)),
which can be measured by the slope of the tangent line to the graph at P (Figure 2.1b).
The relationship between rate of change and slope is illustrate