4-PrincipCalculus Page 91 Friday, March 12, 2004 12:39 PM

I

CHAPTER

4

Principles of Calculus

nvented in the seventeenth century independently by the British physi-

cist Isaac Newton and the German philosopher G.W. Leibnitz, (infini-

tesimal) calculus was a major mathematical breakthrough; it was to

make possible the modern development of the physical sciences. Calcu-

lus introduced two key ideas:

■ The concept of instantaneous rate of change.

■ A framework and rules for linking together quantities and their instan-

taneous rates of change.

Suppose that a quantity such as the price of a financial instrument

varies as a function of time. Given a finite interval, the rate of change of

that quantity is the ratio between the amount of change and the length

of the time interval. Graphically, the rate of change is the steepness of

the straight line that approximates the given curve.^1 In general, the rate

of change will vary as a function of the length of the time interval.

What happens when the length of the time interval gets smaller and

smaller? Calculus made the concept of infinitely small quantities precise

with the notion of limit. If the rate of change can get arbitrarily close to

a definite number by making the time interval sufficiently small, that

number is the instantaneous rate of change. The instantaneous rate of

change is the limit of the rate of change when the length of the interval

gets infinitely small. This limit is referred to as the derivative of a func-

tion, or simply, derivative. Graphically, the derivative is the steepness of

the tangent to a curve.

Starting from this definition and with the help of a number of rules

for computing a derivative, it was shown that the instantaneous rate of

(^1) The rate of change should not be confused with the return on an asset, which is the
asset’s percentage price change.
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