Page 14 - Calculus Demystified

CHAPTER 1

Basics

1.0 Introductory Remarks

Calculus is one of the most important parts of mathematics. It is fundamental to all
of modern science. How could one part of mathematics be of such central impor-
tance? It is because calculus gives us the tools to study rates of change and motion.
All analytical subjects, from biology to physics to chemistry to engineering to math-
ematics, involve studying quantities that are growing or shrinking or moving—in
other words, they are changing. Astronomers study the motions of the planets,
chemists study the interaction of substances, physicists study the interactions of
physical objects. All of these involve change and motion.
In order to study calculus effectively, you must be familiar with cartesian geome-
try, with trigonometry, and with functions.We will spend this first chapter reviewing
the essential ideas. Some readers will study this chapter selectively, merely review-
ing selected sections. Others will, for completeness, wish to review all the material.
The main point is to get started on calculus (Chapter 2).

1.1 Number Systems

The number systems that we use in calculus are the natural numbers, the integers,
the rational numbers, and the real numbers. Let us describe each of these:

• The natural numbers are the system of positive counting numbers 1, 2, 3, ….
We denote the set of all natural numbers by N.

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