ONE - PRELIMINARIES

Abstract

This chapter presents some basic information needed for the study of calculus. It discusses the real number system. The real numbers fall into several categories. The positive integers (sometimes called the natural numbers) are the numbers of counting: 1, 2, 3, 4, 5, …, (the three dots indicate that the string of numbers goes on indefinitely). The integers consist of the positive integers, the negative integers, and the number 0. A rational number is a real number that can be written as the quotient of two integers, where the integer in the denominator is not zero. The chapter discusses the properties of the pairs of numbers. Lines are very important in the study of calculus. The chapter discusses lines. An equation of a line is an equation in the variables x and y satisfied by the coordinates of every point on the line. The graph of an equation in x and y is the set of all points drawn in the xy-plane of which coordinates satisfy the equation. If two points on a line are known, then one can calculate the equation of the line and obtain its graph. A circle is defined as the set of all points at a given distance from a given point. The given point is called the center of the circle and the common distance from the center is called the radius. The chapter presentsthe equation of the circle centered at the origin with radius 1. It discusses functions and shows how functions can be added, subtracted, multiplied, and divided. While more advanced methods are needed to obtain the graphs of most functions (without plotting a large number of points), there are some techniques that make it a relatively simple matter to sketch certain functions.