A Concise Introduction to Calculus

The following sections are included:

• Introduction

• Contents

The following sections are included:

• Real Numbers and Measurement

• Variables and Functions

Recall that calculus is a branch of mathematics which provides the basic techniques and general framework for quantitative analysis of the correlations among variable quantities of a given system or phenomenon. Thus, functions are exactly the primary subject of study in calculus. In this chapter, we shall begin our study of functions by analyzing the intuitive background of some basic features as well as some fundamental properties of functions. To avoid unnecessary complication at the very beginning, we shall only consider the simplest case of real value functions of a single real variable, namely, functions of the type y = f(x) with x, y varies over certain subsets of the real number system ℝ.

Approximation is the fundamental methodology which plays an important role throughout the entire realm of calculus while limit is the basic concept that naturally arises from applications of approximation methodology.

The basic idea of approximation is to use a family of simpler or better-known objects to approximate a certain more general or new type of objects so that the "difference" on certain pertinent quantitative aspects can be made to be as small as one wishes. Generally speaking, this is often an effective methodology which enables us to reduce the study of the latter to that of the former. Historically, such an ingenious methodology of approximation was first invented by the great geometer Eudoxus to overcome the difficulty posed by the existence of non-commensurable quantities to the entire foundation of geometry. His basic idea was, in essence, to approximate irrational numbers by the rational numbers!

In this chapter we shall summarize and combine the results of the previous three chapters to build the foundational framework and the fundamental theory of calculus. Generally speaking, calculus is a branch of mathematics which provides the general framework as well as the basic tools for the quantitative analysis of problems or systems of variable quantities. In most cases, the variable quantities are of the measurement type, e.g., length, area, weight, time, etc. The mathematical system that one uses to represent and to compute the quantitative aspect of these types of quantities is the real number system, ℝ. Hence, the concept of real numbers and the basic properties of the real number system, ℝ, forms the very foundation of calculus.

Mathematically, one uses symbols such as x, y, Z, s, t, etc. to represent the values of variable quantities (such symbols are called "variables") and uses functions to describe those correlations amongst the variable quantities (cf. Sec. 2, Chapter 1). Therefore, functions are exactly the primary subject of investigation in calculus. Among the basic properties of functions, the most fundamental and also the most useful ones are the monotonicity, the continuity, the rate of change and the sum of total effect that we discussed in Chapter 2. Therefore, the first order of fundamental theory of calculus is the analytical formulation (i.e., definition) of the above basic properties, thus transforming the intuitive contents of such basic properties into precise mathematical concepts. It is at this junction that the method of approximation and the concept of limit become indispensable (cf. Chapter 3).

Calculus is itself the fundamental theory and one of the basic tools of mathematical analysis. In the quantitative analysis of systems of variable quantities, one uses functions to express the interlocking correlations amongst the variable parameters. Differentiation and integration are themselves the foundational theory and the basic operations in the study of functions. We shall conclude this concise introduction of calculus by a brief discussion on some pertinent properties of those useful elementary functions such as polynomial functions, trigonometric functions, exponential function and logarithmic function as well as some typical examples of applications of calculus. The purpose of this chapter is to provide the reader with some preliminary experience on the power of calculus and to offer some of the initial exercises on the type of computations needed in carrying out mathematical analysis.