Department of Mathematical Sciences
Northern Illinois University

An Introduction to Calculus

by John Beachy

All of nature is in a state of constant motion and change. The branch of mathematics that provides methods for the quantitative investigation of various processes of change, motion, and dependence of one quantity on another is called mathematical analysis, or simply analysis. A first course in calculus establishes some of the basic methods of analysis, done in relatively simple cases.

The development of the methods of analysis was stimulated by problems in physics. During the 16th century the central problem of physics was the investigation of motion. The expansion of trade, and the accompanying explorations, made it necessary to improve the techniques of navigation, and these in turn depended to a large extent on developments in astronomy. In 1543 Copernicus published the ground-breaking work "On the revolution of the heavenly bodies". The "New astronomy" of Kepler, containing his first and second laws for the motion of planets around the sun, appeared in 1609. The third law was published by Kepler in 1618 in his book "Harmony of the world". Galileo, on the basis of his study of Archimedes and his own experiments, laid the foundations for the new mechanics, an indispensable science for the newly arising technology.

During the Renaissance, Europeans became acquainted with Greek mathematics, by way of the Arabic translations. This finally occurred after a period of almost one thousand years of scientific stagnation. The books of Euclid, Ptolemy, and al-Khwarizmi were translated in the 12th century from Arabic into Latin, the common scientific language of Western Europe. At the same time the earlier Greek and Roman system of calculation was gradually replaced by the vastly superior Indian method, which also reached Europe via the Arabs.

It was not until the 16th century that European mathematicians finally surpassed the achievements of their predecessors, with the solution by the Italians Tartaglia and Ferrari of the general cubic equation and (later) of the general equation of the fourth degree. The concepts of variable magnitude and function arose gradually, as a result of the interest in laws of motion, as, for example, in the work of Kepler and Galileo. Galileo discovered the law of falling objects by establishing that the distance fallen increases proportionally to the square of the time.

The appearance in 1637 of the new "geometry" of Descartes marked the first definite step toward a mathematics of variable magnitudes. This combined algebraic and geometric techniques, and is now known as "analytic geometry". The main content of the new geometry was the theory of conic sections: the ellipse, hyperbola, and parabola. This theory had been developed extensively by the ancient Greeks, in geometric form, and the combination of their knowledge, together with algebraic techniques and the general idea of a variable magnitude, produced analytic geometry.

For the Greeks, the conic sections were a subject of purely mathematical interest, but by the time of Descartes they were of practical importance for astronomy, mechanics, and technology. Kepler discovered that the planets move around the sun in elliptical orbits, and Galileo established that an object thrown into the air travels along a parabolic path. (Of course, both of these models are only first approximations of the actual paths.) These discoveries made it necessary to calculate various magnitudes associated with the conic sections, and it was the method of Descartes that solved this problem.

The next decisive steps were taken by Newton and Leibnitz during the second half of the 17th century, and resulted in the founding of differential and integral calculus. The Greeks and later mathematicians had studied the geometric problems of drawing tangents to curves and finding areas and volumes of irregular figures. The remarkable discovery of the relation of the problems to the problem of the new mechanics and the formulation of general methods for solving them was brought to completion in the work of Newton and Leibnitz. This relationship was discovered because of the possibility, through the use of analytic geometry, of making a graphical representation of the dependence of one variable on another. In short, what is involved is the construction of a geometric model to describe relationships involving variable magnitudes.

We next discuss the construction of a mathematical model for a particularly simple kind of motion. We will consider the motion of a car along a straight road. The language of sets and functions provides a general way of describing relationships between quantities, and in this case we can use this language in talking about a function from one set of real numbers to another. In order to define a numerical relationship, we can select a reference point on the road and a reference point in time. Then we can express the position of the car in terms of a real number, the distance of the car from the reference point, letting one direction be positive and the other negative. This position only requires one number since the car is moving along a straight road, and similarly the time can be given by a single number, the elapsed time.

We can abstract the situation a little further, by representing the car as a point moving along a straight line, whose position is given by a number expressing the distance and direction from the point to a fixed reference point, usually called the origin. Specifying the position of the point at each instant in time is thus equivalent to defining a function from the set of all real numbers (representing time) to the set of all real numbers (representing position).

The following questions are some of those which arise in this situation. If you know the function giving the position of car at each instant, can you give the function which describes its velocity at each instant? If you know only the velocity at each instant, can you tell the distance traveled during a particular interval of time? If you know only the function giving the velocity at each instant, can you reconstruct the function giving the position at each instant?

Answering the first question would be equivalent to giving a function listing the speedometer readings at each instant. Here we assume that we have a speedometer which gives both positive and negative readings, depending on the direction of travel. The function describing the rate of change or velocity at each instant is called the derived function or simply the derivative of the original function. Sometimes information about the motion of the car can be obtained more easily from the derived function than it can from the original function. For example, you could find out when the car is stationary by simply finding out when the derived function is zero. A positive value for the derivative indicates forward motion and a negative value indicates the reverse, so if you know that in a particular time interval the derivative is positive, then zero, and then negative, this tells you that the car was moving forward, then stopped and started moving backwards. The point of farthest advance during this interval can then be found by solving the equation obtained by setting the derived function equal to zero.

The second problem was the following: Knowing only the velocity at each instant, find the distance traveled during a given time interval. If the velocity is constant, the problem can be solved rather easily, by multiplying the velocity by the amount of time. But in general situations, the velocity will be changing all the time, so this method will not work. If we could find an average value for the velocity, then we could just multiply this average value by the amount of time. The problem lies in the fact that there are infinitely many readings of the speedometer involved, as given by the function describing the velocity, and familiar methods deal only with finding the average of a finite number of values.

In physical processes depending on time, there are normally only relatively small changes in the process during short intervals of times. Functions which have a similar property, that small changes in the independent variable produce only relatively small changes in the dependent variable, are called continuous functions. (We must make precise what we mean by ``relatively".) It can be shown that an average can be found for any continuous function, so that the methods we will develop will work win almost all physical situations. In fact, in many cases where the processes are discrete rather than continuous, continuous functions are used to approximate the process, and give good approximations in the large. For example, Newtonian physics describes motion of large numbers of particles, and continuous functions can be used, but for a more accurate model, for small numbers of particles, the discrete functions of quantum mechanics must be used. Unfortunately we will meet continuous processes for which it is impossible to talk about an instantaneous rate of change at certain points.

Integral calculus deals with this second problem. If the rate at which a process is being carried out is known, and described analytically by a function, then the number which gives the total outcome of the process during a particular time interval is called the definite integral of this function, over the given interval of time.

The third problem, where we are given a function describing the velocity of the car and are then asked to find a function giving its position at each instant, is investigated in the branch of analysis know as differential equations. Of course, if we know the answer to this question, we can answer the second one quite easily. This is a difficult area of study, but it is very important, since many physical situations can be described by giving simply equations involving rates of change. An equation involving derivatives of a function is called a differential equation, and such equations often give the simplest statements of physical laws. For example, by solving a differential equation expressing the assumption that the only force acting on a planet is the gravitational attraction of the sun, and that this is inversely proportional to the square of the distance between them, it is possible to show that the planet must follow an elliptical path. This was one of the early triumphs of the techniques of calculus and differential equations.

The Fundamental Theorem of Calculus connects the two areas of differential and integral calculus. It says that finding the instantaneous rates of change of a function and then averaging them gives the average rate of change of the function. It shows that the infinite processes used to define the instantaneous rate of change and average of a function lead to good definitions, at least in the sense that you would certainly expect the above kind of connection. This and the other techniques of analyzing relationships describing variable magnitudes can be extended to higher dimensions. Functions of more than one variable arise naturally in industrial applications, where describing rates of change and finding maximum and minimum values are extremely important.

A general statement on mathematics

Mathematics involves the construction and study of abstract models of physical situations. The construction of a model involves the selection of a finite number of explicitly stated and precisely formulated premises. These assumptions are called axioms, and the study of the model then involves drawing conclusions from these fundamental assumptions, using as high a degree of logical rigor as possible. The rigor of mathematics is not absolute, but is rather in the process of continual development. Euclid's axiomatization of geometry and his study of this model of our spatial surroundings was accepted as completely rigorous for over two thousand years, even though a modern geometer could point out serious flaws in the logical development of the theory.

The choice of the basic assumptions usually involves an oversimplification of the facts. Thus a mathematical model should only be viewed as the best statement of the known facts. In many cases a model should be viewed as merely the most efficient, incorporating only enough assumptions to give a desired degree of accuracy in prediction. In an area small in comparison to the total surface of the earth, plane geometry gives a good approximation for questions involving relationships of figures. As soon as the problems involve large distances, spherical geometry must be used as the model. Newtonian physics is good enough for many problems in mechanics, and it is necessary to introduce the additional assumptions of quantum mechanics only if much greater accuracy is needed.

If any distinction at all is to be made between applied mathematics and theoretical mathematics, it is perhaps at this point. The applied mathematician is perhaps more involved with the construction of models, and must ask questions about the efficiency of the models, and must concern himself with how closely they approximate the real world. The theoretical mathematician is concerned with developing the model, by investigating the implications of the basic assumptions or axioms. This is done by proving theorems. Of course, if a theorem is proved which is obviously contrary to nature, it is clear to everyone concerned that the basic assumptions do not coincide with reality. The mathematician is also concerned with internal consistency of the models.

In order to make logical deductions from the basic axioms, the language used must be extremely precise. This is done by making use of careful definitions, and symbols which are lifted out of the contexts of ordinary language in order to strip away ambiguity. Much of the particular precision and clarity of mathematics is made possible by its use of formulas. The modern reader is usually unaware that this is an achievement only of the past few centuries. For example, the signs + and - appeared in manuscripts for the first time in 1481: parentheses first appeared in 1544; brackets and braces appeared essentially for the first time in 1593 in the works of Vieta; the sign = appeared in 1557; the modern way of writing powers was first used in 1637 by Decartes, but Gauss still wrote xx instead of x2 in 1801.

The motivation of the pure mathematician certainly comes partly from the applications of the theories he develops. But perhaps more than this, it comes from the joy of creating a theory of particular simplicity, elegance and broad scope. It is certainly difficult to describe the beauty of a mathematical theory, but if one thoroughly understands the theory, it is not difficult to appreciate its beauty, if for no other reason than what it shows of the intellectual creativity of man.

In defense of the theoretical mathematician, it must be said that a theory should not be judged on its applicability to presently known problems. The history of mathematics is filled with examples of particular theories which seemed at the time to be mere intellectual exercises devoid of any relationship to physical problems, and which later were discovered to have important applications. One particularly impressive example is provided by non-Euclidean geometry, which arose from the efforts, extending for two thousand years from the time of Euclid, to prove the parallel axiom from Euclid's other, more obvious axioms. This seemed to be a matter of interest only to mathematicians. Even Lobacevskii, the founder of the new geometry, was careful to label it ``imaginary", since he could not see any meaning for it in the actual world. In spite of this, his ideas laid the foundation for a new development of geometry, namely the creation of theories of various non-Euclidean spaces. These ideas later became, in the hands of Einstein, the basis of the general theory of relativity, in which the mathematical model consists of a form of non-Euclidean geometry of four-dimensional space.

The generalizations and abstractions of mathematics often seem at first to be strange and difficult. But with the very general expansion of knowledge and technology we are currently experiencing, it becomes necessary to identify and elucidate general underlying principles, in order to tie this information together. The language and concepts of mathematics help to fill this need.