The main goal of this course is to introduce the student to the subject of Calculus. In other words, we continue the study of functions of a real variable which was begun in algebra and pre-calculus courses. There are two key ideas in calculus. The first deals with the notion of the rate of change of a function. If the function is linear; that is, if it's graph is a straight line, then the rate of change is nothing more than the familiar notion of its slope. The main goal of the so-called differential calculus is to generalize this notion to a wider class of functions. In other words we generalize slopes to apply to curves as well as to straight lines. This work was begun by Fermat, Barrow, & Newton, and culminates in the idea of the derivative of a function.
The second main idea in calculus is much older (the chief exponent was Archimedes), and is the problem of measuring the accumulation of values of a function, or what amounts to the same thing, the area under the graph of the function. Again, if the graph is a straight line, this is an easy problem which can be solved with basic algebra & geometry. However, most quantities in the real world vary in complex ways and so their graphs will not generally be straight lines. So to be sufficiently general, one must be able to find the area under curves as well as straight lines. This problem is solved by the notion of an integral of a function.
These problems are seemingly quite different. Why are they grouped together under a single subject known as calculus? There are two reasons for this, the second reason being much more important and significant than the first. The first reason is that both problems are solved by a similar technique: since we already know the answers for linear functions, one approximates the curve by a linear function which is very "close" to it in some sense. Then the solution to either problem (the slope problem or the area problem) for the linear approximation yields an approximate solution to our problem for the curve. Finally, the approximation is made more and more accurate by focusing our attention to smaller and smaller intervals on the real number line. The "limiting" value of these approximations are then the exact solution we seek.
The second reason why these problems are treated together is far more significant, and frankly is rather a surprise: these two problems are in a sense "inverse problems" of one another! That is, whatever method you use to solve the slope problem (i.e., find the derivative of a function), then you can solve the accumulation or integral problem by doing the reverse (i.e., find the antiderivative of the function.). This result is so important it is called the Fundamental Theorem of Calculus. Newton and, independently, Leibniz, discovered this fact and it is for this reason that they are credited with the invention of calculus, even though people worked on, and solved, both problems before they did.
A few comments are in order. First, the common method of solving both problems involved finding "limiting values" of a function, so the notion of a limit is a basic and powerful tool for calculus. For example, we all have an intuitive idea of what is meant by a continuous function. It turns out that a logically rigorous definition of continuity also depends on this notion. (To summarize, the notions of continuity, derivative, and integral all depend on the notion of a limit.) Secondly, we have been speaking of functions of a single real variable all along. But most things in the real world depend on more than one variable. It turns out that this entire edifice of calculus generalizes to functions of several variables, as well as to variables which are not necessarily real. Such ideas are covered in later courses. (See below if you are interested.)
In our course, we stick to single-valued functions of a single real variable. Our goal is to get to the fundamental theorem of calculus, which means you will be exposed to both of the main ideas of caclulus (differential calculus and integral calculus). However, in the first semester, most of the time is devoted to the differential calculus. Integrals are introduced only toward the end of the course in order to obtain a few basic results, including the fundamental theorem. Finally, our text espouses the philosophy that many functions in the real world are given not by an algebraic formula but by data; either numerical data (tables of values) or pictorial data (graphs of functions). This means that when we are learning the basic notions calculus, we should make every effort to understand how to interpret them from each of the three points of view: algebraic, numeric, graphical. This is to encourage you to think about what the concepts mean in each case, rather than rotely memorize formulas.
These goals will be accomplished by covering roughly Chapters 1-5 in our text. The first chapter is a review of functions, presented from the three points of view. In Chapter 2 one learns about the derivative of a function, and what it measures. Of course, along the way, one encounters the necessary preliminary notions such as limits and continuity. Chapter 3 continues the study of derivatives, developing short cuts for finding them, and introducing some of the applications. Chapter 4 is devoted to more applications of the derivative. Finally in Chapter 5, the integral is introduced, and the fundamental theorem of calculus is covered.
For those who are interested in what happens after MA111, here is a rough outline:
While calculus is not the only type of mathematics there is, the above examples illustrate how significant this mathematical development is - at least 10 courses in the MCS dept. are directly related to calculus! It is not before one has completed several of these course before one begins to appreciate the genius of Newton & Leibniz - Calculus is truly one of the crowning intellectual achievements of western civilization.
