0.2 What Is Calculus and Why do we Study it?
Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.
I have been around for a while, and know how things change, more or less. What can calculus add to that?
I am sure you know lots about how things change. And you have a qualitative notion of calculus. For example the concept of speed of motion is a notion straight from calculus, though it surely existed long before calculus did and you know lots about it.
So what does calculus add for me?
It provides a way for us to construct relatively simple quantitative models of change, and to deduce their consequences.
To what end?
With this you get the ability to find the effects of changing conditions on the system being investigated. By studying these, you can learn how to control the system to do make it do what you want it to do. Calculus, by giving engineers and you the ability to model and control systems gives them (and potentially you) extraordinary power over the material world.
The development of calculus and its applications to physics and engineering is probably the most significant factor in the development of modern science beyond where it was in the days of Archimedes. And this was responsible for the industrial revolution and everything that has followed from it including almost all the major advances of the last few centuries.
Are you trying to claim that I will know enough about calculus to model systems and deduce enough to control them?
If you had asked me this question in 1990 I would have said no. Now it is within the realm of possibility, for some non-trivial systems, with your use of your laptop or desk computer.
OK, but how does calculus models change? What is calculus like?
The fundamental idea of calculus is to study change by studying "instantaneous " change, by which we mean changes over tiny intervals of time.
And what good is that?
It turns out that such changes tend to be lots simpler than changes over finite intervals of time. This means they are lots easier to model. In fact calculus was invented by Newton, who discovered that acceleration, which means change of speed of objects could be modeled by his relatively simple laws of motion.
This leaves us with the problem of deducing information about the motion of objects from information about their speed or acceleration. And the details of calculus involve the interrelations between the concepts exemplified by speed and acceleration and that represented by position.
So what does one study in learning about calculus?
To begin with you have to have a framework for describing such notions as position speed and acceleration.
Single variable calculus, which is what we begin with, can deal with motion of an object along a fixed path. The more general problem, when motion can take place on a surface, or in space, can be handled by multivariable calculus. We study this latter subject by finding clever tricks for using the one dimensional ideas and methods to handle the more general problems. So single variable calculus is the key to the general problem as well.
When we deal with an object moving along a path, its position varies with time we can describe its position at any time by a single number, which can be the distance in some units from some fixed point on that path, called the origin of our coordinate system. (We add a sign to this distance, which will be negative if the object is behind the origin.)
The motion of the object is then characterized by the set of its numerical positions at relevant points in time.
The set of positions and times that we use to describe motion is what we call a function. And similar functions are used to describe the quantities of interest in all the systems to which calculus is applied.
The course here starts with a review of numbers and functions and their properties. You are undoubtedly familiar with much of this, so we have attempted to add unfamiliar material to keep your attention while looking at it.
I will get bogged down if I read about such stuff. Must I?
I would love to have you look at it, since I wrote it, but if you prefer not to, you could undoubtedly get by skipping it, and referring back to it when or if you need to do so. However you will miss the new information, and doing so could blight you forever. (Though I doubt it.)
And what comes after numbers and functions?
A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the "derivative") of various functions. (The process of doing so is called "differentiation".)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called "integration".)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.
There are a few other standard topics in such a course. These include description of functions in terms of power series, and the study of when an infinite series "converges " to a number.
So where does this empower me to do what?
It doesn't really do so. The problem is that such courses were first designed centuries ago, and they were aimed not at empowerment (at that time utterly impossible) but at familiarizing their audience with ideas and concepts and notations which allow understanding of more advanced work. Mathematicians and scientists and engineers use concepts of calculus in all sorts of contexts and use jargon and notations that, without your learning about calculus, would be completely inscrutable to you. The study of calculus is normally aimed at giving you the "mathematical sophistication" to relate to such more advanced work.
So why this nonsense about empowerment?
This course will try to be different and to aim at empowerment as well as the other usual goals. It may not succeed, but at least will try.
And how will it try to perform this wonder?
Traditional calculus courses emphasize algebraic methods for performing differentiating and integrating. We will describe such methods, but also show how you can perform differentiation and integration (and also solution of ordinary differential equations) on a computer spreadsheet with a tolerable amount of effort. We will also supply applets which do the same automatically with even less effort. With these applets, or a spreadsheet, you can apply the tools of calculus with greater ease and flexibility than has been possible before.
(There are more advanced programs that are often available, such as MAPLE and Mathematica, which allow you to do much more with similar ease. With them you can deduce the consequences of models of various kinds in a wide variety of contexts. Once you understand calculus they can make its use much easier, but they provide answers given inputs, which does not provide understanding of how they do it.)
Also, we will put much greater emphasis on modeling systems. With ideas on modeling and methods for solving the differential equations they lead to, you can achieve the empowerment we have claimed.
And I will be able to use this to some worthwhile end?
Okay, probably not. But you might. And also you might be provoked to learn more about the systems you want to study or about mathematics, to improve your chances to do so. Also you might be able to understand the probable consequences of models a little better than you do now. Also you may get to love the concepts and ideas of calculus.
Well, what is in the introductory chapter on numbers?
We start with the natural numbers \((1,2,3,...)\) and note how the operations of subtraction, division and taking the square root lead us to extending our number system to include negative numbers, fractions (called rational numbers) and complex numbers. We also describe decimal expansions (which describe "real numbers") and examine the notion of countability. We mutter about complex numbers as well.
And in the chapter about functions?
We start with an abstract definition of a function (as a set of argument-value pairs) and then describe the standard functions. These are those obtained by starting with the identity function (value=argument) and the exponential function, and using various operations on them.
Operations, what operations?
These are addition, subtraction, multiplication, division, substitution and inversion.
But what is the exponential function, and what are substitution and inversion?
Here are one sentence answers: if you want to know more read the chapter!
The exponential function is mysteriously defined using calculus: it is the function that is its own derivative, defined to have the value 1 at argument 0. It turns out, however, to be something you have seen before. And it turns out to bear a close relation to the sine function of trigonometry.
Substitution of one function f into another g produces a new function, the function defined to have, at argument x, the value of f at an argument which is the value of g at argument x. This is simpler than it sounds. Suppose, for example \( y = x^2 \) and \( x = 2z \) then \( y(x(z)) \) is \( (2z)^2 \).
An inverse of a function is a function obtained by switching its values with its arguments. For example the square function, usually written as \( x^2 \) has the square root function as an inverse.
In the immortal words of Father William to his nephew, as penned by Lewis Carroll, who was a mathematician:
I have answered three questions and that is enough,
Said the sage, don't give yourself airs.
Do you think I can listen all day to such stuff?
Be off or I'll kick you downstairs!