# Topics in Calculus

**Author:**Lee Lady

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In my opinion, calculus is one of the major intellectual achievements of Western civilization --- in fact of world civilization. Certainly it has had much more impact in shaping our world today than most of the works commonly included in a Western Civilization course -- books such as Descartes's Discourse on Method or The Prince by Machiavelli.

But at most universities, we have taken this magnificent accomplishment of the human intellect and turned it into a boring course.

We have been so concerned with presenting calculus in a rigorous way that is satisfying to us as mathematicians that we have completely failed to give students any intuitive concept of what the subject is really about. The textbook by Salas & Hille that we currently use here at the University of Hawaii really embodies this attitude. I would much rather see us teaching calculus in the spirit of some of the older texts such as Sawyer's little book What Is Calculus About? (Another book in the same vein, but more recent, is The Hitchhiker's Guide to Calculus by Michael Spivak.)

For many of us mathematicians, calculus is far removed from what we see as interesting and important mathematics. It certainly has no obvious relevance to any of my own research, and if it weren't for the fact that I teach it, I would long ago have forgotten all the calculus I ever learned.

But we should remember that calculus is not a mere "service course." For students, calculus is the gateway to further mathematics. And aside from our obligation as faculty to make all our courses interesting, we should remember that if calculus doesn't seem like an interesting and worthwhile subject to students, then they are unlikely to see mathematics as an attractive subject to pursue further.

The importance of calculus is that most of the laws of science do not provide direct information about the values of variables which can be directly measured. In other words, if you are lost, then physics will not help you find your way home, because there are no laws of physics that provide direct information about position. Most laws in physics don't even give immediate information about velocity.

Some scientific principles give information relating that values of variables at a given instant, for instance Ohm's Law E=IR, or the Boyle-Charles Law for ideal gasses, pV=kT. Calculus is not relevant for these rules. But many of the most important principles in science are rules for the way variables change. For instance, physics tells you how velocity will change in various situations -- i.e. it tells you about acceleration.

This is why it's important to have a mathematical way of talking about change. That's why you see the concept of the derivative used throughout science -- in physics, chemistry, biology, economics, even psychology.

The purpose of learning differential calculus is not to be able to compute derivatives. In fact, computing derivatives is usually exactly the opposite of what one needs to do in real life or science. In a calculus course, one starts with a formula for a function, and then computes the rate of change of that function. But in the real world, you usually don't have a formula. The formula, in fact, is what you would like to have: the formula is the unknown. What you do have is some information, given by the laws of science, about the way in which the function changes.

In other words, the primary reason for learning differential calculus is in order to be able to understand differential equations. (An integral, in many practical contexts, is simply the simplest case of a differential equation.)

Taking differential calculus without studying differential equations is a lot like studying two years of a foreign language. It may be an interesting intellectual challenge, but it usually doesn't give a student much of permanent value.

It is a mistake to think of calculus, or mathematics in general, as primarily a tool for finding answers (although it is also a mistake to think, as many graduate students do, that calculating is an inferior, unworthy aspect of mathematics). The primary importance of calculus in the hard sciences is that it provides a language, a conceptual framework for describing relationships that would be difficult to discuss in any other language.

What is worthwhile for students to gain from a calculus course is the ability to read books that use the language of calculus and, at least to some extent, follow the derivations in those books. Unfortunately, being being proficient at the sort of chickenshit skills required to get a good grade in a calculus course is not a lot of help in this respect.

I tell my calculus students that their grades are probably not a very good indicator of being able to do well in future courses. What's more important is whether they make the effort to follow the reasoning given in class and in the text. The most important part of the course (at least when I teach it) is the part that's never tested on.

In my opinion, calculus is one of the major intellectual achievements of Western civilization --- in fact of world civilization. Certainly it has had much more impact in shaping our world today than most of the works commonly included in a Western Civilization course -- books such as Descartes's Discourse on Method or The Prince by Machiavelli.

But at most universities, we have taken this magnificent accomplishment of the human intellect and turned it into a boring course.

We have been so concerned with presenting calculus in a rigorous way that is satisfying to us as mathematicians that we have completely failed to give students any intuitive concept of what the subject is really about. The textbook by Salas & Hille that we currently use here at the University of Hawaii really embodies this attitude. I would much rather see us teaching calculus in the spirit of some of the older texts such as Sawyer's little book What Is Calculus About? (Another book in the same vein, but more recent, is The Hitchhiker's Guide to Calculus by Michael Spivak.)

For many of us mathematicians, calculus is far removed from what we see as interesting and important mathematics. It certainly has no obvious relevance to any of my own research, and if it weren't for the fact that I teach it, I would long ago have forgotten all the calculus I ever learned.

But we should remember that calculus is not a mere "service course." For students, calculus is the gateway to further mathematics. And aside from our obligation as faculty to make all our courses interesting, we should remember that if calculus doesn't seem like an interesting and worthwhile subject to students, then they are unlikely to see mathematics as an attractive subject to pursue further.

The importance of calculus is that most of the laws of science do not provide direct information about the values of variables which can be directly measured. In other words, if you are lost, then physics will not help you find your way home, because there are no laws of physics that provide direct information about position. Most laws in physics don't even give immediate information about velocity.

Some scientific principles give information relating that values of variables at a given instant, for instance Ohm's Law E=IR, or the Boyle-Charles Law for ideal gasses, pV=kT. Calculus is not relevant for these rules. But many of the most important principles in science are rules for the way variables change. For instance, physics tells you how velocity will change in various situations -- i.e. it tells you about acceleration.

This is why it's important to have a mathematical way of talking about change. That's why you see the concept of the derivative used throughout science -- in physics, chemistry, biology, economics, even psychology.

The purpose of learning differential calculus is not to be able to compute derivatives. In fact, computing derivatives is usually exactly the opposite of what one needs to do in real life or science. In a calculus course, one starts with a formula for a function, and then computes the rate of change of that function. But in the real world, you usually don't have a formula. The formula, in fact, is what you would like to have: the formula is the unknown. What you do have is some information, given by the laws of science, about the way in which the function changes.

In other words, the primary reason for learning differential calculus is in order to be able to understand differential equations. (An integral, in many practical contexts, is simply the simplest case of a differential equation.)

Taking differential calculus without studying differential equations is a lot like studying two years of a foreign language. It may be an interesting intellectual challenge, but it usually doesn't give a student much of permanent value.

It is a mistake to think of calculus, or mathematics in general, as primarily a tool for finding answers (although it is also a mistake to think, as many graduate students do, that calculating is an inferior, unworthy aspect of mathematics). The primary importance of calculus in the hard sciences is that it provides a language, a conceptual framework for describing relationships that would be difficult to discuss in any other language.

What is worthwhile for students to gain from a calculus course is the ability to read books that use the language of calculus and, at least to some extent, follow the derivations in those books. Unfortunately, being being proficient at the sort of chickenshit skills required to get a good grade in a calculus course is not a lot of help in this respect.

I tell my calculus students that their grades are probably not a very good indicator of being able to do well in future courses. What's more important is whether they make the effort to follow the reasoning given in class and in the text. The most important part of the course (at least when I teach it) is the part that's never tested on.