Differential Calculus: From Practice to Theory

Like the authors, your instructor is probably a highly trained, professional mathematician. As mathematicians we solve problems and we build logical structures. This is what we were trained to do.

The structures we build are what makes mathematics useful but that is not why we build them. Most of the time we create simply for the love of the creative act itself. Usefulness, as such, is often a secondary consideration. In this, mathematicians are more like poets or artists than scientists or engineers. By and large, we consider the act of creation to be its own reward. It is a remarkable fact that sometimes the logical structures we build turn out to be applicable to problems in the real world.

When we explain our mathematical works to each other we mathematicians only display the finished product in much the same way that an artist or an architect will display their work without all of the behind–the–scenes sketches and pencil lines (and mistakes), that they made along the way. We start with simple ideas and bit by bit, piece by piece, we assemble our mathematics like we would a puzzle. When we explain it to others we ignore all of the mistakes we made along the way for the same reason that artists do. They are not part the finished product. That is how we understand, and think about our mathematics, and that is how we talk to each other about it. Displayed in this manner a mathematical structure is truly as beautiful as any artistic creation.

But in this text we are not talking to other mathematicians. We are talking to you, a mathematics student. And you are (presumably) encountering Calculus for the first time. This is not the time to build Calculus up from its logical foundations and expect you to appreciate its beauty. This is the time to show you where Calculus came from, how it was built, what problems it was invented to solve, and, perhaps most importantly, how well Calculus addresses those problems. Once that is done you will appreciate it, or not, as your own sense of aesthetics allows.

As you proceed through this text know that it took several centuries for some of the most brilliant persons who have ever lived to polish the ideas of Calculus to their current luster. But just as a painting is unappealing before it is finished, Calculus is not lustrous until you can look back on it as a whole. Until then it takes a lot of calculation and thought to see how the ideas work together. In the end, we hope you will come to see that it is worth all of the work and frustration.

In this chapter we will be exploring the ad hoc techniques used by the mathematical pioneers who were trying to solve some very real and very specific problems using only the tools that you possess now: Geometry, Algebra, and Trigonometry. The pre–Calculus techniques they came up with substantially influenced the form that Calculus eventually took. If the pioneers had been different people, or if they had begun with different tools Calculus would likely have taken a different form. Or it might not have been invented at all. The pre–Calculus methods we will see in this chapter represent the first ideas that eventually led to Calculus. The supporting logical structure came later, much later.

Thus we will begin with the intuitive notions that preceded Calculus. But we will not hide the problems inherent in this approach. Quite the reverse. It is our intention to highlight them as much as possible. The underlying difficulties are real and they need to be understood. We want you to be aware of them so that in Chapter 13 when we begin to address them, the necessity of the rather severe formalism we will be forced to adopt will be clear to you.

For example, we will soon start talking about “the line tangent to a curve.” You have a very clear image of what is meant by “tangent line” and that will suffice to begin. But, it is actually very hard to define precisely what we mean by “the line tangent to a curve.”

Later we will encounter a similar difficulty with the idea of a continuous curve. The phrase “continuous curve” surely conjures up a very clear image in your mind. Nothing could be clearer really. Continuous curves are unbroken curves, right? This is so intuitively obvious that no one bothered to define the notion of continuity formally for millennia. Most likely you don’t see the need for a formal definition either. Yet. But continuity is very hard to define in a mathematically rigorous manner. The first person to give a rigorous definition in the modern sense was Bernhard Bolzano

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https://mathshistory.st-andrews.ac.uk/Biographies/Bolzano/

in 1817.

One of our goals is to set up the conditions under which it will become very clear to you why your intuitive image of, for example, a “tangent line,” or a “continuous curve” is not sufficient; why these are hard concepts to define. We want you to bump into the difficulties that come with the intuitive understanding you have right now so that you will see — and appreciate — why the definitions we will eventually be offering are better. Or at least more useful.

But have some sympathy for the poor instructor! Everything we do in the first part of this text, especially in this section, goes directly against all of the instructor’s training as a mathematician. Instructors would be much happier with us if we would define the terms “the line tangent to a curve” and “continuous curve” before we use them because that is the way they see and appreciate the beauty of our topic.

The problem is that this is not (in our opinion) a good way to learn Calculus. So instead, we will be following in the footsteps of our forebears. We will proceed using intuitive ideas until they become too unwieldy to use. Only after we have discerned the properties we need from a concept (like “tangent”) will we offer formal definitions. Hopefully it will then be clear to you why the definitions are needed, and why they are stated as they are.

So to the instructor, or anyone else of a mathematical bent who might be reading this, we offer our apologies for proceeding in what we know looks like the wrong way around. As mathematicians ourselves we certainly feel the same discomfort you do. But in this text we are not writing as mathematicians for other mathematicians, we are writing as teachers for students. Please bear with us. “Though this be madness, yet there is method in’t.”