Models

Functions cover the landscape of Calculus. Once you arrive there, their ubiquitous presence will provide a pleasant reminder that you really haven't travelled too far from what you already know.

In fact, you will recognize many things in Calculus. The traditions of Arithmetic, Algebra, Geometry, and Trigonometry flourish there.

It is those two large peaks — the Derivative and the Integral — that may seem cold and intimidating. You may wonder if you'll ever be able to scale them and cross that bridge in the sky, the Fundamental Theorem of Calculus.

Don't worry. You'll do it.

The peaks only look big from where you are now. Their size is actually something of a mirage. As you get closer, you'll see them begin to diminish.

Eventually, you will discover one of Calculus' great secrets: the Derivative and the Integral are actually only mathematical models. Models — like in a toy store! They may resemble the Alps from a distance, but, like the model mountains of a toy train set, they're not as big as we imagine.

Mere models probably deserve names that aren't quite so intimidating. Let's call the peaks "derivative" and "integral", just as they're called in the land of calculus. (That's right, "calculus", not "Calculus".) And that bridge? You may have already guessed: On the other side, it's called only "precalculus".

How curious a place, this calculus! Are the mountains there big, or are they small?

It's important to have proper perspective about the human scale of things in calculus. We wouldn't want to be frightened away before our trip has even begun.

Just as importantly, however, we wouldn't want to confuse calculus' accessibility with any sort of diminished significance. It took tremendous craftsmanship to build the derivative and the integral, and they are two of the most ingeniously useful mathematical models ever devised. They do indeed tower above all of mathematics.

The usefulness of these models comes from their ability to reflect, like great works of art, universal aspects of our everyday experience. We don't have to fully appreciate this ponderous achievement right now, but understanding what goes into the construction of these models is an important part of our preparation for calculus.