Khan Academy admirably provides access to mathematics lectures to anyone with the desire to learn. Flash videos are a good way to slowly walk through examples; they may not, however, help you grasp the ideas that make Calculus (and much of modern mathematics) possible. Four such fundamental concepts are in the images below. At the end is a nearly magical application.

**i)** There are no gaps in the number line. Between any two numbers, lies another.

**ii)** An ongoing string of numbers all decreasing with a floor below must get closer and closer to the floor. [Likewise, increasing with a ceiling.]

**iii)** All curves dealt with in Calculus look like lines at the right zoom and…

**iv)** can be drawn without lifting your pencil off the page.

After some details about these four ideas, I will mention their most important consequences, including the remarkable “Taylor Formula.”

The properties of real numbers (i) and (ii) everyone is familiar with. Together they imply there are no gaps or jumps on the real number line. Another way to say (iii) is: If I stand on a “Calculus-kosher” curve, no matter which direction I turn my head, I can take a step small enough so that I walk essentially on a flat surface. The size of this step will obviously depend on the curve, but for any level of “flatness” I desire, I can find a step tiny enough. Continuity (iv) can be simply stated as: no holes or jumps on the graph.

**The “ruler” of Calculus: the Derivative**

The **derivate** = the slope of the line connecting P and Q as Q comes closer and closer to P.

The different values for the slope, keeping P fixed and bringing Q closer and closer to the left, will be a string of **real numbers,** on which I apply (ii), saying that if this string gets smaller and smaller (or larger and large), then it will eventually settle down to a single value. I know this will happen because of (iii), which tells me that the graph gets flatter and flatter as I zoom in. I repeat the procedure, starting with Q to the left of P. If the two values are the same [they will be because of (iii) and (iv) together], I call that number the **Derivative of f** at x and denote it f'(x).

It would be a lot of trouble if we had to find the derivative by constructing a string of slopes each time we encountered a new function. Luckily, however, the very ideas (i-iv) above give this **derivative** some properties that make it easy to work with, namely that is a **linear operator**! [“Smooth operator” would do just as well, no?] Every trick we could do with lines, like add/subtract, or multiply by a constant, we can do with derivatives; so this new mathematical object is surprisingly easy to work with.

**The “compass” of Calculus: the Integral**

**Area under the curve** between a on the left and b on the right = **the Integral of f** between a and b.

As the image illustrates, the area can be approximated by adding up the areas of the rectangles underneath the curve. With an increasingly finer mesh, adding up the areas of the rectangles will generate a–guess what?–**string of real numbers** to which we can apply (i) and (ii). You can either think of rectangles (as in the image) whose sum gets larger and larger but under the target area, or construct rectangles whose sum gets smaller and smaller but above the target area. Either way, (ii) tells me (with some help from (iii)) that the sum exists and is equal to the area.

All well and good, but this is rather messy. Who wants to draw rectangles and add up areas over and over, isn’t there a shortcut? Yes fortunately, the Fundamental Theorem of Calculus below.

**The two most important theorems**

**Rolle’s Theorem**

If a curve hits an “elevation” on the y-axis at two different places along the x-axis, then the graph was either flat its entire path between them or turned around and was thus momentarily flat (as in the picture.) It follows from the (i) (iii) and (iv) because if the curve was going up then back down (or vise versa) it must have gradually turned around since it does not jump nor does it have holes.

**The Fundamental Theorem of Calculus**

“Fundamental” because it relates “ruler” and “compass,” making Calculus ten times more useful: we get rid of the step with the pesky rectangles!

Without formal “proof” the result can be anticipated in the image. The “rate of change” of the total shaded area under the curve at the point x (**the derivative of A** at x) is proportional to the **area in red ** on the right, which happens to correspond to the height of the curve there as we bring the right endpoint in closer to x by (iii) and (iv).

And now, voila!

All “smooth” curves [functions that satisfy (iii) and (iv) under increasing magnification] can be approximated by a polynomial **as accurately as we like** on an enclosed region. Here, the **blue curve** is our goal, the others successive “Taylor” approximations.

Repeated application of rules (i) & (iv) “no jumps” and (iii) “as flat as you like at appropriate scale” are essentially what Calculus is about. An application of this method *par excellence* gives the elegant, surprising, stunning Taylor’s Theorem. It represents the genie and magic of Calculus, dealing with the complex by a clever combination of the simple.

Any “smooth” function can be approximated as well as we like, by an appropriate polynomial, i.e., a sum of powers of x and constants. “Smooth” meaning repeatedly differentiable, which is not at all a strict requirement, if you remember that when we take a derivative we usually simplify. “As well as you like” here means to within **any desired level** of accuracy in a **finite neighborhood**.

Nothing short of amazing! Every Calculus teacher should bow down to it, pray over it, meditate over it, praise nature for it, and make sure that his or her students witness this awe. The worst a math teacher could do is let something look easy, that is in fact quite brilliant and inspiring. Taylor’s formula should be unveiled from behind a satin curtain in a candlelit room to the score of hypnotic chants, Gregorian, Slavic, or Armenian, as the case may be.

In my experience tutoring, I found the above principles (derived from the properties of the real numbers and found in most “Analysis” texts) help students wrap their heads around the material and make learning the techniques more meaningful and authentic. This also happens to exemplify applied philosophy: from the foundations and basic techniques of a discipline, to trace its trajectory and anticipate its further development. In a subsequent post, I may talk more about the “ideals” behind the four ideas, share a “proof in words” of Taylor’s Theorem, as well as introduce the fifth (grand and significant) concept of Calculus/Analysis, “presque partout” or “almost everywhere,” that takes this approximation business to the next level.

I end with a quote from an awkward immigrant who became a memorable and influential Justice of the Supreme Court.

“**The ultimate function of education is to make men vibrate with wonder**.”

Felix Frankfurter

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I like the way you break this down into more basic mathematical concepts rather than relying on formal proofs. I also find it useful to use examples and applications (like from physics or economics) to help the concepts take root. What is the philosophical term for that approach?

Thanks for sharing this.

I agree wholeheartedly. Examples from real-world applications bring math to life; I may add some here if I can keep it short. For Taylor’s Formula, for ex., I mention to SS’s that it lets us tell a calculator how to compute Natural Logs, Trig functions, etc…

There was a huge error (that I did not notice for an entire day!) in the write-up of the Fundamental Theorem, now corrected I believe.

From a birds-eye view, I feel I could do more to explain the concept of a limit.

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