Yesterday Steven Strogatz shared an unpublished appendix to his book Infinite Powers:

I had originally included two appendices in Infinite Powers, but they felt too mathy and got cut. Still, some of you might find them interesting. Here's a rough draft of appendix 1, about a clever way that Fermat integrated x^n with his bare hands [PDF]: https://t.co/gutuj0R38A

I read it and thought it would be terrific to share with my older son who took calculus last year. This year we’ve been working on Linear Algebra – so not a lot of polynomial calculations (yet!) – so I also thought Strogatz’s appendix would be a terrific review.

I had him read the note first and when he was ready to discuss it we began:

At the end of the last video my son had drawn the picture showing Fermat’s approach to calculating the area under the curve . Now we began calculating. He was able to write down the expression for the approximate area without too much difficulty:

The next step in working through the problem involved some work with a geometric series. Here my son was a little rusty, but I let him spend some time trying to get unstuck:

I just turned the camera off and on at the end of the last video and he continued to struggle with how to manipulate the geometric series into the form we wanted. After a few more minutes of struggle he found the idea, which was really nice to see.

Once he understood the simplification, the rest of Fermat’s proof was easy!

I’m really happy that Strogatz shared his unpublished note yesterday. It is terrific to share with kids who have already had calculus, and would, I think, also be terrific to share with kids studying Riemann sums.