MOOC update #3
/I confess: I am too far behind at this point to watch all the lectures. A week off at the beach followed two weeks later by a week at a conference in Germany has left me unable to catch up. I did watch the BONUS material in Chapter 3, which was very cool (more on that below), but I skipped most of the rest. Now, to be honest, I don't feel too bad about it since those are the lectures about techniques of integration and that's pretty standard stuff. Still, this isn't quite what I promised to do, is it?
And, that, it seems to me, is the issue with MOOCs. You make a promise, but really only to yourself. It's kind of like a new year's resolution--no harm if you don't keep it. If you are conscientious, then you may really gain something, but otherwise you just maintain the status quo. In 2012 I resolved to read Moby Dick and I did it (although I found this version last month, which in many ways is just as good) . This year I promised myself I would read War and Peace ; I quit after 30 pages. Meh.
I've also stopped doing the homework. To really take this seriously requires about an hour per day of work: watch the lecture and do the homework. And, frankly, I've done enough calculus problems in my life. I still enjoy them, but I don't really need the practice. I take the quizzes, though, and I plan to take the final, but that's mostly out of intellectual curiosity.
And really, it's a shame. This course is spectacular . I cannot praise it enough. The lectures are superb. The production value on the videos is top-notch. Rob gives excellent examples, explains things well, etc. In short, it's almost perfect and I would recommend it highly to anyone. Well, anyone with the self-discipline to see it through.
One of the Chapter 3 videos I did watch was the bonus material about the Fundamental Theorem of Calculus. I assumed (correctly) that this is where we would see a proof of the theorem. I once had an idea for a book called Where's the Mean Value Theorem ? It would be a Where's Waldo? style math book in which the reader would be invited to find where the Mean Value Theorem is lurking in the proofs of the theorems. I often assert to my students that the real Fundamental Theorem of Calculus is the Mean Value Theorem--it drives almost everything that comes after it. So, in the case of this course, I was wondering how Rob was going to pull off the proof of the FTC without using the Mean Value Theorem, which had not appeared in any of the lectures.
In case you've forgotten, the Fundamental Theorem of Calculus states the following. Suppose \( f \) is a continuous function defined on some interval \( [a,b] \) and that \( F \) is an antiderivative of \( f \) on the interval. Then \[ \int_a^b f(x)\, dx = F(b) - F(a). \] Calculus students love this theorem. Finally, I can stop dealing with those ridiculous Riemann sums and compute integrals easily (well, assuming I can find an antiderivative).
The standard proof of this involves first proving that the function \[ F(x) = \int_a^x f(t)\, dt \] is an antiderivative of \( f \) on the interval. This is really the tricky part and it is here that we typically invoke the Mean Value Theorem. I won't write the proof or the MVT out explicitly; you can look them up in any calculus text. What Rob does here, though, is use the idea that the derivative measures the first order variation of a function and then use the Taylor expansion of \( f \) to show that this function \( F(x) \) is indeed an antiderivative of \( f \). This requires an assumption, of course, that \( f \) has a Taylor expansion. Now, most reasonable functions do have such representations, but there are plenty of continuous functions that don't. Still, this is a reasonable approach in keeping with the overall theme of the course. And, in practice, most of the functions we're interested in do have such expansions.
So, on to Chapter 4, which covers applications of integration. I will try to watch most of these lectures as I am always interested in seeing good examples. More updates to come.