Feb 4 2008

MAIN POINTS
The first couple of pages are again a restating of calculus theorems. The first is Rolle's Theorem, and the second is the Intermediate Value Theorem. The intermediate value theorem is sort of an obvious result but apparently the proof is difficult (but not shown here). I was less clear on Rolle's theorem. The IVT leads off by saying an efficient way to find the intermediate solution would be given in chapter 2, which is where the next assignment is. In the second part, there is an example with a differential equation (which I didn't really get), and then the text discusses the Bisection Method. This seems to be a divide-and-conquer method of root-finding. This takes the form of an algorithm. The section also discusses divergence, and trying to find optimal bounds.

CHALLENGES
I was not clear on how they constructed the statement of Rolle's theorem, and how this theorem is useful. I would also like to go over the concept of a differential equation, because, though I've seen them before, I've forgotten how the work. As for the bisection method, it would be good to see an example where it converges and an example where it diverges.

REFLECTIONS
It's interesting that the Babylonians used methods like this without computers (and it would blow my mind to use a base-60 number system). I just realized that I used this bisection method in my computer science class in high school to calculate square roots. It was an improvement on the previous method we had used, which was to take fixed steps until we crossed over the value, and then decrease the step value and change directions.