Tuesday, June 15, 2010

Fundamental Theorem, continued

I spent an hour yesterday morning, preparing for my tutoring session with Artemis. That's the first time I've done that. I figured there were 3 questions that followed from what we'd done last week:
  1. Why does the sum of the first n squares turn out to be n(n+1)(2n+1)/6? (I tried to think about it with drawings, but got nowhere. I looked up "sum of first n squares visual" and found this discussion at a blog called Understanding. This pdf was linked to, and it's the best thing I found. But Jason Dyer pointed me to this much better visual. Now I get it.)
  2. What if we wanted that same shape, but could afford a little extra weight, and wanted to find the area out to a variable right edge, labeled R? That was a way to rehash what we'd done the week before, with a little bit more generality.
  3. A car part that's under some crazy function.
 When I explained my three questions to Artemis, he said he didn't want to do any of them. I said I was so excited by this stuff, I could stand to do the first question some other time, but I really wanted to look at the second question. I think he was concerned it would be hard. But once we started, he was very excited about it. We figured out that it would be R^3/3. As we took the limit as n (number of slices) approached infinity, he got a kick out of pointing out the parts that would go to 0.

We haven't done many derivatives yet, so he didn't notice that R^3/3 was the anti-derivative of x^2 (at x=R). I asked him to find the area under a triangle with angled side y=x and right side at x=r. That was easy. I wondered what the area under y=x^3 would be, and he saw the pattern, and guessed it. We haven't proved it yet.

We tried to find the area under y=2^x, but we'd never done derivatives of exponential functions yet, so we weren't able to finish that one.

We have one more session before I go on vacation. Maybe we'll tie up some loose ends, or maybe we'll play with something less strenuous. I'll let him decide.

2 comments:

  1. re: sum of first n squares, I like this visual proof:

    http://mathoverflow.net/questions/8846/proofs-without-words/8851#8851

    ReplyDelete
  2. That is way better than the one I linked to! I totally see it now. Lovely...

    And it starts to give me more of a feel for why cones and pyramids have the 1/3 in their volume formulas. If what I'm thinking is right, then a 4D pyramid or cone (whatever that is) would have a 1/4 in its volume formula! Wow! Thanks, Jason! (No, I'm not excited, I always jump up and down like this in the morning.)

    ReplyDelete

Comments with links unrelated to the topic at hand will not be accepted. (I'm moderating comments because some spammers made it past the word verification.)

 
Math Blog Directory