Paul's Homework 3

3. Good! You should distinguish that the delta associated with f(x)'s epsilon need not be the same associated with h(x)'s epsilon, but then again, we can always choose the smaller delta as the fixed delta. You got the key step (the hint), just be sure to rewrite in ||(g(x)-l||< epsilon for ||x-a||< delta form in the last step.

4. Multiplication....I remember this tricky bastard. One of the best moments of my life is when I realized this super unintuitive trick (took me about a week). Add zero: ||k(x)f(x)-cl|||= ||k(x)f(x)-k(x)l+k(x)l-cl|| Now use triangle inequality in a smart way along with the given epsilon limits.

Physics...read rest later...3AM.


 * Update******

Decided to demonstrate question 4 since it is very pretty and not obvious:

http://www.youtube.com/watch?v=a-CpIK6UuWI http://www.youtube.com/watch?v=NhVwHHBby_g

For p.117 1,2,3,4: Essentially these are all chain rule questions in which you simply apply the given facts. Great job on 2, 3, and 4, but you made a silly error on # 1. You tested for orthogonality here (dot product is 0), but it suffices just to check ||g|| is a constant so it a constant distance from the origin. No biggie though.

Ah,. question 11, the right of passage Frenet formula questions. Was confused about the substitution of N in the first part, but forgot to check the definition given on p. 115 (dammit shifrin)!. So I see you use the identity ||A x B|| = ||A|| ||B|| sin theta but how do we know theta = 90 degrees? for b, and d I do see some very correct parts, but not sure how some steps were made (though like the thing I looked up in a, they are probably correct). I know I bombed parts of this question in college (hate hate hate physics like questions). You can find the proofs somewhere here http://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas#Proof