| Date | Due | Problems | Comments |
|---|---|---|---|
| Jan 10 | Jan 19 | 4-1 #1(a)(b) | |
| Jan 12 | Jan 21 | Prove Theorem 4-2 (the Squeeze Theorem), using the epsilon- delta definition of limit |
|h(x)-L|\leq |h(x)-f(x)|+|f(x)-L| \leq |g(x)-f(x)|+|f(x)-L| (why?) \leq |g(x)-L|+|L-f(x)|+|f(x)-L| =|g(x)-L|+2|f(x)-L| or for another approach, obtain L-epsilon < f(x) \leq h(x) \leq g(x) < L + epsilon |
| Jan 12 | Jan 21 | Prove Theorem 4-2 (the Squeeze Theorem), using Theorem 4-1 and problem 2-1 #9(b) | |
| Jan 19 | Jan 26 | 4-1 #2(a)(b)(e), 4-1 #12 | |
| Jan 21 | Jan 28 | 4-1 #7 | use epsilon-delta (or sequences) to check that f(x)=x is continuous, then use Thm 4-3 and induction |
| Jan 26 | Feb 2 | 4-1 #8 | |
| Jan 26 | Feb 2 | 4-1 #6 | [ 1/pi, 2/pi] |
| Jan 26 | Feb 2 | 4-1 #15 | IVT |
| Jan 31 | Feb 7 | 4-1 #24, 25(a),(b),(c) | |
| Feb 4 | Feb 11 | 4-2 Prove that lim_{x \to 0} ln(x) = -infinity | |
| Feb 4 | Feb 11 | 4-2 #2(b)(c) | for (b), M = 2/epsilon will work for (c), note that when x < 0, sqrt (3) = - sqrt (3x^2) / x |
| Feb 4 | Feb 11 | 4-2 #2(d), 8 | |
| Feb 16 | Feb 23 | 5-1 #9 | |
| Feb 18 | Feb 25 | Let f(x) = x^2 if x is rational and x^4 if x is irrational. Prove f is continuous only at x= -1, 0, and 1, and differentiable only at x=0 | Thm 4-1 and Coro 4-2 can be used |
| Feb 18 | Feb 25 | 5-1 #7 | |
| Feb 21 | Feb 28 | 5-2 #2 | for (b), sqrt(x) is an example (use a theorem to verify that it is uniformly continuous). You can actually find an example differentiable on [0,1] by altering the example x^2 sin(1/x). |
| Feb 25 | Mar 3 | Ch 6 handout problems 1, 2, 3 | |
| Feb 28 | Mar 6 | Ch 6 handout problem 4 | |
| Mar 22 | Mar 29 | 6-2 #1 | |
| Mar 24 | Mar 31 | 6-2 #2, 4 | |
| Mar 27 | Apr 3 | 6-2 #7 | take g=f and show F=0 |
| Mar 29 | Apr 5 | 7-1 #3 | 2-9 |
| Apr 7 | Apr 14 | 8-1 #1, 4, 5(b), 8 | |
| Apr 14 | Apr 24 | 8-2 #1(a)(b) |