Jan. 16 | Jan. 23 | Jan. 30 | Feb. 6 | Feb. 13 | Feb. 20 | Feb. 27 | Mar. 5 | Mar. 12 | Mar. 19 | Apr. 2 | Apr. 9 | Apr. 16 | Apr. 23 | Apr. 30

Final Exam

- Jan. 17:

- Show that given a sequence of real numbers {
*a*}, if_{n}*a*, then |_{n}L*a*||_{n}*L*|. - Show how the prior problem proved that the absolute value function is continuous for all
*x*. - Prove that if
*a*is a limit point of set*A*, then there exists a sequence {*x*} of points in_{n}*A*such that*x*a for all_{n}*n*, and*x*._{n}a - Jan. 19: Read chapter 5.2 and do 5.2 #1ce, 3ab (note the example cited), 5a, 13,15.

- Jan. 24:
- do 5.2 #12 (go ahead and read the hint in the "answers", I did, #16, 19 (add to #19: Prove that f '(0) exists),20
__Use the Inverse Function Theorem__to prove that the derivative of*x*^{1/n}is (1/*n*)*x*^{1/n -1}, for*n*and*x*>0.- Use the chain rule and the previous problem to prove that the derivative of
*x*is^{p}*px*^{p}^{-1}for*p*rational and*x*>0. - Show that if
*f*is continuous and strictly increasing on an interval*I*, and*a,b**I*, then*f*( [*a*,*b*] ) = [*f*(*a*),*f*(*b*)]. Note one must both prove the set in question actually is an interval,*and*that it has endpoints as shown. - As a corollary to the previous, show that if
*f*is continuous and strictly increasing on an interval*I*, and*a,b**I*, then*f*( (*a*,*b*) ) = (*f*(*a*),*f*(*b*) ). - Jan. 26:
- For next Tuesday, Turn in the 1/19,1/24 homework, and rewrite 1/17 homework as needed to get the rest of the points.
- Group oral presentations for Tuesday 1/31: Consider the function
*L*:(0,) such that*L*'(*x*)=1/*x*for all*x*(0,) and*L*(1)=0. - a) Show that
*a*,*b*>0,*L*(*ab*)=*L*(*a*)+*L*(*b*). Hint: Consider the derivative of*L*(*ax*) for a fixed*a*>0, and use some of our theorems. - b) Show that
*b*>0,*L*(1/*b*) =*-L*(*b*). Hint: Consider the derivative of*L*(1/*x*). - b') Show that
*a*,*b*>0,*L*(*a*/*b*)=*L*(*a*)-*L*(*b*).

- c) Show that
*b*>0,*r*rational,*L*(*b*) =^{r}*rL*(*b*). Hint: First prove*L*(*b*) =^{n}*nL*(*b*) for*n*(induction?), then find*L*(*b*^{1/}) for^{n}*n*by considering*L*((*b*^{1/})^{n}). Note the^{n}*rational*restriction. In fact, this holds for all real*r*, but you needn't prove that. - d) Show that
*L*(*e*) = 1, where*e*is the limit as*n*goes to infinity of (1+1/*n*). Hint: (1+1/^{n}*n*)^{n}*e*so*L*((1+1/*n*))? (and why?). But this limit is tricky to evaluate. Further hint:^{n}*L*'(1) exists, write out its definition in the*x*0 form and simplify. Finally, 1/*n*0, apply the continuity of a function built from*L*.

- e) Show Range(
*L*)=. Hint: We know*L*is continuous and strictly increasing (why?), and it includes 1. What's the alternative to the range being ?

- Jan. 31: (Jumping back to finish continuity): Read 4.3
*Uniform Continuity*, and do 4.3 #2ab,3a,4a. Lots of hints in the back, here's another for 4a: You can do cases:*x**y*,*x>*y, or explain why,*WOLOG*,*x**y.* - Feb. 2:

- Feb. 7:

- Feb. 9:

- Feb. 14:
- Feb. 16:

- Feb. 21:
- Feb. 23:

- Feb. 28:
- Mar. 1: Do the integration problems (NOT ALL OF THEM)! #1,2,3,5, and 6.2. Start to think about 6.3.

- Mar. 6:
- Mar. 8:

- Mar. 13:
- Mar. 15:

- Mar. 20:
- Mar. 22:

Spring Break Mar. 26 - 30

- Apr. 3: Read 8.1, 8.2, do 8.1 #1-3,5

- Apr. 5: Read 8.3, do 8.2 #2,5,8a,c,11,13 and 8.3 #4 (use
the results from class when they help, even if they are out of
sequence!)

- Apr. 10:
**Turn in Thursday:** - 8.1 #5, 8.2 #2(ab at least, try for
*c*. Hint: at least one of*f,g*must be unbounded on*E*by part (b)),

- 8.2 #11,13
- New: Prove that if
*A,B*are subsets of , and*f*uniformly on_{n}f*A*and*f*uniformly on_{n}f*B*then*f*uniformly on_{n}f*AB*. - Use the previous problem to do 8.2 #3 as a mostly-easy corollary
- New: Do 8.3 #3

- Apr. 12: Do 8.5 #3,6bcd

- Apr. 17:Read 8.7, especially the examples, do 8.7 #1ab,3,5,11,15 (#15 through
*n*=3) and - Prove that if the power series
*a*(_{n}*x-x*)_{o}converges to^{n}*f*(*x*) with radius of convergence*R*, then*a*/(_{n}*n*+1)*(*x-x*)_{o}^{n}^{+1}converges to*F*(*x*) for some function*F*on an interval with the same radius of convergence (and same center, yes) such that*F*'(*x*)=*f*(*x*) for all*x*in (*x*-_{o}*R,**x*+_{o}*R*).

- Apr. 19: Do 9.1 #1,2 (use theorem 9.1.4), and #4,6a. Note with glee that you've already proven #13. Read 9.1, 9.2

- Apr. 24:Do 9.1 #5, 9.2 #1 (almost off-topic, that one is), 9.2 #2, 3 (use!! 9.1.7 & the fact that this is
*given*to be an orthogonal system) 9.3 #1,3b,c (use #1 for 3c, and use Maple for all of #3)

- Apr. 26:

- May 1 (Last day of class)

- Friday May 4, 2-5 p.m..
**Final Exam**

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