Jan. 19 | Jan. 26 | Feb. 2 | Feb. 9 | Feb. 16 | Feb. 23 | Mar. 2 | Mar. 9 | Mar. 16 | Mar. 30 | Apr. 6 | Apr. 13 | Apr. 20 | Apr. 27

- Jan. 20: The five-old-familiar-proofs handout - turn in all five on Thursday, with at least one of the five in LaTeX.

- Jan. 22:

- Read chapter 4.2, 4.3. Do 4.2 #11a, 4.3 #2b,3a,11a,5 (recommended in that order)
- Optional lemma that may prove useful: If f is continuous on [a,b) and f is uniformly continuous on (a,b) then in fact f is uniformly continuous on [a,b).

- Jan. 27:

- Do 4.3 #4a, 6a, 10; 10 might be tricky, but think about the graph of f in the way we did with f(x)=(1-x
^{2}). - Read 4.4, do 4.4 #2a, d (here "[y]" means the greatest integer less-than-or-equal-to y, i.e. round down to the nearest integer), 10

- Jan. 29:
- Review problem: Give an - proof that [f(x)+g(x)] = f(x) + g(x), provided the latter limits exist. Hint: Start by assuming the "provided" part, and give 'em names. For simplicity, you may assume f & g are defined on .
- Using the text notation on p. 148 for one-sided limits, prove the assertion there that:

- Assuming f is defined on an interval I with interior point a, f(x) exists iff [ f(a+) and f(a-) both exist AND are equal ].

- Do 4.4 #13 (easy), 15 (hard: use a really big theorem from Real I)
- Do p. 162 #1 (this will take some work. Draw a simple example first and draw where the c
_{n}'s are for the first few values of n. It's fairly easy to show {c_{n}} converges by Cauchiness.

Then show |f(c)-0|< for arbitrary , using |f(c)-0||f(c)-f(c_{n})|+|f(c_{n})-0| and a carefully-chosen c_{n}.) P.S. - this is called the midpoint method of root finding; it is easy to code a computer program to find roots using this algorithm.

- Feb. 3:
- Give a formal, - proof of the lemma: Suppose F : E with E and E containing some neighborhood of a. Then F(x) = L if and only if lim
_{h->0 }F(a+h) = L. [The last limit is the limit as h goes to 0 of F(a+h); this is something like our F&G lemma from class today; little computation is needed, but be very precise with the quantifiers.] - Given x
_{n}= 1+(-1)^{n}/n and c_{n}=1/2^{n}n, and defining f on [-1,2] by f(x) = c_{n}I(x-x_{n}), sketch the graph of f on [-1,2] at least for points not very close to 1. Make sure to include at least a few precise pieces or segments of the graph, and try to convey the overall shape. ALSO, using the theorem from class/text that discussed this construction, tell where f is continuous (no need to prove this beyond correctly citing the theorem).

- Read 5.1, do 5.1 #1abc,2 [for #2, you can use the text hint, or use theorem 5.1.5 and induction]
- State a correct variation of today's F&G lemma for one-sided limits. No proof needed if the lemma is correct.
- Do 5.1 #14. At some point, it may be useful to use the lemma that lim
_{ h->0}F(h) = lim_{ h->0}F(-h) [it matters here that h is going to zero!] - On Thursday, turn in: 4.3 #3a, 11a, the first problem listed for Feb. 3, and 4.4 #15

- Feb. 5:
- Read 5.2.
- Give a formal proof of the lemma: Suppose G:(a,b)→ with (a,b) some interval in , and for some δ>0 and some constant M∈, G(x)≥M ∀x∈(a,a+δ).

a.Prove that lim_{x→a+ }G(x)≥M, if this limit exists.

b.State a mirror-image result involving a left-sided limit (no proof required). - Do 5.1 #5a, 9 (theorem 4.1.9, the squeeze theorem, may prove useful).
- Examine this list of
presentation problems and send me preferences, if any, for problems and
partners. Groups will be assigned soon via email, presentations
to start on Feb. 12.

- Catch up!

- Feb. 10: [Prepare your assigned presentations]

- Feb. 12: Presentations from this list

- Feb. 17: do 5.2 #3a,b, 6,13,15 [Hint for 6: Think about |f '(c)| for c∈(a,b), using the definition of f '. ]
- Feb. 19:
- Prove the lemma: lim
_{x→0}+ f(1/x) = L iff lim_{x→}f(x) = L

(assuming f defined on (0,), and the latter limit defined by 4.1.11 or as defined in class; use either definition as you prefer, they are equivalent by setting M=1/.) - Do 5.2 #20
- Read 6.1 through 6.1.5 and skim 6.1.6.
- Do 6.1 #1a
- For next Tuesday, turn in:

- The second bullet item from Feb. 5, i.e. lim
_{x→a+ }G(x)≥M - 5.1 #5a (be rigorous, one way to do this uses the item above.)
- 5.2 #6,13 (use the Mean Value theorem to do #13)

- Feb. 24: Exam I
- Feb. 26:

- Read the rest of 6.1,

- do 6.1 #2a (see example 6.1.6b), 3a, 6,7,13
- Read 6.2

- Mar. 3:
- Here's the useful lemma I mentioned in class today; we'll prove a simplified version. Suppose
*f*is integrable on [*a,b*] and*g*(*x*)=*f*(*x*) for*x*∈(*a,b*], but*g*(*a*) =*f*(*a*)+*k*for some real number*k*.

Show that*g*is also integrable on [*a,b*] and that*f*=*g*. You may restrict your proof to the case that*k*>0, just to save work.

Hints: Riemann's integrability criterion (6.1.7) will be invaluable in both directions here. Since*f*__is__integrable, what does it tell us (for starters)?

Consider M_{1}and m_{1}for*f*and M'_{1}and m'_{1 }for*g*, and compare them (there are two cases). Also consider refining the partition by adding a point near a. - Do 6.1 #4c, 17,21

- Mar. 5:
- Read 6.2 and 6.3,
- Do 6.2 #1,3a,b,4,10. Note: Use the theorems of this chapter whenever possible, [not for #1, it is a theorem in the chapter] and the lemma above (Mar. 3)
- Do 6.3 #1. You don't get to use 6.3.4 here (why not?), but looking at its proof will help; you may assume f is bounded.

- Mar. 10
- Do 6.3 #2ab,4,5(esp. 5d), 8,11ac, 17ab

- Mar. 12
- A short (?) take-home exam, due a week from tomorrow.

- Mar. 17
- Mar. 19

Spring Break Mar. 23 - 27

- Mar. 31: Prepare group presentations from this handout for Thursday.

- Apr. 2: Read chapter 7.1, do 7.1 #1,2acdfghkn, 4abc, 10

- Apr. 7:

- Do 7.1 #9,12,11 (suggested in that order, note the terms are not rational functions of k), AND 14 (oops, I misspoke, Becca), and

- 7.1 #8 (this one is tricky; read the hypotheses carefully, and write out a partial sum of b
_{k}) - Read 7.2
- Optional: Victor Borge's Phonetic Punctuation System.

- Apr. 9
- Do 7.2

- #1,2 (you will need to add quite a bit to the hints in the text),

- #3 (careful, this one is tricky. Hint: It makes a big difference whether the terms are non-negative or not.)
- #5a,c,d,e,f
- Read 7.3

- Apr. 14
- Apr. 16
- Do 7.3 #2(you'll need to work with partial sums here),3,5,6a,b,c,f,g,9,10,11
- Read 8.1, do 8.1 #1a,b,c (try to be rigorous here), 2
- Read 8.2

- Apr. 21
- Do 8.1 #5 (and think what this example illustrates about the different types of convergence we have studied so far)

- Do 8.2 #1 (a variant of something proved in class), #2ab, and think about 2c.
- Do 8.2 #5, 8abc (8.2.7 will work on all of these, though sometimes that will take some real work)
- Do 8.2 #12
- Read 8.3

- Apr. 23: Quiz at the end of the period

- Apr. 28
- Read 8.4, skim 8.5, noting the theorem we proved today.

- Do 8.3 #4, 8.4 #5 (note typo in text, the lower limit of all the integrals should be "a", not "0".) You will need important inequalities we have proved for integrals.
- Suppose f is continuous on [0,1] and 0<b<1, show that {f(x
^{n})} converges uniformly to the constant function f(0) on [0,b]. - Do 8.4 #6; the previous problem will help, if we break the integral on [0,1] into integrals on [0,b] and [b,1], and choose b carefully
- Do 8.4 #9, 8.5 #1,3
- [At least a few of these are likely to turn into takehome problems]

- Apr. 30

- May 5 (Last day of class)
- Friday May 8, 2-5 p.m..
**Final Exam**

Prof. Janeba's Home Page |

Department of Mathematics | Willamette University Home Page