F 8/24 -- New class organization distributed. We finish section
2, begin section 3. Additional material on algebraic integers which
will be summarized in a handout on Monday.
M 8/27 -- We discussed the axioms for fields and for ordered
fields, and defined maximum and minimum. Mr. Hamza Yesilyurt visited
and announced his office hours: W 11-12 and Th 3:10 - 4:0 in room B1A,
Coble Hall. His email address is yesilyur@math.uiuc.edu. The first
homework assignment was distributed, due W 9/5.
W 8/29 -- We almost finished section 4, on the real line as an
ordered field plus. The extra property is that every non-empty set
which is bounded above has a supremum. There will be a short handout
on Friday with some remarks about the day's material.
F 8/31 -- We finished sections 4 and 5, and had a lively
discussion on countable and uncountable sets and theorems thereon.
This material will be covered in a handout, to be distributed in
class on 9/5.
W 9/5 -- Homework 1 was collected, and the solutions were
discussed. Here is a link: PDF file of
HW 1 Solutions. We began to talk about sequences, and I
mentioned my brush with greatness (Erdos number = 1), as it relates to
sequences. Homework 2 will be distributed on Friday.
F 9/7 -- Homework 1 returned and discussed. Homework 2
distributed, due 9/14, not 9/16 of course. Topics: more on convergence
and proofs. On Monday, we start with theorems so we don't have to use
epsilons all the time.
M 9/10 -- We went through about half of Section 9. Theorems on
convergence whereby we can combine limits we already know to find new
ones.
W 9/12 -- Following Chancellor Cantor's MASS-MAIL, we had a
discussion in class on the tragic events of 9/11. The due date of
Homework 2 has been delayed to 9/17.
F 9/14 -- Back to the syllabus. We finished most of section 9,
saw a typed-up version of the handout 3 on countability, and the new
example of iterated square roots, which will be on handout 5 on 9/17.
M 9/17 -- Homework 2 collected, and PDF file of
HW 2 Solutions and PDF file of
HW 3 Questions were discussed . We
finished up section 9 and began to talk about monotone sequences,
limsups and liminfs.
W 9/19 -- More on liminfs, limsups and Cauchy sequences.
The significance of Flannery O'Connor in real analysis was discussed.
F 9/21 -- HW2 returned and discussed. Cauchy sequences. The
Whitman (I contain multitudes), which contains subsequences converging
to every number in [0,1]. The beginning of the Bolzano-Weierstrass
Theorem. Homework 3 is due on Monday 9/24.
M 9/24 , W 9/26, F 9/28, M 10/1, W 10/3, F 10/5-- Well, it's
been a while. Hoemworks 3 and 4 were returned and discussed. We
completed the text through the middle of section 14
on series. (Some of section 13 on metric spaces was skipped.) In the
midst of all this were Bonus Notes 6 (proof that the Euclidean metric is,
in fact a metric) and 7 (some notes on topology and its connection to
sequences.). Also, PDF file of
HW 3 Solutions, PDF file of
HW 4 Questions, PDF file of
HW 4 Solutions, PDF file of
HW 5 Questions were passed out and discussed. A couple of
corrections for hw 5 were noted on the newsgroup, regarding the
optionality of #5 and a silly misprint in the second equation of #10.
M 10/8, W 10/10, F 10/12. We completed sections 14 and 16 and
were in the middle of 17. Homework 5 was returned and discussed. A
hiatus for a test in class on
10/17. Bonus Notes 8 (on series, a non-integral test proof of the
p-test, and an explanation of 13.10 (by request) and Bonus Notes 9 (a
general test for mixed geometric-rational function series.) Also, PDF file of
HW 5 Solutions, PDF file of
HW 6 Questions were passed out.
M 10/15 -- Review for test. Passed out PDF file of
HW 6 Solutions. See you Wednesday.
W 10/17 -- Test 1. Not everybody saw it, but PDF file of
HW 7 Questions was available.
F 10/19 -- Test 1 graded and returned.
The score distribution follows
| Scores | #Undergrads | #Grads |
|---|---|---|
| 100 | 1 | 0 |
| 90s | 5 | 6 |
| 80s | 1 | 2 |
| 70s | 1 | 5 |
| <70 | 3 | 4 |
Full-throttle onto continuous functions
M 10/22 -- Intermediate value theorem and uniform continuity.
You should read about inverse functions on your own. Two hints for
the homework. In 18.6, you can assume that cos[x] is continuous
without having to prove it. In 19.6, LEMMA: If x > y > 0, then
Sqrt[x - y] > Sqrt[x] - Sqrt[y]. (This is an improvement on the
hint I gave in class, but equivalent to it.) To get full credit on the
problem, you have to prove the lemma. HW 7 due Wed.
W 10/24, F 10/26, M 10/29 -- I fell behind in updating, but
we did get PDF file of
HW 7 Solutions, PDF file of
HW 8 Questions passed out and homework 7 was returned. We've
finished the discussion of uniform continuity and are getting into
power series.
W 10/31, F 11/2 -- Sailing along through power series and
arriving at the Weierstrass M-Test. PDF file of
HW 8 Solutions, PDF file of
HW 9 Questions passed out.
M 11/5, W 11/7, F 11/9, M 11/12, W 11/14, F 11/16 -- Sorry for
the delays in getting things updated; I will point out in my defense
that nobody
complained, however. In this two week stretch, we passed out
PDF file of
HW 9 Solutions, PDF file of
HW 10 Questions, PDF file of
HW 10 Solutions, PDF file of
HW 11 Questions, as well as (unlinked, but available from me
during class): 11th and 12th Bonus Notes, More Comments on HW 9 (but
not HW 10), and, for the brave and linked , PDF file of
13th Bonus Notes. The material from these notes (and from HW11
#10) is way too computationally involved to appear on any of my tests.
We've finished off power series and
differentiation, and are headed into integration.
For Homework 11, which is due on 11/26, it is important always to
remember the formal definition of the derivative. Several problems
are designed to probe the continuity of the derivative. Also, you can
use all the
trigonometric differentiations you remember.
There will
be a "virtual" Homework 12 distributed on 11/26 or 11/28 and covering
integration. This will not be collected, and it will not be a topic
for the second hour exam, although something about integration will
appear on the final. I've put a rough sketch of the rest of the
semester at the dates below:
M 11/26 -- HW 11 due and discussed. PDF file of
HW 11 Solutions distributed.
W 11/28 -- HW 11 returned, review for second test.
There will be a review session Thursday 5-6 in 141 AH, run by
Mr. Yesilyurt. I will be there briefly at the beginning.
F 11/30 -- Second test, in class. Everything since the
first test, but no integration.
M 12/3 -- Test 2 graded and returned.
The score distribution follows
| Scores | #Undergrads | #Grads |
|---|---|---|
| 90s | 3 | 4 |
| 80s | 5 | 9 |
| 70s | 0 | 2 |
| <70 | 3 | 1 |
More on integration. PDF file of HW 12 Questions distributed. This is "virtual" and will not be collected. Solutions will be distributed on Friday.
W 12/5. F 12/7 The end. PDF file of
HW 12 Solutions distributed. Summing up.
Sat. 12/8 -- Open office hours, 1:30 -> 4:00 in my office
243 Illini Hall. Enter by the south door.
Tue. 12/11 -- Final Exam (7:00 -- 10:00 pm)