12/24/15, 3 a.m.: I have just submitted the grades. You must be able
to see them online within 24 hours. Thank you for a great
semester. Happy holidays!
12/24/15, 3 a.m.: I have just submitted the grades. You must be able
to see them online within 24 hours. Thank you for a great
semester. Happy holidays!
12/16/15: Extra office hours before the Final: Thursday 1:30 to 2:25
and 3:30 to 4:30 p.m.
12/15/15: The mean on Midterm II was 28.4, which corresponds to 71%,
and the median was 27.5. This is slightly better than on the first
test.
12/13/15: I have finally managed to post the second part of the Sample
Final.
12/13/15: I had problems with internet connection all day yesterday
until now, 12:30 p.m., and have not been able to post the second part
of the Sample Final. The internet seems to be working now, so expect
Part II of the Sample Final soon, within an hour.
12/12/15: I have posted Part I of the Sample Final. Sorry about the
delay. I will post Part II later tonight.
12/12/15: I have added a solution to Problem 1(3) on Sample Midterm
II.
12/9/15: I have posted solutions to Problems 1(2) and 3 on Sample
Midterm II. Just click on the link to the Sample Midterm.
12/7/15: I have also recalled that at some point I said ζ(1) =
-1/12, whereas it is actually ζ(-1) = 1 + 2 + 3 + ... = -1/12.
12/7/15: Today I called Abel's theorem what was supposed to be called
Dirchlet's test, see Theorem 6.4.5. I apologize: I confused it with
Abel's lemma, which is what 6.4.4 is.
12/6/15: The Sample Midterm Exam II is
posted. The best thing to do is to solve it on your own before we
discuss it on Wednesday. See more on how to get ready for the test
below.
12/6/15: Good news: I have removed Section 6.3 from the coverage on
both Midterm II and the Final.
11/30/15: There is a typo in Problem 5.7.27: you should assume that
the limit of g(x) as x approaches b from the left is equal to 0.
11/29/15: I have removed the material following 5.7.12 in Section 5.7
from the coverage on Midterm II and the Final. We will not cover it in
class. However, in doing the homework due December 2, you will need to
read the definitions, the wordings of theorems, and examples
5.7.13-18, skipping the proofs of theorems.
11/24/15: James has pointed out another error in the textbook to me:
In Problem 5.5.1, the a and b in front of the integrals should have
square roots around them. I have also posted an updated Errata file
for the textbook on our class web page. The link through the library
class page goes to the Errata file on the author's homepage, and that
file will be updated only after Thanksgiving.
11/23/15: Wes has pointed out another error in the textbook to me:
Theorem 5.7.11 makes sense only for b = +∞.
11/22/15: Coverage for the Friday, December 18, Final
Exam (10:30-12:30 in our regular classroom) is basically all we
will have studied from the beginning of the semester through the
second midterm. This will be Chapters 1 through 6, skipping the
starred sections, Theorem 3.1.11, the material following 5.7.12 in
Section 5.7 and Section 6.3. Pay attention to
the Errata. Apply
the same preparation gidelines as for the upcoming midterm, see below.
11/22/15: Coverage for the Friday, December 11, Midterm Exam
II (in our regular classroom) is basically all we will have
studied between the two midterms through Monday, December 7. This will
be Sections 3.5 through 6.4, skipping the starred sections, the
material following 5.7.12 in Section 5.7 and Section 6.3. Pay
attention to
the Errata. How
to get ready for the test? The best way would be to try to solve
as many problems as you can on your own (opening the textbook when
needed, and not timing yourself) from
the sample midterm exam, which will be
posted soon, the homeworks, and the textbook. For those problems you
cannot solve, it may be a good idea to discuss them with your friends,
i.e., your study group. Come to my office hours to get hints or other
help. Go over homework problems again without consulting your
solutions and look for problems from exams in similar classes on the
web: search for Analysis tests, exams, midterms, finals, etc.,
but make sure the class you find is for undergraduates. When you know
you can get an idea of how you might approach most of the problems
right away, you may feel more sure that you will do well on our
exam. Good luck!
11/22/15:
Homework 10, due Wednesday, December 2, has been
posted. This will be our last homework.
11/19/15: In Problem 5.3.2, when stating that f' is integrable on
[a,b], what is meant is that its continuous extension to [a,b] as per
Theorem 3.5.9 is automatically integrable as any continuous
function. Please ignore the first part of the problem, which asks you
to prove that f' is integrable. This question does not make sense
otherwise, because for a function to be integrable on [a,b], it has to
be defined at all points of [a,b] -- see the boxed statement in the
beginning of Section 5.1 on integrability.
11/19/15: Another hint for Problem 5.3.14: It is not at all necessary
to use L'Hôpital's rule. There is a pretty simple solution of
this problem (which I learned from Raj) that deals with finding a
large enough M, so that f(x) is sufficiently close to its limit at
∞, and breaking the integral of f(y)dy from 0 to na into
the integrals from 0 to N and from N to na. Since you are
concerned with the limits of the integrals divided by n, the
contribution of the first integral will be negligible for large
n. [This all is pretty frivolous math speak -- of course, when you
write up your solution, all these vague words should turn into
epsilons, M's and alike.]
11/18/15: A hint for Problem 5.3.14: If you are using
L'Hôpital's rule, do not forget to make sure that the integrals
of f(y)dy from 0 to na have a limit ±∞ as n
--> ∞. This actually may not be the case, if lim f(x) = 0 as x
--> +∞. In this case, estimate the integral with ± the
integral of |f(y)|dy and note that the integral of |f(y)|dy from 0
to na always has a limit, finite or infinite. Find the required
limit of the integral of |f(nx)|dx from 0 to a as n -->
∞ in both cases separately.
11/16/15: I have made a misprint in the proof of the change of
variables theorem today in class. When computing G(φ(b)) as the
integral of f(y)dy, I wrote the limits of integration as c to b,
whereas it was supposed to be c to d = φ(b). I should also have
mentioned explicitly that the image of [a,b] under φ was assumed
to be [c,d] in the wording of the theorem.
11/13/15, 1:52
a.m.:
Homework 9, due Friday, November 20, has been posted.
11/9/15: Saurav showed me a simpler way of proving that for an
integrable f and a constant c, the function cf is integrable and its
integral is c times the integral of f. If you use the fact that for c
> 0 and A a subset in R, sup (cA) = c sup A and inf (cA) = c
inf A, then sup _S(cf,P) = sup c _S(f,P) = c sup _S(f,P) = c
(integral(f dx)). And similarly, for the upper Darboux sums. My
argument combined this with the proof of the equation sup(cA) = c
sup(A), which is proven using arbitrary ε > 0, whence there
were all those ε's.
11/7/15, 12:52
a.m.:
Homework 8, due Friday, November 13, has been posted.
11/6/15: Thinking it over on homework Problem 4.6.4, I believe the
only correct way meant to show that the limit in the problem is 0 if
Pn = Tn by the author was to apply
L'Hôpital's rule, as I presented first in class today. However,
I asked the grader not to take points off for an incomplete solution
that uses the remainder, should he be grading that problem. BTW, I am
still working on the new homework.
11/4/15: Homework Problem 4.5.21 contains a misprint not caught on the
Errata list: The interval (0,1) should be changed to (-1,1).
10/31/15:
Homework 7, due Friday, November 6, has been posted. Sorry for the
delay.
10/26/15: I made two errors in that example of a function whose
derivative at 0 is positive but which was not increasing on any
neighborhood of 0. The function must be f(x) = x2
sin(1/x) + x/2 for x ≠ 0 (I did not divide by
2 in class), and the suggested values of x and y which
show that the function is not increasing must be like 1/2πn and
1/(2πn ± π), rather than 2πn and 2πn ± π. It is
also easy to see that the function is not increasing near 0 by
computing the derivative f'(x) = 2x sin(1/x) - cos(1/x) + 1/2 at each
x ≠ 0 and noticing that around small x where the cosine is near
± 1, there will be intervals on which f' > 0 and f' < 0,
respectively. Thus, by the MVT, the function f will be increasing and,
respectively, decreasing on those intervals.
10/23/15:
Homework 6, due Friday, October 30, has been posted.
10/19/15: The mean on Midterm I was 26.1 and the median was 27, which
corresponds to 67.5%. On the average, you have done a good job for an
honors class.
10/19/15: The wording of the same Problem 8 in Section 3.4 needs to be
corrected to talking about solutions x in (0, +∞)).
10/18/15: In Problem 8 in Section 3.4, it is meant use Calculus tools,
such as the First or Second Derivative tests. Compare this to the
theorem that a function with a bounded derivative is uniformly
continuous at the end of the section on uniform continuity in the
text. (We will talk about it on Monday, time permitting.) It sounds
like we are making a closed circle by using the material we have not
studied yet. Sometimes, we do such things, when the focus of the
problem is pretty far away from the tools you would like to use.
10/17/15:
Homework 5, due Friday, October 23, has been posted.
10/12/15: I have removed stuff after Theorem 3.5.6 from the
coverage on Midterm I. I do not plan to cover it in class at
any time but will leave it for your reading after the midterm.
10/11/15: The Sample Midterm exam is
posted. The best thing to do is to solve it on your own before we
discuss it on Wednesday. See more on how to get ready for the test
below.
10/11/15: I have added a paragraph on Cardinality (page 508) to
the coverage for Midterm I. Chapters 1-3 assume its knowledge,
anyway, for instance in the theorem on the countability of the set of
discontinuities of a monotone function in Section 3.3.
10/10/15: I have removed Theorem 3.1.11 from the coverage for
Midterm I. Even though it is now corrected in the Errata, it does not
make sense a Cauchy Criterion and can safely be skipped.
10/9/15: I have adjusted the coverage for Midterm I once again,
removing Chapter 4 from it whatsoever.
10/8/15: I have adjusted the coverage for Midterm I, removing
Section 4.2 from it.
10/8/15: Hint to Problem 3.1.2: Try to use sequences. If q is
an accumulation point for a set of reals, then there is a sequence of
distinct points in the set converging to q. Try to analyze how many
terms 2/m this sequence may have and how many terms 3/n it might
have. Play with subsequences of that sequence to restrict the places
they may converge to.
10/6/15: Coverage for the Friday, October 16, Midterm Exam
I (in our regular classroom) is basically all we will have studied
through Monday, October 12. This will be Chapters 1-3 through 3.5.6
and Cardinality (class notes and page 508), skipping Theorem 3.1.11
and Section 3.2. How to get ready for the test? The best way
would be to try to solve as many problems as you can on your own
(opening the textbook when needed, and not timing yourself) from
the sample midterm exam, which will be
posted soon, the homeworks, and the textbook. For those problems you
cannot solve, it may be a good idea to discuss them with your friends,
i.e., your study group. Come to my office hours to get hints or other
help. Go over homework problems again without consulting your
solutions and look for problems from exams in similar classes on the
web: search for Analysis tests, exams, midterms, finals, etc.,
but make sure the class you find is for undergraduates. When you know
you can get an idea of how you might approach most of the problems
right away, you may feel more sure that you will do well on our
exam. Good luck!
10/5/15: I have added a link to the list
of errata
for the text to our main course page.
10/2/15: Warning: the Cauchy Criterion for functions, Theorem
3.1.11, is not true! Here is a challenge: find an error in the
proof. Moreover, you may wish to find a counterexample. Then show me
what you have got.
9/25/15:
Homework 3, due Friday, October 2, has been posted.
9/23/15: Starting from this Friday, I have decided to modify my office
hours slightly and start them at 11:15 on Wednesday and at 10:10 on
Friday. Thus, my office hours from now on will be Mon 11:15-12:05, Wed
11:15-12:05, Fri 10:10-11:00, or by appointment.
9/21/15: If you have a question to the grader, you may send a message
to him at yeoxx038@umn.edu. If you need to see him, make an
appointment. His name is Yao-Rui Yeo.
9/18/15:
Homework 2, due Friday, September 25, has been posted. Reminder:
If you turn in your homework after the beginning, penalty will be
assessed after the second occurrence.
9/17/15: For homework Problem 1.2.1(e) assume also that c ≠
0, something just overlooked by the author of the text.
9/16/15: Some problems ask to prove a statement related to supremum
and infimum, and the proofs for each case are very similar. You may
just prove one of them, say, the case of the supremum, and then write
a single sentence saying that the proof for the infimum is similar,
without repeating the details of the proof.
9/16/15: I have expanded the comment before Problems A
and B on Homework 1, so as to explain more clearly why it makes
sense to impose that nasty constraint on you that you are not allowed
to use the reals and in particular
√ 2 .
9/14/15: Another student has pointed out to me that the square
root
√ 2
of 2 was printing on Homework 1 as just 2. I have corrected the
typo.
9/14/15: One of the students has pointed out to me that my argument
that p
√ 2 /q
is irrational had to assume that p/q ≠ 0. This is certainly
true, as it involved division by p/q. If it happens
that p/q such that a/
√ 2
< p/q < b/
√ 2
is actually zero, use density of the rationals again and find another
rational r such that a/
√ 2
< 0 < r < b/
√ 2 .
9/14/15: I have posted the
first
Homework, due Friday, September 18. It will be collected or
graded. Making sure to do homework on your own will be the best way to
get ready for the exams. Getting together with other students (that
is, in study groups) is a very effective way to do homework. Your
friend is your TA. :-)
9/6/15: I am out of town, participating in a workshop
on Braids,
Configuration Spaces and Quantum Topology at the University of
Tokyo, Japan, during the first week of classes at the U, through
September 12. I have asked Professor Westerland to substitute for me
in my Wednesday and Friday classes. No office hours during the first
week, either: if you need to contact me, please write me at
voronov@umn.edu . I apologize about starting the term with being
away. I look forward to meeting you and teaching the class!
9/6/15: I recommend the following way to study for this class. Attend
each class, take notes, participate in class actively. After each
class review your notes and study the corresponding part of the
text. You can find out which part of the text at
the
Class Outlines page. Then do the assigned homework problems
pertinent to that material. Some students find it helpful to read the
material before it is covered in class, some prefer to do reading
after class.
9/6/15: If you need to register for this class, please, send a message
to Ms. Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
Last modified: (2018-10-07 23:46:40 CDT)