05/17/07: I have just submitted your grades to the registrar. The
system says you will be able to see them in at most 24 hours. The mean
on the final was 40.57 out of 80. I feel like you have really learned
something in this class, and your life will never be the same! Thank
you for the interesting semester! Good luck!
05/04/07: Schedule of my office hours for the finals week, all in
Vincent Hall 324: Tue, May 8, 2:30-3:30 (Section 8.5), Wed, May 9,
1:30-2:30 (sample final problems), Thu, May 10, 3:30-4:30 (sample
final problems), Fri, May 11, 3:30-4:30 (sample final problems).
05/03/07: I am sorry I have to cancel my office hours today. I will
hold office hours tomorrow right after the class till 1:45 pm. If you
have a HW question before tomorrow, please, ask me by e-mail or call
me up at home at (651) 224-5634.
05/02/07: In response to popular demand, I have posted solutions to
our actual midterms on the class web page.
05/01/07: Final exam practice problems are posted. The final
exam is coming in less than two weeks on Saturday, May 12,
1:30-3:30 pm, in Vincent Hall 2. Coverage: Section 1.1 (skip the
subsection on Cross Products), Sections 1.2-8, 2.1-9 (You do not need
to memorize the integral forms of the remainder in Taylor's theorems,
nor need you to know Euler's theorem on page 68.), 3.1, 7.1, 7.3,
8.1-2 (skip Abel summability), 8.3-5. You do not need to memorize the
answers for u(x,t) from Section 8.5, but you should be able to obtain
them, as done in 8.5.
Hints on getting ready for the Final. A structured approach to getting ready for the exam may be as follows. Review all the homework problems anew. If you see that you know how to solve a homework problem, move on to the next one. If you do not, sit down and solve it. If it does not crack, look for a similar example worked out in the text or in your class notes. Solve the other exercises at the end of the relevant sections in the text. Then do the problems from the sample midterm exams and final practice problem sets. Do not forget to time yourself: if it takes more than 12 minutes to solve a problem, it might be a good reason to worry. Then search for sample exams, such as old or sample finals which might be available in the libraries and on the web for Advanced Calculus or Analysis courses at other schools. When you know you may solve any of them earlier than your time is up, you may rest assured you are ready for the exam.
04/27/07: I have to change the time of my office hours Wednesday, May
2 to 2:00-3:30 pm.
04/26/07: The Smart Learning Commons is seeking to hire experienced
undergraduate students to be Peer-Assisted Learning facilitators for
Math 1031 for Fall 2007. Click here for the job
description.
04/25/07: I have added the answer for Ex. 8.1.3 to the posted
Solutions to the previous homework. This answer is needed to solve
Exercise 8.2.4a on the current homework.
04/25/07: My PhD student will be having a thesis defense at 10 am on
Friday, April 27. I will have to be there on his thesis committee, and
I am afraid, I will not be able to make it to our Friday class. I have
asked Professor Gulliver to substitute for me, so the class will go as
usual. In particular, hand in the homework to him before the
class.
04/12/07: Statistics for the second midterm: mean = 16.54, median =
17, and the maximal score was 34, all out of 40. These are more
pleasing results than those from the first test, but they could have
been better!
04/08/07: I forgot that in Problem 4 on the Other Sample Exam 2, there
was a condition x, y, z > 0, which I totally missed in the
solution. The correct solution is actually simpler: you just need to
use a plus instead of a plus-minus. I have reposted a corrected
version.
04/08/07: Mariam has pointed out an error in the posted solutions of
the Other Sample Exam 2. In Problem 2, I should have used the 7th
derivative of sine, rather than cosine. Sorry. I have corrected that
and reposted.
You do not need to memorize the integral forms of the remainder for Taylor's polynomial on pages 85-87 in the book. These will be given on the exam, if needed. You do not need to know Euler's theorem on page 68, either.
04/07/07: Typing up solutions to the other sample exam 2 late last
night, I started aflling asleep at the computer and the math formulas
stopped making sense to me any more. So, I wrapped things up, cleaned
up the directory, and went to bed. Getting back to work the next
morning, I discovered that I had deleted all the pdf files linked to
our class web page! I restored them quite quickly, so that most of you
probably have not noticed anything. However, that delayed my posting
complete solutions to the sample exam, which are now (1:30 pm) finally
posted.
04/06/07: As promised, I have posted solutions to homework problems
from Section 1.7, last homework (HW #9), and one of the sample
exams. I have got to run, but I will post solutions to the other
sample exam quite late tonight.
04/04/07: Kari and Mariam pointed out an error in the answer to 2.9.15
in the first two printings. The correct answer is (2,0,2).
03/31/07: Another Sample Midterm Exam 2 has been posted.
03/30/07: The second midterm exam is coming in about a week on
Monday, April 9. Coverage: Sections 1.7-8, 2.1-9. You do not
need to memorize the integral forms of the remainder in Taylor's
theorems, nor need you to know Euler's theorem on page 68. Use the
last semester's Midterm Exam 2 which is posted on our class web page
as a sample test. Note that Problem 1 on Sample Exam 1 was covered by
our Exam 1. I will post another sample test for you as well.
Hints on getting ready for Midterm Exam 2. A structured approach to getting ready for the exam may be as follows. Review all the homework problems anew. If you see that you know how to solve a homework problem, move on to the next one. If you do not, sit down and solve it. If it does not crack, look for a similar example worked out in the text or in your class notes. Solve the other exercises at the end of the relevant sections in the text. Then do the problems from the sample exams. Do not forget to time yourself. Then search for sample exams, such as old or sample finals which might be available in the libraries and on the web for Advanced Calculus or Analysis courses at other schools. When you know you may solve any of them earlier than your time is up, you may rest assured you are ready for the exam. In my turn, I can reassure you it is not going to be simple:-) However, I plan to make it simpler than the first midterm.
03/30/07: I got confused by the logic of our solution at the end of
the class on Friday (today). What we did in class was showing that
there was only one local extremum (x0, y0),
which turned out to be a local minimum. It implies there is no
absolute maximum, because if one existed, it must also be a local
maximum. However, the local minimum (x0, y0)
would be an absolute minimum, only if we knew one existed. We still
have to show that. Here is how to show that the local minimum
(x0, y0) is an absolute minimum.
As I said in class, the argument is similar to the proof of Theorem 2.83a. Consider the curvilinear triangle xy ≤ 100, 3/x ≤ 100, 4/y ≤ 100 in the first quadrant. (We could have taken any constant instead of 100, just needed to make sure that it was greater than the value f(x0, y0) of our function at the critical point.) Since our function is f(x,y) = xy + 3/x + 4/y, its value outside the triangle in the first quadrant will be strictly greater than 100 (since at least one of the terms in f will be greater than 100). Since the value of f at the critical point is less than 100, the critical point (x0, y0) will be inside the triangle. The triangle is compact and f is continuous, therefore the function has a minimum value on this triangle. This minimum must be attained either in the interior of the triangle, or on its boundary. However, it may not be on the boundary, because the value of f on the boundary will be at least 100, and the value of f at (x0, y0) is less than that. Thus the minimum of f on the triangle must be at (x0, y0). It will be an absolute minimum in the whole quadrant, because (x0, y0) is a minimum in the triangle, while outside of the triangle the value of the function is greater than 100, thereby greater than the f(x0, y0).
03/29/07: I planned to be posting homework solutions on the web each
Monday, but this week I simply forgot to do that. Only now I
remembered that and posted solutions to Homework 7. Sorry. If I am
late with something in the future, I will not mind your tapping on my
shoulder and reminding me.
03/22/07: It seems like the hint in the text for Exercise 2.4.2 (b) is
kind of cryptic. Here is more to it. For the set S take the open
square (-1,1)x(-1,1) in the xy plane and remove the interval {0}x[0,1)
(that is the interval [0,1) on the y axis) from it. It is an open,
connected, but not convex set in R2. Imagine this as a map
of a staircase, on which you are walking on the left-hand side in the
direction of the negative y axis, then make a U-turn about the origin
and continue walking in the direction of the positive y axis. Then if
you take f(x,y) to be the height of this staircase, it will be given
by -y, when you are in the fourth quadrant, y, when you are in the
first quadrant, and 0 when you are in the second and third
quadrants. This function is almost what you want, but not quit, as it
does not have the partial in y along the x axis. But if you use +-y^2,
you will get a better luck. Show that this function is differentiable
on S, but depends on x.
03/09/07: Because of popular demand, I will be posting homework
solutions on the web, instead of going over them in class on
Fridays. However, I will be posting solutions with a slight delay,
namely the following Monday afternoon.
03/08/07: You do not have to show discontinuity of h(x) in Exercise
2.1.4 on Homework 6, as it is very similar to an old homework problem
(Exercise 1.3.6).
02/23/07: The first midterm was like a disaster for so many of you:
the mean was 13.5 out of 40 with a median of 12 and a maximal score of
28. Let us merge our efforts in getting the exams more pleasant: you
will try to put more into preparation (practice problem solving), and
I will try to make the next tests even easier:-)
02/16/07: It seems like the web site http://www.exambot.com, which I
recommended earlier to get sample exam problems, is no longer
working. Sorry about that. But you may still get loads of sample
tests, if you google the phrase "Advanced Calculus exam test" or
something similar.
02/16/07: Solutions to Sample Exam 1 are posted. A solution to Problem
1 from Sample Exam 2 will be posted soon.
02/16/07: I mistakenly thought that Problem 4 on Sample Exam 2 was
covered by our Exam 1. Sorry about that. Only Problem 1 from Sample
Exam 2 is covered by our Midterm Exam 1.
02/12/07: If you are interested in getting a job as a middle or high
school math teacher in Los Angeles, click here.
02/09/07: The first midterm exam is coming in about a week on
Monday, February 19. Coverage: Sections 1.1-6, skipping the
subsection on Cross Products from Section 1.1. I have posted last
semester's midterm exams as sample exams on our class web page. Note
that our Midterm Exam 1 covers Sample Exam 1 and Problem 1 on Sample
Exam 2.
Hints on getting ready for Midterm Exam 1. A structured approach to getting ready for the exam may be as follows. Review all the homework problems anew. If you see that you know how to solve a homework problem, move on to the next one. If you do not, sit down and solve it. If it does not crack, look for a similar example worked out in the text or in your class notes. Solve the other exercises at the end of the relevant sections in the text. Then do the problems from the sample exams. Do not forget to time yourself. Then search for sample exams, such as old or sample finals which might be available in the libraries and on the web for Advanced Calculus or Analysis courses at other schools. When you know you may solve any of them earlier than your time is up, you may rest assured you are ready for the exam. In my turn, I can reassure you it is not going to be simple:-)
02/07/07: Our grader Hao Feng will hold the following tutoring hours
in Lind Hall 150: Mon 2:30 - 3:20, Wed 8:00 - 8:50 & 2:30 - 3:20, and
Fri 9:05 - 9:55.
02/05/07: Math graduate student tutors begin working in Lind Hall 150
on Wednesday, February 7. Our grader Hao Feng will also hold tutoring
hours there, which are expected to be released today.
01/31/07: Here is an opportunity to participate in the second annual
Undergraduate Symposium on April 18 in Coffman Memorial Union. The
Undergraduate Symposium celebrates the creativity, performance, public
engagement, research, and scholarship of our undergraduate students in
fields from art to zoology. Students can participate by:
01/31/07: Because of a popular demand, I am moving my Friday office
hours for this class to Thursdays, 4:30 to 6:00 pm, starting tomorrow,
February 1. However, tomorrow I will be extremely busy at work all day
long, even after 5:30 pm, so that I will have to cut my office hours
short at 5:30 pm.
01/25/07: The Tutoring Center in Lind Hall has started
working. Although officially the tutors in Lind Hall only cover
courses through the 3xxx level, it does not really matter. All the
graduate tutors there would be able to help you (they might have not
started this week, though), and some of the undergraduate ones would
be able to help you, too. Moreover, when our grader Feng Hao will
start his tutorial hours in Lind Hall 150 (next week, I suppose),
there will be help available specifically for Math 4606 there.
01/24/07: If you are seriously interested in mathematics, I would like
to encourage you to apply to the Mathematics Advanced Study Semesters
(MASS) program and a Research Experience for Undergraduates (REU)
program at Penn State. MASS is a semester-long intensive program for
(mostly) math majors, while REU is a summer program. These program are
among the best in the country. Penn State has all sorts of financial
support for students to make their participation possible. See more
details on a poster on first floor in Vincent Hall (next to Room 115)
or at the MASS web site. I
also have a few brochures with a program description for you, if you
are interested.
01/24/07 (This corrects an earlier office-hour change announcement for
Friday, January 26): I have to move my office hours this Friday,
January 26, from right after the class, 12:15-1:45, to right before
the class, 10:00-11:15. Perhaps, this might be even more convenient
for you, as you will have more office hours before the first homework
is due this week. Please, see me Friday morning or during my regular
Wednesday 12:15-1:45 office hours this week, if you need to.
01/22/07: On Monday, February 12, the University of Minnesota campuses
will partner to host the 2007 University of Minnesota Job and
Internship Fair. The Fair is open to current University of Minnesota
undergraduates and recent graduates from all academic majors. It's
held at the Minneapolis Convention Center from 10 a.m. to 4
p.m. Complete details are available at http://www.umjobfair.org.
The Job and Internship Fair connects students with employers and organizations seeking to recruit them for diverse jobs and internships. It's an excellent opportunity for you. The Fair is free, but does require registration.
01/19/07: I have to move my today's office hours to 2:30-4:00. Please,
see me then, if you would like to.
01/16/07: Welcome to Math 4606! Because of a misscheduling, I will be
giving an oral exam to a PhD student in my department during the first
class meeting on Wednesday, January 17. Professor David Frank will
substitute for me.