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this document in PDF: HW.pdf

**On Term Exam 4. ** It will take place, as
scheduled, during the tutorials on Monday March 21st. You will have an
hour and 50 minutes to solve around 5 questions, with no choice
questions. The material is everything covered in class until Tuesday
March 15th (and not including Thursday March 17th), including
everything in the relevant chapters (18-20) of Spivak's book
(including the appendix to chapter 19). The material in chapters 1-15
is not officially included, though, of course, what chance have you got
answering questions about the Taylor polynomial of (say), if
you aren't yet absolutely fluent with differentiation of and
? Some questions will be taken straight from class, some straight
from homework, and some will be fresh. Calculators will be allowed but
will not be useful beyond emotional support; no devices that can
display text will be allowed. **Good luck!**

**Preparing for Term Exam 4. **

- Re-read your notes and make sure that you understand
*everything*. - Re-read Spivak's chapters 18-20 and make sure that you understand
*everything*. - You may want to prepare a list of all topics touched in class (you may reach 50 or even 100), and you may want to go over this list several times until you are sure you understand everything in full.
- Make sure that you can solve every homework problem assigned or recommended.
- Take a good look at exams, sample exams and exam solutions from previous years. (Scroll down to the bottom of this class' web site and find the relevant links).
- Come to my office hours Friday 3:30-5:30, at the Math Aid Centre, SS 1071.
- It is much more fun to work in a group!

An often-asked question is ``Do we need to know proofs?''. The answer is
**Absolutely**. Proofs are often the deepest form of understanding,
and hence they are largely what this class is about. The ones I show in
class are precisely those that I think are the most important ones,
thus they are the ones you **definitely** need to know.

**Required reading. ** All of Spivak's Chapter 22.

**To be handed in. ** From Spivak Chapter 22: 1 (odd
parts), 2 (odd parts), 5, 13.

**In class review problem(s)** (to be solved in
class on Tuesday March 22). Chapter 22 problem 2(ii): Find

**Recommended for extra practice. ** From Spivak
Chapter 22: 1 (even parts), 2 (even parts), 9, 27, 28, 29.

**Just for fun. ** For some constant number ,
consider the function
. Let be the set of all
pairs so that
and is a limit of a
subsequence of the sequence
,
,
, .... Write a computer program to draw the
set in the plane whose axes are and , and if your program
and picture are nice, they'll find their place on this class' web
site.

It's a hard one, but it's well worth it. The set is way more complex than you would expect, with parts that scream ``structure'' and parts that scream ``mess''. If you've ever heard the word ``chaos'' in a mathematical context before, this is it. And if you've ever seen pictures of the beautiful ``Mandelbrot Set'', our is a close relation.

**Advertisement 1. ** I was asked to distribute
the following message (no free food but it's a good cause):

Date: Wed, 9 Mar 2005 09:40:06 -0500 (EST) * Please circulate to interested undergraduate students * Queen's University is proud to host the 12th annual Canadian Undergraduate Mathematics Conference from July 13th to 17th, 2005. Undergraduate students in all branches of mathematics, including statistics, computer science, and applied math, are invited to attend. The conference provides an opportunity for students to hear talks by both students and professors, as well as to meet like-minded students from across the nation. Students are encouraged to prepare talks in either English or French, but are not required to do so. In the past, topics have ranged from philosophy of math, to the Golden Ratio, to game theory, and many in between. Outside of the talks, there will be planty of opportunities to visit the beautiful city of Kingston and take advantage of the annual Kingston Buskers Rendezvous, which will be taking place at the same time. There is travel support available for students travelling from western Canada as well as from Atlantic Canada. Please encourage your students to take advantage of this opportunity. For more information, please visit our website at cumc.math.ca/, or e-mail us at cumc@cumc.math.ca. We look forward to seeing you! Erica Blom President, CUMC 2005

**Advertisement 2. ** I was asked to advertise
the ``Arts and Sciences Option 2005'' event, March 17-18 2005. No free
food is mentioned at
http://www.artsandscience.utoronto.ca/ps/options/.

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Dror Bar-Natan 2005-03-21