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Homework #1 due Mon. Aug.
31
You should give a proof of your answer for each
problem.
From the book: 1.2, 1.7, 1.13, 1.17, 1.20, 1.21,
1.28
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Writing Assignment
#1 Friday, August 28
Understanding
Mathematical Definitions, done in
class in groups
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Homework #2 due Tues. Sept. 8
From the book: 1.50, 1.51, 2.4,
2.10, 2.21, 2.24, 2.28, 2.35, 2.37, and Practice
with Negations page. Start early and come to office hours!
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Writing Assignment #2, due Friday, Sept. 4
The problem you are assigned depends on the first letter of your last
name. Each of these is a proof by contradiction. Write up a proof
of your assigned statement and bring 3 copies to class on Friday - 1 to turn
in and 2 for others to critique. The proof does not need to be typed at
this point (you will revise and type for later).
Last name A-D: prove that the sum of any rational number and any irrational
number is irrational.
Last name E-K: prove that the square root of 2 is irrational.
Last name L: prove that there are infinitely many prime numbers.
Last name M-Z: prove that log base 2 of 3 is irrational.
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Homework #3 due Mon.
Sept. 14
From the book: 3.5, 3.7, 3.8, 3.9, 3.11, 3.23, 3.26, 3.55
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Writing Assignment #3, due Friday, Sept. 11
1. Write up the problem you worked on for last week (Writing Assignment #2)
and turn it in. Pay special attention to how it is written, using the
questions and suggestions from the peer review sheets you got last Friday.
2. I'd also like you to get access to a LaTeX (or equivalent) mathematical
word processing program. Here are the links to use. (Nothing to turn
in, but you'll need to use this for future writing assignments). I'll
be giving you sample files, so you don't need to start from scratch.
See the course webpage to find these.
* For LaTeX: http://www.latex-project.org/
* For MiKTeX (open source LaTeX): http://miktex.org/
* For WinEdt, an editor which works well with LaTeX or MiKTeX files: http://www.winedt.com/
* Alternatively, you can do your LaTeX entirely
online without downloading anything: see http://monkeytex.bradcater.webfactional.com.
Your files can be stored on the website, or you can type them there and
download to your own computer. Notice there is a button for "see
this document's pdf" - after you type in the source file, that's where
you'll get the nice-looking pdf file to save or print.
*One more option - the math department computer labs all have LaTeX already
installed, and you're welcome to use the labs whenever they're open.
3. There will also be something to do together in class on Friday.
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Writing Assignment #4, due Friday, Sept. 18
Use a version of LaTeX to type a proof that for all real numbers x, the
absolute value of x is non-negative. The proof follows very directly
from the definition of absolute value; this is really an assignment in
learning to use LaTeX. See the notes above. You can begin with
the example file found at www.math.uiuc.edu/~kmortens/348-Fa09/ex1.tex
and alter it. Do not use Word Perfect or similar - the point of this
assignment is to learn about LaTeX, which is the best and most powerful
mathematical word processing program available.
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Homework Assignment #4 -
due Monday, Sept. 28. From the book, exercises 4.5, 4.7, 4.8, 4.10,
4.11, 4.20, 4.22, 4.23, 4.24, 4.25
Note: 4.22 takes more calculation than I had originally thought, so it is no
longer part of the assignment. You can do it for extra credit if you
choose.
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Writing Assignment #5, due Friday, Sept. 25
Give an example of a function f from the real numbers to the real numbers
which is injective (one-to-one) and prove that it is injective. Give an
example of a function g from the plane R^2 to the plane R^2 which is not
injective and prove that it is not injective. Bring 3 copies of a good
first draft to class on Friday. The first draft should have the
mathematics correct and will be graded on that basis. On Monday, Sept.
28, you will turn in a typed version which will be graded on both the mathematics
and the writing. Providing peer review to other students in class on
Friday Sept. 25 is also part of this assignment.
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Homework Assignment #5 - due Monday, Oct. 5. From
the book, exercises 4.33, 4.34, 4.35, 4.36, 4.44, 4.45, 4.47, 4.49
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No writing assignment for Fri., Oct. 2. We will have
class that day, however.
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Homework Assignment #6 - due Monday, Oct. 12.
From the book, exercises 5.2, 5.4, 5.6, 5.8, 5.25, 5.26, 5.31, 5.32, 5.37,
5.39, 5.47
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Writing Assignment #6 - "Understanding
Theorems" - to be done in groups in class on Friday, Oct. 9.
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Homework Assignment #7 - due Friday, Oct.
23. From the book, exercises 6.2, 6.8a, 6.9a, 6.17, 6.19, 6.28, 6.31a,
6.47
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Writing Assignment #7 - due Friday, Oct. 16.
Choose any exercise from Chapter 5 or Chapter 6 which was not assigned as
homework and which is not marked with a (-). Write up a solution of the
exercise, focusing both on having the mathematics correct and on writing as
clearly as possible. Bring 2 typed copies to class on Fri., Oct. 16,
one to turn in and one to give to a partner for feedback. You'll be
turning in a final draft next week. Certainly you can discuss your
exercise with others, but I'd like every person to do a different exercise
(people -> exercises should be injective!)
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Homework Assignment #8 – due Wednesday, Oct.
28. Exercises 7.10, 7.11, 7.12, 7.14, 7.15, 7.17(a, b, or c –
your choice) 7.25, 7.30a
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Next Writing Assignment – please turn in your
final draft of Writing Assignment #7 on Wednesday, Oct. 28.
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Homework Assignment #9 - due Monday Nov.
2. Exercises 8.1, 8.3 (recall exercise 1.20), 8.5, 8.10, 8.12, 8.18,
8.25, 8.26
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Writing Assignment #8 -due Wed. Nov. 4. The assignment can be found
at www.math.uiuc.edu/~kmortens/348-Fa09/audience.html.
It can be done in class on Friday, Oct. 30, or you can do it on your own.
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Homework Assignment #10 - due Monday, Nov. 9. Exercises 13.2, 13.4,
13.8, 13.9, 13.21 (prove LUB property implies GLB property only, rather than
both directions), 13.22 (see definition of bounded set on page 12),
13.23 (you do not need 13.15 for this, just the definitions), 13.24 (pictures
only are ok for this one)
Reminder - Test #3 on Monday, Nov. 16, Chapters 6, 7, 8 and first section of
13.
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Writing Assignment #9 - the next several assignments will comprise the
writing of an article on an advanced mathematical topic, for a general
audience. The topic should be something at the calculus level or beyond
- it can be something you've studied in this class, in another class, or a
new topic you wish to learn a little about.
Choice of topic (just a sentence or two is fine) - due
Monday, Nov. 9
First draft - due Friday, Nov. 20 (earlier is okay!)
Final draft - due Wednesday, December 9
Having trouble thinking of a topic? You might trying browsing around
Wolfram MathWorld, http://mathworld.wolfram.com/.
This site doesn't give extensive detail on each topic, but it could be a good
starting point for finding a topic.
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Homework Assignment #11
- due Wednesday, Nov. 18.
Exercises 13.1, 13.3, 13.11, 13.12, 13.20, 13.29, also one additional
problem: prove lim 6n2(n2+1)-1=6.
Depending how quickly we go in class, I may add more problems - if so, they
will be posted here by Wednesday, Nov. 11.
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No further writing
assignments this semester
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Homework Assignment #12
- due Wednesday, Dec. 2.
Exercises 13.28, 13.30, 13.37, 14.3, 14.8, 14.9, 14.10, 14.15.
Note on 13.30: Use the Monotone Convergence Theorem. To show that
the sequence is bounded above, count how many terms are added to get xn and what
the largest of these terms is. This will allow you to show xn <= 1
for all n.
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Homework Assignment #13
- due Wednesday Dec. 9.
Because it's right before the end of the semester, we'll be going over these
problems together in class on Mon, Tues and Wed (Dec. 7-9).
1. Prove that the limit of x2+x-3 as x approaches 1 is -1, using
the definition of limit.
2. Prove that the limit of sin(1/x) as x approaches 0 does not exist, using
the definition of limit.
3. Prove that if the limit of f(x) as x approaches a is L and the limit of
g(x) as x approaches a is M, then the limit of f(x)+g(x) as x approaches a is
L+M (using definition of limit).
Also, 15.1, 15.2, 15.3, 15.9, 15.10
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