Math 4606, Summer '02
Last changed: August 6, 2002.
This course is a proof-oriented version of one- and
several-variable Calculus.
The course meets 7 hours a
week:
11:15-12:05 MWF, and
11:15-1:05 TuTh (break:
12:05-12:20), always in MechE 108.
Note: MechE 108 is a
UNITE TV studio.
You won't be on the air; just be propared
for air-conditioning!
Our TEXT is Basic
Elements of Real Analysis, by Murray H. Protter.
ISBN
0-387-98479-8
DON'T buy Buck's Advanced
Calculus!
The text will probably be late
arriving, but that doesn't matter much; we will start wih some
material on Logic and Sets that is not in the text anyway.
Office Hours: 12:20 - 1:05 MWF and "on demand": just after
class TuTh. Otherwise, by appointment -- or call 5-3855 to see if
I'm in and available, especially before class.
You can
ask questions by email too: jodeit@math.umn.edu
Please, send
me messages in TEXT mode only! Also, here is a Web link to a list of
special codes for math symbols (most of them start with the \
character).
Announcements, hints, "news" will be here.
Final A scores by code; gradelines
Old News.
Code, then score:
2C 48
2D 59
2H 77
3C 72
3D 63
4D 27
5S 87
7H 75
8C 67
8D 43
8H 48
8S 42
9S 20
TD 52
JH 28
JS 51
QC 54
KD 46
KH 63
KS 34
AD 37
AS 28
Gradelines for Final A: top 95, A 77, B 59, C 39, D 26.
Average: 51.
Gradelines for Quiz 7: top 33, A 20, B 16, C 12, D 8.
Average: about 12.
Gradelines for Quiz 6: top 35, A 22, B 18, C 14, D 10.
Average: about 15.
Gradelines for Quiz 5: top 50, A 30, B 23, C 17, D 14.
Average: about 19.5.
Gradelines for Quizzes 2, 3, 4: top 50, A 36, B 28, C 23, D 19.
Average: about 26.
READINGS For August 6-7: Chapter 8, sections 1 -- 4; Chapter 7,
section 2, especially on Maxima and Minima; "A note on Three Series."
READINGS SO FAR: Chapter 4, sections 1 and 2; Chapter 7, sections 1 -- 5;
Chapter 3, sections 1 -- 7; Chapter 6, sections 1 -- 3;
"A note on Three Series," "The Intermediate Value Theorem";
Chapter 2, sections 1 -- 5; "Inductive Sets"; "Axioms for the Real Numbers".
Hints July 15: Suggestions for 3.1 #2 and #3: there is a hint for
#3 in the back of the book. The general idea for both parts of 2
and for 3 is to use the trick you've learned, about how to do
limits at infinity of rational functions: factor out the highest power.
Then for x large enough (in absolute value) you can
be sure the stuff inside is not very different that the coefficient of
the highest power...
Hints July 3: Suggestions for Special problem
2.
You'll need to use the Recursive Sequence Theorem.
You'll want to use a fact about convergence of
increasing sequences: Axiom C, page 43, in section 2.5. Axiom C
is really a Theorem. Protter prefers to use his Axiom C instead
of The Completeness Axiom. Thus we have to prove Axiom C, using
The Completeness Axiom. Protter proves The Completeness Axiom
using Axiom C, as Theorem 3.4. You should simply u s e
Axiom C. To use Axiom C, you have to show your sequence is
increasing (if your sequence is decreasing, you can multiply it
by -1, and the new sequence will be increasing, so it has a
limit, and the negative of that limit turns out to be the limit
of the original sequence).
To determine whether or not
xn+1 > xn you might want to
prove that for positive numbers x and
y, x > y if and only if x2 >
y2, because it's easier to work with
xn+12 than it is to work with
xn+1.
Next, you'll need to show that a
quadratic polynomial x2 + bx + c that
has distinct roots is negative if and only if x
lies between those roots.
Doing all these things will
help you connect the formula with monotonicity and then enable
you to use the theorems you need from Chapter 3. To find the
limit, once you know it exists, replace both
xn+1 and xn by the
limit (as an unknown), and solve for it.
Hints June 22:
Suggestions for Special problem 1.
You can prove a Lemma
that says that when you multiply polynomials whose coefficients
are in a field, associativity holds. You can even restrict your
Lemma to products of polynomials of degree one or less. The same
can be shown for addition. It's messy but it takes less work than
grinding thru 64 equations!
So you might want to work
with expressions of the form ax + b, where a
and b belong to the field from problem 6.
But when you multiply two of these polynomials you get a
polynomial of degree two, so you have to replace
x2 by an appropriate polynomial of degree one.
This is not unheard of! When we work with complex quantities, we
always replace the square of i by -1.
That means we (by fiat!!) set the polynomial x2
+ 1 = 0. Your task in this problem then becomes figuring out the
right polynomial of degree two to set equal to zero -- the thing
you want is for the multiplication table to be right -- and
you've probably figured out that there can only be one
multiplication table that works!
Homework policies:
Papergrader's Assgt 4 comments.
Papergrader's Assgt 3 comments.
Papergrader's Assgt 2 comments.
HOMEWORK RULES: Homework for submission should be clearly
written on 8.5 x 11 sheets of paper. If you use spiral notebook
paper please trim the edges. Sheets should be fastened with
staples in the upper left-hand corner (if more than one sheet is
used). Clarity, neatness, correct reasoning, correct answers, and
good judgement about how much detail to include are all
important.
LATE POLICY: Each repeated lateness adds a
day of lateness!
Under Regular Circumstances:
no penalty
for papers received within 3 days of initial dropoff.
25%
penalty for papers received more than 3 days after initial
dropoff.
50% penalty for papers received more than 5 days
after initial dropoff.
100% penalty for papers received more
than 7 days after initial dropoff.
Exceptional
Circumstances: (These cases will be determined by Prof.
Jodeit):
If papers are received by me within 7 days of inital
dropoff they will be graded without penalty.
If papers are
received by me more than seven days after inital dropoff, they
will not be graded, but I will assign to that paper the average
of the scores of the papers actually turned in.
System:
Easy problems are 5 pts.
Standard problems are 10
pts.
Difficult problems are 15 pts.
Links, if any, to other Web pages pages related to this course
will go here. They are PDF files (you need Adobe Acrobat Reader to view them).
Follow this link to download the latest Acrobat Reader
from Adobe at no charge.
"Express" URL
Math
4606 Assignments.
A note on Three Series.
A
Sufficient Condition for Differentiability.
The
Intermediate Value Theorem.
The
nth-root Theorem.
Inductive Sets, version 1.
Axioms
for the Real Numbers.
Math 4606
SS '02 Syllabus.
Quiz
7: some solutions.
Quiz
6: some solutions.
Quiz
5: some solutions.
Quiz
4: some solutions.
Quiz
3: some solutions.
Quiz
2: some solutions.
Quiz
1: some solutions.
A
solution for Special Problem 4.
A
solution for Special Problem 3.
A
solution for Special Problem 2.
A
partial solution for Special Problem 1.
Below: background material from a much earlier version of
this course
Introduction
Mathematical
Logic
Sets, variables
and quantifiers
Logic and
sets
Jodeit's Home
Page
Mathematics Home Page