Math 3345-Real Analysis Lecture 01
8/31/05
1. Whats Real Analysis?
For the most part, I would say that real analysis is the study of the concepts needed to talk about dierentiation and integration. Certainly, these concepts would include limits, continuit
MA 3362 Lecture 01 - Casting Out 9s
Monday, August 25, 2008.
Objectives: Review casting out 9s. Use this as an introduction to rings.
Casting Out 9s
In computing, there is something called a parity check. You might have a number or a document which to
a c
MA 3330 Practice Test II Answers in Red
Name
Monday, March 11, 2013 (Note: Test II is Wednesday, March 13th)
t
m
B
l
C
A
1.
Consider the following statements. Which are equivalent to the Parallel Postulate? (1) and (3) are
equivalent (or the same) as the
MA 3330 Practice Test III Answers in Red
Name
Monday, April 15, 2013 (Note: Test III is Wednesday, April 17th)
1.
In a topological context, disk and 2-ball mean the same thing. The same goes for circle and 1-sphere,
and for sphere and 2-sphere. Name the f
MA 3330 Practice Test I Answers in Red
Name
Monday, February 18, 2013.
Note: Test I is Wednesday, February 20, 2013.
1.
Write the name of the person that best ts the statement.
a.
Wrote The Elements. Euclid. (Or possibly some other person or group of peop
MA 3330 Practice Final, Part II Answers in Red
Name
Friday, May 8, 2013.
Note: Final Exam is Monday, May 13th at 3:15PM.
Test I
1.
Write the name of the person that best ts the statement.
a.
Wrote The Elements. Euclid. (Or possibly some other person or gr
MA 3330 Practice Final I Answers in Red
Name
Wednesday, May 8, 2013 (Note: Final Exam is Monday, May 13th at 3:15pm)
1.
Draw a convex pentagon and a non-convex pentagon. Clearly indicate how your non-convex pentagon
is non-convex. The non-convex picture s
MA 3260 Lecture 15 - Fibonacci Sequence
Tuesday, November 13, 2012.
Objectives: Introduce recurrence relations via the Fibonacci sequence.
Heres a problem:
We have this robot soldier that is programmed to build a workstation for itself (this will
take one
MA 3260 Lecture 14 - Multiplication Principle, Combinations, and Permutation
Thursday, November 1, 2012.
Objectives: Basic counting formulas
Heres another way of looking at the fact that a set with n elements has 2n subsets.
If, for example, we have the s
MA 3260 Lecture 13 - Finite Counting Stu
Thursday, October 25, 2012.
Objectives: Basic counting formulas
We spent most of the class going over the Cantor set problem from the Homework. We discussed the
following.
0.9999 . . . = 1
We should occasionally re
MA 3260 Lecture 08 - The Conway Polynomial
Thursday, September 27, 2012.
Objectives: Dene the Conway Polynomial.
1. The Conway Polynomial
We can think of knots as coding information in a geometric manner. Wed like to be able to pull that
information out i
MA 3260 Lecture 10 - Boolean Algebras (cont.)
Thursday, October 4, 2012.
Objectives: Boolean algebras on cfw_ 0, 1 .
Before we move on, I wanted to give you a taste of what the connection between Boolean algebra and
computers might look like.
In simplisti
MA 3260 Lecture 12 - Some Uncountable Stu
Thursday, October 18, 2012.
Objectives: Example of an application of cardinality to continuous stu.
Whenever youre encountering a new concept, I think its a good idea to kind of step over to the other side,
and th
MA 3260 Lecture 09 - Boolean Algebras
Tuesday, October 2, 2012.
Objectives: Introduce binary operations on sets.
You may be familiar with taking the union and intersection of two sets. I would like to point out that union
and intersection are binary opera
MA 3260 Lecture 11 - Set Bigness
Tuesday, October 16, 2012.
Objectives: Introduce measure, probability, and cardinality.
Earlier in the semester, we briey discussed cardinality. Id like to go into this in a bit more detail now. In
particular, I want to lo
MA 3260 Lecture 07 - Binary Operations
Tuesday, September 25, 2012.
Objectives: Continue with binary operations.
Binary Operations
Given a set A, a binary operation is a function from the ordered pairs of elements from A back to A. In
particular, a binary
MA 3260 Lecture 05 - More stu about numbers
Tuesday, September 11, 2012.
Objectives: Investigate basic properties of numbers.
We started talking about linear combinations last time, and given integers a and b, I told you that the
GCD(a, b) is equal to the
MA 3260 Lecture 06 - Clock Arithmetic and Binary Operations
Thursday, September 13, 2012.
Objectives: Introduce binary operations through clock arithmetic.
Quiz 06A
We have a machine that is set to run for x hours, turn itself o for 3 hours, and then rest
MA 3260 Lecture 04 - Stu about numbers
Thursday, September 6, 2012.
Objectives: Investigate basic properties of numbers.
Let me start by naming our basic sets of numbers. In the context of sets, x A will mean that x is an
element of the set A.
Natural/Cou
MA 3260 Lecture 01 - If . . ., then . . . Statements
Tuesday, August 28, 2012.
Objectives: Warm up to thinking about logical statements.
True and False Statements
In some sense, mathematics is a study of statements and whether these statements are true or
MA 3260 Lecture 03 - Little Proofs
Tuesday, September 4, 2012.
Objectives: Proving .
Whats a proof?
Roughly, a proof is a demonstration that a statement is True (or False) based on the assumption that
certain basic statements are True. In other words, a p
MA 3260 Lecture 02 - Ands, Ors, and Nots
Thursday, August 30, 2012.
Objectives: Quantiers and other logical operators.
For convenience, well sometimes want to use shorthand symbols to represent statements. For example, we
might have
(1)
P (x) = The intege
MA 2233 Lecture 10 - Unit Tangent and Unit Normal Vectors (cont.)
Monday, September 24, 2012.
Objectives: Continue T and N with some examples.
v(t)
Note on derivative of
You computed the derivative of v(t) in the quiz last time. You should have gotten
(1)
MA 2233 Lecture 14 - Partial Derivatives and the Gradient
Wednesday, October 3, 2012.
Objectives: Review partial derivatives from labs.
Functions of two variables and partial derivatives
Weve been playing with functions of two variables in the labs. Lets
MA 2233 Lecture 13 - Curvature and the Chain Rule
Monday, October 1, 2012.
Objectives: Introduce arclength parameter and curvature.
The Chain Rule
Working with arclength parametrizations is pretty messy, so what well do is to use the chain rule and do
our
MA 2233 Lecture 15 - Directional Derivatives
Friday, October 5, 2012.
Objectives: Introduce directional derivatives
Directional derivatives
The partial derivatives fx and fy give us the slopes at a point on a surface in the x- and y-directions. Clearly,
t
MA 2233 Lecture 12 - Arclength Parameter
Friday, September 28, 2012.
Objectives: Introduce arclength parameter.
Arclength parameter s
Given a parametrization, we can dene an arclength function
t
(1)
s(t) =
v( )
d
0
that essentially measures distance along
MA 2233 Lecture 11 - Arclength
Wednesday, September 26, 2012.
Objectives: Introduce arclength of a curve.
Computing distance
Example 1. Consider this simple problem. An object is traveling at 12 feet per second for 3.5 seconds.
How far does the object tra
MA 2233 Lecture 09 - Unit Tangent and Unit Normal vectors
Friday, September 21, 2012.
Objectives: Introduce T and N.
Given a parametrization
(1)
r(t) = x(t)i + y(t)j + z(t)k,
we can dierentiate to obtain the velocity and acceleration vectors
(2)
(3)
v(t)
MA 2233 Lecture 07 - Overview of Vector Functions
Friday, September 14, 2012.
Objectives: Collect vector function concepts from labs.
Vector functions in one variable
A vector (valued) function in three dimensions, and in one variable t, will look somethi