Math 2325 Intro to Higher Math -- Assignments

Dr. Duval

Chapter 1, Linear Iteration

Report due Tuesday, September 23.
Revised report due Thursday, October 16. Note update!

Your main goal is to determine when the sequence converges (that is, for which values of a, b, and x_0 does it converge), and, when it does converge, to which limit? These questions are closely related, and working on either one will help you with the other.

The first part of this goal (when does it converge?) involves addressing Question 5 (you may want to, but do not have to, organize your answer along the lines suggested by Question 6). The second part of this goal (about finding the limit) involves addressing Questions 7 and 8.

Chapter 9, Parametric Curve Representation

Report due Thursday, October 9.
Revised report due Thursday, October 23.

Your main goal is to answer the question "Which properties of the parameters p, q, r, s determine the symmetry of the parametric curves

x(t) = sin(pt) + cos(qt)
y(t) = sin(rt) + cos(st)?".
This is, essentially, the last part of Question 2. Question 1 and the first part of Question 2 are good warmups, so do those first; they can also be your first data points. It is also helpful to do Exercises 1-3, some pieces of which can also be data points.

Chapter 3, Euclidean Algorithm

Report due Tuesday, October 28.
Revised report due Thursday, November 13.

Your main goal is to investigate the following:

• The number of steps the Euclidean algorithm takes to compute the GCD of two Fibonacci numbers; and
• the GCD itself of two Fibonacci numbers.
In each case, your answer should depend on the position in the Fibonacci sequence of the original two numbers.

This is largely a restatement of Question 6, which provides a good way of attacking the problem: sorting the Fibonacci numbers by how far apart they are in the Fibonacci sequence, starting with those that are close to each other in the sequence. In particular, it would be more valuable to prove some of the results you get by answering Question 6 than to completely answer without proof the main goal listed above.

Also compare the answers you get to the corresponding answers for non-Fibonacci numbers in Questions 2 and 3.

Chapter 14, Iteration of Quadratic Functions

Report due Thursday, November 6.
Revised report due Thursday, November 20.

Your main goal is to determine when the sequence has:

• attracting fixed points;
• repelling fixed points; and
• 2-cycles.
You should also identify these fixed points and 2-cycles. You do not have to identify if the 2-cycles are attracting or repelling.

In other words, for which values of a do you get attracting fixed points, repelling fixed points, and 2-cycles, and what are the values of these fixed points and 2-cycles?

This goal can be approached by answering Questions 1, 2, 3, and 7. Question 4 may also be helpful in finding values of a to exclude.

Question 6 is also very interesting, but is more of a lead-in to the rest of the lab, which you might consider doing for your optional 7th lab, if you like the idea of chaos, or want to learn more about it.

Chapter 5, The Coloring of Graphs

Report due Tuesday, November 25.
Revised report due Friday, December 12 (last day of finals).

Your main goal is to find the chromatic polynomial of the following two kinds of graphs:

• The path on n vertices; and
• The cycle on n vertices.

This is Questions 7 and 9 (and just a very few parts of Question 5 as warmups). Exercises 4-10 are also good practice with some of the tools you'll need, though these exercises do not directly answer our main questions.

Chapter 6, Randomized Response Surveys

Report due Thursday, December 4.
Revised report Friday, December 12 (last day of finals).

Your main goal is to answer the question: "What should the survey-taker do with the results?" In other words, what is your estimate of the proportion of True Yesses as a function of the proportion of reported yesses? Answer this in the most general setting, where the probabilities of answering the real question (dime lands heads) and the answer to the decoy question being yes (penny lands heads) are variables.

This is some mixture of Questions 1-3, 8, and 11.

Optional 7th Lab

Report due Friday, December 12 (last day of finals).
No official revisions, but I encourage you to consult with me as you write your report.

Talk to me in advance so that we can set up a reasonable main goal for you to pursue.

Also note that, instead of an optional 7th lab, you may turn in a re-revision of any of the first six reports, or you may turn in nothing at all. (See syllabus for how all this affects your grades.)