The October 12 meeting of the SWOSU Math Club focused on examples of what we might call "finite arithmetics." The structures we looked at were the so-called clock arithmetic, the integers modulo five, and the integers modulo six. We examined the operations of addition and multiplication in each system. After that, we looked solving linear equations where the variable cam from the modulo five arithmetic. These equations were all seen to have a unique solution in that system. This situation is contrasted with solving linear equations in the modulo six system. There are linear equations in the modulo six system that have no solution, and there also exist linear equations modulo six that have multiple solution even though they are not identically true.

This gives our Math Club members a preview of what they will see in Modern Algebra. It also lets them get familiar with ideas that some of our Math Concepts classes might be doing. That should be helpful for the Math Club members who work in the tutor room. Things can get tough when Math Concepts students have questions about ideas that are not part of the standard high-school/lower-division college curriculum.

As discussed during last week's meeting, the Math Club will not be meeting this week. Several of our member will be leaving town in the afternoon since Fall Break begins.

For those brave souls that are taking the Putnam Exam this year, make sure you check out the resources that we have been creating here in the department.

1. Putnam Preparation give you a general overview of what to expect on the exam, as well as a plan for getting ready over the next two months. There is a guide to some helpful resources, and you can check out the questions that were on last year's exam.
2. Putnam Practice I is the first of several planned exercise sets focusing on topics that typically appear on the exam. It includes a brief review of some relevant facts as well as a selection of practice problems. There are several "warm-up" exercises included in the set. Thus, everybody should be able to solve at least some of the problems. An important part of your preparation is solving problems regularly, and writing up clear, complete solutions.

Even though some of the problems on the practice sheets are easier than those you will face on the Putnam, do not underestimate the value of working on them. The most important thing a student can take from their exam preparation is a willingness to work on an exercise that does not contain explicit instructions about how to go about solving them.

To clarify, most students are really good at what might be called "exercises." These are routine applications of techniques that students have learned. An exercise then is something like "Differentiate f(x)=... ." While these kinds of exercises are important for mastering basic skills, they are not that helpful for intellectual development. It is much better to tackle a problem that forces you to think about the methods and tools you know, decide which ones might be applicable, propose a solution, and finally check if what you have really does answer the problem. This kind of work is muck more challenging to be sure, but I believe it is the best way to become stronger in mathematics.

The SWOSU Math Club will be venturing out to the bowling alley for this week's meeting (Wednesday, September 7). We will be at Southwestern Lanes, in Weatherford, on the corner of Broadway and Rainey. It will be loads of fun, which is one of the reasons we are going. Another reason is so that we can get some practice in and avenge a loss to the Physics Club from last spring.

Challenge: I will pay for games and shoes if you can be me in a three game series. Ask for details.

The math club's plan for the 2011-2012 school year:

1. Decide on officers. This has been postponed until attendance at the math club goes up somewhat.
2. Practice for the Putnam exam and the Student Competition at the Sectional meeting.
3. Host a cookout. We are holding of on this until the weather cools down somewhat.
4. Spread the word about Kappa Mu Epsilon.
5. Host a workshop that discusses The Shape of Space by J. Weeks.
6. Post Flyers advertising the club.
7. Host our annual Math Field Day.
8. Look into getting some trinkets that advertise the club that we can use at SWOSU Saturday and the Club Fair.
9. Discuss the bast time for the Math Club meetings.
10. Encourage members to volunteer their time on the Allocations Committee.
11. Host some Bowling nights. (So that we get revenge on the Physics Club.)
12. Decide on a new T-shirt design.
13. Have a disk-golf outing
14. Advertise our game meetings.

At the August 31 meeting of the SWOSU Math Club we discussed the ideas behind several classic logic puzzles. The slides contain the statement of the problems; we worked out the solutions over the course of the meeting. Just to give a taste of what we can up with, we present our solution to the first river crossing puzzle.

The statement of the problem:

A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage. There is a boat that can fit himself plus either the wolf, the goat, or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage.

How can the farmer bring the wolf, the goat, and the cabbage across the river?

 Systematically keeping track of what items are on which side, and how many trips across the river have been made, we came up with the following steps to reach our goal. For convenience, we take the river to run south, and label the banks east and west.

At the start we have:

East bank: wolf, goat, cabbage, west bank: empty

Trip 1: Take the goat to the west bank

East: wolf, cabbage, west: goat

Trip 2: Return to east bank, leaving the goat

Trip 3: Pick up the cabbage and take it across to the west side.

East: wolf, west: goat and cabbage, but since the farmer is there he can keep the goat from eating the cabbage

Trip 4: Return to the east side, taking the goat.

East: wolf, goat (farmer is present, so he can stop the wolf from eating the goat), west: cabbage

Trip 5: Bring the wolf to the west side.

Trip 6: Return to the east side, leaving the wolf and the cabbage.

Trip 7: Take the goat to the west side, at which point we have accomplished what we needed, without anything getting eaten by anything else.

An Example Outline

I have mentioned the idea of creating an outline a few times during class. It might be a good idea to present an example, so that we can have something concrete to deal with. Since we covered sections 3.1 and 3.2 in class today, I will give my outline for section 3.1.

I usually try to keep my outline down to three or four key points for each section. For this particular section, I found these points to be the most important:

1. Extrema
2. Critical Point
3. Locating extrema on an interval

This is the start of my outline. There are several criteria to think about when deciding what you will want in your outline. Are there any new words? That is, are the authors using terminology that you haven't seen before? If so, they are probably trying to tell you about a new idea. You should make a note of that, as it might be something you want in your outline. Notice, the authors have not used the words "extrema" or "critical point" before in previous sections. That's why they found their way onto my list. Another idea to keep in mind are procedures and concepts that get used for multiple problems in the exercise set. That indicates that the authors (and the instructor) want you to get practice with these. Probably because they'll show up on a test.

Notice that a the start of the last paragraph, I was careful to say that "This is the start...". If this outline is going to be useful to me later on, such as when I'm studying for a test, I'll need to include more than just the terms themselves. Write down what they mean and give a simple example of each. Looking at my list, I started out with something like this.

You'll notice that my definition does not precisely match the definition that our textbook gives. That's OK. These outlines don't have to be perfect. They should give us an effective way of thinking about the idea, contain enough information so that we can refer to it after few weeks have passed and still be able to understand the ideas.

Continuing with my outline, I might draw pictures that remind me of the different types of critical points a function can have. Lastly, I would write down the procedure for finding the extrema of a function on an interval, writing down an example problem from the notes or from the exercises so that I could go back later an see how each step works.

I would keep my outline separate from the rest of my notes. Further, I would add to this outline as we go through the succesive sections and chapters of the text. Going through the outline each day before you work on your exercises should prove to be quite valuable. It will help fix the important topics in your mind, and let you know where you should concentrate when you are studying for the exams. When you get to the end of the course your outline can serve as a review for the final exam.

"Why does the produt rule look like that?"

One of the more complicated patterns that we have come across while working on derivatives is the product rule. As we saw in class, the derivative of a product is certainly NOT the product of derivatives. That doesn't work, even in the simplest cases. The rule that does work is

[f(x)g(x)]' = f '(x)g(x)+f(x)g'(x)

I think this picture might help us understand why things work this way (if you have trouble seeing this, the link will open a larger version):

In the figure, the area of the large blue rectangle represents f(x)g(x), the product of two functions. Note also that g(x) is given by the base of the rectangle and f(x) tells us the height. We would expect that a change in the area of the rectangle would entail a change in the base and/or the height.

The question becomes: "How does changing these dimensions (base and height) change the area?" I would like you to keep in mind that the change to the base may be different than the change to the height. This is the case in the picture. The base has been increased by a greater amount than the height. So what is the change in the area in terms of the change in each dimension?

Note the red rectangle on top has area Δfg(x) and the green rectangle on the right has area

Δgf(x). Plus, there is that tiny blue piece in the upper right. So the total change in the area is given by

Δfg(x)+Δgf(x)+Δ fΔg

Reasoning informally, when the changes to f and g are really small, they are basically equal to the derivatives of these functions. In other words

Δf≈ f '(x) and Δg≈ g'(x)

Further more, if the changes in f and g are really small, their product will be really, really small. In fact so small that we can safely ignore it. So we have

Change in Area = [f(x)g(x)] ' = Δfg(x)+Δgf(x)+Δ fΔg

≈f '(x)g(x)+f(x)g'(x)

Hopefully, this will help you grasp the basic ideas behind why the product rule (also called the Leibniz rule) works the way it does.