Useful skills for the study of Fractals and Chaos

Like all Liberal Arts Colloquia at UNCA, there is no formal requirement for this course. This is a designated math class, however, so we will do a little math. This page lets you know exactly what level of mathematical knowledge and computational skills a successful student probably has. On the other hand, you can turn this around and say that these are the skills that an eager student might want to develop.

Useful mathematical knowledge

Very roughly, most students in this class should be comfortable with the concepts of precalculus mathematics. Thus, I expect that you're familiar with the concept of a function as a rule that accepts input numbers and returns output numbers and you're familiar with the notation surrounding that concept. Thus, if we write \(f(x)=x^2\), we've defined a function that accepts a number, squares it, and returns the result. For example, \(f(2)=4\) since \(2^2=4\). I also expect you're comfortable with the geometric interpretation of a function as a graph and that you probably know that the graph of \(f(x)=x^2\) is a parabola as shown on the left below. The point \((2,4)\) is on the graph because \(f(2)=4\).

\(f(x) = x^2\)
\(f(x) = 3x(1-x)\)

On the other hand, I won't expect you to come up with the graph of a function on your own. The graph of \(g(x)=3x(1-x)\), for example, is shown on the right above. I think that should be believable to you and you should understand that it tells you some things about the function, but coming up with that particular graph is not part of this class. I think that illustrates the fact that we're focused on concepts here. Similarly, you won't need to factor polynomials, solve complicated equations, or apply trig identities.

One fairly important geometric concept is that of slope, which can be computed as change in \(y\) over change in \(x\). I'd expect that you probably know that the slope of \(y=2x+1\) is \(2\). You can also probably tell which line is steeper in the figure below.

I would guess that some of you have heard of the concept of the derivative of a function and that it is a new function that tells us the slope of the original function. We won't do anything particularly deep with this, but it will be useful in a couple of spots.

You certainly don't need to know what a complex number is - but, you can't really be scared to learn a bit about them as we'll use them a bit near the end of the semester.

Useful computer skills

One reason won't have to do a lot of mathematical computation is because we'll have a computer do it for us. Of course, that means we'll need to know how to use a computer!

Naturally, I expect you have good computer access - ideally, a laptop that you can sometimes bring to class. I also expect that you know how to use it to search and browse the internet and type papers. Mostly, though, you must be willing to learn more and expand on your basic skills

Finally, you do not need to know a programming language but you shouldn't be afraid to learn to type a few lines of code.