Day: October 27, 2008

Arc Length of Lissajous Curves, or Pretty, Pretty Pictures!

In MV Calc today, we were learning about arc length. In 3D, if you have parametric equations defining a curve, you can find the arc length by calculating:

L=\int_a^b \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2+(\frac{dz}{dt})^2}dt

I asked them to calculate the length of this curve, which will repeat itself, over and over and over:

x(t)=\cos(3t), y(t)=\sin(5t), z(t)=\cos(2t)

If you graph it, it looks like this (it’s a 3D Lissajous curve):

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Which is awesome! They had to figure out the limits of integration (the function will go back to it’s original starting point when t=2\pi, so the limits of integration are from 0 to 2\pi. And we had to use our fnInt function on our graphing calculators to actually calculate the length. But it was cool.

During the class, I starting thinking of all the extensions and projects that could come out of this. For example, we could have students study x(t)=\cos(at), y(t)=\sin(bt), z(t)=\cos(ct). If a, b, and c are all rational numbers, we can prove that the curve will repeat itself. However, as soon as we make one of them irrational, we can prove the curve will not repeat itself. Look at this video to see how cool it looks!

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What if all three (a, b, and c) are irrational? What constraints do we need if we want the function to repeat? (The answer won’t be tough, I imagine, but worth exploring.)

Also, on the subway ride home, I wondered if we could come up with an explicit formula (rather than parametric) for a surface containing every point on the curve. (A harder question, for sure.)

And another: what is the smallest volume you can design to enclose a curve which does repeat, like x(t)=\cos(3t), y(t)=\sin(5t), z(t)=\cos(2t). Since sine and cosine bounce between 1 and -1, I will say that your volume had better be better than 2^3.

Terminology, Notation, Ideas

In my (non-AP) calculus class last week, I was teaching my students about continuity of a function. Before we started, I asked them what continuity was, and students in both sections started their answer by saying “well, it’s when you draw a function and you don’t have to lift your pencil.” Some spoke of holes and asymptotes. Others spoke of endpoints.

I then proceeded to wow them by saying — all that they said could be encapsulated mathematically. The act of tracing out a function knowing where you’re going to have to lift it can be rewritten with three rules. They weren’t as impressed with that fact as I was, but I still tried to convey “Think about it! You can translate moving your hand across a page smoothly into mathematical statements.”

What’s needed for continuity of a function f(x) at x=c:

1. f(c) is defined
2. \lim_{x \to c}f(x) exists
3. \lim_{x \to c}f(x)=f(c)

I did the most obvious *you need to memorize this for tomorrow* wink-wink nudge-nudge that I possibly could. I might have even *coughed* the words “pop quiz.”

I just graded the quizzes. Horrible. HORRIBLE.

I got things that show no understanding of the symbols of calculus or what continuity means. Some examples:

(a) function f(x) exists
(b) f(x)=f(c)
(c) \lim_{x \to c}f(c) exists
(d) one value for f(x)
(e) the two-sided limit of c exists
(f) the two-sided limit of x is equal to c
(g) the function has to be continuous (you cannot pick up the pencil)
(h) \lim_{x \to c}f(x)=c

There are some major notational misunderstandings, but also part and parcel, some conceptual misunderstandings. I mean, for example, “the two-sided limit of c exists” doesn’t really mean anything useful to us. First of all, it should be the limit of the function, and second of all, it doesn’t say the limit as x approaches something.

I typed a bunch of these out and we’re going to talk about them in class tomorrow. Hopefully we’ll get to parlay that into a discussion of notation, the precise meaning of math symbols, and the importance of listening to Mr. Shah’s coughs.