Today was a sleepy day. (I wonder what causes those.) I didn’t do any exercises, but ended up watching six videos on voting systems. It seemed an appropriate topic as the US presidential election begins in earnest. I’ve long thought about making a study of this topic, becoming a voting system wonk and joining one of the organizations that advocates for reform. I do think adoption of better systems might benefit the country, especially in the primaries, where it is already common to have more than two candidates. (Elections with more than two candidates being where our current system is most likely to produce perverse results.)
Below is the video I watched that I think is most likely to be interesting to non-wonks. It does a good job explaining some alternative voting systems, though it does not say much about their pros and cons:
Today I worked on exercises from the sections I read yesterday. Due to the day’s other demands, though, my study time was abbreviated and I didn’t get as much done as I would have liked.
Today was dedicated to reading in my calculus textbook. I read the first two sections of chapter two, which introduce the idea of limits, as well as an appendix that covers its rigorous definition. I didn’t have any particularly interesting thoughts about them, unless you count “Hey, look at me! I’m doing calculus again.” And that put me in mind of this fun video I’ve shared before:
Happy International Pi Approximation Day, readers!
Today I did the review exercises that I chose on Saturday and decided to forgo the challenge problems that follow them. This means that I have finally finished the first chapter of my textbook and, with it, the groundwork phase of my review project. I am now ready to start chapter two, which covers the actual calculus topics of limits and derivatives.
Over the weekend, I felt pretty discouraged that it has taken me more than four months to get through this chapter. (I started it on the other Pi Day, in fact.) Today I mostly feel pleased to have done it, though. We’ll see about tomorrow.
I don’t know if there is any good way to pick up the pace. There are a lot of days when my mind simply does not work well, and I have very little control over that. I could perhaps do fewer exercises, but I want to relearn the material thoroughly. As I’ve said before, my ultimate goal is to do some online tutoring, which requires laying out the steps to solve tricky problems quickly and in one’s head. (I do wonder how many days in a month I’m actually going to be able to do that, given my experience working on this review. Not none, hopefully.)
One of the exercises I did today concerned finding parametric equations for a curve called the Cissoid of Dioclese. It is traced by the point $P$ in the diagram below as $\theta$ varies from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. ($P$ is the point on $\overline{OB}$ such that $OP$ is equal to $AB$.)
Source: Single Variable Calculus by James Stewart
I was able to do this problem using the fact that a central angle has twice the measure of an inscribed angle subtended by the same arc, which is to say, that the measure of $\angle ADC$, below, is $2\theta$. (I proved this in the post Angles and Arcs.)
The way the diagram was drawn, however, suggests that mine wasn’t the author’s expected solution. Anyone have an idea about how to find expressions in terms of $\theta$ for the $x$ and $y$ coordinates of $P$ without using $\angle ADC$?
The cissoid looks like this (with the point $P$ renamed to $M$):
Today I was tired. During my study time, I read through the review exercises for the textbook chapter I am wrapping up; I chose which ones I will do, but did not start work on them. I also read more of Infinite Powers, in which Strogatz has begun to discuss calculus in its modern form. In the section I read today, he explained how the discovery of the Fundamental Theorem of Calculus, which links derivatives and integrals, made it possible to calculate many integrals which had previously been beyond reach.
I apologize for not replying to comments for the past month or so. Things have been thingish, as readers of this blog will know. I think I’ve now answered everything.
My brain was not functioning at all well today, but I still ground through almost all of the remaining exercises on parametric curves. It was fairly miserable, but I am proud to have done it. I did skip one exercise that I simply couldn’t get a handle on. I may or may not return to it another day.
I’m now ready to start the review exercises for the chapter on functions. After I have finished those, and possibly some of the challenge problems that follow them, I will be able to start my calculus review in earnest with the chapter on limits and derivatives. Hooray!
The most interesting problem I worked on today concerned the alternative construction of an ellipse that is illustrated in the diagram below.
Source: Single Variable Calculus by James Stewart
It turns out that the figure traced by point $P$ as $\theta$ varies from $0$ to $2\pi$ is the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. (The red right triangle always has legs parallel and perpendicular to the x-axis and changes proportions as $\theta$ varies. It does not tilt in order to maintain its proportions.) That this is so can be shown by using parametric equations to define the curve traced by $P$, then converting them to the Cartesian equation for the ellipse. I thought this was pretty neat, and I wonder if anything interesting would come out of examining the relationship between this construction and the usual one for an ellipse.
(I may demonstrate how to show the figure is an ellipse at some time in the future. Let me know if you are interested.)
Today I worked on but did not finish the last of the exercises on parametric curves. Time just got away from me today, and I didn’t start my study until late in the afternoon despite having an evening commitment. I did get some non-math things done, though, so the day was not a total flop.
I was tired today and struggled to concentrate. I only spent an hour on textbook exercises, even though I had hoped to finish the section on parametric curves. That will have to happen tomorrow. I also spent time on some experiments with parametric curves that I started yesterday. My thoughts about them are too numerous and disorganized for me to write them up here, but I have decided to share the Desmos notebook I’ve been using, to which I’ve added a bit of commentary. I hope some of you may find it interesting. You can view it here.1
Today I worked on more exercises related to parametric curves. I also did some thinking about this blog and how I can keep it useful and interesting. I came up with a new idea to try over the next couple of weeks: on days when I do exercises from my textbook and don’t have any other math to share, I’m going to choose one of the most interesting of the exercises to include in my day’s blog entry. This will give me more of the feeling of communicating about math that I find motivating and will also make posts more fun for readers. (That’ll be fun, right?)
I actually do have some other math to share today, but I have left blogging until too late in the evening, so I’m going to have to keep it for tomorrow. Stay tuned.
Today I worked on exercises in my calculus book concerning parametric curves. I also talked with my father a bit about cycloids, the curves traced by a point on the perimeter of a circle as the circle rolls along a line. (These are simplest to define parametrically.)
Cycloid animation by Zorgit
As part of our discussion, we imagined a related class of curves that turn out to be called called cyclogons, traced by a point at the vertex of a rolling regular polygon. Just as a circle is the limit of a sequence of polygons as the number of sides approaches infinity, so a cycloid is the limit of a sequence of cyclogons as the number of sides of the polygon approaches infinity. Visualizing this can make it more intuitive that the point that traces a cycloid never moves backward as the circle rolls. Check out this demonstration and notice how the point always traces part of the top half of a circle, passing through each horizontal position only once.
(I’d also like to draw readers’ attention to the series of videos on rolling curves that I shared a while ago: The Wonderful World of Weird Wheels by Morphocular)
