Today was another difficult day, though I started to feel somewhat better in the evening. I made a start on the exercises for the section on limit laws, but I didn’t push myself to do more than a couple. Some of the later problems in the section look interesting, so perhaps there will be something for the blog.
Month: July 2024
Bad Weather
The mental weather continues bad. Today I finished reading the section on limit laws, but I am not sure how much I will retain.
Flail
Well, my mental state crashed on Saturday night, and has been more or less bad since then. The only math I did today was some reading from Infinite Powers, and that at the very last minute. 🙁
Another Good Day
Today was another good day. During my study time, I did exercises from the appendix that covers the rigorous definition of limits. I also read a bit of supplemental material on epsilon-delta proofs and part of the next section of the textbook, on limit laws.
Again, the exercises I did today didn’t lend themselves to including in a blog post. They would be routine for math enthusiasts and confusing for others. Faithful readers will have to content themselves with knowing that things are going well.
A Good Day
I had a good day today and spent more than two hours on textbook exercises, finishing two sections. Unfortunately, all the exercises were pretty humdrum, so I don’t have anything fun to share in this blog post. Perhaps there will be something tomorrow, when I work on exercises from the appendix that covers the rigorous definition of limits.
Topical Math
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:
A Boring Post
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.
Back to Calculus
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:
Celebrate
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 $AB$.)
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 $M$):
Preparation
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.