Blog – Page 7 – The Renewal Equation

An Alternative Construction

Jul 19, 2024

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.)

Math Posts

algebra (basic), geometry, trigonometry
Time Management

Jul 18, 2024

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.

Updates
Puzzling Parameters

Jul 17, 2024
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
1. To do. ↩︎
Math Posts

algebra (basic), geometry, interactive model, to-do
Exercise of the Day

Jul 16, 2024

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.

Updates
Rolling Along

Jul 15, 2024

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)

Math Posts

algebra (basic), animation, geometry, video
Perfect Numbers

Jul 8, 2024

Today I read about perfect numbers and some related concepts in an online textbook called Elementary Number Theory that is available from LibreTexts. It was good to have a refresher on some of the finer points.

I will not be studying or blogging Tuesday or Wednesday because I will be busy wrapping up my house sitting and traveling home. I will return either Thursday or Friday, depending on how long it takes me to get settled again.

ETA: I have decided to return Monday, as some new commitments came up after I made this post.

Updates

number theory
Sticktuitive

Jul 6, 2024

Today was hard again, though I did get some needed tasks done. I almost didn’t do any math study, but in the end I read some of Infinite Powers. Not what I aim at, but something.

Updates
The Great Wide Open

Jul 5, 2024

My mental health was poor today, and I spent my study time watching math videos. This included one on open problems that are relatively simple to state but have proven difficult to solve. It was quite interesting, but had several errors (some of which I noticed and some of which I found pointed out in the comments). Therefore I won’t link it here.

My favorite open problem is the first one that caught my imagination: the conjecture that there are infinitely many perfect numbers. This turns out to be closely related to the conjecture that there are infinitely many Mersenne primes, a type of prime number that is one less than a power of two. So a conjecture about summing factors is related to the existence of a type of prime numbers. Cool huh?

Math Posts

number theory
Boom Day

Jul 4, 2024

Happy 4th of July, readers.

Today I finished the exercises and reading from yesterday. The latter explained a way I could use parametric equations to instruct a graphing calculator to draw the inverse of Tuesday’s wild function. Desmos does not require this indirect method, however. It can graph $x=3+y^2+\tan(\frac{\pi y}{2})$ as written. Whatever the benefits for graphing, I don’t think converting the wild function into parametric equations makes it any easier to test whether it has an inverse. That may require calculus, as a reader suggested earlier in the week.

Math Posts

algebra (basic), trigonometry
Day of the Naps

Jul 3, 2024

Today was another of my periodic sleepy days, and my study time was compressed. I did not quite finish the section of exercises I worked on yesterday, but I still spent some time reading the following section, which covers parametric curves. Those were always an afterthought in my mathematical education, and I found them a bit mysterious. They are making more sense this time, I think.

Updates