Today was not a very good day, but I did work for an hour on textbook exercises as well as reading more of Infinite Powers. The exercises didn’t include anything exciting to share, unfortunately.
Blog
Squash!
Today I fixed a bug in the blog that was preventing new posts from being displayed on the main page. The problem seems to have been going on since May, but I only discovered it recently, because it did not occur when I was logged in as blog administrator. Please let me know if this happens again.
Aside from squashing that bug, though, I struggled to make anything happen today. I did finally read some of Infinite Powers very late in the evening (after the very late meal I finally organized myself to eat). I hope to be more on top of things tomorrow.
Circles and a Line Again
Today I read the section of my calculus textbook about continuity, looked up a couple of related proofs in Analysis with an Introduction to Proof, and experimented with a variation on the problem I shared yesterday.
In a comment on yesterday’s post, reader Tim McL asked what would happen if circle $C_2$ in the diagram above were fixed and the radius of $C_1$ were allowed to increase toward infinity. (See this interactive model on Desmos.) How would this affect $R$? And if it caused the $x$-coordinate of $R$ to grow without bound, as seemed likely, why would our intuition be correct in this scenario but not in the one described in the original problem?
I’m not sure I’ve answered the second of those questions, but I did establish that in the scenario Tim McL described, $R=(\sqrt{4t^2-r^2}+2t,0)$, where $t$ is the radius of $C_1$ and $r$ is the (fixed) radius of $C_2$. Notice that the $x$-coordinate does become arbitrarily large as $t$ increases. Notice also, though, that it becomes closer and closer to $4t$. In the original problem, $R$ approached $(4,0)$ and $t$ was equal to $1$. Thus, I think we are seeing the same behavior in both scenarios, with the $x$-coordinate of $R$ dependent on the radii of both circles, and approaching $4t$ as they diverge. It only grows without bound when one of them increases toward infinity, however.
Two Circles and a Line
Today I finished the textbook exercises from the section on limit laws. There were a few interesting ones among the higher numbers. I may discuss more of them in coming days, but today I want to give a sketch of the last problem in the section:
“The figure shows a fixed circle $C_1$ with equation $(x-1)^2+y^2=1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0,r)$, $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ with the $x$-axis. What happens to $R$ as $C_2$ shrinks, that is, as $r\rightarrow 0^+$?”
What do you think? Make a guess.
It turns out the answer is that it approaches the point $(4,0)$. This is not at all in line with my intuition that, since at the limit point $P$ and $Q$ have the same $y$-coordinate, the $x$-coordinate of $R$ would grow without bound as $\overline{PQ}$ approached horizontal.
You can see the actual path of $R$ in this manipulable model of the problem that I made in Desmos after I found the solution. I did that by crunching the algebra arising from the equations of the circles and line in the diagram. I’m not going to give a full account of that here, but the basic procedure was a follows:
- Find the coordinates of the intersections of the two circles.
$(\frac{r^2}{2}, \pm\frac{r}{2}\sqrt{4-r^2})$ - The one with a positive $y$-coordinate is $Q$.
- Use the coordinates of $P$ and $Q$ to find the slope of $\overline{PQ}$.
$\frac{\sqrt{4-r^2}-2}{r}$ - Plug this and $y$-coordinate of $P$ into the slope-intercept form of a line.
$y=\frac{\sqrt{4-r^2}-2}{r}x+r$ - Find the $x$-intercept of this line.
$(\sqrt{4-r^2}+2,0)$ - This is $R$. Find it’s value as $r\rightarrow 0^+$.
It turns out that there is also a trigonometric solution, which you can read about in this thread on Math Stack Exchange. I stumbled on this when I did a Google search for the problem, hoping to check my (to me) surprising answer.
I also did a lot of thinking today about my review project and its ultimate goal of allowing me to do some online tutoring. I may say more about that in the coming days, as well.
Picking Things Up
I’m feeling quite a bit better today, and I spent longer studying than I was able to earlier in the week. The exercises I worked on were all fairly basic computations with limits. I haven’t gotten to the interesting problems yet.
Shuffling Forward
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.
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.