I did more calc reading and exercises this morning. I thought I might fit in a little study this afternoon, as well, but it was a full day by my standards, so that didn’t happen.
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Vacation Day 1
Hello from Elsewhere, readers. This morning I both read the section in my calculus text about calculators and computers and finished the exercises. My more powerful graphing program did not produce the interesting artifacts the book talked about, but I enjoyed using it to graph families of functions and see how changing one element of the equation affected the graph. Desmos has a special feature to facilitate this. Here’s an example.
If It’s Not One Thing
I didn’t do much today because I was (I hope) recovering from a cold that hit me late on Saturday. As far as math goes, I only listened to some of A Brief History of Mathematics.
Also, I am going to take tomorrow off blogging because, if I have recovered enough, I will be driving to visit my mother. I am planning to study and blog while on that visit, though.
Better Day
Things were better today. I finished the section of calculus exercises I have been working on. The next section is about graphing calculators and computers. I will be interested to see how applicable it is to the tools I have available now, nearly 20 years after this edition was published.
Bad Day
I had a pretty miserable day, for multiple reasons. I didn’t manage to do any exercises, but did watch some YouTube videos.
Keeping On
I was mildly depressed today—not enough to incapacitate me, but enough to make everything joyless. Nevertheless, I spent an hour and a half on calculus exercises and some additional time thinking about questions I have raised on the blog. I haven’t got anything fun to share, though. I’m afraid that may be increasingly common as I try to apply myself more consistently to calculus review.
Parabola Magic
I worked on more calculus exercises today, then did some experimenting with graphs using Desmos. What I discovered is that adding a line to a parabola will always produce a congruent parabola. So far, I haven’t been able to find any other type of curve that has this property. For all other curves, adding a line also adds “tiltiness” to the graph, so that the result is not congruent to the original.
Here is an example using a parabola. The original parabola appears in green, the line in red, and their sum in blue. Click here for an interactive version of this figure.
We know that these parabolas are congruent because the coefficient of the $x^2$ term does not change. (It is called $a$ in the interactive graph.) This coefficient is equal to $\frac{1}{4p}$, where $p$ is the distance of the vertex of the parabola from both the focus and the directrix. It is this distance that determines the curvature of the parabola.
(To see that $a$ does equal $\frac{1}{4p}$, try converting the general equation for a parabola $(x-h)^2=4p(y-k)$ into the general equation $y=ax^2+bx+c$. That’s another thing I did today.)
Below is a gallery of some of the other curves I tried. Again, the original graphs are in green, the line in red, and their sum in blue. You can see how each result has greater tiltiness than the original in one way or another.
I haven’t yet been able to explain why parabolas have this congruence property while other types of curve do not nor tried to understand what other implications it may have, but I find it magical.1
- To do. ↩︎
Snoozle
Today was another of my periodic sleepy days, and I spent much of it napping. I did work on calculus for an hour in the evening, and I listened to A Brief History of Mathematics for another half hour or so. Both the calculus exercises and the radio programs emphasized the usefulness of mathematics to natural science. That’s not really something I think about much. For me, the appeal of math is very much inherent in the subject, not tied to what you can do with it. I am glad other people are interested in the latter, though.
Less Fun with Logarithms
Today I did more exercises and reading in my calculus book. I also proved the converse of the proposition from last week’s post Fun with Logarithms. You can see the proof below. As I expected, all that was required was a little algebra. Finally, I listened to some of the BBC radio series A Brief History of Mathematics.
It was struggling to listen to that series at the beginning of this year that inspired me to try doing math with the aid of CBD gummies (as described in Take Drugs, Do Math). The series is intended for a broad audience and is not intellectually demanding. Yet even a short time listening to it left me completely exhausted. That served as strong evidence that the overwhelming difficulties I was still having when I tried to do math were at least partly emotional, as opposed to cognitive. Before, it had never been so clear. The new certainty, combined with an overall improvement in my mental health around that time, led me to seek new ways to overcome the emotional block.
Today I was able to enjoy the series with no ill effects, even without CBD gummies, which I am now using infrequently.
Proposition: Given two natural numbers $a$ and $b$ that are both greater than $1$, if both $a$ and $b$ are powers of a natural number $c$, then $\log_a b$ is rational.
Suppose that $a$ and $b$ are natural numbers greater than $1$ that are both powers of a natural number $c$.
Then, by definition, $a=c^m$ and $b=c^n$ for some natural numbers $m$ and $n$.
Since $a\neq 1$, it follows that $m\neq 0$.
And since $m\neq0$, $a^{\frac{n}{m}}=(c^m)^{\frac{n}{m}}=c^n=b$.
Thus $\log_a b = \frac{n}{m}$, which is rational, and the proposition is proven.
Shopping is Hard (Let’s Do Math)
Today I visited four thrift shops looking for various clothing items I need. It was exhausting. Between that and some other things I needed to do today, I wasn’t up for either calc exercises or explorations.
I ended up reading more of Infinite Powers. It was fun to be doing that with a sharper mind than I often do, especially today, when the section I read covered Archimedes’s Method. I can’t adequately explain it here, but it is a way of doing geometry by imagining shapes as physical objects acting on levers. This allowed Archimedes to discover the relationships between their areas or volumes. It’s pretty wild. “Give me a lever” and I will not only move the Earth, I’ll discover the volume of the sphere, too.