Tag: algebra (basic)

Circles and a Line Again

Olly / / Math Posts

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.

2circles1line Small
Source: Single Variable Calculus by James Stewart

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

Olly / / Math Posts

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 $r\rightarrow 0^+$?”

2circles1line Small
Source: Single Variable Calculus by James Stewart

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:

  1. Find the coordinates of the intersections of the two circles.
    $(\frac{r^2}{2}, \pm\frac{r}{2}\sqrt{4-r^2})$
  2. The one with a positive $y$-coordinate is $Q$.
  3. Use the coordinates of $P$ and $Q$ to find the slope of $\overline{PQ}$.
    $\frac{\sqrt{4-r^2}-2}{r}$
  4. Plug this and $y$-coordinate of $P$ into the slope-intercept form of a line.
    $y=\frac{\sqrt{4-r^2}-2}{r}x+r$
  5. Find the $x$-intercept of this line.
    $(\sqrt{4-r^2}+2,0)$
  6. 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.

Celebrate

Olly / / Math Posts

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 $AB$.)

Cissoid Small
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.)

Cissoid Small Annotated

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 $M$):

Cissoid Of Diocles By Dasha Mic
Source: Dasha Mic

An Alternative Construction

Olly / / Math Posts

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.

Ellipse Construction Small
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.)

Puzzling Parameters

Olly / / Math Posts

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. ↩︎

Rolling Along

Olly / / Math Posts

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 By Zorgit
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)

Boom Day

Olly / / Math Posts

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.

Wild Functions

Olly / / Math Posts

Today went well. I’m nearly finished with the exercises for the section I’m working on.

One of the exercises I did today reminded me what a walled garden I am playing in when working from a textbook. The functions encountered there are always tame ones susceptible to the techniques being taught. Today one of the exercises gave me a glimpse of a wild function, though.

$f(x)=3+x^2+\tan(\frac{\pi x}{2})$ for $-1<x<1$

I was asked to find $f^{-1}(3)$. It’s not too difficult to find that particular value, but I could find no way to do it by the first method I tried: by deriving a formula for the inverse, $f^{-1}(x)$. I don’t know how to solve $x=3+y^2+\tan(\frac{\pi y}{2})$ for $y$. That’s what makes this a wild function. It cannot be bidden the way most of the functions used in textbook exercises can.

(I believe the function does have an inverse, though I would like to come up with a better test for that than I have.1 Right now, I have the word of the textbook and an examination of the graph, which you can see here and which does not show any two $x$ values with the same $y$ value.)

  1. To do. ↩︎

Can’t Live With ‘Em…

Olly / / Math Posts

My struggle to make progress on my review over the past couple of weeks was caused mostly by issues with my medication. Problems with my regimen as prescribed were the main thing, with a little user error thrown in, as well. I’m finally recovered from that, but I complicated today with a little more user error by forgetting to take my medication last night. As a result, I slept a good part of the day and only had time and energy for an hour of study. Yet I did manage both to work on calculus exercises and to explore my question about composition of functions.

I didn’t make any new progress in the explorations. So far, I have only the fairly low-hanging observation that, if $f(x)$ and $g(x)$ each apply a single algebraic operation to $x$ and both operations are of the same type, then $f(g(x))=g(f(x))$. The types are: exponentiation (including roots), multiplication or division, and addition or subtraction. Below are some examples, which you can easily see are interchangeable this way:

Exponentiation
$f(x) = x^2$, $g(x) = \sqrt[3]{x}=x^{\frac{1}{3}}$:
$(x^{\frac{1}{3}})^2=x^{\frac{2}{3}}=(x^2)^{\frac{1}{3}}$

Multiplication and division
$f(x) = 3x$, $g(x) = \frac{x}{5}$:
$3(\frac{x}{5})=\frac{3x}{5}=\frac{(3x)}{5}$

Addition and subtraction
$f(x) = x-5$, $g(x) = x+3$:
$(x+3)-5=x-2=(x-5)+3$

In general, this does not work with algebraic operations of different types:

Exponentiation and multiplication
$f(x) = x^2$, $g(x) = 3x$:
$(3x)^2=9x^2\neq 3(x^2)=3x^2$

I’ve also easily found examples that even simple trigonometric, logarithmic, and exponential functions are not generally interchangeable in composition.

Olly Returns

Olly / / Math Posts

Hello again, faithful readers. I spent today’s study time reading the next section of my calculus book and investigating some of the ideas it presented. In particular, I was interested in what pairs of functions, $f(x)$ and $g(x)$, have the property that $f(g(x))=g(f(x))$.1 I made some minor discoveries, but I am too tired this evening to write them up. Perhaps tomorrow.

  1. To do. ↩︎