Pacing Tracker
- The quizzes have included questions for 10 objectives. Have you passed at least 80% of them? What are you plans to master those you have't mastered yet?
- We've finished units 1 and 2. Have you started your self-directed learning project for each unit? Ideally you've finished your first SDL, and started (or working on a proposal for) the second.
- Remember you can submit only one SDL project per week, so plan ahead and don't let yourself get behind.
Brain Gains
- We know $x=r\cos\theta$. For the curve $r=2\sin\theta$, find $\ds \frac{dr}{d\theta}$ and $\ds \frac{dx}{d\theta}$.
Solution
We have $\frac{dr}{d\theta} = 2\cos\theta$ and $x=(2\sin\theta)(\cos\theta)$, which means $$\begin{align} \frac{dx}{d\theta}&= (2\sin\theta)'(\cos\theta)+(2\sin\theta)(\cos\theta)'\\ &= (2\cos\theta)(\cos\theta)+(2\sin\theta)(-\sin\theta). \end{align}$$
- For the change of coordinates $x=2u+3v^2$ and $y=4u^3+5v$, write the differential $(dx,dy)$ in the form $$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}?\\?\end{pmatrix}du+ \begin{pmatrix}?\\?\end{pmatrix}dv.$$ If needed, start by computing $\frac{dx}{dt}$ and $\frac{dy}{dt}$ in terms of $\frac{du}{dt}$ and $\frac{dv}{dt}$ (implicit differentiation) and then multiply everything by $dt$.
Solution
First, note that $$\begin{align} dx&=2du+6vdv\\ dy&=12u^2du+5dv. \end{align}$$ Rewriting this in vector form gives $$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}2\\12u^2\end{pmatrix}du+ \begin{pmatrix}6v\\5\end{pmatrix}dv. $$
Following the implicit differentiation route instead, our first step is $$\begin{align} \frac{dx}{dt}&=2d\frac{du}{dt}+6v\frac{dv}{dt}\\ \frac{dy}{dt}&=12u^2\frac{du}{dt}+5d\frac{dv}{dt}. \end{align}$$ Multiply both sides by $dt$ and we arrive at the work above.
- For the curve $r=2-2\sin\theta$, graph the curve in the $r\theta$ plane.
Solution
I like to think of this in terms of shifts and stretches. Start with the graph of sine. Reflect about the $x$-axis and double the $y$-values. Then shift it up 2 units.
Here is a Desmos Graph.
- For the curve $r=2-2\sin\theta$, graph the curve in the $xy$ plane.
Solution
Construct a table of values. For my choices for $\theta$, I use points from the previous graph that correspond to maxes, mins, and halfway between, which for this problem means to use $\theta = 0,\frac{1\pi}{2},\frac{2\pi}{2},\frac{3\pi}{2},\frac{4\pi}{2},\frac{5\pi}{2},...$. This gives the table $$ \begin{array}{c|c|l} \theta&r&(x,y) - \text{optional}\\\hline 0&2&(2,0)\\ \pi/2&0&(0,0)\\ 2\pi/2&2&(-2,0)\\ 3\pi/2&4&(0,-4)\\ 4\pi/2&2&(2,0) - \text{where we started}\\ \end{array} $$
Here is a Desmos Graph.
Fourier Transform Video
This video helps answer the question: "Are polar coordinates useful?" Turns out they are the backbone to basically every device in the digital era. Without them, we would not have any modern inventions.
- Fourier Transforms - The first 3 minutes in class is sufficient. Keep watching, on your own, if it spiked your interest.
Group Discussion
- Review: for the equation $z=x^2y+3y^2$, show that $\ds\frac{dz}{dt} = 2xy\frac{dx}{dt}+x^2\frac{dy}{dt}+6y\frac{dy}{dt}$.
- Compute the differential $dA$ for the area function $A=xy$.
- We know $x=r\cos\theta$. Explain why $dx = \cos\theta dr-r\sin\theta d\theta$.
- We know $y=r\sin\theta$. Compute $dy$ in terms of $r,\theta,dr,d\theta$.
- Plot the curve $r=3-3\sin\theta$ in the $r\theta$ plane, and then in the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\cos\theta$ easy to compute. Did you get a heart shaped object?]
- Plot the curve $r=3\cos2\theta$ in the $r\theta$ plane, and then in the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\sin2\theta$ easy to compute (multiples of 45 degrees). Did you get a clover?]
- Plot the curve $r=4-4\cos\theta$ in both the $r\theta$-plane, and the $xy$-plane.
- Plot the curve $r=3\sin2\theta$ in both the $r\theta$-plane, and the $xy$-plane.
Presentations
Quinn had any of 7-9 last time. Have him fill whichever isn't claimed first today.
- 3.7 - __ (new coordinate plot)
- 3.8 - __ (differentials)
- 3.9 - __ (polar differentials)
- 3.10 - __ (polar plot)
- 3.11 - __ (polar plot)
- 3.12 - __ (polar plot)
- 3.13 - __ (Mathematica)
- 3.14 - __ (dy/dx in polar coords)
Future
- 3.15 - __
- 3.16 - __
- 3.17 - __
- 3.18 - __
- 3.19 - __
- 3.20 - __
- 3.21 - __
- 3.22 - __
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