9:00 AM Jamboard Links
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12:45 PM Jamboard Links
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Presenters
9AM
Thanks for sharing things in Perusall. Here are the presenters for today.
- 3.6 - Luke
- 3.7 - Ryan, Chase
- 3.8 - Makenzy
- 3.9 - Olivia
- 3.10 - Ethan
- 3.11 - Spencer B
- 3.12 - Tanner
- 3.13 - Gavin
12:45PM
Thanks for sharing things in Perusall. Here are the presenters for today.
- 3.6 - Aaron R
- 3.7 - Oscar
- 3.8 - Alan
- 3.9 - Forrest
- 3.10 - Trevor
- 3.11 - Nicholas B
- 3.12 -
- 3.13 -
Learning Reminders
- We are in the 13th week of the semester. If you are on track for an A, then ideally you're finishing your SDL project for the 5th unit, and proposed something for the 6th unit.
- There are only 2 weeks left, which means you can submit at most 2 more SDL projects.
- The final SDL project (6th) can be over any topic from the entire semester. You can use it to expand what we do in the 6th unit, or you may choose to revisit something from a prior unit that you would like to spend more time with.
Rapid Recall
- 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
$$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}?\\?\end{pmatrix}du+ \begin{pmatrix}?\\?\end{pmatrix}dv.$$
Solution
First, note that $$\begin{align} dx&=2du+6vdv\\ dy&=12u^2du+5dv. \end{align}$$ Rewriting this in vector form gives $$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}2\\12u^2\end{pmatrix}du+ \begin{pmatrix}6v\\5\end{pmatrix}dv. $$
- For the curve $r=2+2\sin\theta$, graph the curve in the $r\theta$ plane.
Solution
Here is a Desmos Graph.
- For the curve $r=2+2\sin\theta$, graph the curve in the $xy$ plane.
Solution
Here is a Desmos Graph.
Group problems
- Review: for the equation $z=x^2y+3y^2$, briefly discuss why $\ds\frac{dz}{dt} = 2xy\frac{dx}{dt}+x^2\frac{dy}{dt}+6y\frac{dy}{dt}$.
- 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.
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