Rapid Recall 
1. For the curve $r=2+2\sin\theta$, graph the curve in the $r\theta$ plane.
Solution
Here is a Desmos Graph.
2. For the curve $r=2+2\sin\theta$, graph the curve in the $xy$ plane.
Solution
Here is a Desmos Graph.
3. A curve passes through the point $(3,5)$. Differentials at this point are $dx = 7dt$ and $dy=11dt$. Give an equation of the tangent line to the curve at this point.
Solution
One option is to give a vector equation such as $$ \begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}7\\11\end{pmatrix}t+\begin{pmatrix}3\\5\end{pmatrix} \quad\text{or}\quad \begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}7t+3\\11t+5\end{pmatrix} .$$ Another option is to use point-slope form and give $$y-5 = \frac{11}{7}(x-3).$$ Another option is to give parametric equations $$x=7t+3, y=11t+5.$$ All these answers are acceptable. All will generalize to higher dimensions. Some give slightly different objects as we progress.
Group problems
- Plot the curve $r=4-4\cos\theta$ in both the $r\theta$-plane, and 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 $r=3\sin2\theta$ in both the $r\theta$-plane, and the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\sin2\theta$ easy to compute. Did you get a clover?]
- Plot the curve $r=2\theta$ in both the $r\theta$-plane, and the $xy$-plane. [Did you get a spiral?]
- We know $x=r\cos\theta$ and $y=r\sin\theta$, so for $r=2\theta$ we have $x = 2\theta \cos\theta$ and $y=2\theta\sin\theta$. Find $dx$ and $dy$ in terms of $\theta$ and $d\theta$.
- Find the slope $\frac{dy}{dx}$ at $\theta = \pi/2$.
- Let $v=u^2$ and use the coordinates $x=2u+v$, $y=u-2v$.
- Draw the curve in both the $uv$-plane, and the $xy$-plane (make a $(u,v)$ and $(x,y)$ table).
- Find $dx$ and $dy$ in terms of $u$ and $du$.
- Find the slope $dy/dx$ at $u=1$.
- Give a vector equation of the tangent line to the curve in the $xy$ plane at $u=1$.
- Let $v=u^3$ and use the coordinates $x=2u+v$, $y=u-2v$.
- Draw the curve in both the $uv$-plane, and the $xy$-plane (make a $(u,v)$ and $(x,y)$ table).
- Find $dx$ and $dy$ in terms of $u$ and $du$.
- Find the slope $dy/dx$ at $u=1$.
- Give a vector equation of the tangent line to the curve in the $xy$ plane at $u=1$.
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