- For the equation $z=x^2y+3y^2$, explain why $\frac{dz}{dt} = 2xy\frac{dx}{dt}+x^2\frac{dy}{dt}+6y\frac{dy}{dt}$.
- Compute the differential $dA$ for the area function $A=xy$.
- Consider the two vectors $\vec u = (2,1)$ and $\vec v = (-1,1)$. Draw $\vec u$, $\vec v$, $2\vec u$ and $3\vec v$. Then use this information to draw the vector $2\vec u+3\vec v$.
- Let $\vec w = (2,7)$. Notice that $$\begin{pmatrix}2\\7\end{pmatrix} =\begin{pmatrix}2\\1\end{pmatrix}3+\begin{pmatrix}-1\\1\end{pmatrix}4.$$ How do the 3 and 4 help us know how to construct $\vec w$ from $\vec u$ and $\vec v$?
- Plot the curve $r=3-3\sin\theta$ in the $xy$ plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\sin$ easy to compute.]
- Plot the curve $r=3\cos2\theta$ in the $xy$ plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\sin$ easy to compute.]
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