Rapid Recall
1. Compute the differential $dy$ for $y=x^3$.
Solution
We have $\frac{dy}{dx} = 3x^2$, so we have $$dy = 3x^2dx.$$
2. We know $x=r\cos\theta$. For the curve $r=2\sin\theta$, find $\frac{dx}{d\theta}$.
Solution
We have $$\begin{align} x&=(2\sin\theta)\cos\theta\\ \frac{dx}{d\theta}&= (2\cos\theta)\cos\theta-(2\sin\theta)\sin\theta. \end{align}$$ You are welcome to stop here. If you want to simplify this, and use trig identities, then we have $$\begin{align} x&=(2\sin\theta)\cos\theta\\ \frac{dx}{d\theta}&= (2\cos\theta)\cos\theta-(2\sin\theta)\sin\theta\\ &=2(\cos^2\theta-\sin^2\theta)\\ &= 2\cos 2\theta. \end{align}$$
3. 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 $$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}2\\12u^2\end{pmatrix}du+ \begin{pmatrix}6v\\5\end{pmatrix}dv. $$
Group problems
- For the equation $z=x^2y+3y^2$, explain 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 $x=r\cos\theta$ and $y=r\sin\theta$. Compute $dx$ in terms of $r, \theta,dr, d\theta$. (If you need to, assume that everything depends on $t$, compute derivatives, then multiply by $dt$.)
- Consider the two vectors $\vec u = (2,1)$ and $\vec v = (-1,1)$. Draw $\vec u$, $\vec v$, $3\vec u$ and $4\vec v$. Then use this information to draw the vector $3\vec u+4\vec v$.
- 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 $r\theta$ plane, and then in the $xy$-plane.
- Plot the curve $r=3\cos2\theta$ in the $r\theta$ plane, and then in the $xy$-plane.
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