Try FOH - https://www.youtube.com/watch?v=yQq1-_ujXrM
Rapid Recall
1. If we know $r=-4$ and $\theta = \pi$, find $x$ and $y$.
Solution
The solution is $(x,y)=(4,0)$.
You can do this problem visually.
- Start on the $x$-axis and rotate 180 degrees till you are facing west. Then walk backwards (east) 4 units landing you at $(4,0)$.
- Go west 4 units, and then rotate the segment from (0,0) to (-4,0) 180 degrees, fixing the origin, to land at $(4,0)$.
You can also just compute directly
- $x=-4\cos\pi = 4$.
- $4=-4\sin\pi = 0$.
2. For $z=3xy^2+2x$, find the derivative $\frac{dz}{dt}$, provided both $x$ and $y$ are functions of $t$.
Solution
We obtain $\frac{dz}{dt} = 3x(2y)\frac{dy}{dt}+3\frac{dx}{dt}y^2+2\frac{dx}{dt}$.
Note, this means $dz = 3x(2y)dy+3(dx)y^2+2dx$.
3. For the curve $\vec r(t) = (t^2+1, -t+2)$ for $0\leq t\leq 1$, the arc length is $\int_{0}^{1}\sqrt{(2t)^2+(-1)^2}dt.$ If this curve represents a rod lying in the plane, with density (mass per length) of $\delta = xy$, modify the arc length integral to give the total mass of the rod.
Solution
$\int_{0}^{1}(t^2+1)(-t+2)\sqrt{(2t)^2+(-1)^2}dt.$
Group problems
After each problem, or each part, please pass the chalk.
- Consider the curve $C$ parametrized by $\vec r(t) = (t^2, t^3)$ for $0\leq t\leq 2$.
- Set up an integral that gives the length of this curve. Just set it up.
- A wire lies along the curve $C$. The density (mass per length) of the wire at a point $(x,y)$ on the curve is given by $\delta(x,y) = y+2$. Set up an integral formula that gives the total mass of the wire.
- The wire contains charged particles. The charge density (charge per length) at a point $(x,y)$ on the curve is given by the product $q(x,y)=xy$. Set up an integral formula that gives the total charge on the wire.
- Plot the polar points with $(r,\theta)$ given by $(2,0)$, $(4,\pi/2)$, $(-4,\pi/2)$, $(2,\pi/6)$, $(-2,\pi/6)$.
- Give a polar equation of the curve $2x+3y=4$. (So substitute $x=r\cos\theta$ and $y=r\sin\theta$, and then solve for $r$.)
- Give a Cartesian equation of the polar curve $r=\tan\theta\sec\theta$. (Use $x=r\cos\theta$ and $y=r\sin\theta$ to work backwards.)
- 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 $dt$.)
- Plot the curve $r=3-2\sin\theta$.
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