Try FOH (Faculty Office Hours) - https://www.youtube.com/watch?v=yQq1-_ujXrM
Brain Gains
- 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
- $x=-4\cos\pi = 4$.
- $y=-4\sin\pi = 0$.
- 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 $dz = 3x(2y)dy+3(dx)y^2+2dx$.
- Find the differential $dy$ of the function $y = x^3+2x$ in terms of $x$ and $dx$.
Solution
The derivative is $\frac{dy}{dx} = 3x^2+2$. This gives the differential as $$dy = (3x^2+2)dx.$$
- In polar coordinates, we have $x=r\cos\theta$ and $y=r\sin\theta$. Use this to rewrite the equation $y=x^2$ using polar coordinates (so obtain a polar equation of the parabola).
Solution
Since $y=x^2$, we have $(r\sin\theta) = (r\cos\theta)^2$. It's customary to solve for $r$, which gives $\ds r = \frac{\sin\theta}{\cos^2\theta}$.
- Give a Cartesian equation of the polar curve $\ds r = \frac{8}{2\cos\theta+5\sin\theta}$.
Solution
One way to tackle this is to rewrite the above equation in the form $$ 2r\cos\theta+5r\sin\theta = 8.$$ We then substitute $x=r\cos\theta$ and $y=r\sin\theta$ to obtain $$ 2(x)+5(y) = 8.$$ It's a line.
Group problems
After each problem, or each part, remember to let someone else take a turn being scribe.
- 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 $r$.)
- For the equation $z=x^2y+3y^2$, explain why $\ds\frac{dz}{dt} = 2xy\frac{dx}{dt}+x^2\frac{dy}{dt}+6y\frac{dy}{dt}$.
- 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. Start by rewriting the trig functions in terms of sines and cosines.)
- Compute the differential $dA$ for the area function $A=xy$. (Find $dA/dt$ first, if needed, and then multiply by $dt$.)
- 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 $r=3-2\sin\theta$.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |