9:00 AM Jamboard Links
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12:45 PM Jamboard Links
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Try FOH - https://www.youtube.com/watch?v=yQq1-_ujXrM
Presenters
9AM
Thanks for sharing things in Perusall. Here are the presenters for today.
- 3.1 - Jordan C, Parker K (gone)
- 3.2 - Nathan T
- 3.3 - Jae K
- 3.4 - Kallan P
- 3.5 - Zack K
- 3.6 - Luke R
12:45PM
Thanks for sharing things in Perusall. Here are the presenters for today.
- 3.1 - Cheyenne, Chad
- 3.2 - Rick
- 3.3 - Brian
- 3.4 - Joshua
- 3.5 - Alan
- 3.6 - Trevor
Learning Reminders
- We are in the 13th week of the semester. If you are on track for an A, then ideally you're finishing your SDL project for the 5th unit, and proposed something for the 6th unit.
- There are only 2 weeks left, which means you can submit at most 2 more SDL projects.
- The final SDL project (6th) can be over any topic from the entire semester. You can use it to expand what we do in the 6th unit, or you may choose to revisit something from a prior unit that you would like to spend more time with.
Rapid Recall
- 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 $r=3-2\sin\theta$.
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