Reminders
- We are in the 6th week of the semester. If you are on track for an A, then ideally you've finishing your SDL project for the 1st unit, and are working on something for the 2nd unit.
- You may complete at most one SDL project per week for credit. Account for this in your planning.
Brain Gains
- A curve passes through the point $(3,5)$. Differentials at this point are $dx = 7dt$ and $dy=11dt$. Give an equation of the tangent line to the curve at this point.
Solution
One option is to give a vector equation such as $$ \begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}7\\11\end{pmatrix}t+\begin{pmatrix}3\\5\end{pmatrix} \quad\text{or}\quad \begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}7t+3\\11t+5\end{pmatrix} .$$ Another option is to use point-slope form and give $$y-5 = \frac{11}{7}(x-3).$$ Another option is to give parametric equations $$x=7t+3, y=11t+5.$$ All these answers are acceptable. All will generalize to higher dimensions in different ways.
- For the curve $r=5\cos 2\theta$, graph the curve in the $r\theta$ plane.
Solution
We'll do this together.
https://www.desmos.com/calculator/nurgyiojif - The blue curve
- For the curve $r=5\cos 2\theta$, graph the curve in the $xy$ plane.
Solution
We'll do this together.
https://www.desmos.com/calculator/nurgyiojif - The red 4 petaled rose.
- For the curve $r=5\cos 2\theta$, compute $\frac{dx}{d\theta}$.
Solution
We know that $x=r\cos\theta$, and since $r=5\cos 2\theta$, we have $x = 5\cos 2\theta \cos\theta$. From here we just use the product rule to obtain $$\begin{align} \frac{dx}{d\theta} &= \frac{d}{d\theta}(5\cos 2\theta) \cos\theta+5\cos 2\theta \frac{d}{d\theta}(\cos\theta)\\ &=(-10\sin 2\theta) \cos\theta+5\cos 2\theta (-\sin\theta). \end{align}$$
Group problems
- Plot the curve $r=4-4\cos\theta$ in both the $r\theta$-plane, and the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\cos\theta$ easy to compute. Did you get a heart shaped object?]
- Plot the curve $r=3\sin2\theta$ in both the $r\theta$-plane, and the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\sin2\theta$ easy to compute. Did you get a clover?]
- Consider the polar curve $r=\frac{2\theta}{\pi}$.
- Draw the curve in the $xy$-plane for $0\leq \theta\leq 4\pi$. Check your work with Desmos.
- Compute $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$. (Hint: They appear in the integral in the next part.)
- What quantity does the following integral calculate? $$ \int_0^{4\pi}\sqrt{\left(\frac{2}{\pi}\cos\theta-\frac{2\theta}{\pi}\sin\theta\right)^2+\left(\frac{2}{\pi}\sin\theta+\frac{2\theta}{\pi}\cos\theta\right)^2}d\theta$$
- Find the slope $dy/dx$ at $\theta=\pi$.
- Give an equation of the tangent line at $\theta = \pi$.
- Consider the curve $v=u^2$ and use the change-of-coordinates $x=2u+v$, $y=u-2v$.
- Draw the curve in both the $uv$-plane and the $xy$-plane.
- Compute $dx$ and $dy$ in terms of $u$ and $du$ (we know $dv = 2udu$ since $v=u^2$).
- Find the slopes $\frac{dv}{du}$ and $\frac{dy}{dx}$ at $u=-2$.
- Give a vector equation of the tangent line to the curve in the $uv$-plane at $u=-2$. [Check: $(u,v) = (1,-4)t+(-2,4)$.]
- Give a vector equation of the tangent line to the curve in the $xy$-plane at $u=-2$. [Check: $(x,y) = (-2,9)t+(0,-10)$.]
- Let $v=u^3$ and use the coordinates $x=2u+v$, $y=u-2v$.
- Draw the curve in both the $uv$-plane, and the $xy$-plane (make a $(u,v)$ and $(x,y)$ table).
- Find $dx$ and $dy$ in terms of $u$ and $du$.
- Find the slope $dy/dx$ at $u=1$.
- Give a vector equation of the tangent line to the curve in the $xy$ plane at $u=1$.
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