- Construct a graph of $\vec r(t) = (2\cos t, t, 3\sin t)$ for $0\leq t\leq 4 \pi$.
- Construct a graph of $\vec r(a,t) = (a\cos t, a\sin t,9-a^2)$ for $0\leq a\leq 3$ and $0\leq t\leq 2 \pi$ by doing the following:
- Let $a=3$ and graph the curve $\vec r(3,t) = (3\cos t, 3\sin t,9-3^2)$ for $0\leq t\leq 2 \pi$. (It's a circle at what height?)
- Let $a=2$ and graph the curve $\vec r(2,t) = (2\cos t, 2\sin t,9-2^2)$ for $0\leq t\leq 2 \pi$. (More circles?)
- Let $a=1$ and graph the curve $\vec r(1,t) = (1\cos t, 1\sin t,9-1^2)$ for $0\leq t\leq 2 \pi$. (Even more?)
- Let $t=0$ and graph the curve $\vec r(a,0) = (a\cos 0, a\sin 0,9-a^2)$ for $0\leq a\leq 3$. (A parabola on what axes?)
- Let $t=\pi/2$ and graph the curve $\vec r(a,\pi/2) = (a\cos \pi/2, a\sin \pi/2,9-a^2)$ for $0\leq a\leq 3$.
- Consider the function $z=f(x,y)=x^2+y^2-4$. Construct a graph of this function in two ways.
- First make a 3D plot. Pick a variable ($x$, $y$, or $z$) and a constant (like 0, 1, 2, etc.), and then in 3D draw the resultant cross section. Repeat several times till you can tell what the object is. If you see parabolas and circles along the way, you're doing this right.
- Make a plot of several level curves, so pick various constants for the output $z$ and then plot the resultant curve in the $xy$ plane. On each curve, make sure you write the height $z$ which gives this curve. If you just get a bunch of concentric circles, you're doing this right.
- Repeat the previous with the function $z=f(x,y)=x^2-y$.
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