- For the function $\vec r(t) = (3\cos t, 3\sin t, t)$, give an equation of the tangent line at $t=\pi/2$.
- For the function $\vec r(a,t) = (a\cos t, a\sin t, t)$, give an equation of two tangent lines at $(a,t)=(3,\pi/2)$.
- Give an equation of the tangent plane to the surface from the previous problem at $(a,t)=(3,\pi/2)$.
- Consider the parametric surface $\vec r(x,y) = (x, y, x^2+y^2)$. It's a paraboloid.
- Compute the differential $d\vec r$ and write it in both vector and matrix form.
- At the point $\vec r(2,1)=(2,1,5)$ (which is on the surface), state two vectors tangent to the surface.
- Give an equation of the plane tangent to the surface at $\vec r(2,1)$ (hint: recall from chapter 2 how we found the equation of a plane through a point $(a,b,c)$ with normal vector $\vec n$, which we can obtain from two vectors in the plane).
- For the function $f(x,y)=3xy^2+2x^3$, compute the differential $df$. Then state the derivative $Df(x,y)$. Then state the partials $\frac{\partial f}{\partial x}$ and $f_y$.
- For the function $f(u,v)=u^2\cos v$, state both $f_u$ and $\frac{\partial f}{\partial v}$. Then state the derivative $Df(u,v)$. Then state the differential $df$.
- For the function $\vec r(u,v)=(u\cos v, u\sin v, u^2)$, state both partials $\vec r_u$ and $\frac{\partial \vec r}{\partial v}$. Then $D\vec r(u,v)$ and $d\vec r$.
- Repeat the previous two problems with any functions you want. The goal is to get good at finding partial derivatives, derivatives, and differentials, and being able to correctly name each part.
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