- 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 $d\vec r$.
- Consider the function $f(x,y)=3xy^4+4x^2$. Compute $f_x$ and then $f_{xy}$. Then compute $f_y$ and $f_{yx}$.
- Consider the parametric surface $\vec r(u,v) = (u\cos v, u\sin v, u^2+v^2)$.
- Compute the partial derivatives $\frac{\partial\vec r}{du}$ and $\vec r_v$.
- Give an equation of the tangent plane at $(u,v) = (3,\pi/2)$.
- Consider the surface $f(x,y) = x^2+y^2-4$.
- Draw several level curves on the same $xy$ plane.
- Compute the derivative, and then draw it as a vector field on the same plot as your level curves. What do you notice?
- Consider the surface $f(x,y) = 3xy^2+4x^2$.
- Compute the differential $df$.
- At the point $(x,y)=(-1,2)$, a small change in $x$ is $dx=x-(-1)$ and a small change in $y$ is $dy = y-2$. What is a small change in $z$? Use this together with your differential to give an equation of the tangent plane.
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