- 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 surface $z=f(x,y) = 3xy^2+4x^2$.
- 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 $z$? Substitute these values for $dx$, $dy$, and $dz$ into your differential to give an equation of the tangent plane.
- State a vector that is normal to the surface at $(-1,2)$ (you should be able to grab this directly from the equation of the plane).
- We can take the same surface as above and parametrize it using $\vec r (x,y) = (x,y,3xy^2+4x^2)$.
- Compute the partial derivatives $\frac{\partial\vec r}{\partial x}$ and $\frac{\partial\vec r}{\partial y}$.
- Use these partials to get a normal vector for the tangent plane, and then give an equation of the tangent plane. You should see that it matches the equation your got in part 1.
- Now let $f(x,y,z)=3x+4y-7z$. Suppose $x=3t+1$, $y=-2t+6$, and $z=-t-8$.
- Compute $df$ in terms of $x,y,z,dx,dy,dz$.
- Compute $dx$, $dy$, and $dz$ in terms of $t$ and $dt$.
- Use substitution to compute $df$ in terms of $t$ and $dt$. Then state $\frac{df}{dt}$.
- State the derivative $Df(x,y)$.
- State the derivative $D(x,y,z)(t)$.
- How can use the matrices $Df(x,y)$ and $D(x,y,z)(t)$ to obtain the derivative $\frac{df}{dt}$.
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