- The sides of a rectangle $x=3$ ft and $y=2$ ft, with tolerances $dx = .1$ ft and $dy = 0.05$ ft. Use differentials to estimate the tolerance on the area $A=xy$ that results from the given tolerances on $x$ and $y$.
- Let $g(x,y) =x^2y$.
- Compute $dg$.
- State $g_x$ and $\dfrac{\partial g}{\partial y}$. Then state $\vec \nabla g$.
- Find the directional derivative (slope) of $g$ at $P=(3,1)$ in the direction $(-3,2)$.
- Find the directional derivative of $g$ at $P=(3,1)$ in the direction $(2,-5)$.
- Consider the function $z=f(x,y)=x^2+y^2-4$.
- Construct a contour plot of $f$. So let $z=0$ and draw the resulting curve in the $xy$ plane. Then let $z=5$ and draw the resulting curve in the $xy$ plane. Then pick other values for $z$ and draw the resulting curve in the $xy$ plane. If you get a bunch of concentric circles, you're doing this right. On each circle you draw, write the height of that circle.
- Construct a 3D surface plot of the function.
- Consider the function $z=4-y^2$.
- Construct a 2D contour plot.
- Construct a 3D surface plot.
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