Rapid Recall
- A rover, located at $P=(3,-1)$, heads in the northeast $(1,1)$ direction. After moving 3 m, the height drops by 0.2 m. Provided $f$ is the function that gives the elevation of the surface, state $D_{ (1,1) }f(3,-1)$.
Solution
- $D_{ (1,1) }f(3,-1) = \dfrac{-0.2}{3}$, the rise over the run.
- Consider $f(x,y) = 4xy+y^3$. Compute $df$.
Solution
- $df = (4y)dx+(4x+3y^2)dy$.
- Consider $f(x,y) = 4xy+y^3$. State $f_x$, $\dfrac{\partial f}{\partial y}$, and $\vec \nabla f(x,y)$.
Solution
- $f_x = 4y$, $\frac{\partial f}{\partial y}=4x+3y^2$, and $\vec \nabla f(x,y) = (4y,4x+3y^2)$.
Group problems
- The sides of a rectangle are $x=3$ ft and $y=2$ ft, with tolerances $dx = 0.1$ ft and $dy = 0.05$ ft. Use differentials to estimate the error $dA$ in 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 $\vec \nabla g$.
- Discuss together why the slope of $f$ in the direction $(dx,dy)$ at a point $P$ is given by $$D_{ (dx,dy) }f(P) = \frac{dz}{\sqrt{(dx)^2+(dy)^2}}\bigg|_{(x,y)=P}.$$
- 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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