Rapid Recall
- Consider the function $f(x,y)=x^2y+5y$. Compute $\vec \nabla f(x,y)$.
Solution
- $\vec \nabla f(x,y)= (2xy, x^2+5)$.
- For $f(x,y)=x^2y+5y$, compute $D_{ (-2,3) }f(1,1)$, the derivative of $f$ in the direction of $(-2,3)$. Hint: you need $\vec \nabla f(1,1)$ and a unit vector.
Solution
- $D_{ (-2,3) }f(1,1) = \vec \nabla f(1,1)\cdot \dfrac{ (-2,3) }{|(-2,3)|} = (2,6)\cdot \dfrac{ (-2,3) }{\sqrt{4+9}} = \dfrac{14}{\sqrt{13}}$.
- For $f(x,y)=x^2y+5y$, give a cartesian equation of the contour (level curve) that passes through the point $(1,1)$.
Solution
- We know $f(1,1) = 1$, so the equation is $6=x^2y+5y$.
Group problems
- Let $g(x,y) =xy^3$.
- 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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