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• ContourSurfaceGradient.nb
• LagrangeMultipliers.nb
• 2ndDerTest.nb

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2PM

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Brain Gains (Rapid Recall)

1. For the function $f(x,y) = 3x^2+5xy+y^2$, compute $f_y$ and then both $f_{yx}$ and $\frac{\partial^2f}{\partial y^2}$.

Solution

We have

• $f_y = 0+5x+2y$,
• $f_{yx} = 0+5+0$, and
• $\frac{\partial^2f}{\partial y^2} = 0+0+2$.

• Let $f(x,y)=2x^2+4y$, and $g(x,y)=2x+y$. Solve the system $\vec \nabla f = \lambda \vec \nabla g$ together with $g(x,y)=3$.

Solution

We have $\vec \nabla f = (4x,4)$ and $\vec \nabla g = (2,1)$. The equation $\vec \nabla f = \lambda \vec \nabla g$ means $$(4x,4) = \lambda(2,1) = (2\lambda,1\lambda).$$ This gives us the two equations $4x=2\lambda$ and $4 = \lambda$. The second equation tells us $\lambda=4$. The first equation tells us $x=2\lambda/4 = 2$. Substitution of $x=2$ into $2x+y=3$ tells us $y=-1$.

• The surface $x^2+3y^2-4z=-5$ passes through the point $P=(2,1,3)$. Give an equation of the tangent plane to this surface at $P$. Hint: Use differentials.

Solution

Differentials tell us $$2xdx+6ydy-4dz=0.$$ We know $x=2$, $y=1$, and $z=3$. We also know that if $Q=(x,y,z)$ is another point on the plane, then the change from $P$ to $Q$ is $dx = x-2$, $dy=y-1$, and $dz=z-3$. Substitution (plug it in, plug it in) gives the equation of the tangent plane as $$2(2)(x-2)+6(1)(y-1)-4(z-3)=0.$$

• Consider the function $f(x,y,z) = 4x^2+4y^2+z^2$. The level surface that passes through $(1/2,0,\sqrt{3})$ is the ellipsoid $4=4x^2+4y^2+z^2$ (because $f(1/2,0,\sqrt{3})=4$). Draw this ellipsoid, which we can rewrite at $1=x^2+y^2+\frac{z^2}{4}$.

Solution

We start by drawing several cross sections, found by holding a variable constant.

• Let $x=0$ to get an ellipse in the $yz$-plane that opens up and down 2, and left and right 1.
• Let $y=0$ to get an ellipse in the $xz$-plane that opens up and down 2, and left and right 1.
• Let $z=0$ to get a circle of radius 1 in the $xy$-plane.

Putting those three curves together gives an ellipsoid that is 2 units tall in the $z$ direction, with a circle of radius 1 intersecting the ellipsoid in the $xy$-plane.

• A parallelogram has edges $(5-\lambda, 2)$ and $(3, 4-\lambda)$. Find $\lambda$ so that the area of the parallelogram is zero.

Group problems

1. Find the eigenvalues of the following matrices (take turns).
   • $\begin{bmatrix}2&4\\4&2\end{bmatrix}$, $\begin{bmatrix}2&3\\1&4\end{bmatrix}$, $\begin{bmatrix}1&6\\4&3\end{bmatrix}$, $\begin{bmatrix}3&2\\1&2\end{bmatrix}$.
   • Check: 6,-2; 5,1; 7,-3; 4,1
2. Consider the function $f(x,y,z) = 4x^2+4y^2+z^2$. We'll be analyzing the surface at the point $P=(1/2,0,\sqrt{3})$.
   • Compute the gradient $\vec\nabla f(x,y,z)$, and then give $\vec\nabla f(P)$.
   • Compute the differential $df$, and then the differential at $P$. [Check: For the latter, $df = 4dx+0dy+2\sqrt{3}dz$]
   • For a level surface, the output remains constant (so $df=0$). If we let $(x,y,z)$ be a point on the surface really close to $P$, then we have $dx=x-1/2$, $dy=y-0$ and $dz = z-?$. Plug this information into the differential at $P$ to obtain an equation of the tangent plane to the surface.
   • Give an equation of the tangent plane to the level surface of $f$ that passes through $(1,2,-3)$. [Check: $0=8(x-1)+16(y-2)-6(z+3)$.]
   • Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.
3. Consider the function $f(x,y)=2-|x|$.
   • Construct a 2D contour plot. Label your contours with their corresponding height.
   • Construct a 3D surface plot.
   • Construct both the above with software.

Problem Set
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