Presentations
Pacing Tracker
- The quizzes have included questions for 16 objectives. How many have you passed? What are you plans to master those that you haven't mastered yet?
- We've finished units 1 through 3. Have you started your self-directed learning project for each unit?
- Remember you can submit only one SDL project per week. Plan ahead and don't let yourself get behind.
Brain Gains
- 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 as $1=x^2+y^2+\frac{z^2}{4}$.
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.
Solution
The area is $$A=|(5-\lambda)(4-\lambda)-(2)(3)| = |\lambda^2-9\lambda+20-6| = |\lambda^2-9\lambda-14| = |(\lambda - 7)(\lambda - 2)|.$$ This equals zero when $\lambda = 7 $ or $\lambda =2$.
We call these the eigenvalues of the matrix $\begin{bmatrix}5&3\\2&4\end{bmatrix}$. To find the eigenvalues of a 2 by 2 matrix, we
- subtract $\lambda$ from the upper left and lower right entries in the matrix, and
- then we find the values of $\lambda$ that give zero area for a parallelogram formed by the columns of this matrix.
This will always result in solving a quadratic equation.
Group problems
- 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
- 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 $df=0$). Letting $Q=(x,y,z)$ be a point on the tangent plane to the surface, the changes in $x$, $y$, and $z$ from $P$ to $Q$ are $dx=x-1/2$, $dy=y-0$ and $dz = z-?$ (fill in this question mark). Substitute this information into the differential at $P$ to obtain an equation of the tangent plane to the surface at $P$.
- 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 $(a,b,c)$.
- 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.
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