We still have some tasks from Day 21 to finish discussing in class.

Day 21 - Prep

Task 21.1

Let $f(x,y) = 9-x^2-y^2$. Rather than using Cartesian coordinates to examine this function, we could instead use polar coordinates $x=r\cos \theta$ and $y=r\sin\theta$.

  1. Compute the differential $df$ in terms of $x$, $y$, $dx$, and $dy$.
  2. Compute the differentials $dx$ and $dy$ in terms of $r$, $\theta$, $dr$, and $d\theta$.
  3. Use substitution to obtain $df$ in terms of $r$, $\theta$, $dr$, and $d\theta$. Write your answer as the linear combination $df = (?)dr + (?)d\theta$.
  4. State $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$.
  5. We can write the change of coordinates as the function $(x,y) = \vec T(r,\theta) = (r\cos\theta, r\sin\theta)$. Given a polar coordinate $(r,\theta)$, the function $\vec T$ returns the Cartesian (rectangular) coordinate $(x,y)$. Compute $f(\vec T(r,\theta))$.
  6. Compute the differential $d\vec T$ and write is as the linear combination $d\vec T = (?)dr + (?)d\theta$. Note that the questions marks will be vectors, not numbers, because the function $\vec T$ returns a vector (not a number).
  7. State the total derivatives $Df(x,y)$ and $D\vec T(r,\theta)$. How would you interpret $Df(\vec T(r,\theta))$.
  8. Compute the matrix product $Df(\vec T(r,\theta))D\vec T(r,\theta)$. [Hint: the partial derivatives you computed earlier should appear.]

Task 21.2

This task will have you practice using the second derivative test to locate maxima, minima, and/or saddle points for function $f(x,y)$ of two variables.

  1. Consider the function $f(x,y)=x^3-3x+y^2-4y$.
    1. Find the critical points of $f$ by finding when $Df(x,y)$ is the zero matrix.
    2. Find the eigenvalues of $D^2f$ at any critical points. [Hint: First compute $D^2f$. Since there are two critical points, evaluate the second derivative at each point to obtain 2 different matrices. Then find the eigenvalues of each matrix.]
    3. Label each critical point as a local maximum, local minimum, or saddle point, and state the value of $f$ at the critical point.
  2. Consider the function $f(x,y) = 6x^2-2x^3+3y^2+6xy$. The function has two critical points $(0,0)$ and $(1,-1)$. At each of these points, evaluate the second derivative and then find the corresponding eigenvalues. Use these eigenvalues to classify each critical point as the location of a local maximum, local minimum, or saddle point.

The Mathematica Notebook 2ndDerTest.nb can help you check much of your work above.

Task 21.3

To use Lagrange Multipliers, we must (1) identify the function $f(x,y)$ to be optimized along with the constant $c$ and function $g$ in the constraint $g(x,y)=c$, (2) write the system of equations that results from $\vec \nabla f = \lambda \vec \nabla g$ and $g(x,y)=c$, (3) solve this system, and (4) determine which points correspond to maxes and which to mins. The third step, solving a system of equations, can become extremely difficult quite quickly, but luckily modern software can help facilitate this part of the process. Please use the Mathematica notebook LagrangeMultipliers.nb to help you check your work and visual what you're doing in this task.

  1. Let $f(x,y) = 20 x + 2 y^2$. Use Lagrange multipliers to identify the location of any extreme values of $f$ along the line $100=4x+8y$. Complete this by hand, and then check your work with software.
  2. A rover travels along a circle of radius 5, centered at the origin. The elevation of the surrounding hill is give by $z = 4x^2-4xy+y^2$. What are the highest and lowest elevations reached by the rover? [If the system to solve is brutal, then use software to help you.]

Task 21.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.

Day 22 - Prep

Task 22.1

There are three optimization problems below. Each can be solved with a different method (first semester calculus, Lagrange multipliers, and the second derivative test with eigenvalues). Solve each problem below, and explain your choice of the method used.

  1. The elevation near a rover is given by $z=y+x^2$. The rover travels along a path given by $y-2x=5$. Find the $(x,y)$ location of any maxes or mins along the rover's path, and classify the point(s) appropriately.
  2. The elevation near a rover is given by $z=y+x^2$. The rover travels along the path parametrized by $\vec r(t) =(t,2t+5)$. Find the $(x,y)$ location of any maxes or mins along the rover's path, and classify the point(s) appropriately.
  3. The elevation near a rover is given by $f(x,y)=x^2+xy+y^2-2y$. Determine the location of any maxes or mins near the rover, and classify the point(s) appropriately.

Task 22.2

For each problem below, decide if you'll need to use Lagrange multipliers or the second derivative test. If you choose Lagrange multipliers, then state $f$, $g$, and $c$, along with the system of equations that must be solved. If you choose the second derivative test, then state $f$, $Df$, and $D^2f$. Then use the appropriate Mathematica notebook (either LagrangeMultipliers.nb or 2ndDerTest.nb) to solve the problem.

  1. Let $f(x,y)=x^3 + 3xy +y^3$. Find all local extreme values of $f$.
  2. Find the dimensions of the rectangle of largest possible area that will fit inside of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$.
  3. Find three numbers whose sum is 9 and whose sum of squares is minimized.
  4. Find the largest box in the first octant (all variables are positive) that can fit under the paraboloid $z=9-x^2-y^2$. The volume of such a box is given by $V=lwh = xyz = xy(9-x^2-y^2)$.
  5. A rover travels along a circle of radius 5, centered at the origin. The elevation of the surrounding hill is give by $z = 4x^2-4xy+y^2$. What are the highest and lowest elevations reached by the rover.

Task 22.3

In this task we'll derive the version of the second derivative test that is found in most multivariate calculus texts. The test given below only works for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$. The eigenvalue test you have been practicing will work with a function of the form $f:\mathbb{R}^n\to\mathbb{R}$, for any natural number $n$.

Suppose that $f(x,y)$ has a critical point at $(a,b)$.

  1. We know that $D^2f(a,b) = \begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}$, where all partials are evaluated at $(a,b)$. Prove that the eigenvalues of $D^2f(a,b)$ are given by $$\lambda = \frac{(f_{xx}+f_{yy})\pm \sqrt{(f_{xx}+f_{yy})^2 - 4(f_{xx}f_{yy}-f_{xy}^2)}}{2}.$$
  2. Let $D=f_{xx}f_{yy}-f_{xy}^2$.
    1. If $D<0$, explain why the eigenvalues differ in sign.
    2. If $D=0$, explain why zero is an eigenvalue.
    3. If $D>0$, explain why the eigenvalues must have the same sign.
    4. If $D>0$, and $f_{xx}>0$, explain why $f$ has a local minimum at $(a,b)$.
    5. If $D>0$, and $f_{xx}<0$, explain why $f$ has a local maximum at $(a,b)$.
    6. How would you interpret $f_{xx}$ in terms of concavity?
  3. The only critical point of $f(x,y) = x^2+3xy+2y^2$ is at $(0,0)$. Does this point correspond to a local maximum, local minimum, or saddle point? Find $D$ from part 2 to answer the question.

Task 22.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.

Day 22 - In class

Brain Gains (Rapid Recall, Jivin' Generation)

  • Let $f(x,y) = x^2y^3$. Compute both $\frac{\partial f}{\partial x}$ and $\frac{df}{dx}$.

Solution

We have

  • (assume $y$ is constant) $\frac{\partial f}{\partial x} = 2xy^3$ and
  • (don't assume $y$ is constant) $\frac{df}{dx} = \frac{d}{dx}(x^2)y^3+x^2\frac{d}{dx}(y^3) = 2xy^3+3x^2y^2\frac{dy}{dx} = \frac{\partial f}{\partial x}\frac{dx}{dx} +\frac{\partial f}{\partial y}\frac{dy}{dx} $.
  • For the vector field $\vec F(x,y) = (xe^y,x+y^2)$, compute the derivative $D\vec F(x,y)$.

Solution

We compute the two partial derivatives, namely

  • $\vec F_x = (e^y,1)$
  • $\vec F_y = (xe^y,2y)$

These vectors are the columns of the derivative of $\vec F$, which means $$D\vec F(x,y) = \begin{bmatrix}e^y&xe^y\\1&2y\end{bmatrix}. $$

  • For the function $f(x,y)=x^3+3x^2+y^2+10y$, compute the second derivative.

Solution

Note $\frac{\partial}{\partial x} \vec\nabla f= (6x+6,0)$ and $\frac{\partial}{\partial y}\vec\nabla f = (0,2)$. Placing these vectors into the columns of a matrix gives us $$D^2f(x,y) = \begin{bmatrix}\begin{matrix}6x+6\\0\end{matrix}&\begin{matrix}0\\2\end{matrix}\end{bmatrix}.$$

  • Find the critical points of the function $f(x,y)=x^3+3x^2+y^2+10y$.

Solution

There are two critical points, namely $(0,-5)$ and $(-2,-5)$. Note $\vec \nabla f(x,y) = (3x^2+6x,2y+10)$ is the zero matrix precisely when $x=0$ or $x=-2$, and $y=-5$.

  • For the function $f(x,y)=x^3+3x^2+y^2+10y$, determine the location of any maxes, mins, or saddles, and classify each location appropriately using eigenvalues.

Solution

At each critical point, we need to (1) evaluate the second derivative, (2) compute the eigenvalues, and (3) classify the point using the eigenvalues.

  • At the point $(0,-5)$, the second derivative is $D^2f(0,-5) = \begin{bmatrix}\begin{matrix}6\\0\end{matrix}&\begin{matrix}0\\2\end{matrix}\end{bmatrix}$. The eigenvalues are 6 and 2 (gradient point outwards from the point), which means at $(0,-5)$ we have a minimum.
  • At the point $(-2,-5)$, the second derivative is $D^2f(-2,-5) = \begin{bmatrix}\begin{matrix}-6\\0\end{matrix}&\begin{matrix}0\\2\end{matrix}\end{bmatrix}$. The eigenvalues are -6 and 2 (gradient point outwards from the point in one direction, and inwards in another), which means at $(-2,-5)$ we have a saddle point.
  • A rover travels along the curve $x^2+y=3$. The elevation near the rover is given by $z = 4 x - 2 y + 1395$. Locate the $(x,y)$ coordinates provided by Lagrange Multipliers.

Solution

We wish to optimize $f(x,y) = 4 x - 2 y + 1395$ subject to the constraint $g(x,y)=x^2+y=3$. We have $\vec\nabla f = (4,-2)$ and $\vec \nabla g = (2x,1)$. The system we must solve is $$4 = \lambda 2x, -2 = \lambda 1, x^2+y=3.$$ The second equation gives $\lambda = -2$. The first equation then provides $x=-1$. This means $y = 3-1 = 2$. The solution is $(x,y)=(-1,2)$.

Group Problems

  1. Consider the function $f(x,y)= x^3+3xy-y^3$. This function has two critical points, namely $(0,0)$ and $(1,-1)$.
    • Compute the gradient $\vec \nabla f(x,y)$.
    • Compute both $\vec \nabla f(0,0)$ and $\vec \nabla f(1,-1)$. Your work should show that both $(0,0)$ and $(1,-1)$ are critical points. (What value should you obtain, and do you obtain it?)
    • Compute the second derivative $D^2f(x,y)$. Then compute both $D^2f(0,0)$ and $D^2f(1,-1)$, the second derivative at these critical points.
    • Classify each critical point as a maximum, minimum, or saddle point, by computing the eigenvalues of the second derivative at that point.
  2. A rover travels along the curve $4x+y=3$. The elevation near the rover is given by $z=y-x^2$. Use Lagrange multipliers to locate the $(x,y)$ coordinates where the rover reaches maximum height. [Check: $(x,y)= (-2,11)$.]
  3. Find the directional derivative of $f(x,y)=xy^2$ at $P=(4,-1)$ in the direction $(-3,4)$. [Check: $D_{(-3,4)}f(4,-1) = \vec\nabla f(4,-1)\cdot \frac{(-3,4)}{5}=(1,-8)\cdot \frac{(-3,4)}{5}=-35/5=-7$.]
  4. Give an equation of the tangent plane to $xy+z^2=7$ at the point $P=(-3,-2,1)$. [Check: $(-2)(x-(-3))+(-3)(y-(-2))+2(1)(z-1)=0$. ]
  5. Give an equation of the tangent plane to $z=f(x,y)=xy^2$ at the point $P=(4,-1,f(4,-1))$. [Check: $z-4 = (-1)^2(x-4)+2(4)(-1)(y-(-1))$.]
  6. Consider the function $f(x,y,z) = 3xy+z^2$. We'll be analyzing the surface at the point $P=(1,-3,2)$.
    • If $dx=0.1$, $dy=0.2$ and $dz=0.3$, then what is $df$ at $P$.
    • Find the directional derivative of $f$ at $P$ in the direction $(1,-2,2)$.
    • Give an equation of the tangent plane to the level surface of $f$ that passes through $P$.
    • Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.

Day 23 - Prep

We won't have any new tasks today, rather we'll focus on finishing up any tasks we have not yet discussed in class from prior days.


Today

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