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.