Lesson 23 Lecture Notes

23.0.1 Evaluation of Functions of Two Variables

A function of two variables has the form: \[ f(x, y) \] To evaluate the function, simply substitute values for both \(x\) and \(y\).

Example:

If \(f(x, y) = x^2 + y^2\), then: \[ f(2, 3) = 2^2 + 3^2 = 4 + 9 = 13 \] Functions of two variables are often visualized as surfaces in 3D space.

23.0.2 Finding Partial Derivatives

A partial derivative measures the rate of change of a function with respect to one variable while keeping the other constant.

23.0.2.1 Notation:

  • Partial derivative with respect to \(x\):
    \[ f_x(x, y) = \frac{\partial f}{\partial x} \]

  • Partial derivative with respect to \(y\):
    \[ f_y(x, y) = \frac{\partial f}{\partial y} \]

Example:

For \(f(x, y) = x^2y + y^3\):

  • \(f_x = 2xy\)
  • \(f_y = x^2 + 3y^2\)

Higher-order partial derivatives include:

  • \(f_{xx}, f_{yy}, f_{xy}, f_{yx}\)

23.0.3 Clairaut’s Theorem (Equality of Mixed Partials)

Clairaut’s Theorem states that if the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\) are continuous near a point, then:

\[ f_{xy} = f_{yx} \]

This means the order of differentiation does not matter under reasonable conditions (i.e., if the function is sufficiently smooth).

23.0.4 D-Test for Classifying Critical Points

To classify critical points of a function \(f(x, y)\), use the Second Derivative Test (D-test).

23.0.4.1 Steps:

  1. Find critical points by solving:

\[ f_x = 0 \quad \text{and} \quad f_y = 0 \]

  1. Compute second-order partial derivatives:

\[ f_{xx}, f_{yy}, f_{xy} \]

  1. Calculate the discriminant:

\[ D = f_{xx} f_{yy} - (f_{xy})^2 \]

  1. Classify the critical point:
  • If \(D > 0\) and \(f_{xx} > 0\): Local minimum
  • If \(D > 0\) and \(f_{xx} < 0\): Local maximum
  • If \(D < 0\): Saddle point
  • If \(D = 0\): Inconclusive

Example:

Let \(f(x, y) = x^2 + y^2\)

  • \(f_x = 2x, f_y = 2y \Rightarrow \text{critical point at } (0, 0)\)
  • \(f_{xx} = 2, f_{yy} = 2, f_{xy} = 0\)
  • \(D = (2)(2) - 0 = 4 > 0\), and \(f_{xx} = 2 > 0\)

→ Local minimum at (0, 0)

23.1 Example

23.1.1 Functions of Several Variables

Video: Solutions

  1. Let
    \[ f(x, y) = \sqrt{150 - 4x^2 - 2y^2} \] Evaluate \(f(4,3)\).

  2. Given
    \[ f(x, y) = 70x + 60y - 3x^2 - 5y^2 - xy \] Evaluate \(f(2,10)\).

  3. Let
    \[ f(x, y) = \sqrt{y^2 - x^2} \] What is the domain of \(f\)?

  4. Using \(x\) skilled workers and \(y\) unskilled workers, a manufacturer can produce \(Q(x, y) = 10x^2 y\) units per day. Currently there are 10 skilled workers and 20 unskilled workers.

    1. How many units are currently being produced each day?
    2. By how many units will the daily production level change if 2 more skilled workers are added to the current workforce?
    3. By how many units will the daily production level change if 2 more unskilled workers are added to the current workforce?
    4. By how many units will the daily production level change if 1 more skilled worker and 1 more unskilled worker are added to the current workforce?

  5. The daily output at a factory is \(Q(K, L) = 116 K^{\frac{4}{5}} L^{\frac{2}{3}}\) where:

    • \(K\) = capital investment (in units of $1000)
    • \(L\) = size of the labor force (in worker-hours)
    1. Compute the daily output if the capital investment is $130,000 and the size of the labor force is 1491 worker-hours.
    2. What will happen to the output in part (a) if both the level of capital investment and the size of the labor force are cut in half?

23.1.2 Partial Derivatives

Video: Solutions

  1. Given \(f(x, y, z) = \sqrt{4x^2 + y^2 + 2z^2}\), find:
  1. \(f_x(x, y, z)\)
  2. \(f_y(x, y, z)\)
  3. \(f_z(x, y, z)\)
  1. Find \[ \frac{\partial}{\partial x} \left[ 9 e^{4x} \sin(4y) \right] = \quad\_\_\_\_\_ \]

\[ \frac{\partial}{\partial y} \left[ 9 e^{4x} \sin(4y) \right] = \quad\_\_\_\_\_ \]

  1. If \(f(x, y) = \ln\left( \sqrt[3]{x^2 \cdot y} \right)\), find the first partials evaluated at \((5, 4)\). Give answers as reduced fractions.

  2. Find the first partials of \(f(x, y) = \cos\left( x^8 + y^9 \right)\).

  3. Find the first and second partial derivatives of \(f(x, y) = 7y e^{11xy}\).

  4. Given \(f(x, y, z) = -2 e^{5xyz}\), find:

    • \(f_x(x, y, z)\)
    • \(f_y(x, y, z)\)
    • \(f_z(x, y, z)\)

23.1.3 Maximum and Minimum Values

Video: Solutions

  1. Suppose that \(f(x, y) = e^{-2x^2 - 4y^2 + x + 2y}\). Find the maximum.

  2. Suppose that \(f(x, y) = x^2 - xy + y^2 - 3x + 3y\) with \(-3 \leq x, y \leq 3\). Find the absolute maximum and minimum values.

  3. Suppose that \(f(x, y, z) = x + 5y + 5z\) at which \(0 \leq x, y, z \leq 2\). Find the absolute maximum and minimum values.

  4. Suppose that \(f(x, y) = 2x^4 + 2y^4 - 2xy\). Find the extrema.

  5. Let \(f(x, y) = x^2 - y^2 - 30xy\). Find and classify the critical points.

23.2 Practice Problems

  1. Verify Clairaut’s Theorem:

    • \(f(x,y) = 1000 + 5x - 4y - 3xy\)
    • \(f(x,y) = x^2 + y^2\)
    • \(f(x,y) = \dfrac{e^{0.2x}}{1 + e^{-0.1y}}\)

  2. Classify critical points of \(f(x,y) = x^2 + y^2 + 1\)

  3. Classify critical points of \(f(x,y) = x^2 + xy - y^2 + 3x - y\)

  4. Classify critical points of \(f(x,y) = x^2 + y^2 + \dfrac{2}{xy}\)

  5. (Applied) You’re charged $0.03/video and $0.04/audio clip sold by Moneydrain.com. Should you accept their offer of $10/month plus those per-unit costs, or stick with your current $40/month support + $15 hosting, given 400 videos and 600 audios expected?

  6. (Applied) Weekly cost is \(C(x,y) = 24000 + 60x + 20y + 0.3xy\). Find marginal cost of tricycles when producing 10 bikes and 20 trikes.

23.3 Self Assessment

  1. Verify Clairaut’s Theorem:

    • \(f(x,y) = x^4y^2 - x\)
    • \(f(x,y) = x e^{xy}\)

  2. Classify critical points of \(f(x,y) = x^2 + x + y^2 - y - 1\)

  3. Classify critical points of \(f(x,y) = e^{x^2 + y^2}\)

  4. For \(C(x,y) = 24000 + 60x + 20y\), calculate and interpret \(\frac{\partial C}{\partial x}\) and \(\frac{\partial C}{\partial y}\)

23.4 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Calculate partial derivatives of two-variable functions
Find and classify critical points
Solve applied problems involving two-variable functions