Math 223 Multivariable Functions

Purpose

This exercise introduces you to working with multivariable functions. It contributes to the following learning outcomes for this course:

Outcome 4. Compute limits of functions of 2 and 3 variables
Outcome 5. Compute derivatives of functions of 2 and 3 variables
Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.

Background

This exercise is based on material in sections 3.1 through the beginning of 3.3 in our textbook. We talked about those sections in classes between September 30 and October 7. We talked about plotting multivariable functions and their level curves with Mathematica in class on October 3, and about finding their derivatives on October 7.

Activity

Solve each of the following problems.

Problem 1

(Based on exercise 14 in section 3.1E of our textbook.)

Give equations for, and use Mathematica to plot, the level curves for $z = 1$ , $z = 2$ , and $z = 3$ of the function $z (x, y) = y^{2} - x^{2}$ .

Problem 2

(Exercise 18 in section 13.2E of our textbook.)

Determine whether

lim_{(x, y) \to (0, 0)} \frac{x^{2} + y^{2}}{\sqrt{x^{2} + y^{2} + 1} - 1}

exists, and if so what it is.

Problem 3

(Inspired by exercise 28 in section 13.2E of our textbook.)

Part A

Show that

lim_{(x, y) \to (0, 0)} \frac{x y + y^{3}}{x^{2} + y^{2}}

does not exist.

Part B

Use Mathematica to plot the function from Part A near the origin. Be prepared during grading to identify the feature(s) of the plot that correspond to the non-existence of the limit.

Problem 4

Find all the (first) partial derivatives of each of the following functions:

f (x, y) = \sin (x y) - x y

g (x, y, z) = \frac{x}{z} \ln (x + y)

Problem 5

(Inspired by exercise 4 in section 13.1E of our textbook.)

An oxygen tank is formed from a right circular cylinder of radius $r$ and height $h$ , with hemispheres of radius $r$ at the ends of the cylinder.

Part A

Express the tank’s volume as a function of $r$ and $h$ . Remember that the volume of a cylinder of radius $r$ and height $h$ is $V = π r^{2} h$ ; the volume of a hemisphere of radius $r$ is $V = \frac{2}{3} π r^{3}$ . Then use your function to find the volume of a tank with radius 2 and height 5.

Part B

Use Mathematica to plot your volume function over the region $0 \leq r \leq 5, 0 \leq h \leq 5$ .

Part C

The Geneseo Oxygen Tank company makes a standard tank of radius 3 inches and height 10 inches. If they want to change these dimensions slightly in order to hold more oxygen in their standard tank, will they get more “bang for the buck” by increasing the radius or by increasing the height? In other words, does volume change faster with changes in radius or with changes in height when $r = 3$ and $h = 10$ ?

Follow-Up

I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.