Problem Set 8 — Limits and Derivatives of Multivariable Functions

Purpose

This exercise further reinforces your understanding of limits and derivatives of 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 6.2. Apply derivative concepts to solve optimization problems
• 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 mainly based on material in sections 3.1 through 3.4 of our textbook. We covered that material in classes between March 6 and 22. The exercise also asks you to plot multivariable functions and and their level curves with Mathematica. We talked about plotting in class on March 6.

Activity

Solve each of the following problems.

Problem 1

Give equations for, and use Mathematica to plot, the level curves for \(z = 1\), \(z = 3\), and \(z = 5\) of the function \(z(x, y) = 2x^2 + 3y^2\).

Problem 2

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

Part A

Show that

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 3

Suppose \(g(x,y) = \frac{x}{\sqrt{y}}\). Calculate each of the second derivatives of \(g\).

Problem 4

Much as with single-variable functions, multivariable functions have critical points at points where either all of their first partial derivatives equal 0, or at least one of the first partial derivatives doesn’t exist. Also as with single-variable functions, local minimum and maximum values of multivariable functions can only occur at critical points.

For each of the following functions, determine whether the given point is a critical point:

1. \(z = 3x^2 - xy + 2y^2\) at point \((4,1)\).
2. \(g(u,v,w) = uvw\) at point \((0,10,0)\).

Problem 5

The following table gives values for function \(f(x,y)\) for certain values of \(x\) and \(y\):

Estimate the value of \(f(1.9,3.1)\).

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, and should ordinarily last half an hour. 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.