Problem Set 6—Multivariable Functions

Purpose

This problem set ensures that you understand some basic concepts of multivariable functions, in particular their (1) level curves or surfaces, and (2) limits and continuity. This problem set also gives you practice working with (3) curvature of vector-valued functions.

Background

Our textbook introduces multivariable functions in section 14.1, and discusses their limits in section 14.2. We discussed section 14.1 in class on March 1, and limits on March 2.

Curvature is discussed in section 13.4 of our text. We discussed it in class on February 24.

Activity

Solve each of the following problems:

Problem 1

Exercise 10 in section 13.4 of our textbook (find the unit tangent vector, unit normal vector, and curvature of the function ⟨ cost+tsint, sint-tcost, 3 ⟩. Note that all three of these will be functions of t.)

Problem 2

Exercise 17 in section 13.4 of our textbook (analyze the curvature of parabolas of the form y = ax²; see the text for details).

Problem 3

Exercise 32 in section 14.1 of our textbook (match a plot of some level curves to the corresponding surface in 3 dimensions; see textbook for details).

Problem 4

Exercise 62 in section 14.1 of our textbook (find an equation for the level surface of ln( x² + y + z² ) that passes through (-1,2,1)).

Problem 5

Exercise 36a in section 14.2 of our textbook (describe the points at which ln(xyz) is continuous).

Problem 6

Exercise 50 in section 14.2 of our textbook (show that lim_{(x,y)→(1,-1)}( (xy+1) / (x²-y²) ) does not exist).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.