Problem Set 5—Multi-Variable Functions

Purpose

This problem set reinforces your understanding of multi-variable functions. It also gives you a bit of practice working with curvature in vector-valued functions.

Background

We discussed (or will discuss) multi-variable functions, the main focus of this problem set, in lectures on October 9 and 14. Sections 14.1 and 14.2 of our textbook also cover this material.

A second purpose of this exercise is to develop your ability to work with curvature in vector-valued functions, something we discussed in class on October 7. This material is from section 13.4 of the textbook.

Activity

Solve each of the following problems:

Problem 1

Exercise 12 in section 13.4 of our textbook (find the unit tangent vector, principle unit normal, and curvature for r(t) = ⟨ 6sin(2t), 6cos(2t), 5t ⟩).

Problem 2

Exercise 18 in section 14.1 of our textbook (find the domain and range; describe the level curves; find the boundary; determine if the domain is open, closed, both or neither; and decide whether the domain is bounded, for the function f(x,y) = √(y-x)).

Problem 3

Exercise 52 in section 14.1 of our textbook (find an equation for the level curve of (2y-x) / (x+y+1) that passes through the point (-1,1)).

Problem 4

Exercise 70 in section 14.1 of our textbook (plot the surface and several level curves for sinx cosy e^√(x2+y2)/8; see text for additional details). Use muPad or a similar tool.

Problem 5

Exercise 14 in section 14.2 of our textbook (find lim_{(x,y)→(0,0)}( (x²-y²)/(x-y) ); see text for further details).

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.