Problem Set 8—Applications of Partial Derivatives

Purpose

This problem set helps you understand some uses of partial derivatives. In particular, by the time you finish this problem set you should be able to (1) linearize multivariable functions and estimate their values, and (2) find critical points and extreme values of multivariable functions. This exercise also gives you practice finding directional derivatives and gradients.

Background

Most of this problem set is based on material in sections 14.6 and 14.7 of our textbook. We discussed tangents and linearization (section 14.6) in class on March 22 and will start discussing extrema and critical points on March 23.

Directional derivatives are covered in section 14.5 of our textbook, which we discussed in class on March 10.

Activity

Solve each of the following problems:

Problem 1

Exercise 30a in section 14.5 of our textbook (given f(x,y) = (x-y)/(x+y), find the direction in which the directional derivative of f at point ( -1/2, 3/2 ) is largest; also find the value of the directional derivative).

Problem 2

Exercise 32a in section 14.6 of our textbook (use a linearization of a wind chill function to estimate the wind chill for a 49 MPH wind at -22 degrees Fahrenheit).

Problem 3

Exercise 14 in section 14.6 of our textbook (find equations for the line tangent to the surfaces xyz = 1 and x² + 2y² + 3z² = 6 at point (1,1,1)).

Problem 4

Exercise 2 in section 14.7 of our textbook (find the local extrema and saddle points of 2xy - 5x² - 2y² + 4x + 4y - 4).

Problem 5

Exercise 32 in section 14.7 of our textbook (find the absolute extrema of x² - xy + y² + 1 over the closed region bounded by the lines x = 0, y = 4, and x = y).

Problem 6

Exercise 40 in section 14.7 of our textbook (find a and b that maximize the integral from a to b of (24-2x-x²)^1/3)

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.