Problem Set 8—Extreme Values and Optimization

Purpose

This problem set develops your ability to use partial derivatives to find maxima and minima of multivariable functions, including the use of maxima and minima to solve optimization problems.

Background

Section 14.7 of our textbook discusses the basic ideas behind finding maxima and minima, and we will discuss those ideas in class on October 28. Section 14.8 discusses Lagrange multipliers, and key technique for solving optimization problems. We will discuss them in class on October 30.

Activity

Solve each of the following problems:

Problem 1

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 2

Exercise 26 in section 14.7 of our textbook (find any maxima, minima, or saddle points of f(x,y) = e^y - ye^x). Also use muPad or a similar system to plot the function to roughly verify your answer.

Problem 3

Exercise 32 in section 14.7 of our textbook (find the absolute maxima and minima of f(x,y) = x² - xy + y² +  on the closed triangle bounded by the lines x = 0, y = 4 and x = y). Also use muPad or a similar system to plot the function over the given region to roughly verify your answer.

Problem 4

Exercise 2 in section 14.8 of our textbook (find the extreme values of f(x,y) = xy subject to the constraint x² + y² - 10 = 0).

Problem 5

Exercise 10 in section 14.8 of our textbook (find the dimensions of the cylinder of largest surface area that can be inscribed in a sphere, and find the cylinder's surface area; see the textbook for more explanation).

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.