Problem Set 9—Lagrange Multipliers and Multiple Integrals

Purpose

This problem set helps you understand using Lagrange multipliers to optimize constrained multivariable functions, and gives you some initial practice evaluating multiple integrals. When you finish this problem set, you should be able to (1) solve optimization problems with Lagrange mutipliers, (2) manually evaluate multiple integrals over rectangular regions, and (3) use muPad to evaluate multiple integrals.

Background

This problem set is based on material in sections 14.8 and 15.1 through 15.2 of our textbook. Section 14.8 discusses Lagrange multipliers, and we discussed it in lecture on March 29. Sections 15.1 and 15.2 cover double integrals. We discussed (or will discuss) that material in lecture on March 30 and April 5 and 6. We will combine our discussion of double integration with triple integration, even though the book covers triple integrals in section 15.5.

Activity

Solve each of the following problems:

Problem 1

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

Problem 2

Find the point that lies in both of the planes x + 2y + z = 1 and 2x - y + 2z = 2 and that is closest to point (0,3,1). We started this problem in class on March 29, I’m now asking you to finish it.

Problem 3

Exercise 32 in section 14.8 of our textbook (find values of labor and capital that maximize production subject to a budget, per the Cobb-Douglas production model; see the textbook for details). Note: solving this problem involves some not-necesarily-obvious algebra, but the results are elegant once you get them and the problem illustrates a real-world application of multivariable optimization.

Problem 4

Exercise 16 in section 15.1 of our textbook (integrate (√x) / (y²) over a certain region; see book for details). Evaluate the integral both by hand and using muPad (check that you get the same result both ways). Note that muPad’s square root operator is sqrt; for example, √x is written as sqrt(x) in muPad.

Problem 5

Exercise 34 in section 15.1 of our textbook (integrate xe^xy between certain bounds; see the book for details). Evaluate the integral both by hand and using muPad (check that you get the same result both ways). Note that muPad has a function exp that denotes raising e to a power, e.g., e^x would be written exp(x) in muPad.

Problem 6

Exercise 12 in section 15.5 of our textbook (evaluate an integral of x + y + z; see book for details). Evaluate the integral both by hand and using muPad (check that you get the same result both ways).

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.