Calculus III, Own Your Own Function Project
You should choose one of the following four functions of two variables to investigate:

I. $f(x, y) = 2x^2y + xy^2-6xy$ on the rectangle -1 ≤ x ≤ 3 and -1 ≤ y ≤ 3
II. $f(x, y) = 3\sqrt{x+y-4}-2\sqrt{x} - \sqrt{y}$ on the rectangle 0 ≤ x ≤ 6 and 0 ≤ y ≤ 6
III. $f(x, y) = \sin{x} + \sin{y} + \sin(x+y)$ on the rectangle 0 ≤ x ≤ 2π and 0 ≤ y ≤ 2π
IV. $f(x, y) = xe^{-(x^2+y^2)/8}$ on the rectangle -6 ≤ x ≤ 6 and -6 ≤ y ≤ 6

Your goal is to find the:
a. absolute maximum and minimum values,
b. local maximum and minimum points and saddle points,
c. double integral,
d. and average value (see problems 39-40 in section 12.1).

You may use a computer to confirm the values you find, but you should solve for each by hand as well. Anywhere possible your results should be exact numbers. Check and make sure that your results are reasonable using a graph (you may produce the graph with a computer).

Your report should include detailed calculations of each of parts (a)-(d), as well as an explanation of how you found those results. You should also include a graph (which may be computer generated) and a short explanation of how it supports the results you found.

BONUS: Include a 3D model of your graph. This can be handmade (out of whatever materials you prefer to work with), or can be constructed on a 3D printer (visit the Next Gen Tech Bar in Hemmingson to learn about 3D printing options).

Your report may be handwritten or typed. If it is handwritten it should be written neatly (NOT a first draft!). If it is typed, I recommend you use LaTeX to typeset your mathematics (resources for writing in LaTeX are included below). The finished project is due by the end of the day Friday, May 5. You can turn in your project in class, email it to Dr. Kearney, or drop it off in Dr. Kearney's mailbox by the end of the day. You should work in a group of 2-3 students on this project. You may choose your own groups. Your group will receive one grade for all members. Your grade will be based primarily on completeness and mathematical correctness, but will also include points for presentation and clarity of your write up.

This project is adapted from the "Own Your Own Function of Two Variables" project in the book "Problems for Student Investigation" by Michael B. Jackson and John R. Ramsay. The original problem was developed by Eugene Herman and Anita Solow at Grinnell College.

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LaTeX resources
Here are some other resources about working with LaTeX:
• Cloud-based LaTeX computing is available at www.writelatex.com and www.sharelatex.com.
• DeTeXify can help you find a symbol you need to use.
• TeX Stack Exchange is a Q&A site with descriptions of some of the more complicated TeXing techniques.
• The Comprehensive LaTeX symbols list.
• Symbol Search
• A general introduction
• Another brief introduction