Math 115H (Assignments)

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Consider the function $f(x,y) = x^2y+xy^2-2xy$.

1. Sketch the level set of $f$ at height $k=0$. Be as accurate as you can, and be sure to label your axes and their important values. Do not use a computer or graphing calculator to do this for you. You should be able to analyze this situation yourself. Make your plot at least $3\times 3$ inches in size. Label your level set as "$k=0$".

   (Hint: factor out $xy$.)

2. On your plot, shade the region of points where the function's value is positive. Explain how you determined this.

3. On the same diagram, sketch what you think the level set at height $k=.1$ should look like. Note that you will not be able to compute it exactly; you should use reasoning not computation to do this. (Do not plot this using a calculator or computer and then copy that; if you do, you are missing the point of the exercise.) Make sure it is clearly labeled in your drawing and that it can be distinguished from your

4. On the same diagram, sketch what you think the level set at height $k=-.1$ should look like. Make sure it is clearly labeled in your

5. Indicate the (approximate) positions of any critical points for $f$ on your diagram. How did you locate these?

Math 115H (Fall 2017) web pages Created: 25 Sep 2017 Last modified: Sep 25, 2017 9:09:47 PM Comments to: dpvc@union.edu

Assignments