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Homework on 30 September 2013:

[To be turned in on Wednesday]

  1. 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 level set for $k=0$. Explain why you think your level set is correct.

    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 drawing and can be distinguished from your other two level sets. Explain why you think your level set is correct.

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


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Created: 30 Sep 2013
Last modified: Sep 30, 2013 3:45:54 PM
Comments to: dpvc@union.edu
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