Math 223 Homework Chapter 13

  1. The contour diagram for the function

    displaymath35

    displaymath36

    is shown below. At the point (-6,14) what are the signs (positive, negative, or zero) of tex2html_wrap_inline45 , tex2html_wrap_inline47 , tex2html_wrap_inline49 , tex2html_wrap_inline51 , tex2html_wrap_inline53 ?

  2. Estimate values for tex2html_wrap_inline55 , and tex2html_wrap_inline57 using the contour diagram.
  3. Using the analytic form for f(x,y), find tex2html_wrap_inline55 and tex2html_wrap_inline57 exactly.
  4. Use your answer in (3) to find the tangent plane to z=f(x,y) at (-6,14). Use this to estimate f(-7,15). Is this a reasonable approximation?

    Problems 5-10 concerns an environmental disaster. EDIOT Inc. (The Environmentally DIsastrous Oil Tycoons Inc.) has just lost another tanker. You have been sent out to clean the spill by introducing a new algae into the oil. The algae will consume the oil, producing harmless by products. It works best if the algae is introduced where the oil concentration is highest. Let the initial point of the oil spill be at coordinates (0,0). Your boat currently sits at (-6,14) Let

    displaymath37

    displaymath36

    represent the concentration of oil measured in moles per liter.

  5. Interpret what tex2html_wrap_inline55 , tex2html_wrap_inline57 , and tex2html_wrap_inline77 mean physically. (Do not calculate this yet).
  6. What is tex2html_wrap_inline77 (exactly)?
  7. Suppose your boat is equipped with the new Grad-O-Meter 2000 to find the highest concentration of oil. The Grad-O-Meter 2000 measures tex2html_wrap_inline45 and tex2html_wrap_inline47 at the point where the boat is currently. In order to move toward the highest concentration, which direction should you move?
  8. Move the boat 1 unit in the direction you choose in (7). At what point are you now? What is tex2html_wrap_inline85 here?
  9. Compute tex2html_wrap_inline87 (old point) tex2html_wrap_inline89 (new point) tex2html_wrap_inline91 . Repeat (8)-(9) until this quantity is less than 0.05. Note: you should not have to repeat this more than 3 times.
  10. What can you say about the point you are at? Note: the contour diagram and tex2html_wrap_inline85 may help interpret the meaning.