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Final Jeopardy 203

Final Jeopardy 203. If a sequence of level curves are all closed (i.e. form a closed loop) with each one inside the previous one. Then, some point inside the innermost loop is a________. Formulas 100 Answer. A local max or min. Formulas 101. Show that does not exist. Formulas 101 Answer.

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Final Jeopardy 203

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  1. Final Jeopardy 203

  2. If a sequence of level curves are all closed (i.e. form a closed loop) with each one inside the previous one. Then, some point inside the innermost loop is a________.

  3. Formulas 100 Answer A local max or min.

  4. Formulas 101 Show that does not exist.

  5. Formulas 101 Answer The limit along the x-axis is 1/2, but along the y-axis it is 1.

  6. Partial Derivatives 102 Verify Clairault’s Theorem for the function:

  7. Partial Derivatives 102 Answer fxy and fyx

  8. Partial Derivatives 103 Suppose . Find

  9. Partial Derivatives 103 Answer Ans:

  10. Differentials 100 Suppose z= . Find dz.

  11. Differentials Answer Ans

  12. Gradients 200 Find the equation of plane that is tangent to the surface z= at the point (3,-2,15)

  13. Gradients 200 Answer z=15 +8(x-3) - 9(y+2)

  14. Differentials 300 Three positive numbers, each less than 20 are rounded to the nearest natural number and then multiplied together. Estimate the maximum possible error that can occur from rounding?

  15. Differentials 300 Answer The error estimate should be 600

  16. Differentials 400 Find dz/du and dz/dv

  17. Differentials 400 Answer Ans:

  18. Differentials 100 What is the definition of the derivative of f in the direction of unit vector u=<a,b,c> at the point (x0 ,y0 ,z0) Answer with proper notation

  19. Differentials 100 Answer Def: .

  20. Partial Derivatives 200 Find the derivative of at the point (4,1,1) in the direction of the vector <1,2,3> Answer

  21. Partial Derivatives 200 Answer

  22. Gradients 300 At the point (1,1,1), what is the maximum rate of change of the following function and in which direction is it achieved?

  23. Gradients 300 Answer The maximum rate of change is in the <2,3,4> direction Follow-up question: in what direction is the minimum rate of change?

  24. MaxMin 100 Find all the max’s and min’s of the following function on the given domain

  25. MaxMin 100 Answer There is a local min of 4 at (0,0). This is tied with the point (0,-1) for the absolute min . There is no local max. The absolute max of 7 is achieved at (1,1) and (1,-1)

  26. MaxMin 200 Find the size of the largest rectangular box with edges parallel to the coordinate axes that can fit inside 9x² +36y² +4z² = 36

  27. MaxMin 200 Answer The box has corners at and has volume 48

  28. MaxMin 300 An aquarium is to be built from glass and slate. The volume V is given in advance, the builder is free to choose the dimensions. The builder wants to minimize cost. The bottom be made from slate, which costs 5 times as much as the glass that the sides will be made from. What dimensions should the builder choose?

  29. MaxMin 300 Answer The base of the aquarium should be square with side length 2V/5. Then the height must be 25/4V2

  30. MaxMin 100 GiFind the maximum and minimum values of in the domain

  31. MaxMin 100 Answer The max’s and min’s occur where the lines y=x/2 and y=-x/2 meet the boundary of the ellipse .

  32. Double Integrals 200 Find the volume under the surface z=3xy2 over the rectangular region with corners at (2,3) and (4,5).

  33. Double Integrals 200 Answer 588

  34. Double Integrals 300 Find the volume under above the regions determined by the curves and

  35. Double Integrals 300 Answer

  36. Double Integrals 400 Evaluate:

  37. Double Integrals 400 Answer 1

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