1. Final Jeopardy 203

2. Col. 1 100 Differentials 100 MaxMin 100 MaxMin 100 Curves 100 Col. 1 101 Differentials 200 Partial Deriv’s 200 MaxMin 200 Doub. Ints. 200 Col.1 102 Differentials 300 Gradient s 300 MaxMin 300 Doub Ints 300 Col 1. 103 Differentials 400 Double Integrals 400 Empty

3. 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________.

4. A local max or min.

5. Show that does not exist.

6. The limit along the x-axis is 1/2, but along the y- axis it is 1.

7. Verify Clairault’s Theorem for the function:

8. f xy and f yx

9. Suppose. Find

10. Ans:

11. Suppose z=. Find dz.

12. Ans

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

14. z=15 +8(x-3) - 9(y+2)

15. 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?

16. The error estimate should be 600

17. Find dz/du and dz/dv

18. Ans:

19. What is the definition of the derivative of f in the direction of unit vector u= at the point (x 0,y 0,z 0 ) Answer with proper notation

20. Def:.

21. Find the derivative of at the point (4,1,1) in the direction of the vector Answer

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

24. The maximum rate of change is in the direction Follow-up question: in what direction is the minimum rate of change?

25. Find all the max’s and min’s of the following function on the given domain

26. 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)

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

28. The box has corners at and has volume 48

29. 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?

30. The base of the aquarium should be square with side length 2V/5. Then the height must be 25/4V 2

31. GiFind the maximum and minimum values of in the domain

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

33. Find the volume under the surface z=3xy 2 over the rectangular region with corners at (2,3) and (4,5).

34. 588

35. Find the volume under above the regions determined by the curvesFind the volume under above the regions determined by the curves and

37. Evaluate:

38. 1