1. Math 205: Midterm #2 Jeopardy! April 16, 2008

2. Jeopardy ! Definitions Partial Derivatives 1 Partial Derivatives 2 Integration 100 200 300 400

3. Definitions : 100 State Clairaut’s theorem. What part of the second derivative test uses this theorem? Back

4. Definitions : 200 State Fubini’s Theorem and how it applies to the following integral: Back

5. Definitions : 300 Suppose z = f(x,y) is a function in R 3. If (a,b) is a point in the domain, describe the geometric meaning behind f x (a,b). Back

6. Definitions : 400 Define what it means for a function to be continuous at a point (a,b) and give a function in R 3 that is discontinuous. Back

7. Partial Derivatives 1 : 100 Compute the directional derivative of z = 2x 2 - y 3 at the point (0,1) in the direction of the vector u = 2 i - j. Back

8. Partial Derivatives 1 : 200 Compute ∂z/∂t of z = 3xcosy, where x = 3st - t 2 and y = s - 2sint. Back

9. Partial Derivatives 1 : 300 Suppose you are at the point (0, π ) a hill given by the function: z = 15 - x 2 + cos(xy) - y 2. If the positive y-axis represents north, and the positive x-axis represents east, what is your rate of ascent if you head northwest? Suppose you are at the point (0, π ) a hill given by the function: z = 15 - x 2 + cos(xy) - y 2. If the positive y-axis represents north, and the positive x-axis represents east, what is your rate of ascent if you head northwest? Back

10. Partial Derivatives 1 : 400 Find ∂z/∂y of the equation given by: ze xz = y 2 - yz. Back

11. Partial Derivatives 2 : 100 Compute  f of the function z = f(x,y) = 3x 3/2 y 1/2. Back

12. Partial Derivatives 2 : 200 Determine the tangent plane at the point (0, π /2 ) for the function z = 2cos(xy) - x 2. Back

13. Partial Derivatives 2 : 300 Find and classify all critical points of the function z = 1- 3x 2 - y 2 + 2xy. Back

14. Partial Derivatives 2 : 400 Find and classify all critical points of the function z = 2xy-1 subject to the constraint x 2 + y 2 = 1. Back

15. Integration : 100 Reverse the order of integration in the following integral: Back

16. Integration : 200 Use polar coordinates to set up (but not evaluate) an integral to determine the volume under the sphere x 2 + y 2 + z 2 = 4 within the first octant. Back

17. Integration : 300 Set up an integral to describe the area within the curve r = 2cos . Back

18. Integration : 400 Find the volume of the surface that lies under the cone z = 4 - √(x 2 +y 2 ) and between the planes z = 1 and z = 2. Back