Answers to the Math 251 Sample Final:

These are not full worked-out solutions, and mainly point you in the
right direction as you work on these problems.

1.  C
2.  A
3.  E
4.  B
5.  B
6.  D
7.  A
8.  E
9.  D
10. C

11. You need to switch the order of integration.  The answer is
    (e^9 - e)/3.

12. (a) Simply check that (1,2,2) satisfies both defining equations.

    (b) Use the gradient to find a normal vector to the tangent plane
        at P to S_1.  One equation is 4(x-1) + 4(y-2) + 4(z-2) = 0.
        Another is x+y+z=5.

    (c) The cross product of the normal vectors of the two tangent
        planes will be parallel to their intersection.  The equation
        of this intersection can then be found to be x = 1-t, y =
        2+3t, z = 2-2t.

13. The volume is 64*pi*sqrt(2)/3.

14. (a) By Green's Theorem, the line integral is equal to the double
        integral:

           / /
          | |  -4 dA
         / /
           R
 
        where R is the closed unit disk.

    (b) Evaluate the line integral directly.  You need to find a
        parametrization for the unit circle (x=cos(t), y=sin(t), 0 <=
        t <= 2*pi, will do), and use change the line integral into an
        integral in t.  The value is -4*pi.

    (c) Evaluating the double integral can be done in many ways.  The
        simplest is that it is -4 * (area of R).  So the value is
        -4*pi.

15. In this problem it is intended that you use Stokes' Theorem to
    convert the line integral in question into a surface integral over
    S.  So you want to compute the surface integral

           / /
          | |  curl F . n dS
         / /
           S
 
    After using the formula that is given for computing such a surface
    integral, the answer obtained is -2*pi.