Fall 1996 Final Exam

1. Money in a bank account earns interest at a continuous rate, r. The amount of money , , in the account depends on the amount deposited, , and the number of years, t, it has been in the bank according to the formula

   .

   Find and and interpret each in financial terms.

2. Let and let S be the surface z = f(x,y).
   1. Find a normal vector to the surface S at the point .
   2. What is the equation of the tangent plane to the surface S at the point ?
3. Consider the points marked A, B, C in the contour plot below. Which of these appear to be critical points? Classify those that are critical points.
4. Let W be the top half of the unit ball, . Decide whether each of the following integrals is positive, negative or zero. Give reasons for your decision. [Hint: You can answer this question without evaluating the integrals.]
   1. 
   2. 
5. Show that the equations

   

   satisfy the equations x + y + 3z = 6 and x - y - z = 2. What does this tell you about the curve parameterized by these equations?

6. For the following vector field, find the line integral along the straight line segment from the origin to the point (3,3).

   

7. Calculate the flux of

   

   through the disk in the xy-plane, oriented upward.

8. Which of the following vector fields is a gradient field? Find potential functions for the ones that are gradient fields.
   1. 
   2. 
9. Consider the vector field .
   1. Calculate div for .
   2. Find the flux of out of a box of side a centered at the origin with edges parallel to the axes.
10. Find the circulation of the vector field around the circle , z = 2, oriented counterclockwise when viewed from above. [Hint: You can use Stokes's Theorem.]