Fall 1996 Final Exam


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  1. Money in a bank account earns interest at a continuous rate, r. The amount of money , tex2html_wrap_inline79 , in the account depends on the amount deposited, tex2html_wrap_inline81 , and the number of years, t, it has been in the bank according to the formula

    tex2html_wrap_inline85 .

    Find tex2html_wrap_inline87 and tex2html_wrap_inline89 and interpret each in financial terms.

  2. Let tex2html_wrap_inline91 and let S be the surface z = f(x,y).
    1. Find a normal vector to the surface S at the point tex2html_wrap_inline99 .
    2. What is the equation of the tangent plane to the surface S at the point tex2html_wrap_inline103 ?
  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. tex2html_wrap147
  4. Let W be the top half of the unit ball, tex2html_wrap_inline107 . 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. tex2html_wrap_inline109
    2. tex2html_wrap_inline111
  5. Show that the equations

    tex2html_wrap_inline113

    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).

    tex2html_wrap_inline119

  7. Calculate the flux of

    tex2html_wrap_inline121

    through the disk tex2html_wrap_inline123 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. tex2html_wrap_inline127
    2. tex2html_wrap_inline129
  9. Consider the vector field tex2html_wrap_inline131 .
    1. Calculate div tex2html_wrap_inline133 for tex2html_wrap_inline135 .
    2. Find the flux of tex2html_wrap_inline133 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 tex2html_wrap_inline141 around the circle tex2html_wrap_inline143 , z = 2, oriented counterclockwise when viewed from above. [Hint: You can use Stokes's Theorem.]