- 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.
- Let and let S be the surface z = f(x,y).
- Find a normal vector to the surface S at the point .
- What is the equation of the tangent plane to the surface S at the
point ?
- 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.
- 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.]
-
-
- 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?
- For the following vector field, find the line integral along the straight
line segment from the origin to the point (3,3).
- Calculate the flux of
through the disk in the xy-plane, oriented upward.
- Which of the following vector fields is a gradient field? Find potential
functions for the ones that are gradient fields.
-
-
- Consider the vector field .
- Calculate div for .
- Find the flux of out of a box of side a centered at the
origin with edges parallel to the axes.
- 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.]