If and are perpendicular vectors, then . True or False? Justify your
answer.
3.
If f is differentiable at (a,b), fx(a,b) = 0 and fy(a,b) =
0, then the directional derivative of f at (a,b) is zero in any
direction. True or False? Justify your answer.
4.
Which of the following diagrams represents the parametric curve ?
Justify your answer.
5.
Which of the following is a contour diagram for ?
Justify your answer.
6.
The figure below shows a vector field and an oriented curve
C. Is positive, zero, or negative?
Justify your answer.
7.
Is a conservative
vector field? Justify your answer.
8.
The two planes 3z - 2x + y = 5 and -6z + 4x - 2y = 0 are
parallel. True or False? Justify your answer.
9.
The ideal gas law says that
PV = RT
for a certain fixed amount of gas (called a mole of gas), where P
is the pressure (in atmospheres), V is the volume (in cubic meters), T
is the temperature (in degrees Kelvin), and R is a positive constant.
(a)
Find and .
(b)
A mole of a certain gas is at a temperature of 298 degrees Kelvin,
a pressure of 1 atmosphere, and a volume of 0.0245 m3. Calculate
for
this gas. Give the units of your answer and explian what it means in
practical terms.
10.
(a)
Express as an iterated integral, where R is the
region in the xy-plane shown below.
(b)
Evaluate where R is the circle of
radius 2 centered at the origin.
11.
(a)
Write a brief paragraph about Stokes' Theorem. You should include
a statement of the theorem, defining any symbols used, and you should
explain in words what the theorem says.
(b)
Suppose that is parallel to the x-axis and
points in the direction of the positive x-axis at every point in
three-space. Suppose that C is a circle in the yz-plane, oriented
clockwise when viewed from the positive x-axis. Is the circulation of
around C positive, zero, or negative? Explain your answer.
12.
Let .
(a)
Find the critical points of f.
(b)
Classify the critical points as local maxima, minima, or saddle
points.
13.
Consider the vector field
(a)
Calculate the divergence of . Simplify your answer.
(b)
Use your answer in part (a) to calculate the flux of out
through the sphere of radius 1 centered at the point (0,0,2).
14.
Consider the function f(x,y) = 3x ex2y.
(a)
Find .
(b)
In what direction from the point (-1,0) is f increasing most
rapidly? In what direction is the rate of change of f equal to zero?