Two contours of z = f(x,y) corresponding to different values of
z can never cross.
(c)
The two planes 3z - 2x + y = 5 and -6z + 4x - 2y = 0 are
parallel.
(d)
If f(x,y) is a linear function then f(3,4) - f(3,5) = f(4,4) -
f(4,5).
2.
The following are worth 6 points each.
(a)
Given the vectors in figure 1, draw the vector .
(b)
Given that and find:
i.
ii.
iii.
The angle between and .
3.
(20 points) Consider the contour diagram of the linear function shown
in figure 2.
(a)
Find the equation of the plane z = f(x,y) corresponding to this
diagram.
(b)
Find a unit normal vector to this plane.
4.
(20 points) Consider the function z = f(x,y) = exy.
(a)
Draw a contour diagram and label the contours for z = 0.5, 1, 1.5.
(b)
What are the domain and range of f.
(c)
Is f(x,y) increasing or decreasing as a function of y at the
point (1,0)?
5.
(20 points) Fox television uses a linear function to decide what to
do with its programs. A show's score is a function of both its Neilson
Rating, R, and its production cost per episode, C. If a show score is
1000 or
above, then it is placed into a prime time slot. If its score is below
500, then it is cancelled and sold to UPN. Some examples are: (1) a show
with R = 80 and million receives a prime time slot, (2)
for any value of R, a show is kept on the air if its cost is below
$50,000, and (3) a show with a Neilson rating of 80 that costs more than $2
million per episode will be cancelled. Note that the lowest Neilson rating
is R = 0.
(a)
Use the three cases above (1)-(3) to help determine the linear
function that Fox uses to determine the score of a show.
(b)
If the show Married with Pets costs $1.5 million per epsiode
and receives a Neilson rating of 60, what will happen to it?
6.
(20 points) If a burner on a stove is turned off at time t=0, then
the function describing the temperature at a point (x,y) on the burner at
time t is z = e-(x2+y2+t) where the point (0,0) corresponds to
the center of the burner. Draw a contours of constant z (in x and y)
at time t=0 and describe (in words) how the contour diagram wll change as
t increases.