does not exist.
2. Sketch at least four level curves of z = x2 - y, indicating the z value of each.
Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. No actual reindeer were harmed in the making of this exam.
1. Show that
does not exist.
2. Sketch at least four level curves of z = x2
- y, indicating the z value of each.
3. Find the directional derivative of z = x2
- y in the direction of the vector <-3,4> from the point (-2,1,3).
4. Find an equation for the plane tangent to the sphere
x2 + y2 + z2 = 72 at the point
(-3, 6, -2).
5. The organization Reindeer Lovers International is
making an effort to increase the population of reindeer in the world. Their
analysis indicates that changes in the reindeer population (represented
by P, in thousands) are mostly caused by changes in the food supply (represented
by F, in thousands of tons) and changes in the amount of toxic chemicals
(represented by C, in International Toxic Chemical Units, or just I.T.C.U.s)
in the reindeer habitats. Both F and C can change over time. Preliminary
studies also indicate that the rate at which increases in food supply affect
population is about 1.6 reindeer per ton of food, and the rate at which
toxic chemicals affect population is a loss of about 300 reindeer per I.T.C.U.
Give the appropriate version of the chain rule for 
.
If Reindeer Lovers International can provide 400 extra tons of food each
year and keep the increase of toxins down to 2 I.T.C.U.s each year, what
should be the annual change in world reindeer population?
6. Find the maximum rate of change of 
at the point (2, 1). In which direction does it occur?
7. Zeb the mosquito is hovering near the rump of an
elephant (mosquitos are very self-centered, so Zeb thinks of himself as
being at the origin), which from his point of view looks like the surface
5x + 2y - z = 6. Zeb intends to bite the elephant at the point nearest
him. Find the coordinates of that point and the distance Zeb needs to travel
to get there.
8. Biff is a student taking calculus at OSU and he's
a bit confused. Biff says "I don't believe this directional derivative
stuff. I looked at the graph of this function, and you can tell that at
(0,0) the slope along both the x and the y axes is zero, so both fx
and fy are zero. But if you go in a direction like <1,1>,
it drops off. The thing is, that formula for directional derivatives just
puts together the two partial derivatives, and they're both zero at (0,0),
so it'll say the directional derivative is zero too. But from the graph
you can see that it doesn't stay flat when you go that way, so the
directional derivative shouldn't be zero there."
Help Biff out. Either explain why he's right, or clear up his confusion by explaining to him what's wrong with his reasoning.
9. Find all critical points of functions in the family
f(x,y) = xe-ax + ye-by and classify
them as maxima, minima, or saddle points. Graphs of several members of
this family are shown below, but note that the scales are not all the same.
10. Show that the z intercept of a plane tangent to
a sphere of radius r at the point (x0, y0, z0)
is given by 
.
Extra Credit (5 points possible):
We know that for continuous functions fxy = fyx.
Are there functions for which fxx = fyy, but without
fxx or fyy being zero? Either give an example of
one or explain why it couldn't happen.