Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. Careful of that third step, it's a doozy.
 

1. Find  or show that the limit does not exist.
 
 

2. Find the directional derivative of f(x,y) = e^x-y at the point (1,2) in the direction of the vector v = <-3,4>.
 

3. If we have z = f(x,y,z) where x = x(s,t), and y = y(s,t), write out the appropriate chain rule for .
 
 

4. Find an equation for the plane tangent to f(x,y) = x²y - y²x + 7x at the point (-3,1).
 
 

5. Find the maximum rate of change of f(x,y) = xe^-y + 3y at the point (1,0) and the direction in which it occurs.
 
 

6. Find the critical points of the function f(x,y) = x² + 3y² + x²y + 5 and classify them as local maxima, minima, or saddle points.

7. Jebediah is a calculus student at O.S.U. who's having some trouble with level curves. Jeb says "Gosh darn it, I think I just spread a whole load of manure all over my calc test again. There was this one question about those level curve doo-hickys, and said some stuff about how all them level curves of this function was straight lines. I figured the only way that could happen was if the function was just a plane, so I did it like it was a plane. But then some girls was talking after the test and they was saying something totally different. Jeez, I hope I don't fail and have to retake that dang class a third time!"

Help Jeb out by explaining (in a way he can understand!) what you can say about a surface if you know that all of its level curves are straight lines.
 
 

8. Show that the equation of the tangent plane to the ellipsoid x²/a² + y²/b² + z²/c² = 1 at the point (x₀,y₀,z₀) can be written as .
 
 

9. Mike the mountain goat is standing on a pleasant mountaintop shaped exactly like the surface z = 5000 - x² - y². Now Mike especially likes points where the slope in the direction where it's steepest is exactly 2, so he's looking around himself wondering if there are any such points. Tell Mike where to find the points he'll like.
 
 

10. Find a function of the form p(x,y) = ax² + by² + cxy + dx + ey + f (you figure out the values of a, b, c, d, e, and f) which meets the following list of requirements:

p(0,0) = -7
p_x(0,0) = 12
p_y(0,0) = -6
p_xx(0,0) = 6
p_xy(0,0) = 1
p_yy(0,0) = -4
 
 

Extra Credit (5 points possible):

Suppose you know that a function f(x,y) has a critical point which is a local maximum at the point (7,-3). What, if anything, can you say about critical points of g(x,y) = f(x,y) + 1? How about h(x,y) = f(x,y) + x?