Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. "Slug" should not be taken as a pejorative term.

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 (2,1) in the direction of the vector <3,-4>.
 
 

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

4. The function f(x,y) = 2x² -4xy + y⁴ +2 has critical points at (1,1), (0,0) and (-1,-1). Classify each of them as a maximum, minimum, saddle point, or neither.
 
 

5. Sammy the slug is dreaming about a beautiful cabbage leaf. If the leaf resembles the surface and Sammy is standing at the point (1,-2,2), write an equation of the plane tangent to the cabbage at the point where Sammy is standing.
 
 

6. For the function g(x,y) =  at the point (0, /2) find the direction in which the directional derivative is greatest and the value of that directional derivative.
 
 

7. Jebediah is a calculus student as O.S.U. who's having some trouble with directional derivatives. Jeb says "Gosh darn it, I think I just spread a whole load of manure all over my calc test. There was this one question about those directional doo-hickys, and it wanted to know, like, if you had the directional derivative for one direction, then could you get the directional derivative for the opposite direction. I figured maybe it might be just the same, so I put that, but some girls was talking after the test and they was saying something totally different."

Help Jeb out by explaining (in a way he can understand!) what you can say about the directional derivatives in opposite direcctions.
 
 

8. Find the location of the minimum value of the function p(x,y) = x² + y² + ax + by +c.
 
 

9. If f(x,y) = sin x + sin y, what is the largest value D_u f(x,y) can have?
 
 

10. Show that every plane tangent to the cone x² + y² = z² passes through the origin.
 
 

Extra Credit (5 points possible):

A function of two variables whose partial derivatives of all orders are continuous has at most three distinct second order partial derivatives, since f_xy=f_yx. How many distinct third partials might it have? Can you say anything similiar about fourth and higher order partials? [Hint: If nothing else, take a function like f(x,y) = x⁴y³ and find all of its third order partial derivatives, then look at them...]