1. Given the surface f(x,y) = x²y+y²-2y

a) Find the equation of the plane tangent to the surface at the point (1,2).
 
 
 
 
 
 

b) Find the volume of the solid under f in the region R = {(x,y)| 0x2, 0y3}.
 
 

2. Continuing with the function f(x,y) = x²y+y²-2y from problem 1,

a) What is the directional derivative of f in the direction of the vector <1,2>?
 
 

b) In which direction does f increase most rapidly?
 
 

3. Compute , where C is the top half of a circle of radius 2 centered at the origin and oriented counterclockwise.
 
 

4. Find a quadratic approximation for the function f(x,y) = x² cos y near the point (1,).
 
 

5. Calculate  where S is the top half of a sphere of radius 3 along with the disc of radius three (centered at the origin) in the xy-plane.
 
 

6. The mythical blue polar bunnies of Antarctica live secretly on a peninsula shaped like the portion of the curve x = 16 - y² which lies to the right of the y-axis. If the bunnies' population density is given by the function  = 50 + 8y + 10x, what is the total bunny population of the peninsula?
 
 

7. Let D be a region with area A bounded by a simple closed path C in the xy-plane. Use Green's Theorem to prove that the coordinates of the center of mass of the region are

8. Buffy says "Like, Oh my God, my calc final was like, so hard. There was this one problem that was, like, about level curves, you know? And it was like, not even a formula, you know? They just told you, like, some stuff, and you were, you know, s'posed to say what the surface-thingy would be like? Like, for one of them, it said that the level curves were, like, circles, and they were all, like, concentric around the origin. And it said, if one surface with level curves like that had 'em so they got more and more spaced out as you got away from the origin, but another surface had 'em so they got closer and closer together as you got away from the origin, then, like, what are the two surfaces like? Oh my God, I just had no clue at all!"

Give Buffy some clue about what the information she tried to describe would tell us about the two surfaces. Obviously, she needs things explained very clearly.
 
 

9. Consider functions of the form f(x,y) = x² + bxy + y², where b is any constant. For which values of b is (0,0) a local minimum? A local maximum? A saddle point?
 
 

10. Consider the hyperboloid x² + y² - z² = 1. Find the volume enclosed by the hyperboloid and the planes z = -c and z = c.
 
 

Extra Credit (5 points possible):

Look back at the very first problem. Is there some point (a,b) for which the volume under the tangent plane (on the region R) is equal to the volume under f (on R)? Find the point(s), or explain why they can't exist.