Each problem is worth 10 points. Show all work for full credit. Please circle all answers and keep your work as legible as possible. Nothing on this exam shall constitute grounds for litigation.
 

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

2. A flea named Bitsy is sitting at the point (-1,2) on the surface g(x,y) = . Which direction should Bitsy face in order to find the greatest uphill slope? How steep will the slope be in that direction?

3. Find the equation of the plane tangent to the surface h(x,y) = ln(x² + y² - 1) at the point (-1,e).

4. It is regarded by many as a cause for severe national alarm that the price of gizmos has been increasing recently. One very recent study models the wholesale price of a ton of gizmos by the function p(x,y,z)=5xy-3y²z+15000, where the mysterious factors x, y, and z all vary over time with current market research suggesting that  = -1,  = 3, and  = 2, while currently x=75, y=5, and z=9. Find  based on current conditions.
 
 

5. Find all critical points of the function f(x,y) =  and classify them as minima, maxima, saddle points or otherwise (you have to show which is which, the picture doesn't count as proof).

6. Suppose a dormouse is running along the surface z = x² - y² following the line y = -x. Show that the slope of the surface along the path the dormouse is following is zero.

7. The functions f₁(x,y) = cos x cos y and f₂(x,y) = cos³x cos³y look slightly similiar but not identical. Show that at any point (x,y), the gradients on both surfaces point in the same direction.

8. Biff and Buffy are Calculus students at O.S.U. who just got done taking their first exam and are arguing about one of the questions. Biff says "Uh, like, you know that one problem, the one about whether, uh, the level curves could cross? I said no 'cause then where they cross would be two different heights, right?" Buffy responds "Oh my God, like, that was so confusing. I like, said that they could, like, cross, you know? Because, you know, like, if there were, well I don't want to say it, but if there were like two bumps next to each other, then like the outline could be like a figure eight at, like, just the right height, you know? So like that would be, like, the level curve thingy would cross itself, wouldn't it? Oh my God."

Explain as clearly as possible to Buffy and Biff which of them (if either) is right and how they should think about it.
 
 

9. Show that the equation of the tangent plane to the hyperboloid  at the point (x₀,y₀,z₀) can be written as .
 
 

10. Suppose you are given that D_uf(x₀,y₀) = 1 and D_vf(x₀,y₀) = 2 for directional derivatives of a function in two different directions u =  and v = . Find an expression for the gradient of f at (x₀,y₀).
 
 

Extra Credit (5 points possible):

Find the maximum and minimum values, if any, of the function f(x,y)= x² + y² subject to the constraint y = mx + b (in terms of m and b). [Recall that in class we did this with the constraint y=-x+4. If you're having trouble doing it in general, try one particular constraint line like y=2x+5.]