Name: Calculus III
Exam 2
1 November 2000



``... I never went to Cambridge as my brothers did. I had the chance, but I refused it. I wanted to get out into the world. I've always regretted it. I think it would have saved me a lot of mistakes. You learn more quickly under the guidance of experienced teachers. You waste a lot of time going down blind alleys if you have no one to lead you.''

``You may be right. I don't mind if I make mistakes. It may be that in one of the blind alleys I may find something to my purpose.''

from The Razor's Edge by W. Somerset Maugham


Instructions: Show all work to receive full or partial credit. You may use a scientific calculator/graphing calculator. All University rules and guidelines for student conduct are applicable.

1.

[20 pts] If

$$f(x,y) = \cos(x^2 + 3xy + 2y^2),$$

calculate $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2 f}{\partial x \partial y}$.

2.

[20 pts] Prove that

$$\frac{x^2-y^2}{x^2+2y^2}$$

is not continuous at (0,0).

3.

[20 pts] Find the equation of plane tangent to the surface

x² - y² - z² = 1

at the point $(\sqrt{3},1,1)$. Sketch the surface and the tangent plane. (Be sure to clearly label the x, y and z axes.)

4.

[20 pts] Find the critical points of the function

f(x,y) = x² + 5xy + 2y².

5.

[20 pts] Find the maximum and minimum of

f(x,y) = xy

subject to the constraint

x² + y² = 4.