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Name:
Math 206: Winter 2012
Exam 2: March 14
Correct answers accompanied by incorrect or incomplete work will not receive full credit.
Use correct vector notation even in your work. Good Luck!
- (3 points each) Consider the contour plot for the function f (x, y).
-3 -2 -1 1 2 3
1
2
3
c = 2
c = 1
c = 3
c = 0
Determine whether the following quantities are positive, negative, or zero.
(a)
2 f
∂x
2
(b) D ~u
f (− 2 , 1 .5) where ~u = −
i − 2 ˆj.
- (7 points each) Let f (x, y) = 7xy
2 − 11 x
3 y
5
. Compute (in any way you want)
(a)
2 f
∂x
2
(b)
2 f
∂y∂x
(c)
2 f
∂x∂y
- (7 points) The temperature in degrees Celsius on the surface of a metal plate is
T (x, y) = 20 − 4 x
2 − y
2
In what direction from the point (2, −3) does the temperature increase most rapidly?
- (15 points) f (x, y) = 2x
3
3 − 9 xy is a surface in R
3 .
(a) Calculate
∇f.
(b) Write the equation of the line tangent to the level curve 2x
3
3 − 9 xy = 0 at the point (1, 2).
Simplify your answer to the form y = mx + b.
- (15 points)
f (t) =
1 + t, t
2 ,
1
t
, 1 ≤ t ≤ 5 , is the parametric equation for a curve in R
3 .
(a) Calculate
f
′ (t).
(b) Write the parametric equation of the line tangent to this curve at the point where t = 1.
- (15 points) Prove that the following function is continuous at (0, 0).
f (x, y) =
y
4
x
4 +y
2
, if (x, y) 6 = (0, 0);
0 , if (x, y) = (0, 0).
- (1 point) What is your favorite shape?
