Math 2210 Section 1 - Spring 2008
Exam II
Answer the questions in the spaces provided on the question sheets. Show all
work to receive credit. Calculators are not permitted. Each question is worth
10 points.
Name:
Useful formulas:
T (p) = f (p0 ) + ∇f (p0 ) · (p − p0 )
2
D = fxx fyy − fxy
1
2
1. For f (x, y) = (x3 + y 2 )5 , find
∂2f
.
∂x∂y
3
2. Find the slope of the tangent to the curve of intersection of the surface z =
x2 + 11x − 5y 2 with the plane y = 4 at the point (2, 4, −54).
4
3. Find the indicated limit or show that it does not exist:
lim
(x,y)→(0,0)
sin(x2 + y 2 )
sin(3x2 + 3y 2)
5
4. Find the gradient vector of f at (1, 2). Then find the equation of the tangent
plane at (1, 2).
f (x, y) = x3 + 3xy − y 2 .
6
5. Find the directional derivative of f at (1, −1) in the direction −i +
f (x, y) = e−xy
√
3j.
7
6. Find a unit vector in the direction in which f decreases most rapidly at (2, 0, −4).
What is the rate of change of f in this direction?
f (x, y, z) = xeyz
8
7. Let w = x2 y − y 2x. Find
dx
dy
dw
when x = 3, y = 2,
= −3 and
= 8.
dt
dt
dt
9
8. Find all the critical points of f . Indicate whether each such point gives a local
maximum, local minimum or saddle point.
f (x, y) = xy 2 − 6x2 − 3y 2