TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
LAST NAME
FIRST NAME
ID
MATH 251
EXAM 2
Section 504
FALL 2008
O.Shatalov
Show all work.
1. [17%] For f (x, y, z) = x3 + sin(xyz)
(a) find the gradient;
(b) find the directional derivative at the point (1, π2 , 1) in the direction
of the vector h2, 1, 2i;
(c) find the maximum rate of change of f at the point (1, π2 , 1).
2. [13%] Find the equation of the plane which is tangent to the surface
zexyz = 1
at the point (5, 0, 1).
3. [15%] Find all critical points of
f (x, y) = x3 + y 3 − 3xy
and classify them as local maxima, local minima, or saddle points.
4. [15%] Let
f (x, y) = xy − 2x + 5.
Find the absolute maximum and minimum values of f on the set
D which is the closed triangular region with vertices A(0,0), B(1,1),
C(0,1).
5. [15%] For
Z 3Z 9
0
y2
f (x, y) dxdy
(a) sketch the region of integration;
(b) change the order of integration.
6. [15%]Find the volume of the solid that lies under the paraboloid z =
x2 + y 2 , above the xy-plane, and inside the cylinder x2 + y 2 = 4.
7. [15%]Convert the integral
√ 2Z √
Z Z
3
0
9−y
0
0
9−x2 −y 2
(x2 + y 2 + z 2 ) dz dx dy
to an integral in spherical coordinates, but don’t evaluate it.
spherical coordinates: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.