Math 208 (Lutz)
Exam II Practice Problems
11/01/13
1. Explain whether the following statement is true or false: No flow line for the vector field
F! (x, y) = x!i + 2!j has a point where the y-coordinate reaches a relative maximum.
2. A spherical shell centered at the origin has an inner radius of 6 cm and an outer radius of
7 cm. The density, δ, of the material increases linearly with the distance from the center.
At the inner surface, δ = 9 gm/cm3 ; at the outer surface, δ = 11 gm/cm3 . Find the mass of
the shell.
3. Find the local extrema and saddle points of the following function. Determine if the local
extrema are global extrema.
f (x, y) = x2 + y 3 − 3xy.
4. Sketch the vector field
y
!i − ! x
!j.
F! = !
x2 + y 2
x2 + y 2
3
2
1
-3
-2
-1
1
-1
-2
-3
2
3
5. Let R be the triangle in
! 2-space with vertices (−2, 0), (4, 0), and (0, 4). For a continuous
function f (x, y), write R f dA in two different ways.
6. Find the acceleration of a particle moving along the curve x = tet , y = 2et .
7. Explain whether the following statement is true or false: If P is a local maximum for a
function f (x, y, z), then P is also a global maximum for f .
8. Consider the trapezoid in 2-space with vertices (2, 0), (12, 0), (4, 5) and (10, 5). Calculate
the average distance to the y-axis for points in the trapezoid.
9. Use Lagrange multipliers to find the minimum and maximum values of f subject to the given
constraint.
f (x, y) = x3 + y,
x + y ≥ 1.
10. Evaluate the integral:
!
1
0
!
√
0
1−x2
! √x2 +y2
0
(z +
"
x2 + y 2 ) dz dy dx.
11. Give a parameterization for the circle of radius 2 parallel to the xy-plane, centered at (0, 0, 1),
and traversed counterclockwise when viewed from below.
12. Find the volume between the cone z =
!
x2 + y 2 and the sphere x2 + y 2 + z 2 = 4.