MATH 251
Extra Practice (Final Exam)
FALL 2011
xz
1. The equation of the plane tangent to the surface ze y = 1 at (x, y, z) = (0, 12 , 1) is
a.
1
x
2
+z−1=0
b. x = 0
c. 2x + z − 1 = 0
d. 2x + y + z =
3
2
1
2
e. 2x − y + z =
Z
2. Compute
F~ · d~r where F~ (x, y, z) = hx, ey , 2zi and C is any path from (5,1,2) to (-1,1,3).
C
Hint: Find a potential.
a. 7
b. 25
c. −25
d. 0
e. −7
3. For the function f (x, y) = xy 2 + x3 − 2xy the point (x, y) = ( √13 , 1) is
a. a local minimum
b. a local maximum
c. a saddle point
d. not a critical point
e. is a critical point but the Second Derivative Test fails.
4. A particle moves along the curve ~r(t) = hsin t, cos t, ti from the point (0, 1, 0) to the
i. The work done by the force is
point (0, −1, π) due to the force F~ (x, y, z) = hy, −x, 2z
π
a. 1 +
1
π
b. 3π
c. π
d.
1
π
e. 2π
1
5. The integral
y2
x3
) dx + (xy +
+ 3x) dy
2
3
C
where C is the positively oriented triangle with vertices (0, 0), (1, 0),(0, 1).
Z
(x2 y +
Hint: Use Green’s Theorem.
a. − 23
1
b. − 12
c.
3
2
d. 0
e.
1
12
6. Compute
RR
S
~ for the vector field
F~ · dS
F~ (x, y, z) = hx3 + z sin y, y 3 + x2 y, x2 z + 2y 2 zi
over the complete boundary S of the solid cylinder
{(x, y, z) : x2 + y 2 ≤ 4, 0 ≤ z ≤ 5}
with outward normal.
Hint: Use the Divergence Theorem.
a.
b.
400
π
3
80
π
3
c. 100π
d. 200π
e. 40π
7. Let D be the region bounded by the parabola x = 1 − y 2 and the coordinate axes in the
first quadrant. Find the mass of the lamina that occupies the region D if the density
function is ρ(x, y) = y.
a.
1
2
b. 1
c.
d.
e.
1
12
2
3
1
4
2
2
8. Find the volume
p of the solid region which lies above the paraboloid z = x + y and below
2
2
the cone z = x + y .
a.
b.
c.
π
2
π
6
π
3
d. π
e. 2π
2
RR
R
~=
9. Verify Stokes’ Theorem S curlF~ · dS
F~ · d~r for the vector field F~ = hyz 2 , −xz 2 , z 3 i
∂S
2
2
and the cylinder x + y = 9 for 1 ≤ z ≤ 2 oriented out.
10. Find the absolute maximum and minimum values of the function f (x, y) = 1 + xy − x − y
on the region D bounded by the parabola y = x2 and the line y = 4.
RR
11. Evaluate the surface integral S ydS, where S is the part of the plane 3x + 2y + z = 6
that lies in the first octant.
p
12. Find the mass of the conical surface z = x2 + y 2 for z ≤ 2 with density ρ(x, y) = x2 +y 2 .
13. Verify Green’s Theorem for the vector field F(x, y) = h−x2 y, xy 2 i on the region inside
the circle x2 + y 2 = 16.
14. Find the absolute maximum and minimum of the function f (x, y) = 18 x4 − x2 + 2 + y 2 on
the square determined by the four points A(4, 4), B(−4, 4), C(4, −4), D(−4, −4).
15. Find the absolute maximum and minimum of the function f (x, y) = x3 − xy + y 2 − x on
D = {(x, y) : x ≥ 0, y ≥ 0, x + y ≤ 2}.
16. Determine the equation of the plane tangent to the surface ~r(u, v) = hu2 + v, u2 − v, 2uvi
at the point (10, 8, 6).
17. Let F~ (x, y) = hx + y 2 , 2xy + y 2 i.
a) Show that F~ is conservative vector field.
Z
b) Compute
F~ · d~r where C is any path from (-1,0) to (2,2).
C
18. Let F~ (x, y, z) = h4xez , cos y, 2x2 ez i.
a) Show that F~ is conservative vector field.
Z
b) Compute
F~ · d~r where C : r(t) = ht, t2 , t4 i, 0 ≤ t ≤ 1.
C
19. Consider the surface S parameterized by ~r(θ, z) = h3 cos θ, 3 sin θ, zi where 0 ≤ θ ≤
π, 0 ≤ z ≤ 2.
~ (θ, z) to the surface S;
a) Find the normal vector N
~ (θ, z)|;
b) Find |N
ZZ
c) Find
yz dS.
S
ZZ
20. Apply the Divergence Theorem to compute
~ for the vector field F~ (x, y, z) =
F~ · dS
S
hx3 , y 3 , x + yi over the complete boundary S of the solid paraboloid {(x, y, z) : x2 + y 2 ≤
z ≤ 1} with outward normal.
21. Verify that Stokes’ Theorem is true for the vector field F = hx2 , y 2 , z 2 i where S is the
part of the paraboloid z = 1 − x2 − y 2 that lies above the xy-plane, and S has upward
orientation.
3
Z
F · dr, where F = hxy, yz, zxi and C is the triangle
22. Use Stokes’ Theorem to evaluate
C
with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1), oriented counterclockwise as viewed from
above.
ZZ
23. Evaluate
y dS where S is the surface z = x + y 2 , 0 ≤ x ≤ 1, 0 ≤ y ≤ 2.
S
(Hint: parameterize S, find the parameter domain, normal and its magnitude.)
24. Apply the Divergence Theorem to compute
ZZ
F · dS
S
for the vector field
F(x, y, z) = hx3 + sin(yz), y 3 , y + z 3 i
over the complete boundary S of the solid hemisphere
{(x, y, z) : x2 + y 2 + z 2 ≤ 1, z ≥ 0}
with outward normal.
25. Verify Stokes’ Theorem
I
ZZ
F · dr =
∂S
curlF · dS
S
for the vector field F = hy, −x, x2 + y 2 i and the surface S which is the part of the
paraboloid z = x2 + y 2 between z = 0 and z = 1 with normal pointing up and in.
Use the following steps:
(a) Sketch the surface S, indicate its boundary, indicate and explain the orientations
(on S and on ∂S).
(b) Compute the line integral.
H
(Hint: parameterize the boundary circle ∂S and compute ∂S F · dr.)
(c) Compute the surface integral.
(Hint: parametrize the paraboloid, indicate the parameter domain, find the normal and determine
its direction, compute curlF and
RR
S
curlF · dS).
4