Name
1-9
/54 12
/20
Spring 2007
10
/15 13
/6
P. Yasskin
11
/15
ID
MATH 253
Final Exam
Sections 501-503
Solutions
Multiple Choice: (6 points each. No part credit.)
Total
/110
1. Consider the triangle with vertices A = (1, −1, 2), B = (2, 3, 1) and C = (4, 2, 2).
Which vector is perpendicular to the plane of the triangle?
a. (1, 1, 3)
b. (−1, −1, −3)
c. (−1, 1, −3)
d. (1, 1, −3)
e. (1, −1, −3)
Correct Choice
AB = B − A = (1, 4, −1)
î ̂
AB × AC =
k̂
1 4 −1
3 3
AC = C − A = (3, 3, 0)
= î(3) − ̂ (3) + k̂ (3 − 12) = (3, −3, −9) is perpendicular.
0
(1, −1, −3) has the same direction.
a .
2. For the "helix" curve ⃗
r(θ) = (4 cos θ, 4 sin θ, 3θ) find the unit binormal B̂ = ⃗v × ⃗
|v⃗ × ⃗
a|
a. (12 sin θ, 12 cos θ, 16)
b. (−12 sin θ, −12 cos θ, −16)
c.
d.
e.
3 sin θ, 3 cos θ, 4
5
5
5
− 3 sin θ, − 3 cos θ, − 4
5
5
5
3 sin θ, − 3 cos θ, 4
5
5
5
⃗v = (−4 sin θ, 4 cos θ, 3)
⃗v × ⃗
a=
⃗
a = (−4 cos θ, −4 sin θ, 0)
̂
î
−4 sin θ
Correct Choice
k̂
4 cos θ 3
= î(12 sin θ) − ̂ (12 cos θ) + k̂ (16 sin 2 θ + 16 cos 2 θ) = (12 sin θ, −12 cos θ, 16)
−4 cos θ −4 sin θ 0
|v⃗ × ⃗
a| =
144 + 256 = 20
a =
B̂ = ⃗v ⋅ ⃗
|v⃗ ⋅ ⃗
a|
3 sin θ, − 3 cos θ, 4
5
5
5
1
3. Find the equation of the plane tangent to z = xy 2 + x 3 y at (x, y) = (1, 2).
What is the z-intercept?
a. (0, 0, 6)
b. (0, 0, −6)
c. (0, 0, 14)
d. (0, 0, −14)
Correct Choice
e. (0, 0, 26)
f(x, y) = xy 2 + x 3 y
f(1, 2) = 6
2
2
f x (x, y) = y + 3x y
f x (1, 2) = 10
3
f y (x, y) = 2xy + x
f y (1, 2) = 5
z = f(1, 2) + f x (1, 2)(x − 1) + f y (1, 2)(y − 2) = 6 + 10(x − 1) + 5(y − 2)
z-intercept:
x=0
y=0
z = 6 + 10(−1) + 5(−2) = −14
4. The temperature in a room is given by T = 72 + xyz. What is the time rate of change of the
temperature as seen by a fly located at P = (3, 2, 1) with velocity ⃗v = (2, 2, 1)?
a. 4
b. 11
Correct Choice
c. 16
d. 18
e. 22
⃗ T = (yz, xz, xy)
∇
5.
⃗T
∇
= (2, 3, 6)
P
dT = ⃗v ⋅ ∇
⃗T
dt
P
= 4 + 6 + 6 = 16
Find the volume under the surface z = 2x 2 y above
the region bounded by y = x and y = 2 x .
The base is shown at the right.
a.
256
5
b.
320
3
c.
64
7
d.
320
7
e.
64
5
Correct Choice
The curves intersect when x = 2 or x 2 = 4x or x = 0, 4
V=
4
2 x
∫0 ∫x
2x 2 y dy dx =
4
∫0
2 x
[x 2 y 2 ] y=x
dx =
5
∫ 0 (4x 3 − x 4 ) dx = x 4 − x5
4
4
x=0
4
= 4 = 256
5
5
2
Find the mass of the solid apple given in
6.
spherical coordinates by ρ = 1 − cos φ
if the volume mass density is δ = ρ.
a.
2π
5
b.
8π
3
c.
16 π
5
Correct Choice
d. 8π
e.
M=
64 π
15
2π
π
1−cos ϕ
∫∫∫ δ dV = ∫ 0 ∫ 0 ∫ 0
ρ ρ sin ϕ dρ dϕ dθ = 2π ∫
2
π
0
ρ4
4
1−cos ϕ
ρ=0
sin ϕ dϕ = π
2
π
∫ 0 (1 − cos ϕ)
4
sin ϕ dϕ
u = 1 − cos ϕ
du = sin θ dθ
2
5
2
M = π ∫ u 4 du = π u
= 16 π
5
2 0
2 5 0
∫ F⃗ ⋅ ds⃗
7. Compute
⃗ = (1 + yz, 1 + xz, 1 + xy) along the curve
for F
⃗r
⃗r(t) = (ln(1 + t), t ln(1 + t), t 2 ln(1 + t)) between t = 0 and t = 1.
HINT: Find a scalar potential and use the Fundamental Theorem of Calculus for Curves.
a. 3 ln 2 + 3(ln 2) 3
b. 3 ln 2 + (ln 2) 3
Correct Choice
c. 3 ln 2 − 3(ln 2) 3
d. 3 ln 2 − (ln 2) 3
e. −3 ln 2 + 3(ln 2) 3
⃗ =∇
⃗ f for f = x + y + z + xyz
F
A = ⃗r(0) = (0, 0, 0)
B = ⃗r(1) = (ln 2, ln 2, ln 2)
∫ F⃗ ⋅ ds⃗ = ∫ ∇⃗ f ⋅ ds⃗ = f(B) − f(A) = 3 ln 2 + (ln 2) 3
⃗r
⃗r
3
8. Compute
∮ F⃗ ⋅ ds⃗
⃗ = (x 2 y − y 3 , x 3 − xy 2 ) counterclockwise around the circle x 2 + y 2 = 4.
for F
C
HINT: Use Green’s Theorem.
a. 16 π
3
b. 8π
c. 32 π
3
Correct Choice
d. 16π
e. 32π
P = x2y − y3
∂Q
= 3x 2 − y 2
∂x
Q = x 3 − xy 2
∂Q
− ∂P = 2x 2 + 2y 2 = 2r 2
∂x
∂y
∮ F⃗ ⋅ ds⃗ = ∫∫
C
9. Compute
∂Q
− ∂P dA =
∂x
∂y
∫∫ ∇⃗ × F⃗ ⋅ dS⃗
2π
∫0
∂P = x 2 − 3y 2
∂y
4
∫ 0 2r 2 r dr dθ = 2π r2
2
2
0
= 16π
over the paraboloid z = 9 − x 2 − y 2 for z ≥ 0,
P
⃗ = (z + y, z − x, 2z).
oriented up, for the vector field F
HINT: Use Stokes’ Theorem. Parametrize the boundary.
a. −18π
Correct Choice
b. 0
c. 9π
d. 18π
e. 36π
The boundary is the circle x 2 + y 2 = 9 for z = 0.
It may be parametrized as ⃗r(θ) = (3 cos θ, 3 sin θ, 0).
⃗v = (−3 sin θ, 3 cos θ, 0)
Oriented counterclockwise as seen from above.
⃗ (r⃗(θ)) = (3 sin θ, −3 cos θ, 0)
F
2π
∫∫ ∇⃗ × F⃗ ⋅ dS⃗ = ∮ F⃗ ⋅ ds⃗ = ∫ F⃗ ⋅ ⃗v dθ =
P
∂P
0
2π
∫0
−9 dθ = −18π
4
Work Out: (Points indicated. Part credit possible. Show all work.)
z = 27 + 64
x
y
10. (15 points) Find the point (x, y, z) in the first octant on the surface
which is closest
to the origin.
f = x2 + y2 + z2
Minimize the square of the distance to the origin
subject to the constraint that the point lies on the surface
z = 27 + 64 .
x
y
2
Eliminate a constraint:
f x = 2x + 2
Minimize
−27
x2
27 + 64
x
y
Multiply the first equation by
27 + 16 ⋅ 3
x
x
Cross multiply:
x4
and
z = 27 + 64
x
y
27 + 64
x
y
−64
y2
27 + 64
x
y
f y = 2y + 2
x2
54
and the second equation by
27 + 64
y
x
Equate these to obtain
x3 =
27
2
=0
x3 =
27
(1.)
f = x +y +
2
=0
y2
:
128
y3
=
64
27 + 64
y
x
and plug back into (1.) to obtain:
(2.)
x = y
3
4
= 75
x
= 75 ⋅ 27 = 3 4 ⋅ 5 2
So
x=3 5
27
64
25
=
+
=
= 5 5.
3 5
4 5
5
and
y=4 5
11. (15 points) Find the mass and z-component of the center of mass of the "twisted cubic" curve
⃗r(t) = t, t 2 , 2 t 3
3
⃗v = (1, 2t, 2t 2 )
M = ∫ ρ ds =
for 0 ≤ t ≤ 1 if the density is ρ = 3xz + 3y 2 .
|v⃗| =
1
∫0
ρ |v⃗| dt =
1 + 4t 2 + 4t 4 = 1 + 2t 2
5
7
1
∫ 0 5t 4 (1 + 2t 2 ) dt = 5 ∫ 0 (t 4 + 2t 6 ) dt = 5 t5 + 2 t7
1
z-mom = M xy = ∫ z ρ ds =
= 10
3
z̄ =
ρ = 3t 2 t 3 + 3(t 2 ) 2 = 5t 4
3
t 8 + 2 t 10
8
10
1
∫ 0 zρ |v⃗| dt =
1
0
= 10
3
2
3
1
∫ 0 t 3 5t 4 (1 + 2t 2 ) dt =
1 + 1
5
8
10
3
1
0
=5 1 + 2
5
7
= 17
7
1
∫ 0 (t 7 + 2t 9 ) dt
= 13
12
M xy
z-mom
=
= 13 7 = 91
M
M
12 17
204
5
12.
20 points
∫∫ F⃗ ⋅ dS⃗
⃗ dV =
⃗ ⋅F
Verify Gauss’ Theorem ∫∫∫ ∇
V
∂V
⃗ = (xz, yz, −2z 2 ) and
for the vector field F
the volume above the cone C: z =
x2 + y2
for z ≤ 2
and below the disk D: x 2 + y 2 ≤ 4 with z = 2.
Be sure to check and explain the orientations.
Use the following steps.
⃗ and the volume integral
⃗ ⋅F
a. Compute the divergence ∇
⃗ dV.
⃗ ⋅F
∫∫∫ ∇
V
⃗ = z + z − 4z = −2z
⃗ ⋅F
∇
⃗ dV =
⃗ ⋅F
∫∫∫ ∇
V
2π
2
2
2
∫ 0 ∫ 0 ∫ r −2z r dz dr dθ = −2π ∫ 0
2
4
= −2π 2r 2 − r
4
0
z2
2
z=r
2
r dr = −2π ∫ (4 − r 2 ) r dr
0
= −2π(8 − 4) = −8π
b. Parametrize the disk, D, and compute the surface integral:
⃗ (r, θ), ⃗e r , ⃗e θ , N
⃗ , check orientation, F
⃗ R
⃗ (r, θ) ,
Successively find: R
∫∫ F⃗ ⋅ dS⃗.
D
⃗ (r, θ) = (r cos θ, r sin θ, 2)
R
î
̂
k̂
⃗e r = (cos θ,
sin θ, 0)
⃗e θ = (−r sin θ, r cos θ, 0)
⃗ = ⃗e θ × ⃗e z = î(0) − ̂ (0) + k̂ (r cos 2 θ + r sin 2 θ) = (0, 0, r)
N
⃗ has the correct orientation which is up.
N
⃗ R
⃗ (r, θ)
F
= (xz, yz, −2z 2 ) = (2r cos θ, 2r sin θ, −8)
⃗⋅N
⃗ = −8r
F
∫∫ F⃗ ⋅ dS⃗ = ∫∫ F⃗ ⋅ N⃗ dr dθ = ∫ 0 ∫ 0 −8r dr dθ = 2π[−4r 2 ] 0 = −32π
2π
D
2
2
D
6
⃗ (r, θ) = (r cos θ, r sin θ, r).
c. The cone, C, may be parametrize as R
Compute the surface integral:
⃗ , check orientation, F
⃗ R
⃗ (r, θ) ,
Successively find: ⃗e r , ⃗e θ , N
∫∫ F⃗ ⋅ dS⃗.
P
î
k̂
̂
⃗e r = (cos θ,
sin θ, 1)
⃗e θ = (−r sin θ, r cos θ, 0)
⃗ = ⃗e θ × ⃗e z = î(−r cos θ) − ̂ (r sin θ) + k̂ (r cos 2 θ + r sin 2 θ) = (−r cos θ, −r sin θ, r)
N
⃗ needs to be oriented down. So reverse N
⃗:
N
⃗ = (r cos θ, r sin θ, −r)
N
⃗ R
⃗ (r, θ)
F
= (xz, yz, −2z 2 ) = (r 2 cos θ, r 2 sin θ, −2r 2 )
⃗⋅N
⃗ = r 3 cos 2 θ + r 3 sin 2 θ + 2r 3 = 3r 3
F
∫∫ F⃗ ⋅ dS⃗ = ∫∫ F⃗ ⋅ N⃗ dr dθ = ∫ 0 ∫ 0 3r 3 dr dθ = 2π
2π
P
d. Combine
2
P
∫∫ F⃗ ⋅ dS⃗
and
∫∫ F⃗ ⋅ dS⃗
D
P
to get
3r 4
4
2
0
= 24π
∫∫ F⃗ ⋅ dS⃗.
∂V
∫∫ F⃗ ⋅ dS⃗ = ∫∫ F⃗ ⋅ dS⃗ + ∫∫ F⃗ ⋅ dS⃗ = −32π + 24π = −8π
∂V
D
P
which agrees with part (a).
7
13. (6 points) Select the Project that you worked on and then answer the questions in 1 or 2 sentences.
Gauss’ Law and Ampere’s Law
⃗ = 0.
⃗ ⋅E
One of the electric fields produced a charge density of zero: ρ c = 1 ∇
4π
Was there a charge and how did you know? Why was Gauss’ Theorem not violated?
Interpretation of Divergence and Curl
The divergence can be defined using either derivatives or a limit of an integral.
Which Theorem was used to prove their equivalence.
Skimpy Donut
For the minimal donut, what was the relation between a and b?
For the maximal donut, what was the value of b?
Volume Between a Surface and Its Tangent Plane
When minimizing over a square, which tangent point (a, b) minimizes the volume?
Hypervolume of a Hypersphere
The volume enclosed by a sphere of radius R in ℝ n is V n = kπ p R q .
What are the values of p and q when n = 4? What are the values of p and q when n = 5?
Average Temperatures
What was the shape of the probe used to measure the temperature of the water in the pot?
Which Maple command did you use when Maple was unable to compute the integrals?
Center of Mass of Planet X
In computing the mass of the water, how did you ensure you only integrated where the land level
was below sea level.
Which Maple command did you use when Maple was unable to compute the integrals?
8