Test Bank for Thomas' Calculus Early
Transcendentals 15th Edition By Joel
Hass, Christopher Heil, Maurice Weir,
Przemyslaw Bogacki (All Chapters
1-16, 100% Original Verified, A+
Grade)
All Chapters Arranged Reverse:
Chapter 16-1
No Test Questions for Chapter 17, 18
& 19.
This is the Original Test Bank for 15th
Edition, All Other Files in the Market
are Wrong/Old Questions.
All Chapters Test Bank Download Link
at the end of this PDF file.
,Exam
Chapter 16
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the
question.
Match the vector equation with the correct graph.
1) r(t) = (3 - 2t)i + tj; 0 t 3 1)
2
A) Figure 3 B) Figure 1 C) Figure 5 D) Figure 8
2) r(t) = 3i - 2j - tk; -1 t 1 2)
A) Figure 3 B) Figure 6 C) Figure 5 D) Figure 8
3) r(t) = 3 ti +(2 - t)k; 0 t 2 3)
2
A) Figure 3 B) Figure 5 C) Figure 8 D) Figure 1
1
, 4) r(t) = (1 - t2)j + 3tk; -1 t 1 4)
A) Figure 4 B) Figure 7 C) Figure 8 D) Figure 2
5) r(t) = sin tj - cos tk; 0 t 5)
2
A) Figure 7 B) Figure 1 C) Figure 4 D) Figure 2
6) r(t) = tj; -2 t 2 6)
A) Figure 1 B) Figure 5 C) Figure 6 D) Figure 3
7) r(t) = -3ti + 2tj + 2tk; 0 t 1 7)
A) Figure 1 B) Figure 3 C) Figure 8 D) Figure 5
8) r(t) = 2 cos ti + sin tk; 0 t 8)
A) Figure 7 B) Figure 2 C) Figure 6 D) Figure 4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the
question.
Solve the problem.
9) Suppose that the parametrized plane curve C: (f(u), g(u)) is revolved about the x-axis, 9)
where g(u) > 0 and a u b. Show that the surface area of the surface of revolution is 2
b
g(u) g (u) 2 + f (u) 2 du
a
10) In thermodynamics, the differential form of the internal energy of a system is dU = T dS 10)
- P dV, where U is the internal energy, T is the temperature, S is the entropy, P is the
pressure, and V is the volume of the system. The First Law of Thermodynamics asserts
that dU is an exact differential. Using this information, justify the thermodynamic relation
T P
=- .
V S
f f
11) Assuming all the necessary derivatives exist, show that if dx - dy = 0 for all 11)
y x
C
closed curves C to which Green's Theorem applies, then f satisfies the Laplace equation
2f 2f
+ = 0 for all regions bounded by closed curves C to which Green's Theorem
x2 y2
applies.
2 2 2
12) Find a parametrization for the ellipsoid x + y + z = 1. (Recall that the 12)
100 36 64
x2 y2
parametrization of an ellipse + = 1 is x = 10 cos , y = 6 sin , 0 < 2 ).
100 36
2
, y -x
13) Let M = and N = . Show that 13)
2
x +y 2 x + y2
2
N M
M dx + N dy - dx dy , where R is the region bounded by the unit
x y
C R
circle C centered at the origin. Why is Green's Theorem failing in this case?
14) Consider a small region inside an elastic material such as gelatin. As the material "jiggles", 14)
this small region oscillates about its equilibrium position (x0 , y0 , z0 ). The force that tends
to restore the small region to its equilibrium position can be approximated as F = - k(x -
x0 )i - k(y - y0 )j - k(z - z 0 )k Find a potential function f for this force field.
15) For the surface z = f(x,y), show that the surface integral g(x,y,z) d = 15)
g(x, y, f(x,y)) fx(x,y) 2 + fy(x,y) 2 + 1 dxdy.
16) For some inexact differential forms df, a function g(x, y, z) can be found such that dh = 16)
g(x, y, z) df is exact. When it exists, the function g(x,y,z) is called an "integrating factor".
yz 1
Show that g(x, y, z) = is an integrating factor for the inexact differential df = - dx +
x x
1 1
dy + dz.
y z
Parametrize the surface S.
2 2 2
17) S is the portion of the cone x + y = z that lies between z = 4 and z = 9. 17)
16 16 36
Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of
points on the circle x 2 + y2 = 4.
18) F = -xi + yj 18)
Find the equation for the plane tangent to the parametrized surface S at the point P.
19) S is the cylinder r( , z) = 6 cos2 i + 3 sin 2 j + zk; P is the point corresponding to 19)
( , z) = , -6 .
4
Solve the problem.
20) For a surface parametrized in the parameters u and v and a force F, show that F·n d = 20)
F· ru×rv dudv.
21) Imagine a force field in which the force is always parallel to dr. What is special about the 21)
work done in moving a particle in such a field?
3
Transcendentals 15th Edition By Joel
Hass, Christopher Heil, Maurice Weir,
Przemyslaw Bogacki (All Chapters
1-16, 100% Original Verified, A+
Grade)
All Chapters Arranged Reverse:
Chapter 16-1
No Test Questions for Chapter 17, 18
& 19.
This is the Original Test Bank for 15th
Edition, All Other Files in the Market
are Wrong/Old Questions.
All Chapters Test Bank Download Link
at the end of this PDF file.
,Exam
Chapter 16
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the
question.
Match the vector equation with the correct graph.
1) r(t) = (3 - 2t)i + tj; 0 t 3 1)
2
A) Figure 3 B) Figure 1 C) Figure 5 D) Figure 8
2) r(t) = 3i - 2j - tk; -1 t 1 2)
A) Figure 3 B) Figure 6 C) Figure 5 D) Figure 8
3) r(t) = 3 ti +(2 - t)k; 0 t 2 3)
2
A) Figure 3 B) Figure 5 C) Figure 8 D) Figure 1
1
, 4) r(t) = (1 - t2)j + 3tk; -1 t 1 4)
A) Figure 4 B) Figure 7 C) Figure 8 D) Figure 2
5) r(t) = sin tj - cos tk; 0 t 5)
2
A) Figure 7 B) Figure 1 C) Figure 4 D) Figure 2
6) r(t) = tj; -2 t 2 6)
A) Figure 1 B) Figure 5 C) Figure 6 D) Figure 3
7) r(t) = -3ti + 2tj + 2tk; 0 t 1 7)
A) Figure 1 B) Figure 3 C) Figure 8 D) Figure 5
8) r(t) = 2 cos ti + sin tk; 0 t 8)
A) Figure 7 B) Figure 2 C) Figure 6 D) Figure 4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the
question.
Solve the problem.
9) Suppose that the parametrized plane curve C: (f(u), g(u)) is revolved about the x-axis, 9)
where g(u) > 0 and a u b. Show that the surface area of the surface of revolution is 2
b
g(u) g (u) 2 + f (u) 2 du
a
10) In thermodynamics, the differential form of the internal energy of a system is dU = T dS 10)
- P dV, where U is the internal energy, T is the temperature, S is the entropy, P is the
pressure, and V is the volume of the system. The First Law of Thermodynamics asserts
that dU is an exact differential. Using this information, justify the thermodynamic relation
T P
=- .
V S
f f
11) Assuming all the necessary derivatives exist, show that if dx - dy = 0 for all 11)
y x
C
closed curves C to which Green's Theorem applies, then f satisfies the Laplace equation
2f 2f
+ = 0 for all regions bounded by closed curves C to which Green's Theorem
x2 y2
applies.
2 2 2
12) Find a parametrization for the ellipsoid x + y + z = 1. (Recall that the 12)
100 36 64
x2 y2
parametrization of an ellipse + = 1 is x = 10 cos , y = 6 sin , 0 < 2 ).
100 36
2
, y -x
13) Let M = and N = . Show that 13)
2
x +y 2 x + y2
2
N M
M dx + N dy - dx dy , where R is the region bounded by the unit
x y
C R
circle C centered at the origin. Why is Green's Theorem failing in this case?
14) Consider a small region inside an elastic material such as gelatin. As the material "jiggles", 14)
this small region oscillates about its equilibrium position (x0 , y0 , z0 ). The force that tends
to restore the small region to its equilibrium position can be approximated as F = - k(x -
x0 )i - k(y - y0 )j - k(z - z 0 )k Find a potential function f for this force field.
15) For the surface z = f(x,y), show that the surface integral g(x,y,z) d = 15)
g(x, y, f(x,y)) fx(x,y) 2 + fy(x,y) 2 + 1 dxdy.
16) For some inexact differential forms df, a function g(x, y, z) can be found such that dh = 16)
g(x, y, z) df is exact. When it exists, the function g(x,y,z) is called an "integrating factor".
yz 1
Show that g(x, y, z) = is an integrating factor for the inexact differential df = - dx +
x x
1 1
dy + dz.
y z
Parametrize the surface S.
2 2 2
17) S is the portion of the cone x + y = z that lies between z = 4 and z = 9. 17)
16 16 36
Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of
points on the circle x 2 + y2 = 4.
18) F = -xi + yj 18)
Find the equation for the plane tangent to the parametrized surface S at the point P.
19) S is the cylinder r( , z) = 6 cos2 i + 3 sin 2 j + zk; P is the point corresponding to 19)
( , z) = , -6 .
4
Solve the problem.
20) For a surface parametrized in the parameters u and v and a force F, show that F·n d = 20)
F· ru×rv dudv.
21) Imagine a force field in which the force is always parallel to dr. What is special about the 21)
work done in moving a particle in such a field?
3
