vector calculus2

 

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Math 223

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It is not a good idea to rely exclusively on reading through old exam solutions as a
way to prepare for the final exam. In particular, this semester’s course director
may not have written any of the exams available from this page, so the ones he/she
gives will almost certainly have a somewhat different flavor.

Topic Warning:
Because the topics taught differ slightly from semester to semester, it is not a good
idea to use the old exams to gauge the content of the exams this semester.

Math 223 Show Your Work! Page 1 of 1

2

1. (30) Solve the following problems. No partial credit.

(a) If

#„u = 2
#„
i �

#„
j + 3

#„
k , and #„v = a

#„
i + 2

#„
j � 4

#„
k (a is a constant), then #„u · #„v is equal to

A. a � 14.
B. 2(a � 7)
C. 2a � 1

0

D. a + 10

E. 7

(b) Find a normal vector for the plane 7y = z.

(c) Find

@f

@y
, where f(x, y, z) = y2exyz. Simplify your answer as much as possible, factoring

where possible.

Math 223 Show Your Work! Page 2 of 12

(d) If S is the plane x = 0 oriented in the positive x direction, then the surface area

element

# „
dA is which one of the following quantities?

A.

#„
j dxdz.

B.

#„
k dydz

C. �

#„
i dydz

D.

#„
i dydz

E. �
#„
k dydx

(e) The figure below shows the contour diagram of which function f(x, y)?

A. f(x, y) = 6y � 3x + 6
B. f(x, y) = 1�x

2

C. f(x, y) = e�3x�6y+6

D. f(x, y) = �3x � 6y
E. None of the above.

Math 223 Show Your Work! Page 3 of 12

(f) Let R be the two dimensional region shown in the figure below. What is
R
R

f(x, y)dA?

A.

Z 3

0

Z 2y/3

0

f(x, y)dxdy

B.

Z 3
0

Z

3x/2

0

f(x, y)dxdy

C.

Z 3
0
Z 2y/3
0

f(x, y)dydx D.

Z 2

0
Z 3x/2
0

f(x, y)dydx

E.

Z 3
0
Z 2
3x/2
f(x, y)dxdy

Math 223 Show Your Work! Page 4 of 12

2. (20) Consider the integral

Z 0

�1

Z

y

2
0

ex/y
2
dxdy.

(a) Interchange the order of integration. Show your work, including a sketch of the region

of integration.

(b) Evaluate the original integral. Give an exact answer.

Math 223 Show Your Work! Page 5 of 12

3. (30) Let

#„
F = y

#„
i + 2z

#„
j + (1 � z)

#„
k . Evaluate the following:

(a)

Z

C

#„
F ·

#„
dr where C is the straight line from the origin to (1, 3, 1).

(b)

Z

S

#„
F ·

# „
dA where S is the rectangle 0  x  2, 1  y  4, z = 0, oriented upwards.

(c)

Z
S
#„
F ·

# „
dA where S is the sphere of radius 2 centered at the origin, oriented outwards.

Math 223 Show Your Work! Page 6 of 12

4. (20) Let f(x, y) =
p
1 + 4x + y2, and let P be the point (1, 2).

(a) At P , what is the direction of maximal increase for the function f? Give your answer
as a unit vector.

(b) Find the directional derivative of f at P in the direction of 3
#„
i � 4

#„
j .

Math 223 Show Your Work! Page 7 of 12

5. (15) Let H(x, y, z) = x2 + y2 + 2z2, and let S be the level surface H(x, y, z) = 4. Find
the coordinates of a point P on the surface S where the tangent plane to S is parallel to the
plane 2x + 4z = 0.

Math 223 Show Your Work! Page 8 of 12

6. (20) Suppose S is the surface obtained by taking the union of the upper hemisphere of a
sphere of radius 2 centered at (0, 0, 4),

S1 =
�
(x, y, z) such that x2 + y2 + (z � 4)2 = 4 and z � 4

and an open cylinder of radius 2 centered around the z axis,

S2 =
�
(x, y, z) such that 0  z  4 and x2 + y2 = 4

.

The orientation of S is away from the origin.

a.) Sketch the surface S.

b.) Evaluate the integral

Z
S

⇣
curl(

#„
F )

⌘
·

# „
dA, if

#„
F is the vector field

#„
F = y

#„
i � x

#„
j + xy

#„
k .

Hint: It is strongly recommended to use Stokes’ theorem to simplify the
surface integral.

Math 223 Show Your Work! Page 9 of 12

7. (20) Consider the cubic polynomial f(x, y) = 5
2
x2 � xy + 15x + 1

75
y3 � 3y.

(a) Find the critical point(s) of f(x, y).

(b) Use the second derivative test to classify, if possible, the critical point(s) you have

found.

Math 223 Show Your Work! Page 10 of 12

8. (15) Let S be the paraboloid x2 + y2 + z = R2, 0 < z  R2, oriented upward, and let #„ F = x

#„
i + y

#„
j + z2

#„
k . Find the flux of the vector field

#„
F through the surface S.

Math 223 Show Your Work! Page 11 of 12

9. (20) Consider the 2-dimensional force field

#„
F = (4e�2x + 3y3)

#„
i + 9xy2

#„
j .

a) Is

#„
F conservative? If so, find a potential function f(x, y) whose gradient is

#„
F .

b) Find the work done by the force field

#„
F in moving an object from P(0, 1) to Q(1, 2)

along the path y = 1 + sin(⇡x/2) from x = 0 to x = 1.

Math 223 Show Your Work! Page 12 of 12

10. (10) An asteroid is a cylindrical mass of ice, 100 km tall and with radius 5 km. The

density of the asteroid varies linearly along its long dimension, varying from zero at one end

to 10kg/m3 at the other. Set up a triple integral representing the total mass of the asteroid.

• Math 223
• Spring12_223