MA 261 PRACTICE PROBLEMS 1. If the line ` has symmetric equations

| August 30, 2017

Question
MA 261 PRACTICE PROBLEMS
1. If the line ` has symmetric equations
y
x¡1
z+2
2 = ¡3 = 7 ,
¯nd a vector equation for the line `0 that contains the point (2; 1; ¡3) and is parallel to `.
A. ~ = (1 + 2t)~ ¡ 3t~ + (¡2 + 7t)~
r
i
j
k
~ + (1 ¡ 3t)~ + (¡3 + 7t)~
C. ~ = (2 + 2t)i
r
j
k
k
E. ~ = (2 + t)~ + ~ + (7 ¡ 3t)~
r
i j

B. ~ = (2 + t)~ ¡ 3~ + (7 ¡ 2t)~
r
i
j
k
~ + (¡3 + t)~ + (7 ¡ 3t)~
D. ~ = (2 + 2t)i
r
j
k

2. Find parametric equations of the line containing the points (1; ¡1; 0) and (¡2; 3; 5).
A. x = 1 ¡ 3t; y = ¡1 + 4t; z = 5t
C. x = 1 ¡ 2t; y = ¡1 + 3t; z = 5t
E. x = ¡1 + t; y = 2 ¡ t; z = 5

B. x = t; y = ¡t; z = 0
D. x = ¡2t; y = 3t; z = 5t

3. Find an equation of the plane that contains the point (1; ¡1; ¡1) and has normal vector
1~
~
~
2 i + 2j + 3k.
A. x ¡ y ¡ z +

9
2

=0

D. x ¡ y ¡ z = 0

B. x + 4y + 6z + 9 = 0
E.

1
2

C.

x¡1
1
2

=

y+1
2

=

z+1
3

x + 2y + 3z = 1

4. Find an equation of the plane that contains the points (1; 0; ¡1), (¡5; 3; 2), and (2; ¡1;

4).
A. 6x ¡ 11y + z = 5
D. ~ = 18~ ¡ 33~ + 3~
r
i
j
k

B. 6x + 11y + z = 5
E. x ¡ 6y ¡ 11z = 12

C. 11x ¡ 6y + z = 0

j
k
5. Find parametric equations of the line tangent to the curve ~(t) = t~ + t2~ + t3~ at the point (2; 4; 8)
r
i
A. x = 2 + t; y = 4 + 4t; z = 8 + 12t
B. x = 1 + 2t; y = 4 + 4t; z = 12 + 8t
C. x = 2t; y = 4t; z = 8t
D. x = t; y = 4t; z = 12t
E. x = 2 + t; y = 4 + 2t; z = 8 + 3t

6. The position function of an object is
~(t) = cos t~ + 3 sin t~ ¡ t2~
r
i
j
k
Find the velocity, acceleration, and speed of the object when t = ¼.
Velocity
Acceleration p Speed
4
A. ¡~ ¡ ¼ 2~
i
1
k
¡3~ ¡ 2¼~
j
k
p+¼
B. ~ ¡ 3~ + 2¼~
i
j
k
¡~ ¡ 2~ p 10 + 4¼ 2
i
k
2
C. 3~ ¡ 2¼~
j
k
¡~ ¡ 2~
i
k
p9 + 4¼
~ ¡ 2~
D. ¡3~ ¡ 2¼~
j
k
i
k
9 + 4¼ 2
p
E. ~ ¡ 2~
i
k
¡3~ ¡ 2¼~
j
k
5

1

7. A smooth parametrization of the semicircle which passes through the points (1; 0; 5), (0; 1; 5) and
(¡1; 0; 5) is
A. ~(t) = sin t~ + cos t~ + 5~ 0 · t · ¼
r
i
j
k;
~ + sin t~ + 5~ ¼ · t · 3¼
C. ~(t) = cos ti
r
j
k; 2
2
E. ~(t) = sin t + cos t~ + 5~ ¼ · t · 3¼
r
j
k; 2
2

B. ~(t) = cos t~ + sin t~ + 5~ 0 · t · ¼
r
i
j
k;
~ + sin t~ + 5~ 0 · t ·
D. ~(t) = cos ti
r
j
k;

3
3
i 3
j
k,
8. The length of the curve ~(t) = 2 (1 + t) 2~ + 2 (1 ¡ t) 2 ~ + t~ ¡1 · t · 1 is
r
3
p
p
p
p
A. 3
B. 2
C. 1 3
D. 2 3
2

9. The level curves of the function f (x; y) =
A. circles

B. lines

¼
2

E.

p
2

p
1 ¡ x2 ¡ 2y 2 are

C. parabolas

D. hyperbolas

E. ellipses

10. The level surface of the function f (x; y; z) = z ¡x2 ¡y 2 that passes through the point (1; 2; ¡3) intersects
the (x; z)-plane (y = 0) along the curve
A. z = x2 + 8
B. z = x2 ¡ 8
E. does not intersect the (x; z)-plane

C. z = x2 + 5

D. z = ¡x2 ¡ 8

11. Match the graphs of the equations with their names:
(1) x2 + y 2 + z 2 = 4
(a) paraboloid
(2) x2 + z 2 = 4
(b) sphere
(3) x2 + y 2 = z 2
(c) cylinder
(4) x2 + y 2 = z
(d) double cone
(5) x2 + 2y 2 + 3z 2 = 1
(e) ellipsoid
A. 1b, 2c, 3d, 4a, 5e
D. 1b, 2d, 3a, 4c, 5e

B. 1b, 2c, 3a, 4d, 5e
E. 1d, 2a, 3b, 4e, 5c

C. 1e, 2c, 3d, 4a, 5b

12. Suppose that w = u2 =v where u = g1 (t) and v = g2 (t) are di®erentiable functions of t. If g1 (1) = 3,
0
0
g2 (1) = 2, g1 (1) = 5 and g2 (1) = ¡4, ¯nd dw when t = 1.
dt
A. 6

13. If w = euv and u = r + s, v = rs, ¯nd
A. e(r+s)rs (2rs + r2 )
D. e(r+s)rs (1 + s)

C. ¡24

B. 33=2

D. 33

E. 24

@w
@r .

B. e(r+s)rs (2rs + s2 )
E. e(r+s)rs (r + s2 ).

2

C. e(r+s)rs (2rs + r2 )

14. If f (x; y) = cos(xy),

@2f
@x@y

=

A. ¡xy cos(xy)
D. xy cos(xy) + sin(xy)

B. ¡xy cos(xy) ¡ sin(xy)
E. ¡ cos(xy)

C. ¡ sin(xy)

@z
@x .

15. Assuming that the equation xy 2 + 3z = cos(z 2 ) de¯nes z implicitly as a function of x and y, ¯nd
A.

y2
3¡sin(z 2 )

B.

¡y 2
3+sin(z 2 )

C.

y2
3+2z sin(z 2 )

16. If f (x; y) = xy 2 , then rf (2; 3) =
A. 12~ + 9~
i
j

B. 18~ + 18~
i
j

D.

¡y2
3+2z sin(z 2 )

C. 9~ + 12~
i
j

E.

¡y 2
3¡2z sin(z 2 )

D. 21

E.

17. Find the directional derivative of f (x; y) = 5 ¡ 4×2 ¡ 3y at (x; y) towards the origin
A. ¡8x ¡ 3

2

p
B. ¡8×2 3y
2
x +y

C.

¡8x¡3
p
64×2 +9

D. 8×2 + 3y

p
2.

2

8x
E. p 2+3y2 .
x +y

18. For the function f (x; y) = x2 y, ¯nd a unit vector ~ for which the directional derivative D~ f (2; 3) is
u
u
zero.
A. ~ + 3~
i
j

B.

i+3~
p j
10

C. ~ ¡ 3~
i
j

D.

i¡3~
p j
10

E.

3~ ~

p j.
10

19. Find a vector pointing in the direction in which f (x; y; z) = 3xy ¡ 9xz 2 + y increases most rapidly at
the point (1; 1; 0).
A. 3~ + 4~
i
j

B. ~ + ~
i j

C. 4~ ¡ 3~
i
j

D. 2~ + ~
i k

E. ¡~ + ~
i j.

20. Find a vector that is normal to the graph of the equation 2 cos(¼xy) = 1 at the point ( 1 ; 2).
6
p
A. 6~ + ~
i j
B. ¡ 3~ ¡ ~
i j
C. 12~ + ~
i j
D. ~
j
E. 12~ ¡ ~
i j.

21. Find an equation of the tangent plane to the surface x2 + 2y 2 + 3z 2 = 6 at the point (1; 1; ¡1).
A. ¡x + 2y + 3z = 2
D. 2x + 4y ¡ 6z = 0

B. 2x + 4y ¡ 6z = 6
E. x + 2y ¡ 3z = 6.

C. x ¡ 2y + 3z = ¡4

22. Find an equation of the plane tangent to the graph of f (x; y) = ¼ + sin(¼x2 + 2y) when (x; y) = (2; ¼).
A. 4¼x + 2y ¡ z = 9¼
D. 4x + 2¼y ¡ z = 9¼

B. 4x + 2¼y ¡ z = 10¼
E. 4¼x + 2y + z = 9¼.

3

C. 4¼x + 2¼y + z = 10¼

23. The di®erential df of the function f (x; y; z) = xey
A.
B.
C.
D.
E.

df
df
df
df
df

2

2

2

2

2

2

¡z 2

is

2

= xey ¡z dx + xey ¡z dy + xey ¡z dz
2
2
= xey ¡z dx dy dz
2
2
2
2
2
2
= ey ¡z dx ¡ 2xyey ¡z dy + 2xzey ¡z dz
2
2
2
2
2
2
= ey ¡z dx + 2xyey ¡z dy ¡ 2xzey ¡z dz
2
2
= ey ¡z (1 + 2xy ¡ 2xz)

24. The function f (x; y) = 2×3 ¡ 6xy ¡ 3y 2 has
A. a relative minimum and a saddle point
C. a relative minimum and a relative maximum
E. two relative minima.

B. a relative maximum and a saddle point
D. two saddle points

25. Consider the problem of ¯nding the minimum value of the function f (x; y) = 4×2 + y 2 on the curve
xy = 1. In using the method of Lagrange multipliers, the value of ¸ (even though it is not needed) will
be
p
1
A. 2
B. ¡2
C. 2
D. p2
E. 4.
26. Evaluate the iterated integral
A. ¡ 8
9

B. 2

R3Rx
1

0

1
x

dydx.
C. ln 3

D. 0

E. ln 2.

RR
27. Consider the double integral, R f (x; y)dA, where R is the portion of the disk x2 + y 2 · 1, in the upper
half-plane, y ¸ 0. Express the integral as an iterated integral.
R 1 R p1¡x2
R 0 R p1¡x2
A. ¡1 ¡p1¡x2 f (x; y)dydx
B. ¡1 0
f (x; y)dydx
R 1 R p1¡x2
R 1 R p1¡x2
C. ¡1 0
f (x; y)dydx
D. 0 ¡p1¡x2 f (x; y)dydx
p
R 1 R 1¡x2
E. 0 0
f (x; y)dydx.
28. Find a and b for the correct interchange of order of integration:
R 2 R 2x
R4Rb
f (x; y)dydx = 0 a f (x; y)dxdy.
0 x2
p
A. a = y 2 ; b = 2y
B. a = y ; b = y
2
p
E. cannot be done without explicit knowledge of f (x; y).
D. a = y; b = y
2

C. a = y ; b = y
2

RR
29. Evaluate the double integral R ydA, where R is the region of the (x; y)-plane inside the triangle with
vertices (0; 0), (2; 0) and (2; 1).
A. 2

B.

8
3

C.

2
3

D. 1

E.

1
3.

30. The volume of the solid region in the ¯rst octant bounded above by the parabolic sheet z = 1 ¡ x2 ,
below by the xy plane, and on the sides by the planes y = 0 and y = x is given by the double integral
R 1 R 1¡x2
R1 Rx
R1Rx
B. 0 0
x dydx
C. ¡1 ¡x (1 ¡ x2 )dydx
A. 0 0 (1 ¡ x2 )dydx
R 1 R 1¡x2
R1R0
E. 0 x
dydx.
D. 0 x (1 ¡ x2 )dydx

4

31. The area of one leaf of the three-leaved rose bounded by the graph of r = 5 sin 3µ is
A.


6

B.

25¼
12

C.

25¼
6


3

D.

E.

32. Find the area of the portion of the plane x + 3y + 2z = 6 that lies in the ¯rst octant.
p
p
p
p
A. 3 11
B. 6 7
C. 6 14
D. 3 14

25¼
3 .

p
E. 6 11.

33. A solid region in the ¯rst octant is bounded by the surfaces z = y 2 , y = x, y = 0, z = 0 and x = 4. The
volume of the region is
A. 64

B.

64
3

C.

32
3

D. 32

E.

16
3 .

34. An object occupies the region bounded above by the sphere x2 + y 2 + z 2 = 32 and below by the upper
nappe of the cone z 2 = x2 + y 2 . The mass density at any point of the object is equal to its distance
from the xy plane. Set up a triple integral in rectangular coordinates for the total mass m of the object.
R 4 R p16¡x2 R p32¡x2 ¡y2
R 4 R p16¡x2 R p32¡x2 ¡y2
p
p 2 2 z dz dy dx
B. ¡4 ¡p16¡x2 p 2 2
z dz dy dx
A. ¡4 ¡ 16¡x2
¡

C.
E.

x +y

x +y

R 2 R p4¡x2 R p32¡x2 ¡y2
p
p
z dz dy dx
¡2 ¡ 4¡x2 ¡ x2 +y 2
p
p
R 4 R 16¡x2 R 32¡x2 ¡y2
p
p 2 2
xy dz dy dx.
¡4 ¡ 16¡x2
x +y

D.

B.

35. Do problem 34 in spherical coordinates.
R 2¼ R ¼ R p32
A. 0 04 0 ½3 cos ‘ sin ‘ d½ d’ dµ
R 2¼ R ¼ R p32
C. 0 04 0 ½3 sin2 ‘ d½ d’ dµ
R 2¼ R ¼ R p32
E. 0 04 0 ½ cos ‘ d½ d’ dµ.

R 4 R p16¡x2
0 0

R 2¼ R

D.

0

¼
4

R p32¡x2 ¡y2
p 2 2
z dz dy dx

R p32

0
R 2¼ R ¼ R p32
2
0
0
0
0

x +y

½ cos ‘ sin ‘ d½ d’ dµ

½3 cos ‘ sin ‘ d½ d’ dµ

R 1 R p1¡x2 2 2
36. The double integral 0 0
y (x + y 2 )3 dydx when converted to polar coordinates becomes
R¼R1
R ¼ R1
R¼R1
A. 0 0 r9 sin2 µ dr dµ
B. 02 0 r8 sin2 µ dr dµ
C. 0 0 r8 sin µ dr dµ
R ¼ R1
R ¼ R1
D. 02 0 r8 sin µ dr dµ
E. 02 0 r9 sin2 µ dr dµ.
37. Which of the triple integrals converts
R 2 R p4¡x2 Rp
2
p
dz dy dx
¡2 ¡ 4¡x2
x2 +y2
from rectangular to cylindrical coordinates?
R¼R2R2
R 2¼ R 2 R 2
A. 0 0 r r dz dr dµ
B. 0 0 r r dz dr dµ
R¼R2R2
R 2¼ R 2 R 2
D. 0 0 r r dz dr dµ
E. 0 2 ¡2 r r dz dr dµ.
38. If Dp the solid region above p xy-plane that is between z =
is
the
RRR
z = 1 ¡ x2 ¡ y 2 , then
x2 + y 2 + z 2 dV =
D
A.

14¼
3

B.

16¼
3

C.

5

15¼
2

C.

R 2¼ R 2 R 2
0

¡2 r

r dz dr dµ

p
4 ¡ x2 ¡ y 2 and
D. 8¼

E. 15¼.

~
39. Determine which of the vector ¯elds below are conservative, i. e. F = grad f for some function f .
~
1. F (x; y) = (xy 2 + x)~ + (x2 y ¡ y 2 )~
i
j.
~ (x; y) = x ~ + y ~
2. F
y i
x j.
~ (x; y; z) = yez ~ + (xez + ey )~ + (xy + 1)ez ~
i
3. F
j
k.
A. 1 and 2

B. 1 and 3

C. 2 and 3

D. 1 only

E. all three

~
40. Let F be any vector ¯eld whose components have continuous partial derivatives up to second order,
let f be any real valued function with continuous partial derivatives up to second order, and let r =
~ @ + ~ @ + ~ @ . Find the incorrect statement.
i @x j @y k @z
~
B. div(curl F ) = 0

A. curl(grad f ) = ~
0
~
~
D. curl F = r £ F

~
C. grad(div F ) = 0

~
~
E. div F = r ¢ F

41. A wire lies on the xy-plane along the curve y = x2 , 0 · x · 2. The mass density (per unit length) at
any point (x; y) of the wire is equal to x. The mass of the wire is
p
p
p
A. (17 17 ¡ 1)=12
B. (17 17 ¡ 1)=8
C. 17 17 ¡ 1
p
p
D. ( 17 ¡ 1)=3
E. ( 17 ¡ 1)=12
R
~ r
~
j
i
42. Evaluate C F ¢ d~ where F (x; y) = y~ + x2~ and C is composed of the line segments from (0; 0) to (1; 0)
and from (1; 0) to (1; 2).
A. 0

B.

2
3

43. Evaluate the line integral

5
6

C.

Z

D. 2

E. 3

D. ¡ 1
3

E. 0

x dx + y dy + xy dz

C

where C is parametrized by ~(t) = cos t~ + sin t~ + cos t~ for ¡ ¼ · t · 0.
r
i
j
k
2
A. 1

44. Are
1.
2.
3.

B. ¡1

C.

1
3

the following statements true or false?
R
The line integral C (x3 + 2xy)dx + (x2 ¡ y 2 )dy is independent of path in the xy-plane.
R
(x3 + 2xy)dx + (x2 ¡ y 2 )dy = 0 for every closed oriented curve C in the xy-plane.
C
There is a function f (x; y) de¯ned in the xy-plane, such that
grad f (x; y) = (x3 + 2xy)~ + (x2 ¡ y 2 )~
i
j.

A. all three are false
B. 1 and 2 are false, 3 is true
D. 1 is true, 2 and 3 are false
E. all three are true

C. 1 and 2 are true, 3 is false

R
p
45. Evaluate C y 2 dx + 6xy dy where C is the boundary curve of the region bounded by y = x, y = 0 and
x = 4, in the counterclockwise direction.
A. 0

B. 4

C. 8

6

D. 16

E. 32

p
46. If C goes along the x-axis from (0; 0) to (1; 0), then along y = 1 ¡ x2 to (0; 1), and then back to (0; 0)
R
along the y-axis, then C xy dy =
R 1 R p1¡x2
R 1 R p1¡x2
R 1 R p1¡x2
A. ¡ 0 0
y dy dx
B. 0 0
y dy dx
C. ¡ 0 0
x dy dx
R 1 R p1¡x2
x dy dx
E. 0
D. 0 0

R
~ r
~
i
j
47. Evaluate C F ¢ d~, if F (x; y) = (xy 2 ¡ 1)~ + (x2 y ¡ x)~ and C is the circle of radius 1 centered at (1; 2)
and oriented counterclockwise.
A. 2

B. ¼

D. ¡¼

C. 0

E. ¡2

48. Green’s theorem yields the following formula for the area of a simpleRR
region R in terms of a line integral
over the boundary C of R, oriented counterclockwise. Area of R = R dA =
R
R
R
R
R
A. ¡ C y dx
B. C y dx
C. C x dx
D. 1 C y dx ¡ x dy
E. ¡ x dy
2
RR
49. Evaluate the surface integral § x dS where § is the part of the plane 2x + y + z = 4 in the ¯rst octant.
p
p
p
p
p
14
10
A. 8 6
B. 8 6
C. 8 14
D. 3
E. 3
3
3
50. If § is the part of the paraboloid z = x2 + RR with z · 4, ~ is the unit normal vector on § directed
y2
n
~ (x; y; z) = x~ + y~ + z~ then
~ n
upward, and F
i
j
k,
F ¢ ~ dS =
§
A. 0

B. 8¼

C. 4¼

D. ¡4¼

E. ¡8¼

~
51. If F (x; y; z) = cos z~ + sin z~ + xy~ § is the complete boundary of the rectangular solid region bounded
i
j
k,
by the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = ¼ , and ~ is the outward unit normal on §,
n
2
RR
~ n
then § F ¢ ~ dS =
A. 0

1
2

B.

C. 1

D.

¼
2

E. 2

~
n
52. If F (x; y; z) = x~ + y~ + z~ § is the unit sphere x2 + y 2 + z 2 = 1 and ~ is the outward unit normal on
i
j
k,
RR
~ n
§, then § F ¢ ~ dS =
A. ¡4¼

B.


3

C. 0
RR

53. Use Stoke’s theorem to evaluate

S

D.


3

E. 4¼

4
3

E. 2¼

~ ~
curlF ¢ dS , where

~
i
j
F (x; y; z) = x2 eyz~ + y 2 exz~ + z 2 exy~
k;
and S is the hemisphere x2 + y 2 + z 2 = 4, z ¸ 0, oriented upward.
A. ¡¼=3

B. 2¼

54. Use Stoke’s theorem to evaluate

C. 0
R

C

D.

~ r
F ¢ d~, where
~
j
i
k;
F (x; y; z) = x2 z~ + xy 2~ + z 2~

and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y 2 = 9 oriented
counterclockwise as viewed from above.
A.

81¼
2

B.

¼
2

C. 1

7

D.


8

E. 9¼

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