Final Exam Fall 2007 MATH 1120 CALCULUS II FOR ENGINEERS

U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
U A E University, College of Science
Department of Mathematical Sciences
Final Exam Fall 2007
MATH 1120 CALCULUS II FOR ENGINEERS
Student’s Name
Student’s I.D.
Section #
Circle the name of your instructor (with the time of your class)
Dr. Adama Diene - Section 51
Mr. Naim Markos- Section 01
Dr. Adama Diene - Section 52
Dr. Mohamed Hajji -Section 02
Mr. Naim Markos- Section 03
Allowed time is 2 hours.
Please be neat and show all work.
You can use the back of the sheets.
Return this entire booklet to your instructor.
NO BOOKS. NO NOTES. NO PROGRANMING CALCULATORS
Section I
Problem #
Points
Section II
Problem #
1-13
Points
Total
Points
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
Section I: Multiple choice problems [60 Points, 6 each]
(No Partial Credits for this Section)
1. A vector with magnitude 6 in the same direction as, v = i – 2j is
6
12
A)
6 i – 12 j
B)
i–
j
5
5
1
2
i–
j
D)
6i
C)
5
5
2. The graph of the difference a – b; where a = – i + 4 j; and b = –3 i + 3 j is
(in graphs the vectors are in bold black),
A)
B)
C)
D)
2-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
3. The velocity function for the position function r ( t ) = –8cos 5t , –8sin 5t
is
A)
v ( t ) = –40sin 5t , 40 cos 5t
B)
v ( t ) = 8sin 5t , –8cos 5t
C)
v ( t ) = –8sin 5t ,8cos 5t
D)
v ( t ) = 40sin 5t , –40 cos 5t
4. The distance between the parallel planes 2 x − 3 y + z = 6 and 4 x − 6 y + 2 z = 8
is
A) 2
B) 0
C)
2
14
2
7
D)
5. The graph of the plane y + 2z = 6, is
B)
A)
z
z
3
2
y
1
y
–1
6
6
x
x
D)
C)
z
z
y-z plane
1
1
y
–1
y
–1
2
2
x
x
3-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
r
6- The unit tangent vector to the curve r = cos 9t , 2t , sin 9t at t =
1
–9, 2,1
86
1
0, 2,9
85
A)
C)
FALL 2007
π
8
is
1
9, 2, 0
85
1
–9, 2, 0
85
B)
D)
7. The first-order partial derivatives of f ( x, y ) = 7 x3 + 5 x 2 y + 4 y 5 are
A)
f x = 21x 2 + 10 xy + 4 y 5 ; f y = 7 x3 + 5 x 2 + 20 y 4
B)
f x = 3 x 2 + 10 xy; f y = 5 x 2 + 5 y 4
C)
f x = 21x 2 + 10 xy; f y = 5 x 2 + 20 y 4
D)
f x = 21x 2 + 10 xy; f y = 5 x 2 + 20 y 4 ; f xy = 10 x
1 2y
8. Change the order of integration
∫ ∫ f ( x, y) dx dy
0 0
2y 1
A)
∫ ∫
2
f ( x, y ) dy dx
0 x/2
2 1
0 0
2 x/2
C)
∫ ∫ f ( x, y) dy dx
0
1
∫ ∫ f ( x, y) dy dx
B)
∫ ∫ f ( x, y ) dy dx
D)
x/2 0
0
9. Convert the equation x 4 y + y 4 z = 1 into spherical coordinates.
(
)
A)
ρ 5 sin 5 φ cos 4 θ sin θ + sin 4 θ cos θ = 1
B)
ρ 4 sin 4 φ ( cos θ + sin θ ) = 1
C)
ρ 5 sin 4 φ ( cos θ + sin θ ) = 1
D)
ρ 5 sin 5 φ cos 4 θ sin θ + sin 4 φ sin 4 θ cos φ = 1
(
)
10. A constant force of 30, 20 pounds moves an object in a straight line from the
point (0, 0) to the point (24, 10). The work done is
r
r
A) i + j ft.pounds
C)
920
ft.pounds
26
r
r
B) 2i + j ft.pounds
D) 920 ft.pounds
4-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
Section II: Multiple-Step problems [140 Points, 18 each]
ANSWER ONLY 8 QUESTIONS
1. A) Find the distance from the point Q = (1, 2, 0) to the line passing through (0, 1, 2)
and (3, 1, 1)
B) Find a symmetric equation of the line through the point (1, 5, 2) and parallel to the
vector 4, 3, 7 . Also, determine where the line intersects the yz-plane.
5-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
2. In the given figure, two ropes are attached to a 500-pound crate. Rope A exerts a
force of 20, − 130, 200 pounds on the crate, and rope B exerts a force of
− 10, 130, 300 pounds on the crate.
I) If no further ropes are added, find the net force on the crate and the direction it will
move.
ii) If a third rope C is added to balance the crate, what force must this rope exert on the
crate?
6-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
3. A) Show that there are no values of t such that r ( t ) and r ′ ( t ) are parallel,
where r ( t ) = t 2 − 6, t 2 , t . Show your work.
r
B) Find the arc length of curve described by r = t
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3/ 2
, (t − 1), t where 0 ≤ t ≤ 1
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
4. A) Compute the directional derivative of f ( x, y ) = 2 xy − 7 x at the point (1, 5) in the
2
direction of the vector
1 1
,
2 2
.
r
r
r r
r
r
a + b ≤ a + b for any two vectors a and b . Find the relationship
r
r
r r
r
r
that must exist between a and b to have a + b = a + b
B) It is known that
8-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
5. A) Find the volume inside the paraboloid z = 9 − x − y , outside the cylinder
2
2
x 2 + y 2 = 4 and above the xy-plane.
B) Decide if each of the following quantities is a vector, a scalar, or undefined ( write
your answer over the dots)
rr
v .w
a. r r
v×w
r r
u×w
b.
rr
ru .wr r
c. u × (v ⋅ w)
r r r
d. (u × v ).w
..…………………..
…………………….
…………………….
…………………….
9-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
6. A rocket is launched with a constant thrust corresponding to an acceleration of
u ft / s 2 . Ignoring air resistance, the rocket’s height after t seconds is given by
1
f (u , t ) = (u − 32) t 2 feet . Fuel usage for t seconds is proportional to u 2 t and the
2
2
limited fuel capacity of the rocket satisfies the equation u t = 10,000 . Find the value
of u that maximizes the height that the rocket reaches when the fuel runs out.
10-13
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
7. Lamina bounded by y = x and y = x 2 with density ρ ( x, y ) = 4
i) Sketch the region
ii) Find the mass of the lamina
iii) Find, x , the x-coordinate of the center of mass.
3
11-13
FALL 2007
U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
8. A) Use appropriate coordinates to set up and (DON’T EVALUATE) the triple integral
x
∫∫∫ e
2
+ y2
dV , where Q is the region inside x 2 + y 2 = 4 and between z = 1 and z = 2
Q
in the first octant.
B) Draw a picture and describe a situation where the projection of a vector onto a line is
the vector itself.
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U A E U MATH 1120 CALCULUS II FOR ENGINEERS
FINAL EXAM
FALL 2007
9. A) Find rectangular coordinates for the point described by (8, π/4, π/3) in spherical
coordinates.
B) Identify the following objects in 3-space
2x − 3y = 6
i.
ii.
iii.
iv.
v
r
r
by r (t ) = 4 cos t i − 4 sin t j , 0 ≤ t < 2π .
r
r
r
v
r (t ) = (1 + 2t ) i − t j + (2 + 3t ) k
θ = 0 in spherical coordinates
C) Decide if each of the following quantities is a vector or scalar
i.
velocity
ii.
work done
iii.
volume
iv.
acceleration
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