Document 6534162

Department of Mathematical Sciences, UAEU
Spring 2008
Midterm Exam
MATH 1120 Calculus II for Engineers
U A E University, College of Science
Department of Mathematical Sciences
Midterm Exam Spring 2008
MATH 1120 CALCULUS II FOR ENGINEERS
Student’s Name
Student’s I.D.
Section #
Check the Name of Your Instructor
Dr. Nabila Azam - Section 51
Dr. Moh. El Bachraoui - Section 01
Dr. Nabila Azam - Section 52
Dr. Adama Diene -Section 02
Dr. John Abraham - Section 53
Dr. Jae Lee - Section 03
Dr. John Abraham - Section 54
Dr. Adama Diene - Section 04
Allowed time is 1 hours.
You can use the back of the sheets.
NO BOOKS. NO NOTES. NO PROGRAMING CALCULATORS
Section I
Problem #
Points
Section II
Problem #
Section III
Points
1-6
Problem #
Points
Total
Points
Department of Mathematical Sciences, UAEU
Spring 2008
Midterm Exam
MATH 1120 Calculus II for Engineers
Section I: Multiple choice problems [20 Points, 5 each]
(No Partial Credits for this Section)
G
G
K
G
G
G
G
1. The area of the triangle with two sides represented by u = i + 3 j − k and v = i − k , is
A)
12
B) 2 18
C)
1
18
2
D)
18
2. The symmetric equations of the line passing through the points P (1, 2, −1) and
Q ( 5, −3, 4 )
A)
C)
x –1
=
–4
x −1
=
−4
y+2
=
–5
y −2
=
−5
3. Evaluate the given integral.
z +3
7
z +1
5
∫
B)
D)
3
ln t + 9t , t 2 + 8t + c
2
B)
C)
3
ln t + 9t , t 2 + 8t + c
2
D)
z +1
5
z +1
5
x
, then f y is
y
x
B)
f y = e xy + 2
y
x
D)
f y = xe xy − 2
y
1
3
− 2 + 9t , t 2 + 8t + c
2
t
3
ln t + 9, t 2 + 8 + c
2
4. Let f ( x, y ) = e xy +
C)
y −2
=
−5
y+2
=
−5
1
+ 9,3t + 8 dt
t
A)
A)
x −1
=
4
x +1
=
4
x
f y = ye xy + 2
y
2x
f y = xe xy − 2
y
2-6
Department of Mathematical Sciences, UAEU
Spring 2008
Midterm Exam
MATH 1120 Calculus II for Engineers
Section II: Multiple-Step problems [65 Points, 14 each+9 for last]
1. The thrust of an airplane’s engine produces a speed of 500 mph in still air. The wind
velocity is given by 20, 80 . In what direction should the plane head to fly due to east.
G
G
2. Let a = 2 ,1 and b = 3 ,4 . Find
G
G
G
a) 3a − 2b
b) CompbG a
3-6
Department of Mathematical Sciences, UAEU
G
Spring 2008
Midterm Exam
G
MATH 1120 Calculus II for Engineers
G
3. Consider the circle r (t ) = a cos t i + a sin t j with radius a.
a) Find the unit tangent
b) Show that the curvature is the reciprocal of the radius.
4. A cone has the height h and the circular base of radius r. The height and radius are
changing with time. At the instant in which r = 2 cm, h = 3 cm, it was found that
dr
dh
= 0.02 cm / sec and
= −0.03 cm / sec . At what rate is the cone's volume changing
dt
dt
1
at that instant? (Hint: you may use the formula V = πr 2 h )
3
4-6
Department of Mathematical Sciences, UAEU
G
G
G
Spring 2008
Midterm Exam
G
G
MATH 1120 Calculus II for Engineers
G
G
5. Let c = b − a where a = 1 and b = 1 . If θ is the angle between a and c and β is
G
G
the angle between b and c , then find the relation between cos θ and cos β , hence
find the relation between the angles θ and β .
5-6
Department of Mathematical Sciences, UAEU
Spring 2008
Midterm Exam
MATH 1120 Calculus II for Engineers
Section III: Concept problems [15 Points, 5 each]
1. Decide which of the following statements is true or false: (1 point each)
G
a. If u
G
G G
and v are large then, u ⋅ v is also large.
b. For a given vector, there is one unit vector parallel to it.
c. The cross product of two unit vectors is a unit vector.
d. The minimum number of points required to determine an equation
of a plane in 3-space is two points.
K
dr (t )
d K
K
e. If r (t ) is a differentiable vector function, then
r (t ) =
dt
dt
[
]
[
[
]
]
[
]
[
]
2- Decide if each of the following quantities is a vector, a scalar, or undefined (write your
answer over the dots). Give Reasons for your answer.
(1 point each)
a.
G G G
(u ⋅ v ) w
.…………………..
b.
G G G
u (v ⋅ w)
…………………….
c.
G G G
u ⋅v + w
…………………….
d.
GG
G G
(v .w) × (u × w)
……………………..
G
3
w
e. 2v − G
…………………….
3. a) What does the equation x + y = 3 represent in 3-space?
(2 Points)
b) What it is the object in 3-space that can be represented by the vector equation
G
G
G
K
r (t ) = t 2 i + 2t 2 j + 3t 2 k ? Explain your answer. (3 Points)
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