Department of Mathematical Sciences, UAEU
Fall 200 7
Midterm Exam
MATH 1120 Calculus II for Engineers
U A E University, College of Science
Department of Mathematical Sciences
Midterm Exam Fall 2007
MATH 1120 CALCULUS II FOR ENGINEERS
Student’s Name
Student’s I.D.
Section #
Check the Name of Your Instructor
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 1 hours.
You can use the back of the sheets.
NO BOOKS. NO NOTES. NO PROGRANMING CALCULATORS
Section I
Problem #
Points
Section II
Problem #
Section III
Points
1-7
Problem #
Points
Total
Points
Department of Mathematical Sciences, UAEU
Fall 200 7
Midterm Exam
MATH 1120 Calculus II for Engineers
Section I: Multiple choice problems [20 Points, 4 each]
(No Partial Credits for this Section)
r
r
r
r
r
r
r
r
r
r
C) 6 j + 3k
1. Given u = 2i − 3 j and v = i − k , then u × 3v is
r
r
r
r
A) 9i − 6 j + 9k
r
B) 6 j − 3k
r
r
r
D) 9i + 6 j + 9k
2. The distance between the planes x + 8y - z = 45 and x + 8y - z = 41 is
4
A)
4
B)
66
45
C)
0
D)
66
3. The symmetric equations of the line through the two points ( 7,3, –1) and ( 4, –8, –7 )
is
x + 3 y + 11 z + 6
=
=
7
3
–1
x – 7 y – 3 z +1
=
=
–3
–11
–6
A)
C)
v
B)
D)
x + 7 y + 3 z –1
=
=
11
–5
–8
x + 7 y + 3 z –1
=
=
–3
–11
–6
v
v
4. Find t such that r (t ) and r ′(t ) are perpendicular, where r (t ) = 3 cos t , 3 sin t , t
A) t = 0, t =
B) t = 0 ,
C) t = ±
kπ
, k an integer
2
π
2
v
v
D) The vectors r (t ) and r ′(t )
are not perpendicular for any value of t.
5. The first-order partial derivatives of f ( x, y ) =
A)
C)
fx =
fx =
24 y − x 2
x
4
24 y − x 2
x
4
8
; fy = − 3
x
; fy =
x −8
x
3
2-7
x2 − 8 y
x3
are
B)
2
8
fx = ; f y = − 3
x
x
D)
fx =
24 y − x
x
4
8
; fy = − 3
x
Department of Mathematical Sciences, UAEU
Fall 200 7
Midterm Exam
MATH 1120 Calculus II for Engineers
Section II: Multiple-Step problems [60 Points,12 each]
1- Find the volume of the parallelepiped with three adjacent edges formed by the vectors
i + j,
j + k, and
i + j + k.
2- A 100-lb weight is suspended by two cables exerting forces a and b as shown.
If the weight is in equilibrium so that both vertical and horizontal forces balance, find the
vectors a and b.
3-7
Department of Mathematical Sciences, UAEU
Fall 200 7
Midterm Exam
MATH 1120 Calculus II for Engineers
3- You exert a constant force of 22 pounds in the direction of the handle of the wagon
pictured below. If the handle makes an angle of θ =
π
8
with the horizontal and you pull
the wagon along a flat surface for 4 miles, find the work done.
4- Find the equation of the plane containing the point (0, 3, 2) and parallel to the two
vectors 1, 2, 1 and 0, 1, − 1 .
4-7
Department of Mathematical Sciences, UAEU
Fall 200 7
Midterm Exam
MATH 1120 Calculus II for Engineers
r
v
r
5- a) Find the length of the curve described by r (t ) = 4 cos t i − 4 sin t j , 0 ≤ t < 2π .
b) Find a unit tangent to the given vector function at t =
5-7
π
4
.
Department of Mathematical Sciences, UAEU
Fall 200 7
Midterm Exam
MATH 1120 Calculus II for Engineers
Section III: Concept problems [20 Points, 4 each]
1. Decide which of the following statements is true or false:
r r
r
r
a. The dot product a ⋅ b = 0 implies either a = 0 , or b = 0
[
]
r
r r
r
π
[
b. If a ⋅ b > 0 , then the angle between a and b is less than
2
r
r
r r
c. a × b is the unique vector perpendicular to the plane containing a and b [
d. The two vectors 12, − 6, 9
and − 4, 2, − 3 are parallel
[
]
]
]
2- Decide if each of the following quantities is a vector, a scalar, or undefined ( write your
answer over the dots)
r r r
a. (u × v ).w
..…………………..
r r r
b. u × (v ⋅ w)
…………………….
c.
d.
r r
u×w
rr
u .w
rr
v .w
r r
v×w
…………………….
……………………..
3. What is the object in 2 dimensions traced by the tips of all vectors starting at the origin
r
r
that are of the form
cos t i + sin t j , 0 ≤ t < π
6-7
Department of Mathematical Sciences, UAEU
r
Fall 200 7
Midterm Exam
r
MATH 1120 Calculus II for Engineers
r
r
4. Suppose Compbr a = Comp ar b . What does that tell you about a and b ? (There are
two possibilities!)
5. Identify the following objects in 3-space:
a)
y=3
b)
r
r
r
v
r (t ) = (1 + 2t ) i − t j + (2 + 3t ) k
7-7