Final Exam Spring 2007 MATH 1120 CALCULUS II FOR ENGINEERS

U A E University, College of Science
Department of Mathematical Sciences
Final Exam Spring 2007
MATH 1120 CALCULUS II FOR ENGINEERS
Student’s Name
Student’s I.D.
Section #
Check The Name of Your Instructor
John Abraham - Section 51
Nora Merabet - Section 01
John Abraham - Section 52
Nora Merabet - Section 02
Sherif Moussa - Section 53
Naim Markos - Section 03
Nabila Azam - Section 54
Allowed time is 2 hours.
You can use the back of the sheets.
NO BOOKS. NO NOTES. NO PROGRAMING CALCULATORS
Section I
Problem #
Points
Section II
Problem #
1-10
Points
Total
Points
Section I: Multiple choice problems [50 Points, 5 each]
(No Partial Credits for this Section)
r
r
r
r
r
r
1. The projection of the vector u = i + 3 j on the vector v = 4i + 2k , is
r r
i+ j
A)
r
r
B) 2i + j
r
r
r
C) 2i − j
r
D) i + 2 j
2. A unit vector perpendicular to the plane that contains the points P (1, − 1, 0) ,
Q(2, 1, − 1) and R (−1, 1, 2) is
r
r
i
j
+
2
2
A)
r
r
i
j
B)
−
2
2
C)
3. The order of integration of the integral
5 x2
f ( x, y ) dydx
25 5
f ( x, y ) dydx
A)
∫0 ∫0
C)
∫0 ∫
x
B)
5 x2
∫0 ∫0
D)
5 y2
∫0 ∫0
25 5
r
D)
1 r r r
(i + j + k )
2
f ( x, y ) dxdy can be changed into
f ( x, y ) dydx
∫0 ∫
4. The directional derivative of the function
r
r
r
i
k
+
2
2
x
f ( x, y ) dydx
f ( x, y ) = y 2 e 4 x at the point (0, − 2) in
the direction of u , such that u is parallel and in the same direction as 3,−1 is
A)
5
2 10
5.
The maximum rate of change of
A) 1
B)
B) 0
52
10
C)
52
5
D)
52
3 10
f ( x, y ) = x cos(3 y ) at the point (−2, π ) is
C) 6
D) 3
2-10
6. The volume of the solid bounded by the surfaces z = 1 − y ,
and x = 2 is
A) 2
7.
B)
3
2
C)
1
2
z = 0, y = 0 , x =1
D) 3
(
)
3
2
2
2
The form of the equation e x e y e z x 2 + y 2 + z 2 z = 1 in spherical coordinates
is
3 ρ
A) ρ e cos φ = 1
2
6 ρ
B) ρ e cos φ sin θ = 1
7 ρ
C) ρ e cos φ = 1
6 ρ
D) ρ e cos φ cos θ = 1
2
8. The function f ( x, y ) =
A) no critical point
C) only two critical points
2
2
x 3 − 2 y 2 − 2 y 4 + 3x 2 y has
B) only one critical point
D) only three critical points
r
r
v
r = x i + y j be the position vector in 2 dimensions. For the function
1
f ( x, y ) = 9 − ( x 2 + y 2 ) the gradient of f ( x, y ) is
4
1v
1v
v
v
A) − r
B) − 2 r
C) − r
D) − r
2
4
9. Let
r
10. the curl of the vector field F = xy, yz, x
2
is
A) − x, y, − x
B) − 2 y, − 2 x, − x
C) − y, − 2 x, − x
D) − x, − 2 x, − y
3-10
Section II: Multiple-Step problems [150 Points, 15 each]
1. Find the unit vector in 2 dimensional-space that makes angle 60 with the positive
x-axis.
2. Suppose that a box is being towed up an inclined, frictionless, plane as shown in
r
figure. Find the force F needed to make the component of the force parallel to the
inclined plane equal to 2.5 lb.
F
30
15
4-10
3. Determine if the lines
⎧x = 4 + t
⎪
and
⎨y = 2
⎪ z = 3 + 2t
⎩
⎧ x = 2 + 2s
⎪
⎨ y = 2s
⎪ z = −1 + 4s
⎩
are parallel, skew or intersect. If they intersect, then find the point of intersection.
5-10
4. The gas law for a fixed mass m of an ideal gas at absolute temperature T ,
pressure P , and volume V is PV = mRT , where R is the gas constant.
Show that
5.
∂P ∂V ∂T
= −1
∂V ∂T ∂P
Suppose that the temperature of a metal plate is given by
T ( x, y ) = 4 x 2 − 4 xy + y 2 , for points (x, y) on the circular plate defined by
x 2 + y 2 = 25 . Find the maximum and minimum temperatures on the plate.
6-10
6.
Lamina bounded by y = x ( x > 0 ), y = 4 and
2
x = 0 , where the density .
ρ ( x, y ) = distance from y - axis
i) Sketch the region
ii) Find the mass of the lamina
iii) Find, x , the x-coordinate of the center of mass.
7-10
7. Use polar coordinates to evaluate the double integral
∫∫ y dA
where R is the
R
region
r = 2 − 2 cosθ
8. Sketch the box with dimensions 2, 4, 6 units, and then Use the triple
integral to find its volume.
8-10
r
9. Compute the work done by the force F = 2 x, 2 y
2
along the curve C defined
by C: is the quarter-circle from (4, 0) to (0, 4).
10 . Put "T" for the correct statement or relation and "F" for the wrong ones.
r
r
r
r
a) For any two vectors we always have a + b = a + b
b) Any two lines can lie in a plane.
r
v
r
c) For a particle moving in the circle r = cos t i + sin t j , the velocity is always
perpendicular to the position vector.
d) In 2-dimensions the polar coordinate θ = 30 represents a point.
r
r
r
e) The curve r = cos 3t i + sin 3t j can't represent circle
f) Using the cylindrical coordinates, the equation ( x − 2) + y = 4 will take the
form r = 2 cosθ .
2
r
r
g) A vector field F is conservative if ∇ × F = 0 .
r
r
h) For the position vector r , div r =0
9-10
2
BONUS (10 Points)
A rectangular box is to be inscribed in the cone z = 9 − x + y , z ≥ 0 .
Find the dimensions for the box that maximize its volume.
2
10-10
2