FACULTY OF CIVIL AND ENVIROMENTAL ENGINEERING BFC24103/BWM20403 ENGINEERING MATHMATICS III QUIZ 1 Q1
Find the domain and range of each of the following functions a) b)
Q2
Time Duration - 00:00:30
= ln( ln( 2 + 2 ) = � 2 + 2 + 2
Find the following limits if it exist
a)
lim
( , , )→(0,0,0)
3 2 + 2 +
2
4 − 4 4 b) lim ( , )→(0,0) 2 + 2 2 Q3
Determine whether or not lim( , )→(0,0) 2
axis, y-axis and also y = x
exists, by examining the paths along the x 3 + 3
FACULTY OF CIVIL AND ENVIROMENTAL ENGINEERING BFC24103/BWM20403 ENGINEERING MATHMATICS III QUIZ 2
Q1
Time Duration - 00:00:45
By using double integrals, find the area of the regions enclosed by a) curve = − 2 , lines = − 4, = −2, = 2 b) curve = 5 − 2 , line = + 3
Q2
Evaluate the following double integrals. a) b)
∬ 2 2 − 4 3 ℎ = {(, ): 1 ≤ ≤ 2, −1 ≤ ≤ 1} ∬ ℎ = {(, ): 0 ≤ ≤ , 0 ≤ ≤ sin }
FACULTY OF CIVIL AND ENVIROMENTAL ENGINEERING BFC24103/BWM20403 ENGINEERING MATHMATICS III QUIZ 3
Q1
Time Duration - 00:00:45
A lamina with density (, ) = + enclosed by − axis, line = 1 and curve =
√ .
Find
a) its mass. b) moment of mass about y-axis and moment of mass about x-axis. c) its center of mass.
Q2
Find the moment of mass about y-axis and about x-axis for a lamina corresponding to the region enclosed by circle 2 + 2 = 4 in the first quadrant, if the density function is given by (, ) = .
Q3
A lamina enclosed by line y = x + 2 and parabola = 2 has the density function (, ) = 2 . Find a) b) c) d)
its mass the moment of about x-axis, y-axis the center of mass the moment of inertia about z-axis
FACULTY OF CIVIL AND ENVIROMENTAL ENGINEERING BFC24103/BWM20403 ENGINEERING MATHMATICS III QUIZ 4
Time Duration - 00:00:45
Q1
Find a parametric equation of the line passing through the points P(2, 3,-1) and Q(4, -3, 2).
Q2
Express the following parametric equations as a single vector equation of the form () = () + () or () = () + () + (). a) b) c) d) e)
= −3 + 2, = −4 = 2, = −3, = 3 + 1 = 3, = , = 2 − 2 = 3, = 2 cos , = 2 s in = √ , = 2 + 4, for 0 ≤ ≤ 2
FACULTY OF CIVIL AND ENVIROMENTAL ENGINEERING BFC24103/BWM20403 ENGINEERING MATHMATICS III QUIZ 5
Q1
For the given vector-value function below, find its i. ii. iii. iv.
unit tangent vector, principal unit normal vector, binormal vector, curvature.
a)
() = 3 s in + 3 cos + 4 () = 2 c os 2 t + 2 sin 2 − 2
b)
Q2
Find the unit tangent vector T(t), principal unit normal vector N(t), and curvature K, for given vector-valued function below. a) b) c)
Q3
Time Duration - 00:00:45
() = ( cos ) + ( sin ) + () = 3 + 4 sin +4cos () = cos + sin +
Find the velocity, speed, direction and acceleration for the following position vector.
a) b)
Q4
() =< , 2 > at = 0 () = + 2 sin + 2 cos at =
A particle moves with an acceleration () = 9 3 + − +
1
2 . Find its velocity, 4 v(t) and its position vector, r(t). Given (0) = 3 − + 0.5 and (0) = 9 + + 0.25.
FACULTY OF CIVIL AND ENVIROMENTAL ENGINEERING BFC24103/BWM20403 ENGINEERING MATHMATICS III QUIZ 6
Q1
Time Duration - 00:00:45
Find the directional derivatives of function
= √ 12 + √ 12 . b) (, ) = 4 3 2 at (2,1)in the direction of vector = 4 − 3 . c) (, ) = � at (1,4) in the direction of unit vector that makes an angle of π⁄3 a) (1 + )3⁄2 at (3,1) in the direction of unit vector
with the positive x-axis.
Q2
Find the gradient of the following scalar functions. a)
∅(, ) = − � 22+ 2
b) (, ) =
Q3
(−1,1) 2
Find a unit vector in the direction in which (, ) = � 2 + 2 increases most rapidly at point P(4, -3) and find the rate of change of at P in the direction.
Q4
Find the unit normal vector to the surface 2 + 2 − + 3 2 = 7 at P(1,1, -1).
Q5
Find the divergence and curl for the following vector fields. a) (, , ) = 2 + 2 3 + 3 b) (, , ) = − + + at (3, 2, 0)
Q6
Find the curl and divergence of curl F for the vector field (, , ) = 2 + 2 + 2 .
