DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
UNIVERSITY EXAMINATIONS 2021/2022
SECOND YEAR FIRST SEMESTER EXAMINATION FOR THE DEGREEBACHELOR OF
SCIENCE IN ELECTRICAL AND ELECTRONIC & BTECCH ENGINEERING, CIVIL &
BTECH ENGINEERING, MECHATRONIC, MECHANICAL & BTECH ENGINEERING,
CHEMICAL, TIE, AND GEGIS/ GIS ENGINEERING
SMA 2119: CALCULUS III
INSTRUCTIONS:
ANSWER QUESTION ONE (COMPULSORY) AND
ANY OTHER TWO QUESTIONS.
______________________________________________________________________ QUESTI
ON ONE 30 marks
a) Show that the series
n
2n 1 diverges
(3marks)
n 1
b) Show that
lim
4 xy
does not exist
x, y 0,0 x y
c) Find the sum of the series
6
n(n 3)
(3marks)
(4marks)
n2
(4marks)
4
(1) n
e) Find the radius of convergence of the power series
(4marks)
( x 2) n
n
n 1
f) Two objects of masses 1kg and 2kg are located at (1, 2) and (1,3) respectively. Find the
d) Find the Taylor series for f ( x) cos x centered at a
coordinates of the center of mass.
(4marks)
1 2
g) Evaluate
y
2
xdydx
(4marks)
0 0
h) Find f x and f y given that f ( x, y ) sin( x xy )
Page 1 of 3
(4marks)
QUESTION TWO
(20MARKS)
a) Find the directional derivative of F ( x, y, z ) x 2 z y 3 z 2 xyz at point P(1,2,1) in a
direction of u i 3k
(5marks)
a) Find the volume under the plane z 2 x 5 y and over the rectangle :1 x 2,0 y 3
(5marks)
b) Find the mass of the lamina of density ( x, y ) x y occupying the region under the curve
y x 2 in the interval 0 x 2
(5marks)
c) Use double integrals to find the area of the region bounded by the curve y x 2 2, y 0, x 2
(5marks)
QUESTION THREE (20 MARKS)
a) Find a power series representation for the function f ( x)
x
5 x
(5marks)
b) Determine the radius and interval of convergence of the power series
2n
(4 x 8) n (5marks)
n 1 n
c) Find the total differential given that z x 3 y x 2 y 2 xy 3
(5marks)
d) By neglecting x and higher powers of x , show that f ( x ) e x sin x = x x 2
5
3
x
(5marks)
3
QUESTION FOUR (20marks)
a) Show that f xy f yx given that f ( x, y ) ln(2 x 3 y )
y
1
4 2
b) Evaluate
0
y
2
2x y
y
2x y
, v and integrating
dxdy by applying the transformation u
2
2
2
over an appropriate region in the uv plane
c) Determine the convergence or divergence of the following series
i)
k
k 1
2
(5mark)
(10marks)
k
4k 1
( 1) n n
n 1 2 n 5
ii)
QUESTION FIVE (20marks)
a) Evaluate
x
0
dx
4
(3marks)
2
b) Find the domain of the function w 16 x 2 y 2
(3marks)
c) Find the Centre of mass of a triangular lamina with vertices A(0, 0) B(1, 0)C (0, 2) where
( x, y) 1 3x y
(9marks)
Page 2 of 3
2
d) By reversing the order of integration evaluate
y
ydxdy
0 y
Page 3 of 3
(5marks)
0
