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Sma2102-Calculus Ii Question Paper

Sma2102-Calculus Ii

Course:Bachelor Of Science In Computer Science

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2010

Page 1 of 3
KIMATHI UNIVERSITY COLLEGE OF TECHNOLOGY
University Examinations 2010/2011 Academic Year
FIRST YEAR SECOND SEMESTER EXAMINATION FOR THE DEGREE OF
BACHELOR OF SCIENCE IN COMPUTER SCIENCE
SMA2102-CALCULUS II
DATE: 14TH DECEMBER 2010 ___ _____ TIME: 2.00PM – 4.00PM
Instructions: Answer Question ONE and any other two Questions.
================================================================
Question One
a) Briefly explain what you understand by the following terms (i) Implicit
function (ii) Parametric Equations of a curve
(2 Marks)
b) Find the equation of tangent line to the curve described by
2 x ? t ? 2t ?3, y ?1?t at the point (0,4) (5 Marks)
c) Sketch the curve with the equation: y = 3 2 2x ?3x ?12x ?1 .
(4 Marks)
d) Show that 1 2 cosh ln x x 1, x 1 ? ? ? ? ? . Hence or otherwise find
1 (cosh )
d
x
dx
? . (5 Marks)
e) Evaluate
8
2
6
2
1 2
x
dx
? x ? using the simpson’s rule with 8 subdivisions.
(5 Marks)
f) Find the area bounded by the curves 2 y ? x ? 4 and 2 y ? 8? 2x .(5 Marks)
g) Evaluate 2 ? x ln xdx . (4 Marks
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Question Two
a) Evaluate the following integrals
(i)
3
2 8 12
x
dx
x ? x ? ? (6 Marks)
(ii) 2
1
9
dx
x ? ? (4 Marks)
b) Given that 2 2 x ? xy ? y ? 3 show that
2
2
dy x y
dx x y
?
?
?
and
2
2 3
18
( 2 )
d y
dx x y
?
?
(6 Marks)
c) Prove that
3
2
3tan tan
tan 3
1 3tan
? ?
?
?
?
?
?
(4 Marks)
Question Three
a) Find the modulus of
7
3 4
i
i
?
?
and represent it on argand diagram.(5 Marks)
b) Sketch the curve 2
3 9
2
x
y
x x
?
?
? ?
(10 Marks)
c) Find the volume generated when the area enclosed by the x-axis and the
curve 2 3 y ? 3x ? x is rotated around the x – axis. (5 Marks)
Question Four
a) Evaluate (i) 2 3x ? x e dx (4 Marks)
(ii) 2
3 1
( 2)
x
dx
x
?
? ? (4 Marks)
b) Differentiate 2 xsinh(x ?3) (4 Marks)
c) Show that 1 2 1 1
cos sin cos cos n n n n
xdx x x xdx
n n
? ? ?
? ? ? and hence or otherwise
4 ? cos xdx . (8 Marks)
Question Five
a) Evaluate
2
2
0
x e dx ? ? using equal spacing h=0.5 by trapezoidal rule (correct to
4 d.p.) (5 Marks)
Page 3 of 3
b) Find the length of the curve
3
y ? x2 between the point (1,1) and (4,8)
(5 Marks)
c) (i) The rate of change of the voting population in Nyeri Town with respect
to time t(in years) is estimated by 2 2
100
''( )
(1 )
t
N t
t
?
?
where N(t) is the voting
population in thousands at any time t. if N9t) is 60,000 now determine
the voting population in 3 years form now. (5 Marks)
ii) find the general solution of the differential equation
3 2
dy
t y
dx
? ? (5 Marks)

Sma2102-Calculus Ii Question Paper

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