Kenyatta University
Bachelor of Science (Bsc)
Calculus I
Question Paper
Exam Name: Calculus I
Course: Bachelor of Science (Bsc)
Institution/Board: Kenyatta University
Exam Year:2009
KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2008/2009
INSTITUTE OF OPEN LEARNING
EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE
SMA 104 : CALCULUS I
DATE: MONDAY 10TH AUGUST 2009
TIME: 11.00 A.M. – 1.00 P.M.
INSTRUCTIONS:
Answer Question 1 and any other two questions.
Q.1. a)
Evaluate the following limit.
2
Lim
x 1
(4 marks)
x ? 3
2
x + 5x  6
b)
Use the definition of derivative (first principles) to find f (
' x) given that
f(x) = 2x2
–
2x
5.
(5
marks)
c)
Find the constants a and b given that
?ax + b, x > 1
?
f (x) = ?3 , x = 1
?
? 2
x + b, x < 1
is
continuous
at
x
=
1
(4
marks)
dy
d) Find
given that y = xe – ex + 3x
.
(5
marks)
dx
e)
Find the slope (gradient) of the curve x2y – xy2 = 2 at the point (1, 2).
(4 marks)
Page 1 of 3
f) let
g(x) = x2 f(x). Given that f(2) = 3 and f ' (2) = 5, find g (2). (4 marks)
dy
g)
Given
y = ln (t) and x = et , where t is a parameter, find
in terms
dx
of
x
and
y.
(4
marks)
dy
sin(2x)cos x tan3 x
Q.2
a)
Use logarithmic differentiation to evaluate
if y =
dx
1 3x
(6 marks)
b)
Give that y = (sinx) ln(x+1), prove that
2
d y
cos x
sin x
+ y = 2

(7 marks)
2
2
dx
1+ x
1
( + x)
dy
c) Find
if xy +1n (x+y) = 1
(7 marks)
dx
dy
2
d y
Q.3
a)
If x = cost and y = 1 sin2 t, find
and
(8
marks)
dx
2
dx
dy
b)
use logarithmic differentiation to evaluate
if
dx
(x2 + )
1 cot x
y =
(6 marks)
3  cot x
dy
c)
Find
dx
if
i)
y = 23x
(3 marks)
ii)
y = cos (cos x)
(3 marks)
Page 2 of 3
Q.4
a)
Find the stationary points of the curve y = x3 – x2 and distinguish between
them. Find the points of inflection if any and sketch the curve.
(16 marks)
dy
b)
Find
if
y
=
tanh(2x)
(4
marks)
dx
Q.5
a)
A box has a square base and the sum of its height and one side of the base is
20 cm. Find the maximum volume of the box.
(7 marks)
b)
A ball is thrown vertically upwards and its height after t seconds is s meters
where s = 25.2t – 4.9t2 .
Find
i)
its height and velocity after 3 seconds
ii)
when it is momentarily at rest
iii)
the
greatest
height
reached.
(8
marks)
dy
? x ?
e
c)
Find
if y = ln?
?
(5 marks)
dx
?
x
?1+ e ??
Page 3 of 3
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