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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