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Calculus Question Paper

Calculus

Course:Bachelor Of Science In Information Technology

Institution: Kca University question papers

Exam Year:2009

UNIVERSITY EXAMINATIONS: 2008/2009
FIRST YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN INFORMATION TECHNOLOGY
BIT 1201: CALCULUS
DATE: AUGUST 2009 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE [30 Marks] (COMPULSORY)
(a) Find
4
lim 16
2
1 -
-
? x
x
x
[3 Marks]
(b) Using first principles, show that Sinx Cosx
dx
d = [5 Marks]
(c) Show that
( )
log ( ) ( )
/
f x
f x f x
dx
d
e = hence solve for log (x2 -1)
dx
d
e [5 Marks]
(d) A particle moving on a straight line has acceleration a=8-3t2. Given that at time t=0, Velocity, V=0
and distance S=0 and that S (t) is the distance from the origin, find S(5) - S(1). [5 Marks]
(e) Find the equation of the normal to the curve y = 2x3 - 4x2 + 2 at (1,1) [5 Marks]
(f) Evaluate (i) ? x ln xdx [4 Marks]
(ii) ?( )
-
- +
2
1
2x2 2x 1 dx [3 Marks]
QUESTION TWO [20 Marks]
(a) Find the derivatives of the following functions:
(i) y = 2x2ex [3 Marks]
(ii)
x
y x
e log
= 2 [3 Marks]
2
(iii) y = sin x cos x [2 Marks]
(b) Find the slope of the curve x3 - xy3 + x2 y2 - 4 = 0 at (1, -1) [4 Marks]
(c) Find the volume of a sphere of radius a where x2 + y2 = a2 and show that the area rotated about
the x-axis gives a semi sphere of radius a. [5 Marks]
(d) Find ?cos3 sin xdx [5 Marks]
QUESTION THREE [20 Marks]
(a) (i) Find the nature of stationary points to the curve y = 4x - x2 . Hence sketch the graph.
[3 Marks]
(ii) Find the area bounded by the curve y = 4x - x2 and the x-axis [5 Marks]
(b) Evaluate ? x sin xdx by parts [5 Marks]
(c) Express in the form y= a + bi the function
2
1
Z
Z where 2(cos 450 sin 450 )
1 Z = + and
2(cos300 sin300 )
2 z = + without use of calculators and log tables. [4 Marks]
(d) Use substitution method to evaluate ? x(1+ x)10 dx [3 Marks]
QUESTION FOUR [20 Marks]
(a) Derive the formula V r 2h
3
= 1p for the volume of a right circular cone of height h and radius of
base r. [5 Marks]
(b) Given the following parametric equation
x t t
y t
= -
= +
3
2
2
2
Find
dx
dy
at t = 1 [4 Marks]
(c) Find:
(i) ? xexdx [3 Marks]
(ii) dx
x
x ? 1- 2
[4 Marks]
3
(iii) ?sin2 x cos3 xdx [4 Marks]
QUESTION FIVE [20 Marks]
(a) Find the area of the segment cut off from the curve y = x(2 - x) by the line 2y = x . [7 Marks]
(b) A particle moves along a particular path with velocity given by v = 2t 2 + 3t +1, where t is time in
seconds. Given that at t=0 v=a=0. Find:
(i) Its acceleration at time t=4 second [4 Marks]
(ii) Distance covered between time t=1 and t=5 seconds. [4 Marks]
(c) Evaluate ?cos5 xdx [5 Marks]

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