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Calculus Question Paper

Calculus

Course:Bachelor Of Science In Information Technology

Institution: Kca University question papers

Exam Year:2011

UNIVERSITY EXAMINATIONS: 2010/2011
FIRST YEAR EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN INFORMATION TECHNOLOGY
BIT 1201: CALCULUS
DATE: APRIL 2011 TIME: 2 HOURS
INSTRUCTIONS: Answer question ONE and any other TWO questions
QUESTION ONE
a) State whether the following functions are continuous at x = 2 .
i) 2
3, 2
( )
1, 2
x x
f x
x x
? + <
= ?? + =
(4 Marks)
ii)
4 8, 2
( ) 2
2 , 2
x x
h x x
x
? - ??
= - ??
? =
(4 Marks)
b) Evaluate the following limits:
i)
2
3
lim 2 5 3
x 3
x x
?- x
+ -
+
(3 Marks)
ii)
3
0 2
lim 16
x 4
x x
? x x
-
+
(3 Marks)
iii)
3
2
lim 8
x 2
x
? x
-
-
(3 Marks)
2
c) Find the slope of the tangent line to the curve y=x3 - 2x at the points
x = -3, -1.5,0,3. (5 Marks)
d) i) Given that f(x) = 1x , find f '(x) from first principles. (4 Marks)
ii) Evaluate the following integral:
2
1
5 6
dx
x + x+ ? (4 Marks)
QUESTION TWO
Differentiate the following functions
a) 2 1
1
y x
x
-
=
-
(3 Marks)
b) y=x2 sinx (3 Marks)
c) y= cos5x (3 Marks)
d) 4x2y-3y = x3-1 (4 Marks)
e) Find the volume of the largest open box that can be made from a piece of cardboard 24m
square by cutting equal squares from the corners and turning up the sides. (7 Marks)
QUESTION THREE
a) Evaluate the following integrals.
i) ( ) ( ) ? x4+3x304x+3 dx (4 Marks)
ii)
x3 3x2 1dx
x
- + ? (3 Marks)
iii)
2
3
2 5
y dy
y + ? (4 Marks)
iv) ( ) 3
2
2
2x 8 dx
-
? - (4 Marks)
b) Find the area of the region bounded by
y=x3-3x2-x+3, the segment of the x-axis between x = -1
and x = 3. (5 Marks)
3
QUESTION FOUR
a) Evaluate the following limits.
i) lim 2
x 1
x
?8 x -
(3 Marks)
ii) 2 2
lim 1
x y 5 6
x
? x x
+
- +
(3 Marks)
iii)
2
2
lim 2 3
x 7 4 5
x x
?8 x x
- +
+ -
(3 Marks)
b) A farmer has 100 metres of wire fence with which he plans to build two identical adjacent pens.
What are the dimensions of the total enclosure for which its area is maximum? (6 Marks)
c) Given that ( ) 2 , ( ) 2
3
f x x x g x
x
= + =
+
Calculate
i) (g o f) (1)
ii) (f o g) (1) (5 Marks)
QUESTION FIVE
a) Using simpson’s rule and taking n = 10, evaluate:
10
0 ?
3 x dx (10 Marks)
b) Evaluate the following definite integrals:
i)
1
0
? Sinh2x dx (5 Marks)
ii)
1
2
0
2
3 4
dx
+ x ? (5 Marks)

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