Calculus Question Paper
Calculus
Course:Diploma In Information Technology
Institution: Kca University question papers
Exam Year:2009
UNIVERSITY EXAMINATIONS: 2008/2009
STAGE II EXAMINATION FOR DIPLOMA IN INFORMATION
TECHNOLOGY
DIT 201: CALCULUS
DATE: APRIL 2010 TIME: 1½HOURS
INSTRUCTIONS: Answer any THREE questions
Question One
a) If y=cos2t and x=sin t find
i.
dx
dy
[4 Marks]
ii. 2
2
dx
d y
[4 Marks]
b) Find the integral of
i. ? x2 sin xdx
[6 Marks]
ii. ? 1 -
0
2 3 x e x dx
[6 Marks]
Question Two
a) Find the limit of the following:
i.
3
7 12
lim 6 2
2
?
- +
- -
x
x x
x x
[4 Marks]
2
ii.
( )
3
3
lim 5 2
2
?
-
- +
x
x
x
[4 Marks]
iii.
4
2
lim 4
?
-
-
x
x
x
[4 Marks]
b) Given the function f(x)=2x2.Show that it is an even function.
[4 Marks]
c) Given the function g(x)=4x+6. Find:
i. g(4)
ii. g(t)
[4 Marks]
Question Three
a) Given the function y=
2 4
3
x +
.From the first principles, determine the first derivative of y.
[6 Marks]
b) A boy throws a stone in the air from a cliff. The height h meters of the stone above the sea level
at time t seconds after it is thrown is given by: h=32+10-5t2.Find:
i. The height of the cliff
[2 Marks]
ii. The expression of the speed of the stone at t seconds
[2 Marks]
iii. The value of t when the speed is zero
[2 Marks]
iv. The greatest height in meters above the sea level reached by the stone.
[2 Marks]
c) Express 9
2 3
2 -
+
x
x
in partial fractions hence evaluate ? -
+
9
(2 3)
x2
x dx
[6 Marks]
Question Four
a) Find the derivatives in each case:
i. ??
?
??
?
+
+
=
3 4
ln 3
2
x
y x
3
[4 Marks]
ii.
3
3 2 5
ln 4 4 ??
?
??
?
-
-
=
x x
y x
[4 Marks]
b) Find the equation of the tangent and normal to the curve y = x2, at the point A
where x = 1.If the normal meets the curve again at point B,find the coordinates of B. [6 Marks]
c) Find the area of the bounded region between y=x2 and y=2x.
[6 Marks]
Question Five
a) Evaluate the integral ? + x
xdx
4 tan
sec2
[6 Marks]
b) The sum of a number and 3 times another number is 36.Find the maximum product and the two
numbers. [6 Marks]
c) Given the function y=5x2 describe the equation of a curve. Find the area of this curve for the
values of x=1 to x=3 with four strips using the following methods.
i. Trapezium rule [4 Marks]
ii. Simpson’s rule [4 Marks]
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