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Sma 2173: Calculus Ii Question Paper

Sma 2173: Calculus Ii

Course:Bachelor Of Science In Mechatronic Engineering

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2013

SMA 2173
1
DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
UNIVERSITY EXAMINATIONS 2012/2013
YEAR ONE SPECIAL/SUPPLEMENTARY EXAMINATION FOR THE BACHELOR
OF SCIENCE IN MECHATRONIC ENGINEERING,MECHANICAL
ENGINEERING,GEOMATIC ENGINEERING AND GEOSPATIAL INFORMATION
SYSTEMS, GEOSPATIAL INFORMATION SCIENCE,ELECTRICAL AND
ELECTRONIC ENGINEERING , CIVIL ENGINEERING &
INDUSTRIAL CHEMISTRY
SMA 2173: CALCULUS II
DATE: 9TH JULY 2013 TIME: 11.00AM-1.00PM
Instructions: Answer question ONE and any other TWO questions
QUESTION ONE (30 Marks)
a) Using implicit differentiation, show that
2 1
1
(arcsin )
x
x
dx
d
?
? [3 Marks]
b) Find the equation of the tangent line to the curve
x ?1?3sin t , y ? 2 ? 5cos t at
6
?
? t [5 Marks]
c) Prove that ? sec xdx ? ln sec x ? tan x ? c [3 Marks]
d) Evaluate
? ? ? ? ? 2 4 x 1
dx
[6 Marks]
e) Using the t-method evaluate ? ? x
dx
1 cos
[4 Marks]
f) Integrate ? x exp xdx 2 [5 Marks]
g) Use trapezoidal rule to estimate ? ?
2
0
2 1 x
dx
[4 Marks]
QUESTION TWO (20 Marks)
a) Sketch the curve
1
2
2
2
?
?
x
x
y . [12 Marks]
b) Evaluate ? ?dx
x x x
x x
? ? ?
? ?
2 3 2
2 1
2
2
using partial fractions [8 Marks]
SMA 2173
2
QUESTION THREE (20 Marks)
a. Express
x x x
x x x x
2 3
2 3 3
3 2
4 3 2
? ?
? ? ? ?
in terms of partial fractions. Hence prove that
c
x x
x x
dx
x x x
x x x x
?
? ?
? ?
? ?
? ? ? ?
?
2 3
ln
2 3 2
2 3 3
2
2
3 2
4 3 2
[7 marks]
b. After a person takes a pill, the drug contained in the pill is assimilated into the blood stream.
The rate of assimilation t minutes after taking the pill is given by ? ?
? ?
? 1?
201 1
?
?
? ?
t
n t
R t find the
total amount of the drug that is assimilated into the blood stream during the first 10 minutes
after the pill is taken. [6 Marks]
c. Express
15 18
20 5 16
2i i i
i i i
? ?
? ?
in the form ib a ? [4 Marks]
d. If i z 86?? , determine the argument of z and z [3 Marks]
QUESTION FOUR (20 Marks)
a. Given that 6 2 2 x ? xy ? y ? , prove that 2 3
2
( 2 )
36
dx x y
d y
?
? [6 Marks]
b. A company is using Newspaper advertising to introduce a new product to a community of
50,000 people. It was assumed that the rate at which people learn about the new product is
proportional to the product of the number of people who have heard about it and the number
of people who have not heard about it. If 100 individuals were aware of the product initially
and 500 people are aware about it after 10 days of the campaign
i) How many people will be aware of the product after 20 days? [7 Marks]
ii) When will half of the community be aware of the product? [3 Marks]
c. Determine
dx
dy
given y ech x 1 cos ? ? . [4 Marks]

Sma 2173: Calculus Ii Question Paper

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