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INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIAEND OF SEMESTER EXAMINATION

SEMESTER I, 2004/2005 SESSIONKULLIYYAH OF ENGINEERING

Programme : ENGINEERING Level of Study : UG 1

Time : 9:00 am- 12:00 noon Date : 09/10/2004

Duration : 3 Hrs

Course Code : MTH 1121 Section(s) : 1-5

Course Title : Engineering Calculus I

This Question Paper Consists ofFour (4) Printed Pages (Including Cover Page) WithNine (9) Questions.

INSTRUCTION(S) TO CANDIDATESDO NOT OPEN UNTIL YOU ARE ASKED TO DO SO

This question paper consists of two sections, for specific instructions, please refer toappropriate section.

Answer should be clear and intelligible. Justify your answer with simplification of intermediate steps for full marks. No books, notes and programmable calculator are permitted.

Any form of cheating or attempt to cheat is a serious offence whichmay lead to dismissal.

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Engineering Calculus I MTH 1111

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Section A [25 marks]

[Attempt all the questions in this section]

Q1. [10 marks]

(a) If yx = then yoryory ==+= (2 marks)

(b) =+ ax

x

alim (2 marks)

(c) How many time would integration by parts needs to be performed to evaluate

dxxxn ln wheren is a positive integer. (2 marks)

(d) If )(xf exist and is positive for all then )(1 xf exist or not?(2 marks)

(e) Maclaurin series of

==

=

01

1)(

n

nxx

xf . Can this series be used to approximate

)3(f ? (2 marks)

Q2. [15 marks](a) The square rootof the sum of the first 1000 positive integers= ---------------

(3 marks)

(b) = dxx ----------------- (3 marks)

(c) Find all solutions for2x = . (3 marks)

(d) Find )34(tansin 1 (3 marks)

(e) Evaluate )( tan

0

limxx

+

. (3 marks)

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Engineering Calculus I MTH 1111

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Section B [75 marks]

[Attempt any 5 questions, selecting at least one from Q8 and Q9]

Q3. [15 marks](a) Using Intermediate-Value Theorem, show that there is a number between

0 and2

such that =cos . (8 marks)

(b) Using basic definition, show that 2ln2)2( xx

dx

d= (7 marks)

Q4. [15 marks]Evaluate the following integrals;

i. dxxxsectan2

(5 marks)

ii. +

+dx

xx

xxx24

23 1235 (5 marks)

iii. 1

0

ln dxxx (5 marks)

Q5. [15 marks]

(a) If a space shuttles downward acceleration is2/32)( sftta = , find the position

function )(S where initial velocity sec/100)0( fV = , and initial positionfeetS 000,100)0( = . (8 marks)

(b) Using integration, verify that area of a unit circle is . (7 marks)

Q6. [15 marks]

(a) Find equation of the tangent line to the graph of )(1 xfy = at 3= where

5)( 3 = xxf . (8 marks)

(b) Find the length of the curve y cosh= over the interval [ ]a,0 .(7 marks)

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Engineering Calculus I MTH 1111

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Q7. [15 marks](a) Determine radius and interval of convergence for the series

=0 )!2(!

nnx

nn (8 marks)

(b) Use first three non-zero terms of the appropriate series to approximate

1

1

sin

x

x (7 marks)

Q8. [15 marks]

Let1

1)( 2+=

xxxxf . Sketch the graph by identifying intercept(s), asymptote(s),

extrema, intervals of increase or decrease, points of inflection, and concavity.

Q9. [15 marks]

A solid is formed by revolving the region bounded in the first quadrant by2yx= and

2xy= around the line 1= . Find the volume;i. By the method of cross sections. (8 marks)

ii. By the method of cylindrical shells. (7 marks)