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INTERNATIONAL ISLAMIC UNIVERSITY

MALAYSIAEND OF SEMESTER EXAMINATION

SEMESTER I, 2005/2006 SESSIONKUL L I YYAH OF ENGI NEERI NGProgramme : ENGINEERING Level of Study : UG 1

Time : 9.00am 12.00noon Date : 21/10/05

Duration : 3 hours

Course Code : MTH 1112 Section(s) : 1,2,3,4&5

Course Title : Engineering Calculus I

This Question Paper Consists ofFive (5) Printed Pages (Including Cover Page)with Nine (9) Questions.

INSTRUCTION(S) TO CANDIDATESDO NOT OPEN UNTI L YOU ARE ASKED TO DO SO This question paper consists of two sections. For specific instructions,

please refer to appropriate section.

Answers should be clear and intelligible. Justify your answer with simplification of intermediate steps for fullmarks.

No book, notes and programmable calculator are permitted

Any form of cheating or attempt to cheat is a seriousoffence which may lead to dismissal.

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Engineering Calculus I MTH1112

Section A [55 marks ]

[Answer all questions of this section ]

Q1. [10 marks]

(a) If

=1kka converges, then k

ka

lim = .. (2 marks)

(b) Let |x|)x(f =

i. Is it continuous at x=0? (1 mark)

ii. Is it differentiable at x=0? (1 mark)

(c) By LHopital rule, find)ln(

1lim

1 x

x

x

(2 marks)

(d) Find the area below 52 +=xy bounded by x=0, x=3 andy = 0

(2 marks)

(e) Find the derivative of 53+= xey (2 marks)

Q2. [15 marks](a) By usingFundamental Theorem of Calculus (Part 2), find the derivative

)x(f if

+=

22

0

5x

t dt)te()x(f (3 marks)

(b) Is the series

=1 2

1

kk

kk)(

absolutely convergent? (3 marks)

(c) Prove that if )(xg is a differentiable function and n is an integer, then

)()]([)]([ 1 xgxgnxgdx

d nn = (3 marks)

(d) Use basic definition to show that if ,xx)x(f 42 += then

.x)x(f 42 += (3 marks)

(e) Show that ++= cxxdxx |tansec|lnsec where c is a constant(3 marks)

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Engineering Calculus I MTH1112

Q3. [15 marks]

(a) Find the derivative )(xy

(i) 452221 8)ln(tanh + =+ xyxyx (5 marks)

(ii) ))103(ln(sec)2(sinh 3331 ++= xhexy x (5 marks)

(b) Find the derivative 54)(cos += xxy (5 marks)

Q4. [15 marks]

(a) Evaluate the integral dxxx + 923 (8 marks)

(b) Use a Taylor Polynomial with n = 6to approximate

dxxx )(cos2

1

1

2

(7 marks)

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Engineering Calculus I MTH1112

Section B [45 marks]

[Answer Three (3) questions]

Q5. [15 marks]

(a) By using the method of cylindrical shells, sketch and find the volume

of the solid formed by revolving the region bounded by the graphs of

y =3x andy = x2in the first quadrant about the linex = 5. (10 marks)

(b) For each corner of a square piece of sheet metal 18 centimeters on a

side, remove a small square and turn up the edges to form an open box.

What should be the dimensions of the box so as to maximize its volume?(5 marks)

Q6. [15 marks]

(a) Find the Taylor series and determine the interval of convergence and

radius of convergence forx

exxf3)( = about 0=x (5 marks)

(b) If 12 +=x)x(f on ]2,0[ , then find all number ofc in )2,0(

for which

ab

)a(f)b(f)c(f

= (5 marks)

(c) Find the 4th order derivative of 4)102()( += xxf (5 marks)

Q7. [15 marks]

(a) Find the equation of the tangent line of the graph

)(1

xfy= at 4=x , where 14)( 3 += xxxf (8 marks)

(b) By using Simpsons rule approximation with n=4, compute

dxx +1

0

2 4 (7 marks)

Q8. [15 marks]

(a) By using partial fraction, evaluate the following integral

dxxx

x

6

742

(8 marks)

(b) Newtons theory of gravitation states that the weight of a person at

elevation x feet above sea level is

,)/()( 22 xRPRxW +=

where P is a persons weight at sea level and R is the radius of the earth

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(approximately 20,900,000 feet). Find linear approximation of W(x)

at 0=x . (7 marks)Engineering Calculus I MTH1112

Q9. [15 marks]

(b) Evaluate the integral dxxx55 cossin (8 marks)

(b) Use integration by part to calculate

dxxx )(tan 1 (7 marks)