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CONFIDENTIAL CS/OC T 2008/MAT400/575

UNIVERSITI TEKNOLOGI MARAFINAL EXAMINATION

COURSECOURSE CODEEXAMINATIONTIME

INTRODUCTION TO NUMERICAL ANALYSISMAT400/575OCTOBER 20083 HOURS

INSTRUCTIONS TO CANDIDATES1 . This question paper cons ists of five (5) questions.2. Answer ALL questions in the Answer Booklet. Start each answer on a new page.3. Do not bring any material into the examination room unless permission is given by theinvigilator.4. Please check to make sure that this examination pack consists of :

i) the Question Paperii) an Ans we r Booklet - p rovided by the Faculty

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SOThis examination paper consists of 5 printed pages Hak Cipta Universiti Teknologi MARA C O N F ID E N T IA L

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CONFIDENTIAL 2 CS/OCT 2008/MAT400/575

ANSWER ALL QUESTIONSQUESTION 1

3a) Given that f(x) = - x2-4x- 2sin x is defined on the interval (-71, n).a) Show that f has at least one root in the given interval. (5 marks)b) Show that p = 0 is a root of f. (3 marks)

b) i) Given f (x) = 0 in an interval [a,b]. Derive the Newton-Raphson iterationmethod to determine the first approximation of the root of f. Hence, deducethe Newton-Ra phson iteration formula for approximating the root of f.(5 marks)

ii) Given f (x) =cos(x). Use Newton-Raphson method to find the approximateroot of f(x) = 0 in the interval [0, 2]. Use the initial guess, x0 = 1. Give youranswer correct to five decimal places. (7 marks)

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QUESTION 2Consider the data in the following table:

ix,

/ ( * )

00.0-5

11.01

22.010

33.025

a) Con struct the forwa rd divided-difference table up to the third order. (4 marks)b) Using the results in a) above , establish the cubic New ton's forwa rd divided-differenceinterpolation polynomial explicitly in terms of x which represents the given data.

(5 marks)c) Hence, using your result in b) above, evaluate r"(4.5) and give your answer correct tothree decimal places. (2 marks)d) Establish the cubic interpolation polynom ial using the Lagrange interpolatingpolynomial. (5 marks)e) Hence , using your result in d) above , evaluate f (4.5) and give your answer correct tothree decimal places.

(2 marks)f) Com ment on the results obtained in c) and e) above. (2 marks)

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QUESTION 3a) Find a least squares line to approximate the continuous function y = x + x2 on theinterval [ 0 ,2 ] . (11 marks)b) A certain material is tested for cyclic fatigue failure whereb y a stress measured inMPa, is applied to the material. The cycles count, N needed to cause failure isobserved and the results are tabulated as shown below. A log-log plot of stressversus cycles count indicates a linear relationship of the data trend. Use leastsquares app roach to determine the line of best fit for the given data.

Cyclescount, NStress,MPa

11200

10900

100700

1000600

10,000400

100,000300

1,000,000150

(9 marks)QUESTION 4The table below gives the values of y = ex for values of x from 0.0 to 1.0 in steps of 0 .1 .Compute the approximation of the integral [e^dx by using the composite trapezoidal rule.Hence, obtain the absolute error estimate of your approxim ation.

ix,.

y, =e^ix,

y, =e*

00.0

1.0000060.6

1.43333

10.1

1.0100570.7

1.63232

20.2

1.0408180.8

1.89648

30.3

1.0941790.9

2.24791

40.4

1.17351101.0

2.71828

50.5

1.28402

(20 marks)

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QUESTION 5dy .Given an initial value problem = x - x with y(0) = 2.dx

a) Obtain y (0,2) correct to four decimal places usingi) the Euler's method with h = 0 .1 ,

i i) the Taylor's method of order 2 with h = 0.2,

iii) the Runge Kutta method of order four, and

iv) the direct integration of the problem .b) Comm ent on the result obtained in each case and state whichresult as com pared with the result obtained in a)iv) above.

END OF QUESTION PAPER

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(5 Yi marks)

(4 Y2 marks)

(6 marks)

(2 marks)method gives the best

(2 marks)

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MAT400_MAT575