mat575 2009-1

CONFIDENTIAL CS/APR 2009/MAT400/575

UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION

COURSE COURSE CODE EXAMINATION TIME

INTRODUCTION TO NUMERICAL ANALYSIS MAT400/575 APRIL 2009

3 HOURS

INSTRUCTIONS TO CANDIDATES

1. This question paper consists 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 the invigilator.

4. Please check to make sure that this examination pack consists of:

i) the Question Paper ii) an Answer Booklet - provided by the Faculty

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO This examination paper consists of 5 printed pages

Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL 2 CS/APR 2009/MAT400/575

QUESTION 1

a) Consider f(x) = 3 - x - x2 on the interval [0,2].

i) Determine whether f has a root in the given interval. (2 marks)

ii) By using the bisection method, determine the approximate root of f after four iterations. Give your answer correct to four decimal places.

(6 marks)

iii) Find the number of iterations needed to obtain an approximate root of f by using the bisection method with an accuracy of the order 10 3.

(2 marks)

b) Consider the root of f (x) = 3x3 - 5.

i) Show that the Newton-Raphson iteration formula of calculating the root of the given function is written as

P*+i - g r 5 ^ 6P* + - ; JkJ

, /c=0,1,2,....

(4 marks)

ii) By using the result in i) above with an initial guess p0 = 1 . 5 , compute an

approximation of \^>J

correct to four decimal places.

(6 marks)


CONFIDENTIAL CS/APR 2009/MAT400/575

QUESTION 2

Given the following table of data:

/

*,

n*,) 0

1.5 1

1 2.5 10

2 3.5 25

3 4.5 55

a) i)

ii)

b) i)

ii)

Obtain the Newton's forward divided-difference interpolation polynomial of order 3 which represents the given data.

(8 marks) By using your result in a)i) above, calculate the approximation of f(4.0). Give your answer correct to four decimal places.

(2 marks) Obtain the cubic interpolation polynomial representation of the given data by using the Lagrange interpolating polynomial.

(6 marks) By using your result in b)i) above, calculate the approximation of f(4.0). Give your answer correct to four decimal places.

(2 marks)

c) Comment on the results obtained in a)ii) and b)ii) above. (2 marks)


CONFIDENTIAL 4 CS/APR 2009/MAT400/575

QUESTION 3

a) Find a least squares line to approximate the continuous function y = 2ex_1 - 1 on the interval [0,2].

(10 marks) b) The table below gives the continuous assessment marks and the final-examination

marks for 20 numerical methods students. Obtain the least squares line equation for the given data.

i

1 2 3 4 5 6 7 8 9 10

Assignment Marks/50, x,

35 45 20 23 28 43 20 35 45 30

Final-examination Marks/50, y.

45 46 30 36 34 44 30 45 43 34

/

11 12 13 14 15 16 17 18 19 20

Assignment Marks/50, x}

45 27 15 10 15 48 14 33 14 47

Final-examination Marks/50, y,

48 40 35 29 28 50 12 29 26 35

Hence, use the least squares line obtained to determine the final-examination marks given the assignment marks are 45 and 20.

(10 marks) QUESTION 4

A motorist test drives his racing car on a race track for 70 seconds. The speed in m/s of the car is recorded for every 5-second interval starting from the beginning of the drive. The table below records the data collected.

/ Time Speed

0 0

120

1 5

130

2 10 140

3 15 155

4 20 165

5 25 160

6 30 152

7 35 148

8 40 140

9 45 138

10 50 130

11 55 125

12 60 120

13 65 110

14 70 100

a) Approximate the length of the track using: i) the composite trapezoidal rule, and

(9 marks) ii) the composite Simpson's rule.

(9 marks) b) Compare the two results obtained above. Give your comment.

(2 marks) Hak Cipta Universiti Teknologi MARA CONFIDENTIAL

CONFIDENTIAL CS/APR

QUESTION 5

mat575 2009-1

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