PEPERIKSAAN AKHIR
SEMESTER I SESI AKADEMIK 2006/2007 IJAZAH SARJANAMUDA DENGAN KEPUJIAN
NOVEMBER 2006 MASA : 3 JAM
KOD KURSUS : KKKF3113
TAJUK : PENGIRAAN BERANGKA
ARAHAN : 1. Kertas ini mempunyai empat (4) soalan dalam Bahagian A (Section A) dan empat (4) soalan dalam Bahagian B (Section B).
2. Calon dikehendaki menjawab SEMUA soalan dalam Bahagian A dan mana-mana satu (1) soalan dalam Bahagian B.
3. Jawab SEMUA soalan dalam buku jawapan yang diedarkan.
4. Calon dikehendaki menjawab keseluruhannya hanya dalam Bahasa Melayu atau Bahasa Inggeris.
5. Jumlah markah keseluruhan kertas ini 100 markah.
6. Kertas soalan tidak dibenarkan dibawa keluar daripada Dewan Peperiksaan.
No. Pendaftaran :
(dengan perkataan)
Kertas soalan ini mengandungi 6 muka surat bercetak, tidak termasuk muka surat ini.
SECTION A Answer ALL questions in this section.
1. Given the function f x x31.265x1.
(a) Using Newton–Rhapson method, find what is the value of x when f(x) = 0
withxo = -2. (9 marks)
(b) Repeat part (a) with x0 0.8. Limit your iteration up to 5. Why does the method fail with the new initial condition.
(6 marks)
2. (a) Give three (3) methods for finding the roots of linear simultaneous equations.
(3 marks) (b) By using Gauss elimination method and applying partial pivoting, solve the
following set of equations:
2 4x1 x2 x3
4 2
5x1 x2 x3 6
6x1 x2 x2 (12 marks)
3. Table 1 shows the measurements of stream reaeration coefficient Pwhich was related to the water temperatureT. Fit the measurement data with :
(a) linear regression model. (5 marks)
(b) quadratic regression model. (5 marks)
(c) What are the stream reaeration coefficients of 20qC from the above two models and compare them with measured data by calculating the absolute
errors. (5marks)
Table 1
T (qC) P T (qC) P
14 2.89 25 3.58
23 4.20 19 3.06
17 4.17 13 2.41
11 3.69 24 3.30
20 3.78 16 3.05
15 3.56 23 4.18
18 3.35 21 4.08
11 2.69
4. Using a special inhibitor, the size of a colony of bacteria can be reduced according to the function below;
y dt yt
dy 2
where y is the mass area in Pg/mm2, and t is the time in second. If the size of the colony at the initial time (to = 0) of an experiment is 1 Pg/mm2;
(a) Use Euler Method with step size of 0.25 sec to predict the reduce mass area at
0.75 sec. (10 marks)
(b) From the data generated in part (a), use a linear approximation to predict the mass area at y(0.5), using data at y(0.25) and y(0.75). Calculate the relative error between the linear approximation and the Euler Method.
(5 marks)
SECTION B Answer ONLY one (1) question in this section.
1. (a) Given the integral 3 x 2 dx,
0
2
³
(i) evaluate the integral using analytical method. (2marks) (ii) integrate the function numerically using Multi-Step Trapezoidal Method and Simpson38 with n = 3. (8marks) (iii) compute the absolute error for the two methods in (b) with the actual
value given in (i).
(5marks)
(b) Estimate the volume of the solid component in Figure 1 below. The volume can be calculated by integrating numerically the component cross-cut area A.
Choose a suitable numerical integrator and give your reason.
z
2 3 4
A
y
x
Figure 1
(25marks)
(c) The displacement of a structure is defined by the following equation for a damped oscillation :
t e
y 8 kt cosZ
where k = 0.5 and Z = 3 (Z is in radian). Use secant method to determine the root up to the relative error d 0.01 % by assumingt-1 = 0.3 and to = 0.4. (24 marks)
3. The displacement of mass subjected to a force has been recorded and is shown in Table 2.
Table 2
F (kN) d (mm)
0.01 2.0000
1.00 4.4366
1.50 6.7134
2.25 13.9130
If a user pulls the mass with 1.2 kN, determine the estimated displacement using :
(a) Graphical method. (8marks)
(b) Lagrange quadratic method. (8marks)
(c) Newton quadratic and cubic divided difference method. (18 marks) (d) Calculate the absolute errors of all the estimated calculations above if the true
relation of displacement-force is d F 2eF F2. (6marks)
4. During a maintenance exercise, the scale on a boiler wall can be washed off slowly using a special acid-based chemical. The reduction of the scale follow according the mass balanced equation;
1 2 1
dt t
dm m
wherem is the mass per unit area (kg/m2) and t is the time in hour.
The analytical solution of the function is found to be;
3 ) exp(
3
t t
m
If at to = 0, the scale on the boiler wall was 1 kg/m2;
(a) Use Heun’s Method with a single corrector to predict the size of the scale at t= 0.75 hr. Use time step of 0.25 hr. (25 marks) (b) Compare the results obtained in part (a) with the analytical solution by
reporting the relative error.
(10 marks) (c) Plot the results of the Heun’s Method and the analytical method in one graph
paper.
(5 marks)
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