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MTH618 Assignment 2 This is an individual assignment, so it requires individual submission. The date of submission is Friday 13th October 2017, to be submitted before 1pm. Any submission later than this time will incur a penalty of 50% of the marked score per day unless valid reason (with evidence) is provided. The total marks for the assignment is 50 and contributes 5% to the students coursework. The mode of submission is either by handing in a written solution sheet or emailing the scanned assignment to the lecturer/tutors email. All workings to the solutions should be provided in the solution sheet which is submitted. All solutions should be written in blue or black ink (no pencil or red ink allowed) and all writing should be easily legible. Plagiarism is strictly prohibited and the student will be penalized accordingly. The lecturer holds the right to question any student about any solution in the assignment and ask the student to reproduce any portion of the solution at any moment. 1 Exercise. 1. Fit a fourth order Newton’s backward interpolating polynomial to estimate 3 p 10:5 using the following known values: 3 p 6; 3 p 8; 3 p 10; 3 p 12; 3 p 14: [Hint: Study Section 19:3 to answer this question.] (6 marks) 2. Fit a fourth order Newton’s forward interpolating polynomial to estimate sech 5:5 using the following values: sech 2; sech 4; sech 6; sech 8; sech 10: [Hint: Study Section 19:3 to answer this question.] (6 marks) 3. A factory produces two kinds of gaskets, G1, G2, with net profit of $60 a and $30, respectively. Maximise the total daily profit subject to the constraints (xj = number of gaskets Gj produces per day): 40×1 + 40×2 1800 (Machine hours) 200×1 + 20×2 6300 (Labour) x1 0 x2 0 (6 marks) 4. Unconstrained Optimisation. Apply the method of steepest descent to f(x) = 9×21 + x22 + 18×1 􀀀 4×2 for 5 steps. Start from x0 = (2; 4). (8 marks) 2 5. Sturm-Liouville Problem. Find the eigenvalues and eigenfunctions and solve the IVP. Verify orthogonality of the eigenfunctions. y00 + 8y0 + ( + 16)y = 0; y(0) = 0; y() = 0 : (8 marks) 6. Find Fc(f) and Fs(f) of the function f(x) = ( x2; 0 < x 0: (4 + 4 marks) 7. Find the Fourier Transform of the following functions. (a) f(x) = ( jxj; 􀀀1 < x < 1 0; otherwise: (b) f(x) = ( xe􀀀x; 􀀀1 < x < 0 0 otherwise: (4 + 4 marks) 3

MTH618 Assignment 2
This is an individual assignment, so it requires individual submission. The date
of submission is Friday 13th October 2017, to be submitted before 1pm.
Any submission later than this time will incur a penalty of 50% of the marked
score per day unless valid reason (with evidence) is provided.
The total marks for the assignment is 50 and contributes 5% to the students
coursework.
The mode of submission is either by handing in a written solution sheet or emailing
the scanned assignment to the lecturer/tutors email.
All workings to the solutions should be provided in the solution sheet which is
submitted.
All solutions should be written in blue or black ink (no pencil or red ink allowed)
and all writing should be easily legible.
Plagiarism is strictly prohibited and the student will be penalized accordingly.
The lecturer holds the right to question any student about any solution in the
assignment and ask the student to reproduce any portion of the solution at any
moment.
1
Exercise.
1. Fit a fourth order Newton’s backward interpolating polynomial to estimate 3
p
10:5
using the following known values:
3 p
6; 3
p
8; 3
p
10; 3
p
12; 3
p
14:
[Hint: Study Section 19:3 to answer this question.]
(6 marks)
2. Fit a fourth order Newton’s forward interpolating polynomial to estimate sech 5:5
using the following values:
sech 2; sech 4; sech 6; sech 8; sech 10:
[Hint: Study Section 19:3 to answer this question.]
(6 marks)
3. A factory produces two kinds of gaskets, G1, G2, with net profit of $60 a and
$30, respectively. Maximise the total daily profit subject to the constraints (xj =
number of gaskets Gj produces per day):
40×1 + 40×2 1800 (Machine hours)
200×1 + 20×2 6300 (Labour)
x1 0
x2 0
(6 marks)
4. Unconstrained Optimisation. Apply the method of steepest descent to
f(x) = 9×21
+ x22
+ 18×1 􀀀 4×2
for 5 steps. Start from x0 = (2; 4).
(8 marks)
2
5. Sturm-Liouville Problem. Find the eigenvalues and eigenfunctions and solve
the IVP. Verify orthogonality of the eigenfunctions.
y00 + 8y0 + ( + 16)y = 0; y(0) = 0; y() = 0 :
(8 marks)
6. Find Fc(f) and Fs(f) of the function
f(x) =
(
x2; 0 < x < 1;
0; x > 0:
(4 + 4 marks)
7. Find the Fourier Transform of the following functions.
(a) f(x) =
(
jxj; 􀀀1 < x < 1
0; otherwise:
(b) f(x) =
(
xe􀀀x; 􀀀1 < x < 0
0 otherwise:
(4 + 4 marks)
3

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