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NUM701S - NUMERICAL METHODS 1 - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 35BAMS
LEVEL: 6
COURSE CODE: NUM701S
COURSE NAME: NUMERICAL METHODS 1
SESSION:
DURATION:
JUNE 2022
3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Dr S.N. NEOSSI NGUETCHUE
MODERATOR:
Prof S.S. MOTSA
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations. All numerical results must be
given using 4 decimals where necessary unless mentioned otherwise.
3. All written work must be done in blue or black ink and sketches must be done in
pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 2 PAGES (Including this front page)
Attachments
None
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Problem 1 {30 marks]
1.1. If f € C"*[a, b], prove that for any points z and c in [a,b], we have
[12]
mn £(k)(¢
fz= S—s? Fa!
—c)*+R,(x)
where
R,(x) = al1 ld / f? f*) (t)(a — t)"dt
(Hint: use integration by parts f udv = uv — f vdu with appropriate choice of u and v.]
1.2. Consider f(x) = —51t +32-4=0, x € [3.5, 4.5].
Use Newton’s method to approximate the root of the above equation after three iterations. [4]
1.3. The equation « = g(x) = (z? — 1)/3 has a root in [—1, 1].
1.3.1. State the fixed-point Theorem.
[4]
1.3.2. Prove that the sequence (2,)xen with 2,4, = g(v,) converges to the fixed-point of the
equation given above in 1.3. for any choice of a € {—1, 1].
(10}
Problem 2. [40 marks]
2.1. Write down in details the formulae of the Lagrange and Newton’s form of the polynomial
that interpolates the set of data points (Zo, f(20)), (€1, f(£1)),---; (Gn; f(n))-
[7]
2.2. Use the results in 2.1. to determine the Lagrange and Newton’s form of the polynomial
that interpolates the set of data points (0,1), (1,6) and (2,17).
[18]
2.3. Establish the error term for the rule:
[15]
F"(0) & saylBF(w+ h) — 10f(w) + 12f(aw — h) — 6f(w— 2h) + flew - 3h)
Problem 3. [30 marks]
Given the IVP
ystytytt, y(0) =2.
(1)
3.1 Write down in details the fouth-order Runge-Kutta (RK4) algorithm to solve the specific
IVP given by Eq. (1).
[10}
3.2 Given the table below, use the result of question 3.1 to compute the missing values. [20]
kK | te
1 | 0.08
2 | 0.16
3
4
0 | 0.4
ky
ko
2
2.1648
2.50439
2.78488 | 3.0281
3.30856
4.30325
kg
ka
Yk
2.35403 | 2.17369
2.78496
2.62174
3.61874 | 3.94524
4.71963
God bless you !!!
TOTAL: 100 marks