Numerical Analysis 1 (MATS2207)
Mathematics and Computer Science - MCS
Semester: Second Semester
Level: 200
Year: 2018
1
The University of Bamenda Faculty of Science
Continuous Assessment
Department: Mathematics Course Instructors: Tanyu Vivian
Month: June Course Code: MAT S2207
Year: 2018 Course Title: Numerical Analysis
Date: 07/06/2018 Credit Value: Six Credits
Time Allowed: 1.5 hours
Answer all the questions. All necessary work must be shown and must be neatly and
orderly presented.
1. i) Define the following terms: condition number, loss of significance, absolute error,
relative error
ii) Let y = f(x) be the exact value of f at the true value of x and let y + δ = f(x + ) where
is the error introduced in the representation of x
a)
Find the absolute condition number of f?
b)
Find the relative condition number of f?
iii) Explain the difficulty in the evaluation of f( x) = x(
√
+ 1 −
√
) in evaluating f at the
point x = 600. Reformulate f( x ) to avoid the above difficulty (4+2+2+4 marks)
2. Consider the roots of the equation f( x) = 0. Using the first two terms of the Taylor
expansion
of f( x) about x
i
derive the Newton-Raphson iteration formula. Give one advantage and one
disadvantage of the Newton-Raphson method over the bisection method.
ii)Show that the Newton-Raphson iteration scheme for the function
f( x) =
−
where a is a constant is given by
=
(2 −
)
What is the root of f Do two iteration of the Newton-Raphson scheme for a = 3 and starting
with x
0
= 0.3 to find x
1
and x
2
(3+2+5 marks)
3. i) Let f(x ) = ln(x
2
+ 2). Determine the third Taylor polynomial P
3
(x ) for the function f
about x
0
= 1 and the maximum error | f(x ) — P
3
(x)| for 0 ≤ x ≤ 1
ii)The following polynomial can be used to relate the specific heat of dry air f to the
temperature x
f( x) = 0.0074x
4
− 0.284x
3
+ 3.355x
2
− 12.183x + 5
Show that there is a temperature between x
1
= 15 and x
2
− 20 for which the heat is 0
(5+3 marks)
GOOD LUCK
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