MERU UNIVERSITY OF SCIENCE AND TECHNOLOGY
P.O. Box 972-60200 – Meru-Kenya.
Tel: +254 (0)799529958, +254 (0)799529959, +254 (0)712524293
Website: www.must.ac.ke Email: info@must.ac.ke
University Examinations 2020/2021
SUPPLEMENTARY/SPECIAL EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN MATHEMATICS
SME 3350: NUMERICAL METHODS
DATE: SEPTEMBER 2022
TIME: 2 HOURS
INSTRUCTIONS: Answer question one and any other two questions
QUESTION ONE (30 MARKS)
a) Find the root of f ( x ) = x 2 − 2 in the interval [1, 2] using bisection methods
(5 marks)
b) Fit a straight line of the form y = a 0 + a1 x to the data
(5 marks)
x
1
2
3
4
6
8
y
2.4
3.1
3.5
4.2
5.0
6.0
1
c) Evaluate I = ∫
0
1
1
dx correct to 3dp using Simpsons rule with n = 0.25
1+ x
3
(5 marks)
d) Calculate the first derivative of the function described by the values
x
1.0
1.2
1.4
1.6
1.8
2.0
2.2
y
2.7183 3.3201 4.0552 4.9530 6.0496 7.3891 9.0250
e) If y1=4, y3 =12, y4 = 19 and yx =7, find the value of x
(5 marks)
f) Form a table of differences for the function f ( x ) = x 3 + 5 x − 7 for x= -1, 0, 1, 2, 3, 4, 5 and use it
to obtain f (6)
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(5 marks)
Page 1
QUESTION TWO
a) Find the root of the equation 2x = cos x +3 Correct to 3dp
(10 marks)
b) The table below gives the values of tanx for 0.10 ≤ x ≤ 0.30
x
0.10
0.15
0.20
0.25
0.30
y = tanx
0.1003
0.1511
0.2027
0.2553
0.3093
Find the value of tan(0.26)
(10 marks)
QUESTION THREE
a) Use an iteration method to find correct to four significant figures a real root of the equation
x3 + x −1 = 0
(10 marks)
b) Find y 11 (1.6 ) for the function whose values are:
1.2
1.4
1.6
1.8
(10 marks)
x
1.0
2.0
2.2
y
2.7183 3.3201 4.0552 4.9530 6.0496 7.3891 9.0250
QUESTION FOUR
1
a) Apply Trapezoidal method to evaluate I = ∫ 1 − x 2 dx , with n = 10
0
(10 marks)
b) Find the cubic polynomial which takes the values:
x
1
3
5
7
y
24
120
336
720
Hence or otherwise, obtain the value of y(8)
Meru University of Science & Technology is ISO 9001:2015 Certified
Foundation of Innovations
(10 marks)
Page 2
QUESTION FIVE
a) Use an iteration method to obtain a root, correct to 3dp for the equation cosx=3x-1
(10 marks)
b) A reservoir discharging through sluices at a depth h below the water surface has a surface area
A for various values of h as given below
L (in ft)
10
11
12
13
14
∆ (in Sq. ft)
950
1070
1200
1350
1530
If t denoted the time in minutes, the rate of fall of the surface is given by
dh − 48
=
h
dt
A
Use Simpson rule with h = 0.5 to estimate the time taken for the water level to fall from 14 ft
to 10ft above the sluices
Meru University of Science & Technology is ISO 9001:2015 Certified
Foundation of Innovations
Page 3