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This document provides examples of using interpolation to approximate values and find roots of functions. It includes problems involving: 1) Linear interpolation to find function values and roots using a table of x and f(x) values. 2) Finding the root of a function between two x-values using linear interpolation. 3) Using interpolation to find the melting point of an alloy given the percentage of one component and a table of percentages and melting points. 4) Determining the degree of a polynomial and finding it based on a table of x and y values.

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Salem Ghieth
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26 views4 pages

1st Sheet

This document provides examples of using interpolation to approximate values and find roots of functions. It includes problems involving: 1) Linear interpolation to find function values and roots using a table of x and f(x) values. 2) Finding the root of a function between two x-values using linear interpolation. 3) Using interpolation to find the melting point of an alloy given the percentage of one component and a table of percentages and melting points. 4) Determining the degree of a polynomial and finding it based on a table of x and y values.

Uploaded by

Salem Ghieth
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Interpolation Sheet

1- Using the following table to find


i) Best approximation for f(1.3) and f'(3.05)
ii) The root of f(x) = 0.

X 1 2 3 4
f(x) –6 –1 5.625 16

2- Show that the equation has a root in the interval [0.5,


0.7]. Form a table of the values of ( ) at the nodes
* +, then find the root to three decimal places.

3- The following data gives the melting point of an alloy of lead and zinc,
where temperature (toc) and p is the percentage of lead in the alloy
P 40 50 60 70 80 90
T 184 204 226 250 276 304
Using a suitable interpolation formula to find the melting point of the
alloy containing
(i) 84% of lead.
(ii) 65% of zinc.
4- The following data are taken from a polynomial of degree Deduce
its degree and Find it.

X –2 –1 0 1 2 3
Y –5 1 1 1 7 25

Page | 1 Numerical methods Math


5- After some examination, it was found that the distribution of marks
was

Marks 31 – 40 41 – 50 51 – 60 61 – 70 71 – 80
No. of 18 31 42 51 35 8
Students
Using this table to estimate the numbers of students obtained less than or
equal 45 marks and who obtained at least 64 marks.

6- From the data

X 1 2.3 4 5
Y –3 8.167 60 121
Find
i) y when x = 4.5.
ii) x when y = 0.
7- Use the suitable formula to obtain a polynomial that interpolate
the following data

X 1 3 –2 4 5
Y –1 11 11 41 95

8- Use Lagrange's formula and the data

X 0.5 0.6 0.7


y = cos x 0.87758 0.82534 0.76484
to compute cos (0.54) by
i) Linear interpolation.
ii) Quadratic interpolation.
Estimate the error in each case.

Page | 2 Numerical methods Math


9- Using Lagrange formula to find a polynomial that interpolate the
following points (0, – 7), (3, 2) and (4,9).

10- Find a best polynomial that interpolate the following data (1, 154), (2,
1.5) and (3, 1.42). Deduce f(2.5) and compute, if possible, ( ) and
( ). Find, also, the integral of f(x) taken over [1, 3].

11- Let ( ) . Find f(1.3) by

a) Using linear Lagrange interpolation based on the nodes x 0 = 1, x1 =


2.
b) Using quadratic Lagrange interpolation based on the values of
x={0.5, 1, 1.5}.
Compute the percentage error in each case.

12- For a function f, the divided differences are given by

X Y

0 f[x0]
f[x0,x1]
0.4 f[x1] 50/7
10
0.7 6

Determine the missing entries in the table.

Page | 3 Numerical methods Math


13- Show that the Newton general polynomials

P(x) = 3 – 2(x+1) + (x+1) (x) (x-1)


and Q(x) = – 1 + 4(x+2) -3 (x+2)(x+1) + x (x+2) (x+1).

are interpolate the data

X –2 –1 0 1 2
Y –1 3 1 –1 3

Does the part (a) not violate the uniqueness property of interpolating
polynomials?

Page | 4 Numerical methods Math

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