Numerical AnalysisInterpolation Formulae (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 7  14 of 14
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Question number: 7
» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)
Appeared in Year: 2011
Describe in Detail
Use Simpson’s rule with five ordinates to compute an approximation to π with the help of the integration of the function (1 + x ^{2}) ^{1} from 0 to 1.
Explanation
T he rule of the integrand using Simpson’s rule for five ordinates is
where , n = 5
h= (1  0) /5 = 1/5
We divide the range 0 to 1 in five equal part i. e.
Use Simpson’s rule
Question number: 8
» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)
Appeared in Year: 2012
Describe in Detail
An unknown function u _{x} has been tabulated below for some selected values of x. Use Newton’s divided difference formula on these to find an approximate value of u _{3}:
x  0  2  5  10 
u _{x}  3  19  73  223 
Explanation
The Newton’s divided difference formula is used when the xvalues not equally spaced. The Newton’s formula of divided difference for estimating u _{x} corresponding to x is
where , , …
x  y 



0  3  8  2  0.05 
2  19  18  1.5   
5 
Question number: 9
» Numerical Analysis » Interpolation Formulae » Lagrange
Appeared in Year: 2012
Describe in Detail
Solve the equation f (x) = 0 by using a suitable interpolation formula on the following values:
x  3  4  5  6 
f (x)  2.8  1.2  0.3  1.8 
Explanation
To solve the equation f (x) = 0, using Lagrange’s inverse interpolation method
Question number: 10
» Numerical Analysis » Interpolation Formulae » Lagrange
Appeared in Year: 2015
Describe in Detail
Fit the exponential curve y = a + bx to the following data
x: 0 2 4
y: 5.01 10 31.62
Explanation
his is not exponential equation, it is a straight line y = a + bx
S. no  x  y  Xy  x ^{2} 
1  0  5.01  0  0 
2  2  10  20  4 
3  4  31.62  126.48  16 
Sum  6  46.63  146.48  20 
The normal equations is
Using the
Question number: 11
» Numerical Analysis » Interpolation Formulae » NewtonGregory
Appeared in Year: 2010
Describe in Detail
Evaluate log _{e} 7 by Simpson’s 1/3rd rule.
Explanation
We get log _{e} 7, when
So, the seven ordinates of the integrand using Simpson’s rule is
Where h = 1/6 (ba) and y _{k} =f (a + kh) for k = 1, 2, 3, 4, 5, 6, 7
Here, b = 7, a = 1, h =
Question number: 12
» Numerical Analysis » Interpolation Formulae » NewtonGregory
Appeared in Year: 2010
Describe in Detail
Estimate U _{2} from the following table:
x  1  2  3  4  5 
U _{x}  7    13  21  37 
Explanation
To find the missing value, we use binomial expansion method. Here 4 values are known, we would take fourth order finite difference zero. Thus,
Here for x = 1, U _{0} =7, U _{1} =? , U _{2} =13, U _{3} =21, U _{4} =37
Question number: 13
» Numerical Analysis » Interpolation Formulae » Gauss
Appeared in Year: 2009
Describe in Detail
Find the value of
by taking 5 subintervals and using the Trapezoidal rule.
Explanation
Let
First divided the interval into 5 subintervals
x  0  1/4  1/2  3/4  1 
y  1  16/17  4/5  16/25  ½ 
Here a = 0, b = 1, n = 4, h= (ba) /n = 1/4
the Trapezoidal rule is
Question number: 14
» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)
Appeared in Year: 2015
Describe in Detail
By making use of difference table and a suitable interpolation formula, find the number of student who obtained less than 45 marks in an examination, from the following table
Marks  30  40  40  50  50  60  60  70  70  80 
Number of students  31  42  51  35  31 
Explanation
Applying NewtonGregory forward formula for interpolating the number of student who obtained less than 45 marks in an examination.
Marks less than  Number of student  Difference  
∆y _{0}  ∆ ^{2} y _{0}  ∆ ^{3} y _{0}  ∆ ^{4} y _{0}  
x _{0} =40  y _{0} = 31  42  9 
25 