Numerical Analysis (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 8 - 13 of 21

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Question number: 8

» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)

Appeared in Year: 2010

Essay Question▾

Describe in Detail

If f (x) =1/x 2, find the divided differences f (a, b) and f (a, b, c).

Explanation

we can write the Newton’s divided difference formulas as

Equation

Equation

Equation

Equation

Equation

First find f (b, c)

Equation

Equation

Equation

Then, from (1) and (2)

Equation

Equation

Equation

Equation

Equation

Question number: 9

» Numerical Analysis » Summation Formula » Euler-Maclaurin's

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Using Euler’s method, compute the values of y correct upto 4 places of decimal for the differential equation Equation with initial condition x 0 = 0, y 0 = 1, taking h = 0.05.

Explanation

The given differential equation is

Equation

with initial condition x 0 = 0, y 0 = 1, taking h = 0.05.

Using Euler’s method,

Equation

where

Equation

So, putting the initial condition, when n = 0

Equation

Equation

Equation

and

Equation

Equation

Now first modification of y 1

Equation

Equation

… (25 more words) …

Question number: 10

» Numerical Analysis » Interpolation Formulae » Gauss

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Use mathematical induction to prove

Equation

Explanation

Let ∆ i define the i th finite difference which is defined as

∆ =E-1 where E is a shift operator that is

E n f x = f x + nh

Then

Equation

Equation

Equation

Equation

Question number: 11

» Numerical Analysis » Numerical Integration

Appeared in Year: 2012

Essay Question▾

Describe in Detail

The speed y (in km/hr) of a car at different points of time x between 10: 00 a. m. and 10: 40 a. m. on some day was recorded as follows:

Time x (a. m. )

10.00

10.10

10.20

10.30

10.40

Speed y (in km/hr)

24.2

35.0

41.3

42.8

39.2

Calculate the approximate distance covered by the car between 10: 00 a. m. and 10: 40 a. m. on that day using Simpson’s one-third formula for numerical integration.

Explanation

Answer: T he one-third rule of the integrand using Simpson’s rule is

Equation

Here n = 4, b = 10: 40 = 40 minutes, a = 10.00 = 0 minutes, h= (b-a) /n = 10 minutes

The distance covered by the car between 10: 00 a. m. and 10: 40

… (9 more words) …

Question number: 12

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2010

Essay Question▾

Describe in Detail

The following values of the function f (x) for values of x are given:

f (l) = 4, f (2) = 5, f (7) = 5, f (8) = 4.

Find the value of f (6) and also the value of x for which f (x) is maximum or minimum.

Explanation

This is written in the table in this function y = f (x)

x

1

2

7

8

y

4

5

5

4

For find x = 6, we use Lagrange’s interpolation polynomial because the x values is not equal interval. There are four values of x which gives the

… (12 more words) …

Question number: 13

» Numerical Analysis » Interpolation Formulae » Newton (Dividend Difference)

Appeared in Year: 2011

Essay Question▾

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

Equation

where Equation , n = 5

h= (1 - 0) /5 = 1/5

We divide the range 0 to 1 in five equal part i. e.

Equation

Equation

Equation

Equation

Equation

Use Simpson’s rule

Equation

Equation

… (1 more words) …

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