# Numerical Analysis-Interpolation Formulae (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 6 of 14

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## Question number: 1

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2013

### Describe in Detail

Use Simpson’s one-third rule to estimate approximately the area of the cross section of a river 80 feet wide, the depth d (in feet) at a distance x from one bank being given by the following table. :

x | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |

d | 0 | 4 | 7 | 9 | 12 | 15 | 14 | 8 | 3 |

### Explanation

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

The area of the cross section of a river 80 feet wide is

Here n = 8, b = 80, a = 0, h= (b-a) /n = 10

## Question number: 2

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2013

### Describe in Detail

Find the missing value in the table by assuming a polynomial form for y:

X | 0 | 1 | 2 | 3 | 4 |

y | 1 | 2 | 4 | - | 16 |

### Explanation

we have given that there are four values of y is given, then find the missing value of y at x = 3 using Lagrange’s interpolation polynomial because the x values is not equal interval.

The table is

X | 0 | 1 | 2 | 4 |

y | 1 | 2 | 4 | 16 |

## Question number: 3

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2010

### Describe in Detail

The observed-values of a function are respectively 168, 120, 72 and 63 at the four positions 3, 7, 9 and 10 of the independent variable. What is the best estimate you can give for the value of the function at the position 6 of the independent variable?

### Explanation

Using Lagrange’s formula

Given that a = 3, b = 7, c = 9, d = 10 and f (a) =168, f (b) =120, f (c) =72, f (d) =63

Find x = 6, the values of f (x) =?

## Question number: 4

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

Appeared in Year: 2010

### 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

First find f (b, c)

Then, from (1) and (2)

## Question number: 5

» Numerical Analysis » Interpolation Formulae » Gauss

Appeared in Year: 2011

### Describe in Detail

Use mathematical induction to prove

### 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

## Question number: 6

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2010

### 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