Short Questions and Answers

SHORT QUESTIONS AND ANSWERS

1. Give the Newton's divided difference interpolation formula :

Solution :  [A.U Trichy A/M 2010] [A.U A/M 2019 R-13]

f(x) = f(x₀) + (x – x₀) f (x₀,x₁) + (x − x) (x – x₁)f (x₀,x₁,x₂) + ….

 + (x – x₀) (x – x₁)... (x –x_n-1) f (x₀,x₁,...,x_n)

2. What is the nature of nth divided differences of a polynomial of n^th degree?

Solution:

The n^th divided differences of a polynomial of degree n are constants.

3. Find the divided difference of f (x) = x³ + x + 2 for the arguments 1, 3, 6, 11. [A.U CBT N/D 2010], [A.U. A/M 2011] [A.U N/D 2020 (R-17) NM] [A.U A/M 2021 (R-17) NM]

Solution :

The divided difference table is

4. State any two properties of divided differences. [A.U. M/J 2014, R8], [A.U N/D 2016 R13]

Solution :

(i) The divided differences are symmetrical in all their arguments. i.e., the value of any difference is independent of the order of the arguments.

(ii) The divided differences of the sum or difference of two functions is equal to the sum or difference of the corresponding separate divided differences.

5. Find the second divided differences with arguments a, b, c  if f (x) = 1 / x

Solution:

7. If f(x) = 1/x find f(a, b) using divided difference table. [A.U CBT N/D 2011] [A.U Trichy N/D 2011]

Solution:

Given: f(x) = 1/ x