Numerical Differentiation and Integration MCQ Quiz in मराठी - Objective Question with Answer for Numerical Differentiation and Integration - मोफत PDF डाउनलोड करा

First Let f(x)  = 1 

\(\displaystyle \int_0^2\) ​xf(x) dx ≈ αf(0) + βf(1) + γf(2) becomes 

{x²/2} from 0 to 2 = α + β + γ 

α + β + γ  = 2 --- (1)

Let f(x) = x 

{x³/3} from 0 to 2 = α. 0 + β.1 + γ.2

 β + 2γ  = 8/3 ----(2)

Let f(x) = x²

{x⁴/4} from 0 to 2 = α. 0 + β.1 + γ.4

 β+ 4γ = 4 ----(3)

By Solving 2nd and 3rd 

We, get β = 4/3 and γ = 2/3 

So, The Value of 2β - γ = 2(4/3) - (2/3) = 2\

Hence, The Correct Answer is 2.

Explanation:

Using Forward divide difference

Then general Expression is 

P(x) = f(x₀) + (x - x₀) f[x₀, x₁] + (x - x₀)(x - x₁) f[x0, x1, x₂] + (x - x₀)(x - x₁) (x - x₂) f[x0, x1, x2, x₃] + (x - x0)(x - x1) (x - x2)(x - x3) f[x0, x1, x2, x3, x₄]

⇒ P(x) = 2 + (x - (-2)) (-3) + (x -(-2))(x - (-1)) 6 + (x -(-2))(x -(-1)) (x - 0) (-4) + 0 

⇒ P(x) = 2 +  -3(x + 2) + 6(x + 2)(x + 1) + -4(x + 2)(x + 1)x

Compare with given expression we will get the values of 

P(x) = α + β(x + 2) + γ(x + 2)(x + 1) + δ(x + 2)(x + 1)x 

we get

α = 2, β = -3 γ = 6 δ = -4 

Hence α + β + γ + δ = 2 - 3 + 6 - 4 = 1

Hence, The Correct Answer is 1.