Question

If f(1) = 2,f(2) = 4 and f(4) = 16,what is the value of f(3)using lagrange's interpolation formula?

Answer 1

Given :  f(1) = 2,f(2) = 4 and f(4) = 16

To find : value of f(3)using lagrange's interpolation formula

Solution:

f(1) =  2    => x₀ = 1     y₀ = 2

f(2)  = 4   => x₁ = 2     y₁ = 4

f(4) = 16   => x₂ = 4     y₂  = 16

f(3)  => x  = 3

Using lagrange's interpolation formula

y₀ (x - x₁)(x - x₂) /  (x₀ - x₁)(x₀ - x₂)  +  y₁ (x - x₀)(x - x₂) / (x₁ - x₀)(x₁ - x₂)   + y₂ (x - x₀)(x - x₁)/(x₂ - x₀) (x₂ - x₁)

= 2(3 - 2)(3 - 4)/(1 - 2)(1 - 4)   +  4(3 - 1)(3 - 4)/(2 - 1)(2-4)  + 16(3 - 1)(3 - 2)/(4 -1)(4 - 2)

= 2(1)(-1)/(-1)(-3)   + 4(2)(2)/(-1)(-2)  + 16(2)(1)/(3)(2)

= -2/3  + 8  + 16/3

= 14/3  + 8

= (14 + 24)/3

= 38/3

f(3) = 38/3

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Using Lagrange's Mean Value theorem find a point on the curve y ...

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