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Problem: Langrange's Interpolation, 2x^3-4x+1, [2,4], x=2.1, h=0.25 [ Calculator, Method and examples ]

Solution:
Your problem -> Langrange's Interpolation, 2x^3-4x+1, [2,4], x=2.1, h=0.25

Equation is f(x) = 2x^3-4x+1.

The value of table for X and Y

 X Y 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 9 14.7812 22.25 31.5938 43 56.6562 72.75 91.4688 113

Langrange's Interpolating Polynomial
The value of x at you want to find P_n(x) : x = 2.1

Y(x) = ((x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_5)(x - x_6)(x - x_7)(x - x_8))/((x_0 - x_1)(x_0 - x_2)(x_0 - x_3)(x_0 - x_4)(x_0 - x_5)(x_0 - x_6)(x_0 - x_7)(x_0 - x_8)) xx y_0 + ((x - x_0)(x - x_2)(x - x_3)(x - x_4)(x - x_5)(x - x_6)(x - x_7)(x - x_8))/((x_1 - x_0)(x_1 - x_2)(x_1 - x_3)(x_1 - x_4)(x_1 - x_5)(x_1 - x_6)(x_1 - x_7)(x_1 - x_8)) xx y_1 + ((x - x_0)(x - x_1)(x - x_3)(x - x_4)(x - x_5)(x - x_6)(x - x_7)(x - x_8))/((x_2 - x_0)(x_2 - x_1)(x_2 - x_3)(x_2 - x_4)(x_2 - x_5)(x_2 - x_6)(x_2 - x_7)(x_2 - x_8)) xx y_2 + ((x - x_0)(x - x_1)(x - x_2)(x - x_4)(x - x_5)(x - x_6)(x - x_7)(x - x_8))/((x_3 - x_0)(x_3 - x_1)(x_3 - x_2)(x_3 - x_4)(x_3 - x_5)(x_3 - x_6)(x_3 - x_7)(x_3 - x_8)) xx y_3 + ((x - x_0)(x - x_1)(x - x_2)(x - x_3)(x - x_5)(x - x_6)(x - x_7)(x - x_8))/((x_4 - x_0)(x_4 - x_1)(x_4 - x_2)(x_4 - x_3)(x_4 - x_5)(x_4 - x_6)(x_4 - x_7)(x_4 - x_8)) xx y_4 + ((x - x_0)(x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_6)(x - x_7)(x - x_8))/((x_5 - x_0)(x_5 - x_1)(x_5 - x_2)(x_5 - x_3)(x_5 - x_4)(x_5 - x_6)(x_5 - x_7)(x_5 - x_8)) xx y_5 + ((x - x_0)(x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_5)(x - x_7)(x - x_8))/((x_6 - x_0)(x_6 - x_1)(x_6 - x_2)(x_6 - x_3)(x_6 - x_4)(x_6 - x_5)(x_6 - x_7)(x_6 - x_8)) xx y_6 + ((x - x_0)(x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_5)(x - x_6)(x - x_8))/((x_7 - x_0)(x_7 - x_1)(x_7 - x_2)(x_7 - x_3)(x_7 - x_4)(x_7 - x_5)(x_7 - x_6)(x_7 - x_8)) xx y_7 + ((x - x_0)(x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_5)(x - x_6)(x - x_7))/((x_8 - x_0)(x_8 - x_1)(x_8 - x_2)(x_8 - x_3)(x_8 - x_4)(x_8 - x_5)(x_8 - x_6)(x_8 - x_7)) xx y_8

Y(2.1) = ((2.1 - 2.25)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 3.75)(2.1 - 4))/((2 - 2.25)(2 - 2.5)(2 - 2.75)(2 - 3)(2 - 3.25)(2 - 3.5)(2 - 3.75)(2 - 4)) xx 9 + ((2.1 - 2)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 3.75)(2.1 - 4))/((2.25 - 2)(2.25 - 2.5)(2.25 - 2.75)(2.25 - 3)(2.25 - 3.25)(2.25 - 3.5)(2.25 - 3.75)(2.25 - 4)) xx 14.7812 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 3.75)(2.1 - 4))/((2.5 - 2)(2.5 - 2.25)(2.5 - 2.75)(2.5 - 3)(2.5 - 3.25)(2.5 - 3.5)(2.5 - 3.75)(2.5 - 4)) xx 22.25 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.5)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 3.75)(2.1 - 4))/((2.75 - 2)(2.75 - 2.25)(2.75 - 2.5)(2.75 - 3)(2.75 - 3.25)(2.75 - 3.5)(2.75 - 3.75)(2.75 - 4)) xx 31.5938 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 3.75)(2.1 - 4))/((3 - 2)(3 - 2.25)(3 - 2.5)(3 - 2.75)(3 - 3.25)(3 - 3.5)(3 - 3.75)(3 - 4)) xx 43 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.5)(2.1 - 3.75)(2.1 - 4))/((3.25 - 2)(3.25 - 2.25)(3.25 - 2.5)(3.25 - 2.75)(3.25 - 3)(3.25 - 3.5)(3.25 - 3.75)(3.25 - 4)) xx 56.6562 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.75)(2.1 - 4))/((3.5 - 2)(3.5 - 2.25)(3.5 - 2.5)(3.5 - 2.75)(3.5 - 3)(3.5 - 3.25)(3.5 - 3.75)(3.5 - 4)) xx 72.75 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 4))/((3.75 - 2)(3.75 - 2.25)(3.75 - 2.5)(3.75 - 2.75)(3.75 - 3)(3.75 - 3.25)(3.75 - 3.5)(3.75 - 4)) xx 91.4688 + ((2.1 - 2)(2.1 - 2.25)(2.1 - 2.5)(2.1 - 2.75)(2.1 - 3)(2.1 - 3.25)(2.1 - 3.5)(2.1 - 3.75))/((4 - 2)(4 - 2.25)(4 - 2.5)(4 - 2.75)(4 - 3)(4 - 3.25)(4 - 3.5)(4 - 3.75)) xx 113

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