Find Solution using Hermite's formula

x	f(x)
2	0.69315
2.5	0.91629
3	1.09861

x = 2.7
Finding f(2)

Solution:
The value of table for `x`, `f(x)` and `f'(x)`

x	y=f(x)	y'=f'(x)
2	0.6932	0.5
2.5	0.9163	0.4
3	1.0986	0.3333

The Polynomials `I_i(x)` are

`I_0(x)=((x - x_1)(x - x_2))/((x_0 - x_1)(x_0 - x_2))=((x -2.5)(x -3))/((2 -2.5)(2 -3))=((x -2.5)(x -3))/((-0.5)(-1))=(x^2-5.5x+7.5)/(0.5)=2x^2-11x+15`

`I_1(x)=((x - x_0)(x - x_2))/((x_1 - x_0)(x_1 - x_2))=((x -2)(x -3))/((2.5 -2)(2.5 -3))=((x -2)(x -3))/((0.5)(-0.5))=(x^2-5x+6)/(-0.25)=-4x^2+20x-24`

`I_2(x)=((x - x_0)(x - x_1))/((x_2 - x_0)(x_2 - x_1))=((x -2)(x -2.5))/((3 -2)(3 -2.5))=((x -2)(x -2.5))/((1)(0.5))=(x^2-4.5x+5)/(0.5)=2x^2-9x+10`

`I_0'(x)=4x-11`

`I_1'(x)=-8x+20`

`I_2'(x)=4x-9`

`I_0'(x_0)=I_0'(2)=4 xx 2-11=-3`

`I_1'(x_1)=I_1'(2.5)=-8 xx 2.5+20=0`

`I_2'(x_2)=I_2'(3)=4 xx 3-9=3`

Hermite Interpolation Formula is
`H(x)=sum u_i(x)*y_i + sum v_i(x)*y_i'`

where `u_i(x)=[1-2(x-x_i) I_i'(x_i)][I_i(x)]^2` and `v_i(x)=(x-x_i)[I_i(x)]^2`

`u_0(x)=[1-2(x-x_0) I_0'(x_0)][I_0(x)]^2`

`=>u_0(x)=[1-2(x-2) I_0'(2)][I_0(x)]^2`

`=>u_0(x)=[1-2(x-2) * (-3)][I_0(x)]^2`

`=>u_0(x)=[1 +6x-12][I_0(x)]^2`

`=>u_0(x)=(6x-11)(2x^2-11x+15)^2`

`v_0(x)=(x-x_0)[I_i(x)]^2`

`=>v_0(x)=(x-2)(2x^2-11x+15)^2`

`u_1(x)=[1-2(x-x_1) I_1'(x_1)][I_1(x)]^2`

`=>u_1(x)=[1-2(x-2.5) I_1'(2.5)][I_1(x)]^2`

`=>u_1(x)=[1-2(x-2.5) * (0)][I_1(x)]^2`

`=>u_1(x)=[1 +0][I_1(x)]^2`

`=>u_1(x)=(1)(-4x^2+20x-24)^2`

`v_1(x)=(x-x_1)[I_i(x)]^2`

`=>v_1(x)=(x-2.5)(-4x^2+20x-24)^2`

`u_2(x)=[1-2(x-x_2) I_2'(x_2)][I_2(x)]^2`

`=>u_2(x)=[1-2(x-3) I_2'(3)][I_2(x)]^2`

`=>u_2(x)=[1-2(x-3) * (3)][I_2(x)]^2`

`=>u_2(x)=[1 -6x+18][I_2(x)]^2`

`=>u_2(x)=(-6x+19)(2x^2-9x+10)^2`

`v_2(x)=(x-x_2)[I_i(x)]^2`

`=>v_2(x)=(x-3)(2x^2-9x+10)^2`

Hermite Interpolation formula is
`H(x)=u_0(x)*y_0+v_0(x)*y_0'+u_1(x)*y_1+v_1(x)*y_1'+u_2(x)*y_2+v_2(x)*y_2'`

`H(x)=(6x-11)(2x^2-11x+15)^2 * (0.6932) + (x-2)(2x^2-11x+15)^2 * (0.5)+(1)(-4x^2+20x-24)^2 * (0.9163) + (x-2.5)(-4x^2+20x-24)^2 * (0.4)+(-6x+19)(2x^2-9x+10)^2 * (1.0986) + (x-3)(2x^2-9x+10)^2 * (0.3333)`

Putting x=2.7 and simplifying, we obtain
`H(2.7)=0.9933`

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