Find Solution using Hermite's formula

x	f(x)
3	1.0986
3.5	1.2528
4	1.3863

x = 3.2
Finding f(2)

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

x	y=f(x)	y'=f'(x)
3	1.0986	0.3333
3.5	1.2528	0.2857
4	1.3863	0.25

The Polynomials `I_i(x)` are

`I_0(x)=((x - x_1)(x - x_2))/((x_0 - x_1)(x_0 - x_2))=((x -3.5)(x -4))/((3 -3.5)(3 -4))=((x -3.5)(x -4))/((-0.5)(-1))=(x^2-7.5x+14)/(0.5)=2x^2-15x+28`

`I_1(x)=((x - x_0)(x - x_2))/((x_1 - x_0)(x_1 - x_2))=((x -3)(x -4))/((3.5 -3)(3.5 -4))=((x -3)(x -4))/((0.5)(-0.5))=(x^2-7x+12)/(-0.25)=-4x^2+28x-48`

`I_2(x)=((x - x_0)(x - x_1))/((x_2 - x_0)(x_2 - x_1))=((x -3)(x -3.5))/((4 -3)(4 -3.5))=((x -3)(x -3.5))/((1)(0.5))=(x^2-6.5x+10.5)/(0.5)=2x^2-13x+21`

`I_0'(x)=4x-15`

`I_1'(x)=-8x+28`

`I_2'(x)=4x-13`

`I_0'(x_0)=I_0'(3)=4 xx 3-15=-3`

`I_1'(x_1)=I_1'(3.5)=-8 xx 3.5+28=0`

`I_2'(x_2)=I_2'(4)=4 xx 4-13=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-3) I_0'(3)][I_0(x)]^2`

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

`=>u_0(x)=[1 +6x-18][I_0(x)]^2`

`=>u_0(x)=(6x-17)(2x^2-15x+28)^2`

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

`=>v_0(x)=(x-3)(2x^2-15x+28)^2`

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

`=>u_1(x)=[1-2(x-3.5) I_1'(3.5)][I_1(x)]^2`

`=>u_1(x)=[1-2(x-3.5) * (0)][I_1(x)]^2`

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

`=>u_1(x)=(1)(-4x^2+28x-48)^2`

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

`=>v_1(x)=(x-3.5)(-4x^2+28x-48)^2`

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

`=>u_2(x)=[1-2(x-4) I_2'(4)][I_2(x)]^2`

`=>u_2(x)=[1-2(x-4) * (3)][I_2(x)]^2`

`=>u_2(x)=[1 -6x+24][I_2(x)]^2`

`=>u_2(x)=(-6x+25)(2x^2-13x+21)^2`

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

`=>v_2(x)=(x-4)(2x^2-13x+21)^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-17)(2x^2-15x+28)^2 * (1.0986) + (x-3)(2x^2-15x+28)^2 * (0.3333)+(1)(-4x^2+28x-48)^2 * (1.2528) + (x-3.5)(-4x^2+28x-48)^2 * (0.2857)+(-6x+25)(2x^2-13x+21)^2 * (1.3863) + (x-4)(2x^2-13x+21)^2 * (0.25)`

Putting x=3.2 and simplifying, we obtain
`H(3.2)=1.1632`

---

---