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Problem: Numerical Differentiation, 2x^3-4x+1, [2,4], x=2.1, h=0.25 [ Calculator, Method and examples ]

Solution:
Your problem `->` Numerical Differentiation, 2x^3-4x+1, [2,4], x=2.1, h=0.25


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

Numerical differentiation method to find solution.
The value of table for `X` and `Y`

X22.252.52.7533.253.53.754
Y914.781222.2531.59384356.656272.7591.4688113

Newton's forward differentiation table is as follows.
XY(X)`DeltaY``Delta^2Y``Delta^3Y``Delta^4Y`
29
5.7812
2.2514.78121.6875
7.46880.1875
2.522.251.8750
9.34380.1875
2.7531.59382.06250
11.40620.1875
3432.250
13.65620.1875
3.2556.65622.43750
16.09380.1875
3.572.752.6250
18.71880.1875
3.7591.46882.8125
21.5312
4113


The value of `x` at you want to find `f(x) : x_0 = 2.1`

`h = x_1 - x_0 = 2.25 - 2 = 0.25`


`t = (x_0 - 2)/h = (2.1 - 2)/0.25 = 0.4`

`[(dy)/(dx)]_(x=x_0) = 1/h * ( Delta Y_0 + (2t-1)/(2!) * Delta^2 Y_0 + (3t^2-6t+2)/(3!) * Delta^3 Y_0 + (4t^3-18t^2+22t-6)/(4!) * Delta^4 Y_0)`






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