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Fundamental theorem of mathematics

The fundamental theorem of mathematics says that you can always make a function in the form

 


Explanation

It is always possible to define a function, which connects any number of points, which lie in a flat plane. For two points this might be a straight line, but it could also be a hyperbola. Three points may still be on a straight line, but then these points must be on the same line.

The fundamental theorem of mathematics describes this. In short

With this formula you can determine functions by extrapolation.

 


Example 1

The function f(x) = (1 + x)n can be written as

and for x = 0 it gives 1 = a0. The first derivative is

and for x = 0 it gives n = a1. The second derivative is

and for x = 0 it gives n(n − 1) = 2a2. By substitution of a0, a1 and a2 you get

This is a special version of the binomium of Newton.

 


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