Tetration extension for bases between 1 and eta

[dantheman163]

Not bad! How did you come to this formula? And what plot program do you use btw? The formula seems to be restricted to f with derivative 1 at the fixed point, right?

Basically I drew tangent lines to the curve and found the kth iterate for each one at the point of tangentcy. i used the fact that the kth iterate of \( f(x) = ax+b \) is \( a^k{x}+b\frac{1-a^k} {1-a} \).
the line tangent to the function at any point n is \( y=f'(n)x-nf'(n)+f(n) \) then setting \( a=f'(n) \) , \( b=-nf'(n)+f(n) \), and x=n then plunging in and simplifying we come up with the approximation formula.

now since this formula gives nice aproximations near a fixed point you can iterate the function to bring your value near the fixed point, then plug it into the formula, then un-iterate it to bring it back to where it was.

This formula does work with any function with an attracting or repelling fixed point even if the slope through it is not 1.
it may even work with functions with a negative slope (see pic)



\( f^{1/3}(x) \) if \( f(x)=1/x+1 \) (same colors)
   



I checked the values and it does satisfy \( f(f(f(x)))=1/x+1 \)


I use wz grapher to make the pictures. Its a free function grapher that you can download