Infinite power tower approximation

Calculus Level 4

Given $\large \displaystyle f(x) = x^{x^{x^{x^{...}}}}$ :

Using local linear approximation, approximate $f\left(\sqrt{2}\right)$ using $x_{0} =1$ . Please round your answer to the nearest tenths place.

1 solution

First Last
Jun 12, 2017

Rewriting the tower as $\frac{\ln{y}}{y}=\ln{x}$ and taking the derivative with respect to x:

$f^\prime(x)= \frac{y^2}{1-\ln{y}}=\frac{f(x)^2}{1-\ln{f(x)}},\quad f(1) = 1,\quad f^\prime(1) = 1$

The approximation: $y=\bigg(f^\prime (1)(x-1)+f(1)\bigg)\Bigg|_{x=\sqrt{2}}=\boxed{\sqrt{2}}$