Given the equation $ \frac{1}{x^4}+e^{x-100}=10^8 $ that has one positive root > 1, formulate Newton's method for finding the root. Make one iteration with starting value = 1. Try to make another iteration with starting value 0. Explain the different results.
"The two starting values 0 and 1 are too bad to receive good approximation."
Suggest a better starting guess between 0 and 1. Rationalize how you make the assumption. Approximate the error of the starting guess. Clue: Use method-independent error estimation.
My solution
$ f(x)=\frac{1}{x^4}+e^{x-100}-10^8 $ $ f'(x)=-\frac{4}{x^5}+e^{x-100} $ $ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} $ Use starting value 1 $ x_1=1-\frac{1+e^{-99}-10^8}{-4+e^{-99}}=-2.5*10^8 $ Use starting value 0
$x_{1} =$ division by zero!
Starting guess can be rationalized by dropping the small exp term and using 10^-2 as starting guess but that didn't converge for me either. Why not? How do I estimate the error of the starting guess by "method-independent error estimation"?