We have the equation $x^3+2x-1=0$.How many steps we need to find the solution correct up to $5$ decimal places (without to find the solution correct up to $5$ decimal places with bisection method).
I solved a similar exrcise using the bisection method but i am wondering how i can estimate the steps wihout solving it any ideas would be appreciated.
For the bisection method , if $x_1 =\frac {a+b} 2$ is the midpoint of the initial interval, and $x_n$ is the midpoint of the interval in the $n^{th}$ step, the difference between $x_n$ and the solution $x$ is bounded by
$$ |x_{n}-x|\leq {\frac {|b-a|}{2^{n}}} $$
So, for your case $(a=0,b=1)$, you need to solve ${\frac {1}{2^{n}}}=10^{-5} $ which leads to $n=16.61$ then $n=17$ is the number of steps required.