If the error after \( n \) iterations is at most \( (b - a)/2^{n+1} \), what is the smallest number of iterations that guarantees an error below \( 0.001 \) on \( [0, 1] \)?
We need \( 1/2^{n+1} < 0.001 \). Since \( 2^{10} = 1024 > 1000 \), \( n = 9 \) suffices. Choosing \( n = 5 \) leaves a bound of \( 1/2^{6} \approx 0.0156 \), still larger than \( 0.001 \).
The bisection method is an iterative algorithm that approximates a root of a continuous function \( f(x) \) on an interval \( [a, b] \) with a sign change, halving the interval at each step.