The image shows a formula for determining the minimum number of steps, $n$, required for the bisection method to achieve an error tolerance $\varepsilon$, such that $|r - c_n| \leq \varepsilon$, where $r$ is the actual root and $c_n$ is the approximate root after $n$ iterations.

The formula given is:

$n \geq \frac{\log(b_0 - a_0) - \log \varepsilon}{\log 2} - 1$

where:

• $b_0$ and $a_0$ are the initial interval endpoints,
• $\varepsilon$ is the error tolerance.

This expression essentially derives the number of iterations $n$ needed to guarantee that the root approximation error is less than or equal to $\varepsilon$. It arises from the fact that the bisection method halves the interval size with each iteration.

Would you like a more detailed explanation on how this formula is derived or more examples of its application?

Here are 5 related questions you might find useful:

1. What assumptions are required for the bisection method to converge?
2. How does the choice of $\varepsilon$ influence the number of steps in the bisection method?
3. Can you apply this formula for a specific function and interval?
4. How does the bisection method compare with other root-finding methods like Newton’s method in terms of convergence speed?
5. What happens if the initial interval $[a_0, b_0]$ does not contain the root?

Tip: The bisection method guarantees convergence if the function is continuous and changes sign over the initial interval, but it may take many iterations compared to other methods.