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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.

Determine the formula involving b_0 - a_0 and ε for the number of steps that must be taken in the bisection method to guarantee that |r - c_n| ≤ ε.

This guide explains how to calculate the minimum number of steps required in the bisection method for root-finding, ensuring that the error is within a given tolerance. It uses the formula n ≥ (log(b_0 - a_0) - log ε) / log 2 - 1, which is derived from logarithmic properties and the halving behavior of the bisection method.

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