In the bisection method, the selection of an initial interval $[x_0, x_1]$ is crucial. The method is used to find roots of a function $f(x)$, where we iteratively narrow down an interval that contains the root by repeatedly bisecting it. The key requirement is that the function changes sign over the interval, i.e., $f(x_0) \cdot f(x_1) < 0$, meaning $f(x_0)$ and $f(x_1)$ must have opposite signs.

Why $x_0 = 0.9$ Might Be Selected

In your case, $x_0 = 0.9$ could have been selected based on:

1. Preliminary Guess for the Root:

   • $x_0 = 0.9$ might be close to a root based on previous information or analysis of the function.
   • It could be based on visual inspection of the graph, prior estimates, or a known range where the function crosses the x-axis.

2. Sign Change Condition:

   • You would select $x_0 = 0.9$ if there is an indication that the function $f(x)$ changes sign near this value. For the bisection method to work, $f(x_0) \cdot f(x_1) < 0$. This ensures that the root lies between $x_0$ and $x_1$.

3. Neighborhood of Interest:

   • If the problem suggests that the root lies near $x = 0.9$, that would naturally be a starting point for the interval. For instance, if the root is suspected to be around 1, an initial interval such as $[0.9, 1]$ might be chosen.

Example of Selecting the Interval

Suppose you are solving an equation like $f(x) = x^3 - x - 1 = 0$ and want to apply the bisection method.

• Step 1: You evaluate the function at different points to find an interval where the sign changes. For instance:

  • $f(0.9) = 0.9^3 - 0.9 - 1 \approx -0.271$
  • $f(1.1) = 1.1^3 - 1.1 - 1 \approx 0.331$

• Step 2: Since $f(0.9) \cdot f(1.1) < 0$ (i.e., $-0.271 \times 0.331 < 0$), we know the root lies between $x_0 = 0.9$ and $x_1 = 1.1$.

• Step 3: Start applying the bisection method to reduce the interval iteratively.

Summary:

The interval for $x_0 = 0.9$ is likely selected because:

• It is close to where a root is suspected.
• The function $f(x)$ changes sign between $0.9$ and another value $x_1$, making it a suitable starting point for the bisection method.

Would you like further clarification or specific examples?

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Follow-up Questions:

1. What function are you trying to find the root of?
2. How did you determine that $x_0 = 0.9$ was a good starting point?
3. What is the value of $x_1$ in this case?
4. Would you like a step-by-step example of the bisection method?
5. Are you looking to improve the efficiency of the interval selection process?

Tip: The bisection method works best when you first evaluate the function over a broad interval to identify a sign change, and then narrow it down by repeatedly halving the interval.