Explanation:
The bisection method is a root-finding algorithm that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. To apply the bisection method to the equation x^3 - x - 9 = 0, you need to find two initial values a and b such that f(a) and f(b) have opposite signs. This means that there is at least one root of the equation in the interval [a, b].
Let’s take a = 2 and b = 3 as our initial interval. We can see that f(2) = 2^3 - 2 - 9 = -1 and f(3) = 3^3 - 3 - 9 = 18, so f(a) and f(b) have opposite signs.
Now we can apply the bisection method as follows:
Calculate the midpoint c = (a + b)/2 = (2 + 3)/2 = 2.5.
Evaluate f© = 2.5^3 - 2.5 - 9 ≈ 5.625.
Since f© > 0 and f(a) < 0, there must be a root in the interval [a, c]. So we set b = c and repeat the process.
Calculate the new midpoint c = (a + b)/2 = (2 + 2.5)/2 = 2.25.
Evaluate f© ≈ 1.765625.
Since f© > 0 and f(a) < 0, there must be a root in the interval [a, c]. So we set b = c and repeat the process.
We can continue this process until we reach the desired level of accuracy.