Understanding the Bisection Method
The Bisection Method is a root-finding algorithm that repeatedly bisects an interval and then selects a sub-interval in which the function changes sign. Because the function is continuous, the Intermediate Value Theorem guarantees that there is at least one root within the interval if $f(a)$ and $f(b)$ have opposite signs.
The Problem
Find the root of the equation $f(x) = x^3 - x - 4 = 0$ in the interval $[1, 2]$ using the bisection method, accurate to 3 decimal places.
Step-by-Step Solution
Let $f(x) = x^3 - x - 4$. Given $a = 1$ and $b = 2$:
- $f(1) = 1^3 - 1 - 4 = -4$ (Negative)
- $f(2) = 2^3 - 2 - 4 = 2$ (Positive)
Since $f(1) < 0$ and $f(2) > 0$, the root lies between 1 and 2.
Iteration Table
| Iteration | $a$ | $b$ | $x_{mid} = (a+b)/2$ | $f(x_{mid})$ | New Interval |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 1.5 | -1.125 | [1.5, 2] |
| 2 | 1.5 | 2 | 1.75 | 0.609 | [1.5, 1.75] |
| 3 | 1.5 | 1.75 | 1.625 | -0.334 | [1.625, 1.75] |
| 4 | 1.625 | 1.75 | 1.6875 | 0.121 | [1.625, 1.6875] |
| 5 | 1.625 | 1.6875 | 1.65625 | -0.111 | [1.65625, 1.6875] |
| 6 | 1.65625 | 1.6875 | 1.67188 | 0.003 | [1.65625, 1.67188] |
| 7 | 1.65625 | 1.67188 | 1.66407 | -0.054 | [1.66407, 1.67188] |
| 8 | 1.66407 | 1.67188 | 1.66798 | -0.026 | [1.66798, 1.67188] |
| 9 | 1.66798 | 1.67188 | 1.66993 | -0.011 | [1.66993, 1.67188] |
| 10 | 1.66993 | 1.67188 | 1.67091 | -0.004 | [1.67091, 1.67188] |
Continuing this process until the change in $x$ is sufficiently small, we converge to the root.
Final Answer
After several iterations, the value of the root converges to approximately 1.671.