The Bisection Method is a numerical technique used to find the root of a continuous function within a specific interval. This method works by repeatedly halving the interval and selecting the subinterval that contains the root.

Problem:

We are asked to find the real root of the equation: $f(x) = x^3 + 9x + 1 = 0$

Steps of the Bisection Method:

1. Choose an initial interval $[a, b]$ such that $f(a)$ and $f(b)$ have opposite signs, i.e., $f(a) \cdot f(b) < 0$. This ensures that the root lies between $a$ and $b$ by the Intermediate Value Theorem.
2. Compute the midpoint $m$ of the interval: $m = \frac{a + b}{2}$
3. Evaluate $f(m)$. If $f(m) = 0$, then $m$ is the root.
4. If $f(a) \cdot f(m) < 0$, then the root lies between $a$ and $m$. Set $b = m$.
5. If $f(m) \cdot f(b) < 0$, then the root lies between $m$ and $b$. Set $a = m$.
6. Repeat steps 2-5 until the interval is sufficiently small, i.e., $|b - a|$ is less than a given tolerance.

Solving the Equation Using the Bisection Method

Step 1: Choose the interval

Let's try an initial interval $a = -2$ and $b = -1$.

• $f(-2) = (-2)^3 + 9(-2) + 1 = -8 - 18 + 1 = -25$
• $f(-1) = (-1)^3 + 9(-1) + 1 = -1 - 9 + 1 = -9$

Since both $f(-2)$ and $f(-1)$ are negative, we need to adjust the interval. Let’s change the interval to $[-2, 0]$.

• $f(-2) = -25$ (negative)
• $f(0) = 0^3 + 9(0) + 1 = 1$ (positive)

Since the signs of $f(-2)$ and $f(0)$ are opposite, we can proceed with this interval.

Step 2: Compute the midpoint

The midpoint of the interval $[-2, 0]$ is: $m = \frac{-2 + 0}{2} = -1$

Step 3: Evaluate $f(m)$

Now, we evaluate $f(-1)$: $f(-1) = (-1)^3 + 9(-1) + 1 = -1 - 9 + 1 = -9$

Since $f(-1)$ is negative and $f(0)$ is positive, the root lies between $[-1, 0]$.

Step 4: Narrow the interval

The new interval is $[-1, 0]$. We now compute the midpoint of this new interval: $m = \frac{-1 + 0}{2} = -0.5$

Evaluate $f(-0.5)$: $f(-0.5) = (-0.5)^3 + 9(-0.5) + 1 = -0.125 - 4.5 + 1 = -3.625$

Since $f(-0.5)$ is negative and $f(0)$ is positive, the root lies between $[-0.5, 0]$.

Step 5: Continue narrowing the interval

Next, the midpoint of the interval $[-0.5, 0]$ is: $m = \frac{-0.5 + 0}{2} = -0.25$

Evaluate $f(-0.25)$: $f(-0.25) = (-0.25)^3 + 9(-0.25) + 1 = -0.015625 - 2.25 + 1 = -1.265625$

Since $f(-0.25)$ is negative and $f(0)$ is positive, the root lies between $[-0.25, 0]$.

We would continue this process until the interval is sufficiently small (e.g., when $|b - a| < \epsilon$ for some small tolerance $\epsilon$).

Final Answer

By continuing this method, we would obtain the real root to a desired accuracy. Would you like me to calculate a specific number of iterations or further explain the process?

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Further Questions:

1. What are the advantages of using the Bisection Method over other numerical methods like Newton's method?
2. How do you choose the initial interval for the Bisection Method effectively?
3. Can the Bisection Method fail, and if so, how do you address it?
4. How can we improve the convergence rate of the Bisection Method?
5. What is the stopping criterion for the Bisection Method, and how do we select it?

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Tip: Always ensure that the function changes signs at the endpoints of your chosen interval; otherwise, the Bisection Method will not be applicable.