Use newton raphson to determine the real root of F(x)=x^3-2x^2-2x+1. Provide a table

The Newton-Raphson method is an iterative numerical method used to find the roots of a function \( f(x) = 0 \). The iteration formula is given by:

\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

For the function \( f(x) = x^3 - 2x^2 - 2x + 1 \), we first need to compute its derivative:

\[
f'(x) = 3x^2 - 4x - 2
\]

Now, we can apply the Newton-Raphson method using an initial guess \( x_0 \) to compute successive values \( x_1, x_2, x_3, \dots \) until convergence.

Let's assume an initial guess of \( x_0 = 1.5 \), and I'll calculate several iterations using the formula:

\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

Now, I will calculate the values and tabulate the results.Here is the table of values for the Newton-Raphson method, starting with an initial guess of \( x_0 = 1.5 \):

| Iteration | \( x_n \) | \( f(x_n) \) | \( f'(x_n) \) | \( x_{n+1} \) |
|-----------|-----------|--------------|---------------|---------------|
| 0         | 1.5       | -3.125       | -1.25         | -1.0          |
| 1         | -1.0      | 0.0          | 5.0           | -1.0          |

After the first iteration, the method quickly converges to \( x = -1.0 \), which is the root of the equation.

Would you like more details or have any questions?

Here are five follow-up questions:
1. How does the Newton-Raphson method converge so quickly?
2. What would happen if we started with a different initial guess?
3. Can the method fail or diverge?
4. How do the values of \( f'(x) \) affect the method's accuracy?
5. Could the Newton-Raphson method be applied to functions with multiple roots?

**Tip:** The convergence of Newton-Raphson is highly dependent on the initial guess and the function's behavior near the root.

Use Newton-Raphson to determine the real root of F(x)=x^3-2x^2-2x+1. Provide a table.

Learn how to apply the Newton-Raphson method to find the real root of the cubic function x^3 - 2x^2 - 2x + 1. This detailed solution includes a step-by-step explanation, key formulas, and a table of iterations. Ideal for students in Calculus I/II.

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