Use newton raphson to determine the real root of F(x)=x^3-2x^2-2x+1. Provide a table. X0=1

Let's apply the Newton-Raphson method to the function \( f(x) = x^3 - 2x^2 - 2x + 1 \) with an initial guess \( x_0 = 1 \).

### Step-by-Step Process

1. **Given function**:  
   \[
   f(x) = x^3 - 2x^2 - 2x + 1
   \]

2. **Derivative of the function**:  
   \[
   f'(x) = 3x^2 - 4x - 2
   \]

3. **Newton-Raphson Iteration Formula**:  
   \[
   x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
   \]

### Iterations:

#### Iteration 0 (Initial Guess \( x_0 = 1 \)):
\[
f(1) = 1^3 - 2(1)^2 - 2(1) + 1 = -2
\]
\[
f'(1) = 3(1)^2 - 4(1) - 2 = -3
\]
\[
x_1 = 1 - \frac{-2}{-3} = 1 - \frac{2}{3} = \frac{1}{3} = 0.3333
\]

#### Iteration 1:
\[
f(0.3333) = 0.3333^3 - 2(0.3333)^2 - 2(0.3333) + 1 = 0.148
\]
\[
f'(0.3333) = 3(0.3333)^2 - 4(0.3333) - 2 = -3.0
\]
\[
x_2 = 0.3333 - \frac{0.148}{-3.0} = 0.3333 + 0.0493 = 0.3826
\]

#### Iteration 2:
\[
f(0.3826) = 0.0092
\]
\[
f'(0.3826) = -2.8584
\]
\[
x_3 = 0.3826 - \frac{0.0092}{-2.8584} = 0.3826 + 0.0032 = 0.3859
\]

#### Iteration 3:
\[
f(0.3859) \approx 0.0
\]
\[
f'(0.3859) = -2.8499
\]
\[
x_4 = 0.3859
\]
At this point, the method has converged to the root \( x \approx 0.3859 \).

### Table of Iterations:

| Iteration | \( x_n \)  | \( f(x_n) \) | \( f'(x_n) \) | \( x_{n+1} \)  |
|-----------|------------|--------------|---------------|----------------|
| 0         | 1.0000     | -2.0000      | -3.0000       | 0.3333         |
| 1         | 0.3333     | 0.1481       | -3.0000       | 0.3826         |
| 2         | 0.3826     | 0.0092       | -2.8584       | 0.3859         |
| 3         | 0.3859     | 0.0000       | -2.8499       | 0.3859         |

The real root of the equation \( f(x) = 0 \) is approximately \( x = 0.3859 \).

Would you like more details or assistance with another problem?

Here are five follow-up questions:
1. How accurate is the root approximation after the 3rd iteration?
2. What happens if the initial guess \( x_0 \) is farther from the actual root?
3. Can Newton-Raphson fail to converge for some functions?
4. How does the method perform on non-polynomial equations?
5. What are the limitations of the Newton-Raphson method?

**Tip:** Always ensure the derivative \( f'(x) \) does not approach zero, as this can cause divergence in the Newton-Raphson method.

Use Newton-Raphson to determine the real root of F(x)=x^3-2x^2-2x+1. Provide a table. X0=1

Learn how to apply the Newton-Raphson method to determine the real root of the function F(x)=x^3-2x^2-2x+1. This problem demonstrates the step-by-step process, iterations, and key formulas used in numerical methods. Suitable for undergraduate students in calculus and numerical analysis.

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