https://file.aiquickdraw.com/mathbot/file/1726391032972hqz36hga.jpg

The question asks to estimate the root of the equation \( x^3 - 3x^2 + 3x - 4 = 0 \) using the **Newton-Raphson method** and starting with the initial approximation \( x_0 = 2.5 \).

The Newton-Raphson method formula is:

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

Where:
- \( f(x) = x^3 - 3x^2 + 3x - 4 \)
- \( f'(x) \) is the derivative of \( f(x) \).

### Step 1: Compute the derivative \( f'(x) \)
\[
f(x) = x^3 - 3x^2 + 3x - 4
\]
\[
f'(x) = 3x^2 - 6x + 3
\]

### Step 2: Apply the Newton-Raphson formula

1. **Initial guess**: \( x_0 = 2.5 \)

We will first calculate \( f(2.5) \) and \( f'(2.5) \):

\[
f(2.5) = (2.5)^3 - 3(2.5)^2 + 3(2.5) - 4
\]
\[
f(2.5) = 15.625 - 18.75 + 7.5 - 4 = 0.375
\]

\[
f'(2.5) = 3(2.5)^2 - 6(2.5) + 3 = 18.75 - 15 + 3 = 6.75
\]

Now, using the Newton-Raphson formula:

\[
x_1 = 2.5 - \frac{0.375}{6.75} \approx 2.5 - 0.0556 = 2.4444
\]

### Step 3: Refine the approximation (Next iteration)
We can now repeat the process to improve the estimate of the root.

Let me know if you'd like me to continue with additional iterations or provide further details!

### Follow-up Questions:
1. How does the Newton-Raphson method work for estimating roots of nonlinear equations?
2. What are the advantages and disadvantages of using the Newton-Raphson method?
3. How can we tell if the Newton-Raphson method will converge to the correct root?
4. How many iterations are usually sufficient to reach an accurate estimate?
5. What happens if the derivative \( f'(x) \) is close to zero in the Newton-Raphson method?

### Tip:
The Newton-Raphson method can converge very quickly, but it may fail if the initial guess is far from the actual root or if \( f'(x) \) is zero or very small.

Use Newton-Raphson method to estimate the root of the equation x^3 - 3x^2 + 3x - 4 = 0, taking x_0 = 2.5.

Learn how to use the Newton-Raphson method to estimate the root of the equation x^3 - 3x^2 + 3x - 4, starting with the initial guess x = 2.5. Includes step-by-step calculations of function values, derivatives, and refined approximations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation