Calculate two iterations of Newton's Method to approximate zero of the function using the given initial guess. (Round your answers to three decimal places.) f(x) = x5 - 5, x₁ = 1.6 CAN Xn f(xn) f'(x) f'(x) xn- f'(xn)
Calculate two iterations of Newton's Method to approximate zero of the function using the given initial guess. (Round your answers to three decimal places.) f(x) = x5 - 5, x₁ = 1.6 CAN Xn f(xn) f'(x) f'(x) xn- f'(xn)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Problem Statement:
Calculate two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. (Round your answers to three decimal places.)
\[ f(x) = x^5 - 5, \quad x_1 = 1.6 \]
## Iteration Table:
| **n** | \( x_n \) | \( f(x_n) \) | \( f'(x_n) \) | \( \frac{f(x_n)}{f'(x_n)} \) | \( x_n - \frac{f(x_n)}{f'(x_n)} \) |
|------|-----------|--------------|--------------|-------------------------|--------------------|
| 1 | | | | | |
| 2 | | | | | |
## Explanation:
You will need to calculate the following for each iteration:
1. **\( f(x_n) \)** - the value of the function at \( x_n \).
2. **\( f'(x_n) \)** - the value of the derivative of the function at \( x_n \).
3. **\( \frac{f(x_n)}{f'(x_n)} \)** - the ratio of \( f(x_n) \) to \( f'(x_n) \).
4. **\( x_n - \frac{f(x_n)}{f'(x_n)} \)** - the next approximation of the root.
Fill in these values in the provided table for each iteration.
**Note:**
- Make sure your calculations are accurate to three decimal places.
- Use the given initial guess \( x_1 = 1.6 \).
## Submit:
Once you have filled in the table, click the "Submit Answer" button.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc6cf166-7954-4fc5-a62e-be3094a9be9e%2Fdd175954-9be0-4595-8416-b7bc7c301b44%2F8uc2j6_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement:
Calculate two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. (Round your answers to three decimal places.)
\[ f(x) = x^5 - 5, \quad x_1 = 1.6 \]
## Iteration Table:
| **n** | \( x_n \) | \( f(x_n) \) | \( f'(x_n) \) | \( \frac{f(x_n)}{f'(x_n)} \) | \( x_n - \frac{f(x_n)}{f'(x_n)} \) |
|------|-----------|--------------|--------------|-------------------------|--------------------|
| 1 | | | | | |
| 2 | | | | | |
## Explanation:
You will need to calculate the following for each iteration:
1. **\( f(x_n) \)** - the value of the function at \( x_n \).
2. **\( f'(x_n) \)** - the value of the derivative of the function at \( x_n \).
3. **\( \frac{f(x_n)}{f'(x_n)} \)** - the ratio of \( f(x_n) \) to \( f'(x_n) \).
4. **\( x_n - \frac{f(x_n)}{f'(x_n)} \)** - the next approximation of the root.
Fill in these values in the provided table for each iteration.
**Note:**
- Make sure your calculations are accurate to three decimal places.
- Use the given initial guess \( x_1 = 1.6 \).
## Submit:
Once you have filled in the table, click the "Submit Answer" button.
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