Use Newton's method to approximate a zero of the equation 3x + 2x + 2 = 0 as follows. Let #₁ = 2 be the initial approximation. Round all answers to 4 decimal places as needed. a) x₂ b) x3
Use Newton's method to approximate a zero of the equation 3x + 2x + 2 = 0 as follows. Let #₁ = 2 be the initial approximation. Round all answers to 4 decimal places as needed. a) x₂ b) x3
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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![**Title: Using Newton's Method for Approximating Zeros**
**Introduction:**
In this exercise, we will use Newton's method to approximate a zero of the polynomial equation:
\[ 3x^7 + 2x^4 + 2 = 0 \]
**Instructions:**
1. **Initial Approximation:**
- Let \( x_1 = 2 \) be the initial approximation.
2. **Newton's Method Process:**
- Apply Newton’s iterative formula to find successive approximations:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]
- where \( f(x) = 3x^7 + 2x^4 + 2 \) and \( f'(x) \) is the derivative of \( f(x) \).
3. **Rounding:**
- Round all answers to four decimal places as needed.
**Tasks:**
a) Find \( x_2 \approx \) [Enter your answer here]
b) Find \( x_3 \approx \) [Enter your answer here]
**Conclusion:**
By following these steps and applying Newton’s method iteratively, you will be able to approximate the roots of the given polynomial equation effectively. Remember to check your calculations at each step for accuracy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5739d63-6d2f-4eee-96d6-6e1affcc987c%2F7c00658b-e31d-4b5c-aecd-8e99bacada90%2Fvpf5kri_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Using Newton's Method for Approximating Zeros**
**Introduction:**
In this exercise, we will use Newton's method to approximate a zero of the polynomial equation:
\[ 3x^7 + 2x^4 + 2 = 0 \]
**Instructions:**
1. **Initial Approximation:**
- Let \( x_1 = 2 \) be the initial approximation.
2. **Newton's Method Process:**
- Apply Newton’s iterative formula to find successive approximations:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]
- where \( f(x) = 3x^7 + 2x^4 + 2 \) and \( f'(x) \) is the derivative of \( f(x) \).
3. **Rounding:**
- Round all answers to four decimal places as needed.
**Tasks:**
a) Find \( x_2 \approx \) [Enter your answer here]
b) Find \( x_3 \approx \) [Enter your answer here]
**Conclusion:**
By following these steps and applying Newton’s method iteratively, you will be able to approximate the roots of the given polynomial equation effectively. Remember to check your calculations at each step for accuracy.
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