Use Newton's method to approximate a root of the equation Let £₁ = 1 be the initial approximation. The second approximation ₂ is and the third approximation is = 4 x as follows.
Use Newton's method to approximate a root of the equation Let £₁ = 1 be the initial approximation. The second approximation ₂ is and the third approximation is = 4 x as follows.
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: Approximation of a Root Using Newton's Method**
Use Newton's method to approximate a root of the equation \( e^{0.5x^2} = 4 - x \) as follows. Let \( x_1 = 1 \) be the initial approximation.
1. **Second Approximation (\( x_2 \))**:
- [Text Box for Input]
2. **Third Approximation (\( x_3 \))**:
- [Text Box for Input]
**Instructions:**
Using Newton's method involves the following iterative formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
- Define the function \( f(x) = e^{0.5x^2} - 4 + x \).
- Derive the function \( f'(x) \).
- Start with the initial guess \( x_1 = 1 \).
- Calculate \( x_2 \) and \( x_3 \) using the iterative formula. Enter your results in the provided text boxes.
For a detailed understanding, check the solutions after attempting the calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F442cdeaf-d6f5-4c90-9eca-defd0049fe9f%2F3a02274c-994e-4801-848d-080fd5b4f57c%2Fcmmgzhd_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Approximation of a Root Using Newton's Method**
Use Newton's method to approximate a root of the equation \( e^{0.5x^2} = 4 - x \) as follows. Let \( x_1 = 1 \) be the initial approximation.
1. **Second Approximation (\( x_2 \))**:
- [Text Box for Input]
2. **Third Approximation (\( x_3 \))**:
- [Text Box for Input]
**Instructions:**
Using Newton's method involves the following iterative formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
- Define the function \( f(x) = e^{0.5x^2} - 4 + x \).
- Derive the function \( f'(x) \).
- Start with the initial guess \( x_1 = 1 \).
- Calculate \( x_2 \) and \( x_3 \) using the iterative formula. Enter your results in the provided text boxes.
For a detailed understanding, check the solutions after attempting the calculations.
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