Use Newton's method to approximate the indicated root of the equation correct to six decimal places. The positive root of 4 cos x = = XA
Use Newton's method to approximate the indicated root of the equation correct to six decimal places. The positive root of 4 cos x = = XA
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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Question
![**Title: Solving for Roots Using Newton's Method**
**Objective:**
Learn how to apply Newton's method to approximate the indicated root of an equation to six decimal places.
**Problem:**
Find the positive root of the equation:
\[ 4 \cos x = x^4 \]
**Instructions:**
Newton's method is an iterative numerical technique used to find successively closer approximations to the roots of a real-valued function. Follow these steps to apply Newton's method:
1. **Initial Guess:** Start with an initial guess close to the suspected root.
2. **Function and Derivative:**
- Define \( f(x) = 4 \cos x - x^4 \).
- Compute its derivative \( f'(x) = -4 \sin x - 4x^3 \).
3. **Iteration Formula:** Apply the Newton's iteration formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
4. **Convergence:** Continue the iteration until the value converges to six decimal places.
This method requires understanding of calculus concepts such as derivatives and limits. Ensure each iteration is checked thoroughly for precision.
**Diagram Explanation:**
(No diagrams provided in this transcription.)
Remember that convergence depends on choosing a good initial approximation, as poor choices might lead to divergence or slow convergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec5fb306-23dc-410e-b11b-a666d9f99392%2Fb2c6973d-e4ad-4bef-a777-ce8ee15509b9%2F9dktar_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving for Roots Using Newton's Method**
**Objective:**
Learn how to apply Newton's method to approximate the indicated root of an equation to six decimal places.
**Problem:**
Find the positive root of the equation:
\[ 4 \cos x = x^4 \]
**Instructions:**
Newton's method is an iterative numerical technique used to find successively closer approximations to the roots of a real-valued function. Follow these steps to apply Newton's method:
1. **Initial Guess:** Start with an initial guess close to the suspected root.
2. **Function and Derivative:**
- Define \( f(x) = 4 \cos x - x^4 \).
- Compute its derivative \( f'(x) = -4 \sin x - 4x^3 \).
3. **Iteration Formula:** Apply the Newton's iteration formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
4. **Convergence:** Continue the iteration until the value converges to six decimal places.
This method requires understanding of calculus concepts such as derivatives and limits. Ensure each iteration is checked thoroughly for precision.
**Diagram Explanation:**
(No diagrams provided in this transcription.)
Remember that convergence depends on choosing a good initial approximation, as poor choices might lead to divergence or slow convergence.
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