Use Newton's method to approximate a root of the equation cos( s(x² + 3) = x³ as follows. Let x₁ = 2 be the initial approximation. The second approximation is
Use Newton's method to approximate a root of the equation cos( s(x² + 3) = x³ as follows. Let x₁ = 2 be the initial approximation. The second approximation is
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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![**Newton's Method for Approximating Roots**
To approximate a root of the equation \(\cos(x^2 + 3) = x^3\) using Newton's method, follow these steps:
1. **Initial Approximation**: Start with an initial approximation \(x_1 = 2\).
2. **Second Approximation**: We aim to find the second approximation \(x_2\).
In Newton's method, each approximation is refined by using the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \(f(x)\) is the function for which we are trying to find the root, and \(f'(x)\) is its derivative.
### Example Calculation (Hypothetical):
For the given function, solve for \(x_2\) using the formula with the initial value \(x_1 = 2\).
**Note:** Further details and the derivative evaluation should be calculated to obtain \(x_2\), helping to refine the approximation iteratively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F442cdeaf-d6f5-4c90-9eca-defd0049fe9f%2Fe09176db-0806-43cb-8063-d7c463e9f903%2Fdc4wta9_processed.png&w=3840&q=75)
Transcribed Image Text:**Newton's Method for Approximating Roots**
To approximate a root of the equation \(\cos(x^2 + 3) = x^3\) using Newton's method, follow these steps:
1. **Initial Approximation**: Start with an initial approximation \(x_1 = 2\).
2. **Second Approximation**: We aim to find the second approximation \(x_2\).
In Newton's method, each approximation is refined by using the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \(f(x)\) is the function for which we are trying to find the root, and \(f'(x)\) is its derivative.
### Example Calculation (Hypothetical):
For the given function, solve for \(x_2\) using the formula with the initial value \(x_1 = 2\).
**Note:** Further details and the derivative evaluation should be calculated to obtain \(x_2\), helping to refine the approximation iteratively.
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