Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then fine f(x) = x - 6.9x + 10.79x 4.851 Newton's method: Graphing Utility: X = 美= (smallest value) %3= X%3D (largest value) Need Help? Read It Talk to a Tutor
Approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then fine f(x) = x - 6.9x + 10.79x 4.851 Newton's method: Graphing Utility: X = 美= (smallest value) %3= X%3D (largest value) Need Help? Read It Talk to a Tutor
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Approximating Zeros of the Function Using Newton's Method**
To approximate the zero(s) of the function using Newton's Method, follow these instructions. Newton's Method is an iterative root-finding algorithm that produces increasingly accurate approximations of the roots (or zeroes) of a real-valued function.
Given function:
\[ f(x) = x^3 - 6.9x^2 + 10.79x - 4.851 \]
### Newton's Method Steps:
1. Choose an initial guess \(x_0\).
2. Apply the iterative formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \(f'(x)\) is the derivative of \(f(x)\).
3. Continue the process until the difference between successive approximations is less than 0.001.
4. Record and verify the smallest and largest value of \(x\) where \(f(x) = 0\).
### Forms and Values:
Newton's method requires iteration and the entry of \(x\) values:
- \( x = \) _______________
- \( x = \) _______________
- \( x = \) _______________
Graphing Utility (to verify and compare results):
- \( x = \) _______________ (smallest value)
- \( x = \) _______________ (largest value)
### Need Help?
- **Read It**
- **Talk to a Tutor**
This information helps students and learners systematically find and verify zeroes of polynomials using both Newton's method and graphing utilities. The provided boxes to fill in guide users through multiple iterations and comparison of results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d04d9e8-9cf4-4d61-b8ac-39958688637f%2Ff6592a7d-9f4e-4777-963a-6f11631c9fa6%2F9umvse5.jpeg&w=3840&q=75)
Transcribed Image Text:**Approximating Zeros of the Function Using Newton's Method**
To approximate the zero(s) of the function using Newton's Method, follow these instructions. Newton's Method is an iterative root-finding algorithm that produces increasingly accurate approximations of the roots (or zeroes) of a real-valued function.
Given function:
\[ f(x) = x^3 - 6.9x^2 + 10.79x - 4.851 \]
### Newton's Method Steps:
1. Choose an initial guess \(x_0\).
2. Apply the iterative formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \(f'(x)\) is the derivative of \(f(x)\).
3. Continue the process until the difference between successive approximations is less than 0.001.
4. Record and verify the smallest and largest value of \(x\) where \(f(x) = 0\).
### Forms and Values:
Newton's method requires iteration and the entry of \(x\) values:
- \( x = \) _______________
- \( x = \) _______________
- \( x = \) _______________
Graphing Utility (to verify and compare results):
- \( x = \) _______________ (smallest value)
- \( x = \) _______________ (largest value)
### Need Help?
- **Read It**
- **Talk to a Tutor**
This information helps students and learners systematically find and verify zeroes of polynomials using both Newton's method and graphing utilities. The provided boxes to fill in guide users through multiple iterations and comparison of results.
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