Find all the roots of the given function on the given interval. Use preliminary analysis and graphing to find good initial approximations. f(x)=- 18 - sec (X) on [0,20] Continue applying Newton's method to the given values of x as needed. The function has root(s) when x = lod arate answers as needed )
Find all the roots of the given function on the given interval. Use preliminary analysis and graphing to find good initial approximations. f(x)=- 18 - sec (X) on [0,20] Continue applying Newton's method to the given values of x as needed. The function has root(s) when x = lod arate answers as needed )
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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![**Finding the Roots of a Function: An Educational Guide**
---
To determine the roots of the function on a specified interval, follow these steps.
### Problem Statement
**Objective**: Find all the roots of the given function on the given interval using preliminary analysis and graphing to obtain accurate initial approximations.
**Given Function**:
\[ f(x) = \frac{x}{18} - \sec(x) \]
**Interval**:
\[ [0, 2\pi] \]
### Steps for Solution
1. **Preliminary Analysis and Graphing**:
- Plot the function \( f(x) = \frac{x}{18} - \sec(x) \) on the interval [0, 2\pi].
- Identify where the function intersects the x-axis, which gives initial approximations for the roots.
2. **Newton's Method**:
- Once you have the initial approximations \( x_0 \) from the graph, apply Newton's Method iteratively to refine these approximations.
- The iteration formula for Newton's Method is:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
3. **Continue Iterations**:
- Keep applying Newton’s Method until the values converge to the roots (to a specified accuracy).
### Solution
Upon completing the steps above, submit the final roots obtained by rounding to six decimal places. Separate the answers with a comma if there is more than one root.
\[ \text{The function has root(s) when } x = \boxed{\quad\text{(Round to six decimal places as needed. Use a comma to separate answers as needed.)}}\]
### Notes
- Ensure all calculations are accurate and double-check each step if necessary.
- Proper graphing tools or software can significantly aid in identifying good initial approximations.
By following these guidelines and applying Newton's Method, you can accurately determine the roots of the function \( f(x) = \frac{x}{18} - \sec(x) \) on the interval [0, 2\pi].
---
**End of Worksheet**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb02fb262-35a8-4c2d-87d7-48cdf0e8c8b9%2F22e888cc-f2b1-4e2d-9112-44918f186e04%2Fbftkr2d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Roots of a Function: An Educational Guide**
---
To determine the roots of the function on a specified interval, follow these steps.
### Problem Statement
**Objective**: Find all the roots of the given function on the given interval using preliminary analysis and graphing to obtain accurate initial approximations.
**Given Function**:
\[ f(x) = \frac{x}{18} - \sec(x) \]
**Interval**:
\[ [0, 2\pi] \]
### Steps for Solution
1. **Preliminary Analysis and Graphing**:
- Plot the function \( f(x) = \frac{x}{18} - \sec(x) \) on the interval [0, 2\pi].
- Identify where the function intersects the x-axis, which gives initial approximations for the roots.
2. **Newton's Method**:
- Once you have the initial approximations \( x_0 \) from the graph, apply Newton's Method iteratively to refine these approximations.
- The iteration formula for Newton's Method is:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
3. **Continue Iterations**:
- Keep applying Newton’s Method until the values converge to the roots (to a specified accuracy).
### Solution
Upon completing the steps above, submit the final roots obtained by rounding to six decimal places. Separate the answers with a comma if there is more than one root.
\[ \text{The function has root(s) when } x = \boxed{\quad\text{(Round to six decimal places as needed. Use a comma to separate answers as needed.)}}\]
### Notes
- Ensure all calculations are accurate and double-check each step if necessary.
- Proper graphing tools or software can significantly aid in identifying good initial approximations.
By following these guidelines and applying Newton's Method, you can accurately determine the roots of the function \( f(x) = \frac{x}{18} - \sec(x) \) on the interval [0, 2\pi].
---
**End of Worksheet**
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