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 find the zero(s) to three decimal places using a graphing utility and compare the results. (x) - x - cos x Newton's method: Graphing utility:
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 find the zero(s) to three decimal places using a graphing utility and compare the results. (x) - x - cos x Newton's method: Graphing utility:
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Approximating Zeros of a Function Using Newton's Method
**Problem Statement:**
Approximate the zero(s) of the function \( f(x) = x^3 - \cos x \). Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then, find the zero(s) to three decimal places using a graphing utility and compare the results.
**Function:**
\[ f(x) = x^3 - \cos x \]
**Methodologies:**
**Newton's Method:**
\[ x = \text{____} \]
**Graphing Utility:**
\[ x = \text{____} \]
1. **Newton's Method:**
- Start with an initial guess \( x_0 \).
- Use the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \( f'(x) \) is the derivative of \( f(x) \).
- Continue iterating until the difference between successive approximations \( |x_{n+1} - x_n| \) is less than 0.001.
2. **Graphing Utility:**
- Use a graphing utility (such as a graphing calculator or software) to find the zero(s) of the function \( f(x) = x^3 - \cos x \).
- Report the zero(s) accurately to three decimal places.
3. **Comparison:**
- Compare the zero(s) found using Newton's Method with the zero(s) found using the graphing utility to assess accuracy.
Make sure you perform each step carefully and verify your results for consistency.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d04d9e8-9cf4-4d61-b8ac-39958688637f%2F62939095-ec97-4dbb-ad04-1631b08732ee%2Fmo9tup9.png&w=3840&q=75)
Transcribed Image Text:### Approximating Zeros of a Function Using Newton's Method
**Problem Statement:**
Approximate the zero(s) of the function \( f(x) = x^3 - \cos x \). Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. Then, find the zero(s) to three decimal places using a graphing utility and compare the results.
**Function:**
\[ f(x) = x^3 - \cos x \]
**Methodologies:**
**Newton's Method:**
\[ x = \text{____} \]
**Graphing Utility:**
\[ x = \text{____} \]
1. **Newton's Method:**
- Start with an initial guess \( x_0 \).
- Use the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \( f'(x) \) is the derivative of \( f(x) \).
- Continue iterating until the difference between successive approximations \( |x_{n+1} - x_n| \) is less than 0.001.
2. **Graphing Utility:**
- Use a graphing utility (such as a graphing calculator or software) to find the zero(s) of the function \( f(x) = x^3 - \cos x \).
- Report the zero(s) accurately to three decimal places.
3. **Comparison:**
- Compare the zero(s) found using Newton's Method with the zero(s) found using the graphing utility to assess accuracy.
Make sure you perform each step carefully and verify your results for consistency.
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