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We are tasked with using Newton's method to approximate the root of the equation:

\[
5 \sin(x) = x
\]

Newton's method uses the iterative formula:

\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

where \( f(x) = 5 \sin(x) - x \), and \( f'(x) \) is its derivative. Let's find the derivative of \( f(x) \):

\[
f'(x) = 5 \cos(x) - 1
\]

Given that the initial approximation is \( x_1 = 1 \), we will now calculate the second approximation \( x_2 \) and the third approximation \( x_3 \).

1. **Calculate \( f(x_1) \) and \( f'(x_1) \)**:
   - \( f(x_1) = 5 \sin(1) - 1 \)
   - \( f'(x_1) = 5 \cos(1) - 1 \)

2. **Apply Newton's method to get \( x_2 \)**:
   \[
   x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}
   \]

3. **Apply the same process to calculate \( x_3 \)**:
   \[
   x_3 = x_2 - \frac{f(x_2)}{f'(x_2)}
   \]

Let me compute these approximations for you.Using Newton's method, the second and third approximations for the root of the equation \( 5 \sin(x) = x \) are:

- The second approximation, \( x_2 \approx -0.885 \)
- The third approximation, \( x_3 \approx 0.493 \)

Would you like more details or have any questions on the process?

Here are some related questions to explore further:
1. How does Newton's method converge to a root?
2. What are the advantages of Newton's method compared to other root-finding methods?
3. Can Newton's method fail to converge? Under what conditions?
4. How do we interpret the physical meaning of the equation \( 5 \sin(x) = x \)?
5. How sensitive is Newton's method to the choice of the initial guess?

**Tip:** Always check the derivative when using Newton's method—if it's near zero, the method may fail due to division by very small numbers.

Use Newton's method to find the second and third approximation of a root of 5sin(x) = x, starting with x1 = 1 as the initial approximation.

Learn how to use Newton's method to find the second and third approximations for a root of the equation 5sin(x) = x. This problem involves key calculus and numerical methods concepts, including iterative root-finding techniques. Suitable for college-level students.

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