https://file.aiquickdraw.com/mathbot/file/1729905303916qo7wibuj.jpg

To apply Newton's method for this problem, we follow these steps:

Newton's method formula:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

### Analysis and Step-by-Step Solution:

1. **Initial Approximation**:
   - The problem states that the initial approximation \( x_1 = 6 \).

2. **Tangent at \( x_1 = 6 \)**:
   - From the graph, observe the value of \( f(x_1) \) (the y-coordinate of the function at \( x = 6 \)) and estimate \( f'(x_1) \) by examining the slope of the tangent at \( x = 6 \).
   - Draw the tangent line at \( x = 6 \) on the graph and determine where it crosses the x-axis. This intersection is the next approximation \( x_2 \).

3. **Repeating for \( x_2 \)**:
   - Once \( x_2 \) is estimated, use the same process: observe the y-coordinate \( f(x_2) \), approximate the slope \( f'(x_2) \), and determine where the tangent line at \( x_2 \) intersects the x-axis.
   - This intersection will give the next approximation \( x_3 \).

4. **Estimations**:
   - Since this problem is graphical, we’ll use visual approximations for \( f(x) \) and \( f'(x) \) based on the tangent lines drawn.

Please let me know if you would like further clarification or assistance with estimating values directly from the graph, as I can guide you through interpreting the points and tangents.

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### Related Questions:

1. How does Newton's method converge to the root of a function?
2. What are some conditions where Newton's method might fail to converge?
3. Can Newton's method be used if the derivative \( f'(x) \) is zero at any point?
4. How do we find the derivative \( f'(x) \) graphically?
5. What are some alternatives to Newton's method for root-finding?

**Tip:** Newton’s method can be very fast if the initial guess is close to the actual root, but it may diverge if the initial guess is too far.

Suppose that Newton's method is used to approximate the root s of the equation f(x) = 0 with initial approximation x₁ = 6. Draw the tangent lines that are used to find x₂ and x₃, and estimate the numerical values of x₂ and x₃.

Learn how to apply Newton's method to approximate the root of a function. This example explores estimating values of x₂ and x₃ using tangent lines, starting from an initial approximation x₁ = 6. Suitable for high school calculus students.

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