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The image asks for solving the approximation of the root \( r \) of the equation \( f(x) = x^3 - 5x + 4 = 0 \) using Newton's method. Given the starting point \( x_0 = 1 \), we need to calculate the tangent line and the approximation \( x_1 \).

To solve this step-by-step, we can follow Newton's method, which involves the formula:

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

### Step 1: Calculate \( f(x) \) and \( f'(x) \)
Given \( f(x) = x^3 - 5x + 4 \), the derivative \( f'(x) \) is:

\[
f'(x) = 3x^2 - 5
\]

### Step 2: Evaluate at \( x_0 = 1 \)

- \( f(1) = 1^3 - 5(1) + 4 = 1 - 5 + 4 = 0 \)
- \( f'(1) = 3(1)^2 - 5 = 3 - 5 = -2 \)

Since \( f(1) = 0 \), the next approximation \( x_1 \) will be:

\[
x_1 = 1 - \frac{f(1)}{f'(1)} = 1 - \frac{0}{-2} = 1
\]

Thus, the tangent line approximation does not change the point in this iteration.

### Conclusion:
- The tangent line formula at \( x = 1 \) is \( y = -2(x - 1) \) (since the slope is \( f'(1) = -2 \)).
- The next approximation \( x_1 = 1 \) remains the same as the starting point \( x_0 = 1 \) because \( f(1) = 0 \).

Would you like more detailed steps or have any specific questions?

Here are 5 related questions:
1. What happens in Newton's method if \( f(x_0) = 0 \)?
2. Can you apply Newton's method to a non-differentiable function?
3. How does Newton's method perform if the derivative \( f'(x_n) \) is very close to zero?
4. What are the advantages of Newton's method compared to other root-finding algorithms?
5. How many iterations does Newton's method typically take to converge to a root?

**Tip**: Newton's method requires a good initial guess. If the starting point is too far from the actual root, the method may not converge or may converge to a different root.

Solve the approximation for the root of f(x) = x^3 - 5x + 4 = 0 using Newton's method with the initial guess of x_0 = 1.

Learn how to approximate the root of the cubic equation f(x) = x^3 - 5x + 4 = 0 using Newton's method. This step-by-step guide covers key calculus concepts, including the Newton's method formula, derivatives, and the process of root approximation with an initial guess of x_0 = 1. Suitable for advanced high school and college-level students.

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