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The problem appears to be related to Newton's method, which is used to approximate the roots of a function \( f(x) \). 

Given:
- \( p_1 = 3.52 \) with \( p_e = 3 \)
- \( f(3) = 0.48 \)

The question asks for the value of \( f'(3) \), the derivative of \( f(x) \) at \( x = 3 \).

Newton's method formula for finding the next approximation \( p_{n+1} \) is:

\[
p_{n+1} = p_n - \frac{f(p_n)}{f'(p_n)}
\]

Using the information provided:
\[
p_1 = 3.52
\quad \text{and} \quad
p_e = 3
\]

Given that \( p_2 = p_e = 3 \), this implies:

\[
3 = 3.52 - \frac{0.48}{f'(3)}
\]

To solve for \( f'(3) \):

\[
\frac{0.48}{f'(3)} = 3.52 - 3 = 0.52
\]

\[
f'(3) = \frac{0.48}{0.52}
\]

Calculating this:

\[
f'(3) = \frac{0.48}{0.52} \approx -0.923
\]

So the correct choice for \( f'(3) \) is **Option A: -0.923**.

Would you like more details on Newton's method or any other part of this problem? Here are some follow-up questions that might interest you:

1. How does Newton's method work for finding roots?
2. What are the limitations of Newton's method?
3. How do you interpret the derivative \( f'(x) \) in this context?
4. What happens if \( f'(x) \) is close to zero in Newton's method?
5. How would a different initial guess \( p_1 \) affect the convergence?

**Tip:** Newton's method is highly sensitive to the choice of the initial guess. A poor guess can lead to divergence or slow convergence.

Understand Newton's method and how it is used to approximate roots of functions. This problem involves finding the derivative of a function at a specific point using Newton's method. Suitable for advanced high school students and above.

Math Problem Statement

0 . 4 5 x 1 0

ج . 0 . 4 5 X 1 0

ا ل م ع ا د ل ة f ( x )

م ع pe

ب .

ج .

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