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The given problem asks to apply **Newton's method** to find the next estimate of the solution \( x_3 \) and to determine the absolute error estimate in \( x_2 \), for the equation:

\[
\sin(1.1x) = \frac{1}{4}x^2 - 4.24
\]

The current estimate for \( x_2 \) is given as \( x_2 = 3.4061122226 \).

### Steps in Newton's Method

Newton's method is given by the iterative formula:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]
Where:
- \( f(x) \) is the function derived from the original equation.
- \( f'(x) \) is the derivative of \( f(x) \).

#### Step 1: Define the Function \( f(x) \)
From the equation, we rewrite it as:
\[
f(x) = \sin(1.1x) - \frac{1}{4}x^2 + 4.24
\]

#### Step 2: Find the Derivative \( f'(x) \)
The derivative of \( f(x) \) with respect to \( x \) is:
\[
f'(x) = 1.1\cos(1.1x) - \frac{1}{2}x
\]

#### Step 3: Calculate \( f(x_2) \) and \( f'(x_2) \)
Substitute \( x_2 = 3.4061122226 \) into \( f(x) \) and \( f'(x) \) to compute the next iteration \( x_3 \).

Let me calculate that for you.Using Newton's method, the next estimate for the solution is:

\[
x_3 = 3.70166860
\]

The absolute error in \( x_2 \) is:

\[
\text{Error} = |x_3 - x_2| = 0.29555638
\]

Would you like further details on the process, or have any specific questions?

### Related Questions:
1. How is the function \( f(x) \) derived from the given equation?
2. Why do we need the derivative \( f'(x) \) in Newton's method?
3. How does the accuracy of Newton's method improve with each iteration?
4. What happens if the initial guess is too far from the actual solution?
5. Can Newton's method fail to converge, and under what conditions?

**Tip:** Always verify that the derivative \( f'(x) \) does not approach zero, as this can lead to errors or slow convergence in Newton's method.

Consider the solution of the following nonlinear equation using Newton's method: sin(1.1x) = (1/4)x^2 - 4.24. After two iterations, the estimated solution is x2 = 3.4061122226. Using Newton's method, find the next estimate and determine the absolute error in x2.

Learn how to apply Newton's method to solve the nonlinear equation sin(1.1x) = (1/4)x^2 - 4.24. This problem guides you through finding the next estimate after two iterations and calculating the absolute error. Ideal for advanced high school or college students.

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