Use Newton's method (see Example 8 ) to approximate a root...

Use Newton's method (see Example 8 ) to approximate a root for the given function with the specified value of $x_{0}$. Terminate your sequence when $\left|x_{n+1}-x_{n}\right|<0.001$.
$$
f(x)=e^{x}+\sin x, x_{0}=0
$$

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Key Concepts

Newton's Method

Newton's Method is an iterative technique for finding approximate roots of a differentiable function. Starting from an initial guess, the method refines the estimate by using the formula x_(n+1) = x_n - f(x_n)/f'(x_n), which requires evaluating both the function and its derivative at the current approximation.

Derivative Calculation

Calculating the derivative is a key step in Newton's Method. The derivative informs how the function is changing at a given point, and its value is used to adjust the current approximation toward the root, influencing the convergence of the method.

Convergence Criteria

The convergence criterion is essential for determining when to stop the iterative process. In this context, the iterations are terminated when the absolute difference between consecutive approximations becomes smaller than a predetermined tolerance, ensuring that the root has been approximated with sufficient accuracy.

Iterative Process

The iterative process involves repeated application of the method's recursive formula. With each iteration, a new approximation is computed and the process is repeated until the convergence criterion is met, leading to an increasingly accurate estimate of the root.

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