Question

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = sin x + x - 1; x₀ = 0.5

Accepted Answer

Step 1: Understand Newton's Method. Newton's method is an iterative process used to approximate the roots of a real-valued function. The formula for Newton's method is: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. Step 2: Calculate the derivative of the function. For the given function f(x) = sin(x) + x - 1, the derivative f'(x) is cos(x) + 1. Step 3: Set up the iterative formula. Using the initial approximation x₀ = 0.5, apply the Newton's method formula: x_{n+1} = x_n - \frac{sin(x_n) + x_n - 1}{cos(x_n) + 1}. Step 4: Perform the iterations. Start with x₀ = 0.5 and calculate x₁ using the formula. Continue calculating successive approximations x₂, x₃, etc., until two successive approximations agree to five decimal places. Step 5: Create a table to track the iterations. The table should include columns for n (iteration number), x_n (current approximation), f(x_n), f'(x_n), and the difference |x_{n+1} - x_n|. Stop when the difference is less than 0.00001.

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Key Concepts

Newton's Method

Convergence Criteria

Function and Derivative Evaluation