{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) = x ln (x + 1) -1 ; x₀ = 1.7

Verified step by step guidance

1. 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 the next approximation is given by: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.
2. Step 2: Calculate the derivative of the function. Given f(x) = x \ln(x + 1) - 1, we need to find f'(x). Using the product rule and chain rule, f'(x) = \ln(x + 1) + \frac{x}{x + 1}.
3. Step 3: Set up the iterative process. Start with the initial approximation x_0 = 1.7. Use the formula x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} to find the next approximation.
4. Step 4: Perform the iterations. Calculate x_1 using x_0 = 1.7, then calculate x_2 using x_1, and so on. Continue this process until two successive approximations agree to five decimal places.
5. Step 5: Create a table to organize your results. For each iteration, record the values of x_n, f(x_n), f'(x_n), and x_{n+1}. Stop when x_{n+1} and x_n agree to five decimal places after rounding.