An approximate solution of the equation $ x^3 - 17 = 0 $ is to be obtained using the Newton-Raphson method. If the initial guess is $ x_0 = 2 $, the value at the end of the first iteration is $ x_1 = $ ________ (rounded off to two decimal places).

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An approximate solution of the equation $ x^3 - 17 = 0 $ is to be obtained using the Newton-Raphson method. If the initial guess is $ x_0 = 2 $, the value at the end of the first iteration is $ x_1 = $  ________ (rounded off to two decimal places).

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The Newton-Raphson method is given by the formula: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ For the equation $ f(x) = x^3 - 17 $, we have: $$ f'(x) = 3x^2 $$ Starting with the initial guess $ x_0 = 2 $, we can calculate the value of $ x_1 $.  Step 1: Calculate $ f(x_0) $ and $ f'(x_0) $: $$ f(x_0) = 2^3 - 17 = 8 - 17 = -9 $$ $$ f'(x_0) = 3 \times 2^2 = 3 \times 4 = 12 $$  Step 2: Apply the Newton-Raphson formula: $$ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 2 - \frac{-9}{12} = 2 + 0.75 = 2.75 $$ Therefore, the value at the end of the first iteration is $ x_1 = 2.75 $.

An approximate solution of the equation \( x^3 - 17 = 0 \) is to be obtained using the Newton-Raphson method. If the initial guess is \( x_0 = 2 \), the value at the end of the first iteration is \( x_1 = \) ________ (rounded off to two decimal places).

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