{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.


f(x) = e⁻ˣ - x; x₀ = ln 2

Verified step by step guidance

1. Newton's method is an iterative process used to approximate the roots of a real-valued function. The formula for Newton's method is given by: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \).
2. First, we need to find the derivative of the function \( f(x) = e^{-x} - x \). The derivative is \( f'(x) = -e^{-x} - 1 \).
3. Using the initial approximation \( x_0 = \ln 2 \), we calculate \( f(x_0) \) and \( f'(x_0) \). Substitute \( x_0 \) into the function: \( f(x_0) = e^{- ext{ln} 2} - \ln 2 \) and \( f'(x_0) = -e^{- ext{ln} 2} - 1 \).
4. Now, apply Newton's method formula to find \( x_1 \): \( x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \). Substitute the values of \( f(x_0) \) and \( f'(x_0) \) to compute \( x_1 \).
5. Repeat the process to find \( x_2 \) using \( x_1 \): \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \). Calculate \( f(x_1) \) and \( f'(x_1) \) using the function and its derivative, then substitute these into the formula to find \( x_2 \).