Solve exp(x) - x = 0: Newton-Raphson Method & Approximation

To solve the equation exp(x) - x = 0, we can use numerical methods or approximation techniques. One common method is to use the Newton-Raphson method. \n\nLet's define a function f(x) = exp(x) - x. We want to find the value of x for which f(x) is equal to zero. \n\nThe Newton-Raphson method involves iterating the following equation: \n\nx[n+1] = x[n] - f(x[n])/f'(x[n]) \n\nwhere x[n] is the nth approximation of the root, f(x[n]) is the value of the function at x[n], and f'(x[n]) is the derivative of the function at x[n]. \n\nFor our equation, f(x) = exp(x) - x, and f'(x) = exp(x) - 1. \n\nLet's start with an initial guess for x, say x[0] = 1. \n\nIterating the Newton-Raphson equation, we have: \n\nx[1] = x[0] - (exp(x[0]) - x[0]) / (exp(x[0]) - 1) \n\nx[1] = 1 - (exp(1) - 1) / (exp(1) - 1) \n\nx[1] ≈ 0.5671 \n\nWe can continue this process, plugging in the previous approximation x[n] to find the next approximation x[n+1]. The iterations will converge to the root of the equation. \n\nUsing a computer program or calculator, we can continue the iterations until we reach a desired level of accuracy or until the value of x[n+1] doesn't change significantly. \n\nAfter several iterations, we find that the root of the equation exp(x) - x = 0 is approximately x ≈ 0.5671.