Perform three iterations to find approximate solution of xex

Perform three iterations to find approximate solution of x-e-x=0. Start with initial guess of x=0.5 (only perform three iterations and find the final answer).

Solution

x - e^(-x) = 0

let f(x) = x - e^(-x)

f\' \'(x) = 1 + e^(-x)

the initial value is x1 = .5

using the Newton Raphson Method

first iteration

x2 = x1 - f(x1)/f \'(x1)

=> x2 = .5 - [.5 - e^(-.5)]/[1 + e^(-.5)]

x2 = 0.566311

second iteration

x3 = x2 - f(x2)/f \'(x2)

x3 = 0.566311 - [0.566311 - e^(-0.566311)]/[1 + e^(-0.566311)]

x3 = 0.567143

third iteration

x4 = x3 - f(x3)/f \'(x3)

x4 = 0.567143 - [0.567143 - e^(-0.567143)]/[1 + e^(-0.567143)]

x4 = 0.567143