the corrected root for the equation e^x = 6 - 2x using Newton's method with an initial guess of x_0 = 1.0 is approximately 1.2517579.
Use Newton's method to find all roots of the equation correct to six decimal places: e^x = 6 - 2x.
Define the function: The equation is f(x) = e^x - (6 - 2x).
Derive the expression for the derivative f'(x): f'(x) = e^x + 2.
Set up the iteration formula for Newton's method: x_{n+1} = x_n - f(x_n)/f'(x_n).
Perform iterations: Starting with an initial guess, use the iteration formula until convergence.
Iterate until the difference between consecutive values is sufficiently small. For example, starting with an initial guess of x_0 = 1.0, x_1 = x_0 - f(x_0)/f'(x_0), x_2 = x_1 - f(x_1)/f'(x_1), and so on.
Now, use this converged value as a new initial guess and iterate again. The process is repeated until the value stabilizes. After several iterations, the root is found to be 1.2517579.