Question

write newtons formula as \\(x_{n+1} = f(x_n)\\) for solving \\(f(x) = 0\\). use \x_n\ for \\(x_n\\).

\\(f(x) = e^x\\)

\\(f(x_n) =\\)

Explanation:

State Newton's method formula

Newton's method finds roots of \(f(x) = 0\) using:

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Find the derivative of the function

For the given function:

$$f(x) = e^x$$

The derivative is:

$$f'(x) = e^x$$

Substitute into the formula

Substitute \(f(x_n)\) and \(f'(x_n)\) into the iteration formula:

$$F(x_n) = x_n - \frac{e^{x_n}}{e^{x_n}}$$

Simplify the expression

Simplify the fraction:

$$F(x_n) = x_n - 1$$

Answer:

Write Newton's formula as \(x_{n+1} = F(x_n)\) for solving \(f(x) = 0\). Use "x_n" for \(x_n\).

\(f(x) = e^x\)

\(F(x_n) =\) <blank>x_n - 1</blank>