Question

Use newton's method to find an approximate solution of ln(x) = 8 - x

Answer 1

Use Newton's method to approximate the solution of ln(x) = 8 - x

Step-by-step explanation:

Newton's Method:

1. Start with an initial guess for the solution. Let's say x0.
2. Use the equation to calculate f(x0) = ln(x0) - 8 + x0.
3. Compute the derivative of f(x), which is f'(x) = 1/x - 1.
4. Update the guess for the solution using the formula x1 = x0 - f(x0) / f'(x0).
5. Repeat steps 2-4 until the desired level of accuracy is achieved.

Example:

Let's start with an initial guess of x0 = 3.

Using the equation, f(3) = ln(3) - 8 + 3 ≈ -5.0986.

The derivative is f'(3) = 1/3 - 1 ≈ -0.6667.

Updating the guess, x1 = 3 - (-5.0986) / (-0.6667) ≈ 10.2471.

Repeat steps 2-4 with the new guess until the desired level of accuracy is achieved.