Find the cube root of 12 using the Newton Raphson method assuming x0 = 2.5.

Sure, let's use the Newton-Raphson method to find the cube root of 12. The Newton-Raphson method is an iterative method that starts from an initial guess and then improves the guess using the formula:

x_(n+1) = x_n - f(x_n)/f'(x_n)

In this case, we're looking for the cube root of 12, so our function f(x) is x^3 - 12. The derivative of this function, f'(x), is 3x^2.

Let's start with the initial guess x0 = 2.5.

1. Calculate f(x0) = (2.5)^3 - 12 = 2.625
2. Calculate f'(x0) = 3*(2.5)^2 = 18.75
3. Calculate x1 = x0 - f(x0)/f'(x0) = 2.5 - 2.625/18.75 = 2.36

Now, we repeat the process with x1.

1. Calculate f(x1) = (2.36)^3 - 12 = -0.314
2. Calculate f'(x1) = 3*(2.36)^2 = 16.7
3. Calculate x2 = x1 - f(x1)/f'(x1) = 2.36 - (-0.314)/16.7 = 2.38

We can continue this process until we reach a desired level of accuracy. For example, if we want the result accurate to three decimal places, we would stop when the difference between x_(n+1) and x_n is less than 0.001.

Please note that the Newton-Raphson method may not converge to the correct answer for some functions and initial guesses. It's also possible for the method to converge to a root that is not the one you are interested in. However, for this particular problem, the method should work well.