How do you use Newton's method to find the approximate solution to the equation #x^3-10x+4=0, x>1#?

This reference on Newton's Method gives us this equation:

Compute the first derivative:

The equation is:

I am going tell you how to use an Excel spreadsheet to perform the recursion:

Enter the number 2 into cell A1.

Into cell A2, enter the following Excel formula:

=A1 - (A1^3 - 10A1 + 4)/(3A1^2 - 10)

Use the block paste feature of Excel to copy and paste the formula into cells A3 through A20.

The computation converges quickly onto the root 2.939235

You can find the other 2 roots by playing with the seed value in cell A1

To use Newton's method to find an approximate solution to the equation (x^3 - 10x + 4 = 0) with (x > 1), follow these steps:

1. Choose an initial guess, let's say (x_0 = 2).
2. Use the formula (x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}) iteratively until the desired level of accuracy is achieved.
3. Compute (f(x_n) = x_n^3 - 10x_n + 4) and (f'(x_n) = 3x_n^2 - 10).
4. Substitute (x_n) into the formula to find (x_{n+1}).
5. Repeat the process with (x_{n+1}) until the difference between consecutive approximations is within the desired tolerance.

Using (x_0 = 2):

1. (f(2) = 2^3 - 10(2) + 4 = 8 - 20 + 4 = -8)
2. (f'(2) = 3(2)^2 - 10 = 12 - 10 = 2)
3. (x_1 = 2 - \frac{-8}{2} = 2 + 4 = 6)

Repeat the process until the desired accuracy is reached.