Newton's method (intersection points)

1. The problem statement, all variables and given/known data

Given two functions; y= ln(x) and y=(x^2)/8 - 2

Use Newton's Method to approximate all intersection points of the given functions, each with 3 decimal places.

2. Relevant equations

Xn+1 = Xn - f(x) / f`(x)

3. The attempt at a solution

Step 1: I equated the two equations, and I got

f(x) = ln(x) - (x^2)/8 +2

Step 2: I found the derivative as f`(x) = 1/x - x/4

Step 3: then I drew the graphs, and I found that they only intersect at a point near 5.4 (because ln(x) doesn't have any graph in the -x direction)

so I started off x1 = 5.4 and then using the equation given above I found the following values:

X2 = 5.435
X3 = 5.436
X4 = 5.435
X5 = 5.436


Can you guys please tell me if I did the solution right and at which point do I stop ... they keep going ... I'm not going same 3 decimal places for any of them ???
I hope someone would answer soon because I have it due tomorrow early morning. Thanks and can you also tell that whatever I did was right and nothing else was to be done!


Thanks