Suppose that f⁡(x)=x^3+2*⁢x+1 and we use Newton's method to find a solution of f(x)=0.?

Question

... If we use x[0] = 1 as our starting value, find the next two iterates.

x[1]=?
x[2]=?

Note the numbers after the x's are subscripts.

Answer 1

You seek
x^3 + 2x + 1 = 0

2x = -x^3 -1
x = (-x^3 -1)/2

set up:
x[n+1] = (-x[n]^3 -1)/2

begin with x[0] = 0, then

x[1] = (-0 -1)/2 = -0.5
x[2] = (+0.125 -1)/2 = -0.4375
x[3] = (+0.08374 -1)/2 = -0.45813
x[4] = (+0.09615 -1)/2 = -0.451923
etc.

It can be set up in Excel quite easily and you keep copying more lines until you get the accuracy you need.

Or you can set it up as a loop in a program (e.g., MATLAB) that runs until the difference between two successive results gets below a preset limit.