Here's an example showing how to use a TI-85 to carry out Newton's method.

We'll approximate a root to the equation x³-5x+3=0, starting with an initial guess of x=2.

We let f(x)=x³-5x+3, so that f'(x)=3x²-5. The iteration rule for Newton's approximation is

so in this case we get

x_n+1 = x_n - (x³-5x+3)/(3x²-5).

To teach this formula to the calculator, we go to the GRAPH screen, and then into the y(x) editor. We set y1 to

y1=x-(x^3-5x+3)/(3x^2-5).

We then return to the home screen and enter our initial guess:

2 -> x

where "->" is the STO key, which shows up as a right arrow on the screen.

Now to get the next iterate, we need only enter y1 (using 2nd-alpha-y to get the lower-case y) and ENTER. Then we store that value back into x and repeat the process. In fact, we can do all this at once by entering

y1 -> x.

The calculator shows us the next iterate and then stores it into x. And now, thanks to a feature of this particular calculator, if we simply press ENTER again, the calculator behaves as though we had entered y1 -> x again. So to carry out further iterations of Newton's method, we need only press ENTER repeatedly.

Doing this, we get the following sequence of approximations:

1.85714285714

1.83478735005

1.83424350392

1.83424318431

1.83424318431

Since the last two iterates agree to all displayed decimal positions, we assume that they are correct, and to an accuracy of 10^-10, the root is 1.83424318431.