Problem:
Using Newton's method with the initial point x = 2, find the root of the equation

f(x) = x3 + 5 x - 4.

Visualization:

Using the TI-85 graphing calculator:

[Press here to see animation again!]

1. First, enter the equation of the function: y1 = x^3 + 5 x - 4 and press the ENTER key.
2. Next, enter the equation of the derivative: y2 = 3 x^2 + 5 and press the ENTER key. (You could enter y2 = der1(y1, x, x) in place of the derivative.)
3. Enter the formula used in Newton's Method: Z = x - y1 ÷ y2 and press the ENTER key.
4. Enter the initial value for x: 2 STO x : Z and press the ENTER key. The value for Z is displayed.
5. Press STO x : Z and press the ENTER key to obtain the second iteration.
6. Continue by pressing 2nd ENTER followed by the ENTER key to get another iteration.
7. After several repetitions of the previous step, we obtain the following table of values:

nxn
00
11.17647058824
20.79288335372
30.725663700148
40.724076385282
50.724075551387
60.724075551386
70.724075551386