Newton's Method is a method for estimating a solution to f (x) = 0. Producing a sequence of approximations that approach the solution does it.

1.

Guess a first approximation to a root of the equation f (x) = 0. (A graph of y = f (x) will help in this matter.)

2.

Use the first approximation, x ₀, to get the second, x ₁, the second to get the third, x ₂, and so on, using

where f ' (x _n) is the derivative of f at x _n.

EXAMPLE 1:

Use Newton's method to estimate the one real solution of x ³ + 3x + 1 = 0. Start with x ₀ = 0 and find x ₂ and x ₃.

SOLUTION: Let f (x) = x ³ + 3x + 1, then f ' (x) = 3x ² + 3.

f (x) = x ³ + 3x + 1

exact root x =-0.3221854

EXAMPLE 2:

Use Newton's Method to find the positive fourth root of 2 by solving the equation x ⁴ - 2 = 0. Start with x ₀ = 1 and find x ₂ and x ₃.

SOLUTION:

Let f (x) = x ⁴ - 2, then f ' (x) = 4x ³.

exact value x = 1.1892071

f (x) = x ⁴ - 2

Here is a program that you can enter into your calculator to do this method.

 TI - 82

STEP 1

Press PRGM key

STEP 2

Using the right arrow, move the cursor to NEW. Press ENTER. You will see the following: Name =. Enter in the name NEWTON. Press ENTER. You will now see :. Now start to type in the following instructions. I /O contains the commands Disp and Input. CTL contains the commands If, Lbl, and Goto.

:Disp "Enter initial"

:Disp "guess for the"

:Disp "root"

:Input x

:Lbl 1

:x - Y1/nDeriv(Y1, x, x) ® x

:Disp "X = "

:Disp x

:Disp "Do you want"

:Disp "to do it again?"

:Disp "Enter 1 for yes"

:Input B

:If (B = 1)

:Goto 1

Now press 2nd mode to quit.

To execute the program, first type an equation into Y1 in the Y= option. Now press prgm key. Select Newton and then press enter. Press enter again. It will now prompt you for your initial guess. To check your program, let Y1 = x ² - 4, and your initial guess to be 1.5.

 TI - 85

STEP 1

Press PRGM key

STEP 2

Press F2 for edit. You will see Name =, type in Newton. Now you will see :, this is where you will start typing the program in. I /O contains Disp, Input, and ". CTL will contain If, Lbl, and Goto.

:Disp "Enter initial"

:Disp "guess for the"

:Disp "root"

:Input x

:Lbl A

:x - Y1/nDer (Y1, x, x) ® x

:Disp "X = "

:Disp x

:Disp "Do you want"

:Disp "to do it again?"

:Disp "Enter 1 for yes"

:Input B

:If (B = = 1)

:Goto A

Now press 2nd exit to quit.

To execute the program, first type an equation into Y1 in the Y= option. Now press prgm key. Select names, then select Newton and then press enter. Press enter again. It will now prompt you for you initial guess. To check you program let Y1 = x ² - 4, and your initial guess to be 1.5.