**Introduction**

This program finds successive approximations to the solutions of
*f*(*x*) = 0
using Newton's method.

If you have not used one of the programs posted on this website before,
you should read through
the information in the Intro to Programming section first.

**The Program**

:Disp "INITIAL GUESS" | {Disp is in PRGM under I/O} { " is on the + button} |

:Input X | {Input is in PRGM under I/O} {X is the X,T button} |

:Lbl 1 | {Lbl is in PRGM under CTL} |

:XA | {The arrow is STO} |

:X-Y1/Y2X | {Y1 Y2 are in Y-VARS under FUNCTION} |

:Disp X | |

:Pause | {Pause is in PRGM under CTL} |

:If X A | {If is in PRGM under CTL} { is in TEST } |

:Goto 1 | {Goto is in PRGM under CTL} |

**Running the program**

You will need to enter *f*(*x*) and *f '*(*x*)
into Y1 and Y2, respectively. After
entering an initial guess for the solution to *f*(*x*) = 0
, hit
ENTER to obtain each new approximation.
To test the program try the following:

*f*(*x*) = x^{3}-3x^{2}+x-5,
*f '*(*x*) = 3x^{2}-6x+1,
initial guess = 3.

The approximations should be

3.2

3.18019169329

3.17998109582

3.17998107216