Newton's Method on the TI-83+

Given the function: f(x) = X^2 - 3

:ClrHome
:2->A
:0->I
:Repeat I>50
:A-(A^2-3)/(nDerive(B^2-3,B,A))->A
:I+1->I
:Disp I,A
:End

Items in bold can be adjusted to fit your function. Make sure the A's and B's are used.
The 2 in line 2 should be your first guess at the root.

Another example:

Given the function: f(x) = sin(X/2)

:ClrHome
:1->A
:0->I
:Repeat I>50
:A-(sin(A/2))/(nDerive(sin(B/2),B,A))->A
:I+1->I
:Disp I,A
:End

If you don't want the value to display after every calculation, switch the last two lines:

:End
:Disp I,A