NEWTON(473):  The TI-83 uses the nDeriv( function to calculate the numerical derivative of F(X) at X.  nDeriv( is not 100% accurate and the TI-83 can sometimes give a false derivative value at a non-differentiable point.

NEWTONN(496): You must input F'(X) yourself.  This may be time consuming and the equation you put in could be erroneous if you're not careful.  But when input correctly, the derivative output is 100% accurate.

WINDOW RANGE error:
This is the most common type of error in the program.  It is caused when the Xmin and Xmax values become too close to each other.  When the TI-83 displays a graph, it rounds off the Xmin and Xmax values.  If the rounded values are equal, an error occurs because Xmin must always be greater than Xmax and never equal to it.
Example 1:
    F(X)=       X³/3+.5X²+X+1
    F'(X)=      X²+X+1
    X GUESS=    0
A WINDOW RANGE error occurs after trial #4.
    Xmin=       -1.153652866
    Xmax=       -1.153652857
In order to graph the function, the Xmin and Xmax are rounded.  The rounded Xmin and Xmax values become equal.  Thus, the error occurs even though the actual Xmin and Xmax are not equal.

NONREAL ANS or DATA TYPE error:
Example 2:
    F(X)=       sqrt(X)
    F'(X)=      1/(2sqrt(x))
    X GUESS=    any positive real number
The tangent line intercepts the negative X-axis.  The new guess is negative and the square root of any negative number is not real.  If the calculator is in Real mode, there is a NONREAL ANS error.  This happens because only real numbers can be calculated in the Real mode.  If the calculator is in a+bi mode or re^(theta)i mode, there is a DATA TYPE error.  In a logic test using relational operators, the numbers being compared must be the same data type.  They all must be either real or nonreal to be compared not a combination of the two.

DIVIDE BY 0 error:
Example 3:
    F(X)=       (X²-9)/(X-3)
    F'(X)=      1
    X GUESS=    3
F(3) is undefined.  You must first simplify F(X) to (X+3) before you can evaluate F(3).

Example 4:
    F(X)=       X³/3+.5X²+X+1
    X GUESS=    0
After trial #1, the numerical derivative, using nDeriv(Y₁,X,0), equals 1.000000333315
F'(X)=X²+X+1
Thus, F'(0)=1, not 1.000000333315 like nDeriv( says it does.

Example 5:
    F(X)=       abs(X)-10
    X GUESS=    0
The function is not differentiable at X=0 but nDeriv( calculates it as 0.

Example 6:
    F(X)=       X²
    F'(X)=      2X
    X GUESS=    any non-zero, real number
The Newton-Raphson method will never find the root of the equation (which is 0). The new guess will always be one half of the previous one.

Example 7:
    F(X)=       X^(1/3)
    F'(X)=      1/(3X^(2/3))
    X GUESS=    any non-zero, real number
The root will never be found.  The new guess will always be -2 times the value of the previous one.