## Calculus of Functions: Notes on the procedure

The programs *Polynomial Functions* and *Rational Functions* perform roughly the same tasks as the program *Arbitrairy Functions* ,
but differ fundamentally in the procedure.

As well known, zeros, extrema and points of inflection are determined by calculating the zeros of ƒ(x), ƒ'(x), ƒ"(x) and ƒ'"(x).

In the program *Arbitrairy Functions* these are approximately determined by searching the function, or the derivatives, step by step in an interval for a change in sign.
This numerical method can be applied to any function, but it is not a viable method for finding definition gaps.

The programs *Polynomial Functions* and *Rational Functions* are limited to polynomials with rational coefficients.
This makes it possible to exactly calculate the rational zeros of the function or the derivatives and, in the case of fractional rational functions,
to determine the definition gaps via the zeros of the denominator polynomial.

As the following examples show, both methods have advantages and disadvantages.

### Example 1:

x^{2}+ 4·x + 3 (x + 1)·(x + 3) ƒ(x) = ———————— = ————————— x^{4}+ x^{3}- 6·x^{2}x^{2}·(x - 2)·(x + 3) x_{1}= -3 Eliminable Gap L(-3 | 0,0444444 ) x_{2}= 0 Pole without change of sign x_{3}= 2 Pole with change of sign

The definition gaps are correctly determined in the program *Rational Functions* .

In the program *Arbitrairy Functions* however, x_{1} and x_{3} are not recognized and instead are incorrectly displayed as points of inflection.
The reason for this is the change in sign of the second derivative.

### Example 2:

ƒ(x) = 3*x^7 + 3*x^6 + 17*x^5 - 5*x^4 + 34*x^3 - 10*x^2 - 16*x + 8 N_{1}(-0,683 | 0 ) H_{1}(-0,295987 | 10,9025 ) T_{1}( 0,471495 | 1,9943 ) W_{1}( 0,0992583 | 6,34628 )

The polynomial has no rational zeros and the degree of the polynomial is too high to be able to use the formulas of Cardano and Ferrari
(see equations of the 4th degree).

In the program *Polynomial Functions* therefore no zeros, extremes and inflection points are found.

The numerical method in the program *Arbitrairy Functions* can help here.

The linear factor decomposition of ƒ(x) shows whether the results of the programs *Polynomial Functions* and *Rational Functions* have to be checked
with the *Arbitrairy Functions* . If this is not complete and the degree of the remainder polynomial is greater than 4, further irrational zeros, extrema and turning points can exist.

In the programs *Polynomial Functions* and *Rational Functions* , the function term ƒ(x) is automatically copied to the clipboard and can be inserted into the
*Arbitrairy Functions* with *Ctrl V* or the *Paste* option in the local menu.