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 (turning points) and inflection points are determined by calculating the zeros of ƒ(x), ƒ'(x), ƒ"(x) and ƒ'"(x).

In the Arbitrairy Functions program 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 Polynomial Functions  and Rational Functions programs 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 find the definition gaps via the zeros of the denominator polynomial.

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

Example 1:

                 x2 + 4·x + 3               (x + 1)·(x + 3)
ƒ(x) = ———————— = —————————
               x4 + x3 - 6·x2            x2·(x - 2)·(x + 3)
			   
x1 = -3  Eliminable gap L(-3 | 0,0444444 )
x2 =  0  Pole without sign change
x3 =  2  Pole with sign change

The definition gaps are correctly determined in the Rational Functions  program.
In the Arbitrairy Functions  program however, x₁ and x₃ are not recognized and instead are incorrectly displayed as inflection points. This is due to the sign change 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
 
N1(-0,68­3 | 0 )

H1(-0,295987 | 10,9025 )
T1( 0,471495 | 1,9943 )

W1( 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 see 4th degree equations).
Therefor in the Polynomial Functions  program no zeros, extremes and inflection points are found.
The numerical method in the Arbitrairy Functions  program can help here.

The linear factor decomposition of ƒ(x) shows whether the results of the Polynomial Functions  and Rational Functions programs need to be checked with the Arbitrairy Functions  program. If this is not complete and the degree of the remainder polynomial is greater than 4, further irrational zeros, extrema and inflection points can exist.
In the Polynomial Functions  and Rational Functions programs, the function term ƒ(x) is automatically copied to the clipboard and can be inserted into the Arbitrairy Functions  program with Ctrl V or the Paste option in the local menu.