Rational functions

The program carries out the curve discussion for a (broken) rational function. This means that the derivations and the gaps in the domain of definition are determined. The function is examined for zeros, extremes, points of inflection and the behavior for |x|→∞ The diagrams of ƒ, ƒ' and ƒ" are drawn and a table of values is output.

The coefficients of the numerator and denominator polynomial can be entered as fractions, as mixed numbers or as terminating decimal numbers.

All program parts in which the coefficients of a polynomial are entered have a context menu (right mouse button) with which you can copy the entries from one program part to the clipboard and paste them from there into another program part.

Example

Function :
¯¯¯¯¯¯¯¯
             3·x3 + x2 - 4        (x - 1)·(3·x2 + 4·x + 4)  
ƒ(x) = —————— = ———————————
                4·x2 - 16               4·(x - 2)·(x + 2)       

Singularities:
¯¯¯¯¯¯¯¯¯¯¯
x = 2  Pole with change of sign
x =-2  Pole with change of sign

Derivatives:
¯¯¯¯¯¯¯¯¯¯
               3·(x4 - 12·x2)            3·(x2·(x2 - 12))   
ƒ'(x) = ———————— = ————————
            4·(x4 - 8·x2 + 16)       4·(x - 2)2·(x + 2)2 

                      6·(x3 + 12·x)               6·(x·(x2 + 12))  
ƒ"(x) = ——————————— = ———————
              x6 - 12·x4 + 48·x2 - 64       (x - 2)3·(x + 2)3 

Zeros:
¯¯¯¯¯
N( 1 |  0 )                               m = -0,916667

Extrema :
¯¯¯¯¯¯¯¯
H(-3,4641 |-3,64711 )           m =  0
T( 3,4641 | 4,14711 )            m =  0

Pts of inflection:
¯¯¯¯¯¯¯¯¯¯¯¯
W( 0 |  0,25 )                       m =  0

Behavior for  |x|→∞
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Oblique Asymptote  y = 3/4·x + 1/4

With the switches on the right edge, the curves of ƒ, ƒ'and ƒ" as well as the asymptotes or approximation curves can be switched on and off individually.

See also:

Notes on the procedure
Wikipedia: Rational function