Polynomial Functions

The program carries out the curve discussion for a polynomial function. This means that the derivatives and the antiderivative are determined, the function is examined for rational zeros, for extremes, for turning points and for symmetry. The diagrams of ƒ, ƒ' and ƒ" are drawn and a table of values is output.

The coefficients of the polynomial can be entered as fractions, as mixed numbers or as breaking decimal numbers.

If interval limits for an integral are entered, the value of the specific integral over this interval is determined in addition to the antiderivative.

example

Function:
¯¯¯¯¯¯¯¯
ƒ(x) = 3·x4 - 82/3·x2 + 3
       = 1/3·(9·x4 - 82·x2 + 9)
       = 1/3·(3·x - 1)·(3·x + 1)·(x - 3)·(x + 3)

Derivations:
¯¯¯¯¯¯¯¯¯¯
ƒ'(x)  = 12·x3 - 164/3·x
ƒ"(x)  = 36·x2 - 164/3
ƒ'"(x) = 72·x

Antiderivative:
¯¯¯¯¯¯¯¯¯¯¯
F(x) = 3/5·x5 - 82/9·x3 + 3·x + c

Zeros:
¯¯¯¯¯
N1( 1/3 | 0)                             m = -17,7778
N2(-1/3 | 0)                             m =  17,7778
N3( 3 | 0)                                m =  160
N4(-3 | 0)                                m = -160

Extremes:
¯¯¯¯¯¯¯¯
H1( 0 | 3)                                m =  0
T1(-2,13437 |-59,2593)          m =  0
T2( 2,13437 |-59,2593)          m =  0

Pts of inflection:
¯¯¯¯¯¯¯¯¯¯¯¯¯
W1(-1,23228 |-31,5885)        m =  44,9098
W2( 1,23228 |-31,5885)        m = -44,9098

Symmetry:
¯¯¯¯¯¯¯¯¯
Achsensymmetrisch zu a: x =0

The curves of ƒ, ƒ 'and ƒ "can be switched on and off individually using the switches on the right.

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.

See also:

Notes on the procedure
Wikipedia: Polynomial functions