Calculus of Polynomial Functions

The program carries out the curve sketching for a polynomial function. This means that the derivatives and the antiderivative are determined, the function is examined for rational zeros, for extremes (turning points), for inflection points and for symmetry. The graphs of ƒ, ƒ' and ƒ" are plotted 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 are entered for an integral, the value of the specific integral over this interval is determined in addition to the antiderivative.

Example: ƒ(x) = 3·x4 - 82/3·x2 + 3

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)

Derivatives:
¯¯¯¯¯¯¯¯¯¯
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

Inflection points:
¯¯¯¯¯¯¯¯¯¯¯¯¯
W1(-1,23228 |-31,5885)        m =  44,9098
W2( 1,23228 |-31,5885)        m = -44,9098

Symmetry:
¯¯¯¯¯¯¯¯¯
Axial symmetric to a: x =0

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

   ƒ(x) = 3x⁴ - 82/3x² + 3

  x ∈ [ -10; 10 ],  Δx = 2

      x        ƒ(x)              ƒ'(x)            ƒ"(x)
 ——   —————  ————  ————— 
  -10       27269,7     -11453,3      3545,33
   -8        10541,7     -5706,67      2249,33
   -6        2907           -2264           1241,33
   -4        333,667     -549,333      521,333
   -2       -58,3333      13,3333      89,3333
    0         3                  0                -54,6667
    2        -58,3333     -13,3333      89,3333
    4         333,667      549,333      521,333
    6         2907           2264           1241,33
    8         10541,7      5706,67      2249,33
   10        27269,7      11453,3      3545,33

All program parts in which the coefficients of a polynomial are entered have a context menu (right mouse button) that allows you to 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 | Supported Functions | Setting the graphics
Wikipedia: Polynomial functions