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Funktion: f(x) = x3 - 7.5·x2 + 4, Df = R, Wf = R, Polynom 3. Grades, Funktion punktsymmetrisch zum Punkt W(2.5|-27.25), x -> -∞: f(x) -> -∞, x -> +∞: f(x) -> +∞ ->
Wertetabelle: | |||||
x | f(x) | f'(x) | f''(x) | f'''(x) | Besondere Kurvenpunkte |
-10 | -1746 | 450 | -75 | 6 | |
-9.5 | -1530.25 | 413.25 | -72 | 6 | |
-9 | -1332.5 | 378 | -69 | 6 | |
-8.5 | -1152 | 344.25 | -66 | 6 | |
-8 | -988 | 312 | -63 | 6 | |
-7.5 | -839.75 | 281.25 | -60 | 6 | |
-7 | -706.5 | 252 | -57 | 6 | |
-6.5 | -587.5 | 224.25 | -54 | 6 | |
-6 | -482 | 198 | -51 | 6 | |
-5.5 | -389.25 | 173.25 | -48 | 6 | |
-5 | -308.5 | 150 | -45 | 6 | |
-4.5 | -239 | 128.25 | -42 | 6 | |
-4 | -180 | 108 | -39 | 6 | |
-3.5 | -130.75 | 89.25 | -36 | 6 | |
-3 | -90.5 | 72 | -33 | 6 | |
-2.5 | -58.5 | 56.25 | -30 | 6 | |
-2 | -34 | 42 | -27 | 6 | |
-1.5 | -16.25 | 29.25 | -24 | 6 | |
-1 | -4.5 | 18 | -21 | 6 | |
-0.7 | 0 | 11.97 | -19.2 | 6 | Nullstelle N(-0.7|0) |
-0.5 | 2 | 8.25 | -18 | 6 | |
0 | 4 | 0 | -15 | 6 | Schnittpunkt Sy(0|4) = Hochpunkt H(0|4) |
0.5 | 2.25 | -6.75 | -12 | 6 | |
0.77 | 0 | -9.77 | -10.38 | 6 | Nullstelle N(0.77|0) |
1 | -2.5 | -12 | -9 | 6 | |
1.5 | -9.5 | -15.75 | -6 | 6 | |
2 | -18 | -18 | -3 | 6 | |
2.5 | -27.25 | -18.75 | 0 | 6 | Wendepunkt W(2.5|-27.25) |
3 | -36.5 | -18 | 3 | 6 | |
3.5 | -45 | -15.75 | 6 | 6 | |
4 | -52 | -12 | 9 | 6 | |
4.5 | -56.75 | -6.75 | 12 | 6 | |
4.998 | -58.5 | 0 | 14.99 | 6 | Tiefpunkt T(5|-58.5) |
5 | -58.5 | 0 | 15 | 6 | |
5.5 | -56.5 | 8.25 | 18 | 6 | |
6 | -50 | 18 | 21 | 6 | |
6.5 | -38.25 | 29.25 | 24 | 6 | |
7 | -20.5 | 42 | 27 | 6 | |
7.426 | 0 | 54.05 | 29.56 | 6 | Nullstelle N(7.43|0) |
7.5 | 4 | 56.25 | 30 | 6 | |
8 | 36 | 72 | 33 | 6 | |
8.5 | 76.25 | 89.25 | 36 | 6 | |
9 | 125.5 | 108 | 39 | 6 | |
9.5 | 184.5 | 128.25 | 42 | 6 | |
10 | 254 | 150 | 45 | 6 | |
Graph: | |||||
Abkürzungen: Df = (maximaler) Definitionsbereich, f(x) = Funktion, f'(x) = 1. Ableitung, f''(x) = 2. Ableitung, f'''(x) = 3. Ableitung, H = Hochpunkt, N = Nullstelle, P = Polstelle, R = reelle Zahlen, T = Tiefpunkt, W = Wendepunkt, WS = Sattelpunkt, Wf = Wertebereich, {.} = ein-/mehrelementige Menge, [.; .] = abgeschlossenes Intervall, (.; .) = offenes Intervall, [.; .), (.; .] = halboffenes Intervall, ∞ = unendlich.
Bearbeiter: Michael Buhlmann