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Funktion: f(x) = , Df = R\{0}, Wf = R\{2}, gebrochen rationale Funktion als gestreckte, verschobene Hyperbel, Funktion punktsymmetrisch zu Z(0|2), x -> -∞: f(x) -> 2 = y als Grenzkurve, x -> +∞: f(x) -> 2 = y als Grenzkurve ->
Wertetabelle: | |||||
x | f(x) | f'(x) | f''(x) | f'''(x) | Besondere Kurvenpunkte |
-10 | 1.7 | -0.03 | -0.01 | 0 | |
-9.5 | 1.6842 | -0.03 | -0.01 | 0 | |
-9 | 1.6667 | -0.04 | -0.01 | 0 | |
-8.5 | 1.6471 | -0.04 | -0.01 | 0 | |
-8 | 1.625 | -0.05 | -0.01 | 0 | |
-7.5 | 1.6 | -0.05 | -0.01 | -0.01 | |
-7 | 1.5714 | -0.06 | -0.02 | -0.01 | |
-6.5 | 1.5385 | -0.07 | -0.02 | -0.01 | |
-6 | 1.5 | -0.08 | -0.03 | -0.01 | |
-5.5 | 1.4545 | -0.1 | -0.04 | -0.02 | |
-5 | 1.4 | -0.12 | -0.05 | -0.03 | |
-4.5 | 1.3333 | -0.15 | -0.07 | -0.04 | |
-4 | 1.25 | -0.19 | -0.09 | -0.07 | |
-3.5 | 1.1429 | -0.24 | -0.14 | -0.12 | |
-3 | 1 | -0.33 | -0.22 | -0.22 | |
-2.5 | 0.8 | -0.48 | -0.38 | -0.46 | |
-2 | 0.5 | -0.75 | -0.75 | -1.14 | |
-1.5 | 0 | -1.33 | -1.78 | -3.6 | Nullstelle N(-1.5|0) |
-1 | -1 | -3 | -6 | -18.37 | |
-0.5 | -4 | -12 | -48.02 | -300.12 | |
0 | Infinity | 30000 | -Infinity | Infinity | Senkrechte Asymptote/Pol x = 0 mit Vorzeichenwechsel: x -> 0-: f(x) -> -∞, x -> 0+: f(x) -> +∞ |
0.5 | 8 | -12 | 48.02 | -277.03 | |
1 | 5 | -3 | 6 | -17.65 | |
1.5 | 4 | -1.33 | 1.78 | -3.51 | |
2 | 3.5 | -0.75 | 0.75 | -1.11 | |
2.5 | 3.2 | -0.48 | 0.38 | -0.46 | |
3 | 3 | -0.33 | 0.22 | -0.22 | |
3.5 | 2.8571 | -0.24 | 0.14 | -0.12 | |
4 | 2.75 | -0.19 | 0.09 | -0.07 | |
4.5 | 2.6667 | -0.15 | 0.07 | -0.04 | |
5 | 2.6 | -0.12 | 0.05 | -0.03 | |
5.5 | 2.5455 | -0.1 | 0.04 | -0.02 | |
6 | 2.5 | -0.08 | 0.03 | -0.01 | |
6.5 | 2.4615 | -0.07 | 0.02 | -0.01 | |
7 | 2.4286 | -0.06 | 0.02 | -0.01 | |
7.5 | 2.4 | -0.05 | 0.01 | -0.01 | |
8 | 2.375 | -0.05 | 0.01 | 0 | |
8.5 | 2.3529 | -0.04 | 0.01 | 0 | |
9 | 2.3333 | -0.04 | 0.01 | 0 | |
9.5 | 2.3158 | -0.03 | 0.01 | 0 | |
10 | 2.3 | -0.03 | 0.01 | 0 | |
Graph: | |||||
Abkürzungen: Df = (maximaler) Definitionsbereich, f(x) = Funktion, f'(x) = 1. Ableitung, f''(x) = 2. Ableitung, f'''(x) = 3. Ableitung, H = Hochpunkt, L = Lücke, N = Nullstelle, P = Polstelle, R = reelle Zahlen, S = Sprungstelle, T = Tiefpunkt, W = Wendepunkt, WS = Sattelpunkt, Wf = Wertebereich, {.} = ein-/mehrelementige Menge, [.; .] = abgeschlossenes Intervall, (.; .) = offenes Intervall, [.; .), (.; .] = halboffenes Intervall, ∞ = unendlich.
Bearbeiter: Michael Buhlmann