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Funktion: f(x) = , Df = R\{-2}, Wf = [0; +∞), gebrochen rationale Funktion, x -> -∞: f(x) -> 4/9 = y als Grenzkurve, x -> +∞: f(x) -> 4/9 = y als Grenzkurve ->
Wertetabelle: | |||||
x | f(x) | f'(x) | f''(x) | f'''(x) | Besondere Kurvenpunkte |
-20 | 0.6944 | 0.02 | 0 | 0 | |
-19.5 | 0.7024 | 0.02 | 0 | 0 | |
-19 | 0.7109 | 0.02 | 0 | 0 | |
-18.5 | 0.7199 | 0.02 | 0 | 0 | |
-18 | 0.7296 | 0.02 | 0 | 0 | |
-17.5 | 0.74 | 0.02 | 0 | 0 | |
-17 | 0.7511 | 0.02 | 0 | 0 | |
-16.5 | 0.7631 | 0.02 | 0 | 0 | |
-16 | 0.7761 | 0.03 | 0 | 0 | |
-15.5 | 0.7901 | 0.03 | 0 | 0 | |
-15 | 0.8054 | 0.03 | 0.01 | 0 | |
-14.5 | 0.822 | 0.03 | 0.01 | 0 | |
-14 | 0.8403 | 0.04 | 0.01 | 0 | |
-13.5 | 0.8603 | 0.04 | 0.01 | 0 | |
-13 | 0.8825 | 0.05 | 0.01 | 0 | |
-12.5 | 0.907 | 0.05 | 0.01 | 0 | |
-12 | 0.9344 | 0.06 | 0.01 | 0 | |
-11.5 | 0.9652 | 0.07 | 0.02 | 0.01 | |
-11 | 1 | 0.07 | 0.02 | 0.01 | |
-10.5 | 1.0396 | 0.08 | 0.02 | 0.01 | |
-10 | 1.0851 | 0.1 | 0.03 | 0.01 | |
-9.5 | 1.1378 | 0.11 | 0.04 | 0.02 | |
-9 | 1.1995 | 0.13 | 0.05 | 0.02 | |
-8.5 | 1.2728 | 0.16 | 0.06 | 0.03 | |
-8 | 1.3611 | 0.19 | 0.08 | 0.05 | |
-7.5 | 1.4692 | 0.24 | 0.11 | 0.07 | |
-7 | 1.6044 | 0.3 | 0.15 | 0.11 | |
-6.5 | 1.7778 | 0.4 | 0.22 | 0.18 | |
-6 | 2.0069 | 0.53 | 0.34 | 0.31 | |
-5.5 | 2.322 | 0.75 | 0.55 | 0.58 | |
-5 | 2.7778 | 1.11 | 0.96 | 1.19 | |
-4.5 | 3.4844 | 1.79 | 1.89 | 2.85 | |
-4 | 4.6944 | 3.25 | 4.38 | 8.35 | |
-3.5 | 7.1111 | 7.11 | 13.04 | 33.73 | |
-3 | 13.4444 | 22 | 62.01 | 246.06 | |
-2.5 | 44.4444 | 160.12 | 928.6 | 7679.51 | |
-2 | Infinity | Infinity | Infinity | Infinity | Senkrechte Asymptote/Pol x = -2 ohne Vorzeichenwechsel: x -> -2-: f(x) -> +∞, x -> -2+: f(x) -> +∞ |
-1.5 | 28.4444 | -128.11 | 800.55 | -6216.83 | |
-1 | 5.4444 | -14 | 46.01 | -187.23 | |
-0.5 | 1.7778 | -3.56 | 8.3 | -23.3 | |
0 | 0.6944 | -1.25 | 2.38 | -5.18 | Schnittpunkt Sy(0|0.69) |
0.5 | 0.2844 | -0.51 | 0.87 | -1.58 | |
1 | 0.1111 | -0.22 | 0.37 | -0.59 | |
1.5 | 0.0363 | -0.09 | 0.17 | -0.25 | |
2 | 0.0069 | -0.03 | 0.09 | -0.12 | |
2.5 | 0 | 0 | 0.04 | -0.06 | Nullstelle N(2.5|0) = Tiefpunkt T(2.5|0) |
3 | 0.0044 | 0.02 | 0.02 | -0.03 | |
3.5 | 0.0147 | 0.02 | 0.01 | -0.02 | |
4 | 0.0278 | 0.03 | 0 | -0.01 | |
4.5 | 0.0421 | 0.03 | 0 | -0.01 | |
4.75 | 0.0494 | 0.03 | 0 | 0 | Wendepunkt W(4.75|0.05) |
5 | 0.0567 | 0.03 | 0 | 0 | |
5.5 | 0.0711 | 0.03 | 0 | 0 | |
6 | 0.0851 | 0.03 | 0 | 0 | |
6.5 | 0.0984 | 0.03 | 0 | 0 | |
7 | 0.1111 | 0.02 | 0 | 0 | |
7.5 | 0.1231 | 0.02 | 0 | 0 | |
8 | 0.1344 | 0.02 | 0 | 0 | |
8.5 | 0.1451 | 0.02 | 0 | 0 | |
9 | 0.1552 | 0.02 | 0 | 0 | |
9.5 | 0.1647 | 0.02 | 0 | 0 | |
10 | 0.1736 | 0.02 | 0 | 0 | |
10.5 | 0.182 | 0.02 | 0 | 0 | |
11 | 0.19 | 0.02 | 0 | 0 | |
11.5 | 0.1975 | 0.01 | 0 | 0 | |
12 | 0.2046 | 0.01 | 0 | 0 | |
12.5 | 0.2114 | 0.01 | 0 | 0 | |
13 | 0.2178 | 0.01 | 0 | 0 | |
13.5 | 0.2238 | 0.01 | 0 | 0 | |
14 | 0.2296 | 0.01 | 0 | 0 | |
14.5 | 0.2351 | 0.01 | 0 | 0 | |
15 | 0.2403 | 0.01 | 0 | 0 | |
15.5 | 0.2453 | 0.01 | 0 | 0 | |
16 | 0.25 | 0.01 | 0 | 0 | |
16.5 | 0.2545 | 0.01 | 0 | 0 | |
17 | 0.2588 | 0.01 | 0 | 0 | |
17.5 | 0.263 | 0.01 | 0 | 0 | |
18 | 0.2669 | 0.01 | 0 | 0 | |
18.5 | 0.2707 | 0.01 | 0 | 0 | |
19 | 0.2744 | 0.01 | 0 | 0 | |
19.5 | 0.2779 | 0.01 | 0 | 0 | |
20 | 0.2812 | 0.01 | 0 | 0 | |
Graph: | |||||
Graph(en) der Asymptote(n), Grenzkurve(n).
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