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6 6  [[Kompetenzen.K1]] Ich kann die besondere Bedeutung der natürlichen Basis nennen
7 7  [[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann einen Basiswechsel durchführen
8 8  
9 -{{lehrende}}
10 -x im Exponenten
11 -* Welches ist eine Exponentialfunktion, welches nicht? Ordne zu ... (komplexere Terme mit/ ohne x im Exponenten)
12 -
13 -Asymptotischer Verlauf
14 -* Schaubilder , die durch (0|1) verlaufen
15 -* 1, 1 + 2x, 1 + x^2, 2^x, 1/2^x .. 1/x^2 für x<0 weiterzeichnen
16 -* Funktionswert an den Stellen -1, 0, 1, 2
17 -
18 -Warum kommen nur positive Basen in Frage?
19 -* Wertetabelle (-2)^x
20 -
21 -Basiswechsel
22 -* ohne ln
23 -* Auf Potenzgesetz zurückführen
24 -{{/lehrende}}
25 -
26 26  {{lernende}}
27 27  [[GeoGebra-Buch>>https://www.geogebra.org/m/khnsgz5a#material/UcgSUN2M]]
28 28  {{/lernende}}
29 29  
30 -{{aufgabe id="Exponentialfunktion" afb="I" kompetenzen="K5" quelle="Holger Engels" zeit="2"}}
13 +{{aufgabe id="Exponentialfunktion" afb="I" kompetenzen="K5" quelle="Holger Engels" zeit="2" cc="by-sa"}}
31 31  Entscheide jeweils, ob es sich bei dem Funktionsterm um einen Exponentialfunktionsterm handelt.
32 32  (% class="abc" %)
33 -1. {{formula}}f(x) = \frac{1}{8}(2(x-2))^3 + e{{/formula}}
34 -1. {{formula}}f(x) = \frac{1}{8}x^{3(x+1)}-1{{/formula}}
16 +1. {{formula}}f(x) = \frac{1}{8}(2(x-2))^3 + 1{{/formula}}
17 +1. {{formula}}f(x) = \frac{1}{8}2^{3(x+1)}-1{{/formula}}
35 35  {{/aufgabe}}
36 36  
37 -{{aufgabe id="Graphen" afb="I" kompetenzen="K4" quelle="Holger Engels" zeit="8"}}
20 +{{aufgabe id="Graphen" afb="I" kompetenzen="K4" quelle="Holger Engels" zeit="8" cc="by-sa"}}
38 38  Ordne die Funktionsgraphen den Funktionstermen zu und skizziere jeweils im Schaubild den Abschnitt für //x<0//.
39 -{{formula}}f(x)=1+2x{{/formula}} {{formula}}g(x)=1 + x^2{{/formula}} {{formula}}h(x)=(\frac{1}{2})^x{{/formula}} {{formula}}\frac{1}{(x+1)^2}{{/formula}}{{formula}}i(x)=2^x{{/formula}}
40 -[[image:graph f.svg||style="margin: 8px"]] [[image:graph g.svg||style="margin: 8px"]] [[image:graph h.svg||style="margin: 8px"]] [[image:graph p.svg||style="margin: 8px"]] [[image:graph q.svg||style="margin: 8px"]] [[image:graph r.svg||style="margin: 8px"]]
22 +{{formula}}f(x)=1+2x{{/formula}} {{formula}}g(x)=1 + x^2{{/formula}} {{formula}}h(x)=(\frac{1}{2})^x{{/formula}} {{formula}}i(x)=\frac{1}{(x+1)^2}{{/formula}} {{formula}}j(x)=2^x{{/formula}} {{formula}}k(x)=1{{/formula}}
23 +[[image:graph f.svg||style="margin: 8px;width:360px"]] [[image:graph g.svg||style="margin: 8px;width:360px"]] [[image:graph h.svg||style="margin: 8px;width:360px"]] [[image:graph p.svg||style="margin: 8px;width:360px"]] [[image:graph q.svg||style="margin: 8px;width:360px"]] [[image:graph r.svg||style="margin: 8px;width:360px"]]
41 41  (% class="abc" %)
42 42  {{/aufgabe}}
43 43  
44 -{{aufgabe id="Basiswechel" afb="I" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="10"}}
45 -Wandle die gegeben Exponentialfunktion in die entsprechende Basis um, z.B. {{formula}}f(x)=(\frac{1}{2})^x= 2^{-x}{{/formula}} ist eine Umwandlung in die neue Basis {{formula}}b=2{{/formula}}
46 -{{formula}}f(x)=(\frac{1}{4})^x{{/formula}} in die neue Basis {{formula}}b=2{{/formula}}
47 -{{formula}}f(x)=(\frac{3}{18})^x{{/formula}} in die neue Basis {{formula}}b=6{{/formula}}
48 -{{formula}}f(x)=9^x{{/formula}} in die neue Basis {{formula}}b=\frac{1}{3}{{/formula}}
49 -{{formula}}f(x)=5^{2x+1}{{/formula}} in die neue Basis {{formula}}b=25{{/formula}}
50 -{{formula}}f(x)=(\frac{16}{54})^{2x}{{/formula}} in die neue Basis {{formula}}b=\frac{3}{2}{{/formula}}
27 +{{aufgabe id="Negative Basis" afb="II" kompetenzen="K1,K6" quelle="Holger Engels" zeit="4" cc="by-sa"}}
28 +Fülle die Wertetabelle aus soweit wie möglich.
29 +(% class="border slim" %)
30 +|=x|2|1|0|-1|-2|-1,5
31 +|={{formula}}(-2)^x{{/formula}}||||||
32 +
33 +Erläutere, warum wir die Exponentialfunktion nur für positive Basen definieren.
51 51  {{/aufgabe}}
52 52  
53 -{{aufgabe id="Exponentialfunktionen erkennen" afb="II" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="8"}}
36 +{{aufgabe id="Basiswechsel verstehen" afb="II" kompetenzen="K5" quelle="Holger Engels" zeit="4" cc="by-sa"}}
37 +Die Funktion {{formula}}f{{/formula}} ist gegeben mit {{formula}}f(x)=2^x{{/formula}}. Gib die Funktion {{formula}}f{{/formula}} in der Form {{formula}}f(x)=4^{kx}{{/formula}} mit einem geeigneten //k// an.
38 +{{/aufgabe}}
54 54  
55 -Bestimme zu jedem Schaubild eine passende Funktionsgleichung.
56 -
57 -[[image:Exponentialfunktionen.svg||width=600]]
40 +{{aufgabe id="Basiswechel" afb="I" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="10" cc="by-sa"}}
41 +Führe bei folgenden Exponentialfunktionen jeweils einen Basiswechsel durch.
42 +(% class="abc" %)
43 +1. {{formula}}f(x)=(\frac{1}{4})^x{{/formula}}, neue Basis {{formula}}b=2{{/formula}}
44 +1. {{formula}}f(x)=(\frac{3}{18})^x{{/formula}}, neue Basis {{formula}}b=6{{/formula}}
45 +1. {{formula}}f(x)=9^x{{/formula}}, neue Basis {{formula}}b=\frac{1}{3}{{/formula}}
46 +1. {{formula}}f(x)=5^{2x+1}{{/formula}}, neue Basis {{formula}}b=25{{/formula}}
47 +1. {{formula}}f(x)=(\frac{16}{54})^{2x}{{/formula}}, neue Basis {{formula}}b=\frac{3}{2}{{/formula}}
58 58  {{/aufgabe}}
59 59  
60 -{{aufgabe id="e-Funktion im Vergleich" afb="II" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="5"}}
50 +{{aufgabe id="e-Funktion im Vergleich" afb="II" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="5" cc="by-sa"}}
61 61  Gegeben ist der Graph zu {{formula}}f(x)=e^x{{/formula}}. Skizziere deine Vermutung wie die Graphen von {{formula}}g(x)=2^x{{/formula}} und {{formula}}h(x)=3^x{{/formula}} verlaufen.
62 62  (Ohne Taschenrechner, ohne Wertetabelle)
63 63  [[image:EFunktion.svg||width=500]]
64 64  {{/aufgabe}}
65 65  
66 -{{aufgabe id="Eigenschaften der e-Funktion" afb="I" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="5"}}
67 -Erstelle einen Steckbrief für die e-Funktion mit allen dir bekannten Eigenschaften.
56 +{{aufgabe id="Eigenschaften der e-Funktion" afb="I" kompetenzen="K4" quelle="Niklas Wunder, Katharina Schneider" zeit="5" cc="by-sa"}}
57 +Erstelle einen Steckbrief für die e-Funktion {{formula}}f(x)=e^x{{/formula}} mit allen relevanten Eigenschaften.
68 68  {{/aufgabe}}
69 69  
70 70  {{aufgabe id="Eulersche Zahl" afb="II" kompetenzen="K1,K6" quelle="Niklas Wunder, Katharina Schneider" zeit="12"}}
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73 73  {{formula}} a_2=2+\frac{1}{1\cdot 2}{{/formula}}
74 74  {{formula}} a_3=2+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}{{/formula}}
75 75  {{formula}} a_4=2+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{1\cdot 2\cdot 3\cdot 4}{{/formula}}
76 -a) Welchem Muster lässt sich bei der Berechnung erkennen? Führe das Muster fort und berechne {{formula}} a_5, a_6
66 +(% class="abc" %)
67 +1. Welchem Muster lässt sich bei der Berechnung erkennen? Führe das Muster fort und berechne {{formula}} a_5, a_6
77 77  {{/formula}}.
78 -b) Die eulersche Zahl {{formula}} e{{/formula}} ist gegben durch {{formula}} e= 2+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{1\cdot 2\cdot 3\cdot 4}+ ...{{/formula}}, d.h durch Fortsetzung des Musters berechnet man die Zahl {{formula}} e{{/formula}} auf immer mehr Nachkommastellen. Gib die Zahl {{formula}} e{{/formula}} so genau an, wie du sie in a) berechnet hast. Hinweis: Du kannst gerne noch mehr weiter Zahlterme {{formula}} a_7,a_8, usw.{{/formula}}, wenn du eine noch höhere Genauigheit haben willst.
69 +1. Die eulersche Zahl {{formula}} e{{/formula}} ist gegeben durch {{formula}} e= 2+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{1\cdot 2\cdot 3\cdot 4}+ ...{{/formula}}, d.h durch Fortsetzung des Musters berechnet man die Zahl {{formula}} e{{/formula}} auf immer mehr Nachkommastellen. Gib die Zahl {{formula}} e{{/formula}} so genau an, wie du sie in a) berechnet hast.
70 +
71 +**Hinweis:** Für die Zahlterme {{formula}} a_7, a_8, ...{{/formula}} erhältst du eine beliebige Genauigheit.
79 79  {{/aufgabe}}
80 80  
81 81  {{lehrende}}
82 -"[[Kompetenzen.K1]] Ich kann die besondere Bedeutung der natürlichen Basis nennen" wird in den Aufgaben nicht vollständig abgedeckt, da die Bedeutung der Basis e als besondere Basis der Exponentialfunktion erst in der Differentialrechnung eine wichtige Rolle bringt und eine Übungsaufgabe zur stetigen Verzinsung zwar im Unterricht oft behandelt wird aber sich nicht unbedingt zum Verständis der Exponentialfunktion an dieser Stelle benötigt wird.
75 +"Ich kann die besondere Bedeutung der natürlichen Basis nennen" wird in den Aufgaben nicht abgedeckt, da die Bedeutung der Basis //e// als besondere Basis der Exponentialfunktion erst in der Differentialrechnung eine wichtige Rolle spielt. Die stetige Verzinsung bietet sich für den Unterricht an.
76 +K3 wird bewusst weggelassen, weil es in [[BPE 4.6>>BPE_4_6]] behandelt wird.
77 +Zu K2 könnte man sich noch was überlegen.
78 +AFB III muss hier nicht erreicht werden.
83 83  {{/lehrende}}
84 84  
85 -{{seitenreflexion bildungsplan="4" kompetenzen="3" anforderungsbereiche="3" kriterien="4" menge="5"/}}
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Exponentialfunktionen.svg
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