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Zuletzt geändert von Stephanie Wietzorek am 2025/05/21 10:29
Von Version 41.1
bearbeitet von Holger Engels
am 2025/05/20 13:53
Änderungskommentar: Neues Bild Polynome zuordnen f.svg hochladen
Auf Version 62.3
bearbeitet von Holger Engels
am 2025/05/20 16:14
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1 1  {{seiteninhalt/}}
2 2  
3 -=== Kompetenzen ===
4 4  [[Kompetenzen.K4]] [[Kompetenzen.K5]] Ich kann Werte der Tangentensteigung graphisch bestimmen
5 5  [[Kompetenzen.K4]] [[Kompetenzen.K1]] Ich kann aus Werten der Tangentensteigung einen Graphen zeichnen und diesen als Graphen der Ableitungsfunktion deuten
6 6  [[Kompetenzen.K6]] Ich kann Zusammenhänge zwischen den beiden Funktionsgraphen beschreiben
... ... @@ -10,15 +10,11 @@
10 10  **Interaktiv Erkunden:** [[Graphisches Ableiten>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Graphisches%20Ableiten#erkunden]]
11 11  {{/lernende}}
12 12  
13 -* Kann eine Tangente den Funktionsgraphen schneiden?
14 -
15 -* Bereiche mit positiver/ negativer Steigung schraffieren
16 16  * Punktweise graphisch ableiten
17 17  * Qualitativ graphisch ableiten
18 18  * Zusammenhänge HP, TP, SP vorwärts und rückwärts
19 19  
20 20  * Funktionsterm der Ableitungsfunktion aus Tangentensteigungen aufstellen
21 -* Zuordnung Polynomfunktionen
22 22  * Beobachtungen bei e^x
23 23  
24 24  {{aufgabe id="Tangenten einzeichnen" afb="I" kompetenzen="" quelle="Holger Engels" cc="BY-SA" zeit="3"}}
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31 31  [[image:Tangenten einzeichnen 1.svg|| width="350px"]] [[image:Tangenten einzeichnen 2.svg|| width="350px"]] [[image:Tangenten einzeichnen 3.svg|| width="350px"]] [[image:Tangenten einzeichnen 4.svg|| width="350px"]]
32 32  {{/aufgabe}}
33 33  
34 -{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="3"}}
29 +{{aufgabe id="Punkte mit gegebener Steigung finden" afb="?" kompetenzen="" quelle="Stephanie Wietzorek und Simone Kanzler" cc="BY-SA" zeit="?"}}
35 35  Es ist das Schaubild {{formula}}K_f{{/formula}} einer Funktion {{formula}}f{{/formula}} gegeben. Kennzeichne Punkte auf {{formula}}K_f{{/formula}}, für die gilt:
36 - {{formula}}f'(x) =1{{/formula}}
37 - {{formula}}f'(x) =1,5{{/formula}}
38 - {{formula}}f'(x) =0{{/formula}}
39 - {{formula}}f'(x) = -\frac{17}{4}{{/formula}}
31 + die Steigung der Tangente in diesem Punkt ist 1
32 + die Steigung der Tangente in diesem Punkt ist 1,5
33 + die Steigung der Tangente in diesem Punkt ist 0
34 + die Steigung der Tangente in diesem Punkt ist {{formula}}-\frac{17}{4}{{/formula}}
40 40  [[image:Tangentensteigung.svg|| width="700px"]]
41 41  {{/aufgabe}}
42 42  
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48 48  | [[image:Polynome zuordnen i.svg||width=200]] | | | | | [[image:Polynome zuordnen D.svg||width=200]]
49 49  {{/aufgabe}}
50 50  
46 +{{aufgabe id="Skizzieren anhand Eigenschaften" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="4"}}
47 +a) Skizziere eine mögliche Parabel 2. Grades, welche eine waagrechte Tangente an der Stelle {{formula}}x = -2{{/formula}} hat. Welche Gemeinsamkeiten haben diese Parabeln?
48 +
49 +b) Skizziere das Schaubild einer möglichen Funktion, welches drei waagrechte Tangenten besitzt. Welchen minimalen Grad hat die Funktion?
50 +
51 +c) Eine Funktion f hat nur positive Steigungen. Skizziere das Schaubild der Ableitungsfunktion.
52 +
53 +d) Es ist ein zur y-Achse symmetrisches Schaubild einer Funktion 4. Grades gesucht. Folgende Angaben sind bekannt, fülle die Lücken und skizziere das Schaubild der Funktion.
54 +(% class="border" %)
55 +|x|-4|-1|0|1 |4
56 +|Funktionswert|-2,5| |2 |0|
57 +|Tangentensteigung|-2| |0|-1 |
58 +{{/aufgabe}}
59 +
60 +{{aufgabe id="Aussagen Polynomfunktion" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
61 +Prüfe die Aussagen! Welche sind wahr? Eine Polynomfunktion 3. Grades ..
62 +☐ hat immer zwei Extrempunkte!
63 +☐ kann auch mal nur einen Extrempunkt haben!
64 +☐ kann auch mal keinen Extrempunkt haben!
65 +☐ hat immer genau einen Wendepunkt!
66 +☐ hat entweder einen Sattelpunkt oder zwei Extrempunkte!
67 +{{/aufgabe}}
68 +
69 +{{aufgabe id="Aussagen Schaubild" afb="?" kompetenzen="" quelle="Stephanie Wietzorek, Simone Kanzler" cc="BY-SA" zeit="?"}}
70 +Gegeben ist das Schaubild einer Funktion. Nimm Stellung zu folgenden Aussagen und begründe deine Antwort.
71 +[[image:Aussagen.svg|| width="500px"]]
72 +☐ {{formula}}f(-3)=3{{/formula}}
73 +☐ {{formula}}x = 3{{/formula}} ist dreifache Nullstell
74 +☐ die Tangentensteigungen sind negativ für {{formula}}x \in ]2;5[{{/formula}}
75 +☐ die Steigung der Tangente an der Stelle {{formula}}x = 1<-2{{/formula}}
76 +☐ an der Stelle {{formula}}x = 2{{/formula}} liegt eine waagrechte Tangente
77 +☐ die Tangentensteigungen sind negativ für {{formula}}-4 < x < 2{{/formula}}
78 +☐ die Tangentensteigungen haben einen Vorzeichenwechsel bei {{formula}}x=-4{{/formula}} von ⊝ ⇾ ⊕
79 +{{/aufgabe}}
80 +
81 +{{aufgabe id="Aussagen Sattelstelle" afb="I" kompetenzen="" quelle="KMap" cc="BY-SA" zeit="3"}}
82 +Welche Aussagen treffen auf eine Sattelstelle zu?
83 +☐ Eine Sattelstelle hat eine waagrechte Asymptote
84 +☐ An einer Sattelstelle hat die Steigung ein Maximum oder ein Minimum
85 +☐ An einer Sattelstelle gibt es immer auch einen Krümmungswechsel
86 +☐ Eine Sattelstelle ist auch eine Wendestelle
87 +☐ Eine Sattelstelle kann auch eine Maximalstelle sein
88 +{{/aufgabe}}
89 +
51 51  {{seitenreflexion bildungsplan="" kompetenzen="" anforderungsbereiche="" kriterien="" menge=""/}}
Aussagen.ggb
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Polynome zuordnen g.svg
Author
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1 +XWiki.holgerengels
Größe
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1 +29.6 KB
Inhalt
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522 + <path d="M 0.0000 0.0000 L 0.0000 361.00 L 395.00 361.00 L 395.00 0.0000 z"/>
523 +</clipPath>
524 +<g clip-path="url(#clip64)">
525 +<g transform="matrix(1.0000, 0.0000, 0.0000, 1.0000, 182.00, 40.000)">
526 +<text fill-opacity="1.0000" font-style="normal" font-family="Helvetica" font-weight="bold" stroke="none" fill="#000000" font-size="12.000" x="0" y="0">4</text>
527 +</g> <!-- transform -->
528 +</g> <!-- clip64 -->
529 +<clipPath id="clipe3409062-91a3-43bc-9e33-de4d0954404f">
530 + <path d="M 0.0000 0.0000 L 0.0000 361.00 L 395.00 361.00 L 395.00 0.0000 z"/>
531 +</clipPath>
532 +<g clip-path="url(#clip65)">
533 +<g transform="matrix(1.0000, 0.0000, 0.0000, 1.0000, 182.00, 201.00)">
534 +<text fill-opacity="1.0000" font-style="normal" font-family="Helvetica" font-weight="bold" stroke="none" fill="#000000" font-size="12.000" x="0" y="0">0</text>
535 +</g> <!-- transform -->
536 +</g> <!-- clip65 -->
537 +<g id="misc">
538 +</g><!-- misc -->
539 +<g id="layer0">
540 +<clipPath id="clipaca1c80e-8e88-4f30-986b-bb0797fd488b">
541 + <path d="M 0.0000 0.0000 L 0.0000 361.00 L 395.00 361.00 L 395.00 0.0000 z"/>
542 +</clipPath>
543 +<g clip-path="url(#clip66)">
544 +<g stroke-linejoin="round" stroke-width="3.5000" stroke-linecap="round" fill="none" stroke-opacity=".74902" stroke="#009900">
545 + <path d="M 61.719 -8.6233 L 63.262 -2.0105 L 64.805 4.4503 L 66.348 10.761 L 67.891 16.923 L 69.434 22.939 L 70.977 28.809 L 72.520 34.536 L 74.062 40.123 L 75.605 45.569 L 77.148 50.877 L 78.691 56.050 L 80.234 61.088 L 83.320 70.768 L 86.406 79.933 L 89.492 88.595 L 92.578 96.770 L 95.664 104.47 L 98.750 111.71 L 101.84 118.51 L 104.92 124.87 L 108.01 130.82 L 111.09 136.37 L 114.18 141.52 L 117.27 146.30 L 120.35 150.72 L 123.44 154.80 L 126.52 158.54 L 129.61 161.96 L 132.70 165.08 L 135.78 167.91 L 138.87 170.47 L 141.95 172.76 L 145.04 174.80 L 148.12 176.62 L 151.21 178.21 L 154.30 179.60 L 157.38 180.79 L 160.47 181.81 L 163.55 182.67 L 166.64 183.38 L 169.73 183.95 L 172.81 184.41 L 175.90 184.75 L 178.98 185.01 L 182.07 185.19 L 185.16 185.30 L 188.24 185.37 L 191.33 185.39 L 194.41 185.40 L 197.50 185.40 L 200.59 185.41 L 203.67 185.44 L 206.76 185.50 L 209.84 185.62 L 212.93 185.79 L 216.02 186.05 L 219.10 186.39 L 222.19 186.85 L 225.27 187.42 L 228.36 188.12 L 231.45 188.98 L 234.53 190.00 L 237.62 191.19 L 240.70 192.57 L 243.79 194.16 L 246.88 195.97 L 249.96 198.01 L 253.05 200.30 L 256.13 202.85 L 259.22 205.68 L 262.30 208.79 L 265.39 212.21 L 268.48 215.95 L 271.56 220.02 L 274.65 224.43 L 277.73 229.21 L 280.82 234.36 L 283.91 239.90 L 286.99 245.84 L 290.08 252.20 L 293.16 258.99 L 296.25 266.22 L 299.34 273.92 L 302.42 282.08 L 305.51 290.74 L 308.59 299.89 L 311.68 309.57 L 313.22 314.60 L 314.77 319.77 L 316.31 325.07 L 317.85 330.52 L 319.39 336.10 L 320.94 341.82 L 322.48 347.69 L 324.02 353.70 L 325.57 359.86 L 327.11 366.16"/>
546 +<title>Function g</title>
547 +<desc>g(x) = -x³</desc>
548 +
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550 +</g> <!-- clip66 -->
551 +</g><!-- layer0 -->
552 +<defs>
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561 +</font>
562 +
563 +</defs> <!-- font definitions -->
564 +</g> <!-- default stroke -->
565 +</svg> <!-- bounding box -->
566 +