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Änderungen von Dokument BPE 1.4 Lineare Funktionen

Zuletzt geändert von Holger Engels am 2024/11/20 21:57
Von Version 34.1
bearbeitet von Holger Engels
am 2024/09/29 11:17
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 17.7
bearbeitet von sschaefer Sabine Schäfer
am 2023/10/10 18:49
Änderungskommentar: Es gibt keinen Kommentar für diese Version

Zusammenfassung

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Dokument-Autor
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1 -XWiki.holgerengels
1 +XWiki.sschaefer
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9 9  [[Kompetenzen.K5]] Ich kann die Lösungsmengen linearer Ungleichungen mit Äquivalenzumformungen ermitteln
10 10  [[Kompetenzen.K5]] Ich kann die Lagebeziehung zweier Geraden anhand ihrer Gleichungen und der Orthogonalitätsbedingung untersuchen
11 11  
12 -{{lernende}}
13 -[[Interaktive Erkunden>>https://kmap.eu/app/browser/Mathematik/Lineare%20Funktionen/Hauptform#erkunden]]
14 -{{/lernende}}
12 +>> Platz für Links auf Selbstlernmaterial
15 15  
16 -{{aufgabe id="Besondere Geraden" afb="I" kompetenzen="K4, K5" quelle="Holger Engels" cc="BY-SA" zeit="4"}}
17 -[[image:geraden.svg||style="float: left; width: 250px"]]Das Schaubild zeigt vier Geraden. Alle können als Gleichung ausgedrückt werden. Drei stellen auch einen funktionalen Zusammenhang dar.
18 18  
19 -Gib jeweils eine Geradengleichungen und soweit möglich auch einen Funktionsterme an.
15 +{{aufgabe afb="I" kompetenzen="K4" "K5" quelle="Sabine Schäfer" cc="BY-SA" zeit="5"}}
16 +Das Schaubild zeigt die Graphen von linearen Funktionen. Ordnen Sie die folgenden Funktionsvorschriften begründet zu.
17 +
18 + [[image:sb geraden.png]]
20 20  
21 -Begründe, warum die vierte Gerade nicht Graph einer Funktion sein kann.
22 -{{/aufgabe}}
20 +
23 23  
24 -{{aufgabe id="Taxifahrt" afb="I" kompetenzen="K4, K5" quelle="Sabine Schäfer" cc="BY-SA" zeit="5"}}
25 -Stellen Sie folgende Situation grafisch dar und bestimmen Sie eine Gleichung, die den Sachverhalt ebenfalls beschreibt.
22 +a){{formula}}\begin {large}f\left(x\right)=x-1;x\in\mathbb{R}\end{large}{{/formula}} b) {{formula}} f\left(x\right)=1 - x^2;x\in\mathbb{R}{{/formula}} c) {{formula}} f\left(x\right)=\frac23x-2;x\in\mathbb{R}{{/formula}} d) {{formula}}f\left(x\right)=-\frac14x-1;x\in\mathbb{R}{{/formula}} e){{formula}}f\left(x\right)=-0,25 x-2;x\in\mathbb{R} R{{/formula}} f){{formula}}f\left(x\right)=2 + 2x;x\in\mathbb{R}{{/formula}}
26 26  
27 -Für eine Taxifahrt fallen zunächst 5 Euro für die Anfahrt an. Dazu kommen pro angefangener gefahrener Minute 0,75 Euro.
28 -Es werden Fahrten von 5 Minuten, 10 Minuten und 15 Minuten durchgeführt.
24 +
25 + {{/aufgabe}}
26 +
27 +
28 +
29 +{{aufgabe afb="I" kompetenzen="K2, K4, K5" zeit="5" quelle="[[IQB 2019 Analysis gAN Teil 2 CAS>>file:///home/holger/Downloads/Beispielaufgaben_M_grundlegend_B_Analysis_CAS.pdf]]" lizenz="[[CC BY 3.0>>https://creativecommons.org/licenses/by-sa/3.0/deed.de]]" links="[[Interaktiv erkunden>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Mittlere%20%C3%84nderungsrate#erkunden]]"}}
30 +BMX-Fahrräder sind speziell für das Gelände ausgelegte Sportgeräte. Für den profes-
31 +sionellen Einsatz dieser Fahrräder wird auf horizontalem Untergrund eine 3 m breite
32 +Sprungschanze installiert. Im Längsschnitt der Schanze kann deren Profillinie für
33 +{{formula}}x ∈
34 + \in\left[ -8;0 \right]{{/formula}} modellhaft durch die in {{formula}}\mathbb{R}{{/formula}} definierte Funktion f mit
35 +
36 +{{formula}}
37 +f(x)=-\frac{5}{256}x^3-\frac{3}{4}x+2
38 +{{/formula}}
39 +
40 +beschrieben werden. Die Abbildung 1 zeigt den zugehörigen Teil des Graphen von //f//.
41 +Der Startpunkt, von dem aus die Schanze durchfahren wird, wird durch den Punkt
42 +{{formula}}S( -8 | f ( -8 ) ){{/formula}} dargestellt, der Absprungpunkt durch {{formula}}A(0 | f ( 0 ) ){{/formula}}.
43 +
44 +[[Abbildung 1>>image:Schanze.png]]
45 +
46 +Veranschaulichen Sie in Abbildung 1 die mittlere Steigung der Schanze zwischen
47 +Startpunkt und Absprungpunkt. Bestimmen Sie diese Steigung.
29 29  {{/aufgabe}}
30 30  
31 -{{aufgabe id="Funktionsvorschriften zuordnen" afb="I" kompetenzen="K4, K5" quelle="Sabine Schäfer" cc="BY-SA" zeit="5"}}
32 -[[image:sb geraden.png||style="float: right" width="400"]]Das Schaubild zeigt die Graphen von linearen Funktionen. Ordne die folgenden Funktionsvorschriften begründet zu.
50 +{{aufgabe afb="II" kompetenzen="K3, K5" quelle="IQB 2019 Analysis gAN Teil 2 WTR" lizenz="[[CC BY 3.0>>https://creativecommons.org/licenses/by-sa/3.0/deed.de]]"}}
51 +Im Rahmen eines Tests läuft ein Sportler auf einem Laufband. Dabei wird bei ansteigender Geschwindigkeit jeweils die Konzentration sogenannter Laktate im Blut gemessen.
52 +Die Abhängigkeit der Laktatkonzentration von der Geschwindigkeit kann für {{formula}}8,5\leq x \leq 17,5{{/formula}} modellhaft durch die Funktion //k// beschrieben werden mit:
33 33  
34 -a) {{formula}}f\left(x\right)=x-1;x\in\mathbb{R} {{/formula}}
35 -b) {{formula}}f\left(x\right)=1 - x^2;x\in\mathbb{R}{{/formula}}
36 -c) {{formula}}f\left(x\right)=\frac23x-2;x\in\mathbb{R}{{/formula}}
37 -d) {{formula}}f\left(x\right)=-\frac14x-1;x\in\mathbb{R}{{/formula}}
38 -e) {{formula}}f\left(x\right)=-0,25 x-2;x\in\mathbb{R}{{/formula}}
39 -f) {{formula}}f\left(x\right)=2 - 2x;x\in\mathbb{R}{{/formula}}
54 +{{formula}}
55 +k(x) = \frac{1}{40}(x^{3}-30x^{2}+288x-815)
56 +{{/formula}}
57 +
58 +Dabei ist {{formula}}x{{/formula}} die Geschwindigkeit des Sportlers in Kilometer pro Stunde und //k// die Laktatkonzentration in Millimol pro Liter {{formula}}\frac{mmol}{l}{{/formula}}. Berechnen Sie im Modell für den Geschwindigkeitsbereich von 12 bis 17,5 {{formula}}\frac{km}{h}{{/formula}} die mittlere Änderungsrate der Laktatkonzentration.
40 40  {{/aufgabe}}
41 41  
42 -{{aufgabe id="Abstand Graph Koordinatenursprung" afb="I" kompetenzen="K1,K2,K4,K5" quelle="[[IQB>>https://www.iqb.hu-berlin.de/abitur/pools2023/abitur/pools2023/mathematik/grundlegend/2023_M_grundlege_7.pdf]]" niveau="g" tags="iqb"}}
43 -[[image:Graph0,5x+5.PNG||width="220" style="float: right"]]
44 -Die Abbildung zeigt den Graphen der in {{formula}}\mathbb{R}{{/formula}} definierten linearen Funktion {{formula}} f{{/formula}}.
45 -1. Begründe, dass {{formula}}f(x)=\frac{1}{2}x+5{{/formula}} gilt.
46 -1. Berechne den Abstand des Koordinatenursprungs zum Graphen.
61 +{{aufgabe afb="II" kompetenzen="K2, K4, K5" quelle="Abi 2012 Anwendung, modifiziert"}}
62 +Ein Kondensator ist ein Bauteil, das elektrische Ladung speichert. Der Ladevorgang eines Kondensators wird im Labor untersucht. Zum Zeitpunkt t = 0 beginnt der Aufladevorgang. Die Stärke des elektrischen Stroms, der beim Aufladen fließt, wird gemessen. Die Messwerte sind in folgender Tabelle zusammengefasst:
63 +
64 +(% style="width:min-content" %)
65 +|=Zeit [s]|1,0|2,4|4,8|7,2|9,6
66 +|=Stromstärke [mA]|9,0|6,0|3,0|1,5|0,75
67 +
68 +Ermitteln Sie einen Zeitraum beim Ladevorgang, in der die durchschnittliche Änderungsrate der Stromstärke halb so groß ist wie im Zeitraum von 2,4 s bis 4,8 s!
47 47  {{/aufgabe}}
48 48  
49 -{{seitenreflexion kompetenzen="3" anforderungsbereiche="1" kriterien="2" menge="1"/}}
71 +((({{seitenreflexion kompetenzen="3" anforderungsbereiche="1" kriterien="2" menge="1"/}})))
50 50  
73 +>> Hier eventuell ein Abschnitt, der nur für Lehrende sichtbar ist mit Grundvorstellungen, ggf. typischen aufzulösenden Fehlvorstellungen, Unterrichtsideen, ...
51 51  
75 +
76 +
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