From 876e115d152e45729f6b4b5df1d8f24f51b1e15c Mon Sep 17 00:00:00 2001 From: Lars Bogner Date: Tue, 22 Dec 2020 20:06:51 +0100 Subject: [PATCH] Added the first content to the documents. --- Mathematik/Mathematik.md | 52 +++++++++++++++++++++++++++- Mathematik/Mathematik.pdf | Bin 70665 -> 590766 bytes Mathematik/assets/Trigonometrie.png | Bin 0 -> 491145 bytes Physik/Physik.md | 5 ++- Physik/Physik.pdf | Bin 69788 -> 68502 bytes Wirtschaft/Wirtschaft.md | 6 ++-- Wirtschaft/Wirtschaft.pdf | Bin 70626 -> 69596 bytes 7 files changed, 55 insertions(+), 8 deletions(-) create mode 100644 Mathematik/assets/Trigonometrie.png diff --git a/Mathematik/Mathematik.md b/Mathematik/Mathematik.md index 89ba5c0..0097c91 100644 --- a/Mathematik/Mathematik.md +++ b/Mathematik/Mathematik.md @@ -13,11 +13,61 @@ documentclass: report fontsize: 12pt papersize: a4 toc: true +linestretch: 1.25 +copyright: Copyright © 2020 Lars Bogner --- -# Mathematik +# Analysis +## Funktionstypen +### ganzrationale Funktionen +Unter ganzrationalen Funktionen versteht man einen Typ von Funktionen, welcher eine Summe aus Potenzfunktionen mit natürlichem Exponenten sind. Sie sind also wie folgt aufgebaut: $$f(x) = a_n * x^n + .... + a_1 * x^1 + a_0; n \in \mathbb{N}$$. Dabei gibt $n$ an, von welchem Grad diese Funktion ist. $n$ ist immer äquivalent mit dem größten Exponenten. Beispielsweise ist $f(x) = -2x^3+4$ vom Grad 3. + +Eine Funktion vom Grad $n$ kann dabei des Weiteren maximal $n$ Nullstellen besitzen. +Auch ist eine ganzrationale Funktion mit ausschließlich geradzahligen Exponenten immer achsensymmetrisch zur y-Achse und eine mit nur ungeradzahligen Exponenten punktsymmetrisch zum Ursprung. + +### Exponentialfunktionen +Der Begriff Exponentialfunktion bezeichnet eine Funktion, welche die Variable im Exponenten hat. Somit können diese zu $f(x) = a^x$ vereinfacht werden. Im Abitur werden nur natürliche Exponentialfunktionen abgefragt, dass heißt, dass als Basis die eulersche Zahl ($e$) verwendet wird. Somit sind die Exponentialfunktionen welche behandelt werden wie folgt aufgebaut: $$f(x) = e^x$$. + +### trigonometrische Funktionen +Die trigonometrischen Funktionen beschreiben das Verhältnis zwischen der Seitenlängen eines rechtwinkligen Dreiecks. Zu ihnen gehören $\sin, \cos, \tan$. All diese Funktionen sind periodisch und haben in ungestrecktem Zustand eine Periodenlänge von $2 \pi$ bei Sinus und Kosinus, bzw. $\pi$ beim Tangens, im Bogenmaß (`rad`). +Sie beschreiben dabei das Verhältnis der folgenden Seiten: $$\sin{(x)} = \frac{|Gegenkathete|}{|Hypotenuse|}$$ $$\cos{(x)} = \frac{|Ankathete|}{|Hypotenuse|}$$ $$\tan{(x)} = \frac{|Gegenkathete|}{|Ankathete|}$$. +Verallgemeinert können die trigonometrischen Funktionen am Beispiel des Sinus zu folgender Gleichung: $$f(x) = a * \sin{(b * (x-c))} + d$$. Dabei verändern die Parameter die Graphen wie folgt: + +$a$ +: Es kommt zu Streckung ($a>1$), bzw. Stauchung ($a<1$) in y-Richtung. +: > Der Graph der Funktion wurde um den Faktor $a$ in y-Richtung gestreckt/gestaucht. Die entstandene Amplitude entspricht $a$. + +$b$ +: Es kommt zu Streckung ($b<1$), bzw. Stauchung ($a>1$) in x-Richtung. Die Periode der Funktion beträgt dabei immer $p = \frac{2 \pi}{b}$. **-> desto kleiner b, desto größer die Periodenlänge** +: > Der Graph der Funktion wurde um den Faktor $b$ in y-Richtung gestaucht. Die entstandene Periodendauer entspricht $\frac{2 \pi}{b}$. + +$c$ +: Es kommt zur Verschiebung in x-Richtung, nach links ($c<0$) bzw. rechts ($c>0$). +: > Der Graph der Funktion wurde um $c$ nach links/rechts verschoben. + +$d$ +: Es kommt zur Verschiebung in y-Richtung nach oben ($d>1$), bzw. unten ($d<1$). +: > Der Graph der Funktion wurde um $d$ nach oben/unten verschoben. +Abbildung(vgl. Abbildung trigonometrische Funktionen) der trigonometrischen Funktionen im Koordinatensystem. $f(x)= \sin{(x)}; g(x) = \cos{(x)}; h(x) = \tan{(x)}$ + +![trigonometrische Funktionen](assets/Trigonometrie.png) + +### Potenzfunktionen + +# Geometrie + + +# Stochastik + + +# Anhang +## Grundwissen + + +## Abiturrichtlinien +Für genaue Informationen siehe [Dokument](https://rp.baden-wuerttemberg.de/rps/Abt7/Ref75/Fachberater/Documents/Mathe/2021-LF-Konvolut.pdf)^[Regierungspräsidium Baden-Württemberg. Leistungsfach Mathematik Schriftliche Abiturprüfung 2021 Und 2022. 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