\documentclass[a7paper]{kartei} \usepackage[utf8]{inputenc} %UTF8 \usepackage[OT1]{fontenc} \usepackage[scaled]{helvet} \usepackage[ngerman]{babel} % Neue Rechtschreibung \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{nicefrac} % Deutsche Absatzformatierung \setlength{\parindent}{0pt} \setlength{\parskip}{1em} % Oft verwendete spacer \def \bk {\hspace{2mm}} \begin{document} \fach{SBP Mathe Grundkurs 1} \kommentar{by Clifford Wolf} \include{hinweis} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Mengenoperationen} \begin{tabular}{lll} $x \in A$, & $x \notin A$ & $x$ ist (nicht) Element von $A$. \\ $A \subseteq B$, & $A \not\subseteq B$ & $A$ ist (nicht) Teilmenge von $B$. \\ $A \subset B$, & $A \not\subset B$ & $A$ ist (nicht) echte Teilmenge von $B$. \end{tabular} \begin{tabular}{lllll} $A \cap B$ & $=$ & $\left\{ x \mid x \in A \land x \in B \right\}$ & $=$ & Schnittmenge \\ $A \cup B$ & $=$ & $\left\{ x \mid x \in A \lor x \in B \right\}$ & $=$ & Vereinigungsmenge \\ $A \setminus B$ & $=$ & $\left\{ x \mid x \in A \land x \notin B \right\}$ & $=$ & Differenzmenge \\ \end{tabular} \begin{tabular}{lllll} $A \times B$ & $=$ & $\left\{ (x,y) \mid x \in A \land y \in B \right\}$ & $=$ & Produktmenge \end{tabular} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Logische Operationen} \begin{tabular}{lll} $A \Leftrightarrow B$ & \"Aquivalenz & (gleichbedeutind mit) \\ $A \Rightarrow B$ & Implikation & (daraus folgt) \\ $A \land B$ & Konjunktion & (und) \\ $A \lor B$ & Disjunktion & (oder) \\ $A \bar\lor B$ & Antivalenz & (ungleich, entweder-oder) \\ $\neg A$ & Negation & (nicht) \\ $\forall A : B$ & Allquantor & (fuer alle $A$ gilt $B$) \\ $\exists A : B$ & Existenz & (es gibt ein $A$ fuer das $B$ gilt) \end{tabular} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{nat\"urliche Zahlen} \centering \vspace*{15mm} $\mathbb{N} = \left\{ 0, 1, 2, 3, ... \right\}$ ($\mathbb{N}^\star = \left\{ 1, 2, 3, ... \right\}$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{ganze Zahlen} \centering \vspace*{15mm} $\mathbb{Z} = \left\{ ..., -3, -2, -1, 0, 1, 2, 3, ... \right\}$ ($\mathbb{Z}^- = \left\{ ..., -3, -2, -1 \right\}$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{rationale Zahlen} \centering \vspace*{5mm} $$\mathbb{Q} = \left\{ \frac{z}{n} | z \in \mathbb{Z}, n \in \mathbb{N}^\star \right\}$$ $\mathbb{Q}^+ = \left\{ q | q \in \mathbb{Q}, q > 0 \right\}$ \\ $\mathbb{Q}^- = \left\{ q | q \in \mathbb{Q}, q < 0 \right\}$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{reelle Zahlen} \centering \vspace*{15mm} $\mathbb{R} =$ alle Zahlen auf der Zahlengerade Untermengen: $\mathbb{R}^+, \mathbb{R}^-, \mathbb{R}^+_0, \mathbb{R}^-_0$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Assoziativgesetze \\ (Addition und Multiplikation)} \centering \vspace*{15mm} $a + (b + c) = (a + b) + c$ $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Kommutativgesetze \\ (Addition und Multiplikation)} \centering \vspace*{15mm} $a + b = b + a$ $a \cdot b = b \cdot a$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Distributivgesetz \\ der Multiplikation} \centering \vspace*{20mm} $a \cdot (b + c) = a \cdot b + a \cdot c$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{\"Aquivialenzumformungen \\ der Multiplikation} \centering \vspace*{15mm} $$a \cdot b = c \hspace{2mm} \Leftrightarrow \hspace{2mm} a = \frac cb \hspace{2mm} \Leftrightarrow \hspace{2mm} b = \frac ca$$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{\"Aquivialenzumformungen \\ der Addition} \centering \vspace*{15mm} $$a + b = c \hspace{2mm} \Leftrightarrow \hspace{2mm} a = c - b \hspace{2mm} \Leftrightarrow \hspace{2mm} b = c - a$$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: lineare Gleichung} \vspace*{5mm} eine lineare Gleichung ist eine Gleichung mit den Variablen $x_n$ der Gestalt $$ax_1 + bx_2 + ... = k.$$ Die L\"osungsmenge eines linearen Gleichungssystems ist die \\ Schnittmenge der L\"osungsmengen der einzelnen Gleichungen. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Gauss'sches Eliminationsverfahren} \begin{equation*} \begin{split} ax_1 & + bx_2 = c \hspace{5mm} | \cdot d \\ dx_1 & + ex_2 = f \hspace{5mm} | \cdot -a \\ & \Leftrightarrow \\ adx_1 & + bdx_2 = dc \\ -adx_1 & - aex_2 = -af \\ & \Longrightarrow \\ (ad-ad)x_1 + & (bx - ae)x_2 = dc - af \\ & \Leftrightarrow \\ & (bx - ae)x_2 = dc - af \end{split} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Betrag} \vspace*{10mm} \begin{equation*} \left|a\right| = \left\{\begin{aligned} a & \hspace{2mm} | \hspace{2mm} a \ge 0 \\ -a & \hspace{2mm} | \hspace{2mm} a < 0 \end{aligned}\right. \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: lineare Funktion} \centering \vspace*{15mm} $f(x) = kx + d$ \vspace*{10mm} $\Rightarrow f(0) = d$, \hspace{10mm} $\Rightarrow f(x+1) - f(x) = k$ $\Rightarrow$ der Graph von $f$ ist eine Gerade mit der Steigung $k$. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{2-Punkt Formel f\"ur lineare Funktion} \vspace*{2mm} \begin{gather*} f(x) = k \cdot x + d \\ \\ k = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \\ \\ d = f(x) - k \cdot x \end{gather*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: direkte und indirekte Proportionalit\"at} \vspace*{7mm} direkte Proportionalit\"at: \\ \centerline{$f(x) = k \cdot x$} \vspace*{10mm} indirekte Proportionalit\"at: \\ \centerline{$f(x) = \frac k x$} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Konstante Faktoren bei direkter und indirekter Proportionalit\"at} \vspace*{7mm} bei direkter Proportionalit\"at: \\ \centerline{$f(x) = k \cdot x \hspace{2mm} \Rightarrow \hspace{2mm} f(a \cdot x) = a \cdot f(x)$} \vspace*{10mm} bei indirekter Proportionalit\"at: \\ \centerline{$f(x) = \frac k x \hspace{2mm} \Rightarrow \hspace{2mm} f(a \cdot x) = \frac{f(x)}{a}$} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: (streng) monoton steigend/fallend} streng monoton steigend (wachsend): \\ $x_2 > x_1 \Rightarrow f(x_2) > f(x_1)$ \medskip streng monoton fallend (abnehmend): \\ $x_2 > x_1 \Rightarrow f(x_2) < f(x_1)$ \bigskip monoton steigend (wachsend): \\ $x_2 > x_1 \Rightarrow f(x_2) \ge f(x_1)$ \medskip monoton fallend (abnehmend): \\ $x_2 > x_1 \Rightarrow f(x_2) \le f(x_1)$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Graph einer reellen Funktion} \centering \vspace*{10mm} $f: \mathbb{A} \rightarrow \mathbb{R}, \hspace{2mm} \mathbb{A} \in \mathbb{R}$ \vspace*{5mm} $G = \left\{ (x;y) \hspace{2mm} | \hspace{2mm} x \in \mathbb{A}, y = f(x) \right\}$ $G =$ Graph der reellen Funktion $f$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Potenzieren von Ungleichungen} \centering \vspace*{20mm} $a < b \bk \Leftrightarrow \bk a^n < b^n$ wenn $a,b \in \mathbb{R}_0^+$ und $n \in \mathbb{R}^+$. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Monotoniegesetz der Addition} \centering \vspace*{10mm} $Mon+$:\hspace*{50mm} $a < b \bk \Leftrightarrow \bk a + c < b + c$ ($a, b, c \in \mathbb{R}$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Monotoniegesetze der Multiplikation} \centering \vspace*{5mm} $Mon \cdot pos$:\hspace*{50mm} $a < b \bk \Leftrightarrow \bk a \cdot c < b \cdot c \bk | \bk c > 0$ $Mon \cdot neg$:\hspace*{50mm} $a < b \bk \Leftrightarrow \bk a \cdot c > b \cdot c \bk | \bk c < 0$ ($a, b, c \in \mathbb{R}$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Kehrwert und Negation bei Ungleichungen mit 0} \centering \vspace*{5mm} \begin{equation*} \begin{split} 0 < a \bk & \Leftrightarrow 0 < \frac 1 a \\ 0 < a \bk & \Leftrightarrow 0 > -a \\ 0 < a < b & \Leftrightarrow 0 < \frac 1 b < \frac 1 a \end{split} \end{equation*} ($a,b \in \mathbb{R}$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Addition und Multiplikation von Ungleichungen} \centering \vspace*{5mm} $a < b \bk\land\bk c < d \bk \Rightarrow \bk a + c < b + d$ ($a,b,c,d \in \mathbb{R}$) \bigskip $a < b \bk\land\bk c < d \bk \Rightarrow \bk a \cdot c < b \cdot d$ ($a \in \mathbb{R} \bk\land\bk b,c,d \in \mathbb{R}^+$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Transitivgesetz der Ordnungsrelation} \centering \vspace*{20mm} $a < b \bk\land\bk b < c \bk \Rightarrow \bk a < c$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Formeln f\"ur Quadrat-Binome} \centering \vspace*{10mm} $\left(a + b\right)^2 = a^2 + 2ab + b^2$ \bigskip $\left(a - b\right)^2 = a^2 - 2ab + b^2$ \bigskip $\left(a + b\right) \cdot \left(a - b\right) = a^2 - b^2$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{L\"osungsformel und Strategie fuer \\ $x^2 + px + q = 0$} \centering \vspace*{5mm} $x^2 + px + q = 0 \bk\Leftrightarrow$ $\Leftrightarrow\bk x^2 + px = -q \bk\Leftrightarrow$ $\Leftrightarrow\bk x^2 + px + \left(\frac p 2\right)^2 = \left(\frac p 2\right)^2 -q \bk\Leftrightarrow$ $\Leftrightarrow\bk \left(x + \frac p 2\right)^2 = \left(\frac p 2\right)^2 -q \bk\Leftrightarrow\bk ...$ $... \bk\Leftrightarrow\bk x = -\frac p 2 \pm\sqrt{\left(\frac p 2\right)^2 - q}$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Wie viele L\"osungen hat \\ $x^2 + px + q = 0$?} \centering \vspace*{10mm} $x^2 + px + q = 0 \bk\Leftrightarrow\bk x = -\frac p 2 \pm\sqrt{\left(\frac p 2\right)^2 - q}$ \begin{tabular}{rl} 2 L\"osungen in $\mathbb{R}$ wenn & $\left(\frac p 2\right)^2 - q \bk>\bk 0$ \\ 1 L\"osung in $\mathbb{R}$ wenn & $\left(\frac p 2\right)^2 - q \bk=\bk 0$ \\ keine L\"osung in $\mathbb{R}$ wenn & $\left(\frac p 2\right)^2 - q \bk<\bk 0$ \\ \end{tabular} $D \bk=\bk \left(\frac p 2\right)^2 - q \bk=\bk$ Diskriminante \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Satz von VIETA} \vspace*{5mm} Seien $\alpha_1$ und $\alpha_2$ L\"osungen von \bigskip \centerline{$x^2 \bk+\bk px \bk+\bk q \bk=\bk 0$} dann gilt fuer alle $x \in \mathbb{R}$: \bigskip \centerline{$x^2+px+q \bk=\bk (x-\alpha_1) \cdot (x-\alpha_2)$} mit $\alpha_1 + \alpha_2 = -p$ und $\alpha_1 \cdot \alpha_2 = q$. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{L\"osungsformel f\"ur \\ $ax^2 + bx + c = 0$} \centering \vspace*{10mm} $ax^2 + bx + c = 0$ $\Longleftrightarrow$ $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition von Polynomfunktion} \centering \vspace*{15mm} $f(x) \bk=\bk a_nx^n + a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$ \bigskip $n \bk=\bk$ Grad der Polynomfunktion \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Abspalten eines Linearfaktors} Sei $f$ eine Polynomfunktion $n$-ten Grades und $\alpha \in \mathbb{R}$ eine Nullstelle von $f$, dann gibt es eine Polynomfunktion $g$ $(n-1)$-ten Grades, so dass f\"ur alle $x \in \mathbb{R}$ gilt: \bigskip \centerline{$f(x) = (x-\alpha) \cdot g(x)$} \bigskip \bigskip Methoden zur Ermittlung der Koeffizienten von $g$: \begin{itemize} \item Koeffizientenvergleich \item Polynomdivision \end{itemize} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Methode des Koeffizientenvergleichs} Beispiel - allgemeines Polynom dritter Ordnung: \centering \medskip $a_3x^3 + a_2x^2 + a_1x + a_0 \bk=\bk (x - \alpha) \cdot (b_2x^2 + b_1x + b_0) \bk=$ $\begin{array}{llll} =\bk b_2x^3 & + b_zx^2 & + b_0x & \\ & - \alpha b_2x^2 & - \alpha b_1x & - \alpha b_0 \bk= \end{array}$ $=\bk \underbrace{b_2}_{a_3}x^3 + \underbrace{\left(b_1 - \alpha b_2\right)}_{a_2}x^2 + \underbrace{\left(b_0 - \alpha b_1\right)}_{a_1}x + \underbrace{\left(- \alpha b_0\right)}_{a_0} \bk\bk\bk\bk$ \bigskip $\Rightarrow \bk b_2 = a_3, \bk b_1 = a_2 + \alpha b_2, \bk b_0 = a_1 + \alpha b_1, \bk \underbrace{\alpha b_0 = -a_0}_{\mbox{Kontrolle}}$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Satz von Horner} \centering \vspace*{15mm} $a^n-b^n \bk=\bk (a-b)\cdot(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})$ \bigskip Beweis durch Ausmultiplizieren: \\ alle Terme in der Mitte fallen weg. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition der Potenz-Funktion} \vspace*{5mm} Potenz-Funktion = wiederholtes multiplizieren: \begin{equation*} x^n \bk=\bk \underbrace{x\cdot{}x\cdot{}x\cdot{}x\cdots{}x}_{\mbox{$n$ mal}} \end{equation*} \bk \bk $x$ = Basis, \bk $n$ = Exponent \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition der Wurzel-Funktion} \vspace*{5mm} Wurzel-Funktion = Umkehrung der Potenz-Funktion: \begin{equation*} x^n = a \bk\Leftrightarrow\bk x = \sqrt[n]{a} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Wurzeln von Potenzen} \vspace*{10mm} \begin{equation*} \sqrt[n]{a^k} \bk=\bk \left(\sqrt[n]{a}\right)^k \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Potenzen mit Exponenten kleiner 1} \vspace*{5mm} \begin{align*} a^{\frac 1 n} & = \sqrt[n]{a} \\ a^0 & = 1 \\ a^{-n} & = \frac{1}{a^n} \end{align*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Exponenten aus $\mathbb{Q}$} \vspace*{15mm} \begin{equation*} a^{\frac k n} \bk=\bk \sqrt[n]{a^k} \bk=\bk \left(\sqrt[n]{a}\right)^k \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Potenzieren von Potenzen} \vspace*{10mm} \begin{equation*} \left(a^k\right)^n \bk=\bk a^{k \cdot n} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Potenzieren von Produkten} \vspace*{10mm} \begin{equation*} (a \cdot b)^n \bk=\bk a^n \cdot b^n \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Multiplikation von Potenzen gleicher Basis} \vspace*{10mm} \begin{equation*} a^k \cdot a^n \bk=\bk a^{k+n} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Br\"uche von Potenzen gleicher Basis} \vspace*{10mm} \begin{equation*} \frac{a^k}{a^n} = \begin{cases} a^{k-n} & |\bk k > n \\ \frac{1}{a^{n-k}} & |\bk k < n \\ 1 & |\bk k = n \end{cases} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Verzinsung} Beispiel mit 5,5\% Verzinsung im Jahr: $k_0$ = urspr\"unglich eingezahler Betrag \\ $k_1, k_2, k_3, ...$ = Betrag nach 1, 2, 3, ... Jahren \begin{align*} k_1 & \bk=\bk k_0 + 0{,}055 \cdot k_0 \bk=\bk 1{,}055 \cdot k_0 \\ k_2 & \bk=\bk k_1 + 0{,}055 \cdot k_1 \bk=\bk 1{,}055^2 \cdot k_0 \end{align*} \centerline{$k_n \bk=\bk 1{,}055^n \cdot k_0$} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Wurzeln von Produkten und Br\"uchen} \centering \vspace*{5mm} \begin{gather*} \sqrt[n]{a \cdot b} \bk=\bk \sqrt[n]{a} \cdot \sqrt[n]{b} \\ \\ \sqrt[n]{\frac a b} \bk=\bk \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{gather*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Wurzeln von Wurzeln} \centering \vspace*{10mm} \begin{equation*} \sqrt[n]{\sqrt[m]{a}} \bk=\bk \sqrt[m]{\sqrt[n]{a}} \bk=\bk \sqrt[n \cdot m]{a} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition der Exponentialfunktion} \centering \vspace*{15mm} $f(x) \bk=\bk c \cdot a^x$ \bigskip ($c,x \in \mathbb{R}, \bk a \in \mathbb{R}^+$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition der Logarithmusfunktion} \centering \vspace*{15mm} $a^x = y \bk\Leftrightarrow\bk x = \log_a(y)$ \bigskip $\log_a$ = Logarithmus zur Basis $a$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition des nat\"urlichen Logarithmus und der nat\"urlichen Exponentialfunktion} \vspace*{5mm} nat\"urliche Exponentialfunktion: \medskip \centerline{$\exp(x) \bk=\bk e^x$} \bigskip nat\"urlicher Logarithmus: \medskip \centerline{$\ln(x) \bk=\bk \log_e(x)$} \bigskip $e \approx 2.7183\dots$ = die Eulersche Zahl \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Logarithmen beliebiger Basis mit dem nat\"urlichen Logarithmus} \vspace*{15mm} \begin{equation*} log_a \bk=\bk \frac{\ln(x)}{\ln(a)} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Potenzen belibiger Basis mit der nat\"urlichen Exponentialfunktion} \vspace*{5mm} \begin{gather*} a^x \bk=\bk e^{\lambda \cdot x} \\ \\ \lambda \bk=\bk ln(a) \end{gather*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Monotonie von Exponentialfunktionen} \vspace*{10mm} \begin{equation*} a^x \bk \mbox{ist} \bk \begin{cases} \mbox{streng monoton steigend} & | \bk a > 1 \\ \mbox{streng monoton fallend} & | \bk a < 1 \end{cases} \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Logarithmen von Potenzen} \vspace*{15mm} \begin{equation*} log_a(b^x) = x \cdot log_a(b) \end{equation*} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Logarithmen von Produkten} \centering \vspace*{10mm} \begin{equation*} log_a(x \cdot y) \bk=\bk log_a x + log_a y \end{equation*} \bigskip (Prinzip des Rechenschiebers) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Unbeschr\"anktes exponentielles Wachstum} \centering \vspace*{10mm} $$N(t) \bk=\bk N_0 \cdot a^t \bk=\bk N_0 \cdot e^{\lambda t} \bk \bk \mbox{ mit } \lambda = \ln a$$ \begin{tabular}{lll} $\lambda > 0$, & $a > 1$ & = exponentielles Wachstum \\ \\ $\lambda < 0$, & $a < 1$ & = exponentielle Abnahme \end{tabular} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Warum kann $\sqrt 2$ keine \\ rationale Zahl sein?} \small Beweis von $\sqrt 2 \not\in \mathbb{Q}$ durch Widerspruch: $\sqrt{2} \in \mathbb{Q} \bk \Longrightarrow \bk \exists a,b \in \mathbb{N}^\star: \bk \sqrt{2} = \frac a b \bk \land \bk a,b \mbox{ teilerfremd}$ $\Rightarrow \bk \left(\frac a b\right)^2 = 2 \bk \Rightarrow \bk \frac{a^2}{b^2} = 2 \bk \Rightarrow \bk a^2 = 2b^2$ $\Rightarrow \bk a^2 \mbox{ ist gerade} \bk \Rightarrow \bk a \mbox{ ist gerade}$ \bk\bk\bk (denn 2 ist eine Primzahl und muss daher bereits in $a$ als Primfaktor enthalten sein) $\Rightarrow \bk \exists p \in \mathbb{N}^\star: \bk a = 2 p \bk \Rightarrow \bk a^2 = \left(2 p\right)^2 = 4 p^2 = 2 b^2 \bk \Rightarrow \bk 2 p^2 = b^2$ $\Rightarrow \bk b^2 \mbox{ ist gerade} \bk \Rightarrow \bk b \mbox{ ist gerade}$ \vspace*{5mm} \centerline{$\Longrightarrow \bk \mbox{Widerspruch zu } a,b \mbox{ teilerfremd} \bk \Longrightarrow \bk \sqrt 2 \not\in \mathbb{Q} \bk \bk {}_\square$} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Graph einer lineraen Funktion} \centering \vspace*{1mm} \begin{tikzpicture}[domain=-1.9:3.5] \draw[->] (-2,0) -- (5,0) node[right] {$x$}; \draw[->] (0,-1) -- (0,3.5) node[above] {$f(x)$}; \draw (+0.1,1) -- (-0.1,1) node[left] {$d$}; \draw (-1/0.7,+0.1) -- (-1/0.7,-0.1); \draw (-1/0.7,-0.7) node[above] {$\frac{-d}{k}$}; \draw[very thin,color=gray] (0,1.5*0.7+1) -- (1.5,1.5*0.7+1); \draw[very thin,color=gray] (0,2.5*0.7+1) -- (2.5,2.5*0.7+1); \draw (+0.1,1.5*0.7+1) -- (-0.1,1.5*0.7+1) node[left] {$f(a)$}; \draw (+0.1,2.5*0.7+1) -- (-0.1,2.5*0.7+1) node[left] {$f(a+1)$}; \draw[very thin,color=gray] (1.5,0) -- (1.5,1.5*0.7+1) -- (3.2,1.5*0.7+1); \draw[very thin,color=gray] (2.5,0) -- (2.5,2.5*0.7+1) -- (3.2,2.5*0.7+1); \draw (1.5,+0.1) -- (1.5,-0.1); \draw (2.5,+0.1) -- (2.5,-0.1); \draw (1.5,-0.5) node[above] {$a$}; \draw (2.5,-0.53) node[above] {$a+1$}; \draw[<->] (3,1.5*0.7+1) -- (3,2.5*0.7+1); \draw (3,2.0*0.7+1) node[right] {$k$}; \draw[<->] (1.5,1.5) -- (2.5,1.5); \draw (2,1.5) node[above] {$1$}; \draw[color=red] plot[id=x] function{x*0.7 + 1} node[right] {$f(x) = k \cdot x + d$}; \end{tikzpicture} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Graph von $1/x$} \centering \begin{tikzpicture} \draw[->] (-3,0) -- (4,0) node[right] {$x$}; \draw[->] (0,-1.5) -- (0,4) node[above] {$f(x)$}; \draw[very thin,color=gray] (-1.5,-1.5) -- (+3,+3); \draw[very thin,color=gray] (-2.5,+2.5) -- (+1.5,-1.5) node[very near start,sloped,above] {\scriptsize Symmetrisch an} node[very near start,sloped,below] {\scriptsize beiden Medianen}; \draw[color=black] (0,0) node[below right] {\scriptsize Bei $x = 0$ nicht definiert!}; \draw[color=red] (3,1.2) node[above] {$f(x) = \frac1x$}; \draw[color=red] plot[id=x, domain=0.25:4.5] function{1/x}; \draw[color=red] plot[id=x, domain=-3.5:-0.6] function{1/x}; \filldraw[black] (1/3+0.025,3) circle (1.5pt) node[above right, black] {$\nicefrac13;3$}; \filldraw[black] (1/2+0.02,2) circle (1.5pt) node[above right, black] {$\nicefrac12;2$}; \filldraw[black] (1,1) circle (1.5pt) node[above right, black] {$1;1$}; \filldraw[black] (2,1/2) circle (1.5pt) node[above right, black] {$2;\nicefrac12$}; \filldraw[black] (3,1/3) circle (1.5pt) node[above right, black] {$3;\nicefrac13$}; \filldraw[black] (-1,-1) circle (1.5pt) node[below left, black] {$\text{-}1;\text{-}1$}; \filldraw[black] (-2,-1/2) circle (1.5pt) node[below left, black] {$\text{-}1;\text{-}\nicefrac12$}; \end{tikzpicture} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Graph der Exponentialfunktion \\ (Am Beispiel von $2^x$)} \centering \vspace*{1mm} \begin{tikzpicture}[domain=-1.5:3.5] \draw[->] (-2,0) -- (4,0) node[right] {$x$}; \draw[->] (0,-0.1) -- (0,3.5) node[above] {$f(x)$}; \draw[very thin,color=gray] (-1.0,0.5/3) -- (0.4,0.5/3); \draw[very thin,color=gray] (-0.6,1.5/3) -- (0,1.0/3); \draw[very thin,color=gray] (-0.1,2.0/3) -- (1,2.0/3); \draw[very thin,color=gray] (-0.1,4.0/3) -- (2,4.0/3); \draw[very thin,color=gray] (-0.1,8.0/3) -- (3,8.0/3); \draw[very thin,color=black] (0.3,0.5/3+0.05) node[right] {$\nicefrac{1}{2}$}; \draw[very thin,color=black] (-0.5,1.5/3) node[left] {$1$}; \draw[very thin,color=black] (0,2.0/3) node[left] {$2$}; \draw[very thin,color=black] (0,4.0/3) node[left] {$4$}; \draw[very thin,color=black] (0,8.0/3) node[left] {$8$}; \draw[very thin,color=gray] (-1,-0.05) -- (-1,0.5/3); \draw[very thin,color=gray] (1,-0.05) -- (1,2/3); \draw[very thin,color=gray] (2,-0.05) -- (2,4/3); \draw[very thin,color=gray] (3,-0.05) -- (3,8/3); \draw[very thin,color=black] (-1,0) node[below] {$-1$}; \draw[very thin,color=black] (0,0) node[below] {$0$}; \draw[very thin,color=black] (1,0) node[below] {$1$}; \draw[very thin,color=black] (2,0) node[below] {$2$}; \draw[very thin,color=black] (3,0) node[below] {$3$}; \draw[color=red] plot[id=x] function{2**x / 3} node[right] {$f(x) = 2^x$}; \end{tikzpicture} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Graph der Exponential- und Logarithmusfunktion} \centering \begin{tikzpicture} \draw[->] (-3.5,0) -- (+3.5,0) node[right] {$x$}; \draw[->] (0,-2.8) -- (0,+2.8) node[above] {$f(x)$}; \draw (+0.1,1) -- (-0.1,1) node[left] {$1$}; \draw (1,+0.1) -- (1,-0.1) node[below] {$1$}; \draw[very thin,color=gray] (-2,-2) -- (+2,+2) node[very near start,sloped,above] {\scriptsize Gespiegelt an der 1. Mediane}; \draw[domain=-3:+1,color=red] plot[id=exp] function{exp(x)} node[right] {$f(x) = \exp x$}; \draw[domain=0.07:+2.5,color=blue] plot[id=ln] function{log(x)} node[right] {$f(x) = \ln x$}; \end{tikzpicture} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}