\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}} \def \simplecenter {\vspace*{-10mm}\centering\vfil} % dies und das \newcommand*\euler{\mathrm{e}} \newcommand*\leibd{\mathrm{d}} \newcommand*\leibdx{\mathrm{d}x} \newcommand*\bigbar{\!\!\left.\begin{matrix}\,\\\,\end{matrix}\right|} \begin{document} \fach{SBP Mathe Grundkurs 2} \kommentar{by Clifford Wolf} \include{hinweis} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Differentialquotient} Der Differentialquotient ist die \"Anderungsrate von $f(x)$ an einem Punkt. $$ f'(x) \bk = \bk \lim_{z \rightarrow x} \frac{f(z) - f(x)}{z - x} \bk = \bk \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$ Das heisst $f'(x)$ ist die Steigung der Tangente des Funktionsgraphen von $f$ an der Stelle $x$. \vfill Bestimmen der Ableitung: Einsetzen in die obige Definition und umformen, so dass beim Gleichsetzen von $z = x$ (bzw. $\Delta x = 0$) nicht mehr durch 0 dividiert wird. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Namen und Schreibweisen \\ f\"ur Differentialquotienten} $ f'(x) \bk = \bk \lim_{z \rightarrow x} \frac{f(z) - f(x)}{z - x} \bk = \bk \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ \centering \vfill $f'$ \bk = \bk Ableitung von $f$ \bk = \bk Differentialquotient \vspace*{2mm} \\ $f''$ \bk = \bk 2. Ableitung von $f$ \vfill Leibniz'sche Schreibweise: \hspace*{2cm} \vspace*{2mm} \\ $ f'(x) \bk = \bk \frac{\leibd f}{\leibdx} \bk = \bk \frac{\leibd f(x)}{\leibdx} \bk = \bk \frac{\leibd}{\leibdx}f(x) $ \vfill $ f' = \dot f = \frac{\leibd f(x)}{\leibdx}, \bk\bk\bk f'' = \ddot f = \frac{\leibd ^2f(x)}{\leibdx^2} $ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = c$} \simplecenter $$ f(x) = c \bk\bk\bk\text{(konstante Funktion)} $$ \hspace*{5mm} $$ \Longrightarrow \bk f'(x) = 0 $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = kx + d$} \simplecenter $$ f(x) = kx + d \bk\bk \Longrightarrow \bk\bk f'(x) = k $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = x^n$} \simplecenter $$ f(x) = x^n \bk\bk \Longrightarrow \bk\bk f'(x) = n \cdot x^{n-1} $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von Funktionen \\ mit konstanten Faktoren} \simplecenter $$ f(x) = c \cdot g(x) \bk\bk \Longrightarrow \bk\bk f'(x) = c \cdot g'(x) $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = g(x) + h(x)$} \vspace*{0mm}\vfill $$ f(x) = g(x) + h(x) \bk\bk \Longrightarrow \bk\bk f'(x) = g'(x) + h'(x) $$ \vfill Kurze Schreibweise: $(g+h)' = g' + h'$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Satz von der Intervallmonotonie} $f: \mathbb A \mapsto \mathbb R$, \bk $f'$ die Ableitung von $f$, \bk $\mathbb I \subseteq \mathbb A$ ein Intervall: \vfil \centering $f'(x) > 0$ fuer alle inneren Stellen $x$ von $\mathbb I$ \bk\bk \\ \bk\bk $\Longrightarrow$ $f$ ist streng monoton wachsend in $\mathbb I$ \vfil $f'(x) < 0$ fuer alle inneren Stellen $x$ von $\mathbb I$ \bk\bk \\ \bk\bk $\Longrightarrow$ $f$ ist streng monoton abnehmend in $\mathbb I$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Satz vom Ableitungsvorzeichen} $f: \mathbb A \mapsto \mathbb R$, \bk $f'$ die Ableitung von $f$, \bk $\mathbb I \subseteq \mathbb A$ ein Intervall: \hspace*{3mm} \begin{center} \begin{tabular}{lll} \multicolumn{3}{l}{Besitzt $f'$ keine Nullstelle in $\mathbb I$, so ist} \\ \hspace*{3mm} & entweder & $f'(x) > 0$ f\"ur alle $x \in \mathbb I$ \\ \hspace*{3mm} & oder & $f'(x) < 0$ f\"ur alle $x \in \mathbb I$ \end{tabular} \end{center} \begin{center} Das heisst zwischen 2 Nullstellen von $f'$ bleibt das Vorzeichen \\ von $f'$ und damit das Monotonieverhalten von $f$ unver\"andert. \end{center} \vfill Voraussetzung: $f'(x)$ ist in ganz $\mathbb I$ definiert und stetig. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Maximum- \\ und Minimumstelle} $f: \mathbb A \mapsto \mathbb R$, \bk $\mathbb M \subseteq \mathbb A$: \vfil \begin{center} \begin{tabular}{l} $p \in M$ = Maximumstelle von $f$ in M \\ \bk\bk wenn $\forall x \in \mathbb M\!: f(x) \le f(p)$ \\ \\ $p \in M$ = Minimumstelle von $f$ in M \\ \bk\bk wenn $\forall x \in \mathbb M\!: f(x) \ge f(p)$ \end{tabular} \end{center} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Umgebung} $a \in \mathbb R$, \bk $\epsilon \in \mathbb R^+$: \vfil\centering $\left]a-\epsilon;\, a+\epsilon\right[$ = Umgebung von $a$ (mit dem Radius $\epsilon$) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Lokale Maximum- \\ und Minimumstelle} $f: \mathbb A \mapsto \mathbb R$, \bk $p \in \mathbb A$: \vfil\centering $p$ = lokale Maximumstelle von $f$ \\ wenn es ein $\epsilon \in \mathbb R^+$ gibt, so dass \\ $\forall x \in \left]p-\epsilon;\, p+\epsilon\right[\!: f(x) \le f(p)$ \vspace*{5mm} $p$ = lokale Minimumstelle von $f$ \\ wenn es ein $\epsilon \in \mathbb R^+$ gibt, so dass \\ $\forall x \in \left]p-\epsilon;\, p+\epsilon\right[\!: f(x) \ge f(p)$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Extremstelle} \simplecenter Extremstelle = Maximumstelle oder Minimumstelle \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Potentielle lokale Extremstellen \\ von Polynomfunktionen} Sei $f$ eine Polynomfunktion: \vfill $$p \text{ eine lokale Extremstelle von } f \bk \Rightarrow \bk f'(p) = 0$$ \hfil (Jede Extremstelle von $f$ ist eine Nullstelle von $f'$) \vfill Aber: Nicht jede Nullstelle von $f'$ ist auch eine Extremstelle von $f$. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: links- und rechtsgekr\"ummt} $f: \mathbb A \mapsto \mathbb R$, \bk $\mathbb I \subseteq \mathbb A$ ein Intervall: \vfil $f$ ist linksgekr\"ummt in $\mathbb I$ wenn: \vspace*{-5mm}\begin{center}\begin{tabular}{l} $f'$ in $\mathbb I$ streng monoton wachsend ist \\ d.h. wenn $f''(x) > 0$ f\"ur alle inneren Stellen $x$ von $\mathbb I$ \end{tabular}\end{center} \vfil $f$ ist rechtsgekr\"ummt in $\mathbb I$ wenn: \vspace*{-5mm}\begin{center}\begin{tabular}{l} $f'$ in $\mathbb I$ streng monoton abnehmend ist \\ d.h. wenn $f''(x) < 0$ f\"ur alle inneren Stellen $x$ von $\mathbb I$ \end{tabular}\end{center} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Wendestelle (Wendepunkt)} \simplecenter Eine Stelle, an der sich das Kr\"ummungsverhalten einer \\ Funktion \"andert, nennt man Wendestelle oder Wendepunkt. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Lokale Maximum- und Minimumstellen \\ von Polynomfunktionen} Sei $f$ eine Polynomfunktion: \vfill $x$ ist eine lokale Maximumstelle von $f$ wenn: \\ \vspace*{-2mm}$$f'(x) = 0 \bk \land \bk f''(x) < 0$$ \vfill $x$ ist eine lokale Minimumstelle von $f$ wenn: \\ \vspace*{-2mm}$$f'(x) = 0 \bk \land \bk f''(x) > 0$$ \vfill Aber: Nicht alle Extremstellen haben ein $f'' \neq 0$. \\ Beispiel: $x^4$ bei $x=0$: erst die 4. Ableitung ist $\neq$ 0! \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Alle lokalen Maximum- und Minimum- \\ stellen von Polynomfunktionen} Zum finden aller lokalen Maximum- und Minimumstellen der Polynomfunktion $f$ ist wie folgt vorzugehen: \begin{itemize} \item Alle Nullstellen von $f'$ finden. \item Die Werte von $f$ an allen Nullstellen von $f'$ ermitteln. \item Zwei beliebige Werte von $f$ links und rechts von allen \\ Nullstellen von $f'$ ermitteln. \item Aus den ermittelten $f$-Werten das Monotonieverhalten \\ von $f$ zwischen den Nullstellen von $f'$ ablesen. \item Jede Nullstelle von $f'$ ist ein Extremwert, wenn sich \\ das Monotonieverhalten von $f$ an der Stelle \"andert. \end{itemize} \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = \frac{u(x)}{v(x)}$} \vspace*{0mm}\vfill $$ f(x) = \frac{u(x)}{v(x)} \bk\bk \Longrightarrow \bk\bk f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{\left(v(x)\right)^2} $$ \vfill Kurze Schreibweise: $\left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2}$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = u(x) \cdot v(x)$} \vspace*{0mm}\vfill $$ f(x) = u(x) \cdot v(x) \bk\bk \Longrightarrow \bk\bk f'(x) = v(x)u'(x) + u(x)v'(x) $$ \vfill Kurze Schreibweise: $\left(u \cdot v\right)' = vu' + uv'$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = \sqrt[n]{x}$} \simplecenter $$ f(x) = \sqrt[n]{x} = x^{\frac{1}{n}} $$ $$ \Longrightarrow \bk\bk f'(x) = \frac{1}{n} \cdot x^{\frac{1}{n} - 1} = \frac{1}{n \sqrt[n]{x^{n-1}}} $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = h(g(x))$ \\ (Kettenregel)} \vspace*{0mm}\vfill $$ f(x) = h(g(x)) \bk\bk \Longrightarrow \bk\bk f'(x) = h'(g(x)) \cdot g'(x) $$ \vfill Kurze Schreibweise: $\left(h \circ g\right)' = (h' \circ g) \cdot g'$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung der Exponentialfunktion $f(x) = \euler^x$} \vspace*{0mm}\vfill $$ f(x) = \euler^x \bk\bk \Longrightarrow \bk\bk f'(x) = \euler^x $$ \vfill F\"ur die Exponentialfunktion $f(x) = \euler^x$ gilt also $f' = f$. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = \euler^{\lambda x}$} \simplecenter $$ g(x) = \lambda x, \bk h(y) = \euler^y $$ \vspace*{0mm} $$ \Longrightarrow \bk\bk f(x) = \euler^{\lambda x} = h(g(x)) $$ \vspace*{0mm} $$ \Longrightarrow \bk\bk f'(x) = h'(g(x)) \cdot g'(x) = \euler^{\lambda x} \cdot \lambda $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = a^x$} \simplecenter $$ f(x) = a^x = \euler^{\ln(a) \cdot x} $$ \vspace*{0mm} $$ \Longrightarrow \bk\bk f'(x) = \euler^{\ln(a) \cdot x} \cdot \ln(a) = a^x \cdot \ln(a) $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = \ln(x)$} \simplecenter $$ f(x) = \ln(x) \bk\bk \Longrightarrow \bk\bk f'(x) = \frac{1}{x} $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von $f(x) = \log_a(x)$} \simplecenter $$ f(x) = \log_a(x) \bk\bk \Longrightarrow \bk\bk f'(x) = \frac{1}{x \cdot \ln(a)} $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von Sinus, \\ Cosinus und Tangens} \simplecenter $$ \sin' x = \cos x $$ $$ \cos x = \sin\left(\frac{\pi}{2}-x\right) \bk \Rightarrow \bk \cos' x = -\sin x $$ $$ \tan x = \frac{\sin x}{\cos x} \bk \Rightarrow \bk \tan' x = \frac{1}{\cos^2 x} = 1 + \tan^2 x $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Ableitung von Umkehrfunktionen} $f(g(x)) = x$, \bk $g'$ ist bekannt, \bk $f'$ ist gesucht: \vfil $$ f(g(x)) = x \bk\bk \Longrightarrow \bk\bk f'(x) = \frac{1}{g'(f(x))} $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: Stammfunktion} Ist $f$ eine reelle Funktion, dann heisst eine reelle Funktion $F$ eine Stammfunktion von $f$, wenn $F' = f$ gilt. \vfil Ist $F$ eine Stammfunktion von $f$, so ist auch $F + c$ eine Stammfunktion von $f$. \vfil Schreibweise als unbestimmtes Integral: $$ \int\!f = F \bk\bk \text{ bzw. } \bk\bk \int\!f(x) \,\leibdx = F(x) $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Stammfunktion von $f\!: x \mapsto k$} \simplecenter $$ \int\!k\,\leibdx \bk = \bk k x $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Stammfunktion von $f\!: x \mapsto k \cdot g(x)$} \simplecenter $$ \int\!k \cdot g(x) \,\leibdx \bk = \bk k \cdot \int\!g(x)\,\leibdx $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Stammfunktion von $f\!: x \mapsto g(x) + h(x)$} \simplecenter $$ \int\! g(x) + h(x) \,\leibdx \bk = \bk \int\! g(x) \,\leibdx + \int\! h(x) \,\leibdx $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Stammfunktion von $f\!: x \mapsto x^n$} \simplecenter $$ \int\! x^n \,\leibdx \bk = \bk \frac{1}{n+1} \cdot x^{n+1} \bk = \bk \frac{x^{n+1}}{n+1} $$ $$ \text{Spezialfall $n = \text{-}1$: } \bk\bk \int\! x^{\text{-}1} \,\leibdx \bk = \bk \int\! \frac{1}{x} \,\leibdx \bk = \bk \ln x $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Definition: (bestimmtes) Integral} Das (bestimmte) Integral von $f$ in den Grenzen $a$ und $b$: $$ \int_a^bf(x)\,\leibdx \bk = \bk \lim_{n \rightarrow \infty} \sum_{i=1}^n f(\overline{x_i}) \cdot \Delta x $$ Die Funktion wird in $n$ jeweils $\Delta x$ breite Intervalle unterteilt. $\overline{x_i}$ bezeichnet dabei eine Stelle im $i$-ten Intervall. Bei $\lim_{n \rightarrow \infty}$ entspricht diese ``Summe von unendlich vielen jeweils unendlich schmalen Streifen'' dem Integral. \vfill \small Bei durchgehend positiven Funktionswerten: Das Integral ist der Fl\"acheninhalt der Fl\"ache zwischen der 1. Achse und dem Funktionsgraphen. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Hauptsatz der \\ Differential- und Integralrechnung} $f: \mathbb A \mapsto \mathbb R$, \bk $F' = f$, \bk $[a; b] \subseteq \mathbb A$: \vfil $$ \int_a^b f(x) \,\leibdx \bk = \bk F(b) - F(a) \bk = \bk F(x) \bigbar_{x=a}^{x=b} $$ \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Volumen eines Rotationsk\"orpers} Volumen eines Rotationsk\"orpers: \vfill $$ V_\text{K} \bk = \bk \pi \cdot \int_a^b \left( f(x) \right)^2 \,\leibdx $$ \vfill (Denn $\,V_\text{Z} = \pi r^2 h\,$ ist das Volumen eines Zylinders.) \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{L\"ange eines Graphen} L\"ange eines Graphen: \vfill $$ l \bk = \bk \int_a^b \sqrt{ 1 + \left( f'(x) \right)^2} \,\leibdx $$ \vfill Begr\"undung: Die Sekante des $i$-ten Teilabschnitts des Graphen mit der L\"ange $\Delta x$ hat die Steigung $f'(\overline{x_i})$. Das heisst die L\"ange des Teilabschnitts ist die Hypotenuse des rechtwinkligen Dreiecks mit den Katheten $\Delta x$ und $f'(\overline{x_i}) \cdot \Delta x$. Also ist die L\"ange des Teilabschnitts wegen \bk$a^2 + b^2 = c^2 \,\,\Rightarrow\,\, c = \sqrt{a^2 + b^2}$\bk gleich {\scriptsize \bk$\sqrt{ 1 + \left( f'(x) \right)^2} \cdot \Delta x$}. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Partialbruchzerlegung} Ein Bruch zweier Polynome (Nenner von h\"oherem Grad als Z\"ahler) kann in eine Summe einfacher Br\"uche zerlegt werden: $$ \frac{P_1(x)}{P_2(x)} = \frac{a_1}{x - \alpha_1} + \frac{a_2}{x - \alpha_2} + \cdots + \frac{a_n}{x - \alpha_n} $$ wobei $\alpha_i$ die Nullstellen von $P_2(x)$ sind. Doppelte Nullstellen und komplexe Nullstellen k\"onnen behandelt werden, ergeben aber Br\"uche mit einem anderen Aufbau. Zur Ermittlung der $a_i$ wird die rechte Seite auf gemeinsamen Nenner gebracht. Dann k\"onnen die $a_i$ durch Koeffizientenvergleich der Z\"ahler bestimmt werden. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Partielle Integration} \small Die partielle Integration ist ein Verfahren zur Integration mancher Produkte. Dabei wird ein Integral in ein anderes umgewandelt. Das Verfahren kommt dann zur Anwendung, wenn das neue Integral leichter zu l\"osen ist. $$ \int_a^b\! f(x) \cdot g'(x) \,\leibdx \bk = \bk \left( f(x) \cdot g(x) \right) \!\bigbar_{x=a}^{x=b} \,\,-\,\, \int_a^b\! f'(x) \cdot g(x) \,\leibdx $$ Zur Beurteilung, ob die partielle Integration hilfreich ist, ist der Blick auf das vereinfachte unbestimmte Integral oft hilftreich: $$ \int\! f \cdot g' \bk = \bk (\cdots) \,\,-\,\, \int\! f' \cdot g $$ \vfill Die partielle Integration ist Umkehrung der Produktregel $(u \cdot v)' = uv' \cdot vu'$. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{karte}{Substitutionsmethode} \small Die Substitutionsmethode ist ein Verfahren um eine bestimmte Klasse komplizierter Integrale in einfachere Integrale umzuwandeln: $$ \int_a^b\! f(g(t)) \cdot g'(t) \,\leibd t \bk = \bk \int_{g(a)}^{g(b)}\! f(x) \,\leibdx $$ Bei der Anwendung der Substitutionsmethode ist oft erst eine kreative Umformung des Integrals notwendig. Etwa das Einf\"uhren eines konstanten Faktors im Integral und seines Kehrwerts ausserhalb des Integrals, damit ein $g'(t)$ entsteht. Zum Beispiel: $$ \int_a^b\! t \cdot f(t^2) \,\leibd t \bk = \bk \frac{1}{2} \int_a^b\! 2t \cdot f(t^2) \,\leibd t \bk = \bk \frac{1}{2} \int_{a^2}^{b^2}\! f(x) \,\leibdx $$ \vfill Die Substitutionsmethode basiert auf der Kettenregel. \end{karte} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}