From 2c2e25161cfa349381e17141929f264379ab9ac5 Mon Sep 17 00:00:00 2001 From: RobinB27 Date: Mon, 29 Dec 2025 09:46:12 +0100 Subject: [PATCH] [Analysis] ++ Analysis I Cheat-Sheet, Analysis II folder --- README.md | 1 + semester2/analysis-i-rb/degrees_circle.pdf | Bin 0 -> 47446 bytes .../analysis-i-rb/differentialquotient.png | Bin 0 -> 32219 bytes semester2/analysis-i-rb/konvex.png | Bin 0 -> 31317 bytes semester2/analysis-i-rb/main.pdf | Bin 0 -> 467299 bytes semester2/analysis-i-rb/main.tex | 1604 +++++++++++++++++ semester3/analysis-ii-rb/main.tex | 0 7 files changed, 1605 insertions(+) create mode 100644 semester2/analysis-i-rb/degrees_circle.pdf create mode 100644 semester2/analysis-i-rb/differentialquotient.png create mode 100644 semester2/analysis-i-rb/konvex.png create mode 100644 semester2/analysis-i-rb/main.pdf create mode 100644 semester2/analysis-i-rb/main.tex create mode 100644 semester3/analysis-ii-rb/main.tex diff --git a/README.md b/README.md index a439c0e..01d172b 100644 --- a/README.md +++ b/README.md @@ -11,6 +11,7 @@ If you find mistakes, please open an issue or fix them yourself and open a PR. ### Semester 2 - [Analysis Cheat-Sheet (EN & DE)](./semester2/analysis-i/cheat-sheet.pdf) Author: Janis Hutz +- [Analysis Cheat-Sheet (DE)](./semester2/analysis-i-rb/main.pdf) Author: Robin Bacher - [Algorithms & Probability Summary (EN)](./semester2/algorithms-and-probability/aw-summary.pdf) Author: Janis Hutz - [Parallel Programming Compact Summary (EN)](./semester2/parallel-programming-compact/pprog-compact-summary.pdf) Author: Janis Hutz diff --git a/semester2/analysis-i-rb/degrees_circle.pdf b/semester2/analysis-i-rb/degrees_circle.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e4f67ce7e3162c499908d19027dae95d81c76e7d GIT binary patch literal 47446 zcmV)(K#RX6P((&8F)lR4?5av(28Y+-a|L}g=dWMv9IJ_>Vma%Ev{3V58Xy~(aDNwOri_pdlVkcaMOZi8>R zX`lfv*jy4^5J6TK8gUohT=ehLRYX)xJ?_b@s)s@XRaf*_nA@N>>3DqY_kVvJ_y5$Z 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\in \R & |x + y| \leq |x| + |y| \\ + (iv) & \forall x,y \in \R & |x + y| \geq |x| - |y| \\ +\end{array} +$ + +\textbf{Young'sche Ungleichung}\\ +$\forall \epsilon > 0\ \forall x,y \in \R : \quad 2|xy| \leq \epsilon x^2 + \frac{1}{\epsilon}y^2$ + +\textbf{Bernoulli Ungleichung}\\ +$\forall n \in \N, x > -1 \quad \quad (1+x)^n \geq 1 + n \cdot x$ + +\begin{subbox}{Infimum \& Supremum} +\footnotesize +Für $A \subseteq \R$:\\ +\normalsize + $\begin{array}{ccl} + \sup A := & \begin{cases} + \min\{c \in \R \ |\ \forall a \in A:\ a \leq c \} \\ + +\infty\quad \text{\footnotesize falls oben unbeschr.} + \end{cases} \\ + \inf A := & \begin{cases} + \max\{c \in \R \ |\ \forall a \in A: c \leq a\} \\ + - \infty \quad \text{\footnotesize falls unten unbeschr.} + \end{cases} \\ + \end{array}$ +\end{subbox} + +% \textbf{Infimum \& Supremum}\\ +% Für $A$ nach oben / unten beschränkt: + +% $\begin{array}{ccl} +% \sup A := & \min (c) & \quad \text{ s.d. } \quad \forall a \in A a \leq c\\ +% \inf A := & \max (c) & \quad \text{ s.d. } \quad \forall a \in A c \leq a\\ +% \end{array}$ + +% Sonst: $ \quad \sup A = + \infty \quad \inf A = - \infty$ + +\textbf{Monotonie}\\ +Für $f: D \rightarrow \R,\quad D \subset \R, \quad x,y \in D$: + +$\begin{array}{lcl} + \text{mon. wachsend} &\Def& x \leq y \implies f(x) \leq f(y)\\ + \text{str. mon. wachs.} &\Def& x < y \implies f(x) < f(y)\\ + \text{mon. fallend} &\Def& x \geq y \implies f(x) \geq f(y)\\ + \text{str. mon. fallend} &\Def& x < y \implies f(x) < f(y) +\end{array}$ + +\textbf{Intervalle}\\ +Untermengen von $\R$. + +$\begin{array}{lcl} + \text{offenes Intervall} &\Def& (a,b) = \{x \in \R\ |\ a 0: \quad \{ n \in \N\ |\ a_n \notin\ ]l-\epsilon, l+\epsilon [\ \} \text{ ist endlich}\\ +(ii)\quad \forall \epsilon > 0, \ \exists N \geq 1 : \quad |a_n - l| < \epsilon,\ \forall n \geq N$ +\end{subbox} + +\textbf{Konvergenz}\\ +$(a_n)_{n \geq 1}$ ist konvergent, falls $\limn a_n$ existiert. \\ +Konvergente Folgen sind immer beschränkt.\\ +\footnotesize\color{gray} +Nicht umgekehrt: $(-1)^n$ ist beschränkt, aber nicht konvergent. +\normalsize\color{black} + +\textbf{Rechenregeln Limes}\\ +Für konvergente $(a_n)_{n\geq1}, (b_n)_{n\geq1}$: + +$\begin{array}{cccc} + (i) & \limn (a_n + b_n) & = & \limn (a_n) + \limn (b_n) \\ + (ii) & \limn (a_n \cdot b_n) & = & \limn (a_n) \cdot \limn (b_n) \\ + (iii) & \limn (\frac{a_n}{b_n}) & = & \limn (a_n) \setminus \limn (b_n) \\ + \text{wobei} & \forall n \geq 1 (b_n \neq 0) & \land & \limn (b_n) \neq 0 \\ + (iv) & \exists K\ \forall n \geq K (a_n \leq b_n) & \implies & \limn(a_n) \leq \limn (b_n) +\end{array} +$ + +\begin{subbox}{Limes Inferior \& Limes Superior} + Für $(a_n)_{n \geq 1}$ beschränkt:\\ + $b_n := \inf \{ a_k \ | \ k \geq n \} \quad \quad c_n := \sup \{ a_k \ | \ k \geq n \}$ + \begin{align*} + \limi(a_n) := \limn(b_n)\\ + \lims(a_n) := \limn(c_n) + \end{align*} + $\forall n \in \N (b_n \leq c_n) \implies \limi(a_n) \leq \lims(a_n)$ + +\end{subbox} + +\textbf{Komplement-Trick}\\ +Nützlich für einige Grenzwerte: +\small +$$\sqrt{ax + b} -\sqrt{cx+d} = \frac{ax+b - (cx+d)}{\sqrt{ax+b}+\sqrt{cx+d}}$$ +\normalsize + +\newpage +\textbf{Komplexe Folgen}\\ +\color{gray}\footnotesize +$(b_n)_{n\geq1} \in \R,\quad (a_n)_{n\geq1} \in \C$ +\color{black}\normalsize + +In $\C$ gelten die selben Resultate, aber: + +$(i)\quad \ \forall n \in \N :\ |a_n| \leq b_n \land \limn(b_n)=0 \Rightarrow \limn(a_n) = 0\\ +(ii)\quad \liminf(a_n) \text{ und } \limsup(a_n) \text{ existieren nicht.} +$ + +$(a_n)_{n\geq1} \in \C \text{ konv.} \iff (\Re(a_n))_{n\geq1},\ (\Im(a_n))_{n\geq1} \text{ konv.}$ + +\subsection{Konvergenzkriterien} + +\begin{subbox}{Monotoner Konvergenz-Satz} + $(a_n)_{n \geq 1} \text{ mon. fallend}, (b_n)_{n \geq 1} \text{ mon. steigend}$: + + $\begin{array}{lcr} + a_n \text{ unten beschr: } & \implies \limn(a_n) = \inf \{a_n\ |\ n \geq 1 \}\\ + b_n \text{ oben beschr: } & \implies \limn(b_n) = \sup \{b_n\ |\ n \geq 1 \} + \end{array}$ +\end{subbox} + +\begin{subbox}{Sandwich-Satz} + $(a_n), (b_n), (c_n) \text{ s.d. } \forall n \in \N: a_n \leq b_n \leq c_n$\\ + + $\limn(a_n) = \limn(c_n) = A \implies \limn(b_n) = A$ +\end{subbox} + +\textbf{Cauchy Kriterium I}\\ +$(a_n)_{n \geq 1} \text{ beschr. } \land \ \limi(a_n) = \lims(a_n)$ + +\textbf{Cauchy Kriterium II}\\ +$\forall \epsilon > 0\ \exists N \geq 1 : \quad |a_n - a_m| < \epsilon \quad \forall n,m \geq N$ + +\textbf{Bolzano-Weierstrass}\\ +$(a_n)_{n \geq 1}$ beschr. $\implies$ Ex existiert konv. Teilfolge $(b_n)_{n \geq 1}$ + +$\limi(a_n) \leq \limn(b_n) \leq \lims(a_n)$ + +\section{Reihen} + +\textbf{Reihe} $\quad\quad\quad(S_n)_{n\geq1}$ s.d. $S_n := \sum_{k=1}^na_k$ + +\textbf{Konvergenz} $\quad \sum_{k=0}^{\infty}a_k := \limn \sum_{k=0}^n a_k$ + +\textbf{Absolute Konvergenz}\\ +$\sumk | a_k |$ konv. $ \implies \sumk a_k $ konv. + +$\forall (a_k)_{k\geq1}:\quad | \sumk a_k | \leq \sumk | a_k |$ + +\textbf{Rechenregeln}\\ +Für konvergente Reihen $\sumk a_k$, $\sum_{j=1}^\infty b_j$, $\alpha \in \R$: + +$\begin{array}{llll} + (i) & \sumk (a_k + b_k) & = & (\sumn a_k) + (\sum_{j=1}^\infty b_j) \\ + (ii) & \sumk (\alpha \cdot a_k) & = & \alpha \cdot (\sumk a_k) +\end{array}$ + +\textbf{Konvergente Reihen sind Nullfolgen}\\ +$\sumn a_n$ konv $\implies \limn a_n = 0$\\ +\footnotesize\color{gray} +Nicht umgekehrt: $\sumn\frac{1}{n}$ divergiert, obwohl $\limn \frac{1}{n}=0$. +\normalsize\color{black} + +\textbf{Doppelfolge} $(a_{i,j})_{i,j\geq1}$ + +\textbf{Doppelreihe} $\sum_{i=0}^\infty(\sum_{j=0}^\infty a_{i,j})$, $\quad \sum_{j=0}^\infty(\sum_{i=0}^\infty a_{i,j})$\\ +\color{gray}\footnotesize +Die beiden Grenzwerte können verschieden sein. +\color{black}\normalsize + +\textbf{Cauchy} + +$\exists B \geq 0:\quad \sum_{i=0}^m\sum_{j=0}^m|a_{i,j}| \leq B \quad \forall m \geq 0$\\ +$\implies S_i := \sum_{j=0}^\infty a_{ij}$ und $U_j := \sum_{i=0}^\infty a_{i,j}$ konv. abs. + +$\sum_{i=0}^\infty S_i = L_1$ und $\sum_{j=0}^\infty U_j = L_2$ konv. abs. s.d. $L_1 = L_2$. + +Jede Anordnung $\sigma: \N \rightarrow \N \times \N$ s.d. $b_k := a_{\sigma(k)}$ konv. abs. + +\textbf{Cauchy Produkt}\\ +$\sumn(\sum_{j=0}^n a_{n-j}\cdot b_j)\\ += a_ob_0 + (a_ob_1 + a_1b_0) + (a_ 0b_2 + a_1 b_1 + a_2 b_0) + \ldots$ + + +\subsection{Konvergenzkriterien} + +\begin{subbox}{Vergleichssatz} + Für $\sumk a_k$, $\sumk b_k$ mit $\forall k \geq 1 \ (0 \leq a_k \leq b_k)$\\ + oder $\exists K \geq 1 \ (0 \leq a_k \leq b_k) \ \forall k \geq K$ + + $\begin{array}{lll} + \sumk b_k \text{ konv. } & \implies & \sumk a_k \text{ konv. } \\ + \sumk a_k \text{ div. } & \implies & \sumk b_k \text{ div.} + \end{array}$ +\end{subbox} + +\textbf{Cauchy Kriterium}\\ +$\forall \epsilon > 0 \ \exists N \geq 1 \quad \quad | \sum_{k=n}^m a_k | < \epsilon \quad\quad \forall m \geq n \geq N$ + +\textbf{Monotoner Konvergenz-Satz}\\ +$\forall k \in \N^* ( a_k \geq 0) \text{ konv. } \iff \sum_{k=1}^n a_k \text{ oben beschränkt}$ + +\textbf{Leibniz}\\ +Für $(a_n)_{n \geq 1}$ mon. fall. und $\forall n (a_n \geq 0)$, $\limn a_n = 0$:\\ +$\sumk (-1)^{k+1} a_k$ konv. + +%$a_1 - a_2 \quad \leq \quad \sumk (-1)^{k+1} a_k \quad \leq \quad a_1$ + +\textbf{Dirichlet}\\ +$\sumk a_k$ abs. konv.\\ +$\implies$ $\forall \phi:\N^*\rightarrow\N^* \ (\text{bijektiv}): \sumk a_{\phi(k)}$ abs. konv. + +$\sumk a_k = \sumk a_{\phi(k)}$, unabh. von $\phi$. + +\begin{subbox}{Quotientenkriterium} + $\lims \frac{|a_{n+1}|}{|a_n|} < 1 \implies \sumn a_n$ abs. konv. + + $\lims \frac{|a_{n+1}|}{|a_n|} > 1 \implies \sumn a_n$ div.\\\\ + \footnotesize\color{gray} + Wobei $(a_n)_{n \geq 1}$ mit $\forall n (a_n \neq 0)$ + \normalsize\color{black} +\end{subbox} + +\begin{subbox}{Wurzelkriterium} + $\lims |a_n|^\frac{1}{n} < 1 \implies \sumn a_n$ abs. konv. + + $\lims |a_n|^\frac{1}{n} > 1 \implies \sumn a_n$ div. +\end{subbox} + +Beide Kriterien geben keine Aussage bei genau $0$. + +\subsection{Fundamentalreihen} + +\begin{subbox}{Potenzreihen (Konvergenzradius)} + $\sumk c_k z^k$ abs. konv. $\impliedby |z| < \rho$\\ + $\sumk c_k z^k$ div. $\quad\quad\ \ \impliedby |z| > \rho$\\ + $\rho = \begin{cases} + \begin{array}{ll} + + \infty, & \lims |c_k|^\frac{1}{k} = 0\\ + (\lims |c_k|^\frac{1}{k})^{-1}, & \lims |c_k|^\frac{1}{k} > 0 + \end{array} + \end{cases}$ +\end{subbox} + +\textbf{Geometrische Reihe}\\ +$\sumn q^n = \frac{1}{1-q} \iff |q| < 1$ + +\textbf{Harmonische Reihe}\\ +$\sum_{n=1}^\infty \frac{1}{n} = \infty$ + +\textbf{Zeta Funktion}\\ +$\zeta(s) = \sum^\infty_{n=1} \frac{1}{n^s}$ konv. $\iff s > 1$ + +\textbf{Exponentialfunktion}\\ +$\exp(z) := \sumn \frac{z^n}{n!} = e^z$ konv. $\forall z \in \C$ + + +\newpage +\section{Stetige Funktionen} +\footnotesize\color{gray} +$D \subset \R, \quad x_0 \in D, \quad f,g:D\rightarrow\R, \quad \lambda \in \R$ +\normalsize\color{black} + +\textbf{Stetigkeit}\\ +$f \text{ stetg in } x_0 \Def \forall\epsilon>0,\exists\delta>0: \forall x \in D: $\\ +$|x-x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon$ + +$f$ stetig $\Def \forall x \in D:$ $f$ stetig in $x$ + + +\begin{subbox}{Stetigkeit durch Folgen} + $f$ stetig in $x_0$ $\iff \forall (a_n)_{n\geq1} \in D:\\ + \limn a_n = x_0 \implies \limn f(a_n) = f(x_0)$ +\end{subbox} + +\textbf{Rechenregeln}\\ +Für $f,g$ stetig in $x_0$: + +$(i)\ \quad f + g, \quad\quad \lambda \cdot f,\quad\quad f \cdot g$\\ +$(ii)\quad g(x_0) \neq 0 \implies \frac{f}{g}: \{ x \in D \ |\ g(x) \neq 0 \} \rightarrow \R $ + +sind stetig in $x_0$. + +\subsection{Theoreme} + +\begin{subbox}{Zwischenwertsatz} + $\forall c \in \R: f(a) \leq c \leq f(b)\\ + \implies \exists z \in I: a \leq z \leq b \land f(z) = c$\\ + \footnotesize\color{gray} + $I \subset R$ (Intervall), $f: I \rightarrow \R$ (stetig), $a,b \in I$ + \normalsize\color{black} +\end{subbox} + +\textbf{Polynom-Nullstellen}\\ +\footnotesize\color{gray} +Für alle $P: \R \rightarrow \R,\quad P(x) = a_nx^n+\ldots+a_0:$\\ +\normalsize\color{black} +$a_n \neq 0 \land n \equiv_2 1 \implies \exists x \in \R: P(x) = 0$ + + +\begin{subbox}{Min-Max-Satz} + Stetige $f$ sind auf $I$ immer beschränkt.\\ + $\exists u,v \in I:\quad f(u) \leq f(x) \leq f(v)\quad \forall x \in I$ + + \footnotesize\color{gray} + $f:I = [a,b] \rightarrow \R,\quad f \text{ stetig auf }I,\quad I \text{ ist kompakt}$ + \normalsize\color{black} +\end{subbox} + +\textbf{Stetigkeit in Kompositionen}\\ +\footnotesize\color{gray} +$D_1,D_2 \subset \R,\quad f:D_1\rightarrow D_2,\quad g:D_2\rightarrow \R$\\ +\normalsize\color{black} +$f,g$ stetig in $x_0, f(x_0) \implies g \circ f: D_1 \rightarrow \R$ in $x_0$ stetig.\\ +$f,g$ stetig $\quad\quad\quad\quad\quad\ \implies g \circ f: D_1 \rightarrow \R$ stetig + +\textbf{Stetigkeit der Umkehrabbildung}\\ +\footnotesize\color{gray} +$I = [a,b] \subset \R$ ist ein Intervall\\ +\normalsize\color{black} +$f: I \rightarrow \R$ stetig, str. mon.\\ +$\implies f^{-1}: f(I) \subset \R \rightarrow I$ stetig, str. mon.\\ +\& $\ \ \ f(I) = [f(a), f(b)]$ ist ein Intervall. + +\subsection{Funktionenfolgen} +\textbf{Funktionenfolge} +$\quad (f_n)_{n\geq1}$ + +Formal: eine Abbildung $\N \rightarrow \R^D$ s.d. $n \mapsto f(n) =: f_n$ + +\textbf{Punktweise Konvergenz}\\ +$(f_n)_{n\geq1}$ konv. pw. gegen $f: D \rightarrow \R$, wenn:\\ +$\forall x \in D: f(x) = \limn f_n(x)$ + +$f$ muss nicht stetig sein, auch wenn $\forall n \in \N: f_n$ stetig. + +\begin{subbox}{Gleichmässige Konvergenz} + $(f_n)_{n\geq1}$ konv. glm. gegen $f: D \rightarrow \R$, wenn:\\ + $\forall \epsilon > 0, \exists N \geq 1: \forall n \geq N, \forall x \in D: |f_n(x)-f(x)| < \epsilon$\\ + + Wobei $N$ nur von $\epsilon$ abh. (nicht von $x$). + + $f_n$ glm. konv. $\implies f_n$ pw. konv. +\end{subbox} + +\textbf{Altenative Definition für gleichmässige Konvergenz}\\ +$\limn \underset{x \in D}{\sup} |f_n(x)-f(x)| = 0 \iff f_n$ glm. konv. gegen $f$. + +\textbf{Gleichmässige Konvergenz ist stetig}\\ +\footnotesize\color{gray} +$D \subset \R,\quad f_n:D\rightarrow \R,\quad f:D\rightarrow \R$\\ +\normalsize\color{black} +$\forall n \in \N: f_n$ stetig in $D$, glm. konv. gegen $f$\\ +$\implies f$ auch stetig in $D$. + +Es folgt: $f$ nicht stetig $\implies$ $f_n$ nicht glm. konv. + +\textbf{Cauchy Kriterium}\\ +$f_n:D\rightarrow \R$ konv. glm. in $D$, wenn:\\ +$\forall \epsilon > 0, \exists N \geq 1: \forall n,m \geq N, \forall x \in D: |f_n(x)-f_m(x)| < \epsilon$ + +\textbf{Limes-Funktion stetiger glm. konv. Folgen}\\ +$f_n:D\rightarrow\R$ glm. konv. Folge stetiger Funktionen.\\ +$\implies f(x) := \limn f_n(x)$ ist stetig. + +\textbf{Glm. Konvergenz von Funktionenreihen}\\ +$\sum^\infty_{k=0} f_k(x)$ konv. glm. in $D$, falls:\\ +$S_n(x) := \sum^n_{k=0}f_k(x)$ glm. konv. ist. + +\textbf{Vergleichssatz für stetige Funk.-Reihen}\\ +\footnotesize\color{gray} +$D \subset \R,\quad f_n:D\rightarrow \R,\quad \text{alle } f_n \text{ stetig}$\\ +\normalsize\color{black} +$\forall x \in D: |f_n(x)| \leq c_n$ für $(c_n)_{n\geq1}$ s.d. $\sumn c_n$ konv.\\ +$\implies \sumn f_n(x)$ konv., $f(x) := \sumn f_n(x)$ stetig in $D$. + +\textbf{Potenzreihen}\\ +$\sum_{k=0}^\infty c_kx^k$ s.d. $\rho > 0$, $\quad f(x) := \sum_{k=0}^\infty c_kx^k,\quad |x| < \rho$ + +$\implies \forall 0 \leq r < \rho:\quad \sum_{k=0}^\infty c_kx^k$ konv. glm. in $[-r, r]$\\ +\& $\ \ \ \ f: (-\rho, \rho) \rightarrow \R$ ist stetig. + +\subsection{Grenzwerte von Funktionen} +\footnotesize\color{gray} +$D\subset \R,\quad f,g:D\rightarrow\R.\quad x_0 \in \R$ ist Häufungspunkt für $D$ +\normalsize\color{black} + +\textbf{Häufungspunkt}\\ +$x_0 \in \R$ ist ein Häufungspunkt in $D$, falls:\\ +$\forall\delta>0:(\ (x_0-\delta,x_0+\delta)\setminus\{\ x_0\} \ ) \cap D \neq \emptyset$\\ +Man kann (in $D$) beliebig nah zu $x_0$, wobei $x_0 \notin D$ möglich. + +\textbf{Grenzwert für Funktionen}\\ +$\underset{x \to x_0}{\lim}f(x) = L$, wenn für $L$ gilt: + +$\forall \epsilon > 0,\exists \delta > 0:\\ +\forall x \in D \cap (\ (x_0-\delta,x_0+\delta)\setminus\{\ x_0 \} \ ):\quad |f(x)-L|<\epsilon$ + +\textbf{Grenzwert durch Folgen}\\ +$\underset{x \to x_0}{\lim}f(x) = L$ gdw.\\ +$\forall(a_n)_{n\geq1}$ in $D\setminus\{x_0\}$ s.d. $\limn a_n = x_0: \underset{x \to x_0}{\lim}f(a_n)=L$ + +\textbf{Stetigkeit durch Grenzwert}\\ +$f$ stetig in $x_0 \in D \iff \underset{x \to x_0}{\lim}f(x)=f(x_0)$ + +\textbf{Rechenregeln}\\ +Wenn $\underset{x \to x_0}{\lim}f(x),\quad \underset{x \to x_0}{\lim}g(x)$ exisieren: + +$\underset{x \to x_0}{\lim}(f+g)(x)=\underset{x \to x_0}{\lim}f(x) + \underset{x \to x_0}{\lim}g(x)$\\ +$\underset{x \to x_0}{\lim}(f\cdot g)(x) = \underset{x \to x_0}{\lim}f(x) \cdot \underset{x \to x_0}{\lim}g(x)$ + +\textbf{Grenzwerte abschätzen}\\ +$f \leq g \implies \underset{x \to x_0}{\lim}f(x) \leq \underset{x \to x_0}{\lim}g(x)$ falls existent + +\textbf{Sandwich bei Funktionen}\\ +$g_1 \leq f \leq g_2\quad\land\quad \underset{x \to x_0}{\lim}g_1(x)=\underset{x \to x_0}{\lim}g_2(x)$\\ +$\implies \underset{x \to x_0}{\lim}f(x)$ existiert: $\underset{x \to x_0}{\lim}f(x) = \underset{x \to x_0}{\lim}g_1(x)$ + +\textbf{Komposition und Grenzwert}\\ +\footnotesize\color{gray} +Hier: $D, E \subset \R,\quad x_0$ Hf.-P. in $D,\quad f:D\rightarrow E,\quad g:E\rightarrow\R$\\ +\normalsize\color{black} +$\underset{x \to x_0}{\lim}g(f(x))= g(\underset{x \to x_0}{\lim}f(x))$, falls $g$ stetig in $\underset{x \to x_0}{\lim}f(x)$ + +\section{Differenzierbare Funktionen} + +\begin{subbox}{Differenzierbarkeit} + \begin{tabular}{lll} + $f$ diff.-bar in $x_0$ & $\Def$ & $\underset{x \to x_0}{\lim} \frac{f(x)-f(x_0)}{x-x_0}$ existiert.\\ + $f$ diff.-bar in $D$ & $\Def$ & $\forall x_0 \in D:$ $f$ in $x_0$ diff.-bar. + \end{tabular} + + $$f'(x) := \underset{h \to 0}{\lim} \frac{f(x_0+h)-f(x_0)}{h} = \underset{x \to x_0}{\lim}\frac{f(x)-f(x_0)}{x-x_0}$$ +\end{subbox} +\footnotesize\color{gray} +$D \subset \R,\quad f,g:D\rightarrow\R,\quad x_0 $ Häufungspunkt von $D$ +\normalsize\color{black} + +\textbf{Weierstrass}\\ +$f$ diff.-bar in $x_0 \iff\exists c \in \R,\quad r:D\rightarrow\R:\\ +f(x)=f(x_0)+c(x-x_0)+r(x)(x-x_0)\\ +r(x_0)=0$ und $r$ stetig in $x_0$ + +\textbf{Weitere Bedingung}\\ +$f$ in $x_0$ diff.-bar $\iff \exists \phi:D\rightarrow\R,\quad \phi$ stetig in $x_0$ s.d.\\ +$\forall x \in D:\quad f(x) = f(x_0) + \phi(x)(x-x_0),\quad \phi(x_0)=f'(x_0)$ + +\textbf{Differenzierbarkeit impliziert Stetigkeit}\\ +$f$ diff.-bar (in $x_0) \implies f$ stetig (in $x_0$) $\implies f$ integr.\\ +\footnotesize\color{gray} +Nicht umgekehrt: $f(x) = |x|$ ist stetig, aber nicht diff.-bar. +\normalsize\color{black} + +\begin{subbox}{Rechenregeln} + $\begin{array}{lllll} + (i) & (f+g)(x_0) &=& f'(x_0)+g'(x_0)\\ + (ii) & (f \cdot g)'(x_0) &=& f'(x_0)g(x_0)+f(x_0)g'(x_0)\\ + (iii) & (\frac{f}{g})'(x_0) &=& \frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{g(x_0)^2} + \end{array}$ +\end{subbox} +\footnotesize\color{gray} +$f,g$ in $x_0$ diff.-bar, $(iii):\quad g(x_0) \neq 0$ +\normalsize\color{black} + +\textbf{Kompositionen}\\ +\footnotesize\color{gray} +$D, E \subset \R,\quad x_0 \in D $ ist H.-P. in $D$, $\quad f(x_0)$ ist H.-P. in $E$\\$f:D\rightarrow E$ diff.-bar in $x_0,\quad g:E\rightarrow \R$ diff.-bar in $f(x_0)$\\ +\normalsize\color{black} +$g \circ f:D\rightarrow\R$ diff.-bar in $x_0$:\\ +$(g\circ f)'(x_0) = g'(f(x_0))f'(x_0)$ + +\begin{subbox}{Ableitung der Umkehrabbildung} + $y_0$ ist Häufungspunkt in $E$, $f^{-1}$ in $f(x_0)$ diff.-bar und: + + $$(f^{-1})'(f(x_0)) = \frac{1}{f'(x_0)}$$ +\end{subbox} +\footnotesize\color{gray} +$f:D\rightarrow E$ ist bijektiv, $\quad x_0 \in D$ ist H.-P.,\\ +$f$ diff.-bar in $x_0,\quad f^{-1}$ in $f(x_0)$ stetig. +\normalsize\color{black} + +\subsection{Erste Ableitung} + +\textbf{Spezielle Punkte: Lokale Extrema}\\ +\footnotesize\color{gray} +$D\subset\R,\quad f:D\rightarrow\R,\quad x_0 \in D$\\ +\normalsize\color{black} +$x_0$ lokales Minimum, wenn:\\ +$\exists \delta > 0,\quad \forall x \in (x_0-\delta,x_0+\delta)\cap D:\quad f(x) \leq f(x_0)$\\ +$x_0$ lokales Maximum, wenn:\\ +$\exists \delta > 0,\quad \forall x \in (x_0-\delta,x_0+\delta)\cap D:\quad f(x) \geq f(x_0)$ + +Sattelpunkte \& Wendepunkte sind \textit{keine} Extrema. + +\textbf{Lokale Extrema durch Ableitung}\\ +\footnotesize\color{gray} +$f:(a,b)\rightarrow\R,\quad x_0 \in (a,b),\quad f$ in $x_0$ diff.-bar\\ +\normalsize\color{black} +Falls $x_0$ ein lok. Extremum ist: $f'(x_0) =0$. + +$f'(x_0) > 0 \implies \exists\delta>0:\\ +\null\quad\quad f(x) > f(x_0)\quad \forall x \in (x_0, x_0+\delta),\\ +\null\quad\quad f(x) < f(x_0)\quad \forall x \in (x_0-\delta,x_0)$ + +$f'(x_0) < 0 \implies \exists\delta>0:\\ +\null\quad\quad f(x) < f(x_0)\quad \forall x \in (x_0, x_0+\delta),\\ +\null\quad\quad f(x) > f(x_0)\quad \forall x \in (x_0-\delta,x_0)$ + +\textbf{Verhalten von $f$ mittels $f'$}\\ +\footnotesize\color{gray} +$f,g:[a,b]\rightarrow\R$ stetig, in $(a,b)$ diff.-bar\\ +\normalsize\color{black} +$\forall \xi \in (a,b):\ \ldots$\\ +$\null\quad f'(\xi) =0 \implies f$ ist konstant\\ +$\null\quad f'(\xi) = g'(\xi) \implies \exists c \in \R: f(x) = g(x) +c\ \forall x \in [a,b]$\\ +$\null\quad f'(\xi) \geq 0 \implies f$ auf $[a,b]$ mon. wachsend.\\ +$\null\quad f'(\xi) > 0 \implies f$ auf $[a,b]$ str. mon. wachsend.\\ +$\null\quad f'(\xi) \leq 0 \implies f$ auf $[a,b]$ mon. fallend.\\ +$\null\quad f'(\xi) < 0 \implies f$ auf $[a,b]$ str. mon. fallend. + +$\exists M \geq 0:\quad |f'(\xi)| \leq M\quad \forall \xi \in (a,b)$\\ +$\implies \forall x_1,x_2 \in [a,b]:\quad |f(x_1)-f(x_2)| \leq M|x_1-x_2|$ + +\subsection{Höhere Ableitungen} + +\textbf{Definitionen: Höhere Ableitungen}\\ +\footnotesize\color{gray} +$D \subset \R,\quad f:D\rightarrow\R \text{ diff.-bar in } D,\quad \text{jedes } x_0 \in D \text{ ist H.P. von } D,\\ f^{(1)} := f',\quad n \geq 2$\\ +\normalsize\color{black} +$f$ ist $n$-mal diff.-bar in $D \Def f^{(n-1)}$ in $D$ diff.-bar.\\ +$f^{(n)} := (f^{n-1})'$ die $n$-te Ableitung von $f$. + +$f$ ist $n$-mal stetig diff.-bar in $D$ $\Def$ $f^{n}$ stetig in $D$ + +$f$ ist glatt $\Def \forall n \geq 1$ $f$ ist $n$-mal diff.-bar. + +\textbf{Stetigkeit tieferer Ableitungen}\\ +$f$ $n$-mal diff.-bar $\iff$ $f$ $n-1$-mal stetig diff.-bar + +\textbf{Rechenregeln}\\ +\footnotesize\color{gray} +$D \subset \R,\quad n \geq1,\quad f,g:D\rightarrow\R$ $n$-mal diff.-bar in $D$\\ +\normalsize\color{black} +$(i)\ \ \ (f+g)^{(n)} = f^{(n)}+g^{(n)}$\\ +$(ii)\ \ (f \cdot g)^{(n)} = \sum_{k=1}^n\binom{n}{k}f^{(k)}g^{(n-k)}$\\ +$(iii)\ \ \forall x \in D: g(x) \neq 0 \implies \frac{f}{g} \text{ in } D$ $n$-mal diff.-bar + +\textbf{Komposition höherer Ableitungen}\\ +\footnotesize\color{gray} +$E,D \subset \R$ s.d. alle $x_0 \in E, D$ H.-P. sind$,\\ f:D\rightarrow E,\quad g: E \rightarrow \R,\quad f,g$ $n$-mal diff.-bar\\ +\normalsize\color{black} +$g \circ f$ ist $n$-mal diff.-bar. + +$(f\circ g)^{(n)}(x) = \sum^k_{k=1}A_{n,k}(x)(g^{(k)}\circ f)(x)$ + +$A_{n,k}$ ist ein Polynom in $f', f^{(2)},\ldots,f^{(n+1-k)}$. + +\textbf{Extrema mehrfach differenzierbarer $f$}\\ +\footnotesize\color{gray} +$n \geq 0,\quad a < x_0 < b,\quad f:[a,b]\rightarrow\R $ in $(a,b)$ $(n+1)$-mal diff.-bar\\ +\normalsize\color{black} +Wenn: $f'(x_0) = f^{(2)}(x_0) = \ldots = f^ {(n)}(x_0) = 0$: + +$\null\quad n \equiv_2 0,\quad x_0$ lokales Extremum $\implies f^{(n+1)}(x_0) = 0$\\ +$\null\quad n \equiv_2 1,\quad f^{(n+1)}(x_0) > 0 \implies x_0$ str. lokales Minimum\\ +$\null\quad n \equiv_2 1,\quad f^{(n+1)}(x_0) < 0 \implies x_0$ str. lokales Maximum\\ + +\textbf{Extrema zweimal differenzierbarer $f$}\\ +\footnotesize\color{gray} +$f:[a,b]\rightarrow\R,$ stetig, $2$-mal diff.-bar in $(a,b)$\\ +\normalsize\color{black} +$a < x_0 < b,\quad f'(x_0) = 0,$ dann: + +$\null\quad f^{(2)}(x_0) > 0 \implies x_0$ str. lokales Minimum\\ +$\null\quad f^{(2)}(x_0) < 0 \implies x_0$ str. lokales Maximum\\ + +\subsection{Wichtige Theoreme} + +\textbf{Satz von Rolle}\\ +\footnotesize\color{gray} +$f:(a,b)\rightarrow\R$ stetig$,\quad f$ diff.-bar in $(a,b)$\\ +\normalsize\color{black} +$f(a)=f(b)\quad\implies\quad\exists\xi\in(a,b):\quad f'(\xi)=0$ + +\begin{subbox}{Satz von Lagrange} + $f:(a.b) \subset \R \to \R$ in $(a,b)$ diff.-bar: + $$\exists\xi \in (a,b):f(b)-f(a)=f'(\xi)(b-a)$$ +\end{subbox} + +\textbf{Satz von Cauchy}\\ +\footnotesize\color{gray} +$f,g:[a,b]\rightarrow\R$ stetig, in $(a,b)$ diff.-bar\\ +\normalsize\color{black} +$\exists \xi \in (a,b): g'(\xi)(f(b)-f(a)) = f'(\xi)(g(b)-g(a))$ + +Falls $\forall x \in (a,b),\quad g'(x) \neq 0:$\\ +$\implies g(a) \neq g(b),\quad \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(\xi)}{g'(\xi)}$ + +\begin{subbox}{Satz von L'Hôpital} + Falls: $\underset{x \to b^-}{\lim}f(x) = 0, \underset{x \to b^-}{\lim} = 0$, $\underset{x \to b^-}{\lim} \frac{f'(x)}{g'(x)}$ existiert: + + $$\underset{x \to b^-}{\lim} \frac{f(x)}{g(x)}=\underset{x \to b^-}{\lim}\frac{f'(x)}{g'(x)}$$ + + \footnotesize + Auch falls: $b=+\infty,\quad \underset{x \to b^-}{\lim}\frac{f'(x)}{g'(x)} = +\infty,\quad x \to a^+$ +\end{subbox} +\footnotesize\color{gray} +$f,g:(a,b)\rightarrow\R$ diff.-bar in $(a,b),\quad g'(x) \neq 0\quad \forall x \in (a,b)$ +\normalsize\color{black} + +\newpage +\subsection{Konvexe/Konkave Funktionen} +\includegraphics[width=1\linewidth]{konvex.png} + +\textbf{Definition: Konvex}\\ +\footnotesize\color{gray} +$I \subset \R \text{ ist ein beliebiges Intervall,}\quad f:I\rightarrow \R$\\ +\normalsize\color{black} +$f$ konvex auf $I$, falls: $\forall x \leq y \in I,\quad \forall \lambda \in [0,1]:$\\ +$\null\quad f(\lambda x + (1-\lambda)y)\quad\leq\quad\lambda f(x) + (1-\lambda)f(y)$ + +$f$ str. konvex auf $I$, falls: $\forall x < y \in I,\quad \forall \lambda \in (0,1):$\\ +$\null\quad f(\lambda x + (1-\lambda)y)\quad<\quad\lambda f(x) + (1-\lambda)f(y)$ + +\textbf{Summe konvexer Funktionen ist konvex}\\ +\footnotesize\color{gray} +$f:I\rightarrow \R,\quad f$ konvex$,\quad n \geq 1,\quad \{\ x_1,\ldots,x_n\} \subset \R,\\ \lambda_1,\ldots,\lambda_n \in [0,1],\quad\sum_{i=1}^n\lambda_i = 1$ +\normalsize\color{black} + +$f(\sum_{i=1}^n\lambda_ix_i) \leq \sum_{i=1}^n\lambda_if(x_i)$ + +\textbf{Bedingungen für Konvexität}\\ +\footnotesize\color{gray} +$f:I\rightarrow \R,\quad f$ beliebig\\ +\normalsize\color{black} +$f$ (str.) konvex $\iff \forall x_0 < x < x_1 \in I:\\ \frac{f(x)-f(x_0)}{x-x_0} \leq/< \frac{f(x_1)-f(x)}{x_1-x}$ + +\footnotesize\color{gray} +$f:(a,b)\rightarrow \R,\quad f$ diff.-bar in $(a,b)$\\ +\normalsize\color{black} +$f$ (str.) konvex $\iff$ $f'$ (str.) mon. wachsend. + +\footnotesize\color{gray} +$f:(a,b)\rightarrow \R,\quad f$ 2 mal diff.-bar in $(a,b)$\\ +\normalsize\color{black} +$f$ (str.) konvex $\iff f'' \geq 0$ (bzw. $f'' > 0$) auf $(a,b)$. + +\newpage +\subsection{Potenzreihen \& Taylorpolynome} + +\textbf{Gleichmässige konvergenz erhält Differenzierbarkeit}\\ +\footnotesize\color{gray} +$f_n:(a,b)\rightarrow\R$ ist Funktionenfolge$,\\ f_n$ einmal in $(a,b)$ diff.-bar $\forall n \geq 1$\\ +\normalsize\color{black} +$(f_n)_{n\geq1}$ und $(f_n')_{n\geq1}$ glm. konv. in $(a,b)$.\\ +$\implies f:= \limn f_n$ ist stetig diff.-bar s.d. $f' = \limn f_n'$ + +\textbf{Potenzreihen sind differenzierbar}\\ +\footnotesize\color{gray} +$\sum^\infty_{k=1} c_kx^k$ ist Potenzreihe, s.d. $\rho > 0$\\ +\normalsize\color{black} +$f(x) = \sum^\infty_{k=1}c_k(x-x_0)^k$ ist auf $(x_0-\rho, x_0+\rho)$ diff.-bar. + +$f'(x) = \sum_{k=1}^\infty k\cdot c_k (x-x_0)^{k-1}\quad\quad\forall x \in (x_0-\rho, x_0+\rho)$ + +\textbf{Potenzreihen sind glatt}\\ +$f(x) = \sum_{k=1}^\infty c_k (x-x_0)^k$ ist glatt auf $(x_0-\rho,x_0+\rho)$ + +$f^{(j)}(x) = \sum^\infty_{k=j} c_k \frac{k!}{(k-j)!}(x-x_0)^{k-j}$, wobei $c_j = \frac{f^{(j)}(x_0)}{j!}$ + +\textbf{Approximation glatter $f$ durch Polynome}\\ +\footnotesize\color{gray} +$f:[a,b]\rightarrow\R \text{ stetig, } (n+1)\text{-mal diff.-bar in } (a,b)$\\ +\normalsize\color{black} +$\forall a < x \leq b,\quad\exists \xi \in (a,x):$\\ +$f(x) = \sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$ + +\begin{subbox}{Taylor Approximation} + $a \in \R$ s.d. $c < a < d$, $\forall x \in [c,d],\ \exists \xi \in (x,a):$ + + $$f(x) = \underbrace{\sum^n_{k=0} \frac{f^{(k)}(a)}{k!}(x-a)^k}_{=:\ T_n} + \underbrace{\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}}_{=:\ R_n} $$ + + Wobei: $\forall m \geq 1\ \exists n \geq 1: f^{(m)}(x_0) = T_n^{(m)}(x_0)$\\ + \footnotesize + $R_n$ wird als Fehlerabschätzung um $a$ genutzt. + \normalsize +\end{subbox} +\footnotesize\color{gray} +$f:[c,d]\rightarrow\R$ stetig, $(n+1)$ mal diff.-bar in $(c,d)$ +\normalsize\color{black} + + +\footnotesize +\textbf{Taylor-Approximation bei nahen Punkten}\\ +Die Taylor-Approximation bezieht sich wirklich nur auf $x=a$. Beispiel: + +$f(x) = \begin{cases} + 0 & x = 0 \\ + \exp(-\frac{1}{x^2}) & x \neq 0 +\end{cases}$ + +Wenn $a=0$ ist $T_n(x) = 0\ \forall x \in \R\ \forall n \geq 1$.\\ +Aber $\underset{x \to \infty}{\lim} f(x) = \underset{x \to -\infty}{\lim} f(x) = 1$ konvergiert sehr schnell. + +\newpage +\section{Das Riemann Integral} +\footnotesize\color{gray} +$a,b \in \R,\quad a 0,\ \exists P \in \mathcal{P}(I):\quad S(f,P) - s(f,P) < \epsilon$ + +\textbf{Integrierbarkeit als Grenzwert}\\ +\footnotesize\color{gray} +Sei $\mathcal{P}_\delta(I) := \{\text{Partitionen P} \subsetneq [a,b]\ |\ \underset{1 \leq i \leq n}{\max} \delta_i \leq \delta \}$\\ +\normalsize\color{black} +$f:[a,b]\rightarrow\R$ (beschr.) ist integrierbar\\ +$\iff \forall \epsilon > 0,\ \exists \delta > 0: \forall P \in \mathcal{P}_\delta(I):\ \ \ S(f,P)- s(f,P) < \epsilon$ + +\textbf{Polynombrüche sind Integrierbar, ohne Nullstellen}\\ +\footnotesize\color{gray} +$P, Q: \R \rightarrow\R$ sind Polynome\\ +\normalsize\color{black} +$\lnot\exists x \in [a,b]: Q(x)=0\implies \frac{P}{Q}:[a,b]\rightarrow\R$ ist integr. + +\textbf{Stetige Funktionen sind integrierbar}\\ +$f:[a,b]\rightarrow\R$ stetig $\implies f$ ist integrierbar.\\ +\footnotesize\color{gray} +Nicht umgekehrt: Treppenfunktionen sind integr., aber nicht stetig. +\normalsize\color{black} + +\textbf{Monotone Funktionen sind integrierbar}\\ +$f:[a,b]\rightarrow \R$ monoton $\implies f$ ist integrierbar. + +\textbf{Operationen erhalten Integrierbarkeit}\\ +\footnotesize\color{gray} +$f,g:[a,b]\rightarrow \R $ beschr. und intgrierbar$,\quad \lambda \in \R$\\ +\normalsize\color{black} +$f+g,\quad \lambda \cdot f,\quad f \cdot g,\quad |f|,\quad \min(f,g),\quad \max(f,g),\quad \frac{f}{g}$\\ +sind integrierbar.$\quad\quad\quad\quad\quad\quad\quad\quad\quad$ (Für $\frac{f}{g}: |g(x)| > 0$) +\begin{subbox}{Konstanten und Addition} + \footnotesize + $I \text{ kompakt},\quad f_1,f_2:I\rightarrow\R$ beschr. integr.$,\quad \lambda_1,\lambda_2 \in \R$ + \small + $$\int_a^b\lambda_1f_1(x) + \lambda_2f_2(x)\ dx = \lambda_1 \int_a^bf_1(x)\ dx+\lambda_2\int_a^bf_2(x)\ dx$$ +\end{subbox} + \normalsize + +\textbf{Gleichmässige Stetigkeit}\\ +\footnotesize\color{gray} +$D\subset\R,\quad f:D\rightarrow\R$\\ +\normalsize\color{black} +$f$ in $D$ glm. stetig $\Def \forall \epsilon>0,\ \exists\delta >0\quad\forall x,y \in D:\quad |x-y|<\delta \implies |f(x)-f(y)|<\epsilon$ + +\textbf{Gleichmässige Stetigkeit auf kompakten Intervallen}\\ +\footnotesize\color{gray} +$f:[a,b]\rightarrow\R$\\ +\normalsize\color{black} +$f$ Stetig auf kompaktem $[a,b] \implies$ $f$ glm. stetig auf $[a,b]$ + +\textbf{Integrale erhalten Monotonie}\\ +\footnotesize\color{gray} +$f,g:[a,b]\rightarrow\R$ beschr. integr.\\ +\normalsize\color{black} +$\forall x \in [a,b]:\quad f(x) \leq g(x) \implies \int_a^bf(x)\ dx \leq \int^b_a g(x)\ dx$ + +$|\int_a^bf(x)\ dx| \leq \int^b_a|f(x)|\ dx$ + +\newpage +\subsection{Wichtige Theoreme} + +\begin{subbox}{Cauchy-Schwarz} + \footnotesize + $f,g:[a,b]\rightarrow \R$ beschr. integr.\\ + \normalsize + $$\int_a^b |f(x)g(x)\ dx| \leq \sqrt{\int^b_a f^2(x)\ dx}\cdot\sqrt{\int_a^bg^2(x)\ dx}$$ +\end{subbox} + +\textbf{Mittelwertsatz bei Integralen}\\ +\footnotesize\color{gray} +$f:[a,b]\rightarrow \R$ stetig\\ +\normalsize\color{black} +$\exists \xi \in [a,b]:\quad \int_a^bf(x)\ dx = f(\xi)(b-a)$ + +\textbf{Konseqzenz des Mittelwertsatz für Integrale}\\ +\footnotesize\color{gray} +$f,g:[a,b]\rightarrow\R,\quad$ $f$ stetig$,\quad g$ beschr. integr.\\ +\normalsize\color{black} +$\forall x \in [a,b]:\quad g(x) \geq 0$\\ +$\implies \exists\xi \in [a,b]: \int_a^bf(x)g(x)\ dx = f(\xi)\int_a^bg(x)\ dx$ + +\textbf{Integration ist die Umkehrfunktion der Ableitung}\\ +\footnotesize\color{gray} +$a < b,\quad f:[a,b]\rightarrow\R$ stetig\\ +\normalsize\color{black} +$F(x):[a,b]\rightarrow\R,\quad F(x) \mapsto \int_a^xf(x)\ dt,$ + +ist in $[a,b]$ stetig, diff.-bar und $F'(x) = f(x)$. + +\textbf{Stammfunktionen}\\ +\footnotesize\color{gray} +$a < b,\quad f:[a,b]\rightarrow\R$ stetig\\ +\normalsize\color{black} +$F:[a,b]\rightarrow\R$ ist Stammfunktion von $f$\\ +$ \Def F$ stetig diff.-bar in $[a,b]$ und $F' = f$ in $[a,b]$. + +\textbf{Fundamentalsatz der Differentialrechnung}\\ +\footnotesize\color{gray} +$f:[a,b]\rightarrow\R$ stetig\\ +\normalsize\color{black} +Die Stammfunktion $F$ von $f$ existiert s.d.\\ +$\int_a^bf(x)\ dx = F(b) - F(a)$ + +\textbf{Bogenlänge} + +$L = \int_a^b\sqrt{1 + (f'(x))^2}\ dx$ + +\newpage +\subsection{Integrationsmethoden} +\small +Bei rationalen $f$: Polynomdivision \& Partialbruchzerlegung. +\normalsize +\begin{subbox}{Partielle Integration} + \footnotesize + $a a$ \\ +\normalsize\color{black} +Falls $\underset{b \to \infty}{\lim} \int_a^bf(x)\ dx$ existiert:\\ +$\int_a^\infty f(x)\ dx := \underset{b \to \infty}{\lim}\int_a^b f(x)\ dx$ + +Man sagt: $f$ ist auf $[a, \infty)$ integrierbar. + +\textbf{Reihenkonvergenz über Integration}\\ +Sei $f:[1,\infty)\rightarrow[0,\infty)$ mon. fallend. + +$\sum^\infty_{n=1}f(n)$ konv. $\iff \int_1^\infty f(x)\ dx$ konv. + +\textbf{Integration von $f:(a,b]\rightarrow\R$}\\ +Integrierbar, falls $\underset{\epsilon\to0^+}{\lim}\int^b_{a+\epsilon}f(x)\ dx$ existiert.\\ +Man schreibt dann: $\int_a^bf(x)\ dx$. + +\subsection{Konvergente Reihen} + +\textbf{Integration konvergenter Folgen}\\ +\footnotesize\color{gray} +$f_n:[a,b]\rightarrow\R$ Folge beschr. integr. $f$, glm. konv. zu $f:[a,b]\rightarrow\R$\\ +\normalsize\color{black} +$f$ ist beschr. integr. und $\limn\int^b_af_n(x)\ dx = \int_a^b f(x)\ dx$ + +\textbf{Integration konvergenter Reihen}\\ +\footnotesize\color{gray} +$f_n:[a,b]\rightarrow\R$ Folge beschr. integr. $\sumn f_n$, glm. konv. in $[a,b]$\\ +\normalsize\color{black} +$\sumn\int_a^bf_n(x)\ dx = \int_a^b(\sumn f_n(x))\ dx$ + +\textbf{Integration von Potenzreihen}\\ +$f(x) = \sumn c_kx^k$ s.d. $\rho >0$. + +$\forall 0 \leq r < \rho:\quad\ \ $ $f$ auf $[-r,r]$ integr. und\\ +$\forall x \in (-\rho,\rho):\quad\int_0^xf(t)\ dt = \sumn \frac{c^n}{n+1}x^{n+1}$ + +\newpage +\subsection{Approximationsformeln} + +\textbf{Bernoulli Polynome}\\ +Wir nutzen Polynome $P_n$, die erfüllen:\\ +$P_k'=P_{k-1},\quad k \geq1\quad$ und $\null\quad \int_0^1P_k(x)\ dx=0\ \forall k \geq1$ + +Für die Bernoulli-Polynome $B_k$ gilt: $B_k(x) = k!P_k(x)$. + +$B_k$ ist rekursive definiert:\\ +$B_0 = 1,\quad B_{k-1} = \sum_{i=0}^{k-1}\binom{k}{i}B_i = 0$ + +$B_k$ Explizit:\\ +$B_k(x) = \sum_{i=0}\binom{k}{i}B_i ^kx^{k-i}\quad \text{ s.d. }\quad \int_0^1B_k(x)\ dx = 0$ + +Somit:\\ +$B_0(x) = 1,\quad B_1(x) = x-\frac{1}{2},\quad B_2(x) = x^2 - x + \frac{1}{6},\quad +\ldots$ + +$\overset{\sim}{B}_k(x) = \begin{cases} + B_k(x) & \forall x:0 \leq x < 1\\ + B_k(x-n) & \forall x: n \leq x < n+1 +\end{cases}$ + +\textbf{Euler-McLaurin Summationsformel}\\ +\footnotesize\color{gray} +$f:[0,n]\rightarrow\R$ $k$-mal stetig diff.-bar$,\quad k \geq 1$\\ +\normalsize\color{black} +Für $k = 1:$\\ +$\sum_{i=1}^nf(i) = \int^n_0f(x)\ dx + \frac{1}{2}(f(n)-f(0))+\int_0^n\overset{\sim}{B}_1(x)f'(x)\ dx$\\ +Für $k \geq 2:$\\ +$\sum_{i=1}^nf(i)= \int^n_0f(x)\ dx + \frac{1}{2}(f(n)-f(0))\\+\sum_{j=2}^k\frac{(-1)^j B_j}{j!}(f^{(j-1)}(n)-f^{(j-1)}(0))+ \overset{\sim}{R}_k$ + +s.d $\quad\overset{\sim}{R}_k = \frac{(-1)^{k-1}}{k!}\int_0^n\overset{\sim}{B}_k(x)f^{(k)}(x)\ dx$ + +\textbf{Stirling'sche Formel}\\ +Zur Approximation von $n!$ + +$n! \approx \frac{\sqrt{2\pi n}\cdot n^n}{e^n}\quad\quad$ bzw. $\quad \limn\frac{n!}{\frac{\sqrt{2\pi n}\cdot n^n}{e^n}}=1$ + +Via Euler-McLaurin lässt sich präziser beweisen: + +$n! = \frac{\sqrt{2\pi n}\cdot n^n}{e^n}\cdot\exp(\frac{1}{12n}+R_3(n))$ + +s.d. $|R_3(n)| \leq \frac{\sqrt{3}}{216}\cdot\frac{1}{n^2}\quad\forall n \geq 1$ + + +\newpage +\section{Spezifische Funktionen} + +\subsection{Grundfunktionen} + +\textbf{Potenzen}\\ +$f: \R \rightarrow \R:\quad x \mapsto x^n$ + +stetig und glatt $\forall n \in \N$.\\ +$n \equiv_2 1 \iff f $ str. monoton wachsend + +\textbf{Polynome}\\ +$f: \R \rightarrow \R:\quad x \mapsto a_nx^n + \ldots + a_0$ + +stetig und glatt.\\ +$\deg(f) := \underset{0 \leq i \leq n}{\max}\{i \in \N \ |\ a_i \neq 0 \}$ + +Für poly. $f,g \neq 0$, Nullstellen von $g$: $x_1, \ldots , x_m$:\\ +$\frac{f}{g}: \R \setminus \{x_1, \ldots, x_m \} \rightarrow \R:\quad x \mapsto \frac{f(x)}{g(x)}$ ist stetig. + +\textbf{Betragsfunktion}\\ +$f: \R \rightarrow \R:\quad x \mapsto |x|$ + +stetig, in $x_0=0$ nicht diff.-bar.\\ +$g$ stetig $\implies |g|(x) := |g(x)|$ stetig. + +\textbf{Abrundrungsfunktion}\\ +$f:\R \rightarrow \R:\quad x \mapsto \lceil x \rceil := \max \{m \in \Z \ |\ m \leq x \}$ + +$f$ nicht stetig in $x_0 \iff x_0 \in \Z$\\ +Für $D=\R \setminus \Z$ ist $\lceil x \rceil: D\rightarrow\R$ stetig. + +\textbf{Min.-/Max.-Funktionen}\\ +$\max(f,g)(x) := \max(f(x), g(x))$\\ +$\min(f,g)(x)\ := \min(f(x), g(x))$ + +Sind $f,g$ stetig, sind auch $\max(f,g), \min(f,g)$ stetig. + +\newpage +\subsection{Beweisfunktionen} + +\textbf{Indikatorfunktion von $\Q$}\\ +$f(x) = \begin{cases} + 1 & x \in \Q\\ + 0 & x \notin \Q +\end{cases} \quad\quad\quad\ \ f: \R \rightarrow \R$\\ +$\forall x \in \R:$ nicht stetig, nicht integr. in $x$. + +$g(x) = \begin{cases} + x & x \in \Q\\ + 1-x & x \notin \Q +\end{cases}\quad\quad g: \R \rightarrow \R$\\ +Ist nur in $x=\frac{1}{2}$ stetig, sonst nirgends. + +\textbf{Van der Waerden Funktion}\\ +\footnotesize\color{gray} +$x \in \R$\\ +\normalsize\color{black} +Sei $g(x) = \min\{|x-m|\ |\ m \in \Z \}$.\\ +D.h. $g$ gibt die nächste ganze Zahl zu $x$ aus. + +$f(x) = \sumn\frac{g(10^nx)}{10^n}$ + +Die Reihe ist glm. konv. auf ganz $\R$ und $f$ stetig.\\ +$f$ ist nirgendwo diff.-bar. + +\textbf{Glatte Funktion, ohne konv. Potenzreihe}\\ + +$f(x) = \begin{cases} + \exp(-\frac{1}{x^2}) & x \neq 0\\ + 0 & x = 0 +\end{cases}$ + +$f$ ist glatt auf $\R$ s.d. $\forall k \geq 0: f^{(k)}(0) = 0$.\\ +Da $f(x) > 0\quad \forall x \neq 0$ gibt es keine P.-Reihe mit pos. $\rho$. + +\textbf{Funktion mit nicht-stetiger Ableitung}\\ +$f(x) = \begin{cases} + x^2\cdot\sin(\frac{1}{x}) & x \neq 0\\ + 0 & x = 0 +\end{cases}$\\ +Wobei $f'$ nicht stetig in $x_0 = 0$. + +\newpage +\subsection{Exponentialfunktion} + +\begin{subbox}{Definition: Exponentialfunktion} + \footnotesize + $\forall z \in \C: \exp(z)$ konvergiert. + \normalsize + $$\exp: \C \rightarrow \C \quad\quad \exp(z) := \sumn\frac{z^n}{n!}$$ +\end{subbox} + +\textbf{Exponentialfunktion in $\R$}\\ +$\exp: \R \rightarrow (0, +\infty)$ + +str. mon. wachs., stetig, surj. und glatt. + +$\begin{array}{lll} + \exp(x) > 0 & \color{gray} \forall x \in \R \\ + \exp(x) > 1 & \color{gray}\forall x > 0 \\ + \exp(x) \geq 1 + x & \color{gray}\forall x \in \R \\ + \exp(x)\cdot\exp(y) = \exp(x + y) & \color{gray}\forall x,y \in \R +\end{array}$ + +\textbf{Natürlicher Logarithmus}\\ +$\exp^{-1} :=\quad \ln:(0, +\infty) \rightarrow \R$ + +str. mon. wachs., stetig, bijektiv und glatt. + +$\ln(a \cdot b) = \ln(a) + \ln(b) \quad\quad \color{gray}\forall a,b \in (0, +\infty)$ + +\textbf{Allgemeine Potenzen}\\ +$x^a: (0, +\infty) \rightarrow (0, +\infty) :=\quad \exp(a\cdot\ln(x))\quad$\\ +\footnotesize\color{gray} +$x > 0,\quad a \in \R$ +\normalsize\color{black} + +$a > 0 \implies$ stetig, bijektiv, str. mon. wachs.\\ +$a < 0 \implies$ stetig, bijektiv, str. mon. fall. + +\textbf{Potenzgesetze}\\ +\color{gray}\footnotesize +$\forall a,b \in \R,\quad \forall x >0:\quad$\\ +\color{black}\normalsize +$\begin{array}{ll} +\ln(x^a) = a\ln(x) &\quad x^ax^b = x^{a+b} \\ +(x^a)^b=x^{ab} &\quad x^0=1 +\end{array}$ + +\newpage +\subsection{Trigonometrische Funktionen} + +\begin{subbox}{Definition: Trigonometrische Funktionen} + \footnotesize + $\forall z \in \C:\sin(z), \cos(z)$ konv. abs. + \normalsize + \begin{align*} + \sin(z) = & \sumn(-1)^n\frac{z^{2n+1}}{(2n+1)!} &= z - \frac{z^3}{3!}+\frac{z^5}{5!}-\ldots\\ + \cos(z) = & \sumn(-1)^n\frac{z^{2n}}{(2n)!} &= 1-\frac{z^2}{2!}+\frac{z^4}{4!}-\ldots + \end{align*} + \small + \begin{center} + $\tan(z) = \frac{\sin(z)}{\cos(z)}\quad\quad\cot(z)=\frac{\cos(z)}{\sin(z)}$ + \end{center} +\end{subbox} + +\textbf{Trigonometrische Funktionen in $\R$}\\ +$\sin,\cos: \R \rightarrow \R$ sind stetig und glatt. + +$\pi := \inf\{t>0\ |\ \sin(t) = 0\} \in (2,4)$\\ +$\forall0\leq x\leq \sqrt{6}:\quad x \geq \sin(x) \geq x-\frac{x^3}{3!}$ + +Nullstellen von $\sin$ in $\R$: $\{k\cdot\pi\ |\ k \in \Z\}$ + +$\null\quad\sin(x) > 0\quad \forall x \in (2k\pi, (2k+1)\pi)$\\ +$\null\quad\sin(x) < 0\quad\forall x \in ((2k+1)\pi, (2k+2)\pi)$ + +Nullstellen von $\cos$ in $\R$: $\{\frac{\pi}{2}+k\pi\ |\ k \in \Z \}:$ + +$\null\quad\cos(x) > 0\quad\forall x \in (-\frac{\pi}{2}+2k\pi, -\frac{\pi}{2}+(2k+1)\pi)$\\ +$\null\quad\cos(x) < 0\quad\forall x \in (-\frac{\pi}{2}+(2k+1)\pi, -\frac{\pi}{2}+(2k+2)\pi)$ + +\textbf{Hyperbelfunktionen} + +$\begin{array}{llll} +\cosh(x) &: \R \to [1,\infty] &:=& \frac{e^x+e^{-x}}{2}\\ +\sinh(x) &: \R \to \R &:=& \frac{e^x - e^{-x}}{2}\\ +\tanh(x) &: \R \to [-1,1] &:=& \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} +\end{array}$ + +$\sinh,\cosh,\tanh$ sind glatt. + +\newpage + +\section{Tabellen} +\color{gray} +Credits: Einige Tabellen von D. Camenisch \& J. Steinmann +\color{black} +\subsection{Trigonometrische Identitäten} + +\textbf{Trigonometrische Identitäten in $\C$} + +$\begin{array}{lll} +e^{iz} &=& \cos(z)+i\sin(z)\\ +\hline +\cos(z)^2+\sin(z)^2 &=& 1\\ +\hline +\sin(z) &=& \frac{e^{iz}-e^{-iz}}{2i}\\ +\cos(z) &=& \frac{e^{iz}+e^{-iz}}{2}\\ +\hline +\sin(z+w) &=& \sin(z)\cos(w)+\cos(z)\sin(w)\\ +\cos(z+w) &=& \cos(z)\cos(w)+\sin(z)\sin(w)\\ +\hline +\sin(2z) &=& 2\sin(z)\cos(z)\\ +\cos(2z) &=& \cos(z)^2-\sin(z)^2\\ +\end{array}$ + +$e^{i\pi}=-1,\quad e^{2i\pi}=1,\quad e^{\frac{i\pi}{2}}=i$ + +\footnotesize +$\sin(z) = \frac{e^{iz}-e^{-iz}}{2}$ kann hilfreich sein um $\sin(z)^n$ um zuschreiben. +\normalsize + +\textbf{Trigonometrische Identitäten in $\R$} + +$\begin{array}{lll} +\sin(-x) &=& -\sin(x)\\ +\cos(-x) &=& \cos(x)\\ +\tan(-x) &=& -\tan(x)\\ +\hline +\sin(x+\frac{\pi}{2}) &=& \cos(x)\\ +\cos(x+\frac{\pi}{2}) &=& -\sin(x)\\ +\sin(x+\pi) &=& \cos(x)\\ +\sin(x+2\pi) &=& \sin(x)\\ +\hline +\sin(x)^2 &=& \frac{1}{2}(1-\cos(2x))\\ +\cos(x)^2 &=& \frac{1}{2}(1+\cos(2x))\\ +\hline +\sin(\arctan(x)) &=& \frac{x}{\sqrt{x^2+1}} \\ +\cos(\arctan(x)) &=& \frac{1}{\sqrt{x^2+1}} \\ +\hline +\sin(x) &=& \frac{\tan(x)}{\sqrt{1 + \tan(x)^2}} \\ +\cos(x) &=& \frac{1}{\sqrt{1 + \tan(x)^2}} +\end{array}$ + +\newpage +\subsection{Trigonometrische Funktionen: Werte} + +\textbf{Funktionswerte am Winkelkreis}\\ + +\begin{center} + \includegraphics[width=0.8\linewidth]{degrees_circle.pdf} +\end{center} + +\textbf{Trigonometrische Analogien} + +\small +\begin{tabular}{c|c|c|c|c|c|c|c} + & \rotatebox{90}{\(-\alpha\)}& \rotatebox{90}{$90$\(^{\circ}\) \(-\alpha\)}& \rotatebox{90}{$90$\(^{\circ}\) \(+\alpha\)}& \rotatebox{90}{$180$\(^{\circ}\) \(-\alpha\)}& \rotatebox{90}{$180$\(^{\circ}\) \(+\alpha\)}& \rotatebox{90}{$k*360$\(^{\circ}\) \(-\alpha\)}&\rotatebox{90}{$k*360$\(^{\circ}\) \(+\alpha\)}\\ + \hline + \(\sin\)& + \(-\sin{ }\)& \(\cos{ }\)& \(\cos{ }\)& \(\sin{ }\)& + \(-\sin{ }\)& + \(-\sin{ }\)& + \(\sin{ }\)\\ + \hline + \(\cos\)& + \(\cos{ }\)& + \(\sin{ }\)& + \(-\sin{ }\)& + \(-\cos{ }\)& + \(-\cos{ }\)& + \(\cos{ }\)& + \(\cos{ }\)\\ + \hline + \(\tan\)& + \(-\tan{ }\)& + \(\cot{ }\)& + \(-\cot{ }\)& + \(-\tan{ }\)& + \(\tan{ }\)& + \(-\tan{ }\)& + \(\tan{ }\)\\ +\end{tabular} +\normalsize +\\ + +\textbf{Werte der trigonometrischen Funktionen} + +\footnotesize +\begin{tabular}{c|c|c|c|c|c|c|c|c|c} + $\alpha$ & $0$° & $30$° & $45$° & $60$° & $90$° & $120$° & $150$° & $180$° & $270$° \\ + & $0$ & $\frac{\pi}{6}$ & $\frac{\pi}{4}$ & $\frac{\pi}{3}$ & $\frac{\pi}{2}$ & $\frac{2\pi}{3}$ & $\frac{5\pi}{6}$ & $\pi$ & $\frac{3\pi}{2}$\\ + \hline + $\sin$ & $0$ & $\frac{1}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ & $0$ & $-1$\\ + \hline + $\cos$ & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{1}{2}$ & $0$ & $-\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$ & $-1$ & $0$\\ + \hline + $\tan$ & $0$ & $\frac{\sqrt{3}}{3}$ & $1$ & $\sqrt{3}$ & & $-\sqrt{3}$ & $-\frac{\sqrt{3}}{3}$ & $0$ & \\ +\end{tabular} +\normalsize + + +\newpage +\footnotesize +\subsection{Integrale \& Ableitungen} +\begin{center} + \begin{tabular}{c||c||c} + $F(x)$ & $f(x)$ & $f'(x)$ \\ + \hline + \hline + $x$ & $c$ & $0$\\ + $\frac{1}{a+1}\cdot x^{a+1}$ & $x^a$ & $a\cdot x^{a-1}$\\ + $\frac{1}{a\cdot(n+1)}\cdot(ax+b)^{n+1}$ & $(ax+b)^n$ & $n \cdot(ax+b)^{n-1}\cdot a$\\ + $\ln|x|$ & $\frac{1}{x}=x^{-1}$ & $-\frac{1}{x^2} = -x^{-2}$\\ + $\frac{2}{3}\cdot x^{\frac{3}{2}}$ & $\sqrt{x} = x^{\frac{1}{2}}$ & $\frac{1}{2\sqrt{x}} = \frac{1}{2}\cdot x^{-\frac{1}{2}}$\\ + $\frac{n}{n+1}\cdot x^{\frac{1}{n}+1}$ & $\sqrt[\leftroot{0} n]{x} = x^\frac{1}{n}$ & $\frac{1}{n}\cdot x^{\frac{1}{n}-1}$\\ + \hline + $e^x$ & $e^x$ & $e^x$\\ + $\frac{1}{\ln(a)}\cdot a^x$ & $a^x$ & $a^x\cdot \ln(a)$\\ + $x\cdot (\ln|x|-1)$ & $\ln|x|$ & $\frac{1}{x} = x^{-1}$\\ + $\frac{x}{\ln(a)}\cdot(\ln|x|-1)$ & $\log_a(x)$ & $\frac{1}{x\cdot\ln(a)}$\\ + \hline + $-\cos(x)$ & $\sin(x)$ & $\cos(x)$\\ + $\sin(x)$ & $\cos(x)$ & $-\sin(x)$\\ + $-\ln|\cos(x)|$ & $\tan(x)$ & $\frac{1}{\cos(x)^2}= 1 + \tan(x)^2$\\ + $\ln|\sin(x)|$ & $\cot(x)$ & $-\frac{1}{\sin(x)^2}$\\ + $x \cdot\arcsin(x) + \sqrt{1 - x^2}$ & $\arcsin(x)$ & $\frac{1}{\sqrt{1-x^2}}$\\ + $x \cdot \arccos(x)-\sqrt{1-x^2}$ & $\arccos(x)$ & $-\frac{1}{\sqrt{1-x^2}}$\\ + $x \cdot \arctan(x)-\frac{\ln(x^2+1)}{2}$ & $\arctan(x)$ & $\frac{1}{x^2+1}$\\ + \hline + $\sinh(x)$ & $\cosh(x)$ & $\sinh(x)$\\ + $\cosh(x)$ & $\sinh(x)$ & $\cosh(x)$\\ + $\ln|\cosh(x)|$ & $\tanh(x)$ & $\frac{1}{\cosh(x)^2} = 1-\tanh(x)^2$\\ + & $\text{arcsinh}(x)$ & $\frac{1}{\sqrt{x^2+1}}$\\ + & $\text{arccosh}(x)$ & $\frac{1}{\sqrt{x^2}-1}$\\ + & $\text{arctanh}()$ & $\frac{1}{1-x^2}$\\ + \hline + \end{tabular} +\end{center} + +\subsection{Taylorreihen} +\begin{center} + + \begin{tabular}{l||l} + $f(x)$ & $T_n$\\ + \hline + \hline + $\mathrm{e}^x $&$ 1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\mathcal{O}(x^5)$ \\ + $\sin{x} $&$ x-\frac{x^3}{3!}+\frac{x^5}{5!}+\mathcal{O}(x^7)$ \\ + $\sinh(x) $&$ x+\frac{x^3}{3!}+\frac{x^5}{5!}+\mathcal{O}(x^7)$ \\ + $\cos(x) $&$ 1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\mathcal{O}(x^8)$ \\ + $\cosh(x) $&$ 1+\frac{x^2}{2}+\frac{x^4}{4!}+\frac{x^6}{6!}+\mathcal{O}(x^8)$ \\ + $\tan(x) $&$ x+\frac{x^3}{3}+\frac{2x^5}{15}+\mathcal{O}(x^7)$ \\ + $\tanh(x) $&$ x-\frac{x^3}{3}+\frac{2x^5}{15}+\mathcal{O}(x^7)$ \\ + $\log(1+x) $&$ x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\mathcal{O}(x^5)$ \\ + $(1+x)^\alpha $&$ 1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+ \frac{\alpha(\alpha - 1)(\alpha - 2)}{3!}x^3 + \mathcal{O}(x^4)$ \\ + $\sqrt{1+x} $&$ 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \mathcal{O}(x^4)$\\ + \hline + \end{tabular} +\end{center} + +\newpage +\subsection{Weitere Integrale \& Ableitungen} +\begin{center} + \begin{tabular}{c||c} + + $F(x)$ & $f(x)$\\ + \hline + \hline + $\frac{1}{a}\ln|ax+b|$ & $\frac{1}{ax+b}$ \\ + $\frac{ax}{c}-\frac{ad-bc}{c^2}\ln|cx+d|$ & $\frac{a(cx+d) - c(ax+b)}{(cx+d)^2}$\\ + $\frac{x}{2}f(x)+\frac{a^2}{2}\ln|x+f(x)|$ & $\sqrt{a^2+x^2}$ \\ + $\frac{x}{2}f(x)-\frac{a^2}{2}\arcsin(\frac{x}{|a|})$ & $\sqrt{a^2-x^2}$ \\ + $\frac{x}{2}f(x) - \frac{a^2}{2}\ln|x + f(x)|$ & $\sqrt{x^2-a^2}$\\ + $\ln(x + \sqrt{x^2 \pm a^2})$ & $\frac{1}{\sqrt{x^2 \pm a^2}}$\\ + $\arcsin(\frac{x}{|a|})$ & $\frac{1}{\sqrt{a^2-x^2}}$ \\ + $\frac{1}{a} \cdot\arctan(\frac{x}{a})$ & $\frac{1}{x^2+a^2}$\\ + \hline + $-\frac{1}{a}\cos(ax+b)$ & $\sin(ax+b)$ \\ + $\frac{1}{a}\sin(ax+b)$ & $\cos(ax+b)$ \\ + \hline + $x^x$ & $x^x \cdot (1 + \ln|x|)$\\ + $(x^x)^x$ & $(x^x)^x(x+2x\ln|x|)$\\ + $x^{(x^x)}$ & $x^{(x^x)}(x^{x-1}+\ln|x|\cdot x^x(1+\ln|x|)$\\ + \hline + $\frac{1}{2}(x-\frac{1}{2}\sin(2x))$ & $\sin(x)^2$\\ + $\frac{1}{2}(x+\frac{1}{2}\sin(2x))$ & $\cos(x)^2$\\ + \end{tabular} +\end{center} + +%\subsection{Tyler, The Creator} +% \begin{center} +% \includegraphics[width=0.3\linewidth]{tyler2.png} +% \end{center} + +\newpage +\subsection{Grenzwerte: Folgen} +\begin{center} + \begin{tabular}{ l || l } + $\lim_{x\to\infty} \frac{1}{x} = 0$ & $\lim_{x\to\infty} 1 + \frac{1}{x} = 1$ \\ + \hline + $\lim_{x \to \infty} e^x = \infty$ & $\lim_{x \to - \infty} e^x = 0$ \\ + \hline + $\lim_{x \to \infty} e^{-x} = 0$ & $\lim_{x \to - \infty} e^{-x} = \infty$ \\ + \hline + $\lim_{x \to \infty} \frac{e^x}{x^m} = \infty$ & $\lim_{x \to - \infty} xe^x = 0$ \\ + \hline + $\lim_{x \to \infty} \ln(x) = \infty$ & $\lim_{x \to 0} \ln(x) = - \infty$ \\ + \hline + $\lim_{x \to \infty} (1 + x)^{\frac{1}{x}} = 1$ & $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$ \\ + \hline + $\lim_{x\to\infty} \left(1 + \frac{1}{x}\right)^b = 1$ & $\lim_{x\to\infty} \left(1 + \frac{1}{x}\right)^b = 1$ \\ + \hline + $\lim_{x\to\infty} x^aq^x = 0, \; \forall 0 \leq q < 1$ & $\lim_{x\to\infty} n^\frac{1}{n} = 1$ \\ + \hline + $\lim_{x\to \pm \infty} \left(1+\frac{1}{x}\right)^x = \mathrm{e}$ & $\lim_{x\to \infty} \left(1-\frac{1}{x}\right)^x = \frac{1}{\mathrm{e}}$ \\ + \hline + $\lim_{x \to \pm \infty} \left(1+\frac{k}{x}\right)^{mx} = e^{km}$ & $\lim_{x \to 0} \frac{\sin{x}}{x} = 1$ \\ + \hline + $\lim_{x\to 0} \frac{1}{\cos(x)} = 1$ & $\lim_{x\to 0} \frac{\cos{x}-1}{x} = 0$ \\ + \hline + $\lim_{x\to 0} \frac{\log{1-x}}{x} = -1$ & $\lim_{x\to 0} x\log{x} = 0$ \\ + \hline + $\lim_{x\to 0} \frac{1-\cos{x}}{x^2} = \frac{1}{2}$ & $\lim_{x\to 0} \frac{\mathrm{e}^x-1}{x} = 1$ \\ + \hline + $\lim_{x\to 0} \frac{x}{\arctan{x}} = 1$ & $\lim_{x\to\infty} \arctan{x} = \frac{\pi}{2}$ \\ + \hline + $\lim_{x \to \infty} \left(\frac{x}{x + k}\right)^x = e^{-k}$ & $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ \\ + \hline + $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a) \; \forall a > 0$ & $\lim_{x \to 0} \frac{e^{ax} - 1}{x} = a$ \\ + \hline + $\lim_{x \to 0} \frac{\ln(x + 1)}{x} = 1$ & $\lim_{x \to 1} \frac{\ln(x)}{x - 1} = 1$ \\ + \hline + $\lim_{x \to \infty} \frac{\ln(x)}{x} = 0$ & $\lim_{x \to \infty} \frac{\log(x)}{x^a} = 0$ \\ + \hline + $\lim_{x \to \infty} \sqrt[x]{x} = 1$ & $\lim_{x \to \infty} \frac{2x}{2^x} = 0$ \\ + \hline + $\lim_{x \to \frac{\pi^-}{2}} \tan{x} = +\infty$ & $\lim_{x \to \frac{\pi^+}{2}} \tan{x} = -\infty$ \\ + \hline + $\lim_{x \to \infty} \frac{\sin{x}}{x} = 0$ & $\lim_{x \to 0^+} x\ln{x} = 0$\\ + \hline + \end{tabular} +\end{center} +\subsection{Grenzwerte: Reihen} +\begin{center} + \begin{tabular}{ l || l } + $\sum_{i = 1}^{n} i = \frac{n(n+1)}{2}$ & $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ \\ + \hline + $\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}$ & $\sum_{i = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ \\ + \hline + $\sum_{i = 1}^{\infty} \frac{1}{n(n+1)} = 1$ & $\sum_{i = 1}^\infty z^i = \frac{1 - z^{i + 1}}{1 - z}$ \\ + \hline + \end{tabular} +\end{center} + + +\end{document} diff --git a/semester3/analysis-ii-rb/main.tex b/semester3/analysis-ii-rb/main.tex new file mode 100644 index 0000000..e69de29