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-rw-r--r--vorlesungen/slides/10/n-zu-1.tex54
-rw-r--r--vorlesungen/slides/10/taylor.tex195
2 files changed, 249 insertions, 0 deletions
diff --git a/vorlesungen/slides/10/n-zu-1.tex b/vorlesungen/slides/10/n-zu-1.tex
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+%
+% n-zu-1.tex -- Umwandlend einer DGL n-ter Ordnung in ein System 1. Ordnung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt: 2021-04-14, Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Reicht $1.$ Ordnung?}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Beispiel: DGL 3.~Ordnung} \vspace*{-1ex}
+ \begin{align*}
+ x^{(3)} + a_2 \ddot x + a_1 \dot x + a_0 x = 0 \\
+ \Rightarrow
+ x^{(3)} = -a_2 \ddot x - a_1 \dot x - a_0 x
+ \end{align*}
+\end{block}
+\begin{block}{Ziel: Nur noch 1.~Ableitungen}
+ Einführen neuer Variablen:
+ \begin{align*}
+ x_0 &\coloneqq x &
+ x_1 &\coloneqq \dot x &
+ x_2 &\coloneqq \ddot x
+ \end{align*}
+System von Gleichungen 1.~Ordnung
+ \begin{align*}
+ \dot x_0 &= x_1 \\
+ \dot x_1 &= x_2 \\
+ \dot x_2 &= -a_2 x_2 - a_1 x_1 - a_0 x_0
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Als Vektor-Gleichung} \vspace*{-1ex}
+ \begin{align*}
+ \frac{d}{dt}
+ \begin{pmatrix} x_0 \\ x_1 \\ x_2 \end{pmatrix}
+ = \begin{pmatrix}
+ 0 & 1 & 0 \\
+ 0 & 0 & 1 \\
+ -a_0 & -a_1 & -a_2
+ \end{pmatrix}
+ \begin{pmatrix} x_0 \\ x_1 \\ x_2 \end{pmatrix}
+ \end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
new file mode 100644
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+%
+% eindiomensional.tex -- Lösung der eindimensionalen DGL
+%
+% (c) 2021 Roy Seitz, Hochschule Rapperswil
+%
+% !TeX spellcheck = de_CH
+\bgroup
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Beispiel $\sin x$}
+ \vspace{-20pt}
+ %\onslide<+->
+ \begin{block}{Taylor-Approximationen von $\sin x$}
+ \begin{align*}
+ p_n(x)
+ &=
+ \uncover<1->{0}
+ \uncover<2->{+ x}
+ \uncover<3->{+ 0 \frac{x^2}{2!}}
+ \uncover<4->{- 1 \frac{x^3}{3!}}
+ \uncover<5->{+ 0 \frac{x^4}{4!}}
+ \uncover<6->{+ 1 \frac{x^5}{5!}}
+ \uncover<7->{+ \ldots}
+ \uncover<8->{
+ = \sum_{k=0}^{n/2} (-1)^{2k + 1}\frac{x^{2k+1}}{(2k+1)!}
+ }
+ \end{align*}
+ \end{block}
+ \begin{center}
+ \begin{tikzpicture}[>=latex,thick,scale=1.3]
+ \draw[->] (-5.0, 0.0) -- (5.0,0.0) coordinate[label=$x$];
+ \draw[->] ( 0.0,-1.5) -- (0.0,1.5);
+ \clip (-5,-1.5) rectangle (5,1.5);
+ \draw[domain=-4:4, samples=50, smooth, blue]
+ plot ({\x}, {sin(180/3.1415968*\x)})
+ node[above right] {$\sin(x)$};
+ \uncover<1>{
+ \draw[domain=-4:4, samples=2, smooth, red]
+ plot ({\x}, {0})
+ node[above right] {$p_0(x)$};}
+ \uncover<2>{
+ \draw[domain=-1.5:1.5, samples=2, smooth, red]
+ plot ({\x}, {\x})
+ node[below right] {$p_1(x)$};}
+ \uncover<3>{
+ \draw[domain=-1.5:1.5, samples=2, smooth, red]
+ plot ({\x}, {\x})
+ node[below right] {$p_2(x)$};}
+ \uncover<4>{
+ \draw[domain=-3:3, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6})
+ node[above right] {$p_3(x)$};}
+ \uncover<5>{
+ \draw[domain=-3:3, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6})
+ node[above right] {$p_4(x)$};}
+ \uncover<6>{
+ \draw[domain=-3.9:3.9, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6 + \x*\x*\x*\x*\x/120})
+ node[below right] {$p_5(x)$};}
+ \uncover<7>{
+ \draw[domain=-3.9:3.9, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6 + \x*\x*\x*\x*\x/120})
+ node[below right] {$p_6(x)$};}
+ \uncover<8->{
+ \draw[domain=-4:4, samples=50, smooth, red]
+ plot ({\x}, {\x - \x*\x*\x/6 + \x*\x*\x*\x*\x/120 -
+ \x*\x*\x*\x*\x*\x*\x/5040})
+ node[above right] {$p_7(x)$};}
+ \end{tikzpicture}
+ \end{center}
+\end{frame}
+
+
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Taylor-Reihen}
+\vspace{-20pt}
+\onslide<+->
+ \begin{block}{Polynom-Approximationen von $f(t)$}
+ \vspace{-15pt}
+ \begin{align*}
+ p_n(t)
+ &=
+ f(0)
+ + f'(0) t
+ + f''(0)\frac{t^2}{2}
+ + f^{(3)}(0)\frac{t^3}{3!}
+ + \ldots
+ + f^{(n)}(0) \frac{t^n}{n!}
+ =
+ \sum_{k=0}^{n} f^{(k)} \frac{t^k}{k!}
+ \end{align*}
+ \end{block}
+ \begin{block}{Die ersten $n$ Ableitungen von $f(0)$ und $p_n(0)$ sind gleich!}
+ \vspace{-15pt}
+ \begin{align*}
+ p'_n(t)
+ &=
+ f'(0)
+ + f''(0)t
+ + f^{(3)}(0) \frac{t^2}{2!}
+ + \mathcal O(t^3)
+ &\Rightarrow&&
+ p'_n(0) = f'(0)
+ \\
+ p''_n(0)
+ &=
+ f''(0) + f^{(3)}(0)t + \ldots + f^{(n)}(0) \frac{t^{n-2}}{(n-2)!}
+ &\Rightarrow&&
+ p''_n(0) = f''(0)
+ \end{align*}
+ \end{block}
+ \begin{block}{Für unendlich lange Polynome stimmen alle Ableitungen überein!}
+ \vspace{-15pt}
+ \begin{align*}
+ \lim_{n\to \infty} p_n(t)
+ =
+ f(t)
+ \end{align*}
+ \end{block}
+\end{frame}
+
+
+\begin{frame}[t]
+ \setlength{\abovedisplayskip}{5pt}
+ \setlength{\belowdisplayskip}{5pt}
+ \frametitle{Beispiel $\exp x$}
+ \vspace{-20pt}
+ %\onslide<+->
+ \begin{block}{Taylor-Approximationen von $\exp x$}
+ \begin{align*}
+ p_n(x)
+ =
+ 1
+ \uncover<1->{+ x}
+ \uncover<2->{+ \frac{x^2}{2}}
+ \uncover<3->{+ \frac{x^3}{3!}}
+ \uncover<4->{+ \frac{x^4}{4!}}
+ \uncover<5->{+ \frac{x^5}{5!}}
+ \uncover<6->{+ \frac{x^6}{6!}}
+ \uncover<7->{+ \ldots
+ = \sum_{k=0}^{n} \frac{x^k}{k!}}
+ \end{align*}
+ \end{block}
+ \begin{center}
+ \begin{tikzpicture}[>=latex,thick,scale=1.3]
+ \draw[->] (-4.0, 0.0) -- (4.0,0.0) coordinate[label=$x$];
+ \draw[->] ( 0.0,-0.5) -- (0.0,2.5);
+ \clip (-3,-0.5) rectangle (3,2.5);
+ \draw[domain=-4:1, samples=50, smooth, blue]
+ plot ({\x}, {exp(\x)})
+ node[above right] {$\exp(x)$};
+ \uncover<1>{
+ \draw[domain=-4:1.5, samples=10, smooth, red]
+ plot ({\x}, {1 + \x})
+ node[below right] {$p_1(x)$};}
+ \uncover<2>{
+ \draw[domain=-4:1, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2})
+ node[below right] {$p_2(x)$};}
+ \uncover<3>{
+ \draw[domain=-4:1, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6})
+ node[below right] {$p_3(x)$};}
+ \uncover<4>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24})
+ node[below left] {$p_4(x)$};}
+ \uncover<5>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
+ + \x*\x*\x*\x*\x/120})
+ node[below left] {$p_5(x)$};}
+ \uncover<6>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
+ + \x*\x*\x*\x*\x/120
+ + \x*\x*\x*\x*\x*\x/720})
+ node[below left] {$p_6(x)$};}
+ \uncover<7>{
+ \draw[domain=-4:0.9, samples=50, smooth, red]
+ plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6 + \x*\x*\x*\x/24
+ + \x*\x*\x*\x*\x/120
+ + \x*\x*\x*\x*\x*\x/720
+ + \x*\x*\x*\x*\x*\x*\x/5040})
+ node[below left] {$p_7(x)$};}
+ \end{tikzpicture}
+ \end{center}
+\end{frame}
+
+\egroup
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