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-rw-r--r--vorlesungen/08_dgl/MathSem-08-dgl.tex25
-rw-r--r--vorlesungen/slides/10/taylor.tex36
-rw-r--r--vorlesungen/slides/10/vektorfelder.tex14
3 files changed, 48 insertions, 27 deletions
diff --git a/vorlesungen/08_dgl/MathSem-08-dgl.tex b/vorlesungen/08_dgl/MathSem-08-dgl.tex
index 1bcb946..e4ece1b 100644
--- a/vorlesungen/08_dgl/MathSem-08-dgl.tex
+++ b/vorlesungen/08_dgl/MathSem-08-dgl.tex
@@ -7,8 +7,29 @@
\input{common.tex}
\setboolean{presentation}{true}
\begin{document}
-\begin{frame}
-\titlepage
+ \begin{frame}
+ \titlepage
+ \vspace{-1.5cm}
+ \begin{columns}
+ \begin{column}{.48\textwidth}
+ \centering
+ \includegraphics[width=.7\linewidth]{../slides/10/vektorfelder-6.pdf}
+ \end{column}
+ \begin{column}{.48\textwidth}
+ \begin{align*}
+ x(t)
+ &=
+ \exp(At) x_0
+ \\
+ \exp(At)
+ &=
+ 1 + At + \frac{A^2t^2}{2} + \frac{A^3 t^3}{3!} + \ldots
+ \\
+ &=
+ \lim_{n\to \infty} \left(1 + \frac{At}{n}\right)^n
+ \end{align*}
+ \end{column}
+ \end{columns}
\end{frame}
\input{slides.tex}
\end{document}
diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
index 25745f5..8c71965 100644
--- a/vorlesungen/slides/10/taylor.tex
+++ b/vorlesungen/slides/10/taylor.tex
@@ -11,7 +11,7 @@
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Beispiel $\sin(x)$}
- \vspace{-20pt}
+ \ifthenelse{\boolean{presentation}}{\vspace{-20pt}}{\vspace{-8pt}}
\begin{block}{Taylor-Approximationen von $\sin(x)$}
\begin{align*}
p_{
@@ -44,15 +44,15 @@
\draw[domain=-4:4, samples=50, smooth, blue]
plot ({\x}, {sin(180/3.1415968*\x)})
node[above right] {$\sin(x)$};
- \uncover<1>{
+ \uncover<1|handout:0>{
\draw[domain=-4:4, samples=2, smooth, red]
plot ({\x}, {0})
node[above right] {$p_0(x)$};}
- \uncover<2>{
+ \uncover<2|handout:0>{
\draw[domain=-1.5:1.5, samples=2, smooth, red]
plot ({\x}, {\x})
node[below right] {$p_1(x)$};}
- \uncover<3>{
+ \uncover<3|handout:0>{
\draw[domain=-1.5:1.5, samples=2, smooth, red]
plot ({\x}, {\x})
node[below right] {$p_2(x)$};}
@@ -60,19 +60,19 @@
\draw[domain=-3:3, samples=50, smooth, red]
plot ({\x}, {\x - \x*\x*\x/6})
node[above right] {$p_3(x)$};}
- \uncover<5>{
+ \uncover<5|handout:0>{
\draw[domain=-3:3, samples=50, smooth, red]
plot ({\x}, {\x - \x*\x*\x/6})
node[above right] {$p_4(x)$};}
- \uncover<6>{
+ \uncover<6|handout:0>{
\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>{
+ \uncover<7|handout:0>{
\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->{
+ \uncover<8-|handout:0>{
\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})
@@ -85,7 +85,7 @@
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Taylor-Reihen}
- \vspace{-20pt}
+ \ifthenelse{\boolean{presentation}}{\vspace{-20pt}}{\vspace{-8pt}}
\begin{block}{Polynom-Approximationen von $f(t)$}
\begin{align*}
p_n(t)
@@ -135,8 +135,8 @@
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
-% \frametitle{Beispiel $e^t$}
-% \vspace{-20pt}
+ \frametitle{Beispiel $e^t$}
+ \ifthenelse{\boolean{presentation}}{\vspace{-20pt}}{\vspace{-8pt}}
\begin{block}{Taylor-Approximationen von $e^{at}$}
\begin{align*}
p_{
@@ -171,15 +171,15 @@
\draw[domain=-4:1, samples=50, smooth, blue]
plot ({\x}, {exp(\x)})
node[above right] {$\exp(t)$};
- \uncover<1>{
+ \uncover<1|handout:0>{
\draw[domain=-4:4, samples=12, smooth, red]
plot ({\x}, {1})
node[below right] {$p_0(t)$};}
- \uncover<2>{
+ \uncover<2|handout:0>{
\draw[domain=-4:1.5, samples=10, smooth, red]
plot ({\x}, {1 + \x})
node[below right] {$p_1(t)$};}
- \uncover<3>{
+ \uncover<3|handout:0>{
\draw[domain=-4:1, samples=50, smooth, red]
plot ({\x}, {1 + \x + \x*\x/2})
node[below right] {$p_2(t)$};}
@@ -187,22 +187,22 @@
\draw[domain=-4:1, samples=50, smooth, red]
plot ({\x}, {1 + \x + \x*\x/2 + \x*\x*\x/6})
node[below right] {$p_3(t)$};}
- \uncover<5>{
+ \uncover<5|handout:0>{
\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] {$p_4(t)$};}
- \uncover<6>{
+ \uncover<6|handout:0>{
\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] {$p_5(t)$};}
- \uncover<7>{
+ \uncover<7|handout:0>{
\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] {$p_6(t)$};}
- \uncover<8->{
+ \uncover<8-|handout:0>{
\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
diff --git a/vorlesungen/slides/10/vektorfelder.tex b/vorlesungen/slides/10/vektorfelder.tex
index a4612aa..3ba7cda 100644
--- a/vorlesungen/slides/10/vektorfelder.tex
+++ b/vorlesungen/slides/10/vektorfelder.tex
@@ -14,11 +14,11 @@
\begin{columns}[t,onlytextwidth]
\begin{column}{0.48\textwidth}
\vfil
- \only<1>{
+ \only<1|handout:0>{
\includegraphics[width=\linewidth,keepaspectratio]
{../slides/10/vektorfelder-1.pdf}
}
- \only<2>{
+ \only<2|handout:0>{
\includegraphics[width=\linewidth,keepaspectratio]
{../slides/10/vektorfelder-2.pdf}
}
@@ -26,15 +26,15 @@
\includegraphics[width=\linewidth,keepaspectratio]
{../slides/10/vektorfelder-3.pdf}
}
- \only<4>{
+ \only<4|handout:0>{
\includegraphics[width=\linewidth,keepaspectratio]
{../slides/10/vektorfelder-4.pdf}
}
- \only<5>{
+ \only<5|handout:0>{
\includegraphics[width=\linewidth,keepaspectratio]
{../slides/10/vektorfelder-5.pdf}
}
- \only<6->{
+ \only<6-|handout:0>{
\includegraphics[width=\linewidth,keepaspectratio]
{../slides/10/vektorfelder-6.pdf}
}
@@ -51,14 +51,14 @@
\]
\end{block}
- \only<2>{
+ \only<2|handout:0>{
Nach einem Schritt der Länge $t$:
\[
x(t) = x_0 + \dot x t = x_0 + Jx_0t = (1 + Jt)x_0
\]
}
- \only<3>{
+ \only<3|handout:0>{
Nach zwei Schritten der Länge $t/2$:
\[
x(t) = \left(1 + \frac{Jt}{2}\right)^2x_0