diff options
| -rw-r--r-- | vorlesungen/08_dgl/MathSem-08-dgl.tex | 25 | ||||
| -rw-r--r-- | vorlesungen/slides/10/taylor.tex | 36 | ||||
| -rw-r--r-- | vorlesungen/slides/10/vektorfelder.tex | 14 |
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 |