From 588ba7c095b15178deedd7aed4713d66a8cf4da7 Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Wed, 14 Apr 2021 22:29:43 +0200
Subject: =?UTF-8?q?Slides=20f=C3=BCr=20Vorlesung=20DGL=20begonnen.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 vorlesungen/slides/10/taylor.tex | 195 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 195 insertions(+)
 create mode 100644 vorlesungen/slides/10/taylor.tex

(limited to 'vorlesungen/slides/10/taylor.tex')

diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
new file mode 100644
index 0000000..8912cb7
--- /dev/null
+++ b/vorlesungen/slides/10/taylor.tex
@@ -0,0 +1,195 @@
+%
+% 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
-- 
cgit v1.2.1


From bd913de11450cc6294e874f2c38a838b0c1e6cd5 Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Thu, 15 Apr 2021 17:46:09 +0200
Subject: Skalar-zu-Matrix DGL Folie.

---
 vorlesungen/slides/10/taylor.tex | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

(limited to 'vorlesungen/slides/10/taylor.tex')

diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
index 8912cb7..bbd1126 100644
--- a/vorlesungen/slides/10/taylor.tex
+++ b/vorlesungen/slides/10/taylor.tex
@@ -1,7 +1,8 @@
 %
 % eindiomensional.tex -- Lösung der eindimensionalen DGL
 %
-% (c) 2021 Roy Seitz, Hochschule Rapperswil
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
 %
 % !TeX spellcheck = de_CH
 \bgroup
-- 
cgit v1.2.1


From cb562f607b18540df333d6d9a911cf51c91884d0 Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Thu, 15 Apr 2021 23:49:49 +0200
Subject: Mehr Folien und einige Typos.

---
 vorlesungen/slides/10/taylor.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'vorlesungen/slides/10/taylor.tex')

diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
index bbd1126..920470f 100644
--- a/vorlesungen/slides/10/taylor.tex
+++ b/vorlesungen/slides/10/taylor.tex
@@ -1,5 +1,5 @@
 %
-% eindiomensional.tex -- Lösung der eindimensionalen DGL
+% taylor.tex -- Repetition Taylot-Reihen
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 % Erstellt durch Roy Seitz
-- 
cgit v1.2.1


From 8358d9bc031913305a52c6c2ab05184b89f7678f Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Sat, 17 Apr 2021 22:00:36 +0200
Subject: Slides erweitert.

---
 vorlesungen/slides/10/taylor.tex | 176 ++++++++++++++++++++++-----------------
 1 file changed, 98 insertions(+), 78 deletions(-)

(limited to 'vorlesungen/slides/10/taylor.tex')

diff --git a/vorlesungen/slides/10/taylor.tex b/vorlesungen/slides/10/taylor.tex
index 920470f..25745f5 100644
--- a/vorlesungen/slides/10/taylor.tex
+++ b/vorlesungen/slides/10/taylor.tex
@@ -10,12 +10,19 @@
 \begin{frame}[t]
   \setlength{\abovedisplayskip}{5pt}
   \setlength{\belowdisplayskip}{5pt}
-  \frametitle{Beispiel $\sin x$}
+  \frametitle{Beispiel $\sin(x)$}
   \vspace{-20pt}
-  %\onslide<+->
-  \begin{block}{Taylor-Approximationen von $\sin x$}
+  \begin{block}{Taylor-Approximationen von $\sin(x)$}
     \begin{align*}
-      p_n(x)
+      p_{
+      \only<1>{0}
+      \only<2>{1}
+      \only<3>{2}
+      \only<4>{3}
+      \only<5>{4}
+      \only<6>{5}
+      \only<7->{n}
+      }(x)
       &= 
       \uncover<1->{0}
       \uncover<2->{+ x}
@@ -74,121 +81,134 @@
   \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}
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  \frametitle{Taylor-Reihen}
+  \vspace{-20pt}
+    \begin{block}{Polynom-Approximationen von $f(t)$}
+      \begin{align*}
+        p_n(t) 
+        &=
+        f(0)
+        \uncover<2->{ + f' (0) t }
+        \uncover<3->{ + f''(0)\frac{t^2}{2} }
+        \uncover<4->{ + \ldots + f^{(n)}(0) \frac{t^n}{n!} }
+        \uncover<5->{ = \sum_{k=0}^{n} f^{(k)} \frac{t^k}{k!} }
+      \end{align*}
+    \end{block}
+  \uncover<6->{
+    \begin{block}{Erste $n$ Ableitungen von $f(0)$ und $p_n(0)$ sind gleich!}}
     \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)
+      \uncover<6->{ p'_n(t) }
+      &
+      \uncover<7->{
+        = f'(0) 
+        + f''(0)t 
+        + \mathcal O(t^2)
+      }
+      &\uncover<8->{\Rightarrow}&&
+      \uncover<8->{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)
+      \uncover<9->{ p''_n(t) }
+      &
+      \uncover<10->{
+        = f''(0) 
+        + \mathcal O(t)
+      }
+      &\uncover<11->{\Rightarrow}&&
+      \uncover<11->{ p''_n(0) = f''(0) }
     \end{align*}
   \end{block}
+  \uncover<12->{
+    \begin{block}{Für alle praktisch relevanten Funktionen $f(t)$ gilt:}
+      \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 $\exp x$}
+%  \frametitle{Beispiel $e^t$}
+%  \vspace{-20pt}
+  \begin{block}{Taylor-Approximationen von $e^{at}$}
     \begin{align*}
-      p_n(x) 
-      = 
+      p_{
+        \only<1>{0}
+        \only<2>{1}
+        \only<3>{2}
+        \only<4>{3}
+        \only<5>{4}
+        \only<6>{5}
+        \only<7->{n}
+      }(t)
+      &=
       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!}}
+      \uncover<2->{+ a t}
+      \uncover<3->{+ a^2 \frac{t^2}{2}}
+      \uncover<4->{+ a^3 \frac{t^3}{3!}}
+      \uncover<5->{+ a^4 \frac{t^4}{4!}}
+      \uncover<6->{+ a^5 \frac{t^5}{5!}}
+      \uncover<7->{+ a^6 \frac{t^6}{6!}}
+      \uncover<8->{+ \ldots
+                   = \sum_{k=0}^{n} a^k \frac{t^k}{k!}}
+      \\
+      &
+      \uncover<9->{= \exp(at)}
     \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[->] (-4.0, 0.0) -- (4.0,0.0) coordinate[label=$t$];
       \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] {$\exp(x)$};
+        node[above right] {$\exp(t)$};
       \uncover<1>{
-      \draw[domain=-4:1.5, samples=10, smooth, red]
-        plot ({\x}, {1 + \x})
-        node[below right] {$p_1(x)$};}
+      \draw[domain=-4:4, samples=12, smooth, red]
+        plot ({\x}, {1})
+        node[below right] {$p_0(t)$};}
       \uncover<2>{
+      \draw[domain=-4:1.5, samples=10, smooth, red]
+      plot ({\x}, {1 + \x})
+      node[below right] {$p_1(t)$};}
+      \uncover<3>{
       \draw[domain=-4:1, samples=50, smooth, red]
         plot ({\x}, {1 + \x + \x*\x/2})
-        node[below right] {$p_2(x)$};}
-      \uncover<3>{
+        node[below right] {$p_2(t)$};}
+      \uncover<4>{
       \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>{
+        node[below right] {$p_3(t)$};}
+      \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})
-        node[below left] {$p_4(x)$};}
-      \uncover<5>{
+        node[below left] {$p_4(t)$};}
+      \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})
-      node[below left] {$p_5(x)$};}
-      \uncover<6>{
+      node[below left] {$p_5(t)$};}
+      \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})
-      node[below left] {$p_6(x)$};}
-      \uncover<7>{
+      node[below left] {$p_6(t)$};}
+      \uncover<8->{
       \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] {$p_7(x)$};}
+      node[below left] {$p_7(t)$};}
     \end{tikzpicture}
   \end{center}
 \end{frame}
-- 
cgit v1.2.1


From 8a32e72ace1b6442b2601b821e4d7ed24047939e Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Mon, 19 Apr 2021 11:10:24 +0200
Subject: Titleseite und Handout.

---
 vorlesungen/slides/10/taylor.tex | 36 ++++++++++++++++++------------------
 1 file changed, 18 insertions(+), 18 deletions(-)

(limited to 'vorlesungen/slides/10/taylor.tex')

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
-- 
cgit v1.2.1

