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

