From 8358d9bc031913305a52c6c2ab05184b89f7678f Mon Sep 17 00:00:00 2001 From: Roy Seitz 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 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