diff options
Diffstat (limited to 'vorlesungen/slides/10/taylor.tex')
| -rw-r--r-- | vorlesungen/slides/10/taylor.tex | 195 |
1 files changed, 195 insertions, 0 deletions
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 |