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The commands
find . -type f -name '*.md' -exec sed --in-place 's/[[:space:]]\+$//' {} \+
and
find . -type f -name '*.tex' -exec sed --in-place 's/[[:space:]]\+$//' {} \+
were used to do so.
139 lines
5.9 KiB
TeX
139 lines
5.9 KiB
TeX
\chapter{Constant functions}
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\section{Defined on $\mdr$}
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\begin{lemma}
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Let $f:\mdr \rightarrow \mdr$, $f(x) := c$ with $c \in \mdr$ be a constant function.
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Then $(x_P, f(x_P))$ is the only point on the graph of $f$ with
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minimal distance to $P$.
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\end{lemma}
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The situation can be seen in Figure~\ref{fig:constant-min-distance}.
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\begin{figure}[htp]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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legend pos=north west,
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legend cell align=left,
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axis x line=middle,
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axis y line=middle,
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grid = major,
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width=0.8\linewidth,
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height=8cm,
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grid style={dashed, gray!30},
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xmin=-5, % start the diagram at this x-coordinate
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xmax= 5, % end the diagram at this x-coordinate
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ymin= 0, % start the diagram at this y-coordinate
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ymax= 3, % end the diagram at this y-coordinate
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axis background/.style={fill=white},
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xlabel=$x$,
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ylabel=$y$,
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tick align=outside,
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minor tick num=-3,
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enlargelimits=true,
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tension=0.08]
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\addplot[domain=-5:5, thick,samples=50, red] {1};
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\addplot[domain=-5:5, thick,samples=50, green] {2};
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\addplot[domain=-5:5, thick,samples=50, blue, densely dotted] {3};
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\addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)};
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\addplot[blue, mark = *, nodes near coords=$P_{h,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(2, 3)};
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\addplot[green, mark = x, nodes near coords=$P_{g,\text{min}}$,every node near coord/.style={anchor=120}] coordinates {(2, 2)};
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\addplot[red, mark = *, nodes near coords=$P_{f,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(2, 1)};
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\draw[thick, dashed] (axis cs:2,0) -- (axis cs:2,3);
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\addlegendentry{$f(x)=1$}
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\addlegendentry{$g(x)=2$}
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\addlegendentry{$h(x)=3$}
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\end{axis}
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\end{tikzpicture}
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\caption{Three constant functions and their points with minimal distance}
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\label{fig:constant-min-distance}
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\end{figure}
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\begin{proof}
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The point $(x, f(x))$ with minimal distance can be calculated directly:
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\begin{align}
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d_{P,f}(x) &= \sqrt{(x - x_P)^2 + (f(x) - y_P)^2}\\
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&= \sqrt{(x^2 - 2x_P x + x_P^2) + (c^2 - 2 c y_P + y_P^2)} \\
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&= \sqrt{x^2 - 2 x_P x + (x_P^2 + c^2 - 2 c y_P + y_P^2)}\label{eq:constant-function-distance}\\
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\xRightarrow{\text{Theorem}~\ref{thm:fermats-theorem}} 0 &\stackrel{!}{=} (d_{P,f}(x)^2)'\\
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&= 2x - 2x_P\\
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\Leftrightarrow x &\stackrel{!}{=} x_P
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\end{align}
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So $(x_P,f(x_P))$ is the only point with minimal distance to $P$. $\qed$
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\end{proof}
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This result means:
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\[S_0(f, P) = \Set{x_P} \text{ with } P = (x_P, y_P)\]
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\clearpage
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\section{Defined on a closed interval $[a,b] \subseteq \mdr$}
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\begin{theorem}[Solution formula for constant functions]
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Let $f:[a,b] \rightarrow \mdr$, $f(x) := c$ with $a,b,c \in \mdr$ and
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$a \leq b$ be a constant function.
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Then the point $(x, f(x))$ of $f$ with minimal distance to $P$ is
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given by:
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\[\underset{x\in [a,b]}{\arg \min d_{P,f}(x)} = \begin{cases}
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S_0(f,P) &\text{if } S_0(f,P) \cap [a,b] \neq \emptyset \\
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\Set{a} &\text{if } S_0(f,P) \ni x_P < a\\
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\Set{b} &\text{if } S_0(f,P) \ni x_P > b
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\end{cases}\]
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\end{theorem}
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\begin{figure}[htp]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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legend pos=north west,
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legend cell align=left,
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axis x line=middle,
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axis y line=middle,
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grid = major,
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width=0.8\linewidth,
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height=8cm,
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grid style={dashed, gray!30},
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xmin=-5, % start the diagram at this x-coordinate
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xmax= 5, % end the diagram at this x-coordinate
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ymin= 0, % start the diagram at this y-coordinate
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ymax= 3, % end the diagram at this y-coordinate
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axis background/.style={fill=white},
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xlabel=$x$,
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ylabel=$y$,
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tick align=outside,
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minor tick num=-3,
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enlargelimits=true,
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tension=0.08]
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\addplot[domain=-5:-2, thick,samples=50, red] {1};
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\addplot[domain=-1:3, thick,samples=50, green] {1.5};
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\addplot[domain=3:5, thick,samples=50, blue, densely dotted] {3};
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\addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)};
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\addplot[blue, mark = *, nodes near coords=$P_{h,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(3, 3)};
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\addplot[green, mark = x, nodes near coords=$P_{g,\text{min}}$,every node near coord/.style={anchor=120}] coordinates {(2, 1.5)};
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\addplot[red, mark = *, nodes near coords=$P_{f,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(-2, 1)};
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\draw[thick, dashed] (axis cs:2,1.5) -- (axis cs:2,2);
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\draw[thick, dashed] (axis cs:2,2) -- (axis cs:-2,1);
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\draw[thick, dashed] (axis cs:2,2) -- (axis cs:3,3);
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\addlegendentry{$f(x)=1, D = [-5,-2]$}
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\addlegendentry{$g(x)=1.5, D = [-1,3]$}
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\addlegendentry{$h(x)=3, D = [3,5]$}
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\end{axis}
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\end{tikzpicture}
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\caption{Three constant functions and their points with minimal distance}
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\label{fig:constant-min-distance-closed-intervall}
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\end{figure}
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\begin{proof}
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\begin{align}
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\underset{x\in[a,b]}{\arg \min d_{P,f}(x)} &= \underset{x\in[a,b]}{\arg \min d_{P,f}(x)^2}\\
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&=\underset{x\in[a,b]}{\arg \min} \big ((x-x_P)^2 + \overbrace{(y_P^2 - 2 y_P c + c^2)}^{\text{constant}} \big )\\
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&=\underset{x\in[a,b]}{\arg \min} (x-x_P)^2
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\end{align}
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which is optimal for $x = x_P$, but if $x_P \notin [a,b]$, you want
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to make this term as small as possible. It gets as small as possible when
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$x$ is as similar to $x_p$ as possible. This yields directly to the
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solution formula.$\qed$
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\end{proof}
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