%
% wurzel.tex -- Wurzel
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\documentclass[tikz]{standalone}
\usepackage{amsmath}
\usepackage{times}
\usepackage{txfonts}
\usepackage{pgfplots}
\usepackage{csvsimple}
\usetikzlibrary{arrows,intersections,math}
\begin{document}
\def\skala{10}
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{tikzpicture}[>=latex,thick,scale=\skala]

\def\a{0.8}
\def\U{0}

\fill[color=blue!20] (0,\a) rectangle (1.00,1.03);
\draw[line width=0.4pt] (0,1) -- (1,1) -- (1,0);

\draw[->] (0,{-0.01}) -- (0,{1.06}) coordinate[label={right:$y$}];

\begin{scope}
\clip (0,0) rectangle (1,1);
\draw[color=blue,line width=1.4pt] plot[domain=0:1.01,samples=100]
	({\x},{\x*\x});
\end{scope}

%\draw[color=purple,line width=0.5pt] (0.48,-0.01) -- (1,1);
%\fill[color=purple] (1,1) circle[radius=0.008];

\node[color=blue] at (0,{\a}) [left] {$a$};

\def\schritt#1#2{
	\xdef\u{\U}
	\pgfmathparse{0.5*(\a-\u*\u)}
	\xdef\d{\pgfmathresult}
	\pgfmathparse{\u+\d}
	\xdef\U{\pgfmathresult}

	\fill[color=purple!10] (\u,{\u*\u}) -- (\U,\a) -- (\u,\a) -- cycle;

	\node[color=darkgreen] at (\u,0) [below] {$u_#1$};
	\draw[color=darkgreen,line width=0.1pt] (\u,0)--(\u,\a);

	\fill[color=darkgreen] (\u,{\u*\u}) circle[radius=0.006];

	\draw[<->,color=darkgreen] (\u,{\u*\u}) -- (\u,\a);

	\draw[color=purple,shorten <= 0.6mm]
		(\u,{\u*\u}) -- (\U,\a);
}
\def\marke#1#2{
	\node[color=orange] at ({0.5*(\u+\U)},\a) [#2] {$\frac12(a-u_#1^2)$};
	\draw[<->,color=orange,shorten >= 0mm,shorten <= 0mm]
		(\u,\a) -- (\U,\a);
}

\def\hoehe#1{
	\node[color=darkgreen] at ({\u+0.01},{\a-\d-0.01})
		[above,rotate=90] {$a-u_#1^2$};
}

\schritt{0}{1}
\hoehe{0}
\marke{0}{above}

\schritt{1}{2}
\hoehe{1}
\marke{1}{above}
\node[color=darkgreen] at (\u,{\u*\u-0.02}) [above left] {$u_1^2$};

\schritt{2}{3}
\hoehe{2}
%\marke{2}{right,rotate=90}
\marke{2}{above}
\node[color=darkgreen] at (\u,{\u*\u-0.02}) [above left] {$u_2^2$};

\schritt{3}{4}

\draw[color=blue] ({sqrt(\a)},-0.01) -- ({sqrt(\a)},\a);
\node[color=blue] at ({sqrt(\a)-0.02},0) [below right] {$\sqrt{a}$};

\draw[->] (-0.01,0) -- (1.05,0) coordinate[label={$u$}];
\node at (1,0) [below] {$1$};
\node at (0,1) [left] {$1$};
\draw (1,-0.01) -- (1,0.01);

\end{tikzpicture}
\end{document}