fix error in polygonzugverfahren image, add tikz math library to lecture cls

ana16.tex

@@ -1,5 +1,6 @@
 \documentclass{lecture}
 
+\usetikzlibrary{math}
 \begin{document}
 \newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}}
 \newcommand{\graph}{\operatorname{Graph}}
@@ -119,7 +120,7 @@ Notationen: $x' = f(t,x), \dot x = f(t,x), \dv{x}{t} = f(t,x)$ (Dynamischer Proz
                 }
             ]
             \addplot3 (x,y,0);
-            \addlegendentry{$f'(t,x)$}
+            \addlegendentry{$f(t,x)$}
             \addplot{(x-0.25)^2+0.15};
             \addlegendentry{$y(t)$}
           \end{axis}
@@ -236,45 +237,63 @@ Reminder:
     \[y^h(t)\coloneqq y_{n-1}^h + (t-t_{n-1})f(t_{n-1},y_{n-1}^h),\quad t\in [t_{n-1},t_n],\quad \forall n\ge 1\]
     \begin{figure}[h]
         \centering
-        \begin{tikzpicture}[declare function={f1(\x) = 0.5*(2)^(\x-1) + 10/(\x+2);
-            f2(\x) = 0.5*(2)^(\x-1);
-            f3(\x) = 0.5*(2)^(\x-1) - 10/(\x+2);
-            f4(\x) = 0.5*(2)^(\x-1) - 20/(\x+2);}]
+        \begin{tikzpicture}[declare function={
+                    g(\x) = 0.5*exp(\x-2); % base function for y(t)
+                    f(\x) = 0.5*exp(\x-2); % derivative
+                    %g(\x) = 0.5*(\x-2.7)^3 - 2*(\x-2.7)^2; % base function for y(t)
+                    %f(\x) = 1.5*(\x-2.7)^2 - 4*(\x-2.7); % derivative
+                }]
+            \def\h{1} % step length (accuracy of approximation)
+            \def\torig{2} % y_0
+            \def\yorig{1} % t_0
             \begin{axis}%
                 [grid=none,
-                 %minor tick num=4,
                  grid style={line width=.1pt, draw=gray!10},
                  major grid style={line width=.2pt,draw=gray!50},
                  axis lines=middle,
-                 %enlargelimits={abs=0.2},
                  ymax=10, ymin=-1.5,
+                 restrict y to domain=-2:12,
                  xmin=-1, xmax=7,
                  xtick={2,3,4,5},
                  ytick=\empty,
                  xticklabels={$t_0$, $t_1$, $t_2$, $t_3$},
-                 %yticklabels={$y_0$, $y_1^{h}$, $y_2^{h}$},
                  xlabel=$t$,
-                 ylabel=$x$,
+                 ylabel=$y$,
+                 legend pos=outer north east
                 ]
-                \addplot[domain=0:10,samples=50,smooth,green] {f1(x)};
-                \addlegendentry{$y(t,t_0,y_0)$};
-                \addplot[domain=0:10,samples=50,smooth,blue] {f2(x)};
-                \addlegendentry{$y(t, t_1, y_1^{h})$};
-                \addplot[domain=0:10,samples=50,smooth,orange] {f3(x)};
-                \addlegendentry{$y(t, t_2, y_2^{h})$};
-                \addplot[domain=0:10,samples=50,smooth,pink] {f4(x)};
-                \addlegendentry{$y(t, t_3, y_3^{h})$};
-                \draw (2,{f1(2)}) node[circle,fill,inner sep=0.5pt] {}
-                    -- (3,{f2(3)}) node[circle,fill,inner sep=0.5pt] {}
-                    -- (4, {f3(4)}) node[circle,fill,inner sep=0.5pt] {}
-                    -- (5, {f4(5)}) node[circle,fill,inner sep=0.5pt] {};
-                \draw[dashed,green] (2, {f1(2)}) -- (0, {f1(2)}) node[label=left:$y_0$](){};
-                \draw[dashed,blue] (3, {f2(3)}) -- (0, {f2(3)}) node[label=left:$y_1$](){};
-                \draw[dashed,orange] (4, {f3(4)}) -- (0, {f3(4)}) node[label=left:$y_2$](){};
-                \draw[dashed,pink] (5, {f4(5)}) -- (0, {f4(5)}) node[label=left:$y_3$](){};
+                \def\d{0}
+                \def\t{0}
+                \foreach \i/\colour [remember=\d as \dlast (initially \yorig),
+                                     remember=\t as \tlast (initially \torig)]
+                                    in {0/green,1/blue,2/orange,3/pink} {
+                    \tikzmath{\t=\tlast+\h;\d=g(\tlast)+\dlast+\h*f(\tlast)-g(\t);}
+                    \colour
+                    \edef\temp{\noexpand
+                        \addplot[domain=0:10,samples=50,smooth,\colour] {g(x) + \dlast - g(\torig)};
+                    }
+                    \temp
+                    \if\i3
+                        \edef\temp{\noexpand
+                            \draw[dashed,->] (\tlast,{g(\tlast) + \dlast - g(\torig)}) 
+                                -- (\t,{g(\t) + \d - g(\torig)});
+                        }
+                    \else
+                        \edef\temp{\noexpand
+                            \draw (\tlast,{g(\tlast) + \dlast - g(\torig)}) node[circle,fill,inner sep=0.5pt] {}
+                                -- (\t,{g(\t) + \d - g(\torig)}) node[circle,fill,inner sep=0.5pt] {};
+                        }
+                    \fi
+                    \temp
+                    \edef\temp{\noexpand
+                        \draw[dashed,\colour] (\tlast, {g(\tlast) + \dlast - g(\torig)}) -- (0, {g(\tlast) + \dlast - g(\torig)}) node[label=left:$y_{\i}$](){};
+                    }
+                    \temp
+                    \edef\temp{\noexpand\addlegendentry{$y(t,t_{\i},y_{\i})$};}
+                    \temp
+                }
               \end{axis}
         \end{tikzpicture}
-        \caption{Eulersches Polygonzugverfahren}
+        \caption{Eulersches Polygonzugverfahren, Steigung der Tangenten ist $f(t,y)$}
     \end{figure}
     \begin{enumerate}[1)]
         \item \textbf{z.Z.} dass dieses Verfahren durchführbar ist, d.h. $\graph(y^h)\subset D$. Sei $(t,y^h(t))\subset D$ für $t_0 \le t\le t_{k-1}$. Dann gilt 

lecture.cls

@@ -27,7 +27,7 @@
 \RequirePackage{environ}
 \RequirePackage{stackrel}
 
-\usetikzlibrary{quotes, angles}
+\usetikzlibrary{quotes, angles, math}
 \pgfplotsset{compat=1.15} % or \pgfplotsset{compat=newest}
 
 \geometry{