☰

如何绘制柯西中值定理（Cauchy's MVT）的图形说明

tag-icon • tag-icon tikz-pgf draw tikz-graphdrawing
如何绘制柯西中值定理（Cauchy's MVT）的图形说明

我想在我的讲义中插入以下柯西 MVT 的图表说明。 在此处输入图片描述

我不知道如何制作循环以及如何找到特定点的切线。请帮忙。以下代码片段是我用于拉格朗日 MVT 的代码片段，我相信它可以用作 MWE：

\documentclass{article}
\usepackage{tikz}
\usepackage{pgf}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\usetikzlibrary{hobby}
\usetikzlibrary{decorations.pathreplacing, calligraphy}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usepackage{xcolor}
\begin{document}
    \begin{tikzpicture}[xscale=3, yscale=2, >=stealth, declare function={f(\x)=\x*\x*\x+0.5; a=sqrt(0.37); g(\x)=f(a)+1.11*(\x-a); h(\x)=f(0)+1.11*(\x-0.1);}]
        \draw (0,0) -- (1.1,0);
        \draw [smooth, blue, domain=0.1:1, samples=10] plot(\x,{f(\x)});
        \draw [densely dashed, red, thick] (0,{h(0)}) -- (1.1,{h(1.1)});
        \draw [densely dashed] (0.1,0) node[below] {\scriptsize$ a $} -- (0.1,{f(0.1)});
        \draw [pen colour={white}, decorate, decoration={calligraphic brace, raise=1pt, amplitude=5pt}] (0.1,0) -- (0.1,{f(0.1)}) node[midway, left=5pt] {\scriptsize$ f(a) $};
        \draw [densely dashed] (1,0) node[below] {\scriptsize$ b $} -- (1,{f(1)});
        \draw [pen colour={white}, decorate, decoration={calligraphic brace, mirror, raise=1pt, amplitude=5pt}] (1,0) -- (1,{f(1)}) node[midway, right=5pt] {\scriptsize$ f(b) $};
        \draw [yellow, domain=0.3:0.95, samples=2] plot(\x,{g(\x)});
        \draw [yellow, densely dashed] (a,0) node[below] {\scriptsize$ \xi $} -- (a,{f(a)}) node[fill, circle, inner sep=1pt] {};
    \end{tikzpicture}
\end{document}

我实际上是在黑色背景下做讲义的beamer。请原谅 MWE 中的颜色选择。

答案1

经过深思熟虑，并得到了谢布·格拉夫，我终于找到了一个参数化曲线，其图形正是我想要的，如图所示。我使用外螺旋的部分作为参数的正值和负值，并旋转它以获得我想要的方向。我在这里将我的发现作为答案发布。如果有人想编辑问题以使其更易于搜索或答案使其更精确和简洁，请随时这样做。

\documentclass{article}
\usepackage{tikz}
\usepackage{pgf}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\usetikzlibrary{hobby}
\usetikzlibrary{decorations.pathreplacing, calligraphy}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usepackage{xcolor}
\begin{document}
    \begin{figure}
        \pgfarrowsdeclaredouble{<<s}{>>s}{stealth}{stealth}
        \begin{tikzpicture}[rotate=-60, declare function={a=3.5; b=-a; m=0; f(\x)=\x*(sin (\x r)); g(\x)=\x*(cos(\x r));}]
            \draw [decoration={markings, mark=at position 0.5 with {\arrow{>>}}}, postaction={decorate}] (0,-2) -- (0,2);
            \draw [rotate=60] ({f(-a)-3},-2.5) -- ({f(b)+5},-2.5);
            \draw [smooth, domain=a:-a, samples=50, decoration={markings, mark=at position 0.15 with {\arrow{>}}, mark=at position 0.85 with {\arrow{>}}}, postaction={decorate}] plot({f(\x)},{g(\x)});
            \draw [densely dashed, red, thick, decoration={markings, mark=at position 0.5 with {\arrow{>>}}}, postaction={decorate}] ({f(a)},{g(a)-0.5}) -- ({f(b)},{g(b)+0.5});
            \draw ({f(a)},{g(a)}) node[above left] {\scriptsize$ (f(a),g(a)) $};
            \draw ({f(b)},{g(b)}) node[above left] {\scriptsize$ (f(b),g(b)) $};
            \draw (0,0) node[above=-1pt, rotate=30] {\scriptsize$ \frac{g'(\xi)}{f'(\xi)}=\frac{g(b)-g(a)}{f(b)-f(a)} $};
        \end{tikzpicture}
        \caption{Cauchy's MVT}
    \end{figure}
\end{document}

在此处输入图片描述

答案2

我离开了几天，但我有这个：

\documentclass[border=3.14159mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}

\begin{document}

    \begin{tikzpicture}[x=0.7cm,y=1cm]
        \tikzset{arrowsoncurve/.style={
            decoration={markings,
                mark=at position 0.15 with {\arrow{>}},
                mark=at position 0.85 with {\arrow{>}}},
                postaction={decorate}}}
                
        \draw[line width=1pt,->] (-10,0) -- (10,0);
        \draw[line width=1pt,->] (0,-5) -- (0,5);
      
        \draw[line width=1pt,blue,arrowsoncurve] plot[samples=100,domain=-2.815:2.22,smooth,variable=\t] ({\t*\t*\t-4*\t+4},{\t*\t+\t-3.12});
        \draw[line width=1pt,red] plot[samples=100,domain=-10:10,smooth] (\x,0.154*\x+3.08);
        \draw[line width=1pt,red] plot[samples=100,domain=-2:10,smooth] (\x,0.154*\x-4.33);
    
        \coordinate (A) at (-7,2);
        \node [above left] at (A) {$A\left(f(a)\,,\,g(a)\right)$};
        \coordinate (B) at (6,4);
        \node [above left] at (B) {$B\left(f(b)\,,\,g(b)\right)$};
        \coordinate (M) at (6.4,-3.35);
        \node [below right] at (M) {$t \in \,]a\,,\,b[$};
        
        \path (A) -- (B) node[red,pos=0.3,above,sloped] {slope $= \frac{g(b)-g(a)}{f(b)-f(a)}$};
        \path (0,-4.33) -- (M) node[red,pos=0.4,below,sloped] {slope $= \frac{g(b)-g(a)}{f(b)-f(a)}$};
        \foreach \p in {A,B,M} \fill (\p) circle (2pt);  
    \end{tikzpicture}

\end{document}

柯西MVT

相关内容