# Tikz and Secant Line diagram

Hi I am looking for feedback to improve an existing program PLUS advice for a desired diagram in the same direction.

Here is my minimal example:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing}

\begin{document}
\begin{center}
\begin{tikzpicture}[scale=1.75,cap=round]
\tikzset{axes/.style={}}
%\draw[style=help lines,step=1cm, dotted] (-5.25,-5.25) grid (5.25,5.25);
% The graphic
\begin{scope}[style=axes]
\draw[->] (-.5,0) -- (4.5,0) node[below] {$x$};
\draw[->] (0,-.5)-- (0,3) node[left] {$y$};
\foreach \x/\xtext in {1.5/x_{1}, 3/x_{2}}
\draw[xshift=\x cm] (0pt,2pt) -- (0pt,-2pt)
node[below,fill=white,font=\normalsize]
{$\xtext$};
\foreach \y/\ytext in {1/y_{1}=f(x_{1}), 2.125/y_{1}=f(x_{2})}
\draw[yshift=\y cm] (2pt,0pt) -- (-2pt,0pt)
node[left,fill=white,font=\normalsize]
{$\ytext$};
%%%
\draw[domain=.5:3.25,smooth,variable=\x,red,<->,thick] plot ({\x},{.5*(\x-1.5)*(\x-1.5)+1});
%%%
\filldraw[black] (1.5,1) circle (1pt) node[above] {\scriptsize $P$};
\filldraw[black] (3,2.125) circle (1pt) node[left] {\scriptsize $Q$};
\draw[thick,blue!50,shorten >=-.5cm,shorten <=-.5cm] (1.5,1)--(3,2.125)
node[midway,left] {\scriptsize Secant Line};
%%%
\draw[blue!50,thick,dashed] (1.5,1)--(3,1)--(3,2.125);
\draw[blue!50] (3,1.1)--(2.9,1.1)--(2.9,1);
\draw[decoration={brace,mirror,raise=5pt},decorate,blue!50]
(1.5,-.250) -- node[below=6pt] {$x_{2}-x_{1}$} (3,-.250);
\draw[decoration={brace,mirror, raise=5pt},decorate,blue!50]
(3,1) -- node[right=6pt] {$f(x_{2})-f(x_{1})$} (3,2.215);
%%%
\filldraw[black] (1.5,1) circle (1pt) node[above] {\scriptsize $P$};
\filldraw[black] (3,2.125) circle (1pt) node[left] {\scriptsize $Q$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{document}


This will Output

I am trying to go here with the picture:

This is a bit beyond my programming skills I think ? PLease all suggestions welcome

With decorations.markings you can mark coordinates along the path, which then allow you to draw tangents. Note that drawing tangents has already been discussed at length in this nice answer, and I am implicitly using the same approach. However, my code is an attempt to have a unified treatment of both of your requests, i.e. tangent and secants, so at first sight it looks quite different.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{decorations.pathreplacing,decorations.markings,calc,arrows.meta,bending}

\begin{document}
\begin{tikzpicture}[scale=2.5,cap=round,mark pos/.style args={#1/#2}{%
postaction={decorate,decoration={markings,%
mark=at position #1 with {
\coordinate (#2);}}}}]
\tikzset{axes/.style={}}
%\draw[style=help lines,step=1cm, dotted] (-5.25,-5.25) grid (5.25,5.25);
% The graphic
\begin{scope}[style=axes]
%%%
\pgfmathsetmacro{\posP}{0.38}
\draw[red,{Latex[bend]}-{Latex[bend]},thick,mark
pos/.list={\posP-0.005/p-0,\posP/P,\posP+0.005/p-2,0.5/q-4,0.62/q-3,0.74/q-2,0.86/q-1}] plot[domain=.5:3.25,samples=101,variable=\x] ({\x},{.5*(\x-1.5)*(\x-1.5)+1});
\draw[red] let \p1=($(p-2)-(p-0)$),\n1={(\y1/\x1)*(1cm/1pt)}
in ($(P)-1*(1,\n1)$) -- ($(P)+2*(1,\n1)$) node[right,anchor=north
west,font=\scriptsize,text width=1cm]{slope $m$ $=$ instaneous rate \dots};
\fill (P) circle (1pt) node[above,font=\scriptsize] {$P$};
\foreach \X in {1,...,4}
{\fill (q-\X) circle (1pt) node[below right,font=\scriptsize] {$Q_\X$};
\path (P) -- (q-\X) coordinate[pos=-0.5] (L-\X) coordinate[pos={1.2+\X*0.3}] (R-\X);
\draw[cyan,dashed] (L-\X) -- (R-\X) node[right,font=\scriptsize] (m\X) {slope $m_\X$}; }
\draw[line width=2mm,-{Latex[bend]},red!20] ($(m1)+(0.5,0.1)$)
to[out=-90,in=65] ++ (-0.2,-1.2);
%%%
%%%
\end{scope}
\end{tikzpicture}
\end{document}


• No Marmot, you can take them out. I am reviewing the out put . It is hard to read since the points are cluttered but feel free to change these. There is no hurry. I also would not know how to make the red arrow indicating the secants approach the tangent. – MathScholar Nov 18 '18 at 19:12
• In short, make as many changes to the original as you require – MathScholar Nov 18 '18 at 19:16
• @MathScholar Well, I could do that, but this would require me knowing what the target output is. Plus I do not know what " I also would not know how to make the red arrow indicating the secants approach the tangent" means. (Remember, I am just a simple marmot. ;-) – marmot Nov 18 '18 at 19:21
• the target output is exactly the image(picture) above. I just provided a minimal example which I thought could lead there. Your welcome to change as much as you need I appreciate you contribution. – MathScholar Nov 18 '18 at 19:28
• @MathScholar Is that closer now? – marmot Nov 18 '18 at 19:50

I refactored the yesterday answer and added some new features.

\documentclass[pstricks,border=12pt,12pt]{standalone}

\def\f(#1){((#1+3)/3+sin(#1+3))}
\def\fp(#1){Derive(1,\f(#1))}
\psset{unit=2}

\begin{document}
\multido{\r=2.0+-.1}{19}{%
\begin{pspicture}[algebraic](-1.6,-.6)(4.4,3.4)
\psaxes[ticks=none,labels=none]{->}(0,0)(-1.6,-.6)(4.1,3.1)[$x$,0][$y$,90]
\psplot[linecolor=red,linewidth=2pt]{-1}{3.9}{\f(x)}
%
\psplotTangent[linecolor=blue]{1.6}{1}{\f(x)}
\psplotTangent[linecolor=cyan,Derive={-1/\fp(x)}]{1.6}{.5}{\f(x)}
%
\pstGeonode[PosAngle={135,90}]
(*1.6 {\f(x)}){A}
(*{1.6 \r\space add} {\f(x)}){B}
\pstGeonode[PosAngle={-120,-60},PointName={x_1,x_2},PointNameSep=8pt]
(A|0,0){x1}
(B|0,0){x2}
\pstGeonode[PosAngle={210,150},PointName={f(x_1),f(x_2)},PointNameSep=20pt]
(0,0|A){fx1}
(0,0|B){fx2}
\pcline[nodesep=-.5,linecolor=green](A)(B)
%
\psset{linestyle=dashed}
\psCoordinates(A)
\psCoordinates(B)
%
\psset{linecolor=gray,linestyle=dashed,labelsep=4pt,arrows=|*-|*,offset=-16pt}
\pcline(x1)(x2)
\nbput{$x_2-x_1$}
\pcline(fx2)(fx1)
\nbput{$f(x_2)-f(x_1)$}
\end{pspicture}}
\end{document}


Secant, tangent, and normal lines are given free of charge!

• Hey I like it but need the program in Tikz. Thanks for sharing – MathScholar Nov 18 '18 at 19:02
• I can show the tangent but this space is too narrow to contain. – Artificial Odorless Armpit Nov 18 '18 at 19:12
• You can change the original program to allow for your space. Any response is appreciated – MathScholar Nov 18 '18 at 19:13
• Nice animation (+1) – marmot Nov 18 '18 at 19:53
• I really like this animation and will try this tomorrow with TiKz – MathScholar Nov 19 '18 at 1:50

I see that @marmot has already given you the solution. This is just another way of doing it. Just an attempt to do it without using any extra libraries.

\documentclass[border=1cm]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[declare function={func(\y) = 0.1*(\y-5)*(\y-5)+1;}]
\draw[domain=2:15,smooth,variable=\x,thick] plot ({\x},{func(\x)});
\draw[fill] (6.4,{func(6.4)})node[below]{p}circle (2pt)coordinate(p);
\foreach[count=\i] \x in {8.0,9.6,...,14.4}{
\draw[fill] (\x,{0.1*(\x-5)*(\x-5)+1})node[below]{Q$_\i$} circle (2pt)coordinate(Q\i);
\draw[thick,blue!80,dashed,shorten >=-2cm,shorten <=-2cm] (p) -- (Q\i)node[right=0.7cm](m\i){slope m$_\i$};
}
\draw[thick,red!70,shorten >=-9cm,shorten <=-4cm] (p) -- (6.401,{func(6.401)});
\draw[-latex,line width=4mm,red!20] (m4.south east) to[out=-100, in=25] (m2.south east)node[below,anchor=north west,red]{slope $m=\ldots$};
\end{tikzpicture}
\end{document}

• Yes, this looks very good to me! (One reason why I did not go that way is that one may not necessarily plot a known function, but just draw some curve by other means, e.g. with to[out=...,in=...] or .. (...) and (...) ... But as long as you do not go that way, this a very nice and compact way of achieving this.) – marmot Nov 18 '18 at 21:32
• @nidhin Thank you for this. The program has its own merits as well. Thanks for sharing – MathScholar Nov 19 '18 at 1:22