Latex, tkz-tab package. Tables of signs and variations

Introduction

How to draw beautiful tables of signs and variations for math publications using Latex and the powerful tkz-tab package ?

Once we understand the mechanism of the three main macros to know (tkzTabInit, tkzTabLine, tkzTabVar), a table of signs and variations is built in few minutes.

Table skeleton, tkzTabInit

Import the tkz-tab package in the Latex document :

\usepackage{tkz-tab}

To initialize a table, use \tkzTabInit, command to be placed into a tkzpicture environment.

\documentclass[tikz]{standalone}

\usepackage{amsmath,amssymb}
\usepackage{tkz-tab}

\begin{document}

  \begin{tikzpicture}
     
     \tkzTabInit[lgt=3,espcl=1.5]
       {$x$ / 1 , $f'(x)=-\sin(x)$ / 1, $f(x)=cos(x)$ / 2 }
       {$0$, $\dfrac{\pi}{2}$, $\pi$, $\dfrac{3\pi}{2}$, $2\pi$}

  \end{tikzpicture}

\end{document}

The \tkzTabInit command global syntax is :

\tkzTabinit[local options]{e(1)/h(1),...,e(p)/h(p)}{a(1),...,a(n)}

2 mandatory lists and facultative options.

{e(1)/h(1),...,e(p)/h(p)} : left column data. 1 element e(p) \( \implies \) 1 row, so the number of elements define the number of rows of the table. Heights (h(p)) are in centimeters.
{a(1),...,a(n)} : header values. Define the number of columns of the table after the first left column.

In the above example, 2 options applied :

lgt resizes the first column width (in centimeters).
espcl resizes the space between 2 values in the header (in centimeters).

The minimal code for a table tkzTab is the following:

\tkzTabInit{ / 1}{ , }

Use braces to escape "," for numbers :

\tkzTabInit{$x$ / 1, $x^2$ / 1}{ $0$, ${0,5}$, $1$ }

tkz-tab program structure

When the table skeleton is ready using \tkzTabInit, the global program structure is then the following :

\begin{tikzpicture}
  \tkzTabInit{…}{…}
  command for line 1
  command for line 2
  …
\end{tikzpicture}

Commands are (exhaustive list) :

\tkzTabLine : creates a signs line
\tkzTabVar : creates a variations line
\tkzTabIma,\tkzTabVal : adds intermediate values
\tkzTabSlope : adds derivatives values in signs line
\tkzTabTan, \tkzTabTan : adds horizontal tangents

The first 2 commands are the most important.

Signs line, tkzTabLine

The argument of the \tkzTabLine command to create a signs line is a list. For \(n\) values in the header of the table, \tkzTabLine expects a list of \(2n - 1\) elements : \(n\) elements placed below the values and \(n - 1\) elements placed between the values :

  \begin{tikzpicture}
     \tkzTabInit[lgt=3,espcl=1.5]
       {$x$ / 1 , $f'(x)=-\sin(x)$ / 1, $f(x)=cos(x)$ / 2 }
       {$0$, $\dfrac{\pi}{2}$, $\pi$, $\dfrac{3\pi}{2}$, $2\pi$}
     \tkzTabLine
       { , , - , , , , + , , }
  \end{tikzpicture}

Symbols allowed below values are :

z : zero centered on a vertical dotted line
t : vertical dotted line
d : vertical double bar

Notice : they are not allowed between 2 values.

  \begin{tikzpicture}
     \tkzTabInit[lgt=3,espcl=1.5]
       {$x$ / 1 , $f'(x)=-\sin(x)$ / 1, $f(x)=cos(x)$ / 2 }
       {$0$, $\dfrac{\pi}{2}$, $\pi$, $\dfrac{3\pi}{2}$, $2\pi$}
     \tkzTabLine
       {z, , - ,  , z, , + , ,z}
  \end{tikzpicture}

The symbol h draws a forbidden zone (function not defined) : the interval is hatched. Obviously, the symbol h can only be applied for an element in the list between 2 values.

  \begin{tikzpicture}
     \tkzTabInit[lgt=3]
       {$x$ / 1 , $f'(x)$ / 1, $f(x)=\sqrt{\dfrac{x-1}{x+1}}$ / 2 }
       {$-\infty$, $-1$, $1$, $+\infty$}
     \tkzTabLine
       { , + , d , h , d, +, }
  \end{tikzpicture}

Variations line, tkzTabVar

The \tkTabVar macro builds a variation line. The argument is a list like \tkTabLine.

For \(n\) header values, the list will contain \(n\) groups.

\( \pm / e \) to position an element below a value : \(+\) at the top, \(-\) at the bottom of the variation line.
/R if there is nothing to do for a value.

  \begin{tikzpicture}  
     \tkzTabInit[lgt=3]
       {$x$ / 1 , $f'(x)=-\sin(x)$ / 1, $f(x)=cos(x)$ / 2 }
       {$0$, $\dfrac{\pi}{2}$, $\pi$, $\dfrac{3\pi}{2}$, $2\pi$}
     \tkzTabLine
       {z, , - ,  , z, , + , ,z}
     \tkzTabVar
       { +/$1$, R/ , -/$-1$, R/, +/$1$}
  \end{tikzpicture}

About continuity, discontinuity and forbidden zones, symbols are :

Vertical double bar and continuity : -C/x, +C/x
Vertical double bar and discontinuity : -D/x, +D/x, D-/x, D+/x
Forbidden zone : -H/x, +H/x
Combinations: -CH/x, +CH/x, -DH/x, +DH/x

Concrete examples are better than very long explanations about C/D/H symbols :

  \begin{tikzpicture}   
     \tkzTabInit[lgt=3,espcl=1.5]
       {$x$ / 1 , $f'(x)$ / 1, $f(x)=\sqrt{\dfrac{x-1}{x+1}}$ / 2 }
       {$-\infty$, $-1$, $1$, $+\infty$}
     \tkzTabLine
       { , + , d , h , d, +, }
     \tkzTabVar
       { -/1 , +DH/$+\infty$, -C/0, +/1}
  \end{tikzpicture}

  \begin{tikzpicture}  
     \tkzTabInit[lgt=2.75]
       { $x$/1  ,  $f'(x)$/1  ,  $f(x) = tan(x)$/2  }
       { $0$,  $\dfrac{\pi}{2}$,  $\pi$}
     \tkzTabLine
       { , + , d , + , }
     \tkzTabVar
       { -/$0$ ,  +D-/$+\infty$/$-\infty$ , +/$0$}
  \end{tikzpicture}

Intermediate values in variations lines, tkzTabIma, tkzTabVal

tkzTabIma

To add values in arrows of variations for existing header values defined with \tkzTabInit, use \tkzTabIma after \tkzTabVar.

\tkzTabIma {start arrow position}{end arrow position}{header value position}{value to set on line}

  \begin{tikzpicture}     
     \tkzTabInit[lgt=3,espcl=1.5]
       {$x$ / 1 , $f'(x)=-\sin(x)$ / 1, $f(x)=cos(x)$ / 2 }
       {$0$, $\dfrac{\pi}{2}$, $\pi$, $\dfrac{3\pi}{2}$, $2\pi$}
     \tkzTabLine
       {z, , - ,  , z, , + , ,z}
     \tkzTabVar
       { +/$1$, R/ , -/$-1$, R/, +/$1$}
     \tkzTabIma{1}{3}{2}{0}
     \tkzTabIma{3}{5}{4}{0}
  \end{tikzpicture}

tkzTabVal

In a previous section, a table of signs and variations has been built for the function \(f(x) = tan(x)\).

Usually, remarkable values are displayed on variations lines. In the above example, we want to display the value of \(tan(x)\) for \(\dfrac{\pi}{4}\) and \(\dfrac{3\pi}{4}\).

Without redefining \tkzTabInit, \tkzTabLine, \tkzTabVar in order to add \(\dfrac{\pi}{4}\) and \(\dfrac{3\pi}{4}\), values headers and associated values on variation lines can be added using \tkzTabVal. Like \tkzTabIma, \tkzTabVal is called after \tkzTabVar.

\tkzTabVal {start arrow position}{end arrow position}{relative position between start and end [0-1]}{header value}{value to set on line}

  \begin{tikzpicture}
     \tkzTabInit[lgt=2.60]
       { $x$/1  ,  $f'(x)$/1  ,  $f(x) = tan(x)$/2.5  }
       { $0$,  $\dfrac{\pi}{2}$,  $\pi$}
     \tkzTabLine
       { , + , d , + , }
     \tkzTabVar
       { -/$0$ ,  +D-/$+\infty$/$-\infty$ , +/$0$}
     \tkzTabVal {1}{2}{0.5}{$\dfrac{\pi}{4}$}{$1$}
     \tkzTabVal {2}{3}{0.5}{$\dfrac{3\pi}{4}$}{$1$}
  \end{tikzpicture}

The example in this paper which builds the table for the cosinus function in the domain \([0,2\pi]\) is then more easily coded using \tkzTabVal : \(\dfrac{\pi}{2}\) and \(\dfrac{3\pi}{2}\) are indeed no more explicitely defined in \tkzTabInit, \tkzTabLine and \tkzTabVal macros.

  \begin{tikzpicture}
     \tkzTabInit[lgt=3,espcl=2.5]
       {$x$ / 1 , $f'(x)=-\sin(x)$ / 1, $f(x)=cos(x)$ / 2 }
       {$0$, $\pi$, $2\pi$}
     \tkzTabLine
       {z, - , z , + , z}
     \tkzTabVar
       { +/$1$ , -/$-1$ , +/$1$}
     \tkzTabVal {1}{2}{0.5}{$\dfrac{\pi}{2}$}{$0$}
     \tkzTabVal {2}{3}{0.5}{$\dfrac{3\pi}{2}$}{$0$}
  \end{tikzpicture}

In the 2 previous examples, positioning is easy, intermediate values to add are exactly in the middle of the arrow {0.5}.

In the below example, an intermediate value is added on the arrow starting position #1 and ending position #3. The position is qualitatively set to 0.4 to respect the scale.

\begin{tikzpicture}
	\tkzTabInit[espcl=6]
		{$x$/1 , $f'(x)$/1 , $f(x)$/2}
		{$0$ , $\sqrt{e}$ , $+\infty$}
	\tkzTabLine{d,+,0,+,}
	\tkzTabVar{D- / $-\infty$ , R/ , + / $0$ }
	\tkzTabVal[draw]{1}{3}{0.4}{$1$}{$-e$}
\end{tikzpicture}

Notice the draw option added to the tkTabVal macro for ease of reading : a vertical dotted line is displayed between the header value and the intermediate value in the variation line.

Derivatives values, tkzTabSlope

To add derivatives values in the signs line, use tkzTabSlope after tkzTabLine command.

\tkTabSlope { position/left value/right value, position/left value/right value, … }

  \begin{tikzpicture}
     \tkzTabInit[lgt=2.70]
       { $x$/1  ,  $f'(x)=\dfrac{1}{\cos^2x}$/2  ,  $f(x) = tan(x)$/2.5  }
       { $0$,  $\dfrac{\pi}{2}$,  $\pi$}
     \tkzTabLine
       { , + , d , + , }
     \tkzTabSlope { 1//$1$, 2/+\infty/+\infty, 3/1/ }
     \tkzTabVar
       { -/$0$ ,  +D-/$+\infty$/$-\infty$ , +/$0$}
     \tkzTabVal {1}{2}{0.5}{$\dfrac{\pi}{4}$}{$1$}
     \tkzTabVal {2}{3}{0.5}{$\dfrac{3\pi}{4}$}{$1$}
  \end{tikzpicture}

Horizontal Tangents, tkzTabTan

Use tkzTabTan to draw horizontal tangents which show more accurately in the variations extrema, convexity, concavity, inflexion points… :

\tkTabTan { start arrow position, end arrow position , header value position, value to set on tangent }

  \begin{tikzpicture}
     \tkzTabInit[lgt=3]
       {$x$ / 1 , $f'(x)=2x$ / 1, $f(x)=x^2$ / 2 }
       {$-\infty$, $0$, $+\infty$}
     \tkzTabLine
       { , - , z , + , }
     \tkzTabSlope
       { 1//-\infty , 3/+\infty/ }
     \tkzTabVar
       { +/$+\infty$, -/ , +/$+\infty$ }
     \tkzTabTan
       {1}{2}{2}{$0$}
  \end{tikzpicture}

About styles

Default style is usually enough, but it is possible to apply custom styles (colors, background colors,…) on many tables elements using tikzset :

Arrows
Nodes
Double bars
Vertical bars
…

Style options are numerous and it is not the goal of this paper.

In the below example, a custom style is applied for forbidden zones, the default one is somehow "awful".

  \begin{tikzpicture}

     \tikzset{h style/.style = {pattern = north west lines,
		    pattern color = red!30}}
     
     \tkzTabInit[lgt=3]
       {$x$ / 1 , $f'(x)$ / 1, $f(x)=\sqrt{\dfrac{x-1}{x+1}}$ / 2 }
       {$-\infty$, $-1$, $1$, $+\infty$}
     \tkzTabLine
       { , + , d , h , d, +, }
     \tkzTabVar
       { -/1 , +DH/$+\infty$, -C/0, +/1}

  \end{tikzpicture}