\section{Creating a line of variations: \addbs{tkzTabVar}} \subsection{Definition} \begin{NewMacroBox}{tkzTabVar}{\oarg{local options}\var{el($1$),\dots,el($n$)}} with \tkzname{el($i$) = s($i$) / e($i$)} or \tkzname{el($i$) = s($i$) / eg($i$) / ed($i$)}. \noindent\tkzname{s($i$)} is a sequence of symbols chosen from the table below. \tkzname{eg($i$)} and \tkzname{ed($i$)} are mathematical expressions placed on the left (\emph{gauche}) and on the right (\emph{droite}) of the vertical rules. \tkzname{e($i$)} is an expression centered on a rule. \medskip \begin{tabular}{llrl} \toprule Group 1& \emph{with one sign} &&\\ \midrule \tkzname{s($i$)} & Position of the expressions & \tkzname{el($i$)} & \\ \midrule \IargName{tkzTabVar}{$-~~~$}& single expression, centered, at the bottom eg=ed & $- $&$/e $ \\ \IargName{tkzTabVar}{$+~~~$}& single expression, centered, at the top eg=ed & $+ $&$/e $ \\ \IargName{tkzTabVar}{$~R~~$}& nothing, go to the next expression & $~R $&$(/) $ \\ \IargName{tkzTabVar}{$-C$} & continuous extension at the bottom, centered & $-C $&$/e $ \\ \IargName{tkzTabVar}{$+C$} & continuous extension at the top, centered& $+C $&$/e $ \\ \IargName{tkzTabVar}{$-H$} & expression at the bottom, centered, then forbidden zone & $-H $&$/e $ \\ \IargName{tkzTabVar}{$+H$} & expression at the top, centered, then forbidden zone & $+H $&$/e $ \\ \IargName{tkzTabVar}{$+D~~$}& discontinuity, expression at the top left & $+D $&$/e $ \\ \IargName{tkzTabVar}{$-D~~$}& discontinuity, expression at the bottom left & $-D $&$/e $ \\ \IargName{tkzTabVar}{$~D+~$}& discontinuity, expression at the top right & $D+ $&$/e $ \\ \IargName{tkzTabVar}{$~D-~$}& discontinuity, expression at the bottom right & $D- $&$/e $ \\ \IargName{tkzTabVar}{$+DH$} & discontinuity on the left, at the top, then forbidden zone& $+DH $&$/e $ \\ \IargName{tkzTabVar}{$-DH$} & discontinuity on the left, at the bottom, then forbidden zone & $-DH $&$/e $ \\ \IargName{tkzTabVar}{$+CH$} & continuous extension then forbidden zone & $+CH $&$/e $ \\ \IargName{tkzTabVar}{$-CH$} & same, but expression at the bottom left & $-CH $&$/e $ \\ \midrule Group 2& \emph{with two signs}& &\\ \midrule \IargName{tkzTabVar}{$+D-$} & discontinuity,\hfill two expressions & $+D- $&$/eg/ed$ \\ \IargName{tkzTabVar}{$-D+$} & discontinuity, \dots \hfill that are & $-D+ $&$/eg/ed$ \\ \IargName{tkzTabVar}{$+D+$} & discontinuity, \dots \hfill either on the left or on the right &$+D+ $&$/eg/ed$ \\ \IargName{tkzTabVar}{$-D-$} & discontinuity, \dots \hfill either at the top or at the bottom&$-D- $&$/eg/ed$ \\ \IargName{tkzTabVar}{$+CD+$}& continuous extension on the left and & $+CD+$&$/eg/ed$ \\ \IargName{tkzTabVar}{$-CD-$}& \dots \hfill two expressions that are & $-CD-$&$/eg/ed$ \\ \IargName{tkzTabVar}{$+CD-$}& \dots \hfill either on the left or on the right & $+CD-$&$/eg/ed$ \\ \IargName{tkzTabVar}{$-CD+$}& \dots \hfill either at the top or at the bottom & $-CD+$&$/eg/ed$ \\ \IargName{tkzTabVar}{$+DC+$}& continuous extension on the right and & $+DC+$&$/eg/ed$ \\ \IargName{tkzTabVar}{$-DC-$}& \dots \hfill two expressions that are & $-DC-$&$/eg/ed$ \\ \IargName{tkzTabVar}{$+DC-$}& \dots \hfill either on the left or on the right & $+DC-$&$/eg/ed$ \\ \IargName{tkzTabVar}{$-DC+$}& \dots \hfill either at the top or at the bottom & $-DC+$&$/eg/ed$ \\ \IargName{tkzTabVar}{$+V+$} & like a discontinuity but without double bar, and& $+V+ $&$/eg/ed$ \\ \IargName{tkzTabVar}{$-V-$} & \dots \hfill two expressions that are & $-V- $&$/eg/ed$ \\ \IargName{tkzTabVar}{$+V-$} & \dots \hfill either on the left or on the right & $+V- $&$/eg/ed$ \\ \IargName{tkzTabVar}{$-V+$} & \dots \hfill either at the top or at the bottom & $-V+ $&$/eg/ed$ \\ \midrule \tkzname{\textvisiblespace} & leaves the place empty in some cases& & \\ \bottomrule \end{tabular} \medskip \noindent\emph{The macro \tkzcname{tkzTabVar} takes one argument: a list. This list has \tkzname{$n$} elements, matching the \tkzname{$n$} values of the first row. Each element gives the position of one or two expressions in the row: a sign $+$ (at the top) or a sign $-$ (at the bottom). These expressions are values of the function or limits.} \noindent\emph{The elements \tkzname{el($i$)} have the form:\\ either \mbox{\tkzname{\{ s($i$)/ e($i$)\}}} or \mbox{\tkzname{\{ s($i$)/ e($i$) / \}}}, or \mbox{\tkzname{\{ s($i$)/ eg($i$) / ed($i$)\}}}.} \noindent\emph{The first form is for the symbols that have only one sign \tkzname{$+$} or \tkzname{$-$} and place a single expression; the second is for the symbols that have two signs and place two expressions. The expressions are values taken by the function on the left \tkzname{eg($i$)} or on the right \tkzname{ed($i$)}, or limits; they can also be empty. A sign \tkzname{$+$} or \tkzname{$-$} on the left (resp. on the right) of the symbol corresponds to \tkzname{eg($i$)} (resp. \tkzname{ed($i$)}).} \medskip \begin{tabular}{lllc} \toprule \texttt{options} & \texttt{default} & \texttt{definition} \\ \midrule \IoptName{tkzTabVar}{color} & |black| & color of the arrows \\ \IoptName{tkzTabVar}{help} & |false| & shows the structure of a line of variations \\ \bottomrule \end{tabular} \end{NewMacroBox} \medskip A picture is sometimes better than a long speech\dots \begin{center} \begin{tikzpicture} \tkzTabInit[lgt=2,espcl=3]{$x$/1,$f'(x)$/1,$f(x)$/3}% {$0$,$1$,$2$,$+\infty$}% \tkzTabLine{t,-,d,-,z,+,}% \tkzTabVar{}% \node[below =3pt] (FN12) at (N12){}; \node[below =3pt] (FN22) at (N22){}; \node[below =3pt] (FN32) at (N32){}; \node[below =3pt] (FN42) at (N42){}; \node[above =3pt] (FN13) at (N13){}; \node[above =3pt] (FN23) at (N23){}; \node[above =3pt] (FN33) at (N33){}; \node[above =3pt] (FN43) at (N43){}; \node[below right=3pt] (FRN12) at (N12){}; \node[below right=3pt] (FRN22) at (N22){}; \node[below right=3pt] (FRN32) at (N32){}; \node[below right=3pt] (FRN42) at (N42){}; \node[above right=3pt] (FRN13) at (N13){}; \node[above right=3pt] (FRN23) at (N23){}; \node[above right=3pt] (FRN33) at (N33){}; \node[below left=3pt] (FLN22) at (N22){}; \node[below left=3pt] (FLN32) at (N32){}; \node[below left=3pt] (FLN42) at (N42){}; \node[above left=3pt] (FLN13) at (N13){}; \node[above left=3pt] (FLN23) at (N23){}; \node[above left=3pt] (FLN33) at (N33){}; \node[above left=3pt] (FLN43) at (N43){}; \draw[fill=red!50] (FN12) circle(2pt) node[below=2pt,text=red!50] {\small e} node[left=1cm,red] {$+$}; \draw[fill=red!50] (FN22) circle(2pt); \draw[fill=red!50] (FN32) circle(2pt); \draw[fill=red!50] (FN42) circle(2pt); \draw[fill=red!50] (FN13) circle(2pt) node[left=1cm,red] {$-$}; \draw[fill=red!50] (FN23) circle(2pt); \draw[fill=red!50] (FN33) circle(2pt) node[above=2pt,text=red!50] {\small e}; \draw[fill=red!50] (FN43) circle(2pt); \draw[fill=blue!30] (FRN12) circle(2pt); \draw[fill=blue!30] (FRN22) circle(2pt) node[below right=2pt,text=blue!30] {\small ed}; \draw[fill=blue!30] (FRN32) circle(2pt); \draw[fill=blue!30] (FRN13) circle(2pt); \draw[fill=blue!30] (FRN23) circle(2pt); \draw[fill=blue!30] (FRN33) circle(2pt); \draw[fill=green!50] (FLN22) circle(2pt); \draw[fill=green!50] (FLN32) circle(2pt); \draw[fill=green!50] (FLN42) circle(2pt) node[below left =2pt,text=green!50] {\small eg}; \draw[fill=green!50] (FLN23) circle(2pt) node[above left =2pt,text=green!50] {\small eg}; \draw[fill=green!50] (FLN33) circle(2pt); \draw[fill=green!50] (FLN43) circle(2pt); \end{tikzpicture} \end{center} \medskip For some of the tables, I used the following macros: \begin{tkzexample}[code only] \newcommand*{\va}{\colorbox{red!50} {$\scriptscriptstyle V_a$}} \newcommand*{\vb}{\colorbox{blue!50} {$\scriptscriptstyle V_b$}} \newcommand*{\vbo}{\colorbox{blue!50} {$\scriptscriptstyle V_{b1}$}} \newcommand*{\vbt}{\colorbox{yellow!50}{$\scriptscriptstyle V_{b2}$}} \newcommand*{\vc}{\colorbox{gray!50} {$\scriptscriptstyle V_c$}} \newcommand*{\vd}{\colorbox{magenta!50}{$\scriptscriptstyle V_d$}} \newcommand*{\ve}{\colorbox{orange!50} {$\scriptscriptstyle V_e$}} \end{tkzexample} \medskip \begin{center} \begin{tikzpicture} \tkzTabInit[lgt=2,espcl=3]{$x$/1,$f'(x)$/1,$f(x)$/3}% {$0$,$1$,$2$,$+\infty$}% \tkzTabLine{t,-,d,-,z,+,}% \tkzTabVar{+/\va , -D+/\vb/\vc,-/\vd, +D/\ve}% \end{tikzpicture} \end{center} \begin{tkzexample}[code only] \begin{tikzpicture} \tkzTabInit[lgt=2,espcl=3]{$x$/1,$f'(x)$/1,$f(x)$/3}% {$0$,$1$,$2$,$+\infty$}% \tkzTabLine{t,-,d,-,z,+,}% \tkzTabVar{+/\va , -D+/\vb/\vc,-/\vd, +D/\ve}% \end{tikzpicture} \end{tkzexample} Comments: the signs $+$ and $-$ place an end of the arrow at the top or at the bottom of the row. With a single sign, a single expression is needed. Its position with respect to the column is given by the position of the sign with respect to the other symbols (see \tkzname{$+D$}). \tkzname{$-D+$} needs two expressions. \subsection{Using the symbols} \medskip \bgroup\parindent=0pt \begin{minipage}{7cm} \begin{tkzexample}[code only] {+ /\va , -/\vb } \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {+ /\va , -/\vb } \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {-/\va , +/\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {-/\va , +/\vb} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va , +/\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {+/\va , +/\vb} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {-/\va , -/\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {-/\va , -/\vb} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va , -C / \vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {+/\va , -C / \vb} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {-/\va , +C / \vb }\end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {-/\va , +C / \vb } \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+C / \va , -C / \vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {+C / \va , -C / \vb } \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {-C /\va , +C /\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {-C /\va , +C /\vb} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] { D+ /\va , -/\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% { D+ /\va , -/\vb} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] { D- /\va , +/\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar { D- /\va , +/\vb} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va , -D / \vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {+/\va , -D / \vb} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {-/\va , +D / \vb } \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {-/\va , +D / \vb } \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {D+ / \va , -D / \vb } \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {D+ / \va , -D / \vb } \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {D- /\va , +D /\vb} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1]{ /0.5,/2 }{ a , b } \tkzTabVar% {D- /\va , +D /\vb} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/ \va , -/ \vb , +/ \vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+/ \va , -/ \vb ,+/ \vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/ \va ,-C/ \vb , +/ \vc/ } \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+/ \va ,-C/ \vb , +/ \vc/ } \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {- /\va , R , +/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {- /\va , R, +/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {- /\va , R , +/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2}{ a , b , c } \tkzTabVar% {- /\va , R , +/\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va , +DH/\vbo/ , } \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D-/\va , +DH/\vbo/ , } \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va , -DH/\va/\vb , D+/} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D-/\va , -DH/\vbo , D+/} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va , +D-/\vbo/\vbt , +D/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D-/\va , +D-/\vbo/\vbt , +D/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va , +D-/\vbo/\vbt , +D/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D-/\va , -D-/\vbo/\vbt , +D/\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va , -D- / \vbo/\vbt , +/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+/ \va , -D- /\vbo/\vbt,+/\vc } \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {+ /\va,-DC- /\vbo/\vbt,+ /\vc} \end{tkzexample} \begin{tikzpicture}\tikzset{low/.style = {above = 15pt}} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+ /\va ,-DC- /\vbo/\vbt ,+ /\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va, +DC-/\vbo/\vbt, +D/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D-/\va , +DC-/\vbo/\vbt ,+D/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {D+/\va , +DC-/\vbo/\vbt , +D/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D+/\va , +DC-/\vbo/\vbt ,+D/\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va , +CD-/\vbo/\vbt , +D/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D-/\va , +CD-/\vbo/\vbt , +D/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {D-/\va , +CD-/\vbo/\vbt ,+D/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D+/\va , +CD-/\vbo/\vbt , +D/\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va, -DC+ /\vbo/\vbt, - /\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+ /\va ,-DC+ /\vbo/\vbt , -/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {D- /\va, -DC- /\vbo/\vbt,+D/\vc} \end{tkzexample} \begin{tikzpicture}\tikzset{low/.style = {above = 15pt}} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {D- /\va , -DC- /\vbo/\vbt , +D/\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va , -CH /\vbo/\vbt , D+/} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {+/\va , -CH /\vbo/\vbt , D+/} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {+ /\va , -CH/\vb, //} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar% {+ /\va , -CH/\vb, //} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/\va , -V- /\vbo /\vbt, +/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+/\va,-V- /\vbo /\vbt, +/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/ \va ,-V+ / \vbo/ \vbt ,-/ \vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+/ \va ,-V+ / \vbo/ \vbt ,-/ \vc} \end{tikzpicture} \end{minipage} \begin{minipage}{7cm} \begin{tkzexample}[code only] {+/ \va ,+V- /\vbo/ \vbt , -/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {+/ \va ,+V- / \vbo/ \vbt , -/\vc} \end{tikzpicture} \end{minipage} \hfill \begin{minipage}{7cm} \begin{tkzexample}[code only] {-/ \va, +V+ / \vbo/\vbt, -/\vc} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2.5]{ /0.5,/2 }{ a , b , c } \tkzTabVar {-/ \va ,+V+ / \vbo / \vbt, -/\vc} \end{tikzpicture} \end{minipage} \begin{minipage}{16cm} \begin{tkzexample}[code only] {-/ \va ,+H/\vb,-/\vc, +/ \vd} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=3]{ /0.5,/2 }{ a , b , c , d } \tkzTabVar {-/ \va ,+H/\vb,-/\vc, +/ \vd} \end{tikzpicture} \end{minipage} \begin{minipage}{16cm} \begin{tkzexample}[code only] {+/ \va ,-H/\vb,-/\vc, +/ \vd} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=3]{ /0.5,/2 }{ a , b , c , d } \tkzTabVar {+/ \va ,-H/\vb,-/\vc, +/ \vd} \end{tikzpicture} \end{minipage} \begin{minipage}{16cm} \begin{tkzexample}[code only] {-/ \va , R , R , R , +/ \ve} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=3]{ /0.5,/2 }{ a , b , c , d , e} \tkzTabVar {-/ \va ,R,R,R, +/ \ve} \end{tikzpicture} \end{minipage} \begin{minipage}{16cm} \begin{tkzexample}[code only] {-/ \va , +/\vb , -DH/\vc , -/\vd , +/ \ve} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=3]{ /0.5,/2 }{ a , b , c , d , e} \tkzTabVar {-/ \va ,+/\vb ,-DH/\vc,-/\vd, +/ \ve} \end{tikzpicture} \end{minipage} \begin{minipage}{16cm} \begin{tkzexample}[code only] {D-/ \va , +DH/\vb/ , D-/\vc , +/\vd , +D/\ve} \end{tkzexample} \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=3]{ /0.5,/2 }{ a , b , c , d , e} \tkzTabVar {D-/ \va ,+DH/\vb/,D-/\vc,+/\vd, -D/\ve} \end{tikzpicture} \end{minipage} \egroup \medskip Comments \begin{itemize} \item the syntax $s(i)/\ldots/\ldots$ can be used in almost every case, but then the expressions must be placed correctly; \item the empty argument is sometimes used at the end of a row, but then no arrow is drawn; \item $C+$ and $C-$ do not exist: $+C$ and $-C$ are enough because the expressions are centered; \item $D+$ and $D-$ do exist. \end{itemize} \subsection{Using the options} \subsubsection{\texttt{\textcolor{red}{color}}: color of the arrows} The table can be customized with styles. \begin{tkzexample}[vbox, small] \begin{tikzpicture} \tkzTabInit[color,espcl=8]% {$x$ /1,% Sign\\ of $\dfrac{1}{x}$ /1.5, Variations\\ of $\ln$ /1.5}% {$0$,$+\infty$}% \tkzTabLine{d,+,}% \tkzTabVar[color=red]% {D-/ / $-\infty$,+/ $+\infty$ /} \end{tikzpicture} \end{tkzexample} \subsubsection{\texttt{\textcolor{red}{help}}: shows the structure of the table} \Iopt{tkzTabVar}{help} See the section ``Customizing the tables'' (\ref{pers}). \subsection{Using the styles} \subsubsection{Color of a forbidden zone} \Istyle{tkzTabvar}{h style} To color or hatch a zone of the table in another way, change a style. By default, \tkzname{h style} is defined as follows: \begin{tkzexample}[code only] \tikzset{h style/.style = {fill=gray,opacity=0.4}} \end{tkzexample} Another definition could be: \begin{tkzexample}[code only] \tikzset{h style/.style = {fill=red!50}} \end{tkzexample} \begin{tkzexample}[vbox,width=9cm] \begin{tikzpicture} \tikzset{h style/.style = {fill=red!50}} \tkzTabInit[lgt=1,espcl=2]{$x$ /1, $f$ /2}{$0$,$1$,$2$,$3$}% \tkzTabVar{+/ $1$ / , -CH/ $-2$ / , +C/ $5$, -/ $0$ / } \end{tikzpicture} \end{tkzexample} \subsubsection{\texttt{\textcolor{red}{h style}}: hatched forbidden zone} \Istyle{tkzTabVar}{h style} \begin{tkzexample}[code only] \tikzset{h style/.style = {pattern=north west lines}} \end{tkzexample} This code hatches the zone. \begin{tkzexample}[vbox,width=9cm,small] \begin{tikzpicture} \tikzset{h style/.style = {pattern=north west lines}} \tkzTabInit[lgt=1,espcl=2]{$x$ /1, $f$ /2}{$0$,$1$,$2$,$3$}% \tkzTabVar{+/ $1$ / , -CH/ $-2$ / , +C/ $5$, -/ $0$ / } \end{tikzpicture} \end{tkzexample} \subsubsection{\texttt{\textcolor{red}{arrow style}}: style of the arrows} \Istyle{tkzTabVar}{arrow} The style of the arrows is \tkzname{arrow style}; it is defined as follows: \begin{tkzexample}[code only] \tikzset{arrow style/.style = {\cmdTAB@VA@color, ->, > = latex', shorten > = 2pt, shorten < = 2pt}} \end{tkzexample} The arrows stop short of the nodes. Here is a possible change of the style: \begin{tkzexample}[code only] \tikzset{arrow style/.style = {blue, ->, > = latex', shorten > = 6pt, shorten < = 6pt}} \end{tkzexample} The color of the arrows and the gap at their ends are changed. \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tikzset{arrow style/.style = {blue, ->, > = latex', shorten > = 6pt, shorten < = 6pt}} \tkzTabInit[espcl=5]{$x$ /1, $\ln x +1$ /1.5, $x \ln x$ /2}% {$0$ ,$1/\E$ , $+\infty$}% \tkzTabLine{d,-,z,+,} \tkzTabVar% { D+/ / $0$ ,% -/ \colorbox{black}{\textcolor{white}{$\dfrac{-1}{e}$}}/ ,% +/ $+\infty$ / }% \end{tikzpicture} \end{tkzexample} \subsubsection{\texttt{\textcolor{red}{node style}}: style of the nodes} \Istyle{tkzTabVar}{node style} The style of the nodes is \tkzname{node style}; by default it is defined as follows: \begin{tkzexample}[code only] \tikzset{node style/.style = {inner sep = 2pt, outer sep = 2pt, fill = \cmdTAB@tbs@colorT}}\end{tkzexample} To change it while keeping part of this style, you can write: \begin{tkzexample}[code only] \tikzset{node style/.append style = {draw,circle,fill=red!40,opacity=.4}} \end{tkzexample} By default, the nodes are rectangles that are not drawn; here they become disks. \begin{tikzpicture} \tikzset{node style/.append style = {draw,circle,fill=red!40,opacity=.4}} \tkzTabInit[espcl=5]{$x$ /1, $\ln x +1$ /1.5, $x \ln x$ /2}% {$0$ ,$1/\E$ , $+\infty$}% \tkzTabLine{d,-,z,+,} \tkzTabVar% { D+/ / $0$ ,% -/ \colorbox{black}{\textcolor{white}{$\dfrac{-1}{e}$}}/ ,% +/ $+\infty$ / }% \end{tikzpicture} \begin{tkzexample}[code only,small] \begin{tikzpicture} \tikzset{node style/.append style = {draw,circle,fill=red!40,opacity=.4}} \tkzTabInit[espcl=5]{$x$ /1, $\ln x +1$ /1.5, $x \ln x$ /2}% {$0$ ,$1/\E$ , $+\infty$}% \tkzTabLine{d,-,z,+,} \tkzTabVar% { D+/ / $0$ ,% -/ \colorbox{black}{\textcolor{white}{$\dfrac{-1}{e}$}}/ ,% +/ $+\infty$ / }% \end{tikzpicture} \end{tkzexample} \subsection{Some examples} \subsubsection{Reciprocal function} Study of the reciprocal function $i~:~ x \longmapsto \frac{1}{x}$ on $(-\infty, 0) \cup (0, +\infty)$ \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tkzTabInit[lgt=1.5,espcl=6.5]{$x$ /1,$i'(x)$ /1,$i$ /3} {$-\infty$,$0$,$+\infty$}% \tkzTabLine{,-,d,-,} \tkzTabVar{+/ $0$ / ,-D+/ $-\infty$ / $+\infty$ , -/ $0$ /} \end{tikzpicture} \end{tkzexample} \subsubsection{Function with plateaus: the symbol \texttt{\textcolor{red}{R}}} With R, several values can be skipped. \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tkzTabInit[espcl=4]{$x$ /1,$f'(x)$ /1,$f(x)$ /2} {$0$ , $1$ ,$2$, $+\infty$}% \tkzTabLine {d,+ , z,+ , z,+ , } \tkzTabVar{D-/ / $-\infty$,R/ /,R/ /,+/ $+\infty$ /}% \end{tikzpicture} \end{tkzexample} \subsubsection{Forbidden zone} \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2]{$x$ /1, $f$ /2}{$0$,$1$,$2$,$3$}% \tkzTabVar{+/ $1$ / ,-DH/ $-\infty$ / ,D+/ / $+\infty$, -/ $2$ / } \end{tikzpicture} \end{tkzexample} \subsubsection{Forbidden zone and continuous extension} \index{forbidden zone} \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2]{$x$ /1, $f$ /2}{$0$,$1$,$2$,$3$}% \tkzTabVar{+/ $1$ / ,-CH/ $-2$ /, D+/ / $+\infty$,-/ $2$ / } \end{tikzpicture} \end{tkzexample} \subsubsection{Forbidden zone and two continuous extensions} \index{continuous extension} \begin{tkzexample}[vbox] \begin{tikzpicture} \tkzTabInit[lgt=1,espcl=2]{$x$ /1, $f$ /2}{$0$,$1$,$2$,$3$}% \tkzTabVar{+/ $1$ / , -CH/ $-2$ / , +C/ $5$, -/ $0$ / } \end{tikzpicture} \end{tkzexample} \subsubsection{A partly constant function} The option nocadre removes the outer frame. The example also shows that almost anything can be placed in a line of signs. \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tkzTab[nocadre,lgt=3,espcl=4] {$x$ /1, Sign\\ of $f'(x)$ /1.5, Variations\\ of\\ $f$ /2} {$-\infty$, $-2$,$\dfrac{1}{\E}$,$\E$}% {z, <--- 0 --->,d, -, d, \genfrac{}{}{0pt}{0}{\text{sign of}}{ a}, d} {+/ $\dfrac{2}{3}$, +/ $\dfrac{2}{3}$, -D-/ $-\infty$ / $-\infty$,+D/ $+\infty$ } \end{tikzpicture} \end{tkzexample} \subsubsection{Two lines of variations} \begin{tkzexample}[vbox,small] \begin{tikzpicture} \tkzTabInit[espcl=6] {$x$ /1, $f''{x}$ /1,$f'(x)$ /2, $f(x)$ /2}% {$0$ , $1$ , $+\infty$ }% \tkzTabLine{d,+,z,-, }% \tkzTabVar {D-/ /$1$,+/ $\E$ /,-/ $0$ /}% \tkzTabVar {D-/ /$-\infty$ ,R/ $0$ /, +/ $+8$ /} \end{tikzpicture} \end{tkzexample} \endinput