Function sign table, complete with first and second derivative, plus concavity/convexity and asymptotic behaviour - a better way?

Question

There has got to be a better/easier way to do this!! And by the way, how does one format the code [when posting a question] to look nice, compact, and full of colours?

\documentclass{article}

\usepackage[2.5cm]{geometry}% http://ctan.org/pkg/geometry
\usepackage{amsmath}% http://ctan.org/pkg/amsmath

\begin{document}
\begin{tabular}{ l || l c c c c c c c r |}  
  $x$ \vspace{1pt}  
  & $ -\, \infty $  
  & \hspace*{-0.5cm}   
  & \hspace*{-0.5cm} $ -\, 1 $  
  & \hspace*{-0.5cm}  
  & \hspace*{-0.7cm} $0$  
  & \hspace*{-1.0cm}  
  & \hspace*{-0.5cm} $+\, 1$  
  & \hspace*{-0.5cm}  
  & \hspace*{-0.5cm} $ +\, \infty $ \\  
\hline

  \parbox[t][1.0cm]{2.0cm}{\vspace{1pt} $y' \,=\, \dfrac{\,x^{2} \,-\, 1\,}{x^{2}}$ \vspace{1pt}}  
  & $\begin{array}{c} \\ + \quad +\\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ + \quad + \quad +  \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ 0 \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ - \quad - \quad - \\ \end{array}$  
  & \hspace*{-0.7cm} $\begin{array}{c} \\ \vert \\ \end{array}$  
  & \hspace*{-1.0cm} $\begin{array}{c} \\ - \quad - \quad - \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ 0 \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ + \quad + \quad + \\ \end{array}$  
  & \hspace*{-1.0cm} $\begin{array}{c} \\ + \\ \end{array}$ \\  
\hline 

  \parbox[t][1.0cm]{2.0cm}{\vspace{1pt} $y'' \,=\, \dfrac{2}{\,x^{3}\,}$ \vspace{1pt}}  
  & $\begin{array}{c} \\ - \quad - \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ - \quad - \quad - \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ - \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ - \quad - \quad - \\ \end{array}$  
  & \hspace*{-0.7cm} $\begin{array}{c} \\ \vert \\ \end{array}$  
  & \hspace*{-1.0cm} $\begin{array}{c} \\ + \quad + \quad + \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ + \\ \end{array}$  
  & \hspace*{-0.5cm} $\begin{array}{c} \\ + \quad + \quad + \\ \end{array}$  
  & \hspace*{-1.0cm} $\begin{array}{c} \\ + \\ \end{array}$ \\  
\hline 

  \parbox[t][1.0cm]{2.0cm}{ $y \,=\, x \,+\, \dfrac{1}{\,x\,}$ }  
  & $-\, \infty \;\; \nearrow$  
  & \hspace*{-0.5cm} $ \begin{array}{c} \\ \nearrow \quad \nearrow \quad \nearrow \\ \bigcap \end{array} $  
  & \hspace*{-0.5cm} $ -\, 2 $  
  & \hspace*{-0.5cm} $ \begin{array}{c} \\ \searrow \quad \searrow \quad \searrow \\ \bigcap \end{array} $  
  & \hspace*{-0.7cm} $ \;\genfrac{}{}{0.0pt}{}{}{}_{-\,\infty} {\Big |}^{+\, \infty}\; $  
  & \hspace*{-0.7cm} $ \begin{array}{c} \\ \searrow \quad \searrow \quad \searrow \\ \bigcup \end{array} $  
  & \hspace*{-0.5cm} $ +\, 2 $  
  & \hspace*{-0.5cm} $ \begin{array}{c} \\ \nearrow \quad \nearrow \quad \nearrow \\ \bigcup \end{array} \vspace{2pt} $  
  & \hspace*{-1.0cm} $+\, \infty$ \\  
\hline  

\end{tabular}\\
\end{document}

Answer 1

In my original comment, I asked if your code came from a WYSIWYG, because it looked as if it had a lot of unnecessary code- typically you get this kind of code when (attempting) to export from another program to TeX.

Anyway, here's my version of your table:

In the code below I have

• deleted all of the \begin{array}{c}...\end{array}
• deleted all of the \hspace*{...}
• loaded the booktabs package which gives lots of useful features for tables, including \toprule, \midrule, and \bottomrule
• loaded the tabularx package to allow the table to stretch to (in this case) the \textwidth
• made the row separation larger by changing \arraystretch

You'll also notice that I have been quite sparing with my +, -, \searrow, and \nearrow. As the examples get more complicated and you need to consider more intervals, you would necessarily need to delete a lot of them anyway.

If you want to see how wide the table is, you can change showframe=true in the loading of the geometry package. If ever you want your table to be wider than the standard textwidth, then you can enclose it in an adjustwidth environment from the changepage package.

\documentclass{article}

\usepackage[showframe=false]{geometry}
\usepackage{booktabs}
\usepackage{amsmath}
\usepackage{tabularx}

% use this package if you want to temporarily 
% widen the page
%\usepackage{changepage}

% use this columntype if you want to RIGHT justify your columns
\newcolumntype{Y}{>{\raggedleft\arraybackslash}X}


\begin{document}

\renewcommand{\arraystretch}{3}

\noindent\begin{tabularx}{\textwidth}{p{2cm}XXXXXXXXX}
 \toprule
    $x$                         &       $-\infty$   &                  &  $-1$ &                   &   $0$                      &                  &   $+1$  &                   &   $\infty$    \\ \midrule
    $y'=\dfrac{x^2-1}{x^2}$     &       $+$         &      $+$         &  $0$  &       $-$         &   $|$                      &   $-$            &   $0$   &      $+$          &   $+$         \\
    $y''=\dfrac{2}{x^3}$        &       $-$         &      $-$         &  $-$  &       $-$         &   $|$                      &   $+$            &   $+$   &      $+$          &   $+$         \\
    $y=x+\dfrac{1}{x}$          &       $-\infty$   & $\nearrow\bigcap$&  $-2$ & $\searrow\bigcap$ &   ${}_{-\infty}|^{+\infty}$&$\searrow\bigcup$ &   $+2$  & $\nearrow\bigcup$ &   $\infty$\\
 \bottomrule
\end{tabularx}



\end{document}

Answer 2

You are probably looking for the tablor package. Its documentation is in French, but it's rich in examples (the example code is available, on TeX Live, by texdoc tablor.html).

There is also a TikZ based package, tkz-tab, also with many example (and French documentation).

Answer 3

Here's an attempt to simplify cmhughes answer further by dropping down to more primitive level (no packages, every column is defined mathmode in the alignment preamble, and a couple of macros to shorten input):

\documentclass{minimal}
\def\myalign#1{% should be used inside displayed equation
  \def\trule{\noalign{\medskip\hrule\medskip}}
  \def\nebc{\nearrow\bigcup}
  \def\sebc{\searrow\bigcup}
  \def\pminf{{}_{-\infty}|^{+\infty}}
  \let\Inf\infty
  % ^make a couple of macros to shorten the input (scoped inside this macro)
  \vbox{% enclose inside a vbox so it will not be broken across pages
  \openup2\jot% increase interlineskips by a multiple of symbolic unit of
              % measure (jot default is 3pt) for consistency
  \mathsurround0pt
  \tabskip1em plus\displaywidth % tabskip glue gets inserted between columns
  % with "optimal" value of 1em, but with stretchability to displaywidth.
  \halign to\displaywidth{% try to stretch the table to accomodate the whole
                          % equation display width
  &$\displaystyle\hfil##\hfil$\crcr
  % ^alignment preamble; the definition gets repeated for every column by the
  % first ampersand. The column content will be centered (\hfil##\hfil)
  #1% user input
  \crcr}}}
\begin{document}
\[
\myalign{
x                   & -\Inf &       & -1 &       & 0      &       & +1 &       & \Inf \cr
\trule
y'={x^2-1\over x^2} & +     & +     & 0  & -     & |      & -     & 0  & +     & + \cr
y''={2\over x^3}    & -     & -     & -  & -     & |      & +     & +  & +     & + \cr
y=x+{1\over x}      & -\Inf & \nebc & -2 & \sebc & \pminf & \sebc & +2 & \nebc & \Inf \cr
}
\]
\end{document}

Caveats:

• I've seen people say primitive use is discouraged (I'm not sure why, though)
• There is surplus tabskip glues at the beginning and end (I don't think it matters much when the column content is centered, though).

Edit: As per comment, to make it look more similar, it'd need the extra rules, left-align the columns and to drop the surplus tabskip glues. But just to make some differences, I chose not to make it look exactly the same:

\documentclass{article}
\def\myalign#1{%
  \def\trule{\noalign{\smallskip\hrule\medskip}}
  \def\nebc{\nearrow\bigcup}
  \def\sebc{\searrow\bigcup}
  \def\pminf{{}_{-\infty}|^{+\infty}}
  \let\Inf\infty
  \def\amp{&}% props to Bruno; I just love this trick
  \vbox{\mathsurround0pt\openup1\jot
    \halign{%
      &$\displaystyle##\hfil\tabskip0pt$&\amp##\tabskip1em\crcr
      \noalign{\hrule height1pt\smallskip}#1\noalign{\smallskip\hrule height1pt}\crcr}}}
\begin{document}
\[
\myalign{
x                   & -\Inf &       & -1 &       & 0      &       & +1 &       & \Inf \cr
\trule
y'={x^2-1\over x^2} & +     & +     & 0  & -     & |      & -     & 0  & +     & + \cr
y''={2\over x^3}    & -     & -     & -  & -     & |      & +     & +  & +     & + \cr
y=x+{1\over x}      & -\Inf & \nebc & -2 & \sebc & \pminf & \sebc & +2 & \nebc & \Inf \cr
}
\]
\end{document}

Which looks like:

It drops the second caveat of the first version, but introduces another: because it is essentially doubling the number of columns (the \amp in the preamble), it affects commands which are used to span multiple columns.

Now, I'd like to remind that to retain document wide consistency you'd want to keep all those skips and rules and whatnots the same. Packages like booktabs help you achieve that easily.