tableaux_variations.tex

%&plain
%% ========= quelques macros pour tableaux ==============
%% ========== (preconisees par R. Seroul) ===============
\def \tvi {%            trait vertical invisible
   \vrule height 12pt depth 5pt width 0pt}
\def \tv {%             trait vertical taille reglable
   \tvi \vrule }
\def \hfq {%            hfill quad -> un separateur
   \hfill
   \quad }
\def \cc#1{%            case centree
   \hfq
   #1%
   \hfq }
\def \doublevrule{%
   \vrule \hskip.31em \vrule }
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% quelques macros pour les caracteres 'fleches'
%% fichier : arrows.tex
%% format  : plain.tex
%% auteur  : jean-paul Vignault
%% date    : 1 janvier 2000
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\font \tenline = line10         \font \tenlinew = linew10
 
\newdimen \tmpdimen
 
\newbox \tmpbox
 
 
% ============ Série 6 ==============
 
%% rapport largeur/hauteur = 3/5 = 0.6, dp = 0
 
\def \srightuuuparrow{%
   \hbox{\tenline 
      \kern 0.15999pt
      \char20
      \kern -3.7pt \raise 2.6pt \hbox{\char21}
      \kern 0.1pt
}}
 
\def \rightuuuparrow{%
   \hbox{\tenline 
      \kern 0.15999pt
      \char20
      \raise 10pt \hbox{\char20}
      \kern -3.7pt \raise 12.6pt \hbox{\char21}
      \kern 0.1pt
}}
 
\def \mrightuuuparrow{%
   \hbox{\tenline 
      \kern 0.15999pt
      \char20
      \raise 10pt \hbox{\char20}
      \raise 20pt \hbox{\char20}
      \kern -3.7pt \raise 22.6pt \hbox{\char21}
      \kern 0.1pt
}}
 
\def \brightuuuparrow{%
   \hbox{\tenline 
      \kern 0.15999pt
      \char20
      \raise 10pt \hbox{\char20}
      \raise 20pt \hbox{\char20}
      \raise 30pt \hbox{\char20}
      \kern -3.7pt \raise 32.6pt \hbox{\char21}
      \kern 0.1pt
}}
 
\def \bbrightuuuparrow{%
   \hbox{\tenline 
      \kern 0.15999pt
      \char20
      \raise 10pt \hbox{\char20}
      \raise 20pt \hbox{\char20}
      \raise 30pt \hbox{\char20}
      \raise 40pt \hbox{\char20}
      \kern -3.7pt \raise 42.6pt \hbox{\char21}
      \kern 0.1pt
}}
 
\def \suuup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 12.6pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \uuup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 22.6pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \muuup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 32.6pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \buuup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 42.6pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \bbuuup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 52.6pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \suuucenter#1{%
   \vbox to 12.6pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \uuucenter#1{%
   \vbox to 22.6pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \muuucenter#1{%
   \vbox to 32.6pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \buuucenter#1{%
   \vbox to 42.6pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \bbuuucenter#1{%
   \vbox to 52.6pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
% ============= serie 4 ==============
 
%% La encore, le rapport largeur/hauteur doit etre de 3/5
 
\def \srightdddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 2.6pt \hbox{\char84}
      \kern -4pt \char85
      \kern 0.26pt
}}
 
\def \rightdddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 12.6pt \hbox{\char84}
      \raise 2.6pt \hbox{\char84}
      \kern -4pt \char85
      \kern 0.26pt
}}
 
\def \mrightdddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 22.6pt \hbox{\char84}
      \raise 12.6pt \hbox{\char84}
      \raise 2.6pt \hbox{\char84}
      \kern -4pt \char85
      \kern 0.26pt
}}
 
\def \brightdddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 32.6pt \hbox{\char84}
      \raise 22.6pt \hbox{\char84}
      \raise 12.6pt \hbox{\char84}
      \raise 2.6pt \hbox{\char84}
      \kern -4pt \char85
      \kern 0.26pt
}}
 
\def \bbrightdddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 42.6pt \hbox{\char84}
      \raise 32.6pt \hbox{\char84}
      \raise 22.6pt \hbox{\char84}
      \raise 12.6pt \hbox{\char84}
      \raise 2.6pt \hbox{\char84}
      \kern -4pt \char85
      \kern 0.26pt
}}
 
% ============= serie 5 ==============
 
%% rapport largeur/hauteur = 5/3 
 
%      \kern 0.15999pt
%      \char20
%      \kern -3.7pt \raise 2.6pt \hbox{\char21}
%      \kern 0.1pt
 
\def \srightddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 1.56pt \hbox{\char98}
      \kern -8pt 
      \char126
      \kern 0.3pt
}}
 
\def \rightddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 7.56pt \hbox{\char98}
      \raise 1.56pt \hbox{\char98}
      \kern -8pt 
      \char126
      \kern 0.3pt
}}
 
\def \mrightddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 13.56pt \hbox{\char98}
      \raise 7.56pt \hbox{\char98}
      \raise 1.56pt \hbox{\char98}
      \kern -8pt 
      \char126
      \kern 0.3pt
}}
 
\def \brightddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 19.56pt \hbox{\char98}
      \raise 13.56pt \hbox{\char98}
      \raise 7.56pt \hbox{\char98}
      \raise 1.56pt \hbox{\char98}
      \kern -8pt 
      \char126
      \kern 0.3pt
}}
 
\def \bbrightddownarrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \raise 25.56pt \hbox{\char98}
      \raise 19.56pt \hbox{\char98}
      \raise 13.56pt \hbox{\char98}
      \raise 7.56pt \hbox{\char98}
      \raise 1.56pt \hbox{\char98}
      \kern -8pt 
      \char126
      \kern 0.3pt
}}
 
% ============= serie 2 ==============
 
%% rapport largeur/hauteur = 5/3
 
\def \srightuuparrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \char34
      \kern -8pt 
      \raise 0.06pt \hbox{\char62}
      \kern 0.3pt
   }}
 
\def \rightuuparrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \char34
      \raise 6pt \hbox{\char34}
      \kern -8pt 
      \raise 6.06pt \hbox{\char62}
      \kern 0.3pt
   }}
 
\def \mrightuuparrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \char34
      \raise 6pt \hbox{\char34}
      \raise 12pt \hbox{\char34}
      \kern -8pt 
      \raise 12.06pt \hbox{\char62}
      \kern 0.3pt
   }}
 
\def \brightuuparrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \char34
      \raise 6pt \hbox{\char34}
      \raise 12pt \hbox{\char34}
      \raise 18pt \hbox{\char34}
      \kern -8pt 
      \raise 18.06pt \hbox{\char62}
      \kern 0.3pt
   }}
 
\def \bbrightuuparrow{%
   \hbox{\tenline 
      \kern 0.3pt
      \char34
      \raise 6pt \hbox{\char34}
      \raise 12pt \hbox{\char34}
      \raise 18pt \hbox{\char34}
      \raise 24pt \hbox{\char34}
      \kern -8pt 
      \raise 24.06pt \hbox{\char62}
      \kern 0.3pt
   }}
 
\def \suucenter#1{%
   \vbox to 7.56pt{%
      \vfil
      \hbox{#1} 
      \vfil
   }}
 
\def \uucenter#1{%
   \vbox to 13.56pt{%
      \vfil
      \hbox{#1} 
      \vfil
   }}
 
\def \muucenter#1{%
   \vbox to 19.56pt{%
      \vfil
      \hbox{#1} 
      \vfil
   }}
 
\def \buucenter#1{%
   \vbox to 25.56pt{%
      \vfil
      \hbox{#1} 
      \vfil
   }}
 
\def \bbuucenter#1{%
   \vbox to 31.56pt{%
      \vfil
      \hbox{#1} 
      \vfil
   }}
 
\def \suup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 7.56pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \uup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 13.56pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \muup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 19.56pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \buup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 25.56pt \hbox{\tvi height \tmpdimen #1}
}
 
\def \bbuup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 31.56pt \hbox{\tvi height \tmpdimen #1}
}
 
% ============= serie 3 ==============
 
% rapport largeur/hauteur = 4
 
\def \srightdownarrow{%
   \hbox{\tenline 
      \kern 1pt 
      \raise 1.5pt \hbox{\char88}
      \kern -6pt 
      \char113
      \kern 1pt 
}}
 
\def \rightdownarrow{%
   \hbox{\tenline 
      \kern 1pt 
      \raise 4pt\hbox{\char88}
      \raise 1.5pt \hbox{\char88}
      \kern -6pt 
      \char113
      \kern 1pt 
}}
 
\def \mrightdownarrow{%
   \hbox{\tenline 
      \kern 1pt 
      \raise 6.5pt\hbox{\char88}
      \raise 4pt\hbox{\char88}
      \raise 1.5pt \hbox{\char88}
      \kern -6pt 
      \char113
      \kern 1pt 
   }}
 
\def \brightdownarrow{%
   \hbox{\tenline 
      \kern 1pt 
      \raise 9pt\hbox{\char88}
      \raise 6.5pt\hbox{\char88}
      \raise 4pt\hbox{\char88}
      \raise 1.5pt \hbox{\char88}
      \kern -6pt 
      \char113
      \kern 1pt 
   }}
 
\def \bbrightdownarrow{%
   \hbox{\tenline 
      \kern 1pt 
      \raise 11.5pt\hbox{\char88}
      \raise 9pt\hbox{\char88}
      \raise 6.5pt\hbox{\char88}
      \raise 4pt\hbox{\char88}
      \raise 1.5pt \hbox{\char88}
      \kern -6pt 
      \char113
      \kern 1pt 
   }}
 
\def \down#1{%
   \lower 2pt \hbox{#1}}
 
\def \bbucenter#1{%
   \vbox to 14pt{%
      \vfil
      \hbox{#1} 
      \vfil
   }}
 
\def \bbup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 14pt \hbox{\tvi height \tmpdimen #1}
}
 
% ============= serie 1 ==============
 
% rapport largeur/hauteur = 4
 
\def \srightuparrow{%
   \raise 0.2pt \hbox{\tenline 
      \kern 1pt
      \char24
      \kern -6pt
      \raise .46667pt \hbox{\char49}
      \kern 1pt
   }}
 
\def \rightuparrow{%
   \raise 0.2pt \hbox{\tenline 
      \kern 1pt
      \char24
      \raise 2.5pt \hbox{\char24}
      \kern -6pt
      \raise 2.96667pt \hbox{\char49}
      \kern 1pt
   }}
 
\def \mrightuparrow{%
   \raise 0.2pt \hbox{\tenline 
      \kern 1pt
      \char24
      \raise 2.5pt \hbox{\char24}
      \raise 5pt \hbox{\char24}
      \kern -6pt
      \raise 5.46667pt \hbox{\char49}
      \kern 1pt
   }}
 
\def \brightuparrow{%
   \raise 0.2pt \hbox{\tenline 
      \kern 1pt
      \char24
      \raise 2.5pt \hbox{\char24}
      \raise 5pt \hbox{\char24}
      \raise 7.5pt \hbox{\char24}
      \kern -6pt
      \raise 7.96667pt \hbox{\char49}
      \kern 1pt
   }}
 
\def \bbrightuparrow{%
   \raise 0.2pt \hbox{\tenline 
      \kern 1pt
      \char24
      \raise 2.5pt \hbox{\char24}
      \raise 5pt \hbox{\char24}
      \raise 7.5pt \hbox{\char24}
      \raise 10pt \hbox{\char24}
      \kern -6pt
      \raise 10.46667pt \hbox{\char49}
      \kern 1pt
   }}
 
\def \sucenter#1{%
   \vbox to 4pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \mucenter#1{%
   \vbox to 6.5pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \ucenter#1{%
   \vbox to 9pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \bucenter#1{%
   \vbox to 11.5pt{%
         \vfil
         \hbox{#1} 
         \vfil
   }}
 
\def \sup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 4pt \hbox{\tvi height \tmpdimen #1}}
 
\def \up#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 6.5pt \hbox{\tvi height \tmpdimen #1}}
 
\def \mup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 9pt \hbox{\tvi height \tmpdimen #1}}
 
\def \bup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 11.5pt \hbox{\tvi height \tmpdimen #1}}
 
\def \bbup#1{%
   \setbox \tmpbox \hbox{#1}
   \tmpdimen = \ht \tmpbox
   \advance \tmpdimen by 2pt
   \raise 14pt \hbox{\tvi height \tmpdimen #1}}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%            exemples d'utilisation                  %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\newpage{\vfill\eject}
\nopagenumbers
 
%%%%%% exemple 1 %%%%%
 
$$\vcenter {\offinterlineskip 
   \def \cc#1{ \hfil #1 \hfil }
   \halign {
   % preamble
      \cc {$#$}& #\tv && \cc {$#$}
   \cr
      x&& 0&& 2&& +\infty
   \cr
   \noalign {\hrule height 1pt }
      x-2&& & -& 0& +
   \cr
   \noalign {\hrule }
      x&& 0& +& \tv & +
   \cr
   \noalign {\hrule height 1pt}
      f' (x)&& \doublevrule & -& 0& +
   \cr
   \noalign {\hrule height 1pt}
      \buucenter {$f (x)$}&& 
      \doublevrule &
      \brightddownarrow & \down{$\matrix {1-4\ln 2 \cr \approx -1, 77\cr }$}& 
      \brightuuparrow & \buup {\phantom {1}}
   \cr
}}
$$
 
\newpage
%%%%%% exemple 2 %%%%%
 
$$\vcenter {\offinterlineskip
   \def \cc#1{ \hfil #1 \hfil }
   \halign {
   % preamble
      \cc {$#$}& #\tv && \cc {$#$}
   \cr
      x&& -\infty && 3/7
   \cr
   \noalign {\hrule height 1pt }
      3-7x&& & +& 0
   \cr
   \noalign {\hrule }
      -7&& & -
   \cr
   \noalign {\hrule height 1pt}
      f' (x)&& & -& \doublevrule 
   \cr
   \noalign {\hrule height 1pt}
      \bucenter {$f (x)$}&&
      \bup {\phantom {1}} &
      \brightdownarrow & \doublevrule
   \cr
}}
$$
 
\newpage
%%%%%% exemple 3 %%%%%
 
$$\vcenter {\offinterlineskip
   \def \cc#1{ \hfil #1 \hfil }
   \halign {
   % preamble
      \cc {$#$}& #\tv && \cc {$#$}
   \cr
      x&& 0&&& e^{1/2}&& +\infty
   \cr
   \noalign {\hrule height 1pt }
      1 - 2\ln x&& && +& 0& -
   \cr
   \noalign {\hrule }
      x^2&& 0&& +& \tv & +
   \cr
   \noalign {\hrule height 1pt}
      f' (x)&& \doublevrule && +& 0& -
   \cr
   \noalign {\hrule height 1pt}
      \buucenter {$f (x)$}&&
      \doublevrule &
      \down {$-\infty $}&
      \brightuuparrow & \buup {$2(e^{-1/2}-1) \approx -0, 79$}& 
      \brightddownarrow & \down {$-2$}
   \cr
}}
$$
 
\newpage
%%%%%% exemple 4 %%%%%
 
$$\vcenter {\offinterlineskip
   \def \cc#1{ \hfil #1 \hfil }
   \halign {
   % preamble
      \cc {$#$}& #\tv && \cc {$#$}
   \cr
      x&& -\infty && 0&& +\infty
   \cr
   \noalign {\hrule height 1pt }
      f' (x)&& &-& 0& +
   \cr
   \noalign {\hrule height 1pt}
      \buucenter {$f (x)$}&& &
      \brightddownarrow & \down{$0$}& 
      \brightuuparrow & \buup {\phantom {1}}
   \cr
}}
$$
 
\newpage
%%%%%% exemple 5 %%%%%
 
$$\vcenter {\offinterlineskip
   \def \cc#1{ \hfil #1 \hfil }
   \halign {
   % preamble
      \cc {$#$}& #\tv && \cc {$#$}
   \cr
      x&& -\infty && 0&& +\infty
   \cr
   \noalign {\hrule height 1pt }
      f' (x)&& &+& 0& +
   \cr
   \noalign {\hrule height 1pt}
      \buucenter {$f (x)$}&& &
      \brightuparrow & \buucenter{$2$}& 
      \bup {\brightuparrow }& \buup {\phantom {1}}
   \cr
}}
$$
 
\newpage
%%%%%% exemple 6 %%%%%
 
$$\vcenter {\offinterlineskip
   \def \cc#1{ \hfil #1 \hfil }
   \halign {
   % preamble
      \cc {$#$}& #\tv && \cc {$#$}
   \cr
      x&& -3&& -1&& 1&& 3&
   \cr
   \noalign {\hrule height 1pt}
      \buucenter {$f (x)$}&& \down {?}&
      \brightuuparrow & \buup {$3$}&
      \brightddownarrow & \down{$-1$}& 
      \brightuuparrow & \buup {?}
   \cr
}}
$$
 
\newpage
%%%%%% exemple 7 %%%%%
 
$$
\vcenter{
   \def \hfq{\hfil \ }
   \offinterlineskip
   \halign{
   % preamble
      &\hfq #\hfq
   \cr
      $t$& \vrule depth 5pt & 
      $-1$&& $0$&& $1$&& $2$&
   \cr
   \noalign{\hrule}
      $x' (t)$& \vrule height 10pt depth 3pt &
      $-2$& $-$& $0$& $+$& $2$& $+$& $4$&
   \cr
   \noalign{\hrule}
      \bbuucenter{$x (t)$}& \vrule& 
      \bbup{$2$}&
      \brightdownarrow & \down{$1\strut $}& 
      \brightuparrow & \bup{$2$}& 
      \bup{\brightuparrow} & \bup{\bup{$5$}}
   \cr
   \noalign{\hrule}
      $y' (t)$& \vrule height 10pt depth 3pt &
      $-4$& $-$& $-2$& $-$& $0$& $+$& $2$&
   \cr
   \noalign{\hrule}
      \bbuucenter{$x (t)$}& \vrule& 
      \bbuup{$5$}&
      \bbup{\brightdownarrow} & \bup{$2$}& 
      \brightdownarrow & \down{$1$}& 
      \bbrightuparrow & \bbup{$2$}
   \cr
}}
$$
 
\newpage
%%%%%% exemple 8 %%%%%
 
$$
\vcenter {
   \def \hfq {\hfil \ }
   \offinterlineskip
   \halign {
   % preamble
      &\hfq #\hfq
   \cr
      $t$& \vrule depth 5pt & 
      $-\infty $&& $-7/3$&& $0$&& $1$&& $+\infty $&
   \cr
   \noalign{\hrule}
      $x' (t)$& \vrule height 10pt depth 3pt &
      & $-$& $\approx -9, 33$& $-$& $0$& $+$& $1$& $+$&&
   \cr
   \noalign{\hrule}
      \bbuucenter{$x (t)$}& \vrule& 
      \bbuup{$+\infty $}&
      \bbup{\brightdownarrow} & \bup{$\approx 14, 9$}& 
      \brightdownarrow & \down{$4\strut $}& 
      \bbrightuparrow & \bbup{$6$}& \bbup {\bbrightuparrow }& \bbup {\bbup {$+\infty $}}
   \cr
   \noalign{\hrule}
      $y' (t)$& \vrule height 10pt depth 3pt &
      & $+$& $0$& $-$& $-7$& $-$& $0$& $+$&&
   \cr
   \noalign{\hrule}
      \bbuucenter{$y (t)$}& \vrule& 
      \down {$-\infty $}&
      \brightuuparrow & \bbuup {$\approx 18, 52$}& 
      \bbup {\brightdownarrow }& \bbup {$4$}& 
      \brightdownarrow & \down{$0$}& 
      \bbrightuuparrow & \bbuup{$+\infty $}
   \cr
}}
$$
 
 
\bye