% This file:
%   http://anggtwu.net/LATEX/2024-1-C2-mv-defs.tex.html
%   http://anggtwu.net/LATEX/2024-1-C2-mv-defs.tex
%          (find-angg "LATEX/2024-1-C2-mv-defs.tex")
% Author: Eduardo Ochs <eduardoochs@gmail.com>
%
% (defun e () (interactive) (find-angg "LATEX/2024-1-C2-mv-defs.tex"))
% (defun o () (interactive) (find-angg "LATEX/2023-2-C2-mv-defs.tex"))
% (find-MM-aula-links "2024-1-C2-mv-defs" "2" "c2m241mvdefs" "c2mvd")

% «.title»		(to "title")
% «.cfnames»		(to "cfnames")
% «.aligneqs»		(to "aligneqs")
% «.wide-and-thin»	(to "wide-and-thin")
% «.mv-defaults»	(to "mv-defaults")
% «.mv-threecolors»	(to "mv-threecolors")
% «.mv-casoA»		(to "mv-casoA")
% «.mvs»		(to "mvs")
% «.TFC2-and-defdif»	(to "TFC2-and-defdif")

% «title»  (to ".title")
% "\def"s para mudança de variáveis.
%
% Exemplos de uso:
%   (c2m241dip 6 "MVDs-e-MVIs-color")
%   (c2m241dia   "MVDs-e-MVIs-color")
%   (c2m241dip 7 "caso-A")
%   (c2m241dia   "caso-A")

\def\P#1{\left( #1 \right)}

% «cfnames»  (to ".cfnames")
\sa  {[MVD1]}{\CFname{MVD1}{}}
\sa  {[MVI1]}{\CFname{MVI1}{}}
\sa  {[MVD4]}{\CFname{MVD4}{}}
\sa  {[MVI3]}{\CFname{MVI3}{}}
\sa    {[S1]}{\CFname{S}{1}}
\sa    {[S2]}{\CFname{S}{2}}
\sa    {[S3]}{\CFname{S}{3}}
\sa    {[S4]}{\CFname{S}{4}}
\sa  {[TFC2]}{\CFname{TFC2}{}}
\sa{[defdif]}{\CFname{defdif}{}}
\sa    {[RC]}{\CFname{RC}{}}
\sa    {[II]}{\CFname{II}{}}
\sa    {[IP]}{\CFname{IP}{}}
\sa   {[IPD]}{\CFname{IPD}{}}

% «aligneqs»       (to ".aligneqs")
% «wide-and-thin»  (to ".wide-and-thin")
% (c2m241mvdefsp 1 "wide-and-thin")
% (c2m241mvdefsa   "wide-and-thin")
\def\aligneqswidearray#1{\begin{array}{rcll}#1\end{array}}
\def\aligneqsthinarray#1{\begin{array}{ll}  #1\end{array}}
\def\aligneqspor      #1{}

\def\aligneqswide{
  \def\aligneqsarray{\aligneqswidearray}
  \def\aligneqseq      {&=&}
  \def\aligneqsnleq {\\ &=&}
  }
\def\aligneqsthin{
  \def\aligneqsarray{\aligneqsthinarray}
  \def\aligneqseq   {\\ = \;\;}
  \def\aligneqsnleq {\\ = \;\;}
  }

\aligneqswide

\def\aligneqsone#1#2#3{         % one equality, two expressions
  \aligneqsarray{
    #1 \aligneqseq #2 \aligneqspor{#3} \\
  }}
\def\aligneqsthree#1#2#3#4#5#6#7{   % three equalities, four expressions
  \aligneqsarray{
    #1 \aligneqseq #2 \aligneqspor{#3}
     \aligneqsnleq #4 \aligneqspor{#5}
     \aligneqsnleq #6 \aligneqspor{#7} \\
  }}
\def\aligneqsfour#1#2#3#4#5#6#7#8#9{   % four equalities, five expressions
  \aligneqsarray{
    #1 \aligneqseq #2 \aligneqspor{#3}
     \aligneqsnleq #4 \aligneqspor{#5}
     \aligneqsnleq #6 \aligneqspor{#7}
     \aligneqsnleq #8 \aligneqspor{#9} \\
  }}


% «mv-defaults»  (to ".mv-defaults")
\def\mvdefaults{
  \def\mvf ##1{\ColorBlack{f (##1)}}
  \def\mvfp##1{\ColorBlack{f'(##1)}}
  \def\mvg ##1{\ColorBlack{g (##1)}}
  \def\mvgp##1{\ColorBlack{g'(##1)}}
  \sa            {.}{\cdot}
  \sa          {.fg}{}
  \sa            {a}{\ColorBlack{a}}
  \sa            {b}{\ColorBlack{b}}
  \sa            {u}{\ColorBlack{u}}
  \sa            {x}{\ColorBlack{x}}
  \sa         {f(u)}{\mvf {\ga{u}}}
  \sa        {f'(u)}{\mvfp{\ga{u}}}
  \sa         {g(a)}{\mvg {\ga{a}}}
  \sa         {g(b)}{\mvg {\ga{b}}}
  \sa         {g(x)}{\mvg {\ga{x}}}
  \sa        {g'(x)}{\mvgp{\ga{x}}}
  \sa      {f(g(a))}{\mvf {\ga{g(a)}}}
  \sa      {f(g(b))}{\mvf {\ga{g(b)}}}
  \sa      {f(g(x))}{\mvf {\ga{g(x)}}}
  \sa     {f'(g(x))}{\mvfp{\ga{g(x)}}}
  \sa{f'(g(x))g'(x)}{\ga{f'(g(x))} \ga{.fg} \ga{g'(x)}}
  }

% «mv-threecolors»  (to ".mv-threecolors")
% (c2m241mvdefsa         "mv-threecolors")
% (c2m241dip 7 "MVDs-e-MVIs-color")
% (c2m241dia   "MVDs-e-MVIs-color")
\def\mvthreecolors{
  \mvdefaults
  \def\mvf ##1{\ColorGreen {f (##1)}}
  \def\mvfp##1{\ColorGreen {f'(##1)}}
  \def\mvg ##1{\ColorOrange{g (##1)}}
  \def\mvgp##1{\ColorOrange{g'(##1)}}
  \sa      {a}{\ColorRed   {a}}
  \sa      {b}{\ColorRed   {b}}
  }

% «mv-casoA»  (to ".mv-casoA")
% (c2m241mvdefsa   "mv-casoA")
% (c2m241dip 8 "caso-A")
% (c2m241dia   "caso-A")
\def\mvcasoA{
  \mvdefaults
  \def\mvf ##1{\ColorGreen {\sen(##1)}}
  \def\mvfp##1{\ColorGreen {\cos(##1)}}
  \def\mvg ##1{\ColorOrange{2·##1}}
  \def\mvgp##1{\ColorOrange{2}}
  \sa     {a}{\ColorRed   {3}}
  \sa     {b}{\ColorRed   {4}}
  \sa   {.fg}{\cdot}
  }


% «mvs»  (to ".mvs")
%
\sa{(MVD4)}{
  \P{\aligneqsfour
      { \Intx {\ga{a}} {\ga{b}} {\ga{f'(g(x))g'(x)}} }
      { \Difx {\ga{a}} {\ga{b}} {\ga{f(g(x))}}       } {}
      { \ga{f(g(b))} - \ga{f(g(a))}                  } {}
      { \Difu {\ga{g(a)}} {\ga{g(b)}}  {\ga{f(u)}}   } {}
      { \Intu {\ga{g(a)}} {\ga{g(b)}} {\ga{f'(u)}}   } {}
    }
  }
\sa{(MVD1)}{
  \P{\aligneqsone
      { \Intx {\ga{a}} {\ga{b}} {\ga{f'(g(x))g'(x)}} }
      { \Intu {\ga{g(a)}} {\ga{g(b)}} {\ga{f'(u)}}   } {}
    }
  }
\sa{(MVI3)}{
  \P{\aligneqsthree
      { \intx {\ga{f'(g(x))g'(x)}} }
      {        \ga{f(g(x))}        } {}
      {        \ga{f(u)}           } {}
      { \intu {\ga{f'(u)}}         } {}
    }
  }
\sa{(MVI1)}{
  \P{\aligneqsone
      { \intx {\ga{f'(g(x))g'(x)}} }
      { \intu {\ga{f'(u)}}         } {}
    }
  }


% «TFC2-and-defdif»  (to ".TFC2-and-defdif")
%
\sa{(TFC2)}{
  \P{ \Intx{a}{b}{F'(x)} = \Difx{a}{b}{F(x)} }
  }
\sa{(defdif)}{
  \P{ \Difx{a}{b}{F(x)} = F(b)-F(a) }
  }

\sa{(RC)}{
  \P{ \ddx f(g(x)) = f'(g(x))g'(x) }
  }

\sa{(II)}{
  \P{ \intx{F'(x)} = F(x) }
  }
\sa{(IP)}{
  \P{ \intx{fg'} = fg - \intx{f'g} }
  }




% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3mvd"
% ee-tla: "c3m241mvdefs"
% End:
