% This file:
%   http://anggtwu.net/LATEX/2023-2-C2-mv-defs.tex.html
%   http://anggtwu.net/LATEX/2023-2-C2-mv-defs.tex
%          (find-angg "LATEX/2023-2-C2-mv-defs.tex")
% Author: Eduardo Ochs <eduardoochs@gmail.com>
%
% (defun d () (interactive) (find-angg "LATEX/2023-2-C2-mv-defs.tex"))
% Used by: 
%   (find-LATEX "2023-2-C2-DFI.tex")
%    (c2m231dfip 3 "introducao")
%    (c2m231dfia   "introducao")
%   (find-LATEX "2023-2-C2-integracao-por-partes.tex")
%    (c2m232ipp 7 "justifique-o-TFC2")
%    (c2m232ipa   "justifique-o-TFC2")
%    (c2m232ipp 8 "expandindo-sen-cos")
%    (c2m232ipa   "expandindo-sen-cos")
%   (find-LATEX "2023-2-C2-mudanca-de-variaveis.tex")
%    (c2m232mvp 3 "introducao")
%    (c2m232mva   "introducao")
%    (c2m232mvp 10 "caso-1-MVD")
%    (c2m232mva    "caso-1-MVD")


% «.reset»		(to "reset")
% «.wide-and-thin»	(to "wide-and-thin")
% «.MVI-and-MVD»	(to "MVI-and-MVD")
% «.MVI3-and-MVD4»	(to "MVI3-and-MVD4")
% «.substs»		(to "substs")
% «.expands»		(to "expands")
% «.caso-1»		(to "caso-1")
% «.caso-2»		(to "caso-2")
% «.caso-3»		(to "caso-3")
% «.DFI»		(to "DFI")

\sa  {[TFC2]}{\CFname{TFC2}{}}
\sa   {[MVI]}{\CFname{MVI}{}}
\sa  {[MVI3]}{\CFname{MVI3}{}}
\sa   {[MVD]}{\CFname{MVD}{}}
\sa  {[MVD4]}{\CFname{MVD4}{}}
\sa  {[DFI2]}{\CFname{DFI2}{}}
\sa  {[DFI6]}{\CFname{DFI6}{}}
\def\P#1{\left( #1 \right)}


% «reset»  (to ".reset")
\sa{reset}{
  \sa    {a}      {a}
  \sa    {b}      {b}
  \sa  {g(a)}   {g(a)}
  \sa  {g(b)}   {g(b)}
  \sa  {g(x)}   {g(x)}
  \sa {g'(x)}  {g'(x)}
  \sa{.}      {}
  \sa{F(g(a))}{F(g(a))}
  \sa{F(g(b))}{F(g(b))}
  \sa{F(g(x))}{F(g(x))}
  \sa{F(u)}   {F(u)}
  \sa{f(g(x))}{f(g(x))}
  \sa{f(u)}   {f(u)}
  }
\ga{reset}

% «wide-and-thin»  (to ".wide-and-thin")
\def\oneeq      {\oneeqthin}
\def\threeeqs{\threeeqsthin}
\def\foureqs  {\foureqsthin}

\def\oneeq      {\oneeqwide}
\def\threeeqs{\threeeqswide}
\def\foureqs  {\foureqswide}

\def\oneeqwide#1#2{
  \begin{array}{rcl}
    #1 &=& #2 \\
  \end{array}
  }
\def\threeeqswide#1#2#3#4{
  \begin{array}{rcl}
    #1 &=& #2 \\
       &=& #3 \\
       &=& #4 \\
  \end{array}
  }
\def\foureqswide#1#2#3#4#5{
  \begin{array}{rcl}
    #1 &=& #2 \\
       &=& #3 \\
       &=& #4 \\
       &=& #5 \\
  \end{array}
  }

\def\oneeqthin#1#2{
  \begin{array}{l}
    #1 \\
    = \;\; #2 \\
  \end{array}
  }
\def\threeeqsthin#1#2#3#4{
  \begin{array}{l}
    #1 \\
    = \;\; #2 \\
    = \;\; #3 \\
    = \;\; #4 \\
  \end{array}
  }
\def\foureqsthin#1#2#3#4#5{
  \begin{array}{l}
    #1 \\
    \;\; = \; #2 \\
    \;\; = \; #3 \\
    \;\; = \; #4 \\
    \;\; = \; #5 \\
  \end{array}
  }


% «MVI-and-MVD»  (to ".MVI-and-MVD")
\sa{MVI}{
 \oneeq
  { \intx{\ga{f(g(x))}\ga{.}\ga{g'(x)}} }
  {   \intu{\ga{f(u)}}                  }
 }
\sa{MVD}{
 \oneeq
  { \Intx{\ga{a}}{\ga{b}}{\ga{f(g(x))}\ga{.}\ga{g'(x)}} }
  {   \Intu{\ga{g(a)}}{\ga{g(b)}}{\ga{f(u)}}            }
 }

% «MVI3-and-MVD4»  (to ".MVI3-and-MVD4")
\sa{MVI3}{
 \threeeqs
  { \intx{\ga{f(g(x))}\ga{.}\ga{g'(x)}}   }
  {   \ga{F(g(x))}                        }
  {   \ga{F(u)}                           }
  {   \intu{\ga{f(u)}}                    }
 }
\sa{MVD4}{
 \foureqs
  { \Intx{\ga{a}}{\ga{b}}{\ga{f(g(x))}\ga{.}\ga{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)}}            }
 }

% «substs»  (to ".substs")
\sa{subst g}{\bmat{
  g(x) :=\ga{g(x)} \\
  g'(x):=\ga{g'(x)} \\
  }}
\sa{subst g,f}{\bmat{
  g(x) :=\ga{g(x)} \\
  g'(x):=\ga{g'(x)} \\
  f(u) :=\ga{f(u)} \\
  }}
\sa{subst g,f,F}{\bmat{
  g(x) :=\ga{g(x)} \\
  g'(x):=\ga{g'(x)} \\
  f(u) :=\ga{f(u)} \\
  F(u) :=\ga{F(u)} \\
  }}
\sa{subst vars}{\bmat{
  x :=\ga{x} \\
  u :=\ga{u} \\
  }}

% «expands»  (to ".expands")
\sa{expand MVD}{
  \begin{array}{lcl}
    \ga{[MVD]}                                  &=&                \P{\ga{MVD}} \\
    \ga{[MVD]}  \ga{set g}     \ga{subst g}     &=& \ga{set g}     \P{\ga{MVD}} \\ \\[-9pt]
    \ga{[MVD]}  \ga{set g,f}   \ga{subst g,f}   &=& \ga{set g,f}   \P{\ga{MVD}} \\
    \\
    \ga{[MVD4]}                                 &=&                \P{\ga{MVD4}} \\ \\[-9pt]
    \ga{[MVD4]} \ga{set g}     \ga{subst g}     &=& \ga{set g}     \P{\ga{MVD4}} \\ \\[-9pt]
    \ga{[MVD4]} \ga{set g,f}   \ga{subst g,f}   &=& \ga{set g,f}   \P{\ga{MVD4}} \\ \\[-9pt]
    \ga{[MVD4]} \ga{set g,f,F} \ga{subst g,f,F} &=& \ga{set g,f,F} \P{\ga{MVD4}} \\
  \end{array}
  }

\sa{expand MVI}{
  \begin{array}{lcl}
    \ga{[MVI]}                                  &=&                \P{\ga{MVI}} \\
    \ga{[MVI]}  \ga{set g}     \ga{subst g}     &=& \ga{set g}     \P{\ga{MVI}} \\ \\[-9pt]
    \ga{[MVI]}  \ga{set g,f}   \ga{subst g,f}   &=& \ga{set g,f}   \P{\ga{MVI}} \\
    \\
    \ga{[MVI3]}                                 &=&                \P{\ga{MVI3}} \\ \\[-9pt]
    \ga{[MVI3]} \ga{set g}     \ga{subst g}     &=& \ga{set g}     \P{\ga{MVI3}} \\ \\[-9pt]
    \ga{[MVI3]} \ga{set g,f}   \ga{subst g,f}   &=& \ga{set g,f}   \P{\ga{MVI3}} \\ \\[-9pt]
    \ga{[MVI3]} \ga{set g,f,F} \ga{subst g,f,F} &=& \ga{set g,f,F} \P{\ga{MVI3}} \\
  \end{array}
  }

\sa{expand MVI-}{
  \begin{array}{lcl}
    \ga{[MVI]}                                  &=&                \P{\ga{MVI}} \\
    \ga{[MVI]}  \ga{set g}     \ga{subst g}     &=& \ga{set g}     \P{\ga{MVI}} \\ \\[-9pt]
    \ga{[MVI]}  \ga{set g,f}   \ga{subst g,f}   &=& \ga{set g,f}   \P{\ga{MVI}} \\
    \\
    \ga{[MVI3]}                                 &=&                \P{\ga{MVI3}} \\ \\[-9pt]
    \ga{[MVI3]} \ga{set g}     \ga{subst g}     &=& \ga{set g}     \P{\ga{MVI3}} \\ \\[-9pt]
    \ga{[MVI3]} \ga{set g,f}   \ga{subst g,f}   &=& \ga{set g,f}   \P{\ga{MVI3}}%\\ \\[-9pt]
   %\ga{[MVI3]} \ga{set g,f,F} \ga{subst g,f,F} &=& \ga{set g,f,F} \P{\ga{MVI3}} \\
  \end{array}
  }

% «caso-1»  (to ".caso-1")
% (c2m232mvda    "caso-1")
% (c2m232mvp 10 "caso-1-MVD")
% (c2m232mva    "caso-1-MVD")
\sa{caso sen(2x)*2}{
  \sa{g(x) := 2x}{
    \sa{f(g(x))}{f(2x)}
    \sa{F(g(x))}{F(2x)}
    \sa{F(g(a))}{F(2a)}
    \sa{F(g(b))}{F(2b)}
    \sa  {g(x)}   {2x}
    \sa  {g(a)}   {2a}
    \sa  {g(b)}   {2b}
    \sa {g'(x)}   {2}
    \sa{.}{·}
    }
  \sa{f(u) := sen(u)}{
    \sa{f(g(x))}{\sen(2x)}
    \sa{f(u)}{\sen u}
    }
  \sa{F(u) := -cos(u)}{
    \sa{F(g(a))}{(-\cos 2a)}
    \sa{F(g(b))}{(-\cos 2b)}
    \sa{F(g(x))}{(-\cos 2x)}
    \sa{F(u)}{(-\cos u)}
    }
  %
  \sa{set g}{
    \ga{g(x) := 2x}
    }
  \sa{set g,f}{
    \ga{g(x) := 2x}
    \ga{f(u) := sen(u)}
    }
  \sa{set g,f,F}{
    \ga{g(x) := 2x}
    \ga{f(u) := sen(u)}
    \ga{F(u) := -cos(u)}
    }
  }

% «caso-2»  (to ".caso-2")
% (c2m232mvda    "caso-2")
% (c2m232mvp 12  "caso-2-MVD")
% (c2m232mva     "caso-2-MVD")
\sa{caso sen(x^2)*2x}{
  \sa{g(x) := x^2}{
    \sa{f(g(x))}{f(x^2)}
    \sa{F(g(x))}{F(x^2)}
    \sa{F(g(a))}{F(a^2)}
    \sa{F(g(b))}{F(b^2)}
    \sa  {g(x)}   {x^2}
    \sa  {g(a)}   {a^2}
    \sa  {g(b)}   {b^2}
    \sa {g'(x)}   {2x}
    \sa{.}{·}
    }
  \sa{f(u) := sen(u)}{
    \sa{f(g(x))}{\sen(x^2)}
    \sa{f(u)}{\sen u}
    }
  \sa{F(u) := -cos(u)}{
    \sa{F(g(a))}{(-\cos a^2)}
    \sa{F(g(b))}{(-\cos b^2)}
    \sa{F(g(x))}{(-\cos x^2)}
    \sa{F(u)}{(-\cos u)}
    }
  %
  \sa{set g}{
    \ga{g(x) := x^2}
    }
  \sa{set g,f}{
    \ga{g(x) := x^2}
    \ga{f(u) := sen(u)}
    }
  \sa{set g,f,F}{
    \ga{g(x) := x^2}
    \ga{f(u) := sen(u)}
    \ga{F(u) := -cos(u)}
    }
  }

% «caso-3»  (to ".caso-3")
% (c2m232mvda    "caso-3")
% (c2m232mvp 14  "caso-3-MVI")
% (c2m232mva     "caso-3-MVI")
\sa{caso senth^3 costh^5}{
  \sa{g(x) := sen x}{
    \sa{f(g(x))}{f(\sen x)}
    \sa{F(g(x))}{F(\sen x)}
    \sa{F(g(a))}{F(a^2)}
    \sa{F(g(b))}{F(b^2)}
    \sa  {g(x)}   {\sen x}
    \sa  {g(a)}   {a^2}
    \sa  {g(b)}   {b^2}
    \sa {g'(x)}   {\cos x}
    \sa{.}{}
    }
  \sa{f(u) := u^2(1-u^2)^2}{
    \sa{f(g(x))}{(\sen x)^3(1-(\sen x)^2)^2}
    \sa{f(u)}{u^2(1-u^2)^2}
    }
  \sa{F(u) := Bla(u)}{
    \sa{F(g(a))}{Bla(\sen a)}
    \sa{F(g(b))}{Bla(\sen b)}
    \sa{F(g(x))}{Bla(\sen x)}
    \sa{F(u)}{Bla(u)}
    }
  %
  \sa{set g}{
    \ga{g(x) := sen x}
    }
  \sa{set g,f}{
    \ga{g(x) := sen x}
    \ga{f(u) := u^2(1-u^2)^2}
    }
  \sa{set g,f,F}{
    \ga{g(x) := sen x}
    \ga{f(u) := u^2(1-u^2)^2}
    \ga{F(u) := Bla(u)}
    }
  }



% «DFI»  (to ".DFI")
% (find-LATEX "2022-1-C2-der-fun-inv.tex" "defs-DFIs")

\sa {DFI2}{
  \begin{array}{lrcl}
    \text{Se:}    & f(g(x))       &\eqnp{1}& x \\
    \text{Então:} & g'(x)         &\eqnp{6}& \D \frac{1}{f'(g(x))} \\
  \end{array}}

\sa {DFI6}{
  \begin{array}{lrcl}
    \text{Se:}    & f(g(x))       &\eqnp{1}& x \\
    \text{Então:} & \ddx f(g(x))  &\eqnp{2}& \ddx x \\
                                 &&\eqnp{3}& 1 \\
                  & \ddx f(g(x))  &\eqnp{4}& f'(g(x))g'(x) \\
                  & f'(g(x))g'(x) &\eqnp{5}& 1 \\
                  & g'(x)         &\eqnp{6}& \D \frac{1}{f'(g(x))} \\
  \end{array}}



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