% (find-LATEX "2023-2-C2-ref-subst.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-2-C2-ref-subst.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-2-C2-ref-subst.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023-2-C2-ref-subst.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-2-C2-ref-subst.pdf"))
% (defun e () (interactive) (find-LATEX "2023-2-C2-ref-subst.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C2-ref-subst.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-2-C2-ref-subst"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-2-C2-ref-subst.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2023-2-C2-ref-subst")
% (find-pdf-page   "~/LATEX/2023-2-C2-ref-subst.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C2-ref-subst.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C2-ref-subst.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2023-2-C2-ref-subst.pdf")
%   file:///home/edrx/LATEX/2023-2-C2-ref-subst.pdf
%               file:///tmp/2023-2-C2-ref-subst.pdf
%           file:///tmp/pen/2023-2-C2-ref-subst.pdf
%  http://anggtwu.net/LATEX/2023-2-C2-ref-subst.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Maxima2 CME2")
% (find-Deps1-cps   "Caepro5 Maxima2 CME2")
% (find-Deps1-anggs "Caepro5 Maxima2 CME2")
% (find-MM-aula-links "2023-2-C2-ref-subst" "C2" "c2m232refsubst" "c2rs")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.defs-caepro»	(to "defs-caepro")
% «.defs-pict2e»	(to "defs-pict2e")
% «.defs-maxima»	(to "defs-maxima")
% «.defs-cme2»		(to "defs-cme2")
% «.title»		(to "title")
% «.links»		(to "links")
% «.EDOs-RC-TFC2»	(to "EDOs-RC-TFC2")
%
% «.djvuize»		(to "djvuize")



% <videos>
% Video (not yet):
% (find-ssr-links     "c2m232refsubst" "2023-2-C2-ref-subst")
% (code-eevvideo      "c2m232refsubst" "2023-2-C2-ref-subst")
% (code-eevlinksvideo "c2m232refsubst" "2023-2-C2-ref-subst")
% (find-c2m232refsubstvideo "0:00")

\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
\usepackage{colorweb}                  % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx21}               % (find-LATEX "edrx21.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex            % (find-LATEX "edrx21chars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
\begin{document}

% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

\def\drafturl{http://anggtwu.net/LATEX/2023-2-C2.pdf}
\def\drafturl{http://anggtwu.net/2023.2-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2023-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2023-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% «defs-T-and-B»  (to ".defs-T-and-B")
\long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}

% «defs-caepro»  (to ".defs-caepro")
%L dofile "Caepro5.lua"              -- (find-angg "LUA/Caepro5.lua" "LaTeX")
\def\Caurl   #1{\expr{Caurl("#1")}}
\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca      #1{\Cahref{#1}{#1}}

% «defs-pict2e»  (to ".defs-pict2e")
%L --dofile "Piecewise2.lua"         -- (find-LATEX "Piecewise2.lua")
%L --dofile "Escadas1.lua"           -- (find-LATEX "Escadas1.lua")
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt

% «defs-maxima»  (to ".defs-maxima")
%L dofile "Maxima2.lua"              -- (find-angg "LUA/Maxima2.lua")

% «defs-cme2»  (to ".defs-cme2")
% (find-LATEXgrep "grep --color=auto -niH --null -e 'elpeg-cme' 2023*.tex")
%L dofile "CME2.lua"               -- (find-angg "LUA/CME2.lua")
\def\CME#1{\expr{CME("#1")}}
\pu

% (find-LATEX "edrxgac2.tex" "C2-substnames")
% 2dT13 EDOs por chutar e testar
% (c2m212introp 12 "EDOs-chutar-testar")
% (c2m212introa    "EDOs-chutar-testar")
\def\redname#1{{\color{Red3}\text{#1}}}
\sa  {RC}{\redname{[RC]}}
\sa{TFC2}{\redname{[TFC2]}}
\sa   {4}{\redname{[4]}}
\sa   {5}{\redname{[5]}}
\sa   {6}{\redname{[6]}}
\sa   {7}{\redname{[7]}}
\sa   {8}{\redname{[8]}}
\sa  {II}{\redname{[II]}}




%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
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%
% «title»  (to ".title")
% (c2m232refsubstp 1 "title")
% (c2m232refsubsta   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo C2 - 2023.2}

\bsk

Aula de reforço sobre substituição

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2023.2-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m232refsubstp 2 "links")
% (c2m232refsubsta   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{9cm}\firstcol{


}\anothercol{
}}

\newpage

%  _____ ____   ___          ____   ____     _____ _____ ____ ____  
% | ____|  _ \ / _ \ ___    |  _ \ / ___|   |_   _|  ___/ ___|___ \ 
% |  _| | | | | | | / __|   | |_) | |         | | | |_ | |     __) |
% | |___| |_| | |_| \__ \_  |  _ <| |___ _    | | |  _|| |___ / __/ 
% |_____|____/ \___/|___( ) |_| \_\\____( )   |_| |_|   \____|_____|
%                       |/              |/                          
%
% «EDOs-RC-TFC2»  (to ".EDOs-RC-TFC2")
% (c2m232refsubstp 99 "EDOs-RC-TFC2")
% (c2m232refsubsta    "EDOs-RC-TFC2")
% (c2m231macacop 5 "EDOs-RC-TFC2")
% (c2m231macacoa   "EDOs-RC-TFC2")

\def\bigeq#1{\Bigl(#1\Bigr)}
\def\bigeq#1{\Bigl(#1\Bigr)}

{\bf O macaco substituidor: EDOs, RC, TFC2}

\scalebox{0.55}{\def\colwidth{14cm}\firstcol{

Sejam:
%
$$\begin{array}{rcl}
  \ga{4}    &=& \bigeq{ f'(x) = x^4               } \\
  \ga{5}    &=& \bigeq{ f'(x) = 2f(x)             } \\
  \ga{6}    &=& \bigeq{ f''(x) + f'(x) = 6f(x)    } \\
  \ga{7}    &=& \bigeq{ f'(x) = -\frac{1}{f(x)}   } \\
  \ga{8}    &=& \bigeq{ f'(x) = -\frac{x}{f(x)}   } \\
  \ga{RC}   &=& \bigeq{ f(g(x))' = f'(g(x)) g'(x) } \\
  \ga{TFC2} &=& \bigeq{ \Intx{a}{b}{f'(x)} = f(b) - f(a) } \\
  \end{array}
$$

Note que as expressões $\ga{4}$, $\ga{5}$, $\ga{6}$, $\ga{7}$,
$\ga{8}$, são as EDOs deste problema aqui:

\ssk

\Ca{2dT13} EDOs por chutar e testar.

\bsk

{\bf Exercício}

Calcule o resultado de cada uma das substituições à direita. Lembre
que o resultado de uma substituição é sempre uma {\sl expressão} --
{\sl não simplifique ela}. Deixa eu fazer uma comparação com C: o
resultado de substituir cada ocorrência do caracter {\tt 'a'} pelo
caracter {\tt '2'} no string {\tt "a+5"} é o string {\tt "2+5"}, não o
string {\tt "7"}, e nem o número {\tt 7}.

}\anothercol{



$\begin{tabular}[t]{rl}
  a) & $f(g(x)) \CME{[x := 42]}$ \\
  b) & $f(g(x)) \CME{[g(x) := 200*x]}$ \\
  c) & $f(g(x)) \CME{[f(y) := y^2+y^3]}$ \\
  d) & $f(g(x)) \CME{[f(y) := e^{y}]}$ \\
  e) & $f(g(x)) \CME{[g(x) := 4*x]}$ \\
  f) & $f(g(x)) \CME{.[f(y) := e^{y} ;; g(x) := 4*x]}$ \\
  g) & $f(g(x)) \CME{[f(y) := y^{1/2}]}$ \\
  h) & $f(g(x)) \CME{[f(y) := sqrt{y}]}$ \\
  i) & $f(g(x)) \CME{[f(y) := sqrt{y}]}$ \\
  j) & $f(g(x)) \CME{.[g(x) := e^x ;; g'(x) := e^x]}$ \\
  k) & $f(g(x))' \CME{.[g(x) := e^x ;; g'(x) := e^x]}$ \\
  l) & $\ga{RC}  \CME{.[f(y) := e^y ;; f'(y) := e^y]}$ \\
  m) & $\ga{RC}  \CME{.[f(y) := y^{1/2} ;; f'(y) := {1//2} mul y^{-1/2}]}$ \\
  n) & $\ga{RC}  \CME{.[f(y) := sqrt{y} ;; f'(y) := 1 // 2 mul sqrt{y}]}$ \\
  o) & $\ga{6}   \CME{.[f  (x) :=       e^{2 mul x} ;;
                        f' (x) := 2 mul e^{2 mul x} ;;
                        f''(x) := 4 mul e^{2 mul x}]}$ \\
  p) & $\ga{6}   \CME{.[f  (x) :=       e^{3 mul x} ;;
                        f' (x) := 3 mul e^{3 mul x} ;;
                        f''(x) := 9 mul e^{3 mul x}]}$ \\
  q) & $\ga{TFC2}\CME{.[f  (x) := {1//2} mul x^2 ;;
                        f' (x) := x ;;
                        a      := 0 ;;
                        b      := 2 ]}$ \\
  r) & $\ga{8}   \CME{.[f  (x) := sqrt{1 - x^2} ;;
                        f' (x) := -1 // x mul sqrt{1 - x^2}]}$ \\
  \end{tabular}
$

}}

\newpage


$$\begin{array}{lll}
  (f(u)=e^u) [u:=200]  &=& (f(200)=e^{200}) \\
  (f(u)=e^u) [u:=a+b]  &=& (f(a+b)=e^{a+b}) \\
  (f(u)=e^u) [u:=g(x)] &=& (f(a+b)=e^{g(x)}) \\
  \\[-5pt]
  (f(y)=y^2+y^3) [y:=200]  &=& (f(200)=200^2 + 200^3) \\
  (f(y)=y^2+y^3) [y:=a+b]  &=& (f(a+b)=(a+b)^2 + (a+b)^3) \\
  (f(y)=y^2+y^3) [y:=g(x)] &=& (f(g(x))=g(x)^2 + g(x)^3) \\
  \end{array}
$$



\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}


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