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Warning: this is an htmlized version!
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% (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"))
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% (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")
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% (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")
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% (find-Deps1-cps "Caepro5 Maxima2 CME2")
% (find-Deps1-anggs "Caepro5 Maxima2 CME2")
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% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.defs-caepro» (to "defs-caepro")
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% «.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")
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%
% (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")
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\def\T(Total: #1 pts){{\bf(Total: #1)}}
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\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")
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% «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]}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «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}
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2rs"
% ee-tla: "c2m232refsubst"
% End: