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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2021-1-C2-P2.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2021-1-C2-P2.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2021-1-C2-P2.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2021-1-C2-P2.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2021-1-C2-P2.pdf"))
% (defun e () (interactive) (find-LATEX "2021-1-C2-P2.tex"))
% (defun o () (interactive) (find-LATEX "2021-1-C2-P2.tex"))
% (defun u () (interactive) (find-latex-upload-links "2021-1-C2-P2"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2021-1-C2-P2.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2021-1-C2-P2")
% (find-pdf-page "~/LATEX/2021-1-C2-P2.pdf")
% (find-sh0 "cp -v ~/LATEX/2021-1-C2-P2.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2021-1-C2-P2.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2021-1-C2-P2.pdf")
% file:///home/edrx/LATEX/2021-1-C2-P2.pdf
% file:///tmp/2021-1-C2-P2.pdf
% file:///tmp/pen/2021-1-C2-P2.pdf
% http://angg.twu.net/LATEX/2021-1-C2-P2.pdf
% (find-LATEX "2019.mk")
% (find-CN-aula-links "2021-1-C2-P2" "2" "c2m211p2" "c2p2")
%
% Video (not yet):
% (find-ssr-links "c2m211p2" "2021-1-C2-P2" "{naoexiste}")
% (code-video "c2m211p2video" "$S/http/angg.twu.net/eev-videos/2021-1-C2-P2.mp4")
% (find-c2m211p2video "0:00")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.title» (to "title")
% «.regras-e-dicas» (to "regras-e-dicas")
% «.vamos-usar» (to "vamos-usar")
% «.questao-1» (to "questao-1")
% «.questao-1-cont» (to "questao-1-cont")
% «.questao-2» (to "questao-2")
% «.questao-2-cont» (to "questao-2-cont")
% «.gabarito» (to "gabarito")
% «.questao-1a-gab» (to "questao-1a-gab")
% «.questoes-1bc-gab» (to "questoes-1bc-gab")
% «.questoes-1def-gab» (to "questoes-1def-gab")
% «.questoes-2ab-gab» (to "questoes-2ab-gab")
% «.questoes-2cd-gab» (to "questoes-2cd-gab")
% «.questao-2e-gab» (to "questao-2e-gab")
% «.questoes-2fg-gab» (to "questoes-2fg-gab")
%
% «.djvuize» (to "djvuize")
\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")
%
%\usepackage[backend=biber,
% style=alphabetic]{biblatex} % (find-es "tex" "biber")
%\addbibresource{catsem-slides.bib} % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
]{geometry}
%
\begin{document}
%\catcode`\^^J=10
%\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
% %L dofile "edrxtikz.lua" -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua" -- (find-LATEX "edrxpict.lua")
% \pu
% «defs» (to ".defs")
% (find-LATEX "edrx15.sty" "colors-2019")
%\long\def\ColorRed #1{{\color{Red1}#1}}
%\long\def\ColorViolet#1{{\color{MagentaVioletLight}#1}}
%\long\def\ColorViolet#1{{\color{Violet!50!black}#1}}
%\long\def\ColorGreen #1{{\color{SpringDarkHard}#1}}
%\long\def\ColorGreen #1{{\color{SpringGreenDark}#1}}
%\long\def\ColorGreen #1{{\color{SpringGreen4}#1}}
%\long\def\ColorGray #1{{\color{GrayLight}#1}}
%\long\def\ColorGray #1{{\color{black!30!white}#1}}
%\long\def\ColorBrown #1{{\color{Brown}#1}}
%\long\def\ColorBrown #1{{\color{brown}#1}}
%\long\def\ColorOrange#1{{\color{orange}#1}}
%
%\long\def\ColorShort #1{{\color{SpringGreen4}#1}}
%\long\def\ColorLong #1{{\color{Red1}#1}}
%
%\def\frown{\ensuremath{{=}{(}}}
%\def\True {\mathbf{V}}
%\def\False{\mathbf{F}}
%\def\D {\displaystyle}
\def\frown{\ensuremath{{=}{(}}}
\def\True {\mathbf{V}}
\def\False{\mathbf{F}}
\def\D {\displaystyle}
\def\Rd#1{\ColorRed{#1}}
\def\Rdq{{\Rd{?}}}
\def\eqq{\overset{\Rdq}{=}} % equal with a question mark
\def\eqq{\overset{?}{=}} % equal with a question mark
\def\eqvs#1#2{\ph{#1}\rotl{=}\ph{#2}} % vertical equal with spaces
\def\dydx{\frac{dy}{dx}}
\def\pfo #1{\ensuremath{\mathsf{[#1]}}}
\def\pfot#1{\ensuremath{\textsf{[#1]}}}
\def\EDOVSGone{
\left(
\begin{array}{rcl}
\D \frac{dy}{dx} &=& \D \frac{f(x)}{g(y)} \\ [10pt]
g(y) \, dy &=& f(x) \, dx \\ [5pt]
∫ g(y) \, dy &=& ∫ f(x) \, dx \\
\rotl{=}\ph{mm} & & \ph{mm}\rotl{=} \\[-5pt]
G(y) + C_1 & & F(x) + C_2 \\
[5pt]
G(y) + C_1 &=& F(x) + C_2 \\ [5pt]
G(y) &=& F(x) + C_2 - C_1 \\
&=& F(x) + C_3 \\
[5pt]
G^{-1}(G(y)) &=& G^{-1}(F(x) + C_3) \\
\rotl{=}\ph{mm} & & \\[-5pt]
y\ph{mm} & & \\
\end{array}
\right)
}
\def\EDOVSGtwo{
\left(
\begin{array}{rcl}
\D \frac{dy}{dx} &=& \D \frac{f(x)}{g(y)} \\ [10pt]
y &=& G^{-1}(F(x) + C_3) \\
\end{array}
\right)
}
\def\drafturl{http://angg.twu.net/LATEX/2021-1-C2.pdf}
\def\drafturl{http://angg.twu.net/2021.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% «defs-T-and-B» (to ".defs-T-and-B")
% (c3m202p1p 6 "questao-2")
% (c3m202p1a "questao-2")
\long\def\ColorOrange#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){\ColorOrange{\bf(#1 pts)}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m211p2p 1 "title")
% (c2m211p2a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2021.1}
\bsk
P2 (segunda prova)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2021.1-C2.html}
\end{center}
\newpage
% «regras-e-dicas» (to ".regras-e-dicas")
% (c2m211p2p 2 "regras-e-dicas")
% (c2m211p2a "regras-e-dicas")
% (c2m211p1p 2 "regras-e-dicas")
% (c2m211p1a "regras-e-dicas")
As regras e dicas são as mesmas dos mini-testes:
\ssk
\url{http://angg.twu.net/LATEX/2020-2-C2-MT1.pdf}
\url{http://angg.twu.net/LATEX/2020-2-C2-MT2.pdf}
\ssk
exceto que a prova vai ser disponibilizada às 21:20 do dia
16/setembro/2021 e deve ser entregue até as 21:20 do dia
18/setembro/2021.
\newpage
{\bf Avisos}
Durante a duração da prova eu (praticamente só) vou responder
perguntas sobre a prova que 1) sejam feitas nos canais das
turmas no Telegram e que 2) possam ser respondidas com links
pros slides, vídeos e livros que usamos durante o curso ou
com links pros logs das aulas.
\msk
Durante a prova eu vou deixar os logs das aulas disponíveis aqui:
\msk
{\scriptsize
\url{http://angg.twu.net/tmp/C2-C1-RCN-PURO-2021.1.pdf}
\url{http://angg.twu.net/tmp/C2-C1-RCN-PURO-2021.1-2.pdf}
\url{http://angg.twu.net/tmp/C2-C1-RCN-PURO-2021.1-3.pdf}
\msk
\url{http://angg.twu.net/tmp/C2-E1-RCN-PURO-2021.1.pdf}
\url{http://angg.twu.net/tmp/C2-E1-RCN-PURO-2021.1-2.pdf}
\url{http://angg.twu.net/tmp/C2-E1-RCN-PURO-2021.1-3.pdf}
\url{http://angg.twu.net/tmp/C2-E1-RCN-PURO-2021.1-4.pdf}
\url{http://angg.twu.net/tmp/C2-E1-RCN-PURO-2021.1-5.pdf}
}
Vou tirar esses logs do meu site logo depois do fim da prova.
\newpage
% «vamos-usar» (to ".vamos-usar")
% (c2m211p2p 3 "vamos-usar")
% (c2m211p2a "vamos-usar")
Vamos usar muita coisa daqui:
{\footnotesize
% (c2m211edovsp 12 "duas-formulas")
% (c2m211edovsa "duas-formulas")
% http://angg.twu.net/LATEX/2021-1-C2-edovs.pdf#page=12
\url{http://angg.twu.net/LATEX/2021-1-C2-edovs.pdf}
}
% (c2m211edovsp 25 "numeracao")
% (c2m211edovsa "numeracao")
\def\numeracao#1{\myvcenter{
% (find-latexscan-links "C2" "20210916_EDOVSG1_numeracao")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210916_EDOVSG1_numeracao.pdf")
\includegraphics[height=#1]{2021-1-C2/20210916_EDOVSG1_numeracao.pdf}
}}
Em particular:
%
$$\begin{array}{rclc}
\pfo{EDOVSG1} &=& \scalebox{0.7}{$\EDOVSGone$}
& \left( \numeracao{4.5cm} \right) \\
\\[-5pt]
\pfo{EDOVSG2} &=& \scalebox{0.7}{$\EDOVSGtwo$} \\
\end{array}
$$
\newpage
% «questao-1» (to ".questao-1")
% (c2m211p2p 4 "questao-1")
% (c2m211p2a "questao-1")
% (c2m211edovsp 23 "exercicio-6")
% (c2m211edovsa "exercicio-6")
{\bf Questão 1.}
\T(Total: 5.0 pts)
\ssk
\def\SubstQuestaoUm{
\bmat{ f(x) := 2x+3 \\
F(x) := x^2+3x \\
g(y) := y^4 \\
G(y) := y^5 + 2 \\
G^{-1}(x) := \sqrt[5]{x - 2} \\
}
}
a) \B(2.0 pts) Escreva o resultado da substituição
%
$$\pfo{EDOVSG1} \;
\scalebox{0.8}{$
\SubstQuestaoUm
$}
$$
e escreva ``$= \pfo{Q1}$'' à direita do seu resultado
pra indicar que nós vamos usar a expressão $\pfo{Q1}$
pra nos referir a essa expressãozona.
\bsk
Dicas:
{\footnotesize
% (c2m211edovsp 24 "exercicio-6-dica")
% (c2m211edovsa "exercicio-6-dica")
% http://angg.twu.net/LATEX/2021-1-C2-edovs.pdf#page=24
\url{http://angg.twu.net/LATEX/2021-1-C2-edovs.pdf#page=24}
% (c2m211substp 9 "igual-depois-de-subst")
% (c2m211substa "igual-depois-de-subst")
% http://angg.twu.net/LATEX/2021-1-C2-subst.pdf#page=9
\url{http://angg.twu.net/LATEX/2021-1-C2-subst.pdf#page=9}
}
\newpage
% «questao-1-cont» (to ".questao-1-cont")
% (c2m211p2p 3 "questao-1-cont")
% (c2m211p2a "questao-1-cont")
{\bf Questão 1 (cont.)}
Aqui vou usar `(1)', `(2)', ..., `(10)' pra me referir às
igualdades da expressãozona \pfo{Q1} que você definiu no item a.
\msk
b) \B(1.0 pts) Mostre que $y=\sen x$ não é uma solução da EDO
(1).
\ssk
c) \B(0.5 pts) Mostre que (4) é falsa.
\ssk
d) \B(0.5 pts) Mostre que (5) é verdadeira.
\ssk
e) \B(0.5 pts) Digamos que $x=-2$, $y=1$, $C_1=10$.
Encontre o único valor de $C_2$ que faz a (6) ser verdadeira.
\ssk
f) \B(0.5 pts) Mostre que a (10) é verdadeira.
\newpage
% «questao-2» (to ".questao-2")
% (c2m211p2p 7 "questao-2")
% (c2m211p2a "questao-2")
{\bf Questão 2.}
\T(Total: 6.0 pts)
\msk
Seja $(*)$ esta EDO daqui:
%
$$\D \frac{dy}{dx} = \frac{x}{y}$$
a) \B(1.0 pts) Desenhe o campo de direções dela. Obs:
Faça tracinhos nos pontos com $x,y∈\{-2,-1,0,1,2\}$.
\ssk
b) \B(1.0 pts) Encontre a solução geral dela e teste-a.
\ssk
c) \B(1.0 pts) Encontre a solução da $(*)$ que passa pelo
ponto $(3,2)$. Diga qual é o domínio dela e teste-a.
\ssk
d) \B(1.0 pts) Encontre a solução da $(*)$ que passa pelo
ponto $(-1,-3)$. Diga qual é o domínio dela e teste-a.
% Dica:
% (c2m202p2p 20 "questao-2-gab")
% (c2m202p2a "questao-2-gab")
% (c2m202edovsa "videos")
\newpage
% «questao-2-cont» (to ".questao-2-cont")
% (c2m211p2p 8 "questao-2-cont")
% (c2m211p2a "questao-2-cont")
{\bf Questão 2 (cont.)}
\msk
e) \B(0.0 pts) Agora \ColorRed{defina, usando a sintaxe certa,} as
funções $f_b(x)$, $f_c(x)$, $f_d(x)$ que correspondem às funções
que você obteve nos itens b, c e d. Este item vale 0 pontos
\ColorRed{mas se você errar ele todos os próximos itens desta questão
vão ser anulados}.
\bsk
\ColorRed{\bf MUITO IMPORTANTE:} aqui a sua resposta
\ColorRed{\underline{\bf TEM QUE}} começar com um ``sejam'' --- se você
não escrever o ``sejam'' eu vou considerar que
a sua resposta está errada.
\bsk
Obs: na definição da sua $f_b(x)$ você pode usar `$\pm$',
mas nas outros não.
\newpage
{\bf Questão 2 (cont.)}
\msk
f) \B(1.0 pts) Descubra qual é a substituição da forma
%
$$\pfo{EDOVSG2} \;
\scalebox{0.8}{$
\bmat{
f(x) := \Rdq \\
F(x) := \Rdq \\
g(y) := \Rdq \\
G(y) := \Rdq \\
G^{-1}(x) := \Rdq \\
C_3 := \Rdq \\
}
$}
$$
que resolve a EDO $(*)$ e dá a sua solução $y=f_c(x)$.
Escreva ela por extenso como você fez no item 1a ---
mas note que agora estamos usando a \pfo{EDOVSG2}.
\msk
g) \B(1.0 pts) Faça a mesma coisa para a $y=f_d(x)$.
\newpage
% «gabarito» (to ".gabarito")
% (c2m211p2p 10 "gabarito")
% (c2m211p2a "gabarito")
\thispagestyle{empty}
\begin{center}
\vspace*{2.0cm}
{\bf \Large Gabarito}
\end{center}
\newpage
% «questao-1a-gab» (to ".questao-1a-gab")
% (c2m211p2p 11 "questao-1a-gab")
% (c2m211p2a "questao-1a-gab")
{\bf Questão 1a}
\sa{f(x)}{f(x)}
\sa{F(x)}{F(x)}
\sa{g(y)}{g(y)}
\sa{G(y)}{G(y)}
\sa{GiGy}{G^{-1}(G(y))}
\sa{GiFx}{G^{-1}(F(x) + C_3)}
\sa{Gix}{foo}
\sa{f(x)}{2x+3}
\sa{F(x)}{x^2 + 3x}
\sa{g(y)}{y^4}
\sa{G(y)}{y^5 + 2}
\sa {Gix}{\sqrt[5]{x - 2}}
\sa{GiGy}{\sqrt[5]{(y^5 + 2) - 2}}
\sa{GiFx}{\sqrt[5]{(x^2 + 3x + C_3) - 2}}
\def\SubstQuestaoUm{
\bmat{ f(x) := 2x+3 \\
F(x) := x^2+3x \\
g(y) := y^4 \\
G(y) := y^5 + 2 \\
G^{-1}(x) := \sqrt[5]{x - 2} \\
}
}
\def\EDOVSGoneTmp{
\left(
\begin{array}{rcl}
\D \frac{dy}{dx} &=& \D \frac{\ga{f(x)}}{\ga{g(y)}} \\ [10pt]
\ga{g(y)} \, dy &=& (\ga{f(x)}) \, dx \\ [5pt]
∫ \ga{g(y)} \, dy &=& ∫ \ga{f(x)} \, dx \\
\rotl{=}\ph{mm} & & \ph{mm}\rotl{=} \\[-5pt]
\ga{G(y)} + C_1 & & \ga{F(x)} + C_2 \\
[5pt]
\ga{G(y)} + C_1 &=& \ga{F(x)} + C_2 \\ [5pt]
\ga{G(y)} &=& \ga{F(x)} + C_2 - C_1 \\
&=& \ga{F(x)} + C_3 \\
[5pt]
\ga{GiGy} &=& \ga{GiFx} \\
\rotl{=}\ph{mm} & & \\[-5pt]
y\ph{mm} & & \\
\end{array}
\right)
}
\def\SubstQuestaoUmTmp{
\bmat{ f(x) := \ga{f(x)} \\
F(x) := \ga{F(x)} \\
g(y) := \ga{g(y)} \\
G(y) := \ga{G(y)} \\
G^{-1}(x) := \ga{Gix} \\
}
}
$$\scalebox{0.6}{$
\EDOVSGone
\SubstQuestaoUmTmp
=
\EDOVSGoneTmp
=
\pfo{Q1}
$}
$$
\newpage
% «questoes-1bc-gab» (to ".questoes-1bc-gab")
% (c2m211p2p 12 "questoes-1bc-gab")
% (c2m211p2a "questoes-1bc-gab")
{\bf Questões 1b e 1c}
\def\ddy{\frac{d}{dy}}
1b) \qquad
$
\begin{array}[t]{rcl}
y & = & \sen x \\
f(x) & = & \sen x \\
\dydx &\eqq& \frac{2x+3}{y^4} \\
\eqvs{}{ii} & & \eqvs{m}{} \\[-5pt]
\ddx f(x) & & \frac{2x+3}{(\sen x)^4} \\
\eqvs{}{m} & & \\[-5pt]
\cos x & & \\
\cos x &\neq& \frac{2x+3}{(\sen x)^4} \\
\end{array}
$
\bsk
1c) \qquad
$
\begin{array}[t]{rcl}
\inty{y^4} &\eqq& y^5 + 2 + C_1 \\
y^4 &\eqq& \ddy(y^5 + 2 + C_1) \\
& =& 5y^4 \\
y^4 &\neq& 5y^4 \\
\inty{y^4} &\neq& y^5 + 2 + C_1 \\
\end{array}
$
\newpage
% «questoes-1def-gab» (to ".questoes-1def-gab")
% (c2m211p2p 13 "questoes-1def-gab")
% (c2m211p2a "questoes-1def-gab")
{\bf Questões 1d até 1f}
\msk
1d) \qquad
$
\begin{array}[t]{rcl}
\intx{2x+3} &\eqq& x^2 + 3x + C_2 \\
2x+3 &\eqq& \ddx(x^2 + 3x + C_2) \\
& =& 2x + 3 \\
\intx{2x+3} & = & x^2 + 3x + C_2 \\
\end{array}
$
\bsk
1e) A igualdade (6) é $y^5 + 2 + C_1 = x^2 + 3x + C_2$.
Se $x=-2$, $y=1$, $C_1=10$, podemos reescrevê-la
como: $1^5 + 2 + 10 = (-2)^2 + 3(-2) + C_2$; e aí
$C_2 = 1^5 + 2 + 10 - ((-2)^2 + 3(-2)) = 13 - (-2) = 15$.
\bsk
1f) \qquad
$\begin{array}[t]{rcl}
y &\eqq& \sqrt[5]{(y^5 + 2) - 2} \\
& =& \sqrt[5]{y^5} \\
& =& y \\
\end{array}
$
\newpage
% «questoes-2ab-gab» (to ".questoes-2ab-gab")
% (c2m211p2p 14 "questao-2-gab")
% (c2m211p2a "questao-2-gab")
{\bf Questões 2a e 2b}
% (find-latexscan-links "C2" "20210920_gab_1b")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_gab_1b.pdf")
%\includegraphics[height=8cm]{2021-1-C2/20210920_gab_1b.pdf}
2a) \quad
% (find-latexscan-links "C2" "20210920_gab_2a")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_gab_2a.pdf")
$\myvcenter{
\includegraphics[height=3cm]{2021-1-C2/20210920_gab_2a.pdf}
}$
%
\qquad
%
2b) \quad
% (find-latexscan-links "C2" "20210920_gab_2b")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_gab_2b.pdf")
$\myvcenter{
\includegraphics[height=5cm]{2021-1-C2/20210920_gab_2b.pdf}
}$
(Falta o teste da 2b)
\newpage
% «questoes-2cd-gab» (to ".questoes-2cd-gab")
% (c2m211p2p 15 "questoes-2cd-gab")
% (c2m211p2a "questoes-2cd-gab")
{\bf Questões 2c e 2d}
\msk
2c)
\quad
%
% (find-latexscan-links "C2" "20210920_gab_2c")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_gab_2c.pdf")
$\myvcenter{
\includegraphics[height=3cm]{2021-1-C2/20210920_gab_2c.pdf}
}$
%
\quad
%
$\begin{array}{l}
\text{Domínio:} \\
\setofst{x∈\R}{x^2 - 5 ≥ 0} \\
= \;\; \setofst{x∈\R}{x^2 ≥ 5} \\
= \;\; \setofst{x∈\R}{|x| ≥ \sqrt{5}} \\
= \;\; (-∞,\sqrt{5}] ∪ [\sqrt{5},+∞) \\
\end{array}
$
(Falta o teste)
\bsk
2d)
%
$
% (find-latexscan-links "C2" "20210920_2d_solucao")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_2d_solucao.pdf")
\myvcenter{
\includegraphics[height=3cm]{2021-1-C2/20210920_2d_solucao.pdf}
}
%
\quad
%
\begin{array}{c}
% (find-latexscan-links "C2" "20210920_2d_dominio")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_2d_dominio.pdf")
\myvcenter{
\includegraphics[height=1.5cm]{2021-1-C2/20210920_2d_dominio.pdf}
}
%
\quad
%
% (find-latexscan-links "C2" "20210920_2d_teste")
% (find-xpdf-page "~/LATEX/2021-1-C2/20210920_2d_teste.pdf")
\myvcenter{
\includegraphics[height=2.0cm]{2021-1-C2/20210920_2d_teste.pdf}
}
\end{array}
$
\newpage
% «questao-2e-gab» (to ".questao-2e-gab")
% (c2m211p2p 16 "questao-2e-gab")
% (c2m211p2a "questao-2e-gab")
{\bf Questão 2e}
\msk
2e) Sejam:
%
$\begin{array}[t]{rcl}
f_b(x) &=& \pm \sqrt{x^2 + C_4}, \\
f_c(x) &=& \sqrt{x^2 - 5}, \\
f_d(x) &=& - \sqrt{x^2 + 8}. \\
\end{array}
$
\bsk
\bsk
\bsk
Truque pra 2f e pra 2g:
Lembre que na 2b nós fizemos essa passagem aqui:
%
$$\begin{array}{rcl}
\frac12 y^2 &=& \frac12 x^2 + C_3 \\
y^2 &=& x^2 + C_4 \\
\end{array}
$$
Nela ficava implícito que $2C_3 = C_4$.
\newpage
% «questoes-2fg-gab» (to ".questoes-2fg-gab")
% (c2m211p2p 17 "questoes-2fg-gab")
% (c2m211p2a "questoes-2fg-gab")
\bsk
2f) Para $f_b(x) = \pm \sqrt{x^2 + C_4}$:
\ph{2f)} (Ou: $f_b(x) = \pm \sqrt{x^2 + 2C_3}$ ...)
%
\def\SubstQuestaoUmTmp{
\bmat{ f(x) := x \\
F(x) := \frac12 x^2 \\
g(y) := y \\
G(y) := \frac12 y^2 \\
G^{-1}(x) := \pm \sqrt{2x} \\
C_3 := \frac12 C_4 \\
}
}
\def\EDOVSGtwoTmp{
\left(
\begin{array}{rcl}
\D \frac{dy}{dx} &=& \D \frac{x}{y} \\ [10pt]
y &=& \pm \sqrt{2(\frac12 x^2 + \frac12 C_4)} \\
\end{array}
\right)
}
%
$$\scalebox{0.7}{$
\EDOVSGtwo
\SubstQuestaoUmTmp
=
\EDOVSGtwoTmp
$}
$$
\bsk
\bsk
2g) Para $f_c(x) = \sqrt{x^2 - 5}$:
%
\def\SubstQuestaoUmTmp{
\bmat{ f(x) := x \\
F(x) := \frac12 x^2 \\
g(y) := y \\
G(y) := \frac12 y^2 \\
G^{-1}(x) := \sqrt{2x} \\
C_3 := - \frac52 \\
}
}
\def\EDOVSGtwoTmp{
\left(
\begin{array}{rcl}
\D \frac{dy}{dx} &=& \D \frac{x}{y} \\ [10pt]
y &=& \sqrt{2(\frac12 x^2 + (-\frac52))} \\
\end{array}
\right)
}
%
$$\scalebox{0.7}{$
\EDOVSGtwo
\SubstQuestaoUmTmp
=
\EDOVSGtwoTmp
$}
$$
%\printbibliography
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
% ____ _ _
% | _ \(_)_ ___ _(_)_______
% | | | | \ \ / / | | | |_ / _ \
% | |_| | |\ V /| |_| | |/ / __/
% |____// | \_/ \__,_|_/___\___|
% |__/
%
% «djvuize» (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-fline "~/2021.1-C2/")
# (find-fline "~/LATEX/2021-1-C2/")
# (find-fline "~/bin/djvuize")
cd /tmp/
for i in *.jpg; do echo f $(basename $i .jpg); done
f () { rm -v $1.pdf; textcleaner -f 50 -o 5 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf; textcleaner -f 50 -o 10 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf; textcleaner -f 50 -o 20 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 15" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 30" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 45" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 75" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.5" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.25" $1.pdf; xpdf $1.pdf }
f () { cp -fv $1.png $1.pdf ~/2021.1-C2/
cp -fv $1.pdf ~/LATEX/2021-1-C2/
cat <<%%%
% (find-latexscan-links "C2" "$1")
%%%
}
f 20210920_2d_dominio
f 20210920_2d_solucao
f 20210920_2d_teste
% __ __ _
% | \/ | __ _| | _____
% | |\/| |/ _` | |/ / _ \
% | | | | (_| | < __/
% |_| |_|\__,_|_|\_\___|
%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2021-1-C2-P2 veryclean
make -f 2019.mk STEM=2021-1-C2-P2 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2p2"
% ee-tla: "c2m211p2"
% End: