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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2022-2-C3-VR.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2022-2-C3-VR.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2022-2-C3-VR.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2022-2-C3-VR.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2022-2-C3-VR.pdf"))
% (defun e () (interactive) (find-LATEX "2022-2-C3-VR.tex"))
% (defun o () (interactive) (find-LATEX "2022-2-C3-P2.tex"))
% (defun u () (interactive) (find-latex-upload-links "2022-2-C3-VR"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2022-2-C3-VR.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2022-2-C3-VR")
% (find-pdf-page "~/LATEX/2022-2-C3-VR.pdf")
% (find-sh0 "cp -v ~/LATEX/2022-2-C3-VR.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2022-2-C3-VR.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2022-2-C3-VR.pdf")
% file:///home/edrx/LATEX/2022-2-C3-VR.pdf
% file:///tmp/2022-2-C3-VR.pdf
% file:///tmp/pen/2022-2-C3-VR.pdf
% http://angg.twu.net/LATEX/2022-2-C3-VR.pdf
% (find-LATEX "2019.mk")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Piecewise1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Pict3D1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v C2Subst1.lua C2Formulas1.lua ~/LATEX/")
% (find-CN-aula-links "2022-2-C3-VR" "3" "c3m222vr" "c3vr")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.title» (to "title")
% «.links» (to "links")
% «.questao-1» (to "questao-1")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c3m222vr" "2022-2-C3-VR")
% (code-eevvideo "c3m222vr" "2022-2-C3-VR")
% (code-eevlinksvideo "c3m222vr" "2022-2-C3-VR")
% (find-c3m222vrvideo "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")
%\usepackage{emaxima} % (find-LATEX "emaxima.sty")
%
% (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 "Piecewise1.lua" -- (find-LATEX "Piecewise1.lua")
% %L dofile "QVis1.lua" -- (find-LATEX "QVis1.lua")
% %L dofile "Pict3D1.lua" -- (find-LATEX "Pict3D1.lua")
% %L dofile "C2Formulas1.lua" -- (find-LATEX "C2Formulas1.lua")
% %L Pict2e.__index.suffix = "%"
% \pu
% \def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
% \def\pictaxesstyle{\linethickness{0.5pt}}
% \def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
% \celllower=2.5pt
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\u#1{\par{\footnotesize \url{#1}}}
\def\drafturl{http://angg.twu.net/LATEX/2022-2-C3.pdf}
\def\drafturl{http://angg.twu.net/2022.2-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% «defs-T-and-B» (to ".defs-T-and-B")
\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")
% (c3m222vrp 1 "title")
% (c3m222vra "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 3 - 2022.2}
\bsk
Prova de reposição (VR)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.2-C3.html}
\end{center}
\newpage
% «links» (to ".links")
\newpage
% ___ _ _
% / _ \ _ _ ___ ___| |_ __ _ ___ / |
% | | | | | | |/ _ \/ __| __/ _` |/ _ \ | |
% | |_| | |_| | __/\__ \ || (_| | (_) | | |
% \__\_\\__,_|\___||___/\__\__,_|\___/ |_|
%
% «questao-1» (to ".questao-1")
% (c3m222vrp 2 "questao-1")
% (c3m222vra "questao-1")
{\bf Questão 1.}
\scalebox{0.45}{\def\colwidth{12cm}\firstcol{
\vspace*{-0.5cm}
\T(Total: 10.0 pts)
\msk
Lembre que ``$∂C$'' é ``a fronteira do conjunto $C$'' e que se $F$ é
uma função de $\R^2$ em $\R$ então $F^{-1}$ é a ``imagem inversa de
$F$'', que é definida de um jeito quando o argumento é um número e de
outro jeito quando o argumento é um conjunto. Se $a∈\R$ e $C⊂\R$,
então:
%
$$\begin{array}{rcl}
F^{-1}(a) &=& \setofxyst{F(x,y)=a} \\
F^{-1}(C) &=& \setofxyst{F(x,y)∈C} \\
\end{array}
$$
Sejam:
%
$$\begin{array}{rcl}
F(x,y) &=& x^2 + y^2, \\
G(x,y) &=& xy, \\
C_1 &=& F^{-1}(4) \\
C_2 &=& F^{-1}([-1,4]) \\
C_3 &=& G^{-1}(1) \\
C_4 &=& G^{-1}([-1,1]) \\
C_5 &=& C_2∩C_4 \\
C_6 &=& ∂(C_5) \\
D &=& C_5 \\
(x_0,x_1) &=& (1,0) \\
H(x,y) &=& (x-x_0)^2 - (y-y_0)^2 \\
M &:& \hspace*{-0.25cm}
\begin{array}[t]{rcl} D & \to& \R \\
(x,y) &\mto& H(x,y) \\
\end{array} \\
\end{array}
$$
}\anothercol{
a) \B (1.0 pts) Desenhe a região $D$.
\msk
b) \B (1.0 pts) Dê as coordenadas aproximadas dos ``bicos'' de $∂D$.
\msk
c) \B (1.0 pts) Desenhe as curvas de nível da função $M$.
\msk
d) \B (1.0 pts) Diga quais são os mínimos locais, máximos locais e
pontos de sela da função $M$.
\msk
e) \B (1.0 pts) Chame os pontos do item (d) de
$P_1, P_2, \ldots, P_{N}$ e dê as coordenadas aproximadas de cada um
deles. Repare que eu não estou dizendo quem é $N$ --- você vai ter que
descobrir.
\msk
f) \B (1.0 pts) Para cada um dos pontos $P_i=(x_i,y_i)$ do item
anterior calcule $∇H(x_i,y_i)$ e represente graficamente
$(x_i,y_i)+∇H(x_i,y_i)$.
\bsk
\bsk
Pra conseguir fazer os itens acima você vai ter que fazer um monte de
``passos intermediários'' --- como desenhar alguns conjuntos,
diagramas de numerozinhos e curvas de nível. Eu não vou dizer quais
são esses passos intermediários --- você vai ter que descobrir quais
eles são, e escrever cada um deles de um jeito legível.
\bsk
\bsk
g) \B (4.0 pts) Descubra quais são esses ``passos intermediários'' e
faça-os!
\bsk
{\bf Dica 1:} é quase impossível fazer os itens acima em ordem, tipo
primeiro o a, depois o b, etc, e por último o g. Faça eles na ordem
que conseguir.
\ssk
{\bf Dica 2:} alguns itens podem te ajudar a descobrir que você fez um
outro item errado. Se isso acontecer ou conserte o item errado ou
explique em português o que você descobriu --- respostas incoerentes
umas com as outras podem fazer você perder pontos.
}}
\newpage
\vspace*{-0.7cm}
% (find-xpdf-page "~/LATEX/2022-2-C3/C3-VR-Esboco_de_gabarito.pdf")
\includegraphics[height=8cm]{2022-2-C3/C3-VR-Esboco_de_gabarito.pdf}
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
f2(x) := - sqrt(x^2 + 2*C3);
diff(f2(x), x);
estrela : dydx = -x / y;
subst([dydx=diff(f2(x),x), y=f2(x)], estrela);
* (eepitch-maxima)
* (eepitch-kill)
* (eepitch-maxima)
** (find-es "maxima" "2022-2-C3-P2-Q4")
F1(x,y) := (x-1) * (y-1);
F1(x,y) := (x-1) * (y-0);
F1(x,y) := (x-1)^2 - (y-0)^2;
F0(x,y) := x * y;
G(x,y) := x^2 - y^2;
H(x,y) := x^2 + 4*y^2;
[minx,miny,maxx,maxy] : [-4,-4,4,4];
[minx,miny,maxx,maxy] : [-2,-2,2,2];
clevel (clr, eq) := [color=clr, implicit(eq, x,minx,maxx, y,miny,maxy)];
clevels(clr, F, zs) := map(lambda([z], clevel(clr, F=z)), zs);
clevels : append(clevels(black, x*y, [1,-1]),
clevels(black, x^2+y^2, [4]),
clevels(red, (x-1)^2-y^2, [0,1,2,3,4,5,6,7]),
clevels(violet, (x-1)^2-y^2, [-4,-3,-2,-1,0])
);
draw2d(clevels);
clevels : [clevel(black, F0(x,y)=1),
clevel(black, F0(x,y)=-1),
clevel(red, F1(x,y)=-2),
clevel(red, F1(x,y)=-1.5),
clevel(red, F1(x,y)=-1),
clevel(red, F1(x,y)=-0.5),
clevel(red, F1(x,y)=0),
clevel(red, F1(x,y)=0.5),
clevel(red, F1(x,y)=1),
clevel(red, F1(x,y)=1.5),
clevel(red, F1(x,y)=2),
clevel(red, F1(x,y)=2.5),
clevel(red, F1(x,y)=3),
clevel(black, x^2+y^2=4)
];
draw2d(clevels);
clevels(red, x^2, [0,1,2,3]);
append([1,2], [3,4]);
clevels : [clevel(red, F(x,y)=1),
clevel(red, F(x,y)=2),
clevel(red, F(x,y)=3),
clevel(red, F(x,y)=4),
clevel(green, G(x,y)=1),
clevel(violet, H(x,y)=1)
];
clevels : [Eclevel(black, 16),
Hclevel(red, -4),
Hclevel(orange,-3),
Hclevel(green, 0),
Hclevel(blue, 3),
Hclevel(violet, 4)
];
draw2d(clevels);
Hlevel(z) := implicit(H(x,y)=z, x,-4,4, y,-4,4);
Elevel(z) := implicit(E(x,y)=z, x,-4,4, y,-4,4);
Hclevel(clr,z) := [color=clr, Hlevel(z)];
Eclevel(clr,z) := [color=clr, Elevel(z)];
% ____ _ _
% | _ \(_)_ ___ _(_)_______
% | | | | \ \ / / | | | |_ / _ \
% | |_| | |\ V /| |_| | |/ / __/
% |____// | \_/ \__,_|_/___\___|
% |__/
%
% «djvuize» (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-fline "~/2022.2-C3/")
# (find-fline "~/LATEX/2022-2-C3/")
# (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 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 ~/2022.2-C3/
cp -fv $1.pdf ~/LATEX/2022-2-C3/
cat <<%%%
% (find-latexscan-links "C3" "$1")
%%%
}
f 20201213_area_em_funcao_de_theta
f 20201213_area_em_funcao_de_x
f 20201213_area_fatias_pizza
% __ __ _
% | \/ | __ _| | _____
% | |\/| |/ _` | |/ / _ \
% | | | | (_| | < __/
% |_| |_|\__,_|_|\_\___|
%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2022-2-C3-VR veryclean
make -f 2019.mk STEM=2022-2-C3-VR pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3vr"
% ee-tla: "c3m222vr"
% End: