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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2023-2-C3-P1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-2-C3-P1.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-2-C3-P1.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2023-2-C3-P1.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-2-C3-P1.pdf"))
% (defun e () (interactive) (find-LATEX "2023-2-C3-P1.tex"))
% (defun o () (interactive) (find-LATEX "2022-2-C3-P1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-2-C3-P1"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-2-C3-P1.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2023-2-C3-P1")
% (find-pdf-page "~/LATEX/2023-2-C3-P1.pdf")
% (find-sh0 "cp -v ~/LATEX/2023-2-C3-P1.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2023-2-C3-P1.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2023-2-C3-P1.pdf")
% file:///home/edrx/LATEX/2023-2-C3-P1.pdf
% file:///tmp/2023-2-C3-P1.pdf
% file:///tmp/pen/2023-2-C3-P1.pdf
% http://anggtwu.net/LATEX/2023-2-C3-P1.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps "Caepro5 Piecewise2 Maxima2 Cabos3 Numerozinhos1")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2023-2-C3-P1" "C3" "c3m232p1" "c3p1")
% (find-MM-aula-links "2023-2-C3-P1" "3" "c3m232p1" "c3p1")
% «.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")
% «.title» (to "title")
% «.links» (to "links")
% «.questao-1» (to "questao-1")
% «.questao-2» (to "questao-2")
% «.questao-1-grids» (to "questao-1-grids")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c3m232p1" "2023-2-C3-P1")
% (code-eevvideo "c3m232p1" "2023-2-C3-P1")
% (code-eevlinksvideo "c3m232p1" "2023-2-C3-P1")
% (find-c3m232p1video "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}
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\drafturl{http://anggtwu.net/LATEX/2023-2-C3.pdf}
\def\drafturl{http://anggtwu.net/2023.2-C3.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")
%L V = MiniV
%L v = V.fromab
\pu
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c3m232p1p 1 "title")
% (c3m232p1a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 3 - 2023.2}
\bsk
P1 (primeira prova)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://anggtwu.net/2023.2-C3.html}
\end{center}
\newpage
% «links» (to ".links")
% (c3m232p1p 2 "links")
% (c3m232p1a "links")
\newpage
% ___ _ _
% / _ \ _ _ ___ ___| |_ __ _ ___ / |
% | | | | | | |/ _ \/ __| __/ _` |/ _ \ | |
% | |_| | |_| | __/\__ \ || (_| | (_) | | |
% \__\_\\__,_|\___||___/\__\__,_|\___/ |_|
%
% «questao-1» (to ".questao-1")
% (c3m232p1p 2 "questao-1")
% (c3m232p1a "questao-1")
{\bf Questão 1}
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
\vspace*{-0.5cm}
\T(Total: 5.0 pts)
O diagrama de numerozinhos da última folha da prova corresponde a uma
superfície $z=F(x,y)$ que tem 5 faces. Também é possível interpretá-lo
como uma superfície com 6 ou mais faces, mas vamos considerar que a
superfície com só 5 faces é que é a correta.
\msk
a) \B (1.0 pts) Mostre como dividir o plano em 5 polígonos que são as
projeções destas faces no plano do papel.
\msk
b) \B (1.0 pts) Chame estas faces de face NW (``noroeste''), N (``norte''),
W (``oeste''), C (``centro''), R (``resto''), e chame as equações dos
planos delas de $F_{NW}(x,y)$, $F_{N}(x,y)$, $F_{W}(x,y)$, $F_{C}(x,y)$, e
$F_R(x,y)$. Dê as equações destes planos.
\msk
c) \B (1.0 pts) Sejam:
%
$$\begin{array}{rcl}
P_N &=& \setofxyzst{z = F_N(x,y)}, \\
P_C &=& \setofxyzst{z = F_C(x,y)}, \\
r &=& P_N ∩ P_C. \\
\end{array}
$$
Represente a reta $r$ graficamente como numerozinhos.
}\anothercol{
d) \B (0.5 pts) Dê uma parametrização para a reta do item anterior.
Use notação de conjuntos.
\msk
e) \B (0.5 pts) Seja
%
$$A \;=\; \{0,1,\ldots,10\} × \{0,1,\ldots,10\};$$
note que os numerozinhos do diagrama de numerozinhos estão todos
sobre pontos de $A$. Para cada ponto $(x,y)∈A$ represente
graficamente $(x,y)+\frac13 \vec∇F(x,y)$.
\ssk
Obs: quando $\vec∇F(x,y)=0$ desenhe uma bolinha preta sobre o ponto
$(x,y)$, e quando $\vec∇F(x,y)$ não existir faça um `$×$' sobre o
numerozinho que está no ponto $(x,y)$.
\msk
f) \B (1.0 pts) Sejam
%
$$\begin{array}{rcl}
Q(t) &=& (0,2) + t\VEC{1,1}, \\
(x(t),y(t)) &=& Q(t), \\
h(t) &=& F(x(t),y(t)). \\
\end{array}
$$
Faça o gráfico da função $h(t)$. Considere que o domínio dela é o
intervalo $[0,6]$.
}}
\newpage
% ___ _ ____
% / _ \ _ _ ___ ___| |_ __ _ ___ |___ \
% | | | | | | |/ _ \/ __| __/ _` |/ _ \ __) |
% | |_| | |_| | __/\__ \ || (_| | (_) | / __/
% \__\_\\__,_|\___||___/\__\__,_|\___/ |_____|
%
% «questao-2» (to ".questao-2")
% (c3m232p1p 3 "questao-2")
% (c3m232p1a "questao-2")
{\bf Questão 2}
\scalebox{0.55}{\def\colwidth{10cm}\firstcol{
\vspace*{-0.5cm}
\T(Total: 3.0 pts)
Sejam
%
$$\begin{array}{rcl}
u(x,y) &=& y-2x, \\
v(x,y) &=& x+y, \\
F(x,y) &=& u(x,y)v(x,y) \\
&=& 2x^2 -xy -y^2. \\
\end{array}
$$
Nesta questão você vai ter que fazer várias cópias do diagrama de
numerozinhos da função $F(x,y)$ para os pontos com
$x,y∈\{-2,-1,0,1,2\}$. Os numerozinhos vão ser estes aqui:
%
$$\begin{array}{rrrrr}
0 & 4 & 4 & 0 & -8 \\
-5 & 0 & 1 & -2 & -9 \\
-8 & -2 & 0 & -2 & -8 \\
-9 & -2 & 1 & 0 & -5 \\
-8 & 0 & 4 & 4 & 0 \\
\end{array}
$$
a) \B (1.0 pts) Desenhe o ``campo gradiente'' da função $F$ nestes
pontos, mas multiplicando cada $\vec∇F(x,y)$ por $\frac{1}{10}$ pros
vetores não ficarem uns em cima dos outros. Deixa eu traduzir isso pra
termos mais básicos: faça uma cópia do diagrama de numerozinhos da
$F(x,y)$, e sobre cada $(x,y)$ com $x,y∈\{-2,-1,0,1,2\}$ desenhe a
seta $(x,y)+\frac{1}{10}\vec∇F(x,y)$.
}\anothercol{
b) \B (2.0 pts) Faça uma outra cópia desse diagrama de numerozinhos
e desenhe sobre ela as curvas de nível da função $F(x,y)$ para
$z=0$, $z=-2$, $z=-5$, $z=1$ e $z=2$.
\bsk
{\bf Dicas:}
1) O vetor gradiente num ponto $(x,y)$ é sempre ortogonal à curva de
nível que passa pelo ponto $(x,y)$.
2) Faça quantos rascunhos quiser. Eu só vou corrigir seus desenhos
pros itens (a) e (b) que disserem ``versão final'', e eles têm que
ser os mais caprichados possíveis.
}}
\newpage
{\bf Questão 3}
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
\vspace*{-0.5cm}
\T(Total: 3.0 pts)
Se $F:\R^2→\R$, a matriz hessiana de $F$ é definida desta forma:
%
$$HF(x,y) \;=\; \pmat{F_{xx}(x,y) & F_{xy}(x,y) \\
F_{xy}(x,y) & F_{yy}(x,y) \\
}
$$
Sejam:
%
$$\begin{array}{rcl}
u &=& x+y-5 \\
v &=& y-2 \\
s &=& 3+uv \\
p &=& 4+u^2+v^2 \\
S(x,y) &=& 3+u(x,y)v(x,y) \\
P(x,y) &=& 4+u(x,y)^2+v(x,y)^2 \\
(x_0,y_0) &=& (3,2) \\
A &=& \{-1,0,1\} \\
B &=& \setofst{(x_0+Δx,y_0+Δy)}{Δx,Δy∈B} \\
\end{array}
$$
a) \B (1.0 pts) Calcule $HS$ e $HP$.
b) \B (2.0 pts) Desenhe os diagramas de numerozinhos de $u$, $v$, $s$
e $p$ ``nos 9 pontos em torno de $(x_0,y_0)$'' -- ou seja, nos pontos
de $B$.
}\anothercol{
}}
\newpage
% «barranco-defs» (to ".barranco-defs")
% (c3m222p1p 2 "barranco-defs")
% (c3m222p1p 5 "barranco-defs")
% (c3m222p1a "barranco-defs")
% (find-angg "GNUPLOT/2023-2-C3-P1.dem")
% (find-anggfile "GNUPLOT/2023-2-C3-P1.dem" "bgprocess")
% (find-eepitch-intro "3.3. `eepitch-preprocess-line'")
% (setq eepitch-preprocess-regexp "")
% (setq eepitch-preprocess-regexp "^%?%L ?")
%
%%L * (eepitch-lua51)
%%L * (eepitch-kill)
%%L * (eepitch-lua51)
%%L Path.prependtopath "~/LUA/?.lua"
%L require "Cabos3"
%L require "Numerozinhos1"
%L PictBounds.setbounds(v(0,0), v(11,11))
%L
%L bigstr1 = [[
%L 6 6 6 6 4 2 0 0 0 0 0
%L 6 6 6 6 4 2 0 0 0 0 0
%L 6 6 6 6 4 2 0 0 0 0 0
%L 5 5 5 5 4 2 0 0 0 0 0
%L 4 4 4 4 3 2 0 0 0 0 0
%L 3 3 3 3 2 1 0 0 0 0 0
%L 2 2 2 2 1 0 0 0 0 0 0
%L 1 1 1 1 0 0 0 0 0 0 0
%L 0 0 0 0 0 0 0 0 0 0 0
%L 0 0 0 0 0 0 0 0 0 0 0
%L 0 0 0 0 0 0 0 0 0 0 0
%L ]]
%L bigstr2 = [[
%L 6 - 6 - 6 - A - 4 - 2 - B - 0 - 0 - 0 - 0
%L | . | . | . | . | . | . | . | . | . | . |
%L 6 - 6 - 6 - 6 - 4 - 2 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | . | . | . | . | . | . | . |
%L C - 6 - 6 - D - 4 - 2 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | \ | . | . | . | . | . | . |
%L 5 - 5 - 5 - 5 - 4 - 2 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | . | \ | . | . | . | . | . |
%L 4 - 4 - 4 - 4 - 3 - 2 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | . | . | \ | . | . | . | . |
%L 3 - 3 - 3 - 3 - 2 - 1 - E - 0 - 0 - 0 - 0
%L | . | . | . | . | . | / | . | . | . | . |
%L 2 - 2 - 2 - 2 - 1 - 0 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | . | / | . | . | . | . | . |
%L 1 - 1 - 1 - 1 - 0 - 0 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | / | . | . | . | . | . | . |
%L F - 0 - 0 - G - 0 - 0 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | . | . | . | . | . | . | . |
%L 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0
%L | . | . | . | . | . | . | . | . | . | . |
%L 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0
%L ]]
%L clabels = CabosNaDiagonal.from(bigstr2)
%L lbls = clabels.strgrid:labels()
%L spec = lbls:subst("A--D--C B--E--G--F D--E D--G")
%L ns = Numerozinhos.from(0, 0, bigstr1)
%L p1 = ns:show0 {u="25pt"}:sa("barranco")
%L ns:setspec(spec)
%L p2 = ns:show0():sa("barranco 2")
%L p3 = Pict { p1, p2 }
%L p4 = Pict { p1, p2, [[\ga{barranco} \ga{barranco com linhas}]] }
%L p3:output()
%%L = p4:show("")
%%L = Show.bigstr
%%L * (etv)
\pu
\newpage
% «questao-1-grids» (to ".questao-1-grids")
\def\barra{\scalebox{0.35}{\ga{barranco}}}
\def\barras{\barra \quad \barra \quad \barra}
$\begin{array}{l}
\barras \\ \\[-5pt]
\barras \\
\end{array}
$
\newpage
\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 "~/2023.2-C3/")
# (find-fline "~/LATEX/2023-2-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 ~/2023.2-C3/
cp -fv $1.pdf ~/LATEX/2023-2-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=2023-2-C3-P1 veryclean
make -f 2019.mk STEM=2023-2-C3-P1 pdf
% (find-pdfpages2-links "~/LATEX/" "2023-2-C3-P1")
% (find-pdfpages2-links "~/LATEX/" "2023-2-C3-P1" "-pp" "pages=5,fitpaper,landscape=true")
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3p1"
% ee-tla: "c3m232p1"
% End: