% (find-LATEX "2022-1-C3-VS.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2022-1-C3-VS.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2022-1-C3-VS.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2022-1-C3-VS.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2022-1-C3-VS.pdf"))
% (defun e () (interactive) (find-LATEX "2022-1-C3-VS.tex"))
% (defun o () (interactive) (find-LATEX "2022-1-C3-VS.tex"))
% (defun u () (interactive) (find-latex-upload-links "2022-1-C3-VS"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2022-1-C3-VS.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2022-1-C3-VS")
% (find-pdf-page   "~/LATEX/2022-1-C3-VS.pdf")
% (find-sh0 "cp -v  ~/LATEX/2022-1-C3-VS.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2022-1-C3-VS.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2022-1-C3-VS.pdf")
%   file:///home/edrx/LATEX/2022-1-C3-VS.pdf
%               file:///tmp/2022-1-C3-VS.pdf
%           file:///tmp/pen/2022-1-C3-VS.pdf
% http://angg.twu.net/LATEX/2022-1-C3-VS.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 "2022-1-C3-VS" "3" "c3m221vs" "c3vs")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.title»		(to "title")
% «.questao-1»		(to "questao-1")
% «.questao-1-gab»	(to "questao-1-gab")
%
% «.djvuize»		(to "djvuize")



% <videos>
% Video (not yet):
% (find-ssr-links     "c3m221vs" "2022-1-C3-VS")
% (code-eevvideo      "c3m221vs" "2022-1-C3-VS")
% (code-eevlinksvideo "c3m221vs" "2022-1-C3-VS")
% (find-c3m221vsvideo "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")
%
%\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 "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/2022-1-C3.pdf}
\def\drafturl{http://angg.twu.net/2022.1-C3.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)}}



%  _____ _ _   _                               
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%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c3m221vsp 1 "title")
% (c3m221vsa   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 3 - 2022.1}

\bsk

Prova suplementar (VS)

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://angg.twu.net/2022.1-C3.html}

\end{center}

\newpage

%   ___                  _                _ 
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% | | | | | | |/ _ \/ __| __/ _` |/ _ \  | |
% | |_| | |_| |  __/\__ \ || (_| | (_) | | |
%  \__\_\\__,_|\___||___/\__\__,_|\___/  |_|
%                                           
% «questao-1»  (to ".questao-1")
% (c3m221vsp 2 "questao-1")
% (c3m221vsa   "questao-1")

% (c3m221mt2p 2 "questao")
% (c3m221mt2a   "questao")
% (find-angg "LUA/Pict2e1-1.lua" "Numerozinhos-test4")
%
% (c3m221nfp 26 "barranco")
% (c3m221nfa    "barranco")
%
%L Pict2e.bounds = PictBounds.new(v(0,0), v(7,8))
%L barranco_VS = Numerozinhos.from(0, 0, 
%L     [[ 4 4 4 4 4 4 4 4
%L        4 4 4 4 4 4 4 4
%L        3 3 3 4 4 4 4 4
%L        2 2 2 3 4 4 4 4
%L        1 1 1 2 3 4 4 4
%L        0 0 0 1 2 3 4 4
%L        0 0 0 0 1 2 2 2
%L        0 0 0 0 0 0 0 0
%L        0 0 0 0 0 0 0 0 ]])
%L barranco_VS_spec = [[ (0,7)--(2,7)--(6,3)--(7,3)
%L                       (2,3)--(2,7)  (4,1)--(6,3)
%L                       (0,3)--(2,3)--(4,1)--(7,1) ]]
%L barranco_VS:topict(                ):sa("barranco VS")    :output()
%L barranco_VS:topict(barranco_VS_spec):sa("barranco VS gab"):output()
\pu







{\bf Questão 1}


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

\vspace*{-0.4cm}

\T(Total: 10.0 pts)

O diagrama de numerozinhos abaixo corresponde a uma superfície
$z=F(x,y)$ que tem 5 faces. Também é possível interpretá-lo como uma
superfície com 7 ou mais faces, mas vamos considerar que a superfície
com só 5 faces é que é a correta.
%
$$\scalebox{1.0}{$\ga{barranco VS}$}$$

\msk

a) \B (2.0 pts) Mostre como dividir o plano em 5 polígonos que são as
projeções destas faces.

\msk

b) \B (1.0 pts) Chame estas faces de face C (``cima''), E
(``esquerda''), M (``meio''), D (``direita''), e B (``baixo''), e
chame as equações dos planos delas de $F_C(x,y)$, $F_E(x,y)$,
$F_M(x,y)$, $F_D(x,y)$, e $F_B(x,y)$. Dê as equações destes planos.


}\anothercol{

  c) \B (2.0 pts) O segmento de reta que é a fronteira entre as
  regiões M e D ``corresponde à interseção entre os planos $F_M(x,y)$
  e $F_D(x,y)$''. Faça as contas e explique em português o que a
  expressão entre aspas quer dizer.

  \msk

  d) \B (2.0 pts) Para cada um dos 5 polígonos do item (a) escolha um
  ponto com coordenadas inteiras ue esteja dentro desse polígono e não
  na fronteira dele. Chame esses pontos de $P_C$, $P_E$, $P_M$, $P_D$
  e $P_B$. Para cada um destes 5 pontos represente graficamente
  $P + \vec{∇}F(P)$.

  \msk

  e) \B (1.5 pts) Seja $Q(t)$ a seguinte trajetória:
  % 
  $$Q(t) \;=\;
    \begin{cases}
      (t,4)    & \text{quando $t≤6$}, \\
      (6,10-t) & \text{quando $6<t$}. \\
    \end{cases}
  $$

  Represente graficamente $Q(t) + Q'(t)$ para $t=1,3,5,7,9$.

  \msk

  f) \B (1.5 pts) Digamos que $(x,y) = (x(y),y(t)) = Q(t)$ e que
  $z=z(x,y)=z(x(y),y(t))$. Faça o gráfico da função
  $z(t)=z(x(y),y(t))$.


}}


\newpage

\def\rasc{\scalebox{0.9}{$\ga{barranco VS gab}$}}
\def\rasc{\scalebox{0.9}{$\ga{barranco VS}$}}
\def\rasc{\scalebox{0.9}{$\ga{barranco VS}$ \;\;}}

$$\begin{array}{ccc}
  \rasc & \rasc & \rasc \\ \\
  \rasc & \rasc & \rasc \\ \\
  \end{array}
$$


\newpage

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%                                             |___/             
% «questao-1-gab»  (to ".questao-1-gab")
% (c3m221vsp 3 "questao-1-gab")
% (c3m221vsa   "questao-1-gab")

{\bf Gabarito (muito incompleto)}

\bsk

$\scalebox{0.9}{$\ga{barranco VS gab}$}$





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

\end{document}

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%     |__/                       
%
% «djvuize»  (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")



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%                        
% <make>

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2022-1-C3-VS veryclean
make -f 2019.mk STEM=2022-1-C3-VS pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3vs"
% ee-tla: "c3m221vs"
% End:
