% (find-LATEX "2024-1-C3-P1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-1-C3-P1.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-1-C3-P1.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2024-1-C3-P1.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2024-1-C3-P1.pdf"))
% (defun e () (interactive) (find-LATEX "2024-1-C3-P1.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C3-P1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2024-1-C3-P1"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2024-1-C3-P1.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2024-1-C3-P1")
% (find-pdf-page   "~/LATEX/2024-1-C3-P1.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C3-P1.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C3-P1.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-1-C3-P1.pdf")
%   file:///home/edrx/LATEX/2024-1-C3-P1.pdf
%               file:///tmp/2024-1-C3-P1.pdf
%           file:///tmp/pen/2024-1-C3-P1.pdf
%  http://anggtwu.net/LATEX/2024-1-C3-P1.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2024-1-C3-P1" "3" "c3m241p1" "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")
% «.defs-V»		(to "defs-V")
% «.title»		(to "title")
% «.links»		(to "links")
% «.questao-2»		(to "questao-2")
% «.questao-3»		(to "questao-3")
% «.questao-4»		(to "questao-4")
% «.questao-1-grids»	(to "questao-1-grids")
% «.gab-1»		(to "gab-1")
% «.gab-2»		(to "gab-2")
% «.gab-3»		(to "gab-3")
% «.gab-4-diag»		(to "gab-4-diag")



\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")
%
% (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/2024-1-C3.pdf}
\def\drafturl{http://anggtwu.net/2024.1-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2024-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2024-1-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")
\pu

% «defs-V»  (to ".defs-V")
%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\pu


%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c3m241p1p 1 "title")
% (c3m241p1a   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 3 - 2024.1}

\bsk

P1 (primeira prova)

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.1-C3.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c3m241p1p 2 "links")
% (c3m241p1a   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{16cm}\firstcol{

{\footnotesize
\par \url{http://anggtwu.net/LATEX/2024-1-C3-dicas-pra-P1.pdf}
\par
}

}\anothercol{
}}

\newpage

%   ___                  _                _ 
%  / _ \ _   _  ___  ___| |_ __ _  ___   / |
% | | | | | | |/ _ \/ __| __/ _` |/ _ \  | |
% | |_| | |_| |  __/\__ \ || (_| | (_) | | |
%  \__\_\\__,_|\___||___/\__\__,_|\___/  |_|
%                                           
% «questao-1»  (to ".questao-1")
% (c3m232p1p 2 "questao-1")
% (c3m232p1a   "questao-1")

{\bf Questão 1}

\scalebox{0.58}{\def\colwidth{9cm}\firstcol{

\vspace*{-0.5cm}

\T(Total: 4.0 pts)

O diagrama de numerozinhos da última folha da prova corresponde a uma
superfície $z=F(x,y)$ que tem 6 faces. Também é possível interpretá-lo
como uma superfície com 7 ou mais faces, mas vamos considerar que a
superfície com só 6 faces é que é a correta.

\msk

a) \B (1.0 pts) Mostre como dividir o plano em 6 polígonos que são as
projeções destas faces no plano do papel.

\msk

b) \B (0.5 pts) Chame estas faces de face N (``norte''), S (``sul''),
W (``oeste''), E (``leste''), CN (``centro-norte'') e CS
(``centro-sul''), e chame as equações dos planos delas de
$F_{N}(x,y)$, $F_{S}(x,y)$, $F_{W}(x,y)$, $F_{E}(x,y)$, $F_{CN}(x,y)$,
e $F_{CS}(x,y)$. Dê as equações destes planos.

\msk

c) \B (0.5 pts) Sejam:
%
$$\begin{array}{rcl}
  P_{CN} &=& \setofxyzst{z = F_{CN}(x,y)}, \\
  P_{CS} &=& \setofxyzst{z = F_{CS}(x,y)}, \\
  r &=& P_{CN} ∩ P_{CS}. \\
  \end{array}
$$

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
  %
  $$A \;=\; \{0,1,\ldots,7\} × \{0,1,\ldots,10\};$$

  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
  %
  $$\begin{array}{rcl}
    Q(t) &=& (0,3) + t\VEC{1,1}, \\
    (x(t),y(t)) &=& Q(t), \\
    h(t) &=& F(x(t),y(t)). \\
    \end{array}
  $$

  Faça o gráfico da função $h(t)$. Considere que o domínio dela é o
  intervalo $[0,7]$.

}}


\newpage

%   ___                  _                ____  
%  / _ \ _   _  ___  ___| |_ __ _  ___   |___ \ 
% | | | | | | |/ _ \/ __| __/ _` |/ _ \    __) |
% | |_| | |_| |  __/\__ \ || (_| | (_) |  / __/ 
%  \__\_\\__,_|\___||___/\__\__,_|\___/  |_____|
%                                               
% «questao-2»  (to ".questao-2")
% (c3m241p1p 3 "questao-2")
% (c3m241p1a   "questao-2")
% (c3m232p1p 3 "questao-2")
% (c3m232p1a   "questao-2")
% (find-es "maxima" "2024-1-C3-P1")

\scalebox{0.5}{\def\colwidth{10cm}\firstcol{

{\bf Questão 2}

\T(Total: 2.5 pts)

Sejam
%
$$\begin{array}{rcl}
  F(x,y) &=& x^2 + xy - 2y^2, \\
       A &=& \{-2,-1,0,1,2\}, \\
       B &=& A×A. \\
  \end{array}
$$

a) \B (0.2 pts) Faça o diagrama de numerozinhos da função $F(x,y)$.
Desenhe um numerozinho para cada $(x,y)∈B$.

\msk

b) \B (0.8 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 $(x,y)+\frac{1}{10}\vec∇F(x,y)$.

\msk

c) \B (1.5 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.

}\anothercol{

% «questao-3»  (to ".questao-3")
% (c3m241p1p 4 "questao-3")
% (c3m241p1a   "questao-3")
% (find-es "maxima" "2024-1-C3-P1-Q3")

{\bf Questão 3}

\T(Total: 2.5 pts)

Sejam
%
$$\begin{array}{rcl}
  F(x,y) &=& xy(3-x-y), \\
     P_1 &=& (0,3), \\
     P_2 &=& (1,1), \\
     P_3 &=& (3,0). \\
  \end{array}
$$

a) \B (0.5 pts) Mostre que $P_1$, $P_2$ e $P_3$ são pontos críticos da
função $F$.

\ssk

b) \B (2.0 pts) Quais deles são máximos locais? Quais são mínimos
locais? Quais são pontos de sela?

\bsk
\bsk


% «questao-4»  (to ".questao-4")
% (c3m241p1p 3 "questao-4")
% (c3m241p1a   "questao-4")

{\bf Questão 4}

\T(Total: 1.0 pts)

Sejam
%
$$\begin{array}{rcl}
    z &=& z(x,y), \\
    x &=& x(t), \\
    y &=& y(t). \\
  \end{array}
$$

a) \B (0.5 pts) Calcule $z_{tt}$.

b) \B (0.5 pts) Calcule $z_{ttt}$.


}}



\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-angg     "GNUPLOT/2024-1-C3-P1.dem")
% (find-anggfile "GNUPLOT/2023-2-C3-P1.dem" "bgprocess")
% (find-anggfile "GNUPLOT/2024-1-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(8,11))
%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 bigstr1 = [[
%L   4 4 4 4 4 4 4 4
%L   4 4 4 4 4 4 4 4
%L   4 4 4 4 4 4 4 4
%L   4 4 4 2 2 2 2 2
%L   4 4 4 2 0 0 0 0
%L   3 3 3 2 0 0 0 0
%L   2 2 2 1 0 0 0 0
%L   1 1 1 0 0 0 0 0
%L   0 0 0 0 0 0 0 0
%L   0 0 0 0 0 0 0 0
%L   0 0 0 0 0 0 0 0
%L ]]
%L bigstr2 = [[
%L   4 - 4 - 4 - 4 - 4 - 4 - 4 - 4
%L   | . | . | . | . | . | . | . |
%L   4 - 4 - 4 - 4 - 4 - 4 - 4 - 4
%L   | . | . | . | . | . | . | . |
%L   4 - 4 - C - 4 - 4 - 4 - 4 - D
%L   | . | . | \ | . | . | . | . |
%L   4 - 4 - 4 - 2 - 2 - 2 - 2 - 2
%L   | . | . | . | \ | . | . | . |
%L   A - 4 - B - 2 - H - 0 - 0 - I
%L   | . | . | \ | . | . | . | . |
%L   3 - 3 - 3 - 2 - 0 - 0 - 0 - 0
%L   | . | . | . | \ | . | . | . |
%L   2 - 2 - 2 - 1 - G - 0 - 0 - 0
%L   | . | . | . | / | . | . | . |
%L   1 - 1 - 1 - 0 - 0 - 0 - 0 - 0
%L   | . | . | / | . | . | . | . |
%L   E - 0 - F - 0 - 0 - 0 - 0 - 0
%L   | . | . | . | . | . | . | . |
%L   0 - 0 - 0 - 0 - 0 - 0 - 0 - 0
%L   | . | . | . | . | . | . | . |
%L   0 - 0 - 0 - 0 - 0 - 0 - 0 - 0
%L ]]
%L clabels = CabosNaDiagonal.from(bigstr2)
%L lbls    = clabels.strgrid:labels()
%L spec    = lbls:subst("A--B--C--D E--F--G--H--I F--B--G C--H")
%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")
% (c3m241p1p 5 "questao-1-grids")
% (c3m241p1a   "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

% «gab-1»  (to ".gab-1")
% (c3m241p1p 5 "gab-1")
% (c3m241p1a   "gab-1")

{\bf Questão 1: gabarito parcial}

%L PictBounds.setbounds(v(0,0), v(7,5))
%L spec   = "(0,1)--(2,3)--(2.5,3)--(3.5,1)--(5,4)--(7,4)"
%L pws    = PwSpec.from(spec)
%L curve  = pws:topict()
%L p = Pict { curve:prethickness("1pt") }
%L p:pgat("pgatc", {sa="gab 1f"}):output()
\pu

$$\ga{barranco 2} \qquad \ga{gab 1f}$$


\newpage

% «gab-2»  (to ".gab-2")
% (c3m241p1p 6 "gab-2")
% (c3m241p1a   "gab-2")

{\bf Questão 2: gabarito}

%M (%i1) f(x)   := (x+2)*(x-1);
%M (%o1) f\left(x\right):=\left(x+2\right)\,\left(x-1\right)
%M (%i2) expand(f(x));
%M (%o2) x^2+x-2
%M (%i3) F(x,y) := x^2 + x*y - 2*y^2;
%M (%o3) F\left(x , y\right):=x^2+x\,y+\left(-2\right)\,y^2
%M (%i4) F(x,1);
%M (%o4) x^2+x-2
%M (%i5) mkmatrix([x,-2,2], [y,2,-2,-1], [x,y]);
%M (%o5) \begin{pmatrix}\left[ -2 , 2 \right] &\left[ -1 , 2 \right] &\left[ 0 , 2 \right] &\left[ 1 , 2 \right] &\left[ 2 , 2 \right] \cr \left[ -2 , 1 \right] &\left[ -1 , 1 \right] &\left[ 0 , 1 \right] &\left[ 1 , 1 \right] &\left[ 2 , 1 \right] \cr \left[ -2 , 0 \right] &\left[ -1 , 0 \right] &\left[ 0 , 0 \right] &\left[ 1 , 0 \right] &\left[ 2 , 0 \right] \cr \left[ -2 , -1 \right] &\left[ -1 , -1 \right] &\left[ 0 , -1 \right] &\left[ 1 , -1 \right] &\left[ 2 , -1 \right] \cr \left[ -2 , -2 \right] &\left[ -1 , -2 \right] &\left[ 0 , -2 \right] &\left[ 1 , -2 \right] &\left[ 2 , -2 \right] \cr \end{pmatrix}
%M (%i6) mkmatrix([x,-2,2], [y,2,-2,-1], F(x,y));
%M (%o6) \begin{pmatrix}-8&-9&-8&-5&0\cr 0&-2&-2&0&4\cr 4&1&0&1&4\cr 4&0&-2&-2&0\cr 0&-5&-8&-9&-8\cr \end{pmatrix}
%M (%i7) z   : F(x,y);
%M (%o7) -\left(2\,y^2\right)+x\,y+x^2
%M (%i8) z_x : diff(z,x);
%M (%o8) y+2\,x
%M (%i9) z_y : diff(z,y);
%M (%o9) x-4\,y
%M (%i10) define(Fx(x,y), diff(F(x,y), x));
%M (%o10) \mathrm{Fx}\left(x , y\right):=y+2\,x
%L maximahead:sa("levels", "")
\pu

%M (%i11) define(Fy(x,y), diff(F(x,y), y));
%M (%o11) \mathrm{Fy}\left(x , y\right):=x-4\,y
%M (%i12) mkmatrix([x,-2,2], [y,2,-2,-1], [Fx(x,y),Fy(x,y)]);
%M (%o12) \begin{pmatrix}\left[ -2 , -10 \right] &\left[ 0 , -9 \right] &\left[ 2 , -8 \right] &\left[ 4 , -7 \right] &\left[ 6 , -6 \right] \cr \left[ -3 , -6 \right] &\left[ -1 , -5 \right] &\left[ 1 , -4 \right] &\left[ 3 , -3 \right] &\left[ 5 , -2 \right] \cr \left[ -4 , -2 \right] &\left[ -2 , -1 \right] &\left[ 0 , 0 \right] &\left[ 2 , 1 \right] &\left[ 4 , 2 \right] \cr \left[ -5 , 2 \right] &\left[ -3 , 3 \right] &\left[ -1 , 4 \right] &\left[ 1 , 5 \right] &\left[ 3 , 6 \right] \cr \left[ -6 , 6 \right] &\left[ -4 , 7 \right] &\left[ -2 , 8 \right] &\left[ 0 , 9 \right] &\left[ 2 , 10 \right] \cr \end{pmatrix}
%M (%i13) z : F(x,y);
%M (%o13) -\left(2\,y^2\right)+x\,y+x^2
%M (%i14) [xmin,ymin,xmax,ymax] : [-2,-2,2,2];
%M (%o14) \left[ -2 , -2 , 2 , 2 \right] 
%M (%i15) mylevel(eq,[opts]) :=
%M   apply('imp1, append([eq, x,xmin,xmax, y,ymin,ymax], opts))$
%M (%i16) myvec(xy, dxdy) := vector(xy, dxdy, hl(0.1), lw(2), lc(gray))$
%M (%i17) myvecs : create_list(myvec([x,y], [Fx(x,y),Fy(x,y)]/10),
%M                      x, seq(-2,2), y, seqby(2,-2,-1))$
%L maximahead:sa("levels 2", "")
\pu

%M (%i18) myQdraw("2024-1-C3-P1-level",     "height=5cm",
%M         xr(-4,4), yr(-3,3),
%M         more(proportional_axes=xy),
%M         mylevel(z=2,  lk("z=2"),  lc(brown)),
%M         mylevel(z=1,  lk("z=1"),  lc(red)),
%M         mylevel(z=0,  lk("z=0"),  lc(orange)),
%M         mylevel(z=-2, lk("z=-2"), lc(forest_green)),
%M         mylevel(z=-5, lk("z=-5"), lc(blue)),
%M         myvecs
%M         /* myvec([2,0], [1,2]) */
%M        );
%M (%o18) \includegraphics[height=12cm]{2024-1-C3/2024-1-C3-P1-level.pdf}
%L maximahead:sa("levels 3", "")
\pu


\scalebox{0.29}{\def\colwidth{11cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {12cm}
\ga{levels}

\vspace*{-5cm}

}\def\colwidth{15cm}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{levels 2}

}\def\colwidth{14cm}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{levels 3}

}}




\newpage

% «gab-3»  (to ".gab-3")
% (c3m241p1p 8 "gab-3")
% (c3m241p1a   "gab-3")

{\bf Questão 3: gabarito}


%M (%i1) z : x * y * (3-x-y);
%M (%o1) x\,\left(-y-x+3\right)\,y
%M (%i2) gradz : [diff(z,x), diff(z,y)];
%M (%o2) \left[ \left(-y-x+3\right)\,y-x\,y , x\,\left(-y-x+3\right)-x\,y \right] 
%M (%i3) gradz : factor(gradz);
%M (%o3) \left[ -\left(y\,\left(y+2\,x-3\right)\right) , -\left(x\,\left(2\,y+x-3\right)\right) \right] 
%M (%i4) crpts : solve(gradz, [x,y]);
%M (%o4) \left[ \left[ x=0 , y=0 \right]  , \left[ x=0 , y=3 \right]  , \left[ x=3 , y=0 \right]  , \left[ x=1 , y=1 \right]  \right] 
%M (%i5) hessz : hessian(z, [x,y]);
%M (%o5) \begin{pmatrix}-\left(2\,y\right)&-\left(2\,y\right)-2\,x+3\cr -\left(2\,y\right)-2\,x+3&-\left(2\,x\right)\cr \end{pmatrix}
%M (%i6) P1    : [x=0,y=3];
%M (%o6) \left[ x=0 , y=3 \right] 
%M (%i7) P2    : [x=1,y=1];
%M (%o7) \left[ x=1 , y=1 \right] 
%M (%i8) P3    : [x=3,y=0];
%M (%o8) \left[ x=3 , y=0 \right] 
%M (%i9) GH    : [gradz, hessz];
%M (%o9) \left[ \left[ -\left(y\,\left(y+2\,x-3\right)\right) , -\left(x\,\left(2\,y+x-3\right)\right) \right]  , \begin{pmatrix}-\left(2\,y\right)&-\left(2\,y\right)-2\,x+3\cr -\left(2\,y\right)-2\,x+3&-\left(2\,x\right)\cr \end{pmatrix} \right] 
%M (%i10) GH    : expand(GH);
%M (%o10) \left[ \left[ -y^2-2\,x\,y+3\,y , -\left(2\,x\,y\right)-x^2+3\,x \right]  , \begin{pmatrix}-\left(2\,y\right)&-\left(2\,y\right)-2\,x+3\cr -\left(2\,y\right)-2\,x+3&-\left(2\,x\right)\cr \end{pmatrix} \right] 
%L maximahead:sa("Q3", "")
\pu

%M (%i11) GH1   : at(GH, P1);
%M (%o11) \left[ \left[ 0 , 0 \right]  , \begin{pmatrix}-6&-3\cr -3&0\cr \end{pmatrix} \right] 
%M (%i12) GH2   : at(GH, P2);
%M (%o12) \left[ \left[ 0 , 0 \right]  , \begin{pmatrix}-2&-1\cr -1&-2\cr \end{pmatrix} \right] 
%M (%i13) GH3   : at(GH, P3);
%M (%o13) \left[ \left[ 0 , 0 \right]  , \begin{pmatrix}0&-3\cr -3&-6\cr \end{pmatrix} \right] 
%L maximahead:sa("Q3 2", "")
\pu

%M (%i14) [xmin,ymin,xmax,ymax] : [-1,-1,4,4]$
%M (%i15) mylevel(eq,[opts]) :=
%M   apply('imp1, append([eq, x,xmin,xmax, y,ymin,ymax], opts))$
%M (%i16) myQdraw("2024-1-C3-P1-Q3", "height=10cm",
%M         xr(-1,4), yr(-1,4),
%M         more(proportional_axes=xy),
%M         mylevel(z=0.98, lk("z=0.98"), lc(gray)),
%M         mylevel(z=0.2,  lk("z=0.2"),  lc(brown)),
%M         mylevel(z=0.1,  lk("z=0.1"),  lc(red)),
%M         mylevel(z=0,  lk("z=0"),  lc(orange)),
%M         mylevel(z=-0.1, lk("z=-0.1"), lc(forest_green)),
%M         mylevel(z=-0.2, lk("z=-0.2"), lc(blue))
%M        );
%M (%o16) \includegraphics[height=10cm]{2024-1-C3/2024-1-C3-P1-Q3.pdf}
%L maximahead:sa("Q3 3", "")
\pu

\scalebox{0.3}{\def\colwidth{12.5cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{Q3}

}\def\colwidth{9cm}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{Q3 2}

}\def\colwidth{9cm}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{Q3 3}

}}




\newpage

%{\bf Questão 4: gabarito (com erros)}
%
%\scalebox{0.4}{\def\colwidth{9cm}\firstcol{
%
%$$\begin{array}{rcl}
%  z_t     &=& z_x x_t + z_y y_t \\
%  (z_x)_t &=& z_{xx} x_t + z_{xy} y_t \\
%  (z_y)_t &=& z_{yx} x_t + z_{yy} y_t \\
%  (z_xx)_t &=& z_{xxx} x_t + z_{xxy} y_t \\
%  (z_xy)_t &=& z_{xyx} x_t + z_{xyy} y_t \\
%           &=& z_{xxy} x_t + z_{xyy} y_t \\
%  (z_yx)_t &=& z_{yxx} x_t + z_{yxy} y_t \\
%           &=& z_{xxy} x_t + z_{xyy} y_t \\
%  (z_yy)_t &=& z_{yyx} x_t + z_{yyy} y_t \\
%           &=& z_{xyy} x_t + z_{yyy} y_t \\
%  \\[-10pt]
%  (z_x x_t)_t &=& (z_x)_t x_t + z_x x_{tt} \\
%              &=& (z_{xx} x_t + z_{xy} y_t) x_t + z_x x_{tt} \\
%              &=& (z_{xx} x_t x_t + z_{xy} y_t x_t) + z_x x_{tt} \\
%              &=& z_x x_{tt} + z_{xx} x_t x_t + z_{xy} x_t y_t \\
%  \\[-10pt]
%  (z_y y_t)_t &=& (z_y)_t y_t + z_y y_tt \\
%              &=& (z_yx x_t + z_yy y_t) y_t + z_y y_tt \\
%              &=& (z_yx x_t y_t + z_yy y_t y_t) + z_y y_tt \\
%              &=& z_y y_tt + z_xy x_t y_t + z_yy y_t y_t \\
%  \\[-10pt]
%  (z_t)_t &=& (z_x x_t + z_y y_t)_t \\
%          &=& (z_x x_t)_t + (z_y y_t)_t \\
%          &=& (z_x x_tt + z_xx x_t x_t + z_xy x_t y_t) \\
%          &+& (z_y y_tt + z_xy x_t y_t + z_yy y_t y_t) \\
%          &=& z_x x_tt + z_y y_tt \\
%          &+& z_xx x_t x_t \\
%          &+& 2 z_xy x_t y_t \\
%          &+& z_yy y_t y_t \\
%  \end{array}
%$$
%
%
%}\anothercol{
%
%$$\begin{array}{rcl}
%  (z_{tt})_t &=& (z_x x_{tt} + z_y y_{tt})_t \\
%           &+& (z_{xx} x_t x_t)_t \\
%           &+& 2 (z_{xy} x_t y_t)_t \\
%           &+& (z_{yy} y_t y_t)_t \\
%  \\[-10pt]
%           &=& (z_x)_t x_tt + z_x x_ttt + (z_y)_t y_tt + z_y y_ttt \\
%           &+& (z_xx)_t x_t x_t + z_xx x_tt x_t + z_xx x_t x_tt \\
%           &+& 2 ((z_xy)_t x_t y_t + z_xy x_tt y_t + z_xy x_t y_tt) \\
%           &+& (z_yy)_t y_t y_t + z_yy y_tt y_t + z_yy y_t y_tt \\
%  \\[-10pt]
%           &=& (z_xx x_t + z_xy y_t) x_tt + z_x x_ttt + (z_yx x_t + z_yy y_t) y_tt + z_y y_ttt \\
%           &+& (z_xxx x_t + z_xxy y_t) x_t x_t + z_xx x_tt x_t + z_xx x_t x_tt \\
%           &+& 2 ((z_xxy x_t + z_xyy y_t) x_t y_t + z_xy x_tt y_t + z_xy x_t y_tt) \\
%           &+& (z_xyy x_t + z_yyy y_t) y_t y_t + z_yy y_tt y_t + z_yy y_t y_tt \\
%  \\[-10pt]
%           &=& (z_xx x_t x_tt + z_xy y_t x_tt) + z_x x_ttt + (z_yx x_t y_tt + z_yy y_t y_tt) + z_y y_ttt \\
%           &+& (z_xxx x_t x_t x_t + z_xxy y_t x_t x_t) + z_xx x_tt x_t + z_xx x_t x_tt \\
%           &+& 2 ((z_xxy x_t x_t y_t + z_xyy y_t x_t y_t) + z_xy x_tt y_t + z_xy x_t y_tt) \\
%           &+& (z_xyy x_t y_t y_t + z_yyy y_t y_t y_t) + z_yy y_tt y_t + z_yy y_t y_tt \\
%  \\[-10pt]
%           &=& z_x x_ttt + z_y y_ttt + z_xx x_t x_tt + z_xy y_t x_tt + z_xy x_t y_tt + z_yy y_t y_tt \\
%           &+& 2 z_xx x_t x_tt + z_xxx x_t x_t x_t + z_xxy x_t x_t y_t \\
%           &+& 2 z_xy x_tt y_t + 2 z_xy x_t y_tt + 2 z_xxy x_t x_t y_t + 2 z_xyy y_t x_t y_t \\
%           &+& 2 z_yy y_t y_tt + z_xyy x_t y_t y_t + z_yyy y_t y_t y_t \\
%  \\[-10pt]
%           &=& z_x x_ttt \\
%           &+& z_y y_ttt \\
%           &+& 3 z_xx x_t x_tt \\
%           &+& 3 z_yy y_t y_tt \\
%  \\[-10pt]
%           &+& 3 z_xy x_tt y_t \\
%           &+& 3 z_xxy x_t x_t y_t \\
%           &+& 3 z_xyy x_t y_t y_t \\
%           &+& z_xxx x_t x_t x_t \\
%           &+& z_yyy y_t y_t y_t \\
%  \end{array}
%$$
%
%}}


\newpage

% «gab-4-diag»  (to ".gab-4-diag")
% (c3m241p1p 7 "gab-4-diag")
% (c3m241p1a   "gab-4-diag")

{\bf Questão 4: diagrama}

%D diagram ??
%D 2Dx     100         +20           +30             +20          +20     +20          +20             +30     +20
%D 2D  100                                                        z
%D 2D  +20                                           zx.xt                zy.yt
%D 2D  +20             zx.xtt                                                                          zy.ytt
%D 2D  +10                           zxx.xt.xt                zxy.xt.yt                zyy.yt.yt
%D 2D  +20 zx.xttt                                                                                             zy.yttt
%D 2D  +10 zxx.xt.xtt                              zxy.xtt.yt           zxy.xt.ytt                             zyy.yt.ytt
%D 2D  +10             zxxx.xt.xt.xt zxxy.xt.xt.yt                                 zxyy.xt.yt.yt zyyy.xt.yt.yt
%D 2D
%D ren z             ==> z
%D ren zx.xt         ==> z_{x}x_t
%D ren zy.yt         ==> z_{y}y_t
%D ren zx.xtt        ==> z_{x}x_{tt}
%D ren zy.ytt        ==> z_{y}y_{tt}
%D ren zxx.xt.xt     ==> z_{xx}x_{t}x_t
%D ren zxy.xt.yt     ==> z_{xy}x_{t}y_t
%D ren zyy.yt.yt     ==> z_{yy}y_{t}y_t
%D ren zx.xttt       ==> z_{x}x_{ttt}
%D ren zy.yttt       ==> z_{y}y_{ttt}
%D ren zxx.xt.xtt    ==> z_{xx}x_{t}x_{tt}
%D ren zxy.xtt.yt    ==> z_{xy}x_{tt}y_t
%D ren zxy.xt.ytt    ==> z_{xy}x_{t}y_{tt}
%D ren zyy.yt.ytt    ==> z_{yy}y_{t}y_{tt}
%D ren zxxx.xt.xt.xt ==> z_{xxx}x_{t}x_{t}x_t
%D ren zxxy.xt.xt.yt ==> z_{xxy}x_{t}x_{t}y_t
%D ren zxyy.xt.yt.yt ==> z_{xyy}x_{t}y_{t}y_t
%D ren zyyy.xt.yt.yt ==> z_{yyy}x_{t}y_{t}y_t
%D
%D (( z zx.xt ->
%D    z zy.yt ->
%D    zx.xt zx.xtt    ->
%D    zx.xt zxx.xt.xt ->
%D    zx.xt zxy.xt.yt ->
%D    zy.yt zxy.xt.yt ->
%D    zy.yt zyy.yt.yt ->
%D    zy.yt zy.ytt    ->
%D    zx.xtt    zx.xttt       ->
%D    zx.xtt    zxx.xt.xtt    ->
%D    zx.xtt    zxy.xtt.yt    ->
%D    zxx.xt.xt zxxx.xt.xt.xt ->
%D    zxx.xt.xt zxxy.xt.xt.yt ->
%D    zxx.xt.xt zxx.xt.xtt    ->
%D    zxy.xt.yt zxxy.xt.xt.yt ->
%D    zxy.xt.yt zxy.xtt.yt    ->
%D    zxy.xt.yt zxy.xt.ytt    ->
%D    zxy.xt.yt zxyy.xt.yt.yt ->
%D    zyy.yt.yt zxyy.xt.yt.yt ->
%D    zyy.yt.yt zyyy.xt.yt.yt ->
%D    zyy.yt.yt zyy.yt.ytt    ->
%D    zy.ytt    zxy.xt.ytt    ->
%D    zy.ytt    zyy.yt.ytt    ->
%D    zy.ytt    zy.yttt       ->
%D ))
%D enddiagram
%D
$$\pu
  \scalebox{0.8}{$
  \diag{??}
  $}
$$








                 
                 
                 
                 
                 
                 
                 




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

\end{document}

% (find-pdfpages2-links "~/LATEX/" "2024-1-C3-P1")
% (find-pdfpages2-links "~/LATEX/" "2024-1-C3-P1" "-pp" "pages=5,fitpaper,landscape=true")


% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3p1"
% ee-tla: "c3m241p1"
% End:
