|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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: