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% (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-1» (to "questao-1") % «.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") % (c3m241p1p 3 "questao-1") % (c3m241p1a "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 4 "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: