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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2022-2-C3-P2.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2022-2-C3-P2.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2022-2-C3-P2.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2022-2-C3-P2.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2022-2-C3-P2.pdf"))
% (defun e () (interactive) (find-LATEX "2022-2-C3-P2.tex"))
% (defun o () (interactive) (find-LATEX "2022-2-C3-P2.tex"))
% (defun u () (interactive) (find-latex-upload-links "2022-2-C3-P2"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2022-2-C3-P2.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2022-2-C3-P2")
% (find-pdf-page "~/LATEX/2022-2-C3-P2.pdf")
% (find-sh0 "cp -v ~/LATEX/2022-2-C3-P2.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2022-2-C3-P2.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2022-2-C3-P2.pdf")
% file:///home/edrx/LATEX/2022-2-C3-P2.pdf
% file:///tmp/2022-2-C3-P2.pdf
% file:///tmp/pen/2022-2-C3-P2.pdf
% http://angg.twu.net/LATEX/2022-2-C3-P2.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 "2022-2-C3-P2" "3" "c3m222p2" "c3p2")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.title» (to "title")
% «.links» (to "links")
% «.questao-1» (to "questao-1")
% «.elipse» (to "elipse")
% «.questao-2» (to "questao-2")
% «.questao-3» (to "questao-3")
% «.questao-4» (to "questao-4")
% «.grids» (to "grids")
% «.dicas-diferenciais» (to "dicas-diferenciais")
%
% «.questao-1-gab» (to "questao-1-gab")
% «.questao-2-gab» (to "questao-2-gab")
% «.questao-3-gab» (to "questao-3-gab")
% «.questao-4-gab» (to "questao-4-gab")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c3m222p2" "2022-2-C3-P2")
% (code-eevvideo "c3m222p2" "2022-2-C3-P2")
% (code-eevlinksvideo "c3m222p2" "2022-2-C3-P2")
% (find-c3m222p2video "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}
\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/2022-2-C3.pdf}
\def\drafturl{http://angg.twu.net/2022.2-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% «defs-T-and-B» (to ".defs-T-and-B")
\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)}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c3m222p2p 1 "title")
% (c3m222p2a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 3 - 2022.2}
\bsk
P2 (Segunda prova)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.2-C3.html}
\end{center}
\newpage
% «links» (to ".links")
% (c3m222dicasp2p 6 "abertos-e-fechados")
% (c3m222dicasp2a "abertos-e-fechados")
% (c3m222dicasp2p 5 "maximos-e-minimos")
% (c3m222dicasp2a "maximos-e-minimos")
% (c3m222dicasp2p 6 "notacao-de-fisicos")
% (c3m222dicasp2a "notacao-de-fisicos")
% (c3m222dpp 3 "um-exemplo")
% (c3m222dpa "um-exemplo")
% (c3m222dpp 2 "links")
% (c3m222dpa "links")
% (find-books "__analysis/__analysis.el" "leithold")
% (find-books "__analysis/__analysis.el" "leithold" "reescritas usando")
% (find-leitholdptpage (+ 17 275) "reescritas usando notação de Leibniz")
% ___ _ _
% / _ \ _ _ ___ ___| |_ __ _ ___ / |
% | | | | | | |/ _ \/ __| __/ _` |/ _ \ | |
% | |_| | |_| | __/\__ \ || (_| | (_) | | |
% \__\_\\__,_|\___||___/\__\__,_|\___/ |_|
%
% «questao-1» (to ".questao-1")
% «elipse» (to ".elipse")
% (c3m222p2p 2 "questao-1")
% (c3m222p2a "questao-1")
% (c3m222p2p 2 "elipse")
% (c3m222p2a "elipse")
%L Pict2e.bounds = PictBounds.new(v(-2,-2), v(2,2))
%L spec = "(0,1)--(1,1)--(2,4)--(3,5)--(4,4)o (4,3)c (4,1)o--(6,3)--(7,3)"
%L spec = ""
%L pws = PwSpec.from(spec)
%L pws:topict():prethickness("1pt"):pgat("pgatc"):sa("grid Q1"):output()
\pu
\scalebox{0.5}{\def\colwidth{10.5cm}\firstcol{
%\vspace*{-0.4cm}
{\Large \bf Questão 1}
\ssk
\T(Total: 3.5 pts)
\msk
Sejam:
%
$$\begin{array}{rcl}
P(x,y) &=& x^2 + y^2, \\
H(x,y) &=& xy, \\
E(x,y) &=& x^2 + 4y^2. \\
% A &=& \setofxyst{x,y∈\{-2,-1,0,1,2\}} \\
\end{array}
$$
Represente graficamente:
a) \B (0.1 pts) o diagrama de numerozinhos de $P(x,y)$,
b) \B (0.2 pts) o digrama de numerozinhos de $H(x,y)$,
c) \B (0.2 pts) o diagrama de numerozinhos de $E(x,y)$,
d) \B (0.1 pts) pelo menos 5 curvas de nível de $P(x,y)$,
e) \B (0.2 pts) pelo menos 5 curvas de nível de $H(x,y)$,
f) \B (0.2 pts) pelo menos 5 curvas de nível de $E(x,y)$,
\msk
E os conjuntos abaixo:
g) \B (0.2 pts) $C_1 = E^{-1}(4)$
h) \B (0.2 pts) $C_2 = E^{-1}(1)$
i) \B (0.3 pts) $C_3 = E^{-1}([1,4))$
j) \B (0.3 pts) $C_4 = H^{-1}([-2,1))$
k) \B (0.5 pts) $C_5 = C_3 ∩ C_4$
l) \B (0.5 pts) $C_6 = \Int(C_5)$
m) \B (0.5 pts) $C_7 = \overline{C_5}$
\msk
Use os grids da página 4.
Indique claramente qual desenho é a resposta de cada item e quais
desenhos são rascunhos.
}\anothercol{
% «questao-2» (to ".questao-2")
% (c3m222p2p 2 "questao-2")
% (c3m222p2a "questao-2")
{\Large \bf Questão 2}
\ssk
\T(Total: 2.5 pts)
\msk
Sejam:
%
$$\begin{array}{rcl}
z &=& (x-x_0)^4 (y-y_0)^6, \\
α &=& x+y, \\
β &=& x-y, \\
w &=& (α^3-α)+β^2. \\
\end{array}
$$
Nesta questão eu vou ver principalmente quais dos truques da ``notação
de físicos'' você sabe usar direito.
\msk
A página 5 tem um monte de dicas de ``notação de físicos'' que você
pode usar como referência. A coluna da esquerda dessa página tem um
exemplo grande que nós vimos em aula; a parte de cima da coluna da
direita tem uma tabela que eu copiei da página 275 do Leithold, na
qual ele mostra como reescrever certas regras de derivação usando
diferenciais; e a parte de baixo da coluna da direita é uma versão
adaptada do primeiro exemplo do capítulo XVI do Silvanus Thompson, em
que ele mostra como fazer contas ficarem menores criando variáveis
dependentes novas.
\msk
Calcule:
a) \B (0.2 pts) $\frac{dz}{dx}$,
b) \B (0.3 pts) $z_{xx}$,
c) \B (0.5 pts) $dz$,
d) \B (1.5 pts) $dw$.
\msk
No item c tente chegar até uma expressão da forma $z_xdx + z_ydy$, e
no item d tente chegar até uma expressão da forma forma
$w_xdx + w_ydy$.
}}
\newpage
% «questao-3» (to ".questao-3")
% (c3m222p2p 3 "questao-3")
% (c3m222p2a "questao-3")
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
{\Large \bf Questão 3}
\ssk
\T(Total: 3.0 pts)
\msk
Sejam
%
$$\begin{array}{rcl}
z(x,y) &=& dx^2 + exy + fy^2, \\
h(x) &=& z(x,1). \\
\end{array}
$$
Vou dizer que a função $h(x,y)$ é a ``função homogênea de grau 2
associada a $h(x)$''.
\msk
a) \B (1.5 pts) Digamos que
%
$$h(x) = -2(x-1)(x+1).$$
%
Faça o diagrama de sinais da $h(x)$ (em $\R$), os numerozinhos da
função $z(x,y)$ nos pontos com $y=1$ e $x∈\{-2,-1,0,1,2\}$ (siiiim, só
5 pontos!) e o diagrama de sinais da função $z(x,y)$ (em $\R^2$), e
diga se o ponto $(0,0)$ é um mínimo, máximo, ponto de sela, etc, etc.
\msk
b) \B (1.5 pts) Agora digamos que
%
$$h(x) = (x-i)(x+i) = x^2+1.$$
%
Faça as mesmas coisas para esta função $h(x)$ e para a função $z(x,y)$
associada a ela.
}\anothercol{
% «questao-4» (to ".questao-4")
% (c3m222p2p 3 "questao-4")
% (c3m222p2a "questao-4")
{\Large \bf Questão 4}
\ssk
\T(Total: 3.0 pts)
\msk
Sejam:
%
$$\begin{array}{rcl}
H(x,y) &=& xy, \\
E(x,y) &=& x^2 + 4y^2, \\
D &=& E^{-1}([0,16]), \\
F &:& D \to \R \\
&& (x,y) \mapsto H(x,y) \\
% A &=& \setofxyst{x,y∈\{-2,-1,0,1,2\}} \\
\end{array}
$$
Agora só queremos olhar pro que acontece dentro do ``domínio'' $D$,
que é uma elipse; note que a função $F(x,y)$ só está definida em $D$.
Faça pelo menos 5 curvas de nível de $z=F(x,y)$ (obs: só dentro da
elipse!!!) e mostre no seu gráfico quais dos pontos de $D$ são máximos
locais, mínimos locais ou pontos de sela.
}}
% * (eepitch-maxima)
% * (eepitch-kill)
% * (eepitch-maxima)
% z : (x-x0)^4 * (y-y0^6);
% diff(z,x);
% diff(z,x,2);
% aa : x+y;
% bb : x-y;
\newpage
% «grids» (to ".grids")
% (c3m222p2p 4 "grids")
% (c3m222p2a "grids")
\unitlength=10pt
\def\Gr{\scalebox{1.2}{$\ga{grid Q1}$}}
$\begin{matrix}
\Gr & \Gr & \Gr & \Gr & \Gr \\
\Gr & \Gr & \Gr & \Gr & \Gr \\
\Gr & \Gr & \Gr & \Gr & \Gr \\
\Gr & \Gr & \Gr & \Gr & \Gr \\
\end{matrix}
$
\newpage
% ____ _ _ _ __ __
% | _ \(_) ___ __ _ ___ __| (_)/ _|/ _|___
% | | | | |/ __/ _` / __| / _` | | |_| |_/ __|
% | |_| | | (_| (_| \__ \ | (_| | | _| _\__ \
% |____/|_|\___\__,_|___/ \__,_|_|_| |_| |___/
%
% «dicas-diferenciais» (to ".dicas-diferenciais")
% (c3m222p2p 5 "dicas-diferenciais")
% (c3m222p2a "dicas-diferenciais")
\sa{myexample-body}{
z &=& (x^3 + y^4)^5 \\
\\[-7pt]
\frac{∂z}{∂x} &=& \frac{∂}{∂x}(x^3+y^4)^5 \\
&=& 5(x^3 + y^4)^4 \frac{∂}{∂x}(x^3+y^4) \\
&=& 5(x^3 + y^4)^4 (\frac{∂}{∂x}x^3+\frac{∂}{∂x}y^4) \\
&=& 5(x^3 + y^4)^4 (3x^2) \\
\\[-7pt]
\frac{∂z}{∂y} &=& \frac{∂}{∂y}(x^3+y^4)^5 \\
&=& 5(x^3 + y^4)^4 \frac{∂}{∂y}(x^3+y^4) \\
&=& 5(x^3 + y^4)^4 (\frac{∂}{∂y}x^3+\frac{∂}{∂y}y^4) \\
&=& 5(x^3 + y^4)^4 (4y^3) \\
\\[-7pt]
dz &=& 5(x^3 + y^4)^4 \, d(x^3+y^4) \\
&=& 5(x^3 + y^4)^4 (dx^3+dy^4) \\
&=& 5(x^3 + y^4)^4 (3x^2 \, dx + 4y^3 \, dy) \\
&=& 5(x^3 + y^4)^4 (3x^2) dx + 5(x^3 + y^4)^4 (4y^3) dy \\
\\[-7pt]
dz &=& z_x dx + z_y dy \\
}
\sa{leithold-body}{
\frac{d(c)}{dx} &=& 0 \\
\frac{d(x^n)}{dx} &=& nx^{n-1} \\
\frac{d(cu)}{dx} &=& c\frac{du}{dx} \\
\frac{d(u+v)}{dx} &=& \frac{du}{dx} + \frac{dv}{dx} \\
\frac{d(uv)}{dx} &=& u\frac{dv}{dx} + v\frac{du}{dx} \\
\frac{d(\frac{u}{v})}{dx} &=&
\frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \\
\frac{d(u^n)}{dx} &=& nu^{n-1} \frac{du}{dx} \\
\\[-7pt]
}
\sa{leithold-body2}{
d(c) &=& 0 \\
d(x^n) &=& nx^{n-1}dx \\
d(cu) &=& c\,du \\
d(u+v) &=& du+dv \\
d(uv) &=& u\,dv + v\,du \\
d(\frac{u}{v}) &=&
\frac{v\,du - u\,dv}{v^2} \\
d(u^n) &=& nu^{n-1} du \\
}
\sa{thompson-body}{
y &=& (x^2+a^2)^{3/2} \\
u &=& x^2+a^2 \\
du &=& 2x\,dx \\
dy &=& d((x^2+a^2)^{3/2}) \\
&=& d(u^{3/2}) \\
&=& u^{1/2}\,du \\
&=& u^{1/2}·2x\,dx \\
&=& (x^2+a^2)^{1/2}·2x\,dx \\
}
\scalebox{0.65}{\def\colwidth{9cm}\firstcol{
\msk
$\begin{array}{rcl}
\ga{myexample-body}
\end{array}
%\qquad
\hspace*{-1cm}
\begin{array}{c}
\begin{array}{rcl}
\ga{leithold-body}
\end{array}
\quad
\begin{array}{rcl}
\ga{leithold-body2}
\end{array}
\\
\\
\begin{array}{rcl}
\ga{thompson-body}
\end{array}
\end{array}
$
}\anothercol{
}}
\newpage
% «questao-1-gab» (to ".questao-1-gab")
% «questao-2-gab» (to ".questao-2-gab")
% (c3m222p2p 6 "questao-2-gab")
% (c3m222p2a "questao-2-gab")
\def\dzdx{\frac{dz}{dx}}
\def\und#1#2{\underbrace{#1}_{#2}}
\scalebox{0.55}{\def\colwidth{10.5cm}\firstcol{
{\bf \Large Questão 2: gabarito}
\bsk
$\begin{array}[t]{lrcl}
\text{Temos:} &
z &=& (x-x_0)^4 (y-y_0)^6 \\
\text{Sejam:} &
u &=& x-x_0, \\
& v &=& y-y_0. \\
\\[-5pt]
\text{Então:}
& z &=& u^4 v^6, \\
& \dzdx &=& \ddx(u^4)v^6 + u^4\ddx(v^6) \\
&&=& (4u^3\ddx u)v^6 + u^4(6v^5\ddx v) \\
&&=& (4u^3\ddx(x-x_0))v^6 + u^4(6v^5\ddx(y-y_0)) \\
&&=& 4u^3v^6 \\
&&=& 4(x-x_0)^3(y-y_0)^6, \\
\\[-5pt]
& z_{xx} &=& \ddx \ddx z \\
&&=& \ddx (4u^3v^6) \\
&&=& 4(\ddx(u^3)v^6 + u^3\ddx(v^6)) \\
&&=& 4(\ddx(u^3)v^6) \\
&&=& 4(3u^2\ddx(u)v^6) \\
&&=& 4(3u^2v^6) \\
&&=& 12u^2v^6 \\
&&=& 12(x-x_0)^2(y-y_0)^6, \\
\\[-5pt]
& dz &=& d(u^4v^6) \\
&&=& d(u^4)v^6 + u^4d(v^6) \\
&&=& (4u^3du)v^6 + u^4(6v^5dv) \\
&&=& (4u^3v^6)dx + (6u^4v^5)dy \\
&&=& 4(x-x_0)^3(y-y_0)^6dx + 6(x-x_0)^4(y-y_0)^5dy \\
%\text{Obs:}
\end{array}
$
}\anothercol{
\vspace*{0.5cm}
$\begin{array}[t]{lrcl}
\text{Temos:} &
α &=& x+y, \\
& β &=& x-y, \\
& w &=& (α^3-α)+β^2. \\
\\[-5pt]
\text{Então:} &
dw &=& d(α^3+α) + d(β^2) \\
&&=& (2α+1)dα + 2βdβ \\
&&=& (2α+1)d(x+y) + 2βd(x-y) \\
&&=& (2α+1)(dx+dy) + 2β(dx-dy) \\
&&=& (2α+1+2β)dx + (2α+1-2β)dy \\
&&=& (2(x+y)+1+2(x-y))dx \\
&&+& (2(x+y)+1-2(x-y))dy \\
&&=& (4x+1)dx + (4y+1)dx \\
\end{array}
$
}}
\newpage
% «questao-3-gab» (to ".questao-3-gab")
% «questao-4-gab» (to ".questao-4-gab")
\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 "~/2022.2-C3/")
# (find-fline "~/LATEX/2022-2-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 ~/2022.2-C3/
cp -fv $1.pdf ~/LATEX/2022-2-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=2022-2-C3-P2 veryclean
make -f 2019.mk STEM=2022-2-C3-P2 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3p2"
% ee-tla: "c3m222p2"
% End: