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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2023-1-C4-P1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-1-C4-P1.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-1-C4-P1.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2023-1-C4-P1.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-1-C4-P1.pdf"))
% (defun e () (interactive) (find-LATEX "2023-1-C4-P1.tex"))
% (defun o () (interactive) (find-LATEX "2023-1-C4-P1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-1-C4-P1"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-1-C4-P1.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2023-1-C4-P1")
% (find-pdf-page "~/LATEX/2023-1-C4-P1.pdf")
% (find-sh0 "cp -v ~/LATEX/2023-1-C4-P1.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2023-1-C4-P1.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2023-1-C4-P1.pdf")
% file:///home/edrx/LATEX/2023-1-C4-P1.pdf
% file:///tmp/2023-1-C4-P1.pdf
% file:///tmp/pen/2023-1-C4-P1.pdf
% http://anggtwu.net/LATEX/2023-1-C4-P1.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise1")
% (find-Deps1-cps "Caepro5 Piecewise1")
% (find-Deps1-anggs "Caepro5 Piecewise1")
% (find-MM-aula-links "2023-1-C4-P1" "C4" "c4m231p1" "c4p1")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.defs-caepro» (to "defs-caepro")
% «.defs-pict2e» (to "defs-pict2e")
% «.title» (to "title")
% «.links» (to "links")
% «.questoes-123» (to "questoes-123")
% «.dicas» (to "dicas")
% «.questao-1-gab» (to "questao-1-gab")
% «.questao-2-gab» (to "questao-2-gab")
% «.questao-3-gab» (to "questao-3-gab")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c4m231p1" "2023-1-C4-P1")
% (code-eevvideo "c4m231p1" "2023-1-C4-P1")
% (code-eevlinksvideo "c4m231p1" "2023-1-C4-P1")
% (find-c4m231p1video "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}
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\drafturl{http://anggtwu.net/LATEX/2023-1-C4.pdf}
\def\drafturl{http://anggtwu.net/2023.1-C4.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
\def\Dint{\int\!\!\!\!\int}
\def\P#1{\left( #1 \right)}
% «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)}}
% (find-LATEX "2023-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2023-1-C2-carro.tex" "defs-pict2e")
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
% «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 V = nil -- (find-angg "LUA/Pict2e1.lua" "MiniV")
%L dofile "Piecewise1.lua" -- (find-LATEX "Piecewise1.lua")
%L Pict2e.__index.suffix = "%"
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt
\pu
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c4m231p1p 1 "title")
% (c4m231p1a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo C4 - 2023.1}
\bsk
Primeira prova (P1)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://anggtwu.net/2023.1-C4.html}
\end{center}
\newpage
% «links» (to ".links")
% ___ _ _ ____ _____
% / _ \ _ _ ___ ___| |_ ___ ___ ___ / | |___ \ |___ /
% | | | | | | |/ _ \/ __| __/ _ \ / _ \/ __| | | __) | |_ \
% | |_| | |_| | __/\__ \ || (_) | __/\__ \ | |_ / __/ _ ___) |
% \__\_\\__,_|\___||___/\__\___/ \___||___/ |_( ) |_____( ) |____/
% |/ |/
% «questoes-123» (to ".questoes-123")
% (c4m231p1p 2 "questoes-123")
% (c4m231p1a "questoes-123")
\scalebox{0.65}{\def\colwidth{8.5cm}\firstcol{
{\bf Questão 1.}
\T(Total: 1.0 pts)
Seja $R$ o retângulo da direita.
Calcule:
%
$$\Dint_R x^3 y^5 \, dx \, dy.$$
Dica: neste caso tanto faz você integrar primeiro em $x$ e depois em
$y$ quanto integrar primeiro em $y$ e depois em $x$; os dois modos são
igualmente fáceis.
\bsk
\bsk
{\bf Questão 2.}
\T(Total: 5.0 pts)
Seja $B$ a região tipo ``pedaço de bolo'' à direita -- ela é um quarto de
um bolo de raio 2 com furo de raio 1. Calcule
%
$$\Dint_B x \, dx \, dy.$$
Pra resolver isso você vai ter que mudar a integral pra coordenadas
polares.
}\anothercol{
\vspace*{1cm}
% (find-latexscan-links "C4" "C4-P1-R")
% (find-latexscan-links "C4" "C4-P1-B")
% (find-xpdf-page "~/LATEX/2023-1-C4/C4-P1-R.pdf")
% (find-xpdf-page "~/LATEX/2023-1-C4/C4-P1-B.pdf")
$$\begin{array}{rcc}
R &=& \myvcenter{\includegraphics[height=3cm]{2023-1-C4/C4-P1-R.pdf}} \\
B &=& \myvcenter{\includegraphics[height=2cm]{2023-1-C4/C4-P1-B.pdf}} \\
\end{array}
$$
\vspace*{1cm}
{\bf Questão 3.}
\T(Total: 6.0 pts)
Seja $S$ este retângulo: $S = [1,2]×[1,2]$. Calcule esta integral de
linha:
%
$$\oint_{∂S} \VEC{xy^2,x^3y^4}·\VEC{dx,dy}.$$
}}
\newpage
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% | _ \(_) ___ __ _ ___
% | | | | |/ __/ _` / __|
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% |____/|_|\___\__,_|___/
%
% «dicas» (to ".dicas")
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
{}
{\bf Dica:} Pra ``calcular'' $\difx{23}{45}{x^3}$ basta fazer isso
aqui:
%
$$\difx{23}{45}{x^3} \;=\; 45^3 - 23^3$$
com isso você chega em algo que dá pra transformar num número usando
só uma calculadora que só saiba fazer operações básicas. Você não
precisa destes passos extras:
%
$$45^3 - 23^3 \;=\; 91125 - 12167 \;=\; 78958$$
A explicação está em um dos slides de Cálculo 2 do material anexo --
procure por ``dicas sobre simplificação''.
\bsk
\bsk
{\bf Outra dica.} Nas questões desta prova o que vai contar mais
pontos é você organizar as contas de modo que cada passo seja fácil de
entender, de verificar, e de justificar -- ``chegar no resultado
certo'' vai valer relativamente pouco.
}\anothercol{
}}
\newpage
\scalebox{0.7}{\def\colwidth{8cm}\firstcol{
% «questao-1-gab» (to ".questao-1-gab")
% (c4m231p1p 4 "questao-1-gab")
% (c4m231p1a "questao-1-gab")
{\bf Questão 1: gabarito}
\def\Intxx#1{\Intx{20}{42}{#1}}
\def\Intyy#1{\Inty{99}{200}{#1}}
\def\difxx#1{\difx{20}{42}{#1}}
\def\difyy#1{\difx{99}{200}{#1}}
$$\begin{array}{l}
\int \!\!\! \int_R x^3 y^5 \, dx \, dy \\
=\;\; \Intxx{\Intyy{x^3 y^5}} \\
=\;\; \Intxx{x^3 \P{\Intyy{y^5}}} \\
=\;\; \P{\Intxx{x^3}}
\P{\Intyy{y^5}} \\
=\;\; \P{\difxx{\frac{x^4}{4}}}
\P{\difyy{\frac{y^6}{6}}} \\\\[-8pt]
=\;\; \frac{42^4-20^4}{4} \;
\frac{200^6-99^6}{6}
\end{array}
$$
}\anothercol{
% «questao-2-gab» (to ".questao-2-gab")
% (c4m231p1p 4 "questao-2-gab")
% (c4m231p1a "questao-2-gab")
{\bf Questão 2: gabarito}
\def\vmat#1{\left| \begin{matrix} #1\end{matrix} \right|}
\def\vsm #1{\left| \begin{smallmatrix}#1\end{smallmatrix} \right|}
\def\Intrr#1{\Intr {1}{2}{#1}}
\def\Inttt#1{\Intth{0}{\frac{π}{2}}{#1}}
\def\difrr#1{\difr {1}{2}{#1}}
\def\diftt#1{\difth{0}{\frac{π}{2}}{#1}}
Vamos usar esta mudança de variáveis:
%
$$\begin{array}{rcl}
(x,y) &=& (r\cosθ, r\senθ) \\
\vsm{x_r & x_θ \\ y_r & y_θ}
&=& \vsm{\cosθ & r(-\senθ) \\ \senθ & r\cosθ}
\;=\; r \\
% &=& r \\
dx\,dy &=& \vsm{x_r & x_θ \\ y_r & y_θ} dr\,dθ \\
&=& r\,dr\,dθ \\
\end{array}
$$
Então:
$$\begin{array}{l}
\int \!\!\! \int_B x \, dx \, dy \\
=\;\; \Intrr{\Inttt{(r\cosθ)\,r}} \\
=\;\; \Intrr{\Inttt{r^2\cosθ}} \\
=\;\; \P{\Intrr{r^2}} \P{\Inttt{\cosθ}} \\
=\;\; \P{\difrr{\frac{r^3}{3}}} \P{\diftt{\senθ}} \\
=\;\; \frac{8-1}{3}(1-0) \\
\end{array}
$$
}}
\newpage
% «questao-3-gab» (to ".questao-3-gab")
% (c4m231p1p 5 "questao-3-gab")
% (c4m231p1a "questao-3-gab")
\def\Inttp#1{\Intt{ 1}{ 2}{#1}}
\def\Inttn#1{\Intt{-2}{-1}{#1}}
\def\diftp#1{\dift{ 1}{ 2}{#1}}
\def\diftn#1{\dift{-2}{-1}{#1}}
\scalebox{0.45}{\def\colwidth{6cm}\firstcol{
{\bf Questão 3: gabarito}
Vou decompor $∂S$ em $C_1$, $C_2$, $C_3$, $C_4$ -- as paredes direita,
de cima, esquerda, e de baixo do quadrado -- e vou parametrizar $C_1$,
$C_2$, $C_3$, $C_4$ desta forma:
\msk
Em $C_1$: $(x(t),y(t))=(2,t)$,
com $t$ indo de $1$ até 2;
aqui $(x_t,y_t)=(0,1)$.
\msk
Em $C_2$: $(x(t),y(t))=(-t,2)$,
com $t$ indo de $-2$ até -1;
aqui $(x_t,y_t)=(-1,0)$.
\msk
Em $C_3$: $(x(t),y(t))=(1,-t)$,
com $t$ indo de $-2$ até -1;
aqui $(x_t,y_t)=(0,-1)$.
\msk
Em $C_4$: $(x(t),y(t))=(t,1)$,
com $t$ indo de $1$ até 2;
aqui $(x_t,y_t)=(1,0)$.
\ssk
}\anothercol{
\vspace*{0.2cm}
Temos:
\msk
$\begin{array}{l}
\int_{C_1} (P,Q)·(dx,dy) \\
=\; \Inttp {(P,Q)·(x_t,y_t)} \\
=\; \Inttp {(P,Q)·(0,1)} \\
=\; \Inttp {Q(x(t),y(t))} \\
=\; \Inttp {Q(2,t)} \\
\\
\int_{C_2} (P,Q)·(dx,dy) \\
=\; \Inttn {(P,Q)·(x_t,y_t)} \\
=\; \Inttn {(P,Q)·(-1,0)} \\
=\; \Inttn {-P(x(t),y(t))} \\
=\; \Inttn {-P(-t,2)} \\
\\
\int_{C_3} (P,Q)·(dx,dy) \\
=\; \Inttn {(P,Q)·(x_t,y_t)} \\
=\; \Inttn {(P,Q)·(0,-1)} \\
=\; \Inttn {-Q(x(t),y(t))} \\
=\; \Inttn {-Q(1,-t)} \\
\\
\int_{C_4} (P,Q)·(dx,dy) \\
=\; \Inttp {(P,Q)·(x_t,y_t)} \\
=\; \Inttp {(P,Q)·(1,0)} \\
=\; \Inttp {P(x(t),y(t))} \\
=\; \Inttp {P(t,1)} \\
\end{array}
$
% $$\Inttp{\Inttn{foo}}$$
}\anothercol{
\vspace*{0.2cm}
Como nesse problema temos
$P(x,y) = xy^2$ e $Q(x,y) = x^3y^4$,
isso vira:
\msk
$\begin{array}{l}
\int_{C_1} (P,Q)·(dx,dy) \\
=\; \Inttp {Q(2,t)} \\
=\; \Inttp {2^3t^4} \\
=\; 2^3 \Inttp {t^4} \\
=\; 8 \P{\diftp {\frac{t^5}{5}}} \\
\\
\int_{C_2} (P,Q)·(dx,dy) \\
=\; \Inttn {-P(-t,2)} \\
=\; \Inttn {-(-t)2^2} \\
=\; 4 \Inttn {t} \\
=\; 4 \P{\diftn {\frac{t^2}{2}}} \\
\\
\int_{C_3} (P,Q)·(dx,dy) \\
=\; \Inttn {-Q(1,-t)} \\
=\; \Inttn {-1^3(-t)^4} \\
=\; - \Inttn {t^4} \\
=\; 4 \P{\diftn {\frac{t^5}{5}}} \\
\\
\int_{C_4} (P,Q)·(dx,dy) \\
=\; \Inttp {P(t,1)} \\
=\; \Inttp {t·1^2} \\
=\; \Inttp {t} \\
=\; \P{\diftp {\frac{t^2}{2}}} \\
\end{array}
$
}\anothercol{
\vspace*{0.2cm}
E aí:
\msk
$\begin{array}{l}
\oint_{∂S} (xy^2,x^3y^4)·(dx,dy) \\
= \; \int_{C_1} (xy^2,x^3y^4)·(dx,dy) \\
+ \; \int_{C_2} (xy^2,x^3y^4)·(dx,dy) \\
+ \; \int_{C_3} (xy^2,x^3y^4)·(dx,dy) \\
+ \; \int_{C_4} (xy^2,x^3y^4)·(dx,dy) \\
= \; 8 \P{\diftp {\frac{t^5}{5}}} \\
+ \; 4 \P{\diftn {\frac{t^2}{2}}} \\
+ \; 4 \P{\diftn {\frac{t^5}{5}}} \\
+ \; \P{\diftp {\frac{t^2}{2}}} . \\
\end{array}
$
}}
% (find-es "maxima" "2023-1-C4-P1")
\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 "~/2023.1-C4/")
# (find-fline "~/LATEX/2023-1-C4/")
# (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 ~/2023.1-C4/
cp -fv $1.pdf ~/LATEX/2023-1-C4/
cat <<%%%
% (find-latexscan-links "C4" "$1")
%%%
}
f C4-P1-B
f C4-P1-R
f 20230612_012330
f 20201213_area_em_funcao_de_theta
f 20201213_area_em_funcao_de_x
f 20201213_area_fatias_pizza
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c4p1"
% ee-tla: "c4m231p1"
% End: