% (find-LATEX "2023-1-C2-prova-monitor.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-1-C2-prova-monitor.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-1-C2-prova-monitor.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023-1-C2-prova-monitor.pdf"))
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% (defun u () (interactive) (find-latex-upload-links "2023-1-C2-prova-monitor"))
% (defun v () (interactive) (find-2a '(e) '(d)))
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%          (code-eec-LATEX "2023-1-C2-prova-monitor")
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%  http://anggtwu.net/LATEX/2023-1-C2-prova-monitor.pdf
% (find-LATEX "2019.mk")
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% (find-sh0 "cd ~/LUA/; cp -v C2Subst1.lua C2Formulas1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v Gram2.lua Tree1.lua Caepro5.lua ~/LATEX/")
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\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}
%
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%\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")
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           ]{geometry}
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\begin{document}

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% %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
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% \def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
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\def\u#1{\par{\footnotesize \url{#1}}}

\def\drafturl{http://anggtwu.net/LATEX/2023-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2023.1-C2.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)}}
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\def\B       (#1 pts){\ColorOrange{\bf(#1 pts)}}

\catcode`\^^J=10
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%L dofile "Piecewise1.lua"           -- (find-LATEX "Piecewise1.lua")
%L dofile "C2Formulas1.lua"          -- (find-LATEX "C2Formulas1.lua")
%L dofile "Lazy5.lua"                -- (find-LATEX "Lazy5.lua")
%L dofile "2022-1-C2-P2.lua"         -- (find-LATEX "2022-1-C2-P2.lua")
%L Pict2e.__index.suffix = "%"



%  _____ _ _   _                               
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% «title»  (to ".title")
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% (c2m231pma   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo C2 - 2023.1}

\bsk

Prova para seleção de monitor

(17/abril/2023)

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2023.1-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (find-LATEXfile "2023-C2-monitoria-edital.tex")
% (find-LATEXfile "2023-C2-monitoria-edital.tex" "parciais")
%
% \item Métodos de integração: mudança de variável, substituição
%   trigonométrica, frações parciais.
% \item Somas de Riemann.
% \item Equações diferenciais com variáveis separáveis.

% (c4m231introp 6 "exercicio-3")
% (c4m231introa   "exercicio-3")
% (c4m231introa   "exercicio-3" "L e R")

%L namedang("EDOVSintro", "", [[
%L      <EDOVSG>
%L ]])
%L EDOVSintro:sa("FOO"):output()
\pu



\scalebox{0.55}{\def\colwidth{10cm}\firstcol{

{\bf Questão 1}

\T(Total: 1.0 pts)

\ssk

Sejam:
%
$$\begin{array}{rcl}
  [L] &=& \sum_{i=1}^N f(x_{i-1})(x_i-x_{i-1}) \\{}
  [R] &=& \sum_{i=1}^N f(x_{i})(x_i-x_{i-1}) \\{}
  [sup] &=& \sum_{i=1}^N \sup_{x∈[x_{i-1},x_i]}f(x) (x_i-x_{i-1}) \\{}
  f(x) &=& 4-(x-2)^2 \\
  P &=& \{0, 1, 2.5, 3\} \\
  \end{array}
$$

Faça várias cópias do gráfico da $f(x)$ em $x∈[0,4]$ e desenhe no eixo
$x$ de cada uma delas a partição $P$. Depois represente sobre uma das
cópias o $[L]$ como uma soma de retângulos, sobre outra cópia o $[R]$,
e sobre outra o $[sup]$.

\bsk
\bsk


{\bf Questão 2}

\T(Total: 5.0 pts)

\ssk

Resolva esta integral
%
$$\intx{x^3 \sqrt{1-x^2}}
$$

e teste o seu resultado.




}\anothercol{



% (c2m222p2p 2 "edovs")
% (c2m222p2a   "edovs")

{\bf Questão 3}

\T(Total: 4.0 pts)

O nosso ``método'' para resolver EDOs com variáveis separáveis era
este aqui:

$$\ga{FOO}$$

Digamos que $G(x) = x^4 + 3$ e $H(y) = y^2 + 1$.

Diga qual é a EDO associada a este caso e chame-a de $(*)$. Encontre a
solução geral da $(*)$ e encontre a solução particular que passa pelo
ponto $(3,9)$. Teste tudo.


}}




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

\end{document}

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