% (find-LATEX "2023-1-C2-VR.tex")
% (defun c  () (interactive) (find-LATEXsh "lualatex -record 2023-1-C2-VR.tex" :end))
% (defun C  () (interactive) (find-LATEXsh "lualatex 2023-1-C2-VR.tex" "Success!!!"))
% (defun D  () (interactive) (find-pdf-page      "~/LATEX/2023-1-C2-VR.pdf"))
% (defun d  () (interactive) (find-pdftools-page "~/LATEX/2023-1-C2-VR.pdf"))
% (defun e  () (interactive) (find-LATEX "2023-1-C2-VR.tex"))
% (defun o1 () (interactive) (find-LATEX "2023-1-C2-P1.tex"))
% (defun o2 () (interactive) (find-LATEX "2023-1-C2-P2.tex"))
% (defun u  () (interactive) (find-latex-upload-links "2023-1-C2-VR"))
% (defun v  () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-1-C2-VR.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2023-1-C2-VR")
% (find-pdf-page   "~/LATEX/2023-1-C2-VR.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023-1-C2-VR.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023-1-C2-VR.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2023-1-C2-VR.pdf")
%   file:///home/edrx/LATEX/2023-1-C2-VR.pdf
%               file:///tmp/2023-1-C2-VR.pdf
%           file:///tmp/pen/2023-1-C2-VR.pdf
%  http://anggtwu.net/LATEX/2023-1-C2-VR.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-C2-VR" "C2" "c2m231vr" "c2vr")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.defs-caepro»	(to "defs-caepro")
% «.defs-pict2e»	(to "defs-pict2e")
% «.defs-edovs»		(to "defs-edovs")
% «.title»		(to "title")
%
% «.djvuize»		(to "djvuize")



% <videos>
% Video (not yet):
% (find-ssr-links     "c2m231vr" "2023-1-C2-VR")
% (code-eevvideo      "c2m231vr" "2023-1-C2-VR")
% (code-eevlinksvideo "c2m231vr" "2023-1-C2-VR")
% (find-c2m231vrvideo "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-C2.pdf}
\def\drafturl{http://anggtwu.net/2023.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (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-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)}}

% «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

% «defs-edovs»  (to ".defs-edovs")
%
\sa{(M)}{
  \left(\begin{array}{rcl}
             \D \dydx &=& \D \frac{g(x)}{h(y)} \\
             h(y)\,dy &=& g(x)\,dx \\
          \inty{h(y)} &=& \intx{g(x)} \\
           \mcc{\veq} & & \mcc{\veq} \\
        \mcc{H(y)+C1} & & \mcc{G(x)+C2} \\
             H(y)     &=& G(x)+C2-C1 \\
                      &=& G(x)+C3 \\
         H^{-1}(H(y)) &=& H^{-1}(G(x)+C3) \\
           \mcc{\veq} & & \\
              \mcc{y} & & \\
   \end{array}
   \right)
  }
\sa{(F)}{
  \left(\begin{array}{rcl}
             \D \dydx &=& \D \frac{g(x)}{h(y)} \\
         H^{-1}(H(y)) &=& H^{-1}(G(x)+C3) \\
           \mcc{\veq} & & \\
              \mcc{y} & & \\
   \end{array}
   \right)
  }
\sa{[M]}{\CFname{M}{}}
\sa{[F]}{\CFname{F}{}}
\sa{[S]}{\CFname{S}{}}



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m231vrp 1 "title")
% (c2m231vra   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2023.1}

\bsk

Prova de reposição (VR)

\bsk

Eduardo Ochs - RCN/PURO/UFF

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

\end{center}

\newpage




\scalebox{0.54}{\def\colwidth{6.6cm}\firstcol{

{\bf Questão 1.}

\T(Total: 4.0 pts)

Calcule:
%
$$\intx{\frac{x^3}{(x-4)(x+5)}}$$

\bsk

{\bf Questão 2.}

\T(Total: 4.0 pts)

Calcule
%
$$\intx{x^3 \sqrt{1-4x^2}}$$

e teste o seu resultado.

\bsk

{\bf Questão 3.}

\T(Total: 4.0 pts)

Seja (*) esta EDO:
%
$$\frac{dy}{dx} \;=\; - \frac{1}{4y^3}$$

a) \B (1.0 pts) Encontre a solução geral ``positiva'' de $(*)$ e teste-a.

b) \B (2.0 pts) Encontre a solução geral ``negativa'' de $(*)$ e teste-a.

c) \B (1.0 pts) Encontre a solução particular de $(*)$ que passa pelo
ponto $(4,-3)$ e teste-a.

}\anothercol{

{\bf Dicas}

Pra resolver a questão 2 você vai ter que começar com uma substituição
da forma $u=2x$ -- isso vai transformar aquela integral numa que dá
pra resolver por um dos casos mais simples de substituição
trigonométrica.

\msk

Nas questões 1 e 3 é muito fácil a gente se perder nas contas e chegar
ou a soluções erradas ou a soluções quase ilegíveis que só fazem
sentido pra um leitor com muita, muita, muita boa vontade. O melhor
modo de evitar isso é definir várias funções intermediárias usando o
``seja'' -- lembre que cada uma delas tem que ter um nome diferente!!!
-- e usar as partículas em português pra distinguir as igualdades que
são verdade sempre, as que só são verdade em certas condições, as que
vamos testar se são verdadeiras ou não, e as que são hipóteses pro
chutar-e-testar. A folha em anexo tem exemplos de várias das
partículas em português mais comuns.


}\anothercol{

\vspace*{0.25cm}

A EDO da questão 3 é uma EDO com variáveis separáveis (``EDOVS''). Eu
costumo escrever o ``método'' e a ``fórmula'' para resolver EDOVSs
deste jeito,

\msk

$\scalebox{0.7}{$
  \begin{array}{rcl}
  \ga{[M]} &=& \ga{(M)} \\\\[-5pt]
  \ga{[F]} &=& \ga{(F)} \\
  \end{array}
  $}
$

\msk

Mas você pode organizar as suas contas de outros jeitos se quiser.

}}



\newpage

% «anexo-L»  (to ".anexo-L")
\def\anexoL{

A substituição é:
%
$$\ga{[S]} \;=\;
  \bmat{
    G(x) := x^4 + 5 \\
    H(y) := y^2 + 3 \\
    g(x) := 4x^3 \\
    h(y) := 2y \\
    H^{-1}(x) := \sqrt{x-3} \\
  }
$$

a) Seja:
%
$$\frac{dy}{dx} = \frac{4x^3}{2y} \qquad (*)$$

b)
%
 $\begin{array}[t]{lrcl}
  \text{Seja:}  & H^{-1}(x) &=& \sqrt{x-3}. \\
  \text{Temos:} & H^{-1}(H(y)) &=& \sqrt{H(y)-3} \\
                &              &=& \sqrt{(y^2+3)-3} \\
                &              &=& y. \\
  \end{array}
 $

\msk

c) $\begin{array}[t]{lrcl}
       & y &=& H^{-1}(G(x)+C_3) \\
          &&=& \sqrt{(G(x)+C_3)-3} \\
          &&=& \sqrt{((x^4+5)+C_3)-3} \\
          &&=& \sqrt{x^4+2+C_3} \\
    \text{Seja:} &
      f(x) &=& \sqrt{x^4+2+C_3}. \\
    \end{array}
   $

}


% «anexo-R»  (to ".anexo-R")

\def\anexoR{

d) $\begin{array}[t]{l}
    \text{Será que $f(x)$ obedece $(*)$?} \\
    \text{Temos }
    f'(x) = \frac{2x^3}{\sqrt{x^4 + 2 + C_3}},
    \text{ e com isso:}
    \\
    \\[-5pt]
    \left(
      f'(x) = \frac{4x^3}{2f(x)}
    \right)
    \bmat{
      f(x) = \sqrt{x^4+2+C_3} \\
      f'(x) = \frac{2x^3}{\sqrt{x^4 + 2 + C_3}} \\
    }
    \\
    = \;\;
    \left(
      \frac{2x^3}{\sqrt{x^4 + 2 + C_3}}
      = \frac{4x^3}{2\sqrt{x^4+2+C_3}}
    \right)
    \qquad \smile \\
    \end{array}
   $

\bsk

e) $\begin{array}[t]{lrcl}
    \text{Se}    & f(x_1) &=& y_1, \\
    \text{i.e.,} & f(1)   &=& 2,   \\
    \text{então} & f(1)   &=& \sqrt{1^4+2+C_3} \\
                         &&=& \sqrt{3+C_3} \\
                         &&=& 2 \\
                 & 2^2    &=& \sqrt{3+C_3}^2 \\
                 & 4      &=&       3+C_3    \\
                 & C_3    &=& 1 \\
                 & f(x)   &=& \sqrt{x^4+2+C_3} \\
                 &        &=& \sqrt{x^4+3} \\
    \text{Seja:} & f_1(x) &=& \sqrt{x^4+3}. \\
    \end{array}
   $

\bsk

f) $\begin{array}[t]{lrcl}
    \text{Será que} & f_1(x_1) &=& y_1, \\
    \text{i.e.,}    & f_1(1)   &=& 2?   \\
                & \sqrt{1^4+3} &=& \sqrt{4} \\
                              &&=& 2 \qquad \smile \\
    \end{array}
   $

}

% «anexo»  (to ".anexo")

\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

\vspace*{-0.5cm}

{\bf Anexo: gabarito de uma}

{\bf questão da P2 de 2022.2}

\ssk

\anexoL

}\anothercol{

\anexoR

}}



\newpage

\scalebox{0.6}{\def\colwidth{14cm}\firstcol{

{\bf Mini-gabarito}

1) $\intx{\frac{x^3}{(x-4)(x+5)}}
    = x^2 - x + \frac{64}{9} \ln|x-4| + \frac{125}{9} \ln|x+5|
   $

\msk

2) $\intx{x^3 \sqrt{1-4x^2}}
    = (\frac{1}{5} x^4 - \frac{1}{60} x^2 - \frac{1}{120}) \sqrt{1-4x^2}
   $

\msk

3a) $f_1(x) = \sqrt[4]{-x-C_3}$

3b) $f_2(x) = - \sqrt[4]{-x-C_3}$

3c) $f_3(x) = - \sqrt[4]{-x+85}$

}\anothercol{
}}


\newpage

\def\sa#1#2{\expandafter\def\csname myarg#1\endcsname{#2}}
\def\ga#1{\csname myarg#1\endcsname}

\sa{F1}{\int {x^3\,\sqrt{1-4\,x^2}}{\;dx}}
\sa{F3}{{{\int {u^3\,\sqrt{1-u^2}}{\;du}}\over{16}}}
\sa{F7}{\sqrt{1-4\,x^2}\,\left({{x^4}\over{5}}-{{x^2}\over{60}}-{{1}\over{
 120}}\right)}

$$\begin{array}{l}
  \ga{F1} \\
  =\; \ga{F3} \\
  =\; \ga{F7} \\
  \end{array}
$$


\def\hboxthreewidth{4cm}
\def\hboxthree#1#2#3{\hbox to \hboxthreewidth{\rlap{#1}\hss#2\hss\llap{#3}}}

abc \hboxthree{a}{b}{c} def

abc \hboxthree{dd}{ee}{ff} def











\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-C2/")
# (find-fline "~/LATEX/2023-1-C2/")
# (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-C2/
       cp -fv        $1.pdf ~/LATEX/2023-1-C2/
       cat <<%%%
% (find-latexscan-links "C2" "$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=2023-1-C2-VR veryclean
make -f 2019.mk STEM=2023-1-C2-VR pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2vr"
% ee-tla: "c2m231vr"
% End:

