% (find-LATEX "2024-2-C2-P1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-2-C2-P1.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-2-C2-P1.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2024-2-C2-P1.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2024-2-C2-P1.pdf"))
% (defun e () (interactive) (find-LATEX "2024-2-C2-P1.tex"))
% (defun o () (interactive) (find-LATEX "2024-1-C2-P1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2024-2-C2-P1"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2024-2-C2-P1.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2024-2-C2-P1")
% (find-pdf-page   "~/LATEX/2024-2-C2-P1.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-2-C2-P1.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-2-C2-P1.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-2-C2-P1.pdf")
%   file:///home/edrx/LATEX/2024-2-C2-P1.pdf
%               file:///tmp/2024-2-C2-P1.pdf
%           file:///tmp/pen/2024-2-C2-P1.pdf
%  http://anggtwu.net/LATEX/2024-2-C2-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 "2024-2-C2-P1" "2" "c2m242p1" "c2p1")

% «.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")
% «.dicas»		(to "dicas")
% «.questao-1»		(to "questao-1")
% «.questao-2»		(to "questao-2")
% «.questao-3»		(to "questao-3")
% «.questao-4»		(to "questao-4")
% «.questao-4-grids»	(to "questao-4-grids")
% «.gab-1»		(to "gab-1")
% «.gab-2»		(to "gab-2")



\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-LATEX "dednat7-test1.tex")
%\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/2024-2-C2.pdf}
\def\drafturl{http://anggtwu.net/2024.2-C2.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 "dednat7load.lua"}  % (find-LATEX "dednat7load.lua")
\directlua{dednat7preamble()}         % (find-angg "LUA/DednatPreamble1.lua")
\directlua{dednat7oldheads()}         % (find-angg "LUA/Dednat7oldheads.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")
% (c2m242p1p 1 "title")
% (c2m242p1a   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2024.2}

\bsk

P1 (Primeira prova)

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.2-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m242p1p 2 "links")
% (c2m242p1a   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{16cm}\firstcol{
}\anothercol{
}}

\newpage

% «dicas»  (to ".dicas")
% (c2m242p1p 3 "dicas")
% (c2m242p1a   "dicas")

{\bf Dicas:}

\scalebox{0.5}{\def\colwidth{10.5cm}\firstcol{

1) Nestas questões 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.

\ssk

2) Recomendo que vocês usem o método das ``caixinhas de anotações''
nas mudanças de variável... numa caixinha de anotações a primeira
linha diz a relação entre a variável nova e a antiga, todas as outras
linhas são consequências da primeira, e dentro da caixinha de
anotações você pode usar as gambiarras com variáveis dependentes e
diferenciais, como isto aqui: $dx = 42\,du$...

\ssk

3) ...por exemplo:
%
$$\bmat{
  s = \sen θ \\
  \sqrt{1-s^2} = \cos θ \\
  \frac{ds}{dθ} = \cos θ \\
  ds = \cos θ \, dθ \\
  θ = \arcsen s \\
  }
$$

\msk

4) Façam o requerimento de revisão de prova! Eu vou ser super rígido
na correção mas a banca de revisão não costuma ser!

}\anothercol{

% (c2m241p1p 3 "questoes-12-dicas")
% (c2m241p1a   "questoes-12-dicas")
% (c2m231p1p 2 "questoes-123-dicas")
% (c2m231p1a   "questoes-123-dicas")
{}


}}



\newpage

% «questao-1»  (to ".questao-1")
% (c2m242p1p 3 "questao-1")
% (c2m242p1a   "questao-1")
% (c2m242mvp 18 "simplificando-raizes")
% (c2m242mva    "simplificando-raizes")


\scalebox{0.5}{\def\colwidth{10.5cm}\firstcol{

{}

{\bf Questão 1}

\T(Total: 4.0 pts)

Sejam \ga{(*)} e \ga{(**)} estas integrais:
%
$$\intx{x^3 \sqrt{1-x^2}^3} \qquad \ga{(*)}$$
$$\intx{4x^3 \sqrt{25-x^2}^3} \qquad \ga{(**)}$$

a) \B (2.0 pts) Suponha que a gente sabe integrar \ga{(*)}. Transforme
a \ga{(**)} em algo que a gente sabe integrar.

\ssk

b) \B (1.0 pts) Suponha que a gente não sabe integrar \ga{(*)}.
Resolva \ga{(*)} por subtituição trigonométrica.

\ssk

c) \B (1.0 pts) Teste o resultado que você obteve no item (b).

\bsk
\bsk

% «questao-2»  (to ".questao-2")
% (c2m242p1p 3 "questao-2")
% (c2m242p1a   "questao-2")

{\bf Questão 2}

\T(Total: 2.5 pts)

a) \B (2.0 pts) Transforme
%
$$\frac{x^3+6\,x^2-22\,x-55}{x^2+4\,x-21}$$

em algo fácil de integrar.

\ssk

b) \B (0.5 pts) Integre:
%
$$\intx{\frac{x^3+6\,x^2-22\,x-55}{x^2+4\,x-21}}$$

}\anothercol{

% «questao-3»  (to ".questao-3")
% (c2m242p1p 4 "questao-3")
% (c2m242p1a   "questao-3")

{\bf Questão 3}

\T(Total: 2.5 pts)

a) \B (2.0 pts) Transforme
%
$$(\cos x)(\cos 5x)^2$$

em algo fácil de integrar usando o ``método do $E$'', que na
terminologia do Maxima é o método do \textsf{exponentialize} e do
\textsf{demoivre}. As dicas são essas aqui:

% Algumas definições, fórmulas e substituições:

\msk

$\begin{array}[t]{l}
 c = \cos θ \\
 s = \sen θ \\
 t = \tan θ \\
 z = \sec θ \\
 E = e^{iθ} \\
 \end{array}
 %
 \begin{array}[t]{l}
 c^2+s^2=1 \\
 z^2=t^2+1 \\
 \sqrt{1-s^2} = c \\
 \sqrt{t^2+1} = z \\
 \sqrt{z^2-1} = t \\
 \end{array}
 %
 \begin{array}[t]{l}
 \frac{ds}{dθ} = c \\
 \frac{dc}{dθ} = -s \\
 \frac{dt}{dθ} = z^2 \\
 \frac{dz}{dθ} = zt \\
 \end{array}
 %
 \begin{array}[t]{l}
 E = c+is \\
 c = \frac{E+E^{-1}}{2} \\
 s = \frac{E-E^{-1}}{2i} \\
 e^{ikθ} + e^{-ikθ} = 2 \cos kθ \\
 e^{ikθ} - e^{-ikθ} = 2i \sen kθ \\
 \end{array}
$

\msk

b) \B (0.5 pts) Integre:
%
$$\intx{(\cos x)(\cos 5x)^2}$$

\bsk

% «questao-4»  (to ".questao-4")
% (c2m242p1p 4 "questao-4")
% (c2m242p1a   "questao-4")

{\bf Questão 4
}

\T(Total: 1.0 pts)

\msk

Seja $f(t)$ a função no topo da página seguinte.

Seja
%
$$F(x) \;=\; \Intt{3}{x}{f(t)}.$$

Desenhe o gráfico de $F(x)$ em algum dos grids vazios da próxima
página. Indique claramente qual é a versão final e quais desenhos são
rascunhos.

}}


\newpage

% «questao-4-grids»  (to ".questao-4-grids")
% (c2m242p1p 5 "questao-4-grids")
% (c2m242p1a   "questao-4-grids")
% (c2m232p1p 4 "questao-5-grids")
% (c2m232p1a   "questao-5-grids")
% (c2m231p1p 4 "questao-5-grids")
% (c2m231p1a   "questao-5-grids")
% (c2m222p1p 4 "questao-5-grids")
% (c2m222p1a   "questao-5-grids")

%L -- (find-angg "LUA/Pict2e1-1.lua" "FromYs")

%L fry = FromYs.from {ys={0,-1,1,-2,2,-3,3,-3,2,-2,1,-1,0},  Y0=0} :setall()
%L fry = FromYs.from {ys={0,-1,-3,3,1,0,1,2,1,0,-1,-2,-1,0}, Y0=0} :setall()
%L fry = FromYs.from {ys={2,1,0,1,2,-2,1,-2,0,-2,0,1,2,1,0,-1,-2,-1,0}, Y0=-3} :setall()
%L fry = FromYs.from {ys={2,-1,1,-2,-1,0,1,2,0,1,2,0,-1,0,-2,1,-2}, Y0=-2} :setall()
%L Pict {
%L   fry:ypict()   :sa("fig f"),
%L   fry:ypict()   :prethickness("1pt"):sa("fig f"),
%L   fry:Ypict()   :sa("fig F"),
%L   fry:grid(-5,5):sa("grid F"),
%L } :output()
\pu

\unitlength=8pt

$\scalebox{0.9}{$
 \begin{array}{ll}
 \ga{fig f}  \phantom{mm} & \ga{fig f}  \\ \\
 \ga{grid F} & \ga{grid F} \\ \\
 \ga{grid F} & \ga{grid F} \\
 \end{array}
 $}
$


\newpage

% «gab-1»  (to ".gab-1")
% (c2m242p1p 6 "gab-1")
% (c2m242p1a   "gab-1")

{\bf Questão 1: gabarito}

\scalebox{0.55}{\def\colwidth{9cm}\firstcol{

\vspace*{0.0cm}

$\begin{array}{l}
  \intx{4x^3 \sqrt{25-x^2}^3} \\
  = \; \intx{4x^3 \sqrt{5^2-x^2}^3} \\
  = \; \intx{4x^3 \sqrt{5^2-(5^2)\frac{x^2}{5^2}}^3} \\
  = \; \intx{4x^3 \sqrt{5^2(1-\frac{x^2}{5^2})}^3} \\
  = \; \intx{4x^3 \left(\sqrt{5^2(1-\frac{x^2}{5^2})}\right)^3} \\
  = \; \intx{4x^3 \left(\sqrt{5^2}\sqrt{(1-\frac{x^2}{5^2})}\right)^3} \\
  = \; \intx{4x^3 \sqrt{5^2}^3\sqrt{1-\frac{x^2}{5^2}}^3} \\
  = \; \intx{4x^3 · 5^{\frac{2}{3}}\sqrt{1-\frac{x^2}{5^2}}^3} \\
  = \; 4 · 5^{\frac{2}{3}} \intx{x^3\sqrt{1-\frac{x^2}{5^2}}^3} \\
  = \; 4 · 5^{\frac{2}{3}} \intx{x^3\sqrt{1-(\frac{x}{5})^2}^3} \\
  = \; 4 · 5^{\frac{2}{3}} \intu{(5u)^3\sqrt{1-u^2}^3 · 5}\\
  = \; 4 · 5^{\frac{2}{3}} \intu{5^3u^3 · 5 \sqrt{1-u^2}^3}\\
  = \; 4 · 5^{\frac{2}{3}} \intu{5^4 u^3\sqrt{1-u^2}^3}\\
  = \; 4 · 5^4 · 5^{\frac{2}{3}} \intu{u^3\sqrt{1-u^2}^3}\\
  \end{array}
  \quad
  \lower 2.7cm \hbox{$
  \bmat{
    u = x/5 \\
    x = 5u \\
    dx = 5du \\
  }
  $}
$

}\anothercol{

\vspace*{0.0cm}


$\def\s{{\senθ}}
 \def\c{{\cosθ}}
 \def\sqs{{\sqrt{1-s^2}}}
 \def\sqx{{\sqrt{1-x^2}}}
 %
 \begin{array}{l}
   \intx{x^3\sqrt{1-x^2}^3} \\
   = \; \ints{s^3\sqrt{1-s^2}^3} \\
   = \; \intth{(\s)^3(\c)^3\c} \\
   = \; \intth{(\s)^3(\c)^4} \\
   = \; \intth{(\c)^4(\s)^3} \\
   = \; \intth{(\c)^4(\s)^2\s} \\
   = \; \intth{(\c)^4(1-(\c)^2)\s} \\
   = \; \intc{c^4(1-c^2)(-1)} \\
   = \; \intc{c^4(c^2-1)} \\
   = \; \intc{c^6-c^4} \\
   = \; \frac{c^7}{7} - \frac{c^5}{5} \\
   = \; \frac{(\c)^7}{7} - \frac{(\c)^5}{5} \\
   = \; \frac{(\sqs)^7}{7} - \frac{(\sqs)^5}{5} \\
   = \; \frac{(\sqx)^7}{7} - \frac{(\sqx)^5}{5} \\
   = \; \frac{(\sqx)^2(\sqx)^5}{7} - \frac{(\sqx)^5}{5} \\
   = \; \frac{(1-x^2)(\sqx)^5}{7} - \frac{(\sqx)^5}{5} \\
   = \; \frac{5(1-x^2)(\sqx)^5}{35} - \frac{7(\sqx)^5}{35} \\
   = \; \frac{(\sqx)^5}{35} (5(1-x^2) - 7) \\
   = \; \frac{(\sqx)^5}{35} (5 - 5x^2 - 7) \\
   = \; \frac{(\sqx)^5}{35} (-2 - 5x^2) \\
 \end{array}
 %
 \quad
 \bmat{
  s = \sen θ \\
  \sqrt{1-s^2} = \cos θ \\
  \frac{ds}{dθ} = \cos θ \\
  ds = \cos θ \, dθ \\
  θ = \arcsen s \\
  }
$


}}



\newpage

% «gab-2»  (to ".gab-2")
% (c2m242p1p 7 "gab-2")
% (c2m242p1a   "gab-2")

{\bf Questão 1: gabarito}


\newpage


% «questoes-12»  (to ".questoes-12")
% (c2m241p1p 2 "questoes-12")
% (c2m241p1a   "questoes-12")
% (c2m232p1p 2 "questoes-123")
% (c2m232p1a   "questoes-123")
% (c2m231p1p 2 "questoes-123")
% (c2m231p1a   "questoes-123")
% (c2m222p1p 1 "questao-1")
% (c2m222p1a   "questao-1")
% (c2m222p1p 2 "subst-trig")
% (c2m222p1a   "subst-trig")
% (c2m222mva "title")
% (c2m222mva "title" "Aula 10: Mudança de variáveis")
% (c2m222tudop 49)
% (c2m222striga "title")
% (c2m222striga "title" "Aulas 11 e 12: substituição trigonométrica")
% (find-es "maxima" "subst-trig-questions")
% (find-es "maxima" "subst-trig-questions" "F(2,1)")

%\vspace*{-0.4cm}




\newpage

%  _  _            ____  
% | || |     ___  | ___| 
% | || |_   / _ \ |___ \ 
% |__   _| |  __/  ___) |
%    |_|    \___| |____/ 
%                        
% «questoes-45»  (to ".questoes-45")
% (c2m232p1p 3 "questoes-45")
% (c2m232p1a   "questoes-45")
% (c2m231p1p 3 "questoes-45")
% (c2m231p1a   "questoes-45")



\newpage

% «questao-1-gab»  (to ".questao-1-gab")
% (c2m241p1p 5 "questao-1-gab")
% (c2m241p1a   "questao-1-gab")
% (find-es "maxima" "2024-1-C2-P1")





% (c2m231p1p 5 "questao-1-gab")
% (c2m231p1a   "questao-1-gab")
% (c2m222p1p 5 "questao-1-gab")
% (c2m222p1a   "questao-1-gab")


\newpage

% «questao-2-gab»  (to ".questao-2-gab")
% (c2m241p1p 6 "questao-2-gab")
% (c2m241p1a   "questao-2-gab")
% (c2m232p1p 6 "questao-2-gab")
% (c2m232p1a   "questao-2-gab")
% (c2m231p1p 6 "questao-2-gab")
% (c2m231p1a   "questao-2-gab")
% (c2m222p1p 6 "questao-2-gab")
% (c2m222p1a   "questao-2-gab")

\newpage

% (setq eepitch-preprocess-regexp "^")
% (setq eepitch-preprocess-regexp "^%T ?")
%
%T  (eepitch-maxima)
%T  (eepitch-kill)
%T  (eepitch-maxima)
%T linsolve ([A+B=4, 3*A-2*B=5], [A, B]);
%T linsolve ([A+B=2, 5*A-4*B=3], [A, B]);
%T
%T f : (4*x + 5) / ((x-2)*(x+3));
%T partfrac(f, x);
%T F : integrate(f, x);

\newpage

% «questao-3-gab»  (to ".questao-3-gab")
% (c2m232p1p 8 "questao-4-gab")
% (c2m232p1a   "questao-4-gab")

% (c2m231p1p 8 "questao-4-gab")
% (c2m231p1a   "questao-4-gab")
% (c2m222p1p 7 "questao-3-gab")
% (c2m222p1a   "questao-3-gab")
% (c2m221atisp 21 "1-then-2")
% (c2m221atisa    "1-then-2")


\newpage

% «questao-4-gab»  (to ".questao-4-gab")
% (c2m241p1p 5 "questao-4-gab")
% (c2m241p1a   "questao-4-gab")
% (c2m231p1p 9 "questao-5-gab")
% (c2m231p1a   "questao-5-gab")
% (c2m222p1p 9 "questao-5-gab")
% (c2m222p1a   "questao-5-gab")

{\bf Questão 4: gabarito}

\unitlength=10pt

$$\begin{array}{r}
 f(x) \;=\; \ga{fig f}  \\ \\
 F(x) \;=\; \Intt{3}{x}{f(t)}
      \;=\; \ga{fig F}  \\
 \end{array}
$$




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