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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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
$$\begin{array}{r}
f(x) \;=\; \ga{fig f} \\ \\
F(x) \;=\; \Intt{3}{x}{f(t)}
\;=\; \ga{fig F} \\
\end{array}
$$
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
% (find-pdfpages2-links "~/LATEX/" "2024-2-C2-P1")
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2p1"
% ee-tla: "c2m242p1"
% End: