|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2022-2-C2-VS.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2022-2-C2-VS.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2022-2-C2-VS.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2022-2-C2-VS.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2022-2-C2-VS.pdf"))
% (defun e () (interactive) (find-LATEX "2022-2-C2-VS.tex"))
% (defun o () (interactive) (find-LATEX "2022-1-C2-VSA.tex"))
% (defun o () (interactive) (find-LATEX "2022-2-C2-P2.tex"))
% (defun o () (interactive) (find-LATEX "2022-2-C2-VR.tex"))
% (defun u () (interactive) (find-latex-upload-links "2022-2-C2-VS"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2022-2-C2-VS.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2022-2-C2-VS")
% (find-pdf-page "~/LATEX/2022-2-C2-VS.pdf")
% (find-sh0 "cp -v ~/LATEX/2022-2-C2-VS.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2022-2-C2-VS.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2022-2-C2-VS.pdf")
% file:///home/edrx/LATEX/2022-2-C2-VS.pdf
% file:///tmp/2022-2-C2-VS.pdf
% file:///tmp/pen/2022-2-C2-VS.pdf
% http://angg.twu.net/LATEX/2022-2-C2-VS.pdf
% (find-LATEX "2019.mk")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Piecewise1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Pict3D1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v C2Subst1.lua C2Formulas1.lua ~/LATEX/")
% (find-CN-aula-links "2022-2-C2-VS" "2" "c2m222vs" "c2vs")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.title» (to "title")
% «.links» (to "links")
% «.questao-1» (to "questao-1")
% «.questao-1-maxima» (to "questao-1-maxima")
% «.questao-2» (to "questao-2")
% «.questao-2-maxima» (to "questao-2-maxima")
% «.anexo-L» (to "anexo-L")
% «.anexo-R» (to "anexo-R")
% «.anexo» (to "anexo")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c2m222vs" "2022-2-C2-VS")
% (code-eevvideo "c2m222vs" "2022-2-C2-VS")
% (code-eevlinksvideo "c2m222vs" "2022-2-C2-VS")
% (find-c2m222vsvideo "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}
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
%%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 dofile "2022-1-C2-P2.lua" -- (find-LATEX "2022-1-C2-P2.lua")
%L Pict2e.__index.suffix = "%"
\pu
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\u#1{\par{\footnotesize \url{#1}}}
\def\drafturl{http://angg.twu.net/LATEX/2022-2-C2.pdf}
\def\drafturl{http://angg.twu.net/2022.2-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
\sa{[M]}{\CFname{M}{}}
\sa{[F]}{\CFname{F}{}}
\sa{[S]}{\CFname{S}{}}
\def\Smile{\emoji{slightly-smiling-face}}
% «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)}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m222vsp 1 "title")
% (c2m222vsa "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2022.2}
\bsk
Prova suplementar (VS)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.2-C2.html}
\end{center}
\newpage
% «links» (to ".links")
% (c2m222p1p 5 "questao-1-gab")
% (c2m222p1a "questao-1-gab")
% (c2m221atisp 50 "exemplo")
% (c2m221atisa "exemplo")
\newpage
% ___ _ _
% / _ \ _ _ ___ ___| |_ __ _ ___ / |
% | | | | | | |/ _ \/ __| __/ _` |/ _ \ | |
% | |_| | |_| | __/\__ \ || (_| | (_) | | |
% \__\_\\__,_|\___||___/\__\__,_|\___/ |_|
%
% «questao-1» (to ".questao-1")
% (c2m222vsp 2 "questao-1")
% (c2m222vsa "questao-1")
%
%L namedang("EDOVSintro", "", [[
%L \begin{array}{rcl}
%L \ga{[M]} &=& <EDOVSG> \\ \\[-5pt]
%L \ga{[F]} &=& <EDOVSP> \\
%L \end{array}
%L ]])
%L namedang("metodo", "", [[ <EDOVSG> ]])
%L metodo:sa("FOO"):output()
\pu
\scalebox{0.6}{\def\colwidth{9.5cm}\firstcol{
{\bf \large Questão 1}
\T(Total: 6.0 pts)
\ssk
A primeira EDO com variáveis separáveis (``EDOVS'') que nós vimos
no curso foi $\frac{dy}{dx}=-\frac{x}{y}$. As soluções
particulares dela eram pedaços das curvas de nível de $x^2+y^2=C$
--- e esses pedaços eram ou semicírculos acima do eixo $x$ ou
semicírculos abaixo do eixo $y$.
\msk
Na P2 eu pus uma questão sobre uma EDOVS um pouco mais complicada
do que essa dos semicírculos, e muitas pessoas fizeram erros de
conta horríveis {\sl que eu acho que foram causados por
desorganização na hora de fazer as contas}... por exemplo,
várias pessoas escreveram ``$H(x)=\sqrt{x}$'' num lugar das contas
e ``$H(x)=\sqrt{x+3}$'' em outro --- o que OBVIAMENTE é um erro
conceitual GRAVÍSSIMO, né, {\sl a menos que você explique em
português direitinho que $H(x)$ vai ser $\sqrt{x}$ em um
contexto e $\sqrt{x+3}$ em outro contexto separado...}
\msk
}\anothercol{
\vspace*{0.85cm}
Seja $(*)$ esta EDOVS:
%
$$\frac{dy}{dx} = \frac{4x^3}{4(y+3)^3} \qquad (*)
$$
Encontre as duas soluções gerais da EDO $(*)$ --- uma ``positiva'' e
outra ``negativa'' ---, encontre as soluções particulares que passam
pelos pontos $(1,-1)$ e $(1,-5)$, e teste tudo.
\msk
{\sl Nesta questão eu vou avaliar principalmente se você sabe usar
direito os truques do anexo da página 4.} Ou seja, não vai bastar
você usar o ``método'' para resolver EDOVSs, que é esse aqui:
%
$$\ga{[M]} \;=\; \scalebox{0.85}{$\ga{FOO}$}$$
}}
% «questao-1-maxima» (to ".questao-1-maxima")
% (c2m222vsp 2 "questao-1-maxima")
% (c2m222vsa "questao-1-maxima")
% (setq eepitch-preprocess-regexp "^")
% (setq eepitch-preprocess-regexp "^%T ")
%
%T * (eepitch-maxima)
%T * (eepitch-kill)
%T * (eepitch-maxima)
%T define(G(x), x^4);
%T define(H(y), (y+3)^4);
%T
%T define(g(x), diff(G(x),x));
%T define(h(y), diff(H(y),y));
%T edo : dydx = g(x) / h(y);
%T
%T eq_a : x = H(y);
%T solve(eq_a, y);
%T
%T getsol(n,C31,x1,y1) := (
%T eq_b : solve(eq_a, y)[n],
%T define(Hinv(x), rhs(eq_b)),
%T eq_c : y = Hinv(G(x) + C3),
%T eq_d : y = Hinv(G(x) + C31),
%T define(f (x), rhs(eq_c)),
%T define(f1 (x), rhs(eq_d))
%T );
%T
%T getsol(4,15, 1,-1);
%T f(x);
%T f1(x);
%T f1(1);
%T
%T getsol(3,15, 1,-5);
%T f(x);
%T f1(x);
%T f1(1);
\newpage
% ___ _ ____
% / _ \ _ _ ___ ___| |_ __ _ ___ |___ \
% | | | | | | |/ _ \/ __| __/ _` |/ _ \ __) |
% | |_| | |_| | __/\__ \ || (_| | (_) | / __/
% \__\_\\__,_|\___||___/\__\__,_|\___/ |_____|
%
% «questao-2» (to ".questao-2")
% (c2m222vsp 3 "questao-2")
% (c2m222vsa "questao-2")
\scalebox{0.7}{\def\colwidth{9cm}\firstcol{
{\bf \large Questão 2}
\T(Total: 4.0 pts)
\ssk
Calcule a integral abaixo usando pelo menos três mudanças de
variáveis.
%
$$\intx{ \frac{8e^{4x}\ln(e^{4x}+2)}{e^{4x}+2} }
$$
Obs: no curso nós vimos que qualquer integral que pode ser resolvida
por uma sequência de mudanças de variáveis também pode ser resolvida
por uma mudança de variável só, mas aqui é pra usar pelo menos três!
% «questao-2-maxima» (to ".questao-2-maxima")
% (c2m222vsp 3 "questao-2-maxima")
% (c2m222vsa "questao-2-maxima")
% (setq eepitch-preprocess-regexp "^")
% (setq eepitch-preprocess-regexp "^%T ")
%
%T * (eepitch-maxima)
%T * (eepitch-kill)
%T * (eepitch-maxima)
%T b(x) := exp(x);
%T c(x) := x^4 + 2;
%T d(x) := log(x)^2;
%T F : d(c(b(x)));
%T f : diff(F, x);
%T integrate(f, x);
}\anothercol{
}}
\newpage
% _
% / \ _ __ _____ _____
% / _ \ | '_ \ / _ \ \/ / _ \
% / ___ \| | | | __/> < (_) |
% /_/ \_\_| |_|\___/_/\_\___/
%
% (c2m222p2p 5 "questao-1-gab")
% (c2m222p2a "questao-1-gab")
% «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}
{\bf uma questão da P2}
\ssk
\anexoL
}\anothercol{
\anexoR
}}
\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 "~/2022.2-C2/")
# (find-fline "~/LATEX/2022-2-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 ~/2022.2-C2/
cp -fv $1.pdf ~/LATEX/2022-2-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=2022-2-C2-VS veryclean
make -f 2019.mk STEM=2022-2-C2-VS pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2vs"
% ee-tla: "c2m222vs"
% End: