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% (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: