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Warning: this is an htmlized version!
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% (find-LATEX "2020-1-C2-fracs-parcs.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020-1-C2-fracs-parcs.tex" :end))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2020-1-C2-fracs-parcs.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020-1-C2-fracs-parcs.pdf"))
% (defun e () (interactive) (find-LATEX "2020-1-C2-fracs-parcs.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020-1-C2-fracs-parcs"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page "~/LATEX/2020-1-C2-fracs-parcs.pdf")
% (find-sh0 "cp -v ~/LATEX/2020-1-C2-fracs-parcs.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2020-1-C2-fracs-parcs.pdf /tmp/pen/")
% file:///home/edrx/LATEX/2020-1-C2-fracs-parcs.pdf
% file:///tmp/2020-1-C2-fracs-parcs.pdf
% file:///tmp/pen/2020-1-C2-fracs-parcs.pdf
% http://angg.twu.net/LATEX/2020-1-C2-fracs-parcs.pdf
% (find-LATEX "2019.mk")
% (find-C2-aula-links "2020-1-C2-fracs-parcs" "16" "fracparc")
% «.video» (to "video")
%
% «.defs» (to "defs")
% «.title» (to "title")
% «.2020_deriv_ln.mp4» (to "2020_deriv_ln.mp4")
% «.exercicio-4» (to "exercicio-4")
% «.djvuize» (to "djvuize")
% «video» (to ".video")
% (find-ssr-links "c2m201divcomresto" "2020_div_com_resto")
% (code-eevvideo "c2m201divcomresto" "2020_div_com_resto")
% (code-eevlinksvideo "c2m201divcomresto" "2020_div_com_resto")
% (find-c2m201divcomrestovideo "0:00")
\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
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% (find-dn6 "preamble6.lua" "preamble0")
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%\input diagxy % For 2D diagrams ("%D" lines)
%\xyoption{curve} % For the ".curve=" feature in 2D diagrams
%
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\input edrxaccents.tex % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex % (find-LATEX "edrxchars.tex")
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%
%\usepackage[backend=biber,
% style=alphabetic]{biblatex} % (find-es "tex" "biber")
%\addbibresource{catsem-slides.bib} % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
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%
\begin{document}
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
% %L dofile "edrxtikz.lua" -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua" -- (find-LATEX "edrxpict.lua")
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% (find-LATEX "edrx15.sty" "colors-2019")
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\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% (find-angg ".emacs" "c2q192")
% _____ _ _ _
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% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
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% |_| |___/
%
% «title» (to ".title")
% (c2m201fracparcp 1 "title")
% (c2m201fracparc "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2020.1}
\bsk
Aulas 16 e 17: Frações parciais
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2020.1-C2.html}
\end{center}
\newpage
% «2020_deriv_ln.mp4» (to ".2020_deriv_ln.mp4")
% (c2m201fracparcp 2 "2020_deriv_ln.mp4")
% (c2m201fracparc "2020_deriv_ln.mp4")
% (find-twusfile "eev-videos/")
% (find-twusfile "eev-videos/" "2020_deriv_ln.mp4")
% (find-eevvideo-links "derivln" "2020_deriv_ln" "0:00")
% (code-video "derivlnvideo" "$S/http/angg.twu.net/eev-videos/2020_deriv_ln.mp4")
% (find-derivlnvideo "0:00")
% (find-derivlnvideo "2:48" "aparece em C1 mas ninguém dá muita bola pra ela")
% (find-derivlnvideo "4:15" "pode ser calculada de dois jeitos diferentes")
% (find-derivlnvideo "4:41" "f'(g(x))g'(x) = 1")
% (find-derivlnvideo "5:52" "[DFI]")
% (find-derivlnvideo "7:44" "[DFI][substs] = (...)")
% (find-derivlnvideo "8:30" "d/dx ln x = 1/x")
% (find-derivlnvideo "8:48" "ln x = \\int 1/x dx")
% (find-derivlnvideo "9:11" "num primeiro momento a gente não sabe se isso é verdade")
Neste vídeo nós vimos que $\frac{d}{dx} \ln x = 1/x$...
\ssk
\url{http://angg.twu.net/eev-videos/2020_deriv_ln.mp4}
\ssk
e aí começamos a fazer exercícios de integração...
\bsk
{\bf Exercício 1.}
% (find-fline "~/LATEX/2020-1-C2/20201112_C2_fracoes_parciais_1.pdf")
\includegraphics[height=5cm]{2020-1-C2/20201112_C2_fracoes_parciais_1.pdf}
\newpage
% (find-fline "~/LATEX/2020-1-C2/20201112_C2_fracoes_parciais_2.pdf")
\includegraphics[height=8cm]{2020-1-C2/20201112_C2_fracoes_parciais_2.pdf}
\newpage
{\bf Exercício 2.}
% (find-fline "~/LATEX/2020-1-C2/20201112_C2_fracoes_parciais_3.pdf")
\includegraphics[height=7cm]{2020-1-C2/20201112_C2_fracoes_parciais_3.pdf}
\newpage
{\bf Exercício 3.}
% (find-fline "~/LATEX/2020-1-C2/20201112_C2_fracoes_parciais_4.pdf")
\includegraphics[height=7cm]{2020-1-C2/20201112_C2_fracoes_parciais_4.pdf}
\newpage
{\bf Slogan: contas sem ``vai um'' podem ser traduzidas
pra contas com polinômios.}
\ssk
O que mais nos interessa pra Frações Parciais
é \ColorRed{divisão com resto}. Exemplo:
% (find-fline "~/LATEX/2020-1-C2/20201118_C2_div_com_resto_1.pdf")
\includegraphics[width=11cm]{2020-1-C2/20201118_C2_div_com_resto_1.pdf}
\newpage
...e tradução do exemplo para polinômios:
% (find-fline "~/LATEX/2020-1-C2/20201118_C2_div_com_resto_2.pdf")
\includegraphics[height=4cm]{2020-1-C2/20201118_C2_div_com_resto_2.pdf}
% (find-fline "~/LATEX/2020-1-C2/20201118_C2_div_com_resto_3.pdf")
\includegraphics[height=3cm]{2020-1-C2/20201118_C2_div_com_resto_3.pdf}
\newpage
% «exercicio-4» (to ".exercicio-4")
% (c2m201fracparcp 99 "exercicio-4")
% (c2m201fracparc "exercicio-4")
{\bf Exercício 4.}
\ssk
Use estas idéias para integrar:
$$\intx{\frac{2x^3 + 7x^2 + 7x + 3}{x+2}} \;\; = \;\; ?$$
\newpage
{\bf Exercício 5.}
\ssk
O que acontece nos casos em que ``teria vai um''?
\ssk
a) Tente fazer a divisão com resto de $x^3$ por $x+2$.
Mais precisamente, encontre um polinômios $R(x)$ e $Q(x)$ tais que
$(x^3) = Q(x) · (x+2) + R(x)$ e $R(x)$ é no máximo de grau 1.
Teste a sua resposta!
\ssk
b) Calcule $\intx{\frac{x^3}{x+2}}$ pelo método acima.
Teste a sua resposta derivando a sua antiderivada para $\frac{x^3}{x+2}$.
\ssk
c) Calcule $\intx{\frac{x^3}{x+2}}$ fazendo a substituição $u=x+2$.
Você deve obter o mesmo resultado que na (b).
\bsk
d) Calcule $\intx{\frac{x^2}{(x+1)(x-1)}}$ por frações parciais.
\newpage
{\bf Dica importante}
\ssk
Lembre que uns dos meus slogans é
``eu só vou corrigir os sinais de igual''...
No slide 7 a igualdade mais importante é a da última linha.
Nós vamos usá-la assim, pra transformar a integral original
em algo fácil de integrar:
\msk
% (find-fline "~/LATEX/2020-1-C2/20201119_C2_div_com_resto_4.pdf")
\includegraphics[height=4cm]{2020-1-C2/20201119_C2_div_com_resto_4.pdf}
\end{document}
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% |__/
%
% «djvuize» (to ".djvuize")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-fline "~/2020.1-C2/")
# (find-fline "~/LATEX/2020-1-C2/")
# (find-fline "~/bin/djvuize")
cd ~/2020.1-C2/
rm -v 20201112_C2_fracoes_parciais*.png
rm -v 20201112_C2_fracoes_parciais*.pdf
f () { djvuize WHITEBOARDOPTS="-m 0.5" $* }
f 20201112_C2_fracoes_parciais_1.pdf
f 20201112_C2_fracoes_parciais_2.pdf
f 20201112_C2_fracoes_parciais_3.pdf
f 20201112_C2_fracoes_parciais_4.pdf
cd ~/2020.1-C2/
f () { djvuize WHITEBOARDOPTS="-m 0.5" $* }
rm -v 20201118_C2_div_com_resto*.png
rm -v 20201118_C2_div_com_resto*.pdf
f 20201118_C2_div_com_resto_1.pdf
f 20201118_C2_div_com_resto_2.pdf
f 20201118_C2_div_com_resto_3.pdf
f 20201119_C2_div_com_resto_4.pdf
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
cp -v ~/LATEX/2020-1-C2-fracs-parcs.pdf /tmp/
cd /tmp/
xournalpp 2020-1-C2-fracs-parcs.pdf
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% | |\/| |/ _` | |/ / _ \
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%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020-1-C2-fracs-parcs veryclean
make -f 2019.mk STEM=2020-1-C2-fracs-parcs pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2m201fracparc"
% End: