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Warning: this is an htmlized version!
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% (find-LATEX "2021-1-C2-contas-em-C2.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2021-1-C2-contas-em-C2.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2021-1-C2-contas-em-C2.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2021-1-C2-contas-em-C2.pdf"))
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% (defun u () (interactive) (find-latex-upload-links "2021-1-C2-contas-em-C2"))
% (defun v () (interactive) (find-2a '(e) '(d)))
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% (code-eec-LATEX "2021-1-C2-contas-em-C2")
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% (find-xournalpp "/tmp/2021-1-C2-contas-em-C2.pdf")
% file:///home/edrx/LATEX/2021-1-C2-contas-em-C2.pdf
% file:///tmp/2021-1-C2-contas-em-C2.pdf
% file:///tmp/pen/2021-1-C2-contas-em-C2.pdf
% http://angg.twu.net/LATEX/2021-1-C2-contas-em-C2.pdf
% (find-LATEX "2019.mk")
% (find-CN-aula-links "2021-1-C2-contas-em-C2" "2" "c2m211cc2" "c2c")
% «.video-1» (to "video-1")
% «.video-2» (to "video-2")
%
% «.defs» (to "defs")
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%
% «.djvuize» (to "djvuize")
% «video-1» (to ".video-1")
% (find-ssr-links "c2m211cc2" "2021-1-C2-contas-em-C2" "P3fbFkCI_Bo")
% (code-eevvideo "c2m211cc2" "2021-1-C2-contas-em-C2" "P3fbFkCI_Bo")
% (code-eevlinksvideo "c2m211cc2" "2021-1-C2-contas-em-C2" "P3fbFkCI_Bo")
% (find-c2m211cc2video "0:00" "1º/set/2021")
% «video-2» (to ".video-2")
% (find-ssr-links "c2m211cc2" "2021-1-C2-contas-em-C2-2")
% (code-eevvideo "c2m211cc2" "2021-1-C2-contas-em-C2-2")
% (code-eevlinksvideo "c2m211cc2" "2021-1-C2-contas-em-C2-2")
% (find-c2m211cc2video "0:00" "3/set/2021")
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\usepackage{amssymb}
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\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[backend=biber,
% style=alphabetic]{biblatex} % (find-es "tex" "biber")
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%
% (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")
% \pu
% «defs» (to ".defs")
% (find-LATEX "edrx15.sty" "colors-2019")
%\long\def\ColorRed #1{{\color{Red1}#1}}
%\long\def\ColorViolet#1{{\color{MagentaVioletLight}#1}}
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%
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%\long\def\ColorLong #1{{\color{Red1}#1}}
%
%\def\frown{\ensuremath{{=}{(}}}
%\def\True {\mathbf{V}}
%\def\False{\mathbf{F}}
%\def\D {\displaystyle}
\def\pfo#1{\ensuremath{\mathsf{[#1]}}}
\def\drafturl{http://angg.twu.net/LATEX/2021-1-C2.pdf}
\def\drafturl{http://angg.twu.net/2021.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m211cc2p 1 "title")
% (c2m211cc2a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2021.1}
\bsk
Aula 23: como contas de integração
costumam ser organizadas
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2021.1-C2.html}
\end{center}
\newpage
% «exemplo-1» (to ".exemplo-1")
% (c2m211cc2p 2 "exemplo-1")
% (c2m211cc2a "exemplo-1")
{\bf Exemplo 1.}
\ssk
{\footnotesize
% http://angg.twu.net/eev-videos/2021-1-C2-contas-em-C2.mp4
\url{http://angg.twu.net/eev-videos/2021-1-C2-contas-em-C2.mp4}
(Ele está explicado neste $↑$ vídeo! Assista!)
}
\msk
Temos:
%
$$\begin{array}{rcl}
\ddx(f(x)g(x)) &=& f'(x)g(x) + f(x)g'(x) \\
f(x)g(x) &=& \intx {f'(x)g(x) + f(x)g'(x)} \\
&=& \intx {f'(x)g(x)} + \intx {f(x)g'(x)} \\[10pt]
\end{array}
$$
Então:
%
$$\begin{array}{rcl}
\intx {f'(x)g(x)} + \intx {f(x)g'(x)} &=& f(x)g(x) \\
\intx {f(x)g'(x)} &=& f(x)g(x) - \intx {f'(x)g(x)} \\
\intx {f'(x)g(x)} \phantom{mmmmmmmmi} &=& f(x)g(x) - \intx {f(x)g'(x)} \\
\end{array}
$$
\newpage
% «exemplo-1-cont» (to ".exemplo-1-cont")
% (c2m211cc2p 3 "exemplo-1-cont")
% (c2m211cc2a "exemplo-1-cont")
Sejam:
%
$$\begin{array}{lrcl}
\text{IP1}. & \intx {f(x)g'(x)} &=& f(x)g(x) - \intx {f'(x)g(x)} \\
\text{IP2}. & \intx {f'(x)g(x)} &=& f(x)g(x) - \intx {f(x)g'(x)} \\
\end{array}
$$
Então:
%
$$\begin{array}{rcl}
\intx {x^3 e^x} &=& x^3 e^x - \intx {3 x^2 e^x} \\
&=& x^3 e^x - 3 \intx {x^2 e^x} \\
\intx {x^2 e^x} &=& x^2 e^x - \intx {2 x e^x} \\
&=& x^2 e^x - 2 \intx {x e^x} \\
\intx {x e^x} &=& x e^x - \intx {e^x} \\
&=& x e^x - e^x \\[10pt]
%
\intx {x^3 e^x} &=& x^3 e^x - 3 \intx {x^2 e^x} \\
&=& x^3 e^x - 3 (x^2 e^x - 2 \intx {x e^x}) \\
&=& x^3 e^x - 3 (x^2 e^x - 2 (x e^x - \intx {e^x})) \\
&=& x^3 e^x - 3 (x^2 e^x - 2 (x e^x - e^x)) \\
\end{array}
$$
\newpage
{\bf Derivada da função inversa}
\def\Dom#1{\ColorRed{#1}}
$$\begin{array}{rcl}
\pfo{DFI1} &=&
\left(
\begin{array}{ll}
\text{Se} & \Dom{∀x∈D.\;} f(g(x)) = x \\[2pt]
\text{Então} & \ddx f(g(x)) = \ddx x = 1, \\[2pt]
& \ddx f(g(x)) = f'(g(x))g'(x), \\[2pt]
& f'(g(x))g'(x) = 1, \\[2pt]
& g'(x) = 1 / f'(g(x)) \\
\end{array}
\right)
\\
[40pt]
\pfo{DFI2} &=&
\left(
\begin{array}{ll}
\text{Se} & \Dom{∀x∈D.\;} f(g(x)) = x \\[2pt]
\text{Então} & g'(x) = 1 / f'(g(x)) \\
\end{array}
\right)
\end{array}
$$
$$\pfo{DFI2}
\bmat{f(y) := e^y \\
g(x) := \ln x \\
D := (0,+∞) \\
}
\;\;=\;\;
\left(
\begin{array}{ll}
\text{Se} & \Dom{∀x∈(0,+∞).\;} e^{\ln x} = x \\[2pt]
\text{Então} & \ln'(x) = 1 / e^{\ln x} \\
\end{array}
\right)
$$
\newpage
% «d-ln» (to ".d-ln")
% (c2m211cc2p 5 "d-ln")
% (c2m211cc2a "d-ln")
{\bf Derivada do $\ln$}
%
$$\pfo{DFI2}
\bmat{f(y) := e^y \\
g(x) := \ln x \\
D := (0,+∞) \\
}
\;\;=\;\;
\left(
\begin{array}{ll}
\text{Se} & \Dom{∀x∈(0,+∞).\;} e^{\ln x} = x \\[2pt]
\text{Então} & \ln'(x) = 1 / e^{\ln x} \\
\end{array}
\right)
$$
$$\begin{array}{rcl}
\ln'(x) &=& 1/e^{\ln x} \qquad \text{(pelo \pfo{DFI2})}\\
&=& 1/x \\
\end{array}
$$
\newpage
% «d-arcsen» (to ".d-arcsen")
% (c2m211cc2p 6 "d-arcsen")
% (c2m211cc2a "d-arcsen")
{\bf Derivada do $\arcsen$}
%
$$\scalebox{0.85}{$
\begin{array}{c}
\pfo{DFI2}
\;\;=\;\;
\left(
\begin{array}{ll}
\text{Se} & \Dom{∀x∈D.\;} f(g(x)) = x \\[2pt]
\text{Então} & g'(x) = 1 / f'(g(x)) \\
\end{array}
\right)
%
\\[15pt]
%
\pfo{DFI2}
\bmat{x := s \\
f(θ) := \sen θ \\
g(s) := \arcsen s \\
D := (-1,+1) \\
}
\;\;=\;\;
\left(
\begin{array}{ll}
\text{Se} & \Dom{∀s∈(-1,+1).\;} \sen \arcsen s = s \\[2pt]
\text{Então} & \arcsen'(s) = 1 / (\cos \arcsen s) \\
\end{array}
\right)
\end{array}
$}
$$
$$\scalebox{0.85}{$
\begin{array}{rcl}
\cos^2 θ + \sen^2 θ &=& 1 \\
\cos^2 θ &=& 1 - \sen^2 θ \\
\cos θ &=& \sqrt{1 - \sen^2 θ} \\
\cos (\arcsen s) &=& \sqrt{1 - (\sen(\arcsen s))^2} \\
&=& \sqrt{1 - s^2} \\
\frac{d}{ds} \arcsen(s) &=& 1 / (\cos \arcsen s) \\
&=& 1 / \sqrt{1 - s^2} \\
\arcsen(s) &=& \ints {1 / \sqrt{1 - s^2}} \\
\end{array}
$}
$$
\newpage
% «d-arcsen-2» (to ".d-arcsen-2")
% (c2m211cc2p 7 "d-arcsen-2")
% (c2m211cc2a "d-arcsen-2")
{\bf Derivada do $\arcsen$ (2)}
Se $s = \senθ$ então $\frac{ds}{dθ} = \cos θ$, $ds = \cosθ \, dθ$, e:
%
$$\begin{array}{l}
\ints {1 / \sqrt{1 - s^2}} \\
=\;\; \intth {\frac{1}{\sqrt{1 - (\senθ)^2}} \; \cosθ} \\
=\;\; \intth {\frac{1}{\sqrt{(\cosθ)^2}} \; \cosθ} \\
=\;\; \intth {\frac{1}{\cosθ} \; \cosθ} \\
=\;\; \intth {1} \\
=\;\; θ \\
=\;\; \arcsen \sen θ \\
=\;\; \arcsen s \\
\end{array}
$$
\newpage
% «d-arcsen-3» (to ".d-arcsen-3")
% (c2m211cc2p 8 "d-arcsen-3")
% (c2m211cc2a "d-arcsen-3")
{\bf Derivada do $\arcsen$ (3)}
Se usarmos uma caixa de anotações bem maior
podemos fazer essa conta bem mais rápido...
$$\begin{array}{l}
\ints {1 / \sqrt{1 - s^2}} \\
=\;\; \intth {\frac{1}{\cosθ} \; \cosθ} \\
=\;\; \intth {1} \\
=\;\; θ \\
=\;\; \arcsen s \\
\end{array}
\qquad
\bmat{ s = \sen θ \\
\frac{ds}{dθ} = \frac{d}{dθ} \sen θ = \cos θ \\
ds = \cos θ \, dθ\\
1 - s^2 = \cos^2 θ \\
\sqrt{1 - s^2} = \cos θ \\
θ = \arcsen s \\
}
$$
\bsk
Essa caixa de anotações grande vai ser chamada de
\ColorRed{substituição trigonométrica} (para $s=\sen θ$).
\msk
Outras substituições trigonométricas
famosas: $t=\tanθ$, $z=\secθ$.
\newpage
Normalmente a gente aprende substituições trigonométricas
depois do método pra integrar potências de senos e cossenos...
\msk
Entenda os exemplos 1 e 2 daqui,
{\footnotesize
% (c2m202isp 12 "senos-e-cossenos")
% (c2m202isa "senos-e-cossenos")
% http://angg.twu.net/LATEX/2020-2-C2-int-subst.pdf#page=12
\url{http://angg.twu.net/LATEX/2020-2-C2-int-subst.pdf#page=12}
}
e tente fazer o ``exercício 3'' desse PDF.
Obs: no semestre passado eu usei convenções
um pouco diferentes das de agora pras
caixas de anotações...
%\printbibliography
\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 "~/2021.1-C2/")
# (find-fline "~/LATEX/2021-1-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 ~/2021.1-C2/
cp -fv $1.pdf ~/LATEX/2021-1-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=2021-1-C2-contas-em-C2 veryclean
make -f 2019.mk STEM=2021-1-C2-contas-em-C2 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2c"
% ee-tla: "c2mcc2"
% ee-tla: "c2m211cc2"
% End: