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Warning: this is an htmlized version!
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% (find-pdf-page "~/LATEX/2022-2-C2-TFC1-e-TFC2.pdf")
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% file:///tmp/2022-2-C2-TFC1-e-TFC2.pdf
% file:///tmp/pen/2022-2-C2-TFC1-e-TFC2.pdf
% http://angg.twu.net/LATEX/2022-2-C2-TFC1-e-TFC2.pdf
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% (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-TFC1-e-TFC2" "2" "c2m222tfcs" "c2tf")
% «.defs» (to "defs")
% «.title» (to "title")
% «.links» (to "links")
% «.def-particao» (to "def-particao")
% «.def-inf-e-sup» (to "def-inf-e-sup")
% «.def-integral» (to "def-integral")
% «.exercicio-2» (to "exercicio-2")
% «.exercicio-3» (to "exercicio-3")
% «.exercicio-4» (to "exercicio-4")
%
% <videos>
% Video (not yet):
% (find-ssr-links "c2m222tfcs" "2022-2-C2-TFC1-e-TFC2")
% (code-eevvideo "c2m222tfcs" "2022-2-C2-TFC1-e-TFC2")
% (code-eevlinksvideo "c2m222tfcs" "2022-2-C2-TFC1-e-TFC2")
% (find-c2m222tfcsvideo "0:00")
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%
% (find-dn6 "preamble6.lua" "preamble0")
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%
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\input edrxgac2.tex % (find-LATEX "edrxgac2.tex")
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%
%\usepackage[backend=biber,
% style=alphabetic]{biblatex} % (find-es "tex" "biber")
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%
% (find-es "tex" "geometry")
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%L dofile "Piecewise1.lua" -- (find-LATEX "Piecewise1.lua")
%L dofile "QVis1.lua" -- (find-LATEX "QVis1.lua")
%L dofile "Pict3D1.lua" -- (find-LATEX "Pict3D1.lua")
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\pu
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% «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}}
% (find-LATEX "edrxgac2.tex" "C2")
\def\Rext{\overline{\R}}
\def\into{\overline ∫}
\def\intu{\underline∫}
\def\intou{\overline{\underline∫}}
\def\INTx#1#2#3#4{#1_{x=#2}^{x=#3} #4 \, dx}
\def\INTP #1#2#3{#1_{#2} #3 \, dx}
\def\mname#1{\ensuremath{[\text{#1}]}}
\def\minf{\mname{inf}}
\def\msup{\mname{sup}}
\def\sse {\text{sse}}
\sa{into_P f(x) dx}{\INTP{\into} {P}{f(x)}}
\sa{intu_P f(x) dx}{\INTP{\intu} {P}{f(x)}}
\sa{intou_P f(x) dx}{\INTP{\intou}{P}{f(x)}}
\sa{into_ab2k f(x) dx}{\INTP{\into} {[a,b]_{2^k}}{f(x)}}
\sa{intu_ab2k f(x) dx}{\INTP{\intu} {[a,b]_{2^k}}{f(x)}}
\sa{intou_ab2k f(x) dx}{\INTP{\intou}{[a,b]_{2^k}}{f(x)}}
\sa{into_xab f(x) dx}{\INTx{\into} {a}{b}{f(x)}}
\sa{intu_xab f(x) dx}{\INTx{\intu} {a}{b}{f(x)}}
\sa{intou_xab f(x) dx}{\INTx{\intou}{a}{b}{f(x)}}
\sa{int_xab f(x) dx}{\INTx{\int} {a}{b}{f(x)}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m222tfcsp 1 "title")
% (c2m222tfcsa "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2022.2}
\bsk
Aula 19: o TFC1 e o TFC2.
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.2-C2.html}
\end{center}
\newpage
% % «links» (to ".links")
% % (c2m222tfcsp 2 "links")
% % (c2m222tfcsa "links")
%
% Infs e sups
% % (c2m222srp 25 "na-semana-academica")
% % (c2m222sra "na-semana-academica")
%
% Partições
% % (c2m212somas1p 9 "particoes")
% % (c2m212somas1a "particoes")
%
% Partição preferida
%
% Partições cada vez mais finas
% % (c2m202escadasp 4 "exercicio-2")
% % (c2m202escadas "exercicio-2")
%
% Definição da integral
% % (c2m221isp 19 "definicao-integral")
% % (c2m221isa "definicao-integral")
%
% Funções de Dirichlet (a original e a diagonal)
% % (c2m212somas2p 51 "dirichlet")
% % (c2m212somas2a "dirichlet")
%
% TFC1 no vídeo do Mathologer (localizar)
% % (find-TH "mathologer-calculus-easy" "legendas")
%
% TFC2 no vídeo do Mathologer (localizar)
%
% Revisão de derivadas laterais
% % (c2m221tfc1p 11 "exercicio-4-dicas")
% % (c2m221tfc1a "exercicio-4-dicas" "laterais")
%
% Quando os TFCs não valem
%
% Quais livros usam ai e bi? Quais usam $Δx$?
%
% Animacoes sobre def integral e TFC1
%
% % (find-books "__analysis/__analysis.el" "leithold" "5.5. A integral definida")
% % (find-books "__analysis/__analysis.el" "martins-martins" "Integral Definida")
% % (find-books "__analysis/__analysis.el" "miranda" "7.2. Integral definida")
% (find-books "__analysis/__analysis.el" "thomas" "5.3 The definite integral")
\newpage
Os próximos 3 slides são uma versão melhorada (?)
das definições de partições, de inf e sup, e de
integral definida destes PDFs antigos:
\ssk
Partições:
{\scriptsize
% (c2m212somas1p 9 "particoes")
% (c2m212somas1a "particoes")
% http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf#page=9
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf\#page=9} (até a p.12)
}
\ssk
Infs e sups:
{\scriptsize
% (c2m221isp 2 "uma-figura")
% (c2m221isa "uma-figura")
% http://angg.twu.net/LATEX/2022-1-C2-infs-e-sups.pdf#page=2
\url{http://angg.twu.net/LATEX/2022-1-C2-infs-e-sups.pdf\#page=2} (até a p.15)
}
\ssk
Integral definida como limite -- definições:
{\scriptsize
% (c2m221isp 16 "aproximacoes-por-cima")
% (c2m221isa "aproximacoes-por-cima")
% http://angg.twu.net/LATEX/2022-1-C2-infs-e-sups.pdf#page=16
\url{http://angg.twu.net/LATEX/2022-1-C2-infs-e-sups.pdf\#page=16} (até a p.21)
% (c2m212somas2p 35 "exercicio-13")
% (c2m212somas2a "exercicio-13")
% http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf#page=35
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf\#page=35} -- $[a,b]_n$
% (c2m212somas1p 16 "exercicio-9-dicas")
% (c2m212somas1a "exercicio-9-dicas")
% http://angg.twu.net/LATEX/2021-1-C2-somas-1.pdf#page=16
\url{http://angg.twu.net/LATEX/2021-1-C2-somas-1.pdf\#page=16} -- simplificações
}
\ssk
Integral definida como limite -- uma animação:
{\scriptsize
% (c2m221tfc1p 35 "descontinuidades-2")
% (c2m221tfc1a "descontinuidades-2")
% http://angg.twu.net/LATEX/2022-1-C2-TFC1.pdf#page=35
\url{http://angg.twu.net/LATEX/2022-1-C2-TFC1.pdf\#page=35} (até a p.41)
}
\newpage
% «def-particao» (to ".def-particao")
% (c2m222tfcsp 3 "def-particao")
% (c2m222tfcsa "def-particao")
{\bf A definição de partição}
\scalebox{0.85}{\def\colwidth{11cm}\firstcol{
Se $P$ é um subconjunto \ColorRed{finito} e \ColorRed{não-vazio} de $\R$,
então podemos interpretar $P$ como uma partição...
Por exemplo, se $P=\{200,20,42,99,63,33,20,20\}$
então $P=\{20,33,42,63,99,200\}$, e aí vamos interpretar
esse conjunto de 6 pontos -- ordenados em ordem crescente --
como uma partição do intervalo $I = [a,b] = [20,200]$ em
5 subintervalos (``$N=5$''), assim:
$$\begin{array}{ccccccl}
20 & 33 & 42 & 63 & 99 & 200 \\
x_0 & x_1 & x_2 & x_3 & x_4 & x_5 \\
a_1 & b_1 & & & & & I_1=[a_1,b_1] \\
& a_2 & b_2 & & & & I_2=[a_2,b_2] \\
& & a_3 & b_3 & & & I_3=[a_3,b_3] \\
& & & a_4 & b_4 & & I_4=[a_4,b_4] \\
& & & & a_5 & b_5 & I_5=[a_5,b_5] \\
a & & & & & b & I = [a,b] = [x_0,x_N]\\
\end{array}
$$
}\anothercol{
}}
\newpage
% «def-inf-e-sup» (to ".def-inf-e-sup")
% (c2m222tfcsp 4 "def-inf-e-sup")
% (c2m222tfcsa "def-inf-e-sup")
% (c2m221isp 3 "algumas-definicoes")
% (c2m221isa "algumas-definicoes")
{\bf As definições de inf e sup}
\scalebox{0.9}{\def\colwidth{10cm}\firstcol{
Digamos que $f:\R→\R$ e $B⊂\R$.
Vamos definir $\inf(f(B))$ e $\sup(f(B))$ ---
e também $\inf(D)$ e $\sup(D)$, pra $D⊂\R$ ---
desta forma:
%
$$\begin{array}{rcl}
\Rext &=& \R∪\{-∞,+∞\} \\
C &=& \setofst{(x,f(x))}{x∈B} \\
D &=& \setofst{f(x)}{x∈B} \\
D' &=& \setofst{y∈\R}{∃x∈B.\ f(x)=y} \\
L &=& \setofst{y∈\Rext}{∀d∈D.\;y≤d} \\
U &=& \setofst{y∈\Rext}{∀d∈D.\;d≤y} \\
(α=\inf(D)) &=& α∈L ∧ (∀ℓ∈L.\;ℓ \le α) \\
(β=\sup(D)) &=& β∈U ∧ (∀u∈U.\;β \le u) \\
\end{array}
$$
Com isto podemos definir a integral definida.
A definição formal dela está na próxima página.
}\anothercol{
}}
\newpage
% «def-integral» (to ".def-integral")
% (c2m222tfcsp 5 "def-integral")
% (c2m222tfcsa "def-integral")
\vspace*{-0.25cm}
$$\scalebox{0.44}{$
\begin{array}{rcl}
[a,b]_N &=& \setofst{a+k(\frac{b-a}{N})}{k∈\{0,\ldots,N\}} \\
&=& \{ a+0(\frac{b-a}{N}),
\; a+1(\frac{b-a}{N}),
\; \ldots,
\; a+N(\frac{b-a}{N}) \} \\
&=& \{ a,
\; a + \frac{b-a}{N},
\; a + 2\frac{b-a}{N},
\; a + 3\frac{b-a}{N},
\; \ldots, \; b\} \\
\D \ga{into_P f(x) dx} &=& \msup_P \\[-5pt]
&=& \D \sum_{i=1}^{n} \sup(f([a_i,b_i])) (b_i-a_i) \\
\D \ga{intu_P f(x) dx} &=& \minf_P \\[-5pt]
&=& \D \sum_{i=1}^{n} \inf(f([a_i,b_i])) (b_i-a_i) \\ \\[-5pt]
\D \ga{intou_P f(x) dx} &=& \D \INTP{\into}{P}{f(x)}
- \INTP{\intu}{P}{f(x)} \\ \\[-5pt]
\D \ga{into_xab f(x) dx} &=& \D \lim_{k→∞} \ga{into_ab2k f(x) dx} \\
\D \ga{intu_xab f(x) dx} &=& \D \lim_{k→∞} \ga{intu_ab2k f(x) dx} \\ \\[-5pt]
\D \ga{intou_xab f(x) dx} &=& \D \ga{into_xab f(x) dx}
- \ga{intu_xab f(x) dx} \\ \\[-5pt]
\D \left( \ga{int_xab f(x) dx} \text{\;\;existe} \right)
&=& \D \left( \ga{into_xab f(x) dx}
= \ga{intu_xab f(x) dx} \right) \\ \\[-7pt]
&=& \D \left( \ga{intou_xab f(x) dx} = 0 \right) \\ \\[-7pt]
\D \ga{int_xab f(x) dx} &=& \D \ga{into_xab f(x) dx}
\qquad \text{(se a integral existir)} \\ \\[-7pt]
&=& \D \ga{intu_xab f(x) dx}
\qquad \text{(se a integral existir)} \\
\end{array}
$}
$$
\newpage
{\bf Exercício 1.}
Leia isto aqui:
\ssk
{\scriptsize
% (c2m212somas1p 9 "particoes")
% (c2m212somas1a "particoes")
% http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf#page=9
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf\#page=9} (até a p.12)
}
\msk
a) Seja $P=\{4,2,1,1.5\}$.
Interprete $P$ como uma partição.
Diga quem são o $N$, o $a$ e o $b$ dela e monte
a tabela dos subintervalos dela (p.10 do link acima).
\msk
b) Seja $P=[2,4]_6$.
Diga quem são os pontos da partição $P$.
\msk
c) Seja $P=[2,5]_{2^3}$.
Diga quem são os pontos da partição $P$.
\newpage
% «exercicio-2» (to ".exercicio-2")
% (c2m222tfcsp 6 "exercicio-2")
% (c2m222tfcsa "exercicio-2")
% (c2m221isp 12 "exercicio-5")
% (c2m221isa "exercicio-5")
{\bf Exercício 2.}
%L Pict2e.bounds = PictBounds.new(v(0,0), v(9,7))
%L spec = "(0,3)--(2,1)o (2,3)c (2,5)o--(7,0)"
%L pws = PwSpec.from(spec)
%L curve = pws:topict()
%L p = PictList { curve:prethickness("2pt") }
%L p:addputstrat(v(2.7,5.5), "\\cell{(2,5)}")
%L p:addputstrat(v(7.7,0.5), "\\cell{(7,0)}")
%L p:pgat("pgatc"):preunitlength("17pt"):sa("Exercicio 2"):output()
\pu
\msk
Sejam
%
$f(x) = \scalebox{0.5}{$\ga{Exercicio 2}$}$
e $B=[1,3]$.
\msk
Represente graficamente B, $f(B)$ e os conjuntos abaixo:
%
$$\begin{array}{rcl}
% \Rext &=& \R∪\{-∞,+∞\} \\
C &=& \setofst{(x,f(x))}{x∈B} \\
D &=& \setofst{f(x)}{x∈B} \\
D' &=& \setofst{y∈\R}{∃x∈B.\ f(x)=y} \\
L &=& \setofst{y∈\Rext}{∀d∈D.\;y≤d} \\
U &=& \setofst{y∈\Rext}{∀d∈D.\;d≤y} \\
% (α=\inf(D)) &=& α∈L ∧ (∀ℓ∈L.\;ℓ \le α) \\
% (β=\sup(D)) &=& β∈U ∧ (∀u∈U.\;β \le u) \\
\end{array}
$$
\newpage
% «exercicio-3» (to ".exercicio-3")
% (c2m222tfcsp 8 "exercicio-3")
% (c2m222tfcsa "exercicio-3")
{\bf Exercício 3.}
\scalebox{0.9}{\def\colwidth{9cm}\firstcol{
Sejam
%
$f(x) = \scalebox{0.5}{$\ga{Exercicio 2}$}$ \; .
e $P=\{1,3,4,5\}$.
\msk
Represente graficamente:
\msk
a) $\ga{into_P f(x) dx}$
\msk
b) $\ga{intu_P f(x) dx}$
\msk
c) $\ga{intou_P f(x) dx}$
\msk
d) $\INTP{\intou}{[1,5]_2}{f(x)}$
\msk
e) $\INTP{\intou}{[1,5]_4}{f(x)}$
}\anothercol{
}}
\newpage
% «exercicio-4» (to ".exercicio-4")
% (c2m222tfcsp 9 "exercicio-4")
% (c2m222tfcsa "exercicio-4")
{\bf Exercício 4.}
\scalebox{0.9}{\def\colwidth{12cm}\firstcol{
Dê uma olhada nesses 4 slides sobre a função de Dirichlet:
\ssk
{\scriptsize
% (c2m212somas2p 51 "dirichlet")
% (c2m212somas2a "dirichlet")
% http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf#page=51
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf\#page=51} (até a p.54)
}
\ssk
Sejam:
%
$$\begin{array}{rcl}
f(x) &=&
\begin{cases}
0 & \text{quando $x∈\Q$}, \\
1 & \text{quando $x∈\R∖\Q$} \\
\end{cases} \; , \\
g(x) &=& f(x) + x. \\
\end{array}
$$
Represente graficamente:
\msk
a) $g(x)$
b) $\INTP{\intou}{[0,4]_1}{g(x)}$
c) $\INTP{\intou}{[0,4]_2}{g(x)}$
d) $\INTP{\intou}{[0,4]_4}{g(x)}$
e) $\INTP{\intou}{[0,4]_8}{g(x)}$
}\anothercol{
}}
% (c2m212somas2p 51 "dirichlet")
% (c2m212somas2a "dirichlet")
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
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% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2022-2-C2-TFC1-e-TFC2 veryclean
make -f 2019.mk STEM=2022-2-C2-TFC1-e-TFC2 pdf
% Local Variables:
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% ee-tla: "c2m222tfcs"
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