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Warning: this is an htmlized version!
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% (find-LATEX "2023-2-C2-TFC1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-2-C2-TFC1.tex" :end))
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% (code-eec-LATEX "2023-2-C2-TFC1")
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% (find-sh0 "cp -v ~/LATEX/2023-2-C2-TFC1.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2023-2-C2-TFC1.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2023-2-C2-TFC1.pdf")
% file:///home/edrx/LATEX/2023-2-C2-TFC1.pdf
% file:///tmp/2023-2-C2-TFC1.pdf
% file:///tmp/pen/2023-2-C2-TFC1.pdf
% http://anggtwu.net/LATEX/2023-2-C2-TFC1.pdf
% (find-LATEX "2019.mk")
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% Video (not yet):
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%
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\begin{document}
% «defs» (to ".defs")
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% (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\ColorDarkOrange#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){\ColorDarkOrange{\bf(#1 pts)}}
% «defs-caepro» (to ".defs-caepro")
%L dofile "Caepro5.lua" -- (find-angg "LUA/Caepro5.lua" "LaTeX")
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\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 = "%"
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\pu
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m232tfc1p 1 "title")
% (c2m232tfc1a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo C2 - 2023.2}
\bsk
Aula 16: o TFC1
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://anggtwu.net/2023.2-C2.html}
\end{center}
\newpage
% «links» (to ".links")
% (c2m232tfc1p 2 "links")
% (c2m232tfc1a "links")
{\bf Links}
\scalebox{0.9}{\def\colwidth{12cm}\firstcol{
% (c2m232carrop 4 "exercicio-1")
% (c2m232carroa "exercicio-1")
% (c2m232carrop 9 "exercicio-5")
% (c2m232carroa "exercicio-5")
\par \Ca{2hT27} (2023.2) Exercício 1: faça um gráfico da $G'(x)$
\par \Ca{2hT32} (2023.2) Exercício 5: $G(x) = \Intt{3}{x}{g(t)}$
\ssk
% (find-books "__analysis/__analysis.el" "stewart-pt" "97" "Teorema do confronto")
% (find-books "__analysis/__analysis.el" "stewart-pt" "351" "TFC1")
% (find-books "__analysis/__analysis.el" "stewart-pt" "352" "TFC1, demonstração")
\par \Ca{StewPtCap2p26} (p.97) O teorema do confronto
\par \Ca{StewPtCap5p30} (p.351) TFC1
\par \Ca{StewPtCap5p31} (p.352) TFC1, demonstração
\ssk
% (find-books "__analysis/__analysis.el" "leithold" "114" "2.8. Teorema do confronto ou do sanduíche")
% (find-books "__analysis/__analysis.el" "leithold" "345" "5.8.1. TFC1")
\par \Ca{Leit2p61} (p.114) 2.8 Teorema do confronto ou do sanduíche
\par \Ca{Leit5p62} (p.345) 5.8.1 TFC1
\ssk
% (find-books "__analysis/__analysis.el" "miranda" "29" "Teorema do confronto")
% (find-books "__analysis/__analysis.el" "miranda" "225" "7.5 Teorema Fundamental do Cálculo")
\par \Ca{MirandaP29} Teorema do confronto
\par \Ca{MirandaP225} TFC1
\ssk
% (find-books "__analysis/__analysis.el" "ross" "291" "34 Fundamental Theorem of Calculus")
\par \Ca{RossAp38} (p.291) Fundamental Theorem of Calculus
}\anothercol{
}}
\newpage
% ___ _ _
% |_ _|_ __ | |_ _ __ ___ __| |_ _ ___ __ _ ___
% | || '_ \| __| '__/ _ \ / _` | | | |/ __/ _` |/ _ \
% | || | | | |_| | | (_) | (_| | |_| | (_| (_| | (_) |
% |___|_| |_|\__|_| \___/ \__,_|\__,_|\___\__,_|\___/
%
% «intro-1» (to ".intro-1")
% (c2m221tfc1p 12 "intro-1")
% (c2m221tfc1a "intro-1")
{\bf Introdução (2021.2)}
\scalebox{0.75}{\def\colwidth{12cm}\firstcol{
Digamos que $f:[a,b] \to \R$ é uma função integrável.
Digamos que $c∈[a,b]$.
Digamos que a função $F:[a,b] \to \R$ é \ColorRed{definida} por:
%
$$F(t) \;\; = \Intx{c}{t}{f(x)}.$$
O TFC1 tem duas versões.
A versão mais simples diz o seguinte:
se a função $f$ é contínua então para todo $t∈(a,b)$ vale:
%
$$F'(t) \;\; = f(t). \qquad \qquad (*)$$
A versão mais complicada do TFC1, que vamos ver
depois, não supõe que a função $f$ é contínua.
\msk
Nós vamos ver um argumento visual que mostra que
a igualdade $(*)$ é verdade. Esse argumento visual é
\ColorRed{quase} uma demonstração formal, num sentido que eu
vou explicar depois.
}}
\newpage
% «intro-2» (to ".intro-2")
% (c2m221tfc1p 3 "intro-2")
% (c2m221tfc1a "intro-2")
{\bf Introdução (2)}
\scalebox{0.75}{\def\colwidth{12cm}\firstcol{
Digamos que $f:[a,b] \to \R$ é uma função \ColorRed{contínua}.
Digamos que $c∈[a,b]$.
Digamos que a função $F:[a,b] \to \R$ é \ColorRed{definida} por:
%
$$F(t) \;\; = \Intx{c}{t}{f(x)}.$$
\def\eqq{\overset{\ColorRed{???}}{=}}
Então:
%
$$\begin{array}{rcl}
F'(t) &=& \D \lim_{ε→0} \frac{F(t+ε)-F(t)}{ε} \\
&=& \D \lim_{ε→0} \frac{ \Intx{c}{t+ε}{f(x)} - \Intx{c}{t}{f(x)} }{ε} \\
&=& \D \lim_{ε→0} \frac{ \Intx{t}{t+ε}{f(x)} }{ε} \\[12pt]
&=& \D \lim_{ε→0} \frac{1}{ε} \Intx{t}{t+ε}{f(x)} \\[12pt]
&\eqq& f(t) \\
\end{array}
$$
}}
\newpage
% «intro-3» (to ".intro-3")
% (c2m221tfc1p 4 "intro-3")
% (c2m221tfc1a "intro-3")
{\bf Introdução (3)}
Digamos que $f:[a,b] \to \R$ é uma função \ColorRed{contínua}.
Digamos que $c∈[a,b]$.
Digamos que a função $F:[a,b] \to \R$ é \ColorRed{definida} por:
%
$$F(t) \;\; = \Intx{c}{t}{f(x)}.$$
O nosso argumento visual vai mostrar que:
%
$$\begin{array}{rcl}
\D \lim_{ε→0} \frac{1}{ε} \Intx{t}{t+ε}{f(x)}
&=& f(t). \\
\end{array}
$$
\newpage
% _____ _ _
% | ____|_ _____ _ __ ___ _ __ | | ___ / |
% | _| \ \/ / _ \ '_ ` _ \| '_ \| |/ _ \ | |
% | |___ > < __/ | | | | | |_) | | (_) | | |
% |_____/_/\_\___|_| |_| |_| .__/|_|\___/ |_|
% |_|
%
% «exemplo-1» (to ".exemplo-1")
% (c2m232tfc1p 6 "exemplo-1")
% (c2m232tfc1a "exemplo-1")
% (c2m221tfc1p 15 "exemplo-1")
% (c2m221tfc1a "exemplo-1")
% (find-angg "LUA/Piecewise1.lua" "TFC1-tests")
%
%L Pict2e.bounds = PictBounds.new(v(0,0), v(7,5))
%L tfc1_fig_parabola = function (scale)
%L local f = function (x) return 4*x - x^2 end
%L local tfc1 = TFC1.fromf(f, seqn(0, 4, 64))
%L tfc1:setxts(0,1,4, 5, scale):setpwg()
%L local p = PictList {
%L tfc1:areaify_f():Color("Orange"),
%L tfc1:areaify_g():Color("Orange"),
%L tfc1:lineify_f(),
%L tfc1:lineify_g(),
%L }
%L return p
%L end
%L
%L tfc1_fig_parabola(1/2):pgat("pgat"):sa("TFC1 parabola 1/2"):output()
%L tfc1_fig_parabola(1) :pgat("pgat"):sa("TFC1 parabola 1"):output()
%L tfc1_fig_parabola(2) :pgat("pgat"):sa("TFC1 parabola 2"):output()
%L tfc1_fig_parabola(4) :pgat("pgat"):sa("TFC1 parabola 4"):output()
%L tfc1_fig_parabola(8) :pgat("pgat"):sa("TFC1 parabola 8"):output()
%L tfc1_fig_parabola(16) :pgat("pgat"):sa("TFC1 parabola 16"):output()
%L tfc1_fig_parabola(32) :pgat("pgat"):sa("TFC1 parabola 32"):output()
%L tfc1_fig_parabola(64) :pgat("pgat"):sa("TFC1 parabola 64"):output()
%L tfc1_fig_parabola(-1) :pgat("pgat"):sa("TFC1 parabola -1"):output()
%L tfc1_fig_parabola(-2) :pgat("pgat"):sa("TFC1 parabola -2"):output()
%L tfc1_fig_parabola(-4) :pgat("pgat"):sa("TFC1 parabola -4"):output()
%L tfc1_fig_parabola(-8) :pgat("pgat"):sa("TFC1 parabola -8"):output()
%L tfc1_fig_parabola(-16):pgat("pgat"):sa("TFC1 parabola -16"):output()
%L tfc1_fig_parabola(-32):pgat("pgat"):sa("TFC1 parabola -32"):output()
%L tfc1_fig_parabola(-64):pgat("pgat"):sa("TFC1 parabola -64"):output()
\pu
\unitlength=10pt
\scalebox{1.0}{\def\colwidth{5cm}\firstcol{
{\bf Primeiro exemplo:}
$f(x)$ é a nossa parábola
preferida, e $t=1$.
\msk
Primeira figura: $ε=2$.
Segunda figura: $ε=1$.
Terceira figura: $ε=1/2$.
\msk
À esquerda: $\Intx{t}{t+ε}{f(x)}$.
À direita: $\frac{1}{ε}\Intx{t}{t+ε}{f(x)}$.
\msk
Repare que a área em
laranja à esquerda sempre
tem base $ε$ e a área em
laranja à direita sempre
tem base $ε·\frac{1}{ε}=1$.
}\anothercol{
\unitlength=10pt
$$\ga{TFC1 parabola 1/2}$$
$$\ga{TFC1 parabola 1}$$
$$\ga{TFC1 parabola 2}$$
}}
\newpage
\unitlength=25pt
\def\myint{\Intx{1}{1+ε}{f(x)}}
\def\myinte#1{
$\begin{array}{rl}
\D \myint & \text{e} \\[15pt]
\D \frac{1}{ε} \myint & \text{quando $ε=#1$:} \\
\end{array}
$}
\msk
\myinte{2}
$$\ga{TFC1 parabola 1/2}$$
\newpage
\myinte{1}
$$\ga{TFC1 parabola 1}$$
\newpage
\myinte{1/2}
$$\ga{TFC1 parabola 2}$$
\newpage
\myinte{1/4}
$$\ga{TFC1 parabola 4}$$
\newpage
\myinte{1/8}
$$\ga{TFC1 parabola 8}$$
\newpage
\myinte{1/16}
$$\ga{TFC1 parabola 16}$$
\newpage
\myinte{1/32}
$$\ga{TFC1 parabola 32}$$
\newpage
\myinte{1/64}
$$\ga{TFC1 parabola 64}$$
\newpage
% «exemplo-1-left» (to ".exemplo-1-left")
% (c2m221tfc1p 14 "exemplo-1-left")
% (c2m221tfc1a "exemplo-1-left")
\scalebox{1.0}{\def\colwidth{5cm}\firstcol{
{\bf Agora com $ε$ negativo!...}
\msk
$f(x)$ é a nossa parábola
preferida, e $t=1$.
\msk
Primeira figura: $ε=-1$.
Segunda figura: $ε=-1/2$.
Terceira figura: $ε=-1/4$.
\msk
À esquerda: $\Intx{t}{t+ε}{f(x)}$.
À direita: $\frac{1}{ε}\Intx{t}{t+ε}{f(x)}$.
% \msk
%
% Repare que a área em
%
% laranja à esquerda sempre
%
% tem base $ε$ e a área em
%
% laranja à direita sempre
%
% tem base $ε·\frac{1}{ε}=1$.
}\anothercol{
\unitlength=10pt
$$\ga{TFC1 parabola -1}$$
$$\ga{TFC1 parabola -2}$$
$$\ga{TFC1 parabola -4}$$
}}
\newpage
\myinte{-1}
$$\ga{TFC1 parabola -1}$$
\newpage
\myinte{-1/2}
$$\ga{TFC1 parabola -2}$$
\newpage
\myinte{-1/4}
$$\ga{TFC1 parabola -4}$$
\newpage
\myinte{-1/8}
$$\ga{TFC1 parabola -8}$$
\newpage
\myinte{-1/16}
$$\ga{TFC1 parabola -16}$$
\newpage
\myinte{-1/32}
$$\ga{TFC1 parabola -32}$$
\newpage
\myinte{-1/64}
$$\ga{TFC1 parabola -64}$$
\newpage
% _____ _ _ ____
% | ____|_ _____ _ __ ___(_) ___(_) ___ | ___|
% | _| \ \/ / _ \ '__/ __| |/ __| |/ _ \ |___ \
% | |___ > < __/ | | (__| | (__| | (_) | ___) |
% |_____/_/\_\___|_| \___|_|\___|_|\___/ |____/
%
% «exercicio-5» (to ".exercicio-5")
% (c2m221tfc1p 32 "exercicio-5")
% (c2m221tfc1a "exercicio-5")
% (c2m221tfc1p 22 "exercicio-1")
% (c2m221tfc1a "exercicio-1")
% (find-angg "LUA/Piecewise1.lua" "TFC1-tests")
%
%L Pict2e.bounds = PictBounds.new(v(0,0), v(7,5))
%L exerc_1_spec = "(0,2)--(1,1)--(2,3)--(3,4)--(4,3)"
%L exerc_2_spec = "(0,2)--(1,0)--(2,1)o (2,2)c (2,3)o--(3,4)--(4,3)"
%L
%L tfc1_exercs_1_2 = function (spec, scale)
%L local tfc1 = TFC1.fromspec(spec)
%L tfc1:setxts(0,2,4, 5, scale)
%L local p = PictList {
%L tfc1:areaify_f():Color("Orange"),
%L tfc1.pws:topict(),
%L }
%L return p
%L end
%L tfc1_exerc1 = function (scale) return tfc1_exercs_1_2(exerc_1_spec, scale) end
%L tfc1_exerc2 = function (scale) return tfc1_exercs_1_2(exerc_2_spec, scale) end
%L tfc1_exerc1(1/2) :pgat("pgat"):sa("TFC1 exerc1 1/2"):output()
%L tfc1_exerc1(1) :pgat("pgat"):sa("TFC1 exerc1 1"):output()
%L tfc1_exerc1(2) :pgat("pgat"):sa("TFC1 exerc1 2"):output()
%L tfc1_exerc1(-1/2):pgat("pgat"):sa("TFC1 exerc1 -1/2"):output()
%L tfc1_exerc1(-1) :pgat("pgat"):sa("TFC1 exerc1 -1"):output()
%L tfc1_exerc1(-2) :pgat("pgat"):sa("TFC1 exerc1 -2"):output()
%L tfc1_exerc2(1/2):pgat("pgat"):sa("TFC1 exerc2 1/2"):output()
%L tfc1_exerc2(1) :pgat("pgat"):sa("TFC1 exerc2 1"):output()
%L tfc1_exerc2(2) :pgat("pgat"):sa("TFC1 exerc2 2"):output()
%L tfc1_exerc2(-1/2):pgat("pgat"):sa("TFC1 exerc2 -1/2"):output()
%L tfc1_exerc2(-1) :pgat("pgat"):sa("TFC1 exerc2 -1"):output()
%L tfc1_exerc2(-2) :pgat("pgat"):sa("TFC1 exerc2 -2"):output()
\pu
\scalebox{1.0}{\def\colwidth{6cm}\firstcol{
{\bf Exercício 5.}
Seja $f(x)$ a função à direita.
Seja $t=2$.
\msk
a) Desenhe $\frac{1}{ε}\Intx{t}{t+ε}{f(x)}$
para $ε=2$, $ε=1$, $ε=1/2$.
\msk
b) Desenhe $\frac{1}{ε}\Intx{t}{t+ε}{f(x)}$
para $ε=-2$, $ε=-1$, $ε=-1/2$.
\msk
Dica: comece entendendo as
áreas em laranja à direita!
\msk
c) Quanto você acha que dá
$\lim_{ε→0^+} \frac{1}{ε} \Intx{t}{t+ε}{f(x)}$?
\msk
d) Quanto você acha que dá
$\lim_{ε→0^-} \frac{1}{ε} \Intx{t}{t+ε}{f(x)}$?
}\hspace*{-1cm}\anothercol{
\unitlength=7.5pt
$$\ga{TFC1 exerc1 1/2} \quad \ga{TFC1 exerc1 -1/2}$$
$$\ga{TFC1 exerc1 1} \quad \ga{TFC1 exerc1 -1}$$
$$\ga{TFC1 exerc1 2} \quad \ga{TFC1 exerc1 -2}$$
}}
\newpage
% «exercicio-6» (to ".exercicio-6")
% (c2m221tfc1p 33 "exercicio-6")
% (c2m221tfc1a "exercicio-6")
% (find-LATEX "edrx21defs.tex" "firstcol-anothercol")
\scalebox{1.0}{\def\colwidth{6cm}\firstcol{
{\bf Exercício 6.}
Seja $f(x)$ a função à direita.
Seja $t=2$.
\msk
a) Desenhe $\frac{1}{ε}\Intx{t}{t+ε}{f(x)}$
para $ε=2$, $ε=1$, $ε=1/2$.
\msk
b) Desenhe $\frac{1}{ε}\Intx{t}{t+ε}{f(x)}$
para $ε=-2$, $ε=-1$, $ε=-1/2$.
\msk
Dica: comece entendendo as
áreas em laranja à direita!
\msk
c) Quanto você acha que dá
$\lim_{ε→0^+} \frac{1}{ε} \Intx{t}{t+ε}{f(x)}$?
\msk
d) Quanto você acha que dá
$\lim_{ε→0^-} \frac{1}{ε} \Intx{t}{t+ε}{f(x)}$?
}\hspace*{-1cm}\anothercol{
\unitlength=7.5pt
\def\PPP#1{\ParR{\expr{Pwil(#1)}}}
$$\ga{TFC1 exerc2 1/2} \quad \ga{TFC1 exerc2 -1/2}$$
$$\ga{TFC1 exerc2 1} \quad \ga{TFC1 exerc2 -1}$$
$$\ga{TFC1 exerc2 2} \quad \ga{TFC1 exerc2 -2}$$
}}
\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 "~/2023.2-C2/")
# (find-fline "~/LATEX/2023-2-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 ~/2023.2-C2/
cp -fv $1.pdf ~/LATEX/2023-2-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=2023-2-C2-TFC1 veryclean
make -f 2019.mk STEM=2023-2-C2-TFC1 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2t1"
% ee-tla: "c2m232tfc1"
% End: