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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2022-1-C2-somas-3.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2022-1-C2-somas-3.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2022-1-C2-somas-3.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2022-1-C2-somas-3.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2022-1-C2-somas-3.pdf"))
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% (defun o () (interactive) (find-LATEX "2022-1-C2-somas-3.tex"))
% (defun l () (interactive) (find-LATEX "Piecewise1.lua"))
% (defun l () (interactive) (find-angg "LUA/Piecewise1.lua"))
% (defun u () (interactive) (find-latex-upload-links "2022-1-C2-somas-3"))
% (defun v () (interactive) (find-2a '(e) '(d)))
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% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2022-1-C2-somas-3")
% (find-pdf-page "~/LATEX/2022-1-C2-somas-3.pdf")
% (find-sh0 "cp -v ~/LATEX/2022-1-C2-somas-3.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2022-1-C2-somas-3.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2022-1-C2-somas-3.pdf")
% file:///home/edrx/LATEX/2022-1-C2-somas-3.pdf
% file:///tmp/2022-1-C2-somas-3.pdf
% file:///tmp/pen/2022-1-C2-somas-3.pdf
% http://angg.twu.net/LATEX/2022-1-C2-somas-3.pdf
% (find-LATEX "2019.mk")
% (find-CN-aula-links "2022-1-C2-somas-3" "2" "c2m221somas3" "c2so3")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Piecewise1.lua ~/LATEX/")
% «.defs» (to "defs")
% «.title» (to "title")
% «.links» (to "links")
% «.mountains» (to "mountains")
% «.exercicio-1» (to "exercicio-1")
% «.imagens-figuras» (to "imagens-figuras")
% «.imagens-figuras-2» (to "imagens-figuras-2")
% «.exercicio-2» (to "exercicio-2")
% «.sup-e-inf-informais» (to "sup-e-inf-informais")
% «.exercicio-3» (to "exercicio-3")
% «.exercicio-4» (to "exercicio-4")
% «.exercicio-5» (to "exercicio-5")
% «.exercicio-6» (to "exercicio-6")
% «.metodos-nomes» (to "metodos-nomes")
% «.exercicio-7» (to "exercicio-7")
% «.exercicio-7-figs» (to "exercicio-7-figs")
% «.exercicio-8» (to "exercicio-8")
% «.exercicio-8-figs» (to "exercicio-8-figs")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c2m221somas3" "2022-1-C2-somas-3")
% (code-eevvideo "c2m221somas3" "2022-1-C2-somas-3")
% (code-eevlinksvideo "c2m221somas3" "2022-1-C2-somas-3")
% (find-c2m221somas3video "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")
%
%\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
]{geometry}
%
\begin{document}
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
%L dofile "Piecewise1.lua" -- (find-LATEX "Piecewise1.lua")
%L Pict2e.__index.suffix = "%"
\pu
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
% «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/2022-1-C2.pdf}
\def\drafturl{http://angg.twu.net/2022.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c2m221somas3p 1 "title")
% (c2m221somas3a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 2 - 2022.1}
\bsk
Aula 11: somas de retângulos
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.1-C2.html}
\end{center}
\newpage
% «links» (to ".links")
% (c2m221somas3p 2 "links")
% (c2m221somas3a "links")
{\bf Links}
Este PDF complementa estes três PDFs
do semestre passado:
\ssk
{\footnotesize
% (c2m212somas1a "title")
% (c2m212somas1a "title" "Aula 4: integrais como somas de retângulos (1)")
% (c2m212somas2a "title")
% (c2m212somas2a "title" "Aula 13: integrais como somas de retângulos (2)")
% (c2m212mt1p 1 "title")
% (c2m212mt1a "title")
% http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf}
% http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf}
% http://angg.twu.net/LATEX/2021-2-C2-MT1.pdf
\url{http://angg.twu.net/LATEX/2021-2-C2-MT1.pdf}
}
\ssk
% __ __ _ _
% | \/ | ___ _ _ _ __ | |_ __ _(_)_ __ ___
% | |\/| |/ _ \| | | | '_ \| __/ _` | | '_ \/ __|
% | | | | (_) | |_| | | | | || (_| | | | | \__ \
% |_| |_|\___/ \__,_|_| |_|\__\__,_|_|_| |_|___/
%
% «mountains» (to ".mountains")
% (c2m221somas3p 3 "mountains")
% (c2m221somas3a "mountains")
% (find-angg "LUA/Piecewise1.lua" "Xtoxytoy-test2")
%
%L Pict2e.bounds = PictBounds.new(v(0,0), v(23,9))
%L spec = "(0,1)--(5,6)--(7,4)--(11,8)--(15,4)--(17,6)--(23,0)"
%L xs = { 1,3, 6, 9, 11, 13, 16,19, 21 }
%L labely = -1
%L pws = PwSpec.from(spec)
%L xtos = Xtoxytoy.from(pws:fun(), xs)
%L vlines = xtos:topict("v")
%L curve = pws:topict()
%L labels = PictList {}
%L for i,x in ipairs(xs) do
%L labels:addputstrat(v(x,labely), "\\cell{x_"..(i-1).."}")
%L end
%L p = PictList { vlines, curve:prethickness("2pt"), labels }
%L p:pgat("pA", "mountain"):output()
\pu
\unitlength=8pt
\vspace*{-0.25cm}
\hspace*{-0.5cm}
$\scalebox{0.55}{$
\begin{array}{ccccc}
\mountain && \mountain && \mountain \\[20pt]
\mountain && \mountain && \mountain \\[20pt]
\mountain && \mountain && \mountain \\[20pt]
\mountain && \mountain && \mountain \\[20pt]
\end{array}
$}
$
\newpage
% _____ _ _ _
% | ____|_ _____ _ __ ___(_) ___(_) ___ / |
% | _| \ \/ / _ \ '__/ __| |/ __| |/ _ \ | |
% | |___ > < __/ | | (__| | (__| | (_) | | |
% |_____/_/\_\___|_| \___|_|\___|_|\___/ |_|
%
% «exercicio-1» (to ".exercicio-1")
% (c2m221somas3p 4 "exercicio-1")
% (c2m221somas3a "exercicio-1")
% (find-pdf-page "~/2022.1-C2/C2-quadros-tarde.pdf" 9)
{\bf Exercício 1.}
Seja $f(x)$ a função da página anterior ---
a página anterior tem 12 cópias do gráfico dela.
Represente graficamente cada um dos somatórios abaixo.
Cada item abaixo vai virar 8 retângulos sobre
uma das cópias do gráfico da $f(x)$.
\msk
\def\sumo{\sum_{i=1}^{8}}
\def\sumoo#1{\sumo #1 (x_i - x_{i-1})}
a) $\sumoo{f(x_i)}$
\ssk
b) $\sumoo{f(x_{i-1})}$
\ssk
c) $\sumoo{\max(f(x_{i-1}), f(x_i))}$
\ssk
d) $\sumoo{\min(f(x_{i-1}), f(x_i))}$
\ssk
e) $\sumoo{f(\frac{x_{i-1} + x_i}{2})}$
\ssk
f) $\sumoo{\frac{f(x_{i-1}) + f(x_i)}{2}}$
\newpage
% ___ __ _
% |_ _|_ __ ___ __ _ __ _ ___ _ __ ___ _ / _(_) __ _ ___
% | || '_ ` _ \ / _` |/ _` |/ _ \ '_ \/ __(_) | |_| |/ _` / __|
% | || | | | | | (_| | (_| | __/ | | \__ \_ | _| | (_| \__ \
% |___|_| |_| |_|\__,_|\__, |\___|_| |_|___(_) |_| |_|\__, |___/
% |___/ |___/
%
% «imagens-figuras» (to ".imagens-figuras")
% (c2m221somas3p 5 "imagens-figuras")
% (c2m221somas3a "imagens-figuras")
% (find-angg "LUA/Piecewise1.lua" "Xtoxytoy-test3")
{\bf Imagens de intervalos: figuras}
\bsk
\bsk
%L Pict2e.bounds = PictBounds.new(v(0,0), v(8,4))
%L
%L cthick = "2pt" -- curve
%L dthick = "0.25pt" -- dots
%L sthick = "4pt" -- segments
%L
%L -- Curve:
%L cspec = "(0,2)--(2,4)--(6,0)--(8,2)"
%L cpws = PwSpec.from(cspec)
%L curve = cpws:topict():prethickness(cthick)
%L
%L -- Segments:
%L sspec = "(1,0)c--(2,0)--(4,0)c" ..
%L " (1,3)c--(2,4)--(4,2)c" ..
%L " (0,2)c--(0,4)c"
%L spws = PwSpec.from(sspec)
%L segs = spws:topict():prethickness(sthick):Color("Orange")
%L
%L -- Dots:
%L dotsn = function (nsubsegs)
%L local xs = seqn(1, 4, nsubsegs)
%L local dots0 = Xtoxytoy.from(cpws:fun(), xs)
%L local dots = dots0:topict("vhxpy"):prethickness(dthick):Color("Red")
%L return dots
%L end
%L
%L PictList { curve, dotsn(1) } :pgat("pgatc", "ImageOne") :output()
%L PictList { curve, dotsn(3) } :pgat("pgatc", "ImageThree") :output()
%L PictList { curve, dotsn(6) } :pgat("pgatc", "ImageSix") :output()
%L PictList { curve, dotsn(12) } :pgat("pgatc", "ImageTwelve"):output()
%L PictList { curve, segs } :pgat("pgatc", "ImageSegs") :output()
\pu
$\ImageOne$
$\ImageThree$
$\ImageSix$
$\ImageTwelve$
$\ImageSegs$
\newpage
% «imagens-figuras-2» (to ".imagens-figuras-2")
% (c2m221somas3p 6 "imagens-figuras-2")
% (c2m221somas3a "imagens-figuras-2")
{\bf Imagens de intervalos: figuras $→$ contas}
\scalebox{0.75}{\def\colwidth{12cm}\firstcol{
Algumas pessoas acham que isto é sempre verdade:
%
$$f([a,b]) = [f(a),f(b)].$$
\ColorRed{Não seja como elas!!!}
Nas três figuras à esquerda da página anterior temos:
%
$$\begin{array}{rcl}
f(\{1,4\}) &=& \{f(1),f(4)\} \\
&=& \{3,2\} \\
&=& \{2,3\} \\
f(\{1,2,3,4\}) &=& \{f(1),f(2),f(3),f(4)\} \\
&=& \{2,3,4,3\} \\
&=& \{2,3,4\} \\
f([1,3]) &=& [2,4] \\{}
[f(1),f(3)] &=& [3,2] \\
&=& \setofst{y∈\R}{3≤y≤2} \\
&=& ∅ \\
\end{array}
$$
%}\anothercol{
}}
\newpage
% _____ _ _ ____
% | ____|_ _____ _ __ ___(_) ___(_) ___ |___ \
% | _| \ \/ / _ \ '__/ __| |/ __| |/ _ \ __) |
% | |___ > < __/ | | (__| | (__| | (_) | / __/
% |_____/_/\_\___|_| \___|_|\___|_|\___/ |_____|
%
% «exercicio-2» (to ".exercicio-2")
% (c2m221somas3p 7 "exercicio-2")
% (c2m221somas3a "exercicio-2")
{\bf Exercício 2.}
Seja $f(x)$ esta função:
\msk
%L Pict2e.bounds = PictBounds.new(v(0,0), v(8,4))
%L spec = "(0,2)--(2,4)--(6,0)--(8,2)"
%L pws = PwSpec.from(spec)
%L curve = pws:topict()
%L p = PictList { curve:prethickness("2pt") }
%L p:pgat("pgatc", "falsoseno"):output()
\pu
%
$f(x) = \falsoseno$
\msk
Calcule estas imagens de intervalos:
\msk
\begin{tabular}[t]{l}
a) $f([0,1])$ \\
b) $f([1,2])$ \\
c) $f([0,2])$ \\
d) $f([2,3])$ \\
e) $f([1,3])$ \\
f) $f([0,3])$ \\
\end{tabular}
\qquad
\begin{tabular}[t]{l}
g) $f([0,4])$ \\
h) $f([4,8])$ \\
i) $f([0,8])$ \\
j) $f([1,7])$ \\
\end{tabular}
\newpage
% _ ___ _ __ _
% ___ _ _ _ __ (_) ( _ ) (_)_ __ / _(_)
% / __| | | | '_ \| | / _ \/\ | | '_ \| |_| |
% \__ \ |_| | |_) | | | (_> < | | | | | _| |
% |___/\__,_| .__/|_| \___/\/ |_|_| |_|_| |_|
% |_|
%
% «sup-e-inf-informais» (to ".sup-e-inf-informais")
% (c2m221somas3p 8 "sup-e-inf-informais")
% (c2m221somas3a "sup-e-inf-informais")
{\bf Sup informal e inf informal}
% (find-pdf-page "~/2022.1-C2/C2-quadros-manha.pdf" 7)
\def\sup {\mathsf{sup}}
\def\supi{\mathsf{supi}}
\def\inf {\mathsf{inf}}
\def\infi{\mathsf{infi}}
\scalebox{0.85}{\def\colwidth{12cm}\firstcol{
O sup informal, `$\supi$', é como uma função bugada.
O $\sup$ ``de verdade'' recebe um conjuntos
de números e sempre retorna um número.
O $\supi$ recebe um conjuntos de números e às vezes
retorna um número, mas às vezes ele dá erro.
Quando o $\supi$ recebe um intervalo fechado $[a,b]$ ele
retorna a extremidade superior do intervalo: $\supi([a,b])=b$.
Quando o supi recebe um conjunto que
não é um intervalo fechado ele dá erro.
\msk
O inf informal, `$\infi$', é como o $\supi$, mas ele retorna
a extremidade inferior do intervalo: $\infi([a,b])=a$.
\msk
O sup e o inf de verdade são BEM difíceis de definir.
Nós vamos começar usando o $\supi$ e o $\infi$ e vamos deixar
pra definir o sup e o inf de verdade só quando
já tivermos bastante prática com o $\supi$ e o $\infi$.
%}\anothercol{
}}
\newpage
% _____ _ _ _____
% | ____|_ _____ _ __ ___(_) ___(_) ___ |___ /
% | _| \ \/ / _ \ '__/ __| |/ __| |/ _ \ |_ \
% | |___ > < __/ | | (__| | (__| | (_) | ___) |
% |_____/_/\_\___|_| \___|_|\___|_|\___/ |____/
%
% «exercicio-3» (to ".exercicio-3")
% (c2m221somas3p 9 "exercicio-3")
% (c2m221somas3a "exercicio-3")
{\bf Exercício 3.}
Cada um dos itens do exercício 2 pede pra
você calcular uma expressão da forma $f([a,b])$.
Para cada um dos itens do exercício 2
faça uma cópia do gráfico da função $f(x)$
e desenhe sobre esta cópia estes dois retângulos:
%
$$\begin{array}{c}
\supi(f([a,b]))(b-a), \\
\infi(f([a,b]))(b-a) \\
\end{array}
$$
\newpage
% _____ _ _ _ _
% | ____|_ _____ _ __ ___(_) ___(_) ___ | || |
% | _| \ \/ / _ \ '__/ __| |/ __| |/ _ \ | || |_
% | |___ > < __/ | | (__| | (__| | (_) | |__ _|
% |_____/_/\_\___|_| \___|_|\___|_|\___/ |_|
%
% «exercicio-4» (to ".exercicio-4")
% (c2m221somas3p 10 "exercicio-4")
% (c2m221somas3a "exercicio-4")
{\bf Exercício 4.}
Acrescente estes dois itens extras
no exercício 1 e faça eles:
\bsk
\def\sumo{\sum_{i=1}^{8}}
\def\sumoo#1{\sumo #1 (x_i - x_{i-1})}
g) $\sumoo{\supi(f([x_{i-1},x_i]))}$
\msk
h) $\sumoo{\infi(f([x_{i-1},x_i]))}$
\newpage
% «exercicio-5» (to ".exercicio-5")
% (c2m221somas3p 11 "exercicio-5")
% (c2m221somas3a "exercicio-5")
{\bf Exercício 5.}
\scalebox{0.9}{\def\colwidth{12cm}\firstcol{
Vou me referir a este PDF aqui, do semestre passado,
como ``Somas 1'', ou ``S1'':
%\ssk
{\footnotesize
% http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf}
}
\ssk
Vou usar nomes como ``S1E4'' pra me referir aos
exercícios dele --- S1E4 é o Exercício 4 do ``Somas 1''.
\bsk
% (c2m212somas1p 9 "particoes")
% (c2m212somas1a "particoes")
% (c2m212somas1p 10 "exercicio-4")
% (c2m212somas1a "exercicio-4")
a) Leia as páginas 9 e 10 do S1 e faça o S1E4.
% (c2m212somas1p 11 "exercicio-5")
% (c2m212somas1a "exercicio-5")
b) Leia a página 11 e 12 do S1 e faça o S1E5.
% (c2m212somas1p 23 "metodos-nomes")
% (c2m212somas1a "metodos-nomes")
\def\mname#1{\text{[#1]}}
\msk
Para os próximos itens leia a página
23 do S1 e considere que $f(x) = \scalebox{0.33}{$\falsoseno$}$.
\ssk
c) Seja $P=\{0,1,4,5\}$. Represente num gráfico só $f(x)$ e $\mname{L}$.
d) Seja $P=\{0,1,4,5\}$. Represente num gráfico só $f(x)$ e $\mname{R}$.
e) Seja $P=\{0,1,4,5\}$. Represente num gráfico só $f(x)$ e $\mname{min}$.
}\anothercol{
}}
\newpage
% «exercicio-6» (to ".exercicio-6")
% (c2m221somas3p 12 "exercicio-6")
% (c2m221somas3a "exercicio-6")
{\bf Exercício 6.}
Vou me referir a este outro PDF do semestre
passado como ``Soma 2'', ou S2:
%\ssk
{\footnotesize
% (c2m212somas2p 1 "title")
% (c2m212somas2a "title")
% http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf}
}
\ssk
Vou usar nomes como ``S2E5'' pra me referir aos
exercícios dele --- S2E5 é o Exercício 5 do ``Somas 2''.
\bsk
% (c2m212somas2p 35 "exercicio-13")
% (c2m212somas2a "exercicio-13")
a) Leia as páginas 32 até 35 do S2 e faça o S2E13.
\bsk
Obs: \ColorRed{não faça} o S2E12 da página 34 do S2 ---
nos próximos exercícios você vai fazer algo
parecido com ele, mas com outra notação.
\newpage
% «metodos-nomes» (to ".metodos-nomes")
% (c2m221somas3p 13 "metodos-nomes")
% (c2m221somas3a "metodos-nomes")
{\bf Métodos de integração: nomes}
\scalebox{0.9}{\def\colwidth{12cm}\firstcol{
O S1 e o S2 usam estes nomes pros métodos de integração:
{\footnotesize
% (c2m212somas1p 23 "metodos-nomes")
% (c2m212somas1a "metodos-nomes")
% http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf#page=23
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-1.pdf\#page=23}
% (c2m212somas2p 32 "metodos-nomes")
% (c2m212somas2a "metodos-nomes")
% http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf#page=32
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf\#page=32}
}
\ssk
Ainda não definimos o $\sup$ e o $\inf$ ``de verdade'',
então vamos usar estes nomes aqui...
\def\sumiN#1{\sum_{i=1}^N #1 (b_i-a_i)}
\def\mname#1{\text{[#1]}}
%
$$\scalebox{0.95}{$
\begin{array}{ccl}
\mname{L} &=& \sumiN {f(a_i)} \\[2pt]
\mname{R} &=& \sumiN {f(b_i)} \\[2pt]
\mname{Trap} &=& \sumiN {\frac{f(a_i) + f(b_i)}{2}} \\[2pt]
\mname{M} &=& \sumiN {f(\frac{a_i+b_i}{2})} \\[2pt]
\mname{min} &=& \sumiN {\min(f(a_i), f(b_i))} \\[2pt]
\mname{max} &=& \sumiN {\max(f(a_i), f(b_i))} \\[2pt]
\mname{infi} &=& \sumiN {\infi(f([a_i,b_i]))} \\[2pt]
\mname{supi} &=& \sumiN {\supi(f([a_i,b_i]))} \\
\end{array}
$}
$$
}\anothercol{
}}
\newpage
% «exercicio-7» (to ".exercicio-7")
% (c2m221somas3p 14 "exercicio-7")
% (c2m221somas3a "exercicio-7")
{\bf Exercício 7.}
Faça o S2E12 (página 34 do S2), mas substituindo
os `sup's por `supi's e os `inf's por `infi's.
Use a função do enunciado do S2E12.
(Tem cópias dela na próxima página).
\msk
Link:
{\footnotesize
% (c2m212somas2p 32 "metodos-nomes")
% (c2m212somas2a "metodos-nomes")
% (c2m212somas2p 34 "exercicio-12")
% (c2m212somas2a "exercicio-12")
% http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf#page=34
\url{http://angg.twu.net/LATEX/2021-2-C2-somas-2.pdf\#page=34}
}
\ssk
\newpage
% «exercicio-7-figs» (to ".exercicio-7-figs")
% (c2m221somas3p 15 "exercicio-7-figs")
% (c2m221somas3a "exercicio-7-figs")
% (c2m212somas2p 34 "exercicio-12")
% (c2m212somas2a "exercicio-12")
%L Pict2e.bounds = PictBounds.new(v(0,0), v(11,7))
%L spec = "(0,3)--(3,6)--(8,1)--(11,4)"
%L pws = PwSpec.from(spec)
%L curve = pws:topict()
%L p = PictList { curve:prethickness("2pt") }
%L p:addputstrat(v(3,6.4), "\\cell{(3,6)}")
%L p:addputstrat(v(8,0.4), "\\cell{(8,1)}")
%L p:pgat("pgatc"):preunitlength("15pt"):sa("FOO"):output()
\pu
%
\def\FOO{\scalebox{0.5}{$\ga{FOO}$}}
\def\FOO{\ga{FOO}}
% $f(x) = \FOO \smallfoo$
\vspace*{-0.25cm}
\hspace*{-0.5cm}
$\scalebox{0.40}{$
\begin{array}{ccccccc}
\FOO && \FOO && \FOO && \FOO \\[70pt]
\FOO && \FOO && \FOO && \FOO \\[70pt]
\FOO && \FOO && \FOO && \FOO \\[70pt]
\FOO && \FOO && \FOO && \FOO \\[70pt]
\end{array}
$}
$
\newpage
% «exercicio-8» (to ".exercicio-8")
% (c2m221somas3p 16 "exercicio-8")
% (c2m221somas3a "exercicio-8")
% (c2m211mt1p 4 "questao-1")
% (c2m211mt1a "questao-1")
{\bf Exercício 8.}
\def\mname#1{\text{[#1]}}
%L Pict2e.bounds = PictBounds.new(v(0,0), v(10,5))
%L spec = "(0,3)--(2,1)--(8,4)--(10,0)"
%L pws = PwSpec.from(spec)
%L curve = pws:topict()
%L p = PictList { curve:prethickness("2pt") }
%L p:pgat("pgatc"):preunitlength("15pt"):sa("Exercicio 8 bare"):output()
%L p:addputstrat(v(2,0.4), "\\cell{(2,1)}")
%L p:addputstrat(v(8,4.3), "\\cell{(8,4)}")
%L p:pgat("pgatc"):preunitlength("15pt"):sa("Exercicio 8"):output()
\pu
%
\def\EB{\scalebox{0.6}{$\ga{Exercicio 8 bare}$}}
\def\EC{\scalebox{0.6}{$\ga{Exercicio 8}$}}
Seja:
%
$$f(x) = \EC
$$
\msk
Em cada um dos casos abaixo represente num gráfico só
a função $f$ e os dois somatórios pedidos.
\msk
a) $\mname{supi}_{[1,9]_{2^1}}$, $\mname{infi}_{[1,9]_{2^1}}$
b) $\mname{supi}_{[1,9]_{2^2}}$, $\mname{infi}_{[1,9]_{2^2}}$
c) $\mname{supi}_{[1,9]_{2^3}}$, $\mname{infi}_{[1,9]_{2^3}}$
d) $\mname{max}_{[1,9]_{2^1}}$, $\mname{min}_{[1,9]_{2^1}}$
e) $\mname{max}_{[1,9]_{2^2}}$, $\mname{min}_{[1,9]_{2^2}}$
\newpage
% «exercicio-8-figs» (to ".exercicio-8-figs")
% (c2m221somas3p 15 "exercicio-8-figs")
% (c2m221somas3a "exercicio-8-figs")
%$\ga{Exercicio 8 bare}$
%$\ga{Exercicio 8}$
\def\EB{\scalebox{1.0}{$\ga{Exercicio 8 bare}$}}
\def\EC{\scalebox{1.0}{$\ga{Exercicio 8}$}}
\vspace*{-0.25cm}
\hspace*{-0.5cm}
$\scalebox{0.5}{$
\begin{array}{ccccccc}
\EC && \EB && \EB && \EB \\[60pt]
\EB && \EB && \EB && \EB \\[60pt]
\EB && \EB && \EB && \EB \\[60pt]
\EB && \EB && \EB && \EB \\[60pt]
\end{array}
$}
$
%\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 "~/2022.1-C2/")
# (find-fline "~/LATEX/2022-1-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 ~/2022.1-C2/
cp -fv $1.pdf ~/LATEX/2022-1-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=2022-1-C2-somas-3 veryclean
make -f 2019.mk STEM=2022-1-C2-somas-3 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2so3"
% ee-tla: "c2m221somas3"
% End: