% (find-LATEX "2024-2-C2-TFC1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-2-C2-TFC1.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-2-C2-TFC1.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2024-2-C2-TFC1.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2024-2-C2-TFC1.pdf"))
% (defun e () (interactive) (find-LATEX "2024-2-C2-TFC1.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C2-TFC1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2024-2-C2-TFC1"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2024-2-C2-TFC1.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2024-2-C2-TFC1")
% (find-pdf-page   "~/LATEX/2024-2-C2-TFC1.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-2-C2-TFC1.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-2-C2-TFC1.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-2-C2-TFC1.pdf")
%   file:///home/edrx/LATEX/2024-2-C2-TFC1.pdf
%               file:///tmp/2024-2-C2-TFC1.pdf
%           file:///tmp/pen/2024-2-C2-TFC1.pdf
%  http://anggtwu.net/LATEX/2024-2-C2-TFC1.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2024-2-C2-TFC1" "2" "c2m242tfc1" "c2t1")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.defs-caepro»	(to "defs-caepro")
% «.defs-pict2e»	(to "defs-pict2e")
% «.defs-maxima»	(to "defs-maxima")
% «.defs-V»		(to "defs-V")
% «.title»		(to "title")
% «.links»		(to "links")
% «.links-stewart»	(to "links-stewart")
% «.links-leithold»	(to "links-leithold")
% «.links-miranda»	(to "links-miranda")
% «.links-ross»		(to "links-ross")



\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-LATEX "dednat7-test1.tex")
%\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")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
\begin{document}

% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

\def\drafturl{http://anggtwu.net/LATEX/2024-2-C2.pdf}
\def\drafturl{http://anggtwu.net/2024.2-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2024-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2024-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat7load.lua"}  % (find-LATEX "dednat7load.lua")
\directlua{dednat7preamble()}         % (find-angg "LUA/DednatPreamble1.lua")
\directlua{dednat7oldheads()}         % (find-angg "LUA/Dednat7oldheads.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")
\def\Caurl   #1{\expr{Caurl("#1")}}
\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca      #1{\Cahref{#1}{#1}}

% «defs-pict2e»  (to ".defs-pict2e")
%L dofile "Piecewise2.lua"           -- (find-LATEX "Piecewise2.lua")
%L --dofile "Escadas1.lua"           -- (find-LATEX "Escadas1.lua")
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt

% «defs-maxima»  (to ".defs-maxima")
%L dofile "Maxima2.lua"              -- (find-angg "LUA/Maxima2.lua")
\pu

% «defs-V»  (to ".defs-V")
%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\pu


%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m242tfc1p 1 "title")
% (c2m242tfc1a   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2024.2}

\bsk

Aula 33: o TFC1

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.2-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m242tfc1p 2 "links")
% (c2m242tfc1a   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{16cm}\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

% «links-stewart»  (to ".links-stewart")
% (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

% «links-leithold»  (to ".links-leithold")
% (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

% «links-miranda»  (to ".links-miranda")
% (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

% «links-ross»  (to ".links-ross")
% (find-books "__analysis/__analysis.el" "ross" "291" "34 Fundamental Theorem of Calculus")
%\par \Ca{RossAp38} (p.291) Fundamental Theorem of Calculus
\par  \Ca{RossP304} (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 PictBounds.setbounds(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 = Pict {
%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 PictBounds.setbounds(v(0,0), v(7,5))
%L
%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 = Pict {
%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}

% (find-pdfpages2-links "~/LATEX/" "2024-2-C2-TFC1")


% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2t1"
% ee-tla: "c2m242tfc1"
% End:
