% (find-LATEX "2025-1-C2-numeros-complexos.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2025-1-C2-numeros-complexos.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2025-1-C2-numeros-complexos.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2025-1-C2-numeros-complexos.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2025-1-C2-numeros-complexos.pdf"))
% (defun e () (interactive) (find-LATEX "2025-1-C2-numeros-complexos.tex"))
% (defun o () (interactive) (find-LATEX "2024-2-C2-numeros-complexos.tex"))
% (defun u () (interactive) (find-latex-upload-links "2025-1-C2-numeros-complexos"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2025-1-C2-numeros-complexos.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2025-1-C2-numeros-complexos")
% (find-pdf-page   "~/LATEX/2025-1-C2-numeros-complexos.pdf")
% (find-sh0 "cp -v  ~/LATEX/2025-1-C2-numeros-complexos.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2025-1-C2-numeros-complexos.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2025-1-C2-numeros-complexos.pdf")
%   file:///home/edrx/LATEX/2025-1-C2-numeros-complexos.pdf
%               file:///tmp/2025-1-C2-numeros-complexos.pdf
%           file:///tmp/pen/2025-1-C2-numeros-complexos.pdf
%  http://anggtwu.net/LATEX/2025-1-C2-numeros-complexos.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 "2025-1-C2-numeros-complexos" "2" "c2m251nc" "c2nc")

% «.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-provas»	(to "links-provas")
% «.links-stewart»	(to "links-stewart")
% «.links-boyce»	(to "links-boyce")
% «.links-boyce-eng»	(to "links-boyce-eng")
% «.links-quadros»	(to "links-quadros")
% «.maxima-dots»	(to "maxima-dots")
%
% «.exercicios»		(to "exercicios")
% «.exercicios-2»	(to "exercicios-2")



\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/2025-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2025.1-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")
% (c2m251ncp 1 "title")
% (c2m251nca   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2025.1}

\bsk

Aulas 29 e 30: revisão de números complexos

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2025.1-C2.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c2m251ncp 2 "links")
% (c2m251nca   "links")

{\bf Links}

\scalebox{0.5}{\def\colwidth{16cm}\firstcol{

% «links-provas»  (to ".links-provas")
% 2jT227: (c2m242p1p 10 "gab-3")
%         (c2m242p1a    "gab-3")
% (c2m242ncp 2 "links-provas")
% (c2m242nca   "links-provas")
% (c2m192p1a "gab-3")
% (c2m182p1a "gab-1")
\par Provas:
\par \Ca{2jT227} (2024.2. P1, gabarito)
\par \Ca{2yT12} (2019.2, P1, gabarito)
\par \url{http://anggtwu.net/LATEX/2018-2-C2-P1.pdf\#page=2} Questão 1

\msk

% «links-stewart»  (to ".links-stewart")
% (find-books "__analysis/__analysis.el" "stewart-pt" "1020" "17.1 Equações Lineares de Segunda Ordem")
% (find-books "__analysis/__analysis.el" "stewart-pt" "1034" "subamortecimento")
% (find-books "__analysis/__analysis.el" "stewart-pt" "51" "H Números Complexos")
\par \Ca{StewPtCap17p6} (p.1020) Equações diferenciais de 2ª ordem
\par \Ca{StewPtCap17p20} (p.1034) Caso 3: subamortecimento
\par \Ca{StewPtApendiceHp5} (p.A51) {\bf Apêndice H: Números complexos}

\ssk

% «links-boyce»  (to ".links-boyce")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "105" "3. Equações lineares de segunda")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "111" "operador diferencial")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "113" "princípio da superposição")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "121" "3.3. Raízes complexas")
% (find-books "__analysis/__analysis.el" "boyce-diprima-pt" "123" "Figura 3.3.1")
\par \Ca{BoyceDip3p5} (p.105) Capítulo 3: Equações lineares de 2ª ordem
\par \Ca{BoyceDip3p11} (p.111) Seção 3.2: o operador diferencial $L$
\par \Ca{BoyceDip3p13} (p.113) Teorema 3.2.2: o princípio da superposição
\par \Ca{BoyceDip3p21} (p.121) 3.3. Raízes complexas da equação característica
\par \Ca{BoyceDip3p23} (p.123) Figura 3.3.1

\ssk

% (find-books "__analysis/__analysis.el" "zill-cullen-pt" "173" "4.3" "coeficientes constantes")
%\par \Ca{ZillCullenCap4p33} (p.173) 4.3. Equações lineares homogêneas com coeficientes constantes

% «links-boyce-eng»  (to ".links-boyce-eng")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "103" "3 Second-Order Linear")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "110" "differential operator")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "112" "Theorem 3.2.2" "Superposition")
% (find-books "__analysis/__analysis.el" "boyce-diprima" "120" "3.3 Complex Roots")
\par \Ca{BoyceDipEng3p4} (p.103) Chapter 3: Second-order linear ODEs
\par \Ca{BoyceDipEng3p11} (p.110) Section 3.2: the differential operator $L$
\par \Ca{BoyceDipEng3p13} (p.112) Theorem 3.2.2: principle of superposition
\par \Ca{BoyceDipEng3p21} (p.120) 3.3 Complex Roots of the Characteristic Equation
\par \Ca{BoyceDipEng3p24} (p.123) Figure 3.3.1

% (find-angg ".emacs" "c2q191" "20190524")
% (find-angg ".emacs" "c2q192" "60" "20190920")
% (find-c2q222page 45 "nov23: Números complexos")
% (find-c2q231page 50 "jun23: Oscilações")
% (c2q191 31 "20190524" "E = c + is")

\ssk

% (find-SUBSfile "2021aulas-por-telegram.lua" "14:16")
% http://www.youtube.com/watch?v=-dhHrg-KbJ0 e to the pi i for dummies (Mathologer)
\par \url{http://www.youtube.com/watch?v=-dhHrg-KbJ0} $e^{πi}$ for dummies (Mathologer)
\par \url{https://en.wikipedia.org/wiki/Complex_number} (bom)
\par \url{https://pt.wikipedia.org/wiki/N\%C3\%BAmero_complexo} (ruim, cheio de erros)

% (c2m222srp 4 "somas-de-retangulos")
% (c2m222sra   "somas-de-retangulos")

% (find-LATEXgrep "grep --color=auto -niH --null -e 'reas negativas' 202*.tex")
%\par \Ca{2fT63} ``Áreas negativas não existem''

\bsk

% (find-books "__analysis/__analysis.el" "hernandez" "47" "principais identidades trigonométricas")
\par \Ca{HernandezP57} (p.47) principais identidades trigonométricas

}\anothercol{

% «links-quadros»  (to ".links-quadros")
% 2jQ55: (find-angg ".emacs" "c2q242" "13/11: números complexos")
% 2iQ81: (find-angg ".emacs" "c2q241" "jul31: Revisão de números complexos e séries de Taylor")
% 2hQ66: (find-angg ".emacs" "c2q232" "nov06: Revisão de variáveis complexas")
% 2fQ45: (find-angg ".emacs" "c2q222" "nov23: Números complexos")
% 2yQ63: (find-angg ".emacs" "c2q192" "20190925 peq aula 12: E=c+is e aplicações")
% 2yQ98: (find-angg ".emacs" "c2q192" "20191106 peq aula 20: f''+af'+bf = 0 complexo")
\par Quadros:
\par \Ca{2jQ55} (2024.2)
\par \Ca{2iQ81} (2024.1)
\par \Ca{2hQ66} (2023.2)
\par \Ca{2fQ45} (2022.2)
\par \Ca{2yQ63} (2019.2)

}}

\newpage


% «maxima-dots»  (to ".maxima-dots")
% (c2m242ncp 3 "maxima-dots")
% (c2m242nca   "maxima-dots")
% (c2m241ncp 5 "dots")
% (c2m241nca   "dots")
% (find-es "maxima" "2024.1-intro-complex")
% (find-es "maxima" "2024.2-C2-intro-complex")

%M (%i1) Re(z)    := realpart(z)$
%M (%i2) Im(z)    := imagpart(z)$
%M (%i3) sqhyp(z) := Re(z)^2 + Im(z)^2$
%M (%i4) tom(z)   := matrix([Re(z),-Im(z)], [Im(z),Re(z)])$
%M (%i5) det(M)   := determinant(M)$
%M (%i6) nm(z)    := Re(z) + %i*Im(z)$
%M (%i7) stringdisp : false$
%M (%i8) z : a + %i*b;
%M (%o8) i\,b+a
%M (%i9) w : c + %i*d;
%M (%o9) i\,d+c
%M (%i10) matrix([    z , "+",     w , "=",   nm(z+w)],
%M        [tom(z), "+", tom(w), "=", tom(z)+tom(w)],
%M        [    "",  "",     "",  "",      ""      ],
%M        [    z , "*",     w , "=",   nm(z*w)    ],
%M        [tom(z), ".", tom(w), "=", tom(z).tom(w)]);
%M (%o10) \begin{pmatrix}i\,b+a&\mbox{ + }&i\,d+c&\mbox{ = }&i\,\left(d+b\right)+c+a\cr \begin{pmatrix}a&-b\cr b&a\cr \end{pmatrix}&\mbox{ + }&\begin{pmatrix}c&-d\cr d&c\cr \end{pmatrix}&\mbox{ = }&\begin{pmatrix}c+a&-d-b\cr d+b&c+a\cr \end{pmatrix}\cr &&&&\cr i\,b+a&\mbox{ * }&i\,d+c&\mbox{ = }&i\,\left(a\,d+b\,c\right)-b\,d+a\,c\cr \begin{pmatrix}a&-b\cr b&a\cr \end{pmatrix}&\mbox{ . }&\begin{pmatrix}c&-d\cr d&c\cr \end{pmatrix}&\mbox{ = }&\begin{pmatrix}a\,c-b\,d&-\left(a\,d\right)-b\,c\cr a\,d+b\,c&a\,c-b\,d\cr \end{pmatrix}\cr \end{pmatrix}
%L maximahead:sa("dots", "")
\pu

%M (%i11)  drawzpts (zs,[opts]) := myqdrawp(xyrange(), zpts(zs, opts))$
%M (%i12) [xmin,ymin, xmax,ymax] :  [-5,-5, 5,5]$
%M (%i13) myqdrawp_to_screen()$  myps(size) := ps(size)$
%M (%i15) myqdrawp_to_new_pdf()$ myps(size) := ps(size/5)$
%M (%i17)  as_33     : create_list(x+%i*y, y,seqn(2,0,2), x,seqn(0,2,2));
%M (%o17) \left[ 2\,i , 2\,i+1 , 2\,i+2 , i , i+1 , i+2 , 0 , 1 , 2 \right] 
%M (%i18) as_55     : create_list(x+%i*y, y,seqn(2,0,4), x,seqn(0,2,4))$
%M (%i19) as_22     : create_list(x+%i*y, y,seqn(0,1,1), x,seqn(0,1,1))$
%M (%i20)  D1 : drawzpts(as_33,          myps(3), pc(red))$
%M  D2 : drawzpts(as_55,          myps(3), pc(red))$
%M 
%M (%i21) 
%M (%i22) [D1, D2];
%M (%o22) \left[ \myvcenter{\includegraphics[height=5cm]{2024-2-C2/complex_001.pdf}} , \myvcenter{\includegraphics[height=5cm]{2024-2-C2/complex_002.pdf}} \right] 
%M (%i23)  D3 : drawzpts(as_55 + 1,      myps(3), pc(orange))$
%M 
%M (%i24)  D4 : drawzpts(as_55 + %i,     myps(3), pc(orange))$
%M 
%M (%i25) [D3, D4];
%M (%o25) \left[ \myvcenter{\includegraphics[height=5cm]{2024-2-C2/complex_003.pdf}} , \myvcenter{\includegraphics[height=5cm]{2024-2-C2/complex_004.pdf}} \right] 
%L maximahead:sa("dots 3", "")
\pu

%M (%i26)  D5 : drawzpts(as_55 * 2,      myps(3), pc(forest_green))$
%M (%i27)  D6 : drawzpts(as_55 * (1+%i), myps(3), pc(forest_green))$
%M (%i28) [D2, D5, D6];
%M (%o28) \left[ \myvcenter{\includegraphics[height=4cm]{2024-2-C2/complex_002.pdf}} , \myvcenter{\includegraphics[height=4cm]{2024-2-C2/complex_005.pdf}} , \myvcenter{\includegraphics[height=4cm]{2024-2-C2/complex_006.pdf}} \right] 
%M (%i29) topdf_opts : "height=10cm"$
%M (%i30) as_332    : create_list(z+w, z,as_33, w,as_22*0.2)$
%M (%i31) as_332_sq : makelist(z^2, z, as_332)$
%M (%i32) [xmin,ymin, xmax,ymax] :  [-10,-10, 10,10];
%M (%o32) \left[ -10 , -10 , 10 , 10 \right] 
%M (%i33)  D7 : drawzpts(as_332,    myps(0.5))$
%M (%i34)  D8 : drawzpts(as_332_sq, myps(0.5))$
%M (%i35) [D7, D8];
%M (%o35) \left[ \myvcenter{\includegraphics[height=7cm]{2024-2-C2/complex_007.pdf}} , \myvcenter{\includegraphics[height=7cm]{2024-2-C2/complex_008.pdf}} \right] 
%M (%i36) 
%L maximahead:sa("dots 4", "")
\pu

%M (%i30) as_332    : create_list(z+w, z,as_33, w,as_22*0.2)$
%M (%i31) as_332_sq : makelist(z^2, z, as_332)$
%M (%i32) [xmin,ymin, xmax,ymax] :  [-10,-10, 10,10];
%M (%o32) \left[ -10 , -10 , 10 , 10 \right] 
%M (%i33)  D7 : drawzpts(as_332,    myps(0.5))$
%M (%i34)  D8 : drawzpts(as_332_sq, myps(0.5))$
%M 
%M (%i35) [D7, D8];
%M (%o35) \left[ \myvcenter{\includegraphics[height=10cm]{2024-2-C2/complex_007.pdf}} , \myvcenter{\includegraphics[height=10cm]{2024-2-C2/complex_008.pdf}} \right] 
%M (%i36) 
%L maximahead:sa("dots 5", "")
\pu





\scalebox{0.5}{\def\colwidth{9cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots 2}

}}


\newpage

\scalebox{0.35}{\def\colwidth{9cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots 3}


}\anothercol{
}}

\newpage

\scalebox{0.35}{\def\colwidth{9cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{dots 4}

}\anothercol{
}}

\newpage


% «2019.2»  (to ".2019.2")
% (c2m242ncp 2 "links-provas")
% (c2m242nca   "links-provas")
% (c2m192p1a "gab-3")
% (c2m182p1a "gab-1")
{\bf 2019.2}

\scalebox{0.7}{\def\colwidth{16cm}\firstcol{

\def\E#1 {E^{#1}}

$\begin{array}[t]{lll}
      (\sen 5θ)^2 (\cos 6θ)^2 \\
      = \;\; \left(\frac{\E5 - \E-5 }{2i}\right)^2
             \left(\frac{\E6 + \E-6 }{2}\right)^2 \\
      = \;\; -\frac1{16}(\E10 - 2 +\E-10 )(\E12 + 2 +\E-12 ) \\
      = \;\; -\frac1{16}\pmat{
                          (\E10 - 2 +\E-10 ) \E12 + \\
                          (\E10 - 2 +\E-10 ) · 2 + \\
                          (\E10 - 2 +\E-10 ) \E-12 \\
                        } \\
      = \;\; -\frac1{16}\pmat{
                           \E22 - 2\E12 +\E2 + \\
                          2\E10 - 4  +2\E-10 + \\
                          \E-2 - 2\E-12 +\E-22 \\
                        }  \\
      = \;\; -\frac1{16}( (\E22 + \E-22 )
                          -2 (\E12 + \E-12 )
                          +2 (\E10 + \E-10 )
                          +  (\E2 + \E-2 )
                          - 4
                        ) \\
      = \;\; -\frac1{16}( 2\cos22θ
                          -4\cos12θ
                          +4\cos10θ
                          +2\cos2θ
                          - 4
                        ) \\
      = \;\; -\frac18 \cos22θ
             +\frac14 \cos12θ
             -\frac14 \cos10θ
             -\frac18 \cos2θ
             \frac14 \\
    \end{array}
   $

\bsk

$\begin{array}[t]{rcl}
   \intx {(\sen 5x)^2 (\cos 6x)^2}
   &=& \intx {-\frac18 \cos22x
             +\frac14 \cos12x
             -\frac14 \cos10x
             -\frac18 \cos2x
             +\frac14}  \\
   &=& -\frac1{8·22} \sen22x
             +\frac1{4·12} \sen12x
             -\frac1{4·10} \sen10x
             -\frac1{8·2} \sen2x
             +\frac14x  \\
   \end{array}
$

}\anothercol{
}}


\newpage

% «exercicios»  (to ".exercicios")
% (c2m251ncp 7 "exercicios")
% (c2m251nca   "exercicios")

{\bf Exercícios de ``seja o seu próprio Geogebra''}

\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

{}

Nestes exercícios a gente vai usar algumas técnicas de Cálculo 3 que
eu vou explicar no quadro: o ``seja o seu próprio Geogebra'', a
convenção de escrever todos os labels, a convenção de que um vetor
$\vv$ é um deslocamento, e a convenção de que $A+\vv$ é representado
como \standout{um ponto e uma seta}. Lembre que se \ColorRed{quase
  sempre} se você precisou fazer um conta -- como o
\ColorRed{resultado} de $A+\vv$ -- é porque você está fazendo algo
errado.

Links:

\par \Ca{3jT16} Uma convenção (temporária)
\par \Ca{2kT140} Sobre músculos mentais e perguntar


\bsk


{\bf Exercício 1.}

a) Seja $z=i+1$.

Represente graficamente $\{z^0, z^1, z^2, z^3, z^4\}$.

(Aqui você vai precisar fazer contas).

\ssk

b) Faça o mesmo para $z=i$ (num outro gráfico).

(Aqui você também vai precisar fazer contas).




}\anothercol{

{\bf Exercício 2.}

Sejam:
%
$$\begin{array}{rcl}
  A(z,w) &=& \{z, w, z + \Vec{(w-z)}, z + \frac12 \Vec{(w-z)} \} \\
  B(z,w) &=& \{z, w, z + \Vec{(w-z)}, z + \frac14 \Vec{(w-z)} \} \\
  C(z,w) &=& \{z, w, \frac{z + w}{2} \} \\
  \end{array}
$$

Para cada um dos casos abaixo faça três gráficos -- um pra $A(z,w)$,
um pra $B(z,w)$ e um pra $C(z,w)$.

\ssk

\par a) $z=2+4i$ e $w=-3+4i$
\par b) $z=2+4i$ e $w=2+i$
\par c) $z=1+2i$ e $w=4+3i$.

\bsk

{\bf Exercício 3.}

Lembre que $E = e^{iθ}$ e  $E^k = (e^{iθ})^k = e^{i(kθ)}$.

Seja
%
$$\begin{array}{rcl}
  A(θ) &=& \{E, E^2, E^3, E^0, E^{-1}\} \\
  \end{array}
$$

\par a) Represente graficamente $A(θ)$ para $θ=45°$.
\par b) Represente graficamente $A(θ)$ para $θ=30°$.
\par c) Represente graficamente $A(θ)$ para $θ=60°$.
\par d) Represente graficamente $A(θ)$ para $θ=-30°$.

}}

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{\bf Exercícios de ``seja o seu próprio Geogebra'' (2)}

\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

O {\sl conjugado} é definido assim: $\ovl{a+ib} = a-ib$.

Mais precisamente: $∀a,b∈\R. \, \ovl{a+ib} = a-ib$.

Um exemplo concreto: $\ovl{3+4i} = 3-4i$.

\bsk

{\bf Exercício 4.}

Sejam:
%
$$\begin{array}{rcl}
  A(z) &=& \{z, \ovl{z}, -\ovl{z}, \frac{z+\ovl{z}}{2}, \frac{z+(-\ovl{z})}{2} \} \\
  B(z) &=& \{z, \ovl{z}, -\ovl{z}, \frac{z+\ovl{z}}{2}, \frac{z-\ovl{z}}{2} \} \\
  \end{array}
$$

Em cada um dos itens abaixo faça um gráfico para $A(z)$ e outro para
$B(z)$. Os dois gráficos vão ser muito parecidos, só um label/rótulo
vai mudar.

\msk

\par a) $z=3+2i$
\par b) $z=2+3i$
\par c) $z=2+i$
\par d) $z=2-i$




}\anothercol{

{\bf Exercício 5.}

Lembre que $E = e^{iθ}$.

Seja:
%
$$\begin{array}{rcl}
  A(θ) &=& \{E, E^{-1}, -E^{-1}, \frac{E+E^{-1}}{2}, \frac{E-E^{-1}}{2} \}. \\
  \end{array}
$$

Represente graficamente $A(θ)$ em cada um dos casos abaixo.

\par a) $θ=45°$
\par b) $θ=60°$
\par c) $θ=135°$

}}





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