% (find-LATEX "2024-1-C3-dois-metodos.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-1-C3-dois-metodos.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-1-C3-dois-metodos.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2024-1-C3-dois-metodos.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2024-1-C3-dois-metodos.pdf"))
% (defun e () (interactive) (find-LATEX "2024-1-C3-dois-metodos.tex"))
% (defun o () (interactive) (find-LATEX "2024-1-C3-dois-metodos.tex"))
% (defun u () (interactive) (find-latex-upload-links "2024-1-C3-dois-metodos"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2024-1-C3-dois-metodos.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2024-1-C3-dois-metodos")
% (find-pdf-page   "~/LATEX/2024-1-C3-dois-metodos.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C3-dois-metodos.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-1-C3-dois-metodos.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-1-C3-dois-metodos.pdf")
%   file:///home/edrx/LATEX/2024-1-C3-dois-metodos.pdf
%               file:///tmp/2024-1-C3-dois-metodos.pdf
%           file:///tmp/pen/2024-1-C3-dois-metodos.pdf
%  http://anggtwu.net/LATEX/2024-1-C3-dois-metodos.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 "2024-1-C3-dois-metodos" "3" "c3m241dm" "c3dm")

% «.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")
% «.title»		(to "title")
% «.links»		(to "links")
% «.metodo-1»		(to "metodo-1")
% «.metodo-2»		(to "metodo-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-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")
%
% (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/2024-1-C3.pdf}
\def\drafturl{http://anggtwu.net/2024.1-C3.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 "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")
\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



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c3m241dmp 1 "title")
% (c3m241dma   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 3 - 2024.1}

\bsk

Aula 15: dois métodos

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.1-C3.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c3m241dmp 2 "links")
% (c3m241dma   "links")

{\bf Links}

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

\newpage

% «metodo-1»  (to ".metodo-1")
% (c3m241dmp 3 "metodo-1")
% (c3m241dma   "metodo-1")
% (find-es "maxima" "2024.1-dois-metodos")

{\bf Método 1}

%M (%i1) f(t) := a + (t-b)/c * d;
%M (%o1) f\left(t\right):=a+{\frac{t-b}{c}}\,d
%M (%i2) eq1 : f(12)    = 34;
%M (%o2) {\frac{\left(12-b\right)\,d}{c}}+a=34
%M (%i3) eq2 : f(12+23) = 34+56;
%M (%o3) {\frac{\left(35-b\right)\,d}{c}}+a=90
%M (%i4) solve([eq1, eq2], [a,c]);
%M (%o4) \left[ \left[ a={\frac{56\,b+110}{23}} , c={\frac{23\,d}{56}} \right]  \right] 
%M (%i5) solve([eq1, eq2], [a,d]);
%M (%o5) \left[ \left[ a={\frac{56\,b+110}{23}} , d={\frac{56\,c}{23}} \right]  \right] 
%M (%i6) solve([eq1, eq2], [b,c]);
%M (%o6) \left[ \left[ b={\frac{23\,a-110}{56}} , c={\frac{23\,d}{56}} \right]  \right] 
%M (%i7) solve([eq1, eq2], [b,d]);
%M (%o7) \left[ \left[ b={\frac{23\,a-110}{56}} , d={\frac{56\,c}{23}} \right]  \right] 
%L maximahead:sa("metodo-1", "")
\pu

%M (%i8) eq3 : f(x0) = y0;
%M (%o8) {\frac{d\,\left(\mathrm{x0}-b\right)}{c}}+a=\mathrm{y0}
%M (%i9) eq4 : f(x1) = y1;
%M (%o9) {\frac{d\,\left(\mathrm{x1}-b\right)}{c}}+a=\mathrm{y1}
%M (%i10) solve([eq3, eq4], [a,c]);
%M (%o10) \left[ \left[ a=-\left({\frac{\left(\mathrm{x0}-b\right)\,\mathrm{y1}+\left(b-\mathrm{x1}\right)\,\mathrm{y0}}{\mathrm{x1}-\mathrm{x0}}}\right) , c={\frac{d\,\mathrm{x1}-d\,\mathrm{x0}}{\mathrm{y1}-\mathrm{y0}}} \right]  \right] 
%M (%i11) solve([eq3, eq4], [a,d]);
%M (%o11) \left[ \left[ a={\frac{\mathrm{x0}\,\mathrm{y1}-b\,\mathrm{y1}+\left(b-\mathrm{x1}\right)\,\mathrm{y0}}{\mathrm{x0}-\mathrm{x1}}} , d={\frac{c\,\mathrm{y0}-c\,\mathrm{y1}}{\mathrm{x0}-\mathrm{x1}}} \right]  \right] 
%M (%i12) solve([eq3, eq4], [b,c]);
%M (%o12) \left[ \left[ b={\frac{\mathrm{x0}\,\left(a-\mathrm{y1}\right)+\mathrm{x1}\,\mathrm{y0}-a\,\mathrm{x1}}{\mathrm{y0}-\mathrm{y1}}} , c={\frac{d\,\mathrm{x0}-d\,\mathrm{x1}}{\mathrm{y0}-\mathrm{y1}}} \right]  \right] 
%M (%i13) solve([eq3, eq4], [b,d]);
%M (%o13) \left[ \left[ b={\frac{\mathrm{x0}\,\mathrm{y1}-\mathrm{x1}\,\mathrm{y0}+a\,\mathrm{x1}-a\,\mathrm{x0}}{\mathrm{y1}-\mathrm{y0}}} , d={\frac{c\,\mathrm{y1}-c\,\mathrm{y0}}{\mathrm{x1}-\mathrm{x0}}} \right]  \right] 
%M (%i14) 
%L maximahead:sa("metodo-1-b", "")
\pu

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

Digamos que queremos encontrar $a,b,c,d∈\R$ que obedeçam isto aqui:
%
$$\begin{array}{lcl}
  f(t)     &=& a + ((t-b)/c)·d, \\
  f(12)    &=& 34, \\
  f(12+23) &=& 34+56, \\
  \end{array}
$$

O ``método 1'' é encontrarmos $a,b,c,d$ ``algebricamente'', fazendo
contas como estas daqui...

\vspace*{0.5cm}

\def\hboxthreewidth {14cm}
\scalebox{0.4}{\ga{metodo-1}}

}\anothercol{

...ou fazendo contas como estas, que dão fórmulas mais gerais:

\vspace*{0.5cm}

\def\hboxthreewidth {14cm}
\scalebox{0.6}{\ga{metodo-1-b}}

}}


\newpage

% «metodo-2»  (to ".metodo-2")
% (c3m241dmp 4 "metodo-2")
% (c3m241dma   "metodo-2")

{\bf Método 2}

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

...mas Cálculo 3 é um curso sobre {\sl aprender a visualizar}, não um
curso sobre {\sl fazer contas}. Nos exemplos que a gente vai trabalhar
no curso {\sl quase sempre} o Método 1 vai nos atrapalhar bem mais do
que nos ajudar.

\ssk

A gente fez um monte de exercícios sobre o ``Método 2'' antes da
greve, e eu vou precisar que vocês refaçam eles em casa pra lembrar
como o Método 2 funciona.

\bsk

Alguns links:

% 3iT4: (c3m241introp 3 "pontos-mais-faceis")
%       (c3m241introa   "pontos-mais-faceis")
\Ca{3iT4} Pontos mais fáceis de calcular

% 3iT7: (c3m241introp 5 "traj-em-3-partes")
%       (c3m241introa   "traj-em-3-partes")
\Ca{3iT7} Uma trajetória em três partes (2)

% 3iT50: (c3m241isp 5 "exercicios-aula-10")
%        (c3m241isa   "exercicios-aula-10")
\Ca{3iT50} Exercícios da aula 10 - exercício 5

% 3hT78: (c3m232nfp 23 "regioes")
%        (c3m232nfa    "regioes")
\Ca{3hT78} Regiões 

% 3hT79: (c3m232nfp 24 "regioes-2")
%        (c3m232nfa    "regioes-2")
\Ca{3hT79} Regiões (2)

\bsk

...e essa animação daqui, que é novidade:

{\footnotesize
\url{http://anggtwu.net/LATEX/2024-1-C3-anim-rp1.gif}
}

}\anothercol{

{\bf Exercício}

Reescreva as definições para $F_N, F_W, F_E, F_S$ que você obteve aqui

% 3hT79: (c3m232nfp 24 "regioes-2")
%        (c3m232nfa    "regioes-2")
\Ca{3hT79} Regiões (2)

neste formato:
%
$$\begin{array}{lcl}
  F_N(x,y)  &=& \_\_ + \_\_(x-4) + \_\_(y-3) \\
  F_S(x,y)  &=& \_\_ + \_\_(x-4) + \_\_(y-5) \\
  F_W(x,y)  &=& \_\_ + \_\_(x-3) + \_\_(y-4) \\
  F_E(x,y)  &=& \_\_ + \_\_(x-5) + \_\_(y-4) \\
  F_E(x,y)  &=& \_\_ + \_\_(x-6) + \_\_(y-4) \\
  \end{array}
$$




% (c3m241isp 4 "exercicios-aula-10")
% (c3m241isa   "exercicios-aula-10")



}}




\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}



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