% (find-LATEX "2023-2-C3-gradiente.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-2-C3-gradiente.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-2-C3-gradiente.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023-2-C3-gradiente.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-2-C3-gradiente.pdf"))
% (defun e () (interactive) (find-LATEX "2023-2-C3-gradiente.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C3-gradiente.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-2-C3-gradiente"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-2-C3-gradiente.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2023-2-C3-gradiente")
% (find-pdf-page   "~/LATEX/2023-2-C3-gradiente.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C3-gradiente.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C3-gradiente.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2023-2-C3-gradiente.pdf")
%   file:///home/edrx/LATEX/2023-2-C3-gradiente.pdf
%               file:///tmp/2023-2-C3-gradiente.pdf
%           file:///tmp/pen/2023-2-C3-gradiente.pdf
%  http://anggtwu.net/LATEX/2023-2-C3-gradiente.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Numerozinhos1")
% (find-Deps1-cps   "Caepro5 Numerozinhos1")
% (find-Deps1-anggs "Caepro5 Numerozinhos1")
% (find-MM-aula-links "2023-2-C3-gradiente" "C3" "c3m232gradiente" "c3gr")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.defs-caepro»	(to "defs-caepro")
% «.defs-pict2e»	(to "defs-pict2e")
% «.title»		(to "title")
% «.links»		(to "links")
%
% «.djvuize»		(to "djvuize")



% <videos>
% Video (not yet):
% (find-ssr-links     "c3m232gradiente" "2023-2-C3-gradiente")
% (code-eevvideo      "c3m232gradiente" "2023-2-C3-gradiente")
% (code-eevlinksvideo "c3m232gradiente" "2023-2-C3-gradiente")
% (find-c3m232gradientevideo "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")
%
% (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/2023-2-C3.pdf}
\def\drafturl{http://anggtwu.net/2023.2-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2023-1-C2-carro.tex" "defs-caepro")
% (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")
\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 "Numerozinhos1.lua"        -- (find-LATEX "Numerozinhos1.lua")
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt

\pu



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c3m232gradientep 1 "title")
% (c3m232gradientea   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo C3 - 2023.2}

\bsk

Aulas 12 e 13: o gradiente

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2023.2-C3.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c3m232gradientep 2 "links")
% (c3m232gradientea   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{14cm}\firstcol{

% (find-books "__analysis/__analysis.el" "stewart-pt" "727" "12.4 O Produto Vetorial")
% (find-books "__analysis/__analysis.el" "stewart-pt" "796" "Curvas de Nível")
% (find-books "__analysis/__analysis.el" "stewart-pt" "839" "14.6" "e o Vetor Gradiente")
\par \Ca{StewPtCap12p25} (p.727) 12.4 O produto vetorial
\par \Ca{StewPtCap14p10} (p.796) Curvas de nível
\par \Ca{StewPtCap14p53} (p.839) 14.6 Derivadas direcionais e o vetor gradiente

% (find-books "__analysis/__analysis.el" "apex-calculus" "731" "Definition 12.6.2. Gradient")
\par \Ca{Apexcap12p54} (p.731) Definition 12.6.2: Gradient

% (c4m231dicasp2p 2 "links")
% (c4m231dicasp2a   "links")

\msk

% (c3m222p1p 3 "questao-1")
% (c3m222p1a   "questao-1")
\par \Ca{3fT80} A P1 de 2022.2 (tem questões sobre gradientes)

\msk

% (find-angg ".emacs" "c3q232")
\par \Ca{3hQ36} Quadros da aula 12 (06/out/2023)
\par \Ca{3hQ42} Quadros da aula 13 (11/out/2023)

}\anothercol{
}}

\newpage

{\bf Exercício 1}

\scalebox{0.6}{\def\colwidth{11cm}\firstcol{

Lembre das técnicas do ``Seja o seu próprio GeoGebra'' pra entender o
que certos parâmetros ``querem dizer'':

\url{http://anggtwu.net/LATEX/2023-2-C3-geogebra.pdf}

Seja:
%
$$z(x,y) = a + bx + cy$$

Isto é a equação de um plano. O plano em si é
%
$$S = \setofst{(x,y,z)∈\R^3}{z = a + bx + cy}
$$

ou, equivalentemente,
%
$$S = \setofst{(x,y,a+bx+cy)}{(x,y)∈\R^2}
$$

Cada escolha de $a$, $b$ e $c$ gera um plano $z=z(x,y)$ diferente.

Neste exercício nós vamos tentar entender o que os valores de $a$, $b$
e $c$ ``querem dizer''. Desenhe o diagrama de numerozinhos para cada
um dos planos abaixo; mais precisamente, para cada um dos planos
abaixo desenhe os valores de $z$ nos pontos com $x,y∈\{0,1,2,3\}$ como
numerozinhos:

\msk

\par a) $(a,b,c) = (0,0,0)$
\par b) $(a,b,c) = (1,0,0)$
\par c) $(a,b,c) = (2,0,0)$

}\anothercol{

\par d) $(a,b,c) = (0,1,0)$
\par e) $(a,b,c) = (0,2,0)$
\par f) $(a,b,c) = (0,0,1)$
\par g) $(a,b,c) = (0,0,2)$
\par h) $(a,b,c) = (3,2,1)$

\bsk

Repare que os pontos mais fáceis

de calcular são estes aqui:

$(x,y)=(0,0)$,

$(x,y)=(1,0)$,

$(x,y)=(0,1)$

}}




\newpage

{\bf Exercício 2}

\scalebox{0.6}{\def\colwidth{11cm}\firstcol{

Agora nós vamos considerar que $x_0$ e $y_0$ são constantes, e
que:
%
$$\begin{array}{rcl}
  x &=& x_0+Δx \quad \text{e} \\
  y &=& y_0+Δy,
  \end{array}
$$

e portanto estas duas definições são equivalentes:
%
$$\begin{array}{rcl}
  z(x, y) &=& a + b·(x-x_0) + c·(y-y_0) \\
  z(x_0+Δx, y_0+Δy) &=& a + bΔx + cΔy \\
  \end{array}
$$


Dicas: 1) os três pontos mais fáceis de calcular são os em que
$(Δx,Δy)=(0,0)$, $(Δx,Δy)=(1,0)$ e $(Δx,Δy)=(0,1)$

% 3) os diagramas de numerozinhos estão explicados aqui:
% http://anggtwu.net/LATEX/2022-1-C3-Tudo.pdf#page=44

Represente graficamente os três pontos mais fáceis de calcular de cada um dos planos abaixo.

\par a) $(x_0, y_0, a, b, c) = (3, 2, 3, 0, 0)$
\par b) $(x_0, y_0, a, b, c) = (3, 2, 4, 0, 0)$
\par c) $(x_0, y_0, a, b, c) = (3, 2, 4, 1, 0)$
\par d) $(x_0, y_0, a, b, c) = (3, 2, 4, 2, 0)$
\par e) $(x_0, y_0, a, b, c) = (3, 2, 4, 0, 1)$
\par f) $(x_0, y_0, a, b, c) = (3, 2, 4, 0, 2)$


}\anothercol{
}}



\newpage

% (c3m222dpp 8 "normal-e-gradiente")
% (c3m222dpa   "normal-e-gradiente")



\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-C3/")
# (find-fline "~/LATEX/2023-2-C3/")
# (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-C3/
       cp -fv        $1.pdf ~/LATEX/2023-2-C3/
       cat <<%%%
% (find-latexscan-links "C3" "$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-C3-gradiente veryclean
make -f 2019.mk STEM=2023-2-C3-gradiente pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3gr"
% ee-tla: "c3m232gradiente"
% End:
