% (find-LATEX "2024-2-C3-teste-2a-derivada.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2024-2-C3-teste-2a-derivada.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2024-2-C3-teste-2a-derivada.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2024-2-C3-teste-2a-derivada.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2024-2-C3-teste-2a-derivada.pdf"))
% (defun e () (interactive) (find-LATEX "2024-2-C3-teste-2a-derivada.tex"))
% (defun o () (interactive) (find-LATEX "2024-2-C3-teste-2a-derivada.tex"))
% (defun u () (interactive) (find-latex-upload-links "2024-2-C3-teste-2a-derivada"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2024-2-C3-teste-2a-derivada.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2024-2-C3-teste-2a-derivada")
% (find-pdf-page   "~/LATEX/2024-2-C3-teste-2a-derivada.pdf")
% (find-sh0 "cp -v  ~/LATEX/2024-2-C3-teste-2a-derivada.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2024-2-C3-teste-2a-derivada.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2024-2-C3-teste-2a-derivada.pdf")
%   file:///home/edrx/LATEX/2024-2-C3-teste-2a-derivada.pdf
%               file:///tmp/2024-2-C3-teste-2a-derivada.pdf
%           file:///tmp/pen/2024-2-C3-teste-2a-derivada.pdf
%  http://anggtwu.net/LATEX/2024-2-C3-teste-2a-derivada.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-C3-teste-2a-derivada" "3" "c3m242t2d" "c3t2d")

% «.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")
% «.maxima-Ms»		(to "maxima-Ms")
% «.maxima-1»		(to "maxima-1")
% «.maxima-2»		(to "maxima-2")
% «.maxima-3»		(to "maxima-3")



\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-C3.pdf}
\def\drafturl{http://anggtwu.net/2024.2-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 "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")
% (c3m242t2dp 1 "title")
% (c3m242t2da   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 3 - 2024.2}

\bsk

Aula 22: o teste da segunda derivada

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2024.2-C3.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c3m242t2dp 2 "links")
% (c3m242t2da   "links")

{\bf Links}

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

%            (find-es "maxima" "2024.2-C3-eigenvectors")
\par \texttt{(find-es "maxima" "2024.2-C3-eigenvectors")}
%         http://anggtwu.net/e/maxima.e.html#2024.2-C3-eigenvectors
\par \url{http://anggtwu.net/e/maxima.e.html\#2024.2-C3-eigenvectors}

% (find-es "maxima" "2024.2-C3-eigenvectors")

}\anothercol{
}}

\newpage

% «maxima-Ms»  (to ".maxima-Ms")

%M (%i1) [[r1,r2], k, [zneg,zpos]] : [[-2,-1], 10, [-5,10]];
%M (%o1) \left[ \left[ -2 , -1 \right]  , 10 , \left[ -5 , 10 \right]  \right] 
%M (%i2) [xmin,ymin,xmax,ymax] : [-5,-5, 5,5];
%M (%o2) \left[ -5 , -5 , 5 , 5 \right] 
%M (%i3) 
%M f : k * (x -   r1) * (x -   r2);
%M (%o3) 10\,\left(x+1\right)\,\left(x+2\right)
%M (%i4) F : k * (x - y*r1) * (x - y*r2);
%M (%o4) 10\,\left(y+x\right)\,\left(2\,y+x\right)
%M (%i5) f : expand(f);
%M (%o5) 10\,x^2+30\,x+20
%M (%i6) F : expand(F);
%M (%o6) 20\,y^2+30\,x\,y+10\,x^2
%M (%i7) define(f(x),   f);
%M (%o7) f\left(x\right):=10\,x^2+30\,x+20
%M (%i8) define(F(x,y), F);
%M (%o8) F\left(x , y\right):=20\,y^2+30\,x\,y+10\,x^2
%M (%i9) 
%M solve(f(x)=0, x);
%M (%o9) \left[ x=-2 , x=-1 \right] 
%M (%i10) [r1,r2];
%M (%o10) \left[ -2 , -1 \right] 
%L maximahead:sa("eigens 1", "")
\pu

%M (%i11) drawlevels() :=
%M   [myimp1(F=zpos, lc(red),    lk(z=zpos)),
%M    myimp1(F=0,    lc(orange), lk(z=0)),
%M    myimp1(F=zneg, lc(blue),   lk(z=zneg))]$
%M (%i12) drawroots() := pts([[r1,1], [r2,1]], pc(forest_green), pk("r1,r2"), myps(3))$
%M (%i13) drawunitcircle() := mypara([cos(th),sin(th)], th,0,2*%pi, lc(gray))$
%M (%i14) myqdrawp(xyrange(), drawlevels(), drawroots(), drawunitcircle());
%M (%o14) \myvcenter{\includegraphics[height=17cm]{2024-2-C3/eigenvectors_001.pdf}}
%L maximahead:sa("eigens 2", "")
\pu

%M (%i15) myexth(fth,[opts]) :=
%M   myapply_fl('ex1, fth, th,0,2*%pi, opts)$
%M (%i16) F;
%M (%o16) 20\,y^2+30\,x\,y+10\,x^2
%M (%i17) z1 : subst([y=cos(th), x=sin(th)], F);
%M (%o17) 10\,\left(\sin \mathrm{th}\right)^2+30\,\cos \mathrm{th}\,\sin \mathrm{th}+20\,\left(\cos \mathrm{th}\right)^2
%M (%i18) block(
%M   [xmin,xmax,ymin,ymax],
%M   [xmin,xmax,ymin,ymax] : [0,2*%pi, -40,40],
%M   myqdrawp(xyrange0(), myexth(z1))
%M   );
%M (%o18) \myvcenter{\includegraphics[height=9cm]{2024-2-C3/eigenvectors_002.pdf}}
%L maximahead:sa("eigens 3", "")
\pu

%M (%i19) F_x  : diff(F,x);
%M (%o19) 30\,y+20\,x
%M (%i20) F_y  : diff(F,y);
%M (%o20) 40\,y+30\,x
%M (%i21) F_xx : diff(F,x,2);
%M (%o21) 20
%M (%i22) F_xy : diff(F,x,1,y,1);
%M (%o22) 30
%M (%i23) F_yy : diff(F,y,2);
%M (%o23) 40
%M (%i24) 
%M M : matrix([F_xx, F_xy],
%M            [F_xy, F_yy]);
%M (%o24) \begin{pmatrix}20&30\cr 30&40\cr \end{pmatrix}
%M (%i25) determinant(M);
%M (%o25) -100
%L maximahead:sa("eigens 4", "")
\pu

%M (%i26) [vals,vectors]    : eigenvectors(M)$
%M (%i27) [lambda1,lambda2] : vals[1];
%M (%o27) \left[ 30-10^{{\frac{3}{2}}} , 10^{{\frac{3}{2}}}+30 \right] 
%M (%i28) [v1,v2]           : [vectors[1][1], vectors[2][1]];
%M (%o28) \left[ \left[ 1 , -\left({\frac{\sqrt{10}-1}{3}}\right) \right]  , \left[ 1 , {\frac{\sqrt{10}+1}{3}} \right]  \right] 
%M (%i29) 
%M fpprintprec;
%M (%o29) 0
%M (%i30) fpprintprec : 3;
%M (%o30) 3
%M (%i31) float([lambda1,v1]);
%M (%o31) \left[ -1.62 , \left[ 1.0 , -0.721 \right]  \right] 
%M (%i32) float([lambda2,v2]);
%M (%o32) \left[ 61.6 , \left[ 1.0 , 1.39 \right]  \right] 
%M (%i33) 
%M        lambda1 * lambda2;
%M (%o33) \left(30-10^{{\frac{3}{2}}}\right)\,\left(10^{{\frac{3}{2}}}+30\right)
%M (%i34) expand(lambda1 * lambda2);
%M (%o34) -100
%M (%i35) [M, determinant(M)];
%M (%o35) \left[ \begin{pmatrix}20&30\cr 30&40\cr \end{pmatrix} , -100 \right] 
%L maximahead:sa("eigens 5", "")
\pu

%M (%i36) myvector(v,[opts]) := myapply_fl('vector, [0,0], v, hl(0.2), opts)$
%M (%i37) draweigenvectors() :=
%M   [myvector(v1, lc(gray)),
%M    myvector(v2, lc(gray))]$
%M 
%M (%i38) 
%M myqdrawp(xyrange(), drawlevels(), drawroots(),
%M          drawunitcircle(), draweigenvectors());
%M 
%M (%o38) \myvcenter{\includegraphics[height=16cm]{2024-2-C3/eigenvectors_003.pdf}}
%L maximahead:sa("eigens 6", "")
\pu


% «maxima-1»  (to ".maxima-1")
% (c3m242t2dp 3 "maxima-1")
% (c3m242t2da   "maxima-1")

\scalebox{0.35}{\def\colwidth{13cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {14cm}
\ga{eigens 1}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {19cm}
\ga{eigens 2}

}}

\newpage

% «maxima-2»  (to ".maxima-2")
% (c3m242t2dp 4 "maxima-2")
% (c3m242t2da   "maxima-2")

\scalebox{0.45}{\def\colwidth{14cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {12cm}
\ga{eigens 3}

}\anothercol{

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

}}

\newpage

% «maxima-3»  (to ".maxima-3")
% (c3m242t2dp 5 "maxima-3")
% (c3m242t2da   "maxima-3")

\scalebox{0.35}{\def\colwidth{13cm}\firstcol{

\vspace*{0cm}
\def\hboxthreewidth {12cm}
\ga{eigens 5}

}\anothercol{

\vspace*{0cm}
\def\hboxthreewidth {17cm}
\ga{eigens 6}

}}







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

\end{document}

% (find-pdfpages2-links "~/LATEX/" "2024-2-C3-teste-2a-derivada")


% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3t2d"
% ee-tla: "c3m242t2d"
% End:
