|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2022-1-C3-P2.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2022-1-C3-P2.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2022-1-C3-P2.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2022-1-C3-P2.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2022-1-C3-P2.pdf"))
% (defun e () (interactive) (find-LATEX "2022-1-C3-P2.tex"))
% (defun o () (interactive) (find-LATEX "2022-1-C3-P1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2022-1-C3-P2"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2022-1-C3-P2.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2022-1-C3-P2")
% (find-pdf-page "~/LATEX/2022-1-C3-P2.pdf")
% (find-sh0 "cp -v ~/LATEX/2022-1-C3-P2.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2022-1-C3-P2.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2022-1-C3-P2.pdf")
% file:///home/edrx/LATEX/2022-1-C3-P2.pdf
% file:///tmp/2022-1-C3-P2.pdf
% file:///tmp/pen/2022-1-C3-P2.pdf
% http://angg.twu.net/LATEX/2022-1-C3-P2.pdf
% (find-LATEX "2019.mk")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Piecewise1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v Pict2e1.lua Pict2e1-1.lua Pict3D1.lua ~/LATEX/")
% (find-sh0 "cd ~/LUA/; cp -v C2Subst1.lua C2Formulas1.lua ~/LATEX/")
% (find-CN-aula-links "2022-1-C3-P2" "3" "c3m221p2" "c3p2")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.title» (to "title")
% «.questoes-1-e-2» (to "questoes-1-e-2")
% «.questoes-1-e-2-gab» (to "questoes-1-e-2-gab")
% «.questoes-3-e-4» (to "questoes-3-e-4")
% «.questao-3-gab» (to "questao-3-gab")
% «.questao-4-gab» (to "questao-4-gab")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c3m221p2" "2022-1-C3-P2")
% (code-eevvideo "c3m221p2" "2022-1-C3-P2")
% (code-eevlinksvideo "c3m221p2" "2022-1-C3-P2")
% (find-c3m221p2video "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")
%
%\usepackage[backend=biber,
% style=alphabetic]{biblatex} % (find-es "tex" "biber")
%\addbibresource{catsem-slides.bib} % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
]{geometry}
%
\begin{document}
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
%L dofile "Piecewise1.lua" -- (find-LATEX "Piecewise1.lua")
%L dofile "QVis1.lua" -- (find-LATEX "QVis1.lua")
%L dofile "Pict3D1.lua" -- (find-LATEX "Pict3D1.lua")
%L dofile "C2Formulas1.lua" -- (find-LATEX "C2Formulas1.lua")
%L Pict2e.__index.suffix = "%"
\pu
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
\def\u#1{\par{\footnotesize \url{#1}}}
\def\drafturl{http://angg.twu.net/LATEX/2022-1-C3.pdf}
\def\drafturl{http://angg.twu.net/2022.1-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% «defs-T-and-B» (to ".defs-T-and-B")
% (c3m202p1p 6 "questao-2")
% (c3m202p1a "questao-2")
\long\def\ColorOrange#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){\ColorOrange{\bf(#1 pts)}}
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (c3m221p2p 1 "title")
% (c3m221p2a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 3 - 2022.1}
\bsk
P2 (Segunda prova)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2022.1-C3.html}
\end{center}
\newpage
% (find-LATEX "edrx21defs.tex" "firstcol-anothercol")
\long\def\anothercol#1{\qquad\quad\firstcol{#1}}
\newpage
% «questoes-1-e-2» (to ".questoes-1-e-2")
% (c3m221p2p 2 "questoes-1-e-2")
% (c3m221p2a "questoes-1-e-2")
\scalebox{0.5}{
\def\colwidth{10cm}\firstcol{
{\bf Questão 1}
\T(Total: 1.0 pts)
Digamos que:
%
$$\begin{array}{rcl}
F(x,y) &=& a \\
&+& bx + cy \\
&+& dx^2 + exy + fy^2 \\
\end{array}
$$
a) \B(0.2 pts) Calcule $F_x$, $F_y$, $F_{xx}$, $F_{xy}$ e $F_{yy}$
nos pontos $(x,y)$ e $(0,0)$.
\ssk
b) \B(0.8 pts) Mostre como reescrever $F(x,y)$ como
%
$$\def\uu{\_\_}
\begin{array}{rcl}
F(x,y) &=& \uu \\
&+& \uu x + \uu y \\
&+& \uu x^2 + \uu xy + \uu y^2 \\
\end{array}
$$
onde em cada lacuna você vai pôr uma expressão que depende só das
derivadas parciais de $F(x,y)$ no ponto $(0,0)$.
}\anothercol{
{\bf Questão 2}
\T(Total: 3.0 pts)
Digamos que:
%
$$\begin{array}{rcl}
G(x_0+Δx,y_0+Δy) &=& a \\
&+& bΔx + cΔy \\
&+& d(Δx)^2 + eΔxΔy + f(Δy)^2 \\
\end{array}
$$
a) \B(0.6 pts) Calcule $G_x$, $G_y$, $G_{xx}$, $G_{xy}$ e $G_{yy}$
nos pontos $(x,y)$ e $(0,0)$.
\ssk
b) \B(2.4 pts) Mostre como reescrever $G(x,y)$ como
%
$$\def\uu{\_\_}
\begin{array}{rcl}
G(x,y) &=& \uu \\
&+& \uu Δx + \uu Δy \\
&+& \uu Δx^2 + \uu ΔxΔy + \uu Δy^2 \\
\end{array}
$$
onde em cada lacuna você vai pôr uma expressão que depende só das
derivadas parciais de $G(x,y)$ no ponto $(x_0,y_0)$.
}}
\newpage
% «questoes-1-e-2-gab» (to ".questoes-1-e-2-gab")
% (c3m221p2p 3 "questoes-1-e-2-gab")
% (c3m221p2a "questoes-1-e-2-gab")
{\bf Gabarito das questões 1 e 2}
\scalebox{0.45}{\def\colwidth{9cm}\firstcol{
$$\begin{array}{rcl}
F(x,y) &=& a + bx + cy + dx^2 + exy + fy^2 \\
F_x(x,y) &=& b + 2dx + ey \\
F_{xx}(x,y) &=& 2d \\
F_{xy}(x,y) &=& e \\
F_y(x,y) &=& c + ex + 2fy \\
F_{yx}(x,y) &=& e \\
F_{yy}(x,y) &=& 2f \\
\\
F(0,0) &=& a \\
F_x(0,0) &=& b \\
F_{xx}(0,0) &=& 2d \\
F_{xy}(0,0) &=& e \\
F_y(0,0) &=& c \\
F_{yx}(0,0) &=& e \\
F_{yy}(0,0) &=& 2f \\
\\
F(x,y) &=& F(0,0) \\
&+& F_x(0,0)x + F_y(0,0)y \\
&+& \frac12 F_{xx}(0,0)x^2 + F_{xy}(0,0)xy + \frac12 F_{yy}(0,0)y^2 \\
\\
\multicolumn{3}{l}{\text{Se $(x_0,y_0)=(0,0)$,}} \\
z(x,y) &=& z \\
&+& z_x x + z_y y \\
&+& \frac12 z_{xx}x^2 + z_{xy}xy + \frac12 z_{yy}y^2 \\
\end{array}
$$
}\anothercol{
$$\begin{array}{rcl}
F(x_0+Δx,y_0+Δy) &=& a + bΔx + cΔy + dΔx^2 + eΔxΔy + fΔy^2 \\
F_x(x_0+Δx,y_0+Δy) &=& b + 2dΔx + eΔy \\
F_{xx}(x_0+Δx,y_0+Δy) &=& 2d \\
F_{xy}(x_0+Δx,y_0+Δy) &=& e \\
F_y(x_0+Δx,y_0+Δy) &=& c + eΔx + 2fΔy \\
F_{yx}(x_0+Δx,y_0+Δy) &=& e \\
F_{yy}(x_0+Δx,y_0+Δy) &=& 2f \\
\\
F(x_0,y_0) &=& a \\
F_x(x_0,y_0) &=& b \\
F_{xx}(x_0,y_0) &=& 2d \\
F_{xy}(x_0,y_0) &=& e \\
F_y(x_0,y_0) &=& c \\
F_{yx}(x_0,y_0) &=& e \\
F_{yy}(x_0,y_0) &=& 2f \\
\\
F(x_0+Δx,y_0+Δy) &=& F(x_0,y_0) \\
&+& F_x(x_0,y_0)Δx + F_y(x_0,y_0)Δy \\
&+& \frac12 F_{xx}(x_0,y_0)Δx^2 + F_{xy}(x_0,y_0)ΔxΔy + \frac12 F_{yy}(x_0,y_0)Δy^2 \\
\\
% \multicolumn{3}{l}{\text{Se $(x_0,y_0)=(0,0)$,}} \\
z(x_0+Δx,y_0+Δy) &=& z \\
&+& z_x Δx + z_y Δy \\
&+& \frac12 z_{xx}Δx^2 + z_{xy}ΔxΔy + \frac12 z_{yy}Δy^2 \\
\end{array}
$$
}}
\newpage
% «questoes-3-e-4» (to ".questoes-3-e-4")
% (c3m221p2p 4 "questoes-3-e-4")
% (c3m221p2a "questoes-3-e-4")
\scalebox{0.6}{\def\colwidth{9cm}\firstcol{
{\bf Questão 3}
\T(Total: 5.0 pts)
\ssk
Seja $H(x,y) = \sqrt{x^2 + 3y^2}$ e seja $(x_0,y_0)=(1,1)$.
Encontre as aproximações de Taylor de ordem 1 e 2 para
$H(x_0+Δx,y_0+Δy)$.
\bsk
\bsk
\bsk
{\bf Questão 4}
\T(Total: 1.0 pts)
\ssk
Seja $M(x_0+Δx,y_0+Δy) = Δx(Δx+Δy)$.
Digamos que $(x_0,y_0) = (4,3)$.
Faça o diagrama de numerozinhos da $M(x,y)$ nos pontos com
$Δx,Δy∈\{-2,-1,-0,1,2\}$.
}\anothercol{
}}
\newpage
% «questao-3-gab» (to ".questao-3-gab")
% (c3m221p2p 5 "questao-3-gab")
% (c3m221p2a "questao-3-gab")
% (setq eepitch-preprocess-regexp "^")
% (setq eepitch-preprocess-regexp "^%T ")
%
%T * (eepitch-maxima)
%T * (eepitch-kill)
%T * (eepitch-maxima)
%T H : sqrt(x^2 + 3*y^2);
%T Hx : diff(H, x);
%T Hy : diff(H, y);
%T Hxx : diff(Hx, x);
%T Hxy : diff(Hx, y);
%T Hyy : diff(Hy, y);
%T rat(Hyy);
%T s : sqrt(x^2 + 3*y^2);
%T Hx_ : x / s;
%T Hy_ : 3*y / s;
%T Hxx_ : (s^2 - x^2) / s^3;
%T Hxy_ : -3*x*y / s^3;
%T Hyy_ : (3*s^2 - 9*y^2) / s^3;
%T V : [H, Hx, Hy, Hxx, Hxy, Hyy];
%T V_ : [H, Hx_, Hy_, Hxx_, Hxy_, Hyy_];
%T V-V_;
%T subst([x=1.23, y=4.56], V-V_);
%T rat(V-V_);
%T rat(Hyy-Hyy_);
%T rat([Hxy, Hyy]);
%T rat([Hx, Hx_]);
%T rat([Hxx, Hxx_]);
%T rat(Hxx - Hxx_);
%T subst([x=1.23, y=4.56], [Hxx, Hxx_]);
%T [x0,y0] : [1,1];
%T foo(sym) := [sym, rat(ev(sym)), subst([x=x0,y=y0],ev(sym))];
%T foo('H);
%T foo('Hx);
%T foo('Hy);
%T foo('Hxx);
%T foo('Hxy);
%T foo('Hyy);
%T [H0, Hx0, Hy0, Hxx0, Hxy0, Hyy0] : subst([x=x0,y=y0], [H, Hx, Hy, Hxx, Hxy, Hyy]);
%T H1 : H0 + Hx0*Dx + Hy0*Dy;
%T H2 : H0 + Hx0*Dx + Hy0*Dy + Hxx0*Dx^2/2 + Hxy0*Dx*Dy + Hyy0*Dy^2;
\newpage
{\bf Questão 3: gabarito}
\scalebox{0.6}{\def\colwidth{15cm}\firstcol{
Seja $H(x,y) = \sqrt{x^2 + 3y^2} = S$.
Então:
%
$$\begin{array}{rclrcl}
H(x,y) &=& S & H(x_0,y_0) &=& 2 \\
H_x(x,y) &=& x / S & H_x(x_0,y_0) &=& 1/2 \\
H_y(x,y) &=& 3y / S & H_y(x_0,y_0) &=& 3/2 \\
H_{xx}(x,y) &=& (S^2-x^2) / S^3 & H_{xx}(x_0,y_0) &=& 3/8 \\
H_{xy}(x,y) &=& -3xy / S^3 & H_{xy}(x_0,y_0) &=& -3/8 \\
H_{yy}(x,y) &=& (3S^2-9y^2) / S^3 & H_{yy}(x_0,y_0) &=& 3/8 \\
\end{array}
$$
Aproximação de Taylor de 1a ordem:
%
$$\begin{array}{rcl}
H(x_0+Δx,y_0+Δy) &≈& H(x_0,y_0) \\
&+& H_x(x_0,y_0)Δx + H_y(x_0,y_0)Δy \\
&=& 2 \\
&+& \frac12 Δx + \frac32 Δy \\
\end{array}
$$
Aproximação de Taylor de 2a ordem:
%
$$\begin{array}{rcl}
H(x_0+Δx,y_0+Δy) &≈& H(x_0,y_0) \\
&+& H_x(x_0,y_0)Δx + H_y(x_0,y_0)Δy \\
&+& \frac12 H_{xx}(x_0,y_0)Δx^2 + H_{xy}(x_0,y_0)ΔxΔy + \frac12 H_{yy}(x_0,y_0)Δy^2 \\
&=& 2 \\
&+& \frac12 Δx + \frac32 Δy \\
&+& \frac3{16} Δx^2 - \frac3{8} ΔxΔy + \frac3{16} Δy^2 \\
\end{array}
$$
}\anothercol{
}}
\newpage
% «questao-4-gab» (to ".questao-4-gab")
% (c3m221p2p 6 "questao-4-gab")
% (c3m221p2a "questao-4-gab")
% (c3m221fhp 7 "exercicio-5")
% (c3m221fha "exercicio-5")
{\bf Questão 4: gabarito}
%L Pict2e.bounds = PictBounds.new(v(0,0), v(6,5))
%L x0,y0 = 4,3
%L nff = function (str)
%L return Code.vc("x,y => local Dx,Dy = x-x0,y-y0; return "..str)
%L end
%L p = Numerozinhos.fromf(v(x0-2,y0-2),v(x0+2,y0+2), nff "Dx*(Dx+Dy)")
%L p:pgat("pN"):preunitlength("11pt"):sa("Questao 4 gab"):output()
\pu
$$\ga{Questao 4 gab}$$
% (c3m221fha "title")
% (c3m221fha "title" "Aula 29: funções homogêneas")
% Funções homogêneas:
% fora da origem
% diagrama de numerozinhos
% demonstrar homogeneidade
% Taylor de ordem 2
% (c3m221tudop 2 "parts")
% (c3m221tudoa "parts")
% (find-pdf-page "~/2022.1-C3/C3-quadros.pdf" 23)
\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 "~/2022.1-C3/")
# (find-fline "~/LATEX/2022-1-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 ~/2022.1-C3/
cp -fv $1.pdf ~/LATEX/2022-1-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=2022-1-C3-P2 veryclean
make -f 2019.mk STEM=2022-1-C3-P2 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3p2"
% ee-tla: "c3m221p2"
% End: