|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2021-2-C3-P1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2021-2-C3-P1.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2021-2-C3-P1.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2021-2-C3-P1.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2021-2-C3-P1.pdf"))
% (defun e () (interactive) (find-LATEX "2021-2-C3-P1.tex"))
% (defun o () (interactive) (find-LATEX "2021-2-C3-P1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2021-2-C3-P1"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2021-2-C3-P1.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (code-eec-LATEX "2021-2-C3-P1")
% (find-pdf-page "~/LATEX/2021-2-C3-P1.pdf")
% (find-sh0 "cp -v ~/LATEX/2021-2-C3-P1.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2021-2-C3-P1.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2021-2-C3-P1.pdf")
% file:///home/edrx/LATEX/2021-2-C3-P1.pdf
% file:///tmp/2021-2-C3-P1.pdf
% file:///tmp/pen/2021-2-C3-P1.pdf
% http://angg.twu.net/LATEX/2021-2-C3-P1.pdf
% (find-LATEX "2019.mk")
% (find-CN-aula-links "2021-2-C3-P1" "3" "c3m212p1" "c3p1")
% «.defs» (to "defs")
% «.defs-T-and-B» (to "defs-T-and-B")
% «.title» (to "title")
% «.regras-e-dicas» (to "regras-e-dicas")
% «.questao-1» (to "questao-1")
% «.questao-2-abcde» (to "questao-2-abcde")
% «.questao-2-fghi» (to "questao-2-fghi")
% «.questao-2-jk» (to "questao-2-jk")
% «.gabarito-maxima» (to "gabarito-maxima")
%
% «.djvuize» (to "djvuize")
% <videos>
% Video (not yet):
% (find-ssr-links "c3m212p1" "2021-2-C3-P1")
% (code-eevvideo "c3m212p1" "2021-2-C3-P1")
% (code-eevlinksvideo "c3m212p1" "2021-2-C3-P1")
% (find-c3m212p1video "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 "2022pict2e.lua" -- (find-LATEX "2022pict2e.lua")
%L Pict2e.__index.suffix = "%"
% \pu
% \def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
% \def\pictaxesstyle{\linethickness{0.5pt}}
% % %L dofile "edrxtikz.lua" -- (find-LATEX "edrxtikz.lua")
% % %L dofile "edrxpict.lua" -- (find-LATEX "edrxpict.lua")
% % \pu
% «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/2021-2-C3.pdf}
\def\drafturl{http://angg.twu.net/2021.2-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
\def\derivs{\mathsf{derivs}}
\def\frt#1{\frac{f^{(#1)}(0)}{#1!}}
\def\ddy{\frac{d}{dy}}
% «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")
% (c3m212p1p 1 "title")
% (c3m212p1a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Cálculo 3 - 2021.2}
\bsk
Primeira prova (P1)
\bsk
Eduardo Ochs - RCN/PURO/UFF
\url{http://angg.twu.net/2021.2-C3.html}
\end{center}
\newpage
% «regras-e-dicas» (to ".regras-e-dicas")
% (c3m212p1p 3 "regras-e-dicas")
% (c3m212p1a "regras-e-dicas")
{\bf Regras e dicas}
As regras e dicas são as mesmas dos mini-testes,
\ssk
\url{http://angg.twu.net/LATEX/2020-2-C3-MT1.pdf}
\url{http://angg.twu.net/LATEX/2020-2-C3-MT2.pdf}
\ssk
exceto que a prova vai ser disponibilizada às 22:30 do dia
21/janeiro/2022 e deve ser entregue até as 10:30 do dia
23/janeiro/2022.
\bsk
Pra fazer essa prova você vai precisar de idéias que a gente
viu durante o curso todo. Se você precisar saber onde estão
as idéias necessárias pra resolver algum item pergunte
\ColorRed{no grupo do Telegram da turma} que eu respondo com um
link pros slides, vídeos, ou livros em que aquela idéia
aparece.
\newpage
% «questao-1» (to ".questao-1")
% (c3m212p1p 3 "questao-1")
% (c3m212p1a "questao-1")
% (c3m212tap 15 "exercicio-5")
% (c3m212taa "exercicio-5")
{\bf Questão 1.}
\T(Total: 5.0 pts)
Esta questão é baseada no Exercício 5
do PDF sobre Séries de Taylor.
\ssk
Digamos que $f,g:\R→\R$ são funções suaves,
$x_0∈\R$, $y=y(x)=f(x)$, $z=z(y)=g(y)$, $h=g∘f$.
\bsk
\bsk
Calcule $\derivs_{x_0}^3(h(x))$ em ``notação de físicos''.
\bsk
\bsk
Dica: você pode começar fazendo MUITAS contas pequenas
e traduções simples que você sabe que são verdade ---
por exemplo ``$z=z(y(x))=g(f(x))$'', ``$z_x = \ddx g(f(x))$'',
``$z_y = z_y(y) = \ddy g(y) = g'(y) = g'(y(x))$'', etc.
\newpage
% «questao-2-abcde» (to ".questao-2-abcde")
% (c3m212p1p 4 "questao-2-abcde")
% (c3m212p1a "questao-2-abcde")
{\bf Questão 2.}
\T(Total: 5.0 pts)
Sejam $x=x(t)=f(t)$, $y=y(t)=g(t)$,
$z=z(x,y)=H(x,y)=x+y-5$,
$P(t)=(x(t),y(t))=(4-t^2,3+\ColorRed{t})$,
$t_0=0$.
\bsk
Calcule:
a) \B(0.2 pts) $x_0$, $y_0$, $P(0), P(1), P(-1)$,
b) \B(0.3 pts) $P'(t), P''(t), P'(0), P'(1), P'(-1), P''(0)$.
\msk
Represente graficamente num gráfico só:
c) \B(0.5 pts) $P(0)+P'(0), P(1)+P'(1), P(-1)+P'(-1)$,
d) \B(0.5 pts) A trajetória $P(t)$ entre $t=-1$ e $t=+1$,
e) \B(0.5 pts) $P(0)+P''(0)$.
\newpage
% «questao-2-fghi» (to ".questao-2-fghi")
% (c3m212p1p 5 "questao-2-fghi")
% (c3m212p1a "questao-2-fghi")
{\bf Questão 2 (cont.)}
\msk
f) \B(0.2 pts) Faça o diagrama de numerozinhos de $H(x,y)$
para $x∈\{x_0-1, x_0, x_0+1\}$, $y∈\{y_0-1, y_0, y_0+1\}$.
\msk
g) \B(0.8 pts) Represente graficamente a superfície $z = H(x,y)$
no quadrado com $x∈[x_0-1, x_0+1]$, $y∈[y_0-1, y_0+1]$.
Faça um desenho em perspectiva improvisada com postes
ligados por cabos, como aqui:
\ssk
{\footnotesize
% (c3m212dnp 9 "figuras-3D")
% (c3m212dna "figuras-3D")
% http://angg.twu.net/LATEX/2021-2-C3-diag-nums.pdf#page=9
\url{http://angg.twu.net/LATEX/2021-2-C3-diag-nums.pdf#page=9}
}
\bsk
Desenhe sobre a sua figura do item anterior
as seguintes trajetórias (para $t∈[-1,1]$):
h) \B(0.2 pts) $(x(t), y(t), 0)$,
i) \B(0.8 pts) $(x(t), y(t), z(x,y))$.
\newpage
% «questao-2-jk» (to ".questao-2-jk")
% (c3m212p1p 6 "questao-2-jk")
% (c3m212p1a "questao-2-jk")
{\bf Questão 2 (cont.)}
\ssk
Sejam:
$Q(t) = (x(t), y(t), 0)$,
$R(t) = (x(t), y(t), z(x(t),y(t)))$.
\bsk
j) \B(0.8 pts) Calcule $R(0)$, $R'(0)$, $R''(0)$.
k) \B(0.2 pts) Calcule $Q(0)$, $Q'(0)$, $Q''(0)$.
% Versao original, errada:
%
% i) \B(0.8 pts) Calcule $R(0)$, $R'(0)$, $R''(0)$.
%
% j) \B(0.2 pts) Calcule $Q(0)$, $Q'(0)$, $Q''(0)$.
\newpage
% «gabarito-maxima» (to ".gabarito-maxima")
% (setq eepitch-preprocess-regexp "^")
% (setq eepitch-preprocess-regexp "^%T ")
%
%T * (eepitch-maxima)
%T * (eepitch-kill)
%T * (eepitch-maxima)
%T x : 4-t^2;
%T y : 3+t;
%T z : x+y-5;
%T P : [x,y];
%T Q : [x,y,0];
%T R : [x,y,z];
%T P_t : diff(P, t);
%T P_tt : diff(P_t, t);
%T subst([t= 0], z);
%T subst([t= 1], P);
%T subst([t= 0], P);
%T subst([t=-1], P);
%T subst(t, 0, z);
%T ["a:", subst([t=0],x), subst([t=0],y),
%T subst([t=0],P), subst([t=1],P), subst([t=-1],P)];
%T ["b:", P_t, P_tt,
%T subst([t=0],P_t), subst([t=1],P_t), subst([t=-1],P_t),
%T subst([t=0],P_tt)];
%T ["c:", [subst([t=0],P), "+", subst([t=0],P_t)],
%T [subst([t=1],P), "+", subst([t=1],P_t)],
%T [subst([t=-1],P), "+", subst([t=-1],P_t)]
%T ];
%T ** d:
%T plot2d([parametric, x, y, [t, -1, 1]]);
%T ["e:", [subst([t=0],P), "+", subst([t=0],P_tt)]];
{\bf Questão 2: gabarito parcial}
% ["a:", subst([t=0],x), subst([t=0],y),
% subst([t=0],P), subst([t=1],P), subst([t=-1],P)];
% (%o14) [a:, 4, 3, [4, 3], [3, 4], [3, 2]]
% (%i15) ["b:", P_t, P_tt,
% (%i15) subst([t=0],P_t), subst([t=1],P_t), subst([t=-1],P_t),
% (%i15) subst([t=0],P_tt)];
% (%o15) [b:, [- 2 t, 1], [- 2, 0], [0, 1], [- 2, 1], [2, 1], [- 2, 0]]
% (%i16) ["c:", [subst([t=0],P), "+", subst([t=0],P_t)],
% [subst([t=1],P), "+", subst([t=1],P_t)],
% [subst([t=-1],P), "+", subst([t=-1],P_t)]
% ];
% (%o16) [c:, [[4, 3], +, [0, 1]], [[3, 4], +, [- 2, 1]], [[3, 2], +, [2, 1]]]
% (%i17)
%T x(t) = 4-t^2;
%T y(t) = 3+t;
%T z(x,y) = x+y-5;
%T H(x,y) = x+y-5;
%T zoft(t) = z(x(t),y(t));
%T xyzoft(t) = [x(t), y(t), zoft(t)];
%T xyzoft(0);
a) \B(0.2 pts)
$x_0=4$,
$y_0=3$,
$P(0) = (4, 3)$,
$P(1) = (3, 4)$,
$P(-1) = (3, 2)$
b) \B(0.3 pts)
$P'(t) = \VEC{-2t, 1}$,
$P''(t) = \VEC{-2, 0}$,
$P'(0) = \VEC{0, 1}$,
$P'(1) = \VEC{- 2, 1}$,
$P'(-1) = \VEC{2, 1}$,
$P''(0) = \VEC{- 2, 0}$
\msk
% (find-LATEX "2022pict2e.lua" "Pict2e")
% (find-LATEX "2022pict2e.lua" "Pict2e" "b0show =")
%L s2 = Surface {
%L unitlength="15pt", sw=v(-1,-3), ne=v(13,7),
%L p1=v(2,-0.5), p2=v(0.5,1.5), p3=v(0,0.5),
%L maxx=5, maxy=4, maxz=3,
%L x0=4, y0=3, nsteps=16,
%L -- f = function (x, y) return (x-4)^2 + (y-3)^2 end,
%L f = function (x, y) return (x-3) + (y-2) end,
%L }
%L
%L p = s2:base() + s2:fig()
%L
%L fP = function (t) return v3(4-t^2, 3+t, 0) end
%L fQ = function (t) return v3(4-t^2, 3+t, 0) end
%L fR = function (t) return v3(4-t^2, 3+t, (4-t^2)+(3+t)-5) end
%L trajQ = Points2(map(fQ, seq(-1, 1, 1/16))):Line()
%L trajR = Points2(map(fR, seq(-1, 1, 1/16))):Line()
%L
%L -- print(s2)
%L -- p = (s2:base() + s2:fig() + s2:twoDgrid())
%L -- :setbounds():bepcb():def("FOO")
%L
%L p = (s2:base() + s2:fig() + trajQ + trajR)
%L :setbounds():bep():def("FOO")
%L p:output()
\pu
% \scalebox{1.0}{\def\colwidth{4cm}\firstcol{
% c) \B(0.5 pts)
% d) \B(0.5 pts)
% e) \B(0.5 pts)
% \msk
% f) \B(0.2 pts) (numerozinhos
% g) \B(0.8 pts)
% h) \B(0.2 pts)
% i) \B(0.8 pts)
% }\anothercol{
\unitlength=15pt
$\FOO$
Faltou desenhar:
Diagrama de numerozinhos
Vetores
% }}
\msk
% j) \B(0.8 pts)
% k) \B(0.2 pts)
%T load("/usr/share/emacs/site-lisp/maxima/emaxima.lisp")$
%T display2d:'emaxima$
%\printbibliography
\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 "~/2021.2-C3/")
# (find-fline "~/LATEX/2021-2-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 ~/2021.2-C3/
cp -fv $1.pdf ~/LATEX/2021-2-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=2021-2-C3-P1 veryclean
make -f 2019.mk STEM=2021-2-C3-P1 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3p1"
% ee-tla: "c3m212p1"
% End: