% (find-LATEX "2021-2-C2-VS.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2021-2-C2-VS.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2021-2-C2-VS.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2021-2-C2-VS.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2021-2-C2-VS.pdf"))
% (defun e () (interactive) (find-LATEX "2021-2-C2-VS.tex"))
% (defun o () (interactive) (find-LATEX "2021-2-C2-VS.tex"))
% (defun u () (interactive) (find-latex-upload-links "2021-2-C2-VS"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2021-2-C2-VS.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2021-2-C2-VS")
% (find-pdf-page   "~/LATEX/2021-2-C2-VS.pdf")
% (find-sh0 "cp -v  ~/LATEX/2021-2-C2-VS.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2021-2-C2-VS.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2021-2-C2-VS.pdf")
%   file:///home/edrx/LATEX/2021-2-C2-VS.pdf
%               file:///tmp/2021-2-C2-VS.pdf
%           file:///tmp/pen/2021-2-C2-VS.pdf
% http://angg.twu.net/LATEX/2021-2-C2-VS.pdf
% (find-LATEX "2019.mk")
% (find-CN-aula-links "2021-2-C2-VS" "2" "c2m212vs" "c2vs")

% «.defs»		(to "defs")
% «.defs-T-and-B»	(to "defs-T-and-B")
% «.title»		(to "title")
% «.regras-e-dicas»	(to "regras-e-dicas")
%
% «.djvuize»		(to "djvuize")



% <videos>
% Video (not yet):
% (find-ssr-links     "c2m212vs" "2021-2-C2-VS")
% (code-eevvideo      "c2m212vs" "2021-2-C2-VS")
% (code-eevlinksvideo "c2m212vs" "2021-2-C2-VS")
% (find-c2m212vsvideo "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 "2021pict2e.lua"           -- (find-LATEX "2021pict2e.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-C2.pdf}
\def\drafturl{http://angg.twu.net/2021.2-C2.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)}}
%
% (defun T () (interactive) (insert "\\T(Total: ?.? pts)\n"))
% (defun B () (interactive) (insert "a) \\B(?.? pts)\n"))



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c2m212vsp 1 "title")
% (c2m212vsa   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 2 - 2021.2}

\bsk

Quinta, 24/fev/2022: VS especial

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://angg.twu.net/2021.2-C2.html}

\end{center}

\newpage



% «regras-e-dicas»  (to ".regras-e-dicas")
% (c2m212vsp 2 "regras-e-dicas")
% (c2m212vsa   "regras-e-dicas")

{\bf Regras e dicas}

As regras e dicas são as mesmas dos mini-testes,

exceto que a prova será disponibilizada às 20:00

da quinta, 24/fev/2022, e você deverá entregá-la

até as 20:00 da sexta, 25/fev/2022.


\bsk

As questões desta prova são baseadas no exercício 3

das aulas extras. Você pode consultar ele aqui:

\ssk

{\footnotesize

% (c2m212tudop 161)
%    http://angg.twu.net/LATEX/2021-2-C2-tudo.pdf#page=161
\url{http://angg.twu.net/LATEX/2021-2-C2-tudo.pdf#page=161}

}

\newpage

{\bf Muito importante}

\ssk


Lembre que a operação `$[:=]$' é a operação

mais básica do curso. Um erro como esse aqui
%
$$(2 = 3 + a·4) \, [a:=10]  \;\;=\;\; (2 = 3 + 40)$$

é considerado tão grave quando $2+3=23$

e pode fazer a sua questão levar zero.


\newpage

{\bf Fórmulas:}

% (c2m212fpp 9 "exercicio-3c")
% (c2m212fpa   "exercicio-3c")


\sa{f1}{\frac{A}{x-a} + \frac{B}{x-b}}
\sa{f1.5}{\frac{A(x-b)}{(x-a)(x-b)} + \frac{B(x-a)}{(x-a)(x-b)}}
\sa{f1.6}{\frac{A(x-b) +                    B(x-a)}{(x-a)(x-b)}}
\sa{f2}{\frac {(A+B)x + (-Ab-Ba)} {x^2 + (-a-b)x + ab}}
\sa{f3}{\frac {cx+d} {x^2+ex+f}}
\sa{g1}{\frac {2x+3} {x^2-7x+10}}
\sa{g2}{\frac {2x+3} {(x-2)(x-5)}}
\sa{g3}{\frac {A(x-2)} {(x-2)(x-5)} + \frac {B(x-5)} {(x-2)(x-5)}}


$$\begin{array}{rcl}
  \text{[PFP]} &=&   \left(\ga{f1} \;\;=\;\; \ga{f1.6} \right) \\[10pt]
  \text{[PIP1]} &=& 
    \pmat{ \intx{f'(x)g(x) + f(x)g'(x)} \;\;=\;\; f(x)g(x)     \\
           \intx{f'(x)g(x)} \;\; = f(x)g(x) - \intx{f(x)g'(x)}  
    }
  \\[10pt]
  \text{[PIP2]} &=&
    \pmat{ \intx{f'(x)g(x) + f(x)g'(x)} \;\;=\;\; f(x)g(x) \\
           \intx{f(x)g'(x)} \;\; = f(x)g(x) - \intx{f'(x)g(x)}  
    }
  \end{array}
$$


\newpage

{\bf Questão 1.}

\T(Total: 5.0 pts)

a) \B(1.0 pts) Resolva esta integral:
%
$$\intx{ \frac {2x + 3} {x^2 - 8x + 15} } \;.$$

b) \B(4.0 pts) Mostre qual é o caso particular da [PFP]

que está por trás do passo mais difícil do item (a).


% (find-es "maxima" "partial-fractions")
% (setq eepitch-preprocess-regexp "^")
% (setq eepitch-preprocess-regexp "^%T ")
%
%T * (eepitch-maxima)
%T * (eepitch-kill)
%T * (eepitch-maxima)
%T f : (2*x + 3) / (x^2 - 8*x + 15);
%T g : partfrac(f, x);


%T load("/usr/share/emacs/site-lisp/maxima/emaxima.lisp")$
%T display2d:'emaxima$


\newpage


{\bf Questão 2.}

\T(Total: 5.0 pts)

a) \B(1.0 pts) Resolva esta integral:
%
$$\intx{ x^2 \sen 3x } \;.$$

b) \B(4.0 pts) Pra resolver o item (a) você precisou

usar integração por partes duas vezes. Pra cada uma

dessas vezes mostre qual é o caso particular de um

dos ``[PIP]''s que está por trás dela.













%\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-C2/")
# (find-fline "~/LATEX/2021-2-C2/")
# (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-C2/
       cp -fv        $1.pdf ~/LATEX/2021-2-C2/
       cat <<%%%
% (find-latexscan-links "C2" "$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-C2-VS veryclean
make -f 2019.mk STEM=2021-2-C2-VS pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2vs"
% ee-tla: "c2m212vs"
% End: