% (find-angg "LATEX/2009-2-C4-prova-1.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009-2-C4-prova-1.tex && latex    2009-2-C4-prova-1.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009-2-C4-prova-1.tex && pdflatex 2009-2-C4-prova-1.tex"))
% (eev "cd ~/LATEX/ && Scp 2009-2-C4-prova-1.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (defun d () (interactive) (find-dvipage "~/LATEX/2009-2-C4-prova-1.dvi"))
% (find-dvipage "~/LATEX/2009-2-C4-prova-1.dvi")
% (find-pspage  "~/LATEX/2009-2-C4-prova-1.pdf")
% (find-pspage  "~/LATEX/2009-2-C4-prova-1.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2009-2-C4-prova-1.ps 2009-2-C4-prova-1.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2009-2-C4-prova-1.ps 2009-2-C4-prova-1.dvi && ps2pdf 2009-2-C4-prova-1.ps 2009-2-C4-prova-1.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2009-2-C4-prova-1.pdf" (ee-twupfile "LATEX/2009-2-C4-prova-1.pdf") 'over)
% (ee-cp "~/LATEX/2009-2-C4-prova-1.pdf" (ee-twusfile "LATEX/2009-2-C4-prova-1.pdf") 'over)

\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08}       % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty"  -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex   % (find-dn4ex "edrxheadfoot.tex")
\begin{document}

\input 2009-2-C4-prova-1.dnt

% (find-angg "LATEX/2009-2-C4-prova-1-notas.tex")

%
% (eedn4-51-bounded)

% (find-fline "~/PURO/diarios_de_classe/")

%Index of the slides:
%\msk
% To update the list of slides uncomment this line:
%\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")


\def\sen{\operatorname{sen}}
\def\pmat#1{\begin{pmatrix} #1 \end{pmatrix}}
\def\bmat#1{\left|\begin{matrix} #1 \end{matrix}\right|}
\def\sm#1{\begin{smallmatrix} #1 \end{smallmatrix}}
\def\bsm#1{\left|\begin{smallmatrix} #1 \end{smallmatrix}\right|}
\def\psm#1{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)}
\def\subst#1{\left[\begin{smallmatrix} #1 \end{smallmatrix}\right]}

Cálculo 4 - Primeira Prova (P1)

PURO-UFF - 2009.2

28/outubro/2009

Prof: Eduardo Ochs


\bsk
\bsk

\noindent {\bf (1)} (Total: 4.0 pontos). Seja $B_{xy} =
\sst{(x,y)}{xÝ[1,2], yÝ[x^2,2x^2]}$. Considere a mudança de variáveis:
%
$$\pmat{u \\ v} := \pmat{x \\ y/x^2}$$
%
Ela leva a região $B_{xy}$, contida no plano $(x,y)$, numa região
$B_{uv}$ do plano $(u,v)$.

\ssk

a) (1.0 pts) Represente graficamente as regiões $B_{xy}$ e $B_{uv}$.

\ssk

b) (2.0 pts) Complete:
%
$$\int_{x=1}^{x=2} \int_{y=x^2}^{y=2x^2} f(x,y)\,dy\,dx
  = \int_{u=\ldots}^{u=\ldots} \int_{v=\ldots}^{v=\ldots} \ldots\,dv\,du
$$

\ssk

c) (1.0 pts) Inverta a ordem de integração nas duas integrais do item (b).

\bsk
% \bsk

\noindent {\bf (2)} (Total: 1.0 ponto). Calcule o volume de:
%
$$ S = \sst{(x,y,z)}{x,yÝ[0,1], \, x\ge y,\, zÝ[xy,2xy]}$$

\bsk
% \bsk

\noindent {\bf (3)} (Total: 2.5 pontos). Calcule a área de:
%
$$ P = \sst{(x,y,z)}{x^2+y^2 \le 1, \, z = 1-(x^2+y^2)}$$

\bsk

\noindent {\bf (4)} (Total: 2.5 pontos). Seja:
%
$$ C = \sst{(x,y)}{(x-2)^2 + (y-1)^2 \le 1, \, x \le 2, y \le 1}.$$

a) (1.0 pts) Encontre uma mudança de variáveis que transforme $C$ num
retângulo.

b) (1.5 pts) Use esta mudança para transformar $\int\!\!\int_C
f(x,y)\,dx\,dy$ numa integral sobre um retângulo.


\bsk
\bsk
\bsk


\newpage

{\setlength{\parindent}{0pt}



{\bf Algumas fórmulas:}

Área de uma superfície $z=z(x,y)$ sobre uma região $B \subset \R^2$:
%
$$\int\int_B \sqrt{1 + z_x^2 + z_y^2} \;dx\,dy$$

Coordenadas polares:
%
$$
\begin{array}{rcl}
  x &=& r \cos  \\
  y &=& r \sen  \\
  r^2 &=& x^2 + y^2 \\
\end{array}
\qquad
dx\,dy = \bsm{x_r & x_ \\ y_r & y_} \,dr\,d  = r\,dr\,d
$$


Um modo de escrever a substituição (na integral simples):
%
$$\int_{t=\sqrt\pi}^{t=\sqrt{2\pi}} (\sen t^2)\, 2t \, dt =
  \subst{x = t^2 \\ t = \sqrt x \\ dx = 2t \, dt} \int_{x=\pi}^{x=2\pi} \sen x \, dx
$$

\bsk
\bsk


A prova é para ser feita em duas horas, sem consulta.

Responda claramente e justifique cada passo.

Lembre que a correção irá julgar o que você escreveu, e

que é impossível ler o que você pensou mas não escreveu.

Lembre que a resposta esperada para cada questão não é só

uma fórmula ou um número --- a ``resposta certa'' é um

raciocínio claro e convincente.

Outra dica: {\sl confira as suas respostas!}

\ssk

{\bf Boa prova!}

}


\newpage

{\bf Mini-gabarito:}

(Versão preliminar, incompleta e com erros, 2009nov25)

% (find-kopkadaly4text "\n\\includegraphics[llx,lly]")
% (find-kopkadaly4text "with the graphicx package")

\noindent {\bf (1a)} (1.0 pts):
$B_{xy} = \sm{\includegraphics[scale=1.0]{2009-2-C4-prova-1-a.eps}}$,
$B_{uv} = [1,2]×[1,2]$

\noindent {\bf (1b)} (2.0 pts):

  $\psm{u \\ v} = \psm{x \\ yx^2}$,
  $\psm{x \\ y} = \psm{u \\ vu^2}$,

  $dx\,dy = \bsm{x_u & x_v \\ y_u & y_v} du\,dv
          = \bsm{1 & 0 \\ 2vu & u^2} du\,dv
          = u^2 du \, dv$

  $\int_{x=1}^{x=2} \int_{y=x^2}^{y=2x^2} f(x,y) dy\,dx =
   \int_{u=1}^{u=2} \int_{v=1}^{v=2} f(u,vu^2) u^2\,dv\,du$

\noindent {\bf (1c)} (1.0 pts):

  $\int_{y=1}^{y=2} \int_{x=1}         ^{x=\sqrt{y}} f(x,y) dx\,dy +
   \int_{y=2}^{y=4} \int_{x=\sqrt{y/2}}^{\sqrt{y}}   f(x,y) dx\,dy +
   \int_{y=4}^{y=8} \int_{x=\sqrt{y/2}}^{x=2}        f(x,y) dx\,dy$

\msk

\noindent {\bf (2)} (1.0 pts):
  $S_{xy} = \sst{(x,y)}{xÝ[0,1], yÝ[0,x]}$

  $¯{Vol}(X) = \int_{x=0}^{x=1} \int_{y=0}^{y=x} 2xy-xy\,dy\,dx
             = \int_{x=0}^{x=1} x \int_{y=0}^{y=x} y \, dy\,dx
             = \int_{x=0}^{x=1} x \frac{x^2}{2} \,dx
             = \frac{1}{2} \int_{x=0}^{x=1} x^3 \,dx
             = \frac{1}{2} \frac{x^4}{x} |_{x=0}^{x=1}
             = \frac{1}{8}$

\msk

\noindent {\bf (3)} (2.5 pts):
  $z = 1 - (x^2+y^2)$, 
  $z_x = -2x$, 
  $z_y = -2y$

  $\sqrt{1 + z_x^2 + z_y^2} = \sqrt{1 + 4x^2 + 4y^2} = \sqrt{1+4r^2}$

  $\text{Área}(P) = \int\!\!\int_{P_{xy}} \sqrt{1+4r^2} dx\,dy
                  = \int\!\!\int_{P_{r}} \sqrt{1+4r^2} r\,dr\,d
                  = 2\pi \int_{r=0}^{r=1} \sqrt{1+4r^2} r\,dr
                  = \subst{u=r^2 \\ r=\sqrt{u} \\ du=2r\,dr}
                    2\pi \int_{u=0}^{u=1} \sqrt{1+4u} \frac{1}{2}\,du
                  = ...
                  = \frac{\pi}{6}(5^{3/2}-1)$

\msk

\noindent {\bf (4a)} (1.0 pts):

\noindent {\bf (4b)} (1.5 pts):



\end{document}

% (find-maximacvsnode "Functions and Variables for draw" "Function: set_draw_defaults")
% (find-maximacvsnode "Functions and Variables for draw" "object: polygon")
% (find-maximacvsnode "Functions and Variables for draw" "object: rectangle")
% (find-maximacvsnode "Functions and Variables for draw" "object: bars")
% (find-maximacvsnode "Functions and Variables for draw" "object: parametric")
% http://www.telefonica.net/web2/biomates/maxima/gpdraw/parametric/index.html
% (find-maximacvsnode "Functions and Variables for draw" "option: eps_width")
% (find-maximacvsnode "Functions and Variables for draw" "option: eps_height")

 (eepitch-maximacvs)
 (eepitch-kill)
 (eepitch-maximacvs)
load(draw);
set_draw_defaults();
OptsR : [xrange = [0, 2], yrange = [0, 8]];
OptsG : [grid = true, axis_top =false, axis_right = false];
OptsG : [grid = false, axis_top =false, axis_right = false];
OptsT : [terminal = screen];
OptsEps(w, h, fname) := [terminal = eps, eps_width = w, eps_height = h, file_name = fname];
hif(x)  := 2 * x^2;
lowf(x) :=     x^2;
Curves  : [explicit(hif(x), x, 0, 2), explicit(lowf(x), x, 0, 2)];
Verticals : [parametric(1, y, y, lowf(1), hif(1)), parametric(2, y, y, lowf(2), hif(2))];
OptsE : OptsEps(6 * 0.6, 8 * 0.6, "/tmp/foo");
OptsE : OptsEps(6 * 0.6, 8 * 0.6, "2009-2-C4-prova-1-a");
apply(draw2d, append(OptsR, OptsG, OptsT, Curves, Verticals));
apply(draw2d, append(OptsR, OptsG, OptsE, Curves, Verticals));

;; (find-fline  "/tmp/")
;; (find-pspage "/tmp/foo.eps")
;; (find-pspage "2009-2-C4-prova-1-a.eps")

parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi)

       key   = "This is the parametric one!!",
          parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi))$



set_draw_defaults();
set_draw_defaults(grid = true, axis_top =false, axis_right = false);
set_draw_defaults(terminal = screen);
draw2d(fill_color  = grey,
       filled_func = x^2,
       explicit(2 * x^2, x, 0, 2));


       filled_func = false,
       explicit(lowf(x), x, 0, x2),
       explicit(hif(x),  x, 0, x2)
       );
draw2d(terminal = eps,
       file_name = "2009-2-C2-prova-1",
       eps_width  = 13 * 0.6,
       eps_height = 7  * 0.6,
       /*
        * (find-pspage "2009-2-C2-prova-1.eps")
       */
       axis_top = false,
       axis_right = false,
       xrange = [0, 13],
       yrange = [0, 7],
       fill_color  = grey,
       filled_func = lowf(x),
       explicit(hif(x), x, 0, x2),
       filled_func = false,
       explicit(lowf(x), x, 0, x2),
       explicit(hif(x),  x, 0, x2)
       );

;; (find-maximacvsnode "")
# (find-maximacvsnode "Functions and Variables for draw" "filled_func")





%

\end{document}

% Local Variables:
% coding:           raw-text-unix
% ee-anchor-format: "«%s»"
% End:
