|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-angg "LATEX/2009may08-C2.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009may08-C2.tex && latex 2009may08-C2.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009may08-C2.tex && pdflatex 2009may08-C2.tex"))
% (eev "cd ~/LATEX/ && Scp 2009may08-C2.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (find-dvipage "~/LATEX/2009may08-C2.dvi")
% (find-pspage "~/LATEX/2009may08-C2.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2009may08-C2.ps 2009may08-C2.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -o 2009may08-C2.ps 2009may08-C2.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2009may08-C2.ps 2009may08-C2.dvi && ps2pdf 2009may08-C2.ps 2009may08-C2.pdf")
% (find-pspage "~/LATEX/2009may08-C2.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2009may08-C2.pdf" (ee-twupfile "LATEX/2009may08-C2.pdf") 'over)
% (ee-cp "~/LATEX/2009may08-C2.pdf" (ee-twusfile "LATEX/2009may08-C2.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 2009may08-C2.dnt
%*
% (eedn4-51-bounded)
%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")
Cálculo II
PURO-UFF
Notas sobre duas técnicas de integração:
substituição trigonométrica e frações parciais
Prof: Eduardo Ochs
8/maio/2009
\bsk
% (find-kopkadaly4page (+ 12 635) "Index" "picture")
% (find-kopkadaly4page (+ 12 288) "picture")
% (find-kopkadaly4text "13.1.4 Picture element commands")
% (find-kopkadaly4page (+ 12 301) "13.1.6 Shifting a picture environment")
% (find-kopkadaly4text "13.1.6 Shifting a picture environment")
% (find-kopkadaly4text "\nStraight lines\n")
\def\sen{\operatorname{sen}}
\def\sec{\operatorname{sec}}
\def\bhbox{}
Abreviações: quando $$ é uma variável,
$s = \sen $
$c = \cos $
$t = \tan = \frac{\sen }{\cos } = \frac{s}{c}$
$z = \sec = \frac{1}{\cos } = \frac{1}{c}$
\msk
Identidades:
$t^2 = \frac{s^2}{c^2} = \frac{1 - c^2}{c^2}$
$t^2c^2 = 1 - c^2$
$t^2c^2 + c^2 = 1$
$(1 + t^2)c^2 = 1$
$1 + t^2 = \frac{1}{c^2} = z^2$
$z^2 = 1 + t^2$
$z = \sqrt{1 + t^2}$
$t^2 = z^2 - 1$
$t = \sqrt{z^2 - 1}$
\msk
Derivadas e diferenciais:
$\frac{ds}{d} = \frac{d\sen}{d} = \cos = c$
$\frac{dt}{d} = \frac{d}{d} \frac{s}{c}
= \frac{s'c - sc'}{c^2}
= \frac{c^2 + s^2}{c^2}
= \frac{1}{c^2}
= z^2
= 1 + t^2
$
$\frac{dz}{d} = \frac{d}{d}c^{-1} = -c^{-2}c' = -c^{-2}(-s)
= \frac{1}{c} \frac{s}{c} = zt$
$ds = c \, d = \sqrt{1 - s^2}d$
$dt = z^2 d = (1 + t^2) d$
$dz = zt\, d$
\newpage
Caso 1:
\bhbox{%
\setlength{\unitlength}{1cm}%
\begin{picture}(5,3)(0,-1)
\thicklines
\put(0.8,0.1){$\theta$}
\put(0,0){\line(2,1){4}}
\put(0,0){\line(1,0){4}}
\put(4,0){\line(0,1){2}}
\put(1.9,1.4){1}
\put(4,0.9){$\begin{array}{c}
\sen \\ = s
\end{array}$}
\put(1.4,-.6){$\begin{array}{c}
\cos = c = \\ \sqrt{1-s^2}
\end{array}$}
\end{picture}%
}
Caso 2:
\bhbox{%
\setlength{\unitlength}{1cm}%
\begin{picture}(5.7,3.1)(0,-0.6)
\thicklines
\put(0.8,0.1){$\theta$}
\put(0,0){\line(2,1){4}}
\put(0,0){\line(1,0){4}}
\put(4,0){\line(0,1){2}}
\put(1.1,1.7){$\begin{array}{l}
\frac{1}{\cos } = \\ \sec = z = \\ \sqrt{1 + t^2}
\end{array}$}
\put(4,0.9){$\begin{array}{l}
\frac{\sen }{\cos } = \\ \, \tan = t
\end{array}$}
\put(1.4,-.4){$\begin{array}{c}
\frac{\cos }{\cos } = 1
\end{array}$}
\end{picture}%
}
\msk
Caso 3:
\bhbox{%
\setlength{\unitlength}{1cm}%
\begin{picture}(5.9,2.8)(0,-0.6)
\thicklines
\put(0.8,0.1){$\theta$}
\put(0,0){\line(2,1){4}}
\put(0,0){\line(1,0){4}}
\put(4,0){\line(0,1){2}}
\put(0.9,1.6){$\begin{array}{r}
\frac{1}{\cos } = \\ \sec = z
\end{array}$}
\put(4,0.9){$\begin{array}{l}
\frac{\sen }{\cos } = \\ \, \tan = t \\ \, = \sqrt{z^2 - 1}
\end{array}$}
\put(1.4,-.4){$\begin{array}{c}
\frac{\cos }{\cos } = 1
\end{array}$}
\end{picture}%
}
\newpage
Exemplos (adaptados do Munem, pp.492--493):
\begin{eqnarray*}
\int \frac{s^2}{(1-s^2)^{3/2}} \, ds
& = & \int \frac{s^2}{(1-s^2)^{3/2}} (1-s^2)^{1/2} \, d \\
& = & \int \frac{s^2}{1-s^2} \, d \\
& = & \int \frac{(\sen )^2}{(\cos )^2} \, d \\
& = & \int (\tan )^2 \, d \\
\end{eqnarray*}
\begin{eqnarray*}
\int \frac{1}{t^2 \sqrt{1 + t^2}} \, dt
& = & \int \frac{1}{t^2 \sqrt{1 + t^2}} (1 + t^2) \, d \\
& = & \int \frac{\sqrt{1 + t^2}}{t^2} \, d \\
& = & \int \frac{(1/c)}{(s^2/c^2)} \, d
= \int \frac{1}{c} \frac{c^2}{s^2} \, d
= \int \frac{c}{s^2} \, d \\
\end{eqnarray*}
\begin{eqnarray*}
\int \frac{1}{z^3 \sqrt{z^2 - 1}} \, dz
& = & \int \frac{1}{z^3 \sqrt{z^2 - 1}} zt \, d \\
& = & \int \frac{zt}{z^3 t} \, d \\
& = & \int \frac{1}{z2} \, d \\
& = & \int c^2 \, d \\
\end{eqnarray*}
\newpage
\def\qv#1{\left[\begin{matrix}#1\end{matrix}\right]}
\def\sqv#1{\left[\begin{smallmatrix}#1\end{smallmatrix}\right]}
\def\displayfrac{\displaystyle\frac}
Um exercício de frações parciais:
usando esta notação,
$\sqv{a_2 \\ a_1 \\ a_0} = [a_2,a_1,a_0] = a_2x^2 + a_1x + a_0$,
$\sqv{a_3 \\ a_2 \\ a_1 \\ a_0} = [a_3,a_2,a_1,a_0] = a_3x^3 + a_2x^2 + a_1x + a_0$, etc,
descubra que valores de $k_1, k_0, c_1, c_2$ e $c_3$ fazem a conta abaixo fazer sentido:
$$\begin{array}{rcl}
\multicolumn{3}{l}{
\displaystyle
k_1x + k_0 + \frac{c_1}{x+2} + \frac{c_2}{x+1} + \frac{c_3}{(x+1)^2} =
} \\ \\
\qquad\qquad\qquad
& = & \displaystyle\frac{
k_1 \qv{1 \\ 5 \\ 7 \\ 3 \\ 0}
+ k_0 \qv{0 \\ 1 \\ 5 \\ 7 \\ 3}
+ c_1 \qv{0 \\ 0 \\ 1 \\ 2 \\ 1}
+ c_2 \qv{0 \\ 0 \\ 1 \\ 3 \\ 2}
+ c_3 \qv{0 \\ 0 \\ 0 \\ 1 \\ 2}
}{ [1, 5, 7, 3]
} \\ \\
& = & \displaystyle\frac{
\begin{array}{crl}
& 10000 & x^4 \\
+ & 51000 & x^3 \\
+ & 75110 & x^2 \\
+ & 37231 & x \\
+ & 3122 & \\
\end{array}
}{ x^3 + 5x^2 + 7x + 3 }
\end{array}
$$
E agora encontre uma primitiva para:
%
$$\int \frac{10000 x^4 +
51000 x^3 +
75110 x^2 +
37231 x +
3122}
{ x^3 + 5x^2 + 7x + 3 } \, dx
$$
%*
\end{document}
% Local Variables:
% coding: raw-text-unix
% ee-anchor-format: "«%s»"
% End: