From 12c6be7e0245c944752469acef9c093319e44df7 Mon Sep 17 00:00:00 2001 From: Charlie Root Date: Fri, 27 Feb 2026 20:57:10 -0300 Subject: [PATCH] Derives the sum-sub trigonometric identities. --- calculus.pdf | Bin 802733 -> 817199 bytes calculus.tex | 311 +++++++++++++++------- images/triangle-trig-addition-formula.png | Bin 0 -> 12914 bytes 3 files changed, 221 insertions(+), 90 deletions(-) create mode 100644 images/triangle-trig-addition-formula.png diff --git a/calculus.pdf b/calculus.pdf index c6d5a16b93b9d4676f3ba79e8008e3d33062b44a..25fd59d01b5cb9ffdb2b92197049f8f453092e93 100644 GIT binary patch delta 302770 zcmV(=K-s^o|1hucF_1(7G?$Sl11f)7U2oeq6n*!v;M-bYTSbbWKvAGrw{^gV6>Ybt zJj`+2M1U=2ww$5;^*eMD(pRJ`CeDCj4;2-Ox;*!s%gal;oK;M*x?-9+!yMzaV7y{= zUE`M3+A=Pg(uQ%vbcJ`8San1+iAM)4_Jf+r3Tgkc_xio$oES|#uiUMcCFuQ+bBnQ>66wDM0B zZk*>6ZW&Kl<^A#kfH-A6rBFm!vWV9?AsKLkWG4$5$-W-INV7~D92GD_bx+0>ws54LF=5Jy?n`Dd|)qr?ml+x#T)i)yIk~5 z*M85RvsbTX&(3P2@z0!9RrQO%wEoij%f??if4T6Noxhyb(%iN6{`K>8fZcf*BR9(L 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zSAhA``CYA{?R`#>A#CZA{Ik-qWS6%V`g{z?%JQ>y1&=g+AzEs-* z)tJ8tg*9u;LYFsOKVIzA8`V2REM1r$)eNw8{nd&7M8N)MT^YLw3WBS@9ry!;PrV(W z)7))f4f^L3Ew)Pfw?tenvv1yM`$;W2a4@uM;IAP3QH=GSI%(K1`REUMBw>H+(Vq8d zK5b>~%H*UU?^&u6S!kx-sYdexu+@6~p=lwVOz5JjQJNKlEr6fln?T@=B!pc-J!WUjG+kbOx f|CeIx>-XBnFUaR5R~7?8--- + +%% --8<-------------------------------------------------------->8--- +%% Sometimes I wanna make a section without making it a real section, +%% so let's call it a ghostsection. Because I currently write with no +%% paragraph skip at all, I'm using the math-mode space skip to +%% separate the section from the text (of the previous section). A +%% ghost section is cute---check it out. Notice that the +%% responsibility of adding a period after the bold title is the +%% user's. +\newcommand{\ghostsection}[1]{{\vskip\the\belowdisplayskip\noindent{\bf #1}}} +%% --8<-------------------------------------------------------->8--- + \usepackage{amsmath,amsthm,amssymb} \usepackage{icomma} \allowdisplaybreaks @@ -131,6 +143,7 @@ \DeclareMathOperator{\cm}{cm} \undef\sin \DeclareMathOperator{\sin}{sen} +\DeclareMathOperator{\sen}{sen} \DeclareMathOperator{\ctg}{ctg} \DeclareMathOperator{\imp}{\implies} \DeclareMathOperator{\inv}{inv} @@ -140,6 +153,7 @@ \newcommand{\dy}{\ dy} \newcommand{\dt}{\ dt} \newcommand{\ds}{\ ds} +\undef\d\newcommand{\d}{\frac{d}{\dx}} \DeclareMathOperator{\csch}{csch} \DeclareMathOperator{\sech}{sech} @@ -1215,6 +1229,29 @@ $e=7$ e obteremos $|x^2 - 9| < 7.80$, o que está $0.80$ acima de onde deveria, ilustrando que $d=1$ é o maior delta possível pra $e=7$. \end{exercise*} +\subsection{Como descrever uma elipse} + +Uma elipse é uma generalização de um círculo. A fórmula geral de uma +elipse é +% + $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ +% +Quanto maior é $a^2$, mais larga (horizontalmente) fica a elipse: a +velocidade de varredura do eixo $x$ é atrasada em função do fato +$1/a^2$. Similarmente, quanto maior é $b^2$, mais alta +(verticalmente) fica a elipse. + +Quando a elipse intercepta o eixo $x$ em $(1/9)x^2 + y^2 = 1$? +Coloque $y=0$, obtendo $(1/9)x^2 = 1$, implicando $x = \pm 3$. Onde +ela intercepta o eixo $y$? Coloque $x=0$, obtendo $y = \pm 1$. + +\begin{exercise*} +Descreva o gráfico da equação $9x^2 + 16y^2 = 144$. Note que $144 = +12^2 = (4\times 3)^2 = 16 \times 9$. Logo, a equação é equivalente a +$(1/16)x^2 + (1/9)y^2 = 1$. O gráfico é uma elipse que intercepta o +eixo $x$ em $\pm 4$ e intercepta o eixo $y$ em $\pm 3$. +\end{exercise*} + \subsection{Trigonometria} É assunto difícil. Uma parte do problema é o fato de que as noções @@ -1242,7 +1279,7 @@ três ângulos internos de 60 graus, o que é $\sfrac{\pi}{3}$. Se bisectarmos um desses ângulos, obteremos dois ângulos de $\sfrac{\pi}{6}$. A bisecção divide o lado oposto em duas partes iguais de tamanho $\sfrac{1}{2}$. Logo, a altura do triângulo -satisfaz $1^2 = 1 = \sfrac{1}{4} + h^2$, sendo $h$ a altura. Daí $h = +satisfaz $1^2 = \sfrac{1}{4} + h^2$, sendo $h$ a altura. Daí $h = \sfrac{\sqrt{3}}{2}$, implicando $\sin \sfrac{\pi}{6} = \sfrac{1}{2}$ e $\sin \sfrac{\pi}{3} = \sfrac{\sqrt{3}}{2}$. @@ -1290,6 +1327,12 @@ ctg & ind & $\sqrt3$ & 1 & $\sfrac{1}{\sqrt3}$ & ind & $-\sf \label{tb:trig} \end{table} +\begin{definition*} +Relativo a um dado ângulo, o {\em ângulo de referência} é o menor +ângulo agudo que se pode formar (com o eixo $x$) e, por definição, ele +é sempre positivo em $[0, \sfrac{\pi}{2}]$. +\end{definition*} + \begin{exercise*} Encontre as razões trigonométricas no plano cartesiano relativo ao ângulo $\sfrac{2\pi}{3}$. O ângulo se encontra no segundo quadrante. @@ -1363,70 +1406,6 @@ inversamente proporcional a uma quantidade cujo crescimento é determinado por $m(x)$ se o produto $M(x) m(x)$ é uma constante. \end{definition*} -\subsubsection{Identidades trigonométricas}\label{sec:trig-identities} - -Há uma série de identidades trigonométricas. - -A identidade do círculo unitário é óbvia: aplicando a definição de -$\sin$ e $\cos$ e usando o Teorema de Pitágoras, obtemos $\sin^2 x + -\cos^2 x = 1$. Se divirmos ambos os lados da equação por $\cos^2 x$, -obtemos $$\tg^2 x + 1 = \sec^2 x.$$ Se, em vez disso, dividirmos por -$\sin^2 x$, obtemos $$1 + \ctg^2 x = \csc^2 x.$$ - -A função $\sin x$ é ímpar e $\cos x$ é par, o que significa dizer -$\sin(-x) = - \sin x$ e $\cos(-x) = \cos x$. O período de ambas é -$2\pi$. Logo, $\sin(x + 2\pi) = \sin x$ e $\cos(x + 2\pi) = \cos x$. - -Stewart sugere que todas as identidades que podem aparecer pelo livro -podem ser obtidas das {\em fórmulas da adição}, que são $$\sin(x + y) -= \sin x \cos y + \cos x \sin y \qquad\text{e}\qquad \cos(x + y) = -\cos x \cos y - \sin x \sin y.$$ Por exemplo, colocando $x = y$ nelas, -obtemos as {\em fórmulas dos ângulos duplos} $$\sin(2x) = 2\sin x \cos -x \qquad\text{e}\qquad \cos(2x) = \cos^2 x - \sin^2 x.$$ - -\begin{definition*} -Relativo a um dado ângulo, o {\em ângulo de referência} é o menor -ângulo agudo que se pode formar (com o eixo $x$) e, por definição, ele -é sempre positivo em $[0, \sfrac{\pi}{2}]$. -\end{definition*} - -\begin{exercise*} -Encontre todos os valores de $x$ tal que $\sin(2x) = \sin x$ e $x \in -[0,2\pi]$. Usando a fórmula dos ângulos duplos, sabemos que $2\sin x -\cos = \sin x$, logo $2\sin x \cos - \sin x = 0$ e então $\sin x -(2\cos x - 1) = 0$. Daí estamos interessados em $\sin x = 0$, em que -são soluções $0, \pi, 2\pi$. Estamos também interessados em $2\cos x -- 1 = 0$ em que $\cos x = \sfrac{1}{2}$, em que uma solução é -$\sfrac{\pi}{3}$ --- e a última é $\sfrac{5\pi}{3}$ porque o ângulo de -referência relativo a $\sfrac{5\pi}{3}$ é $\sfrac{\pi}{3}$. Coletando -as soluções, obtemos $\{0, \sfrac{\pi}{3}, \pi, \sfrac{5\pi}{3}, -2\pi\}.$ -\end{exercise*} - -\begin{exercise*} -Se $\sin x = \sfrac{1}{3}$ e $\sec y = \sfrac{5}{4}$, calcule $\sin(x -+ y)$, sendo ambos $x, y$ ângulos agudos positivos. Aplicando a -fórmula da adição, obtemos $\sin(x + y) = \sin x \cos y + \cos x \sin -y$ e daí -% - $$\sin(x + y) = \frac{1}{3}\frac{4}{5} + \cos x \sin y,$$ -% -já que $\sec y = \sfrac{5}{4}$ implica $\cos y = \sfrac{4}{5}$. -Falta-nos conhecer $\cos x$ e $\sin y$. Mas se $\sin x = -\sfrac{1}{3}$, então aplicando $\sin^2 x + \cos^2 x = 1$, obtemos -% - $$\left(\frac{1}{3}\right)^2 + \cos^2 x = 1,$$ -% -que implica $\cos x = \pm \sfrac{2\sqrt{2}}{3}$. Similarmente, $\sin -y = \pm \sfrac{3}{5}.$ Precisamos decidir que sinal nos interessa. -Como $x, y$ estão restritos ao primeiro quadrante, então $\cos x = -\sfrac{2\sqrt{2}}{3}$ e $\sin y = \sfrac{3}{5}$. Logo, -% - $$\sin(x + y) = \frac{1}{3}\frac{4}{5} + \frac{2\sqrt{2}}{3} + \frac{3}{5},$$ -% -como desejado. -\end{exercise*} - \subsubsection{Sobre as palavras ``seno'' e ``cosseno''}\label{sec:sine-cosine} Parece que a palavra ``seno'' origina-se em uma má tradução de grego @@ -1473,29 +1452,176 @@ sobre a hipotenusa, que é $\sqrt{x^2 + 1}$. Conclua então que %% adivinha o que queremos. %% \end{remark*} -\subsection{Como descrever uma elipse} +\subsection{Identidades trigonométricas}\label{sec:trig-identities} -Uma elipse é uma generalização de um círculo. A fórmula geral de uma -elipse é -% - $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ -% -Quanto maior é $a^2$, mais larga (horizontalmente) fica a elipse: a -velocidade de varredura do eixo $x$ é atrasada em função do fato -$1/a^2$. Similarmente, quanto maior é $b^2$, mais alta -(verticalmente) fica a elipse. +Há uma série de identidades trigonométricas. -Quando a elipse intercepta o eixo $x$ em $(1/9)x^2 + y^2 = 1$? -Coloque $y=0$, obtendo $(1/9)x^2 = 1$, implicando $x = \pm 3$. Onde -ela intercepta o eixo $y$? Coloque $x=0$, obtendo $y = \pm 1$. +A identidade do círculo unitário é óbvia: aplicando a definição de +$\sin$ e $\cos$ e usando o Teorema de Pitágoras, obtemos $\sin^2 x + +\cos^2 x = 1$. Se divirmos ambos os lados da equação por $\cos^2 x$, +obtemos $$\tg^2 x + 1 = \sec^2 x.$$ Se, em vez disso, dividirmos por +$\sin^2 x$, obtemos $$1 + \ctg^2 x = \csc^2 x.$$ + +A função $\sin x$ é ímpar e $\cos x$ é par, o que significa dizer +$\sin(-x) = - \sin x$ e $\cos(-x) = \cos x$. O período de ambas é +$2\pi$. Logo, $\sin(x + 2\pi) = \sin x$ e $\cos(x + 2\pi) = \cos +x$. Além disso, se empurrarmos o seno $\pi/2$ pra direita, obtemos o +cosseno, ou seja, $\sin(pi/2 - x) = \cos(x)$ e similarmente +$\cos(\pi/2 - x) = \sin(x)$. + +Stewart sugere que todas as identidades que podem aparecer pelo livro +podem ser obtidas das {\em fórmulas da adição}, que são +% + $$\sin(x + y) = \sin x \cos y + \cos x \sin y \qquad\text{e}\qquad \cos(x + y) = \cos x \cos y - \sin x \sin y.$$ +% +Por exemplo, colocando $x = y$ nelas, obtemos as {\em fórmulas dos + ângulos duplos} +% + $$\sin(2x) = 2\sin x \cos x \qquad\text{e}\qquad \cos(2x) = \cos^2 x - \sin^2 x.$$ +% +Antes de computarmos muitas identidades, vamos trabalhar nas mais +fundamentais primeiro pra ver se conseguímos alguma ordem. Pra +obtermos essas fórmulas da adição, vamos precisar da Lei dos Cossenos +e aproveitamos assim pra incluir a Lei dos Senos também. + +\ghostsection{A lei dos senos}. A lei dos senos é super fácil de +memorizar porque ela diz que o tamanho dos lados de um triângulo são +proporcionais ao seno de seus respectivos ângulos em qualquer +triângulo. Em símbolos, +% + $$\frac{\sen a}{a} = \frac{\sen b}{b} = \frac{\sen c}{c}$$ +% +Note que ela não fala que a o cosseno é. Isso certamente faz a gente +se perguntar por que não. E o legal dela é que é em {\em qualquer} +triângulo. + +\ghostsection{A lei dos cossenos}. A lei dos cossenos é uma espécie +de Teorema de Pitágoras. Considere qualquer triângulo com lados $A, +B, C$. Nomeie os ângulos em letras minúsculas respectivamente. A lei +diz que +% +\begin{align*} + a^2 &= b^2 + c^2 - 2bc\cos a\\ + b^2 &= a^2 + c^2 - 2ac\cos a\\ + c^2 &= a^2 + b^2 - 2ab\cos a. +\end{align*} + +\ghostsection{As fórmulas da adição}. Usando a lei dos cossenos a +gente consegue provar a fórmula da subtração dos cossenos, o que a +gente usa pra provar a fórmula da soma. Os exercícios abaixo mostram +como. +% +\begin{figure}[!htb] + \centering + \includegraphics[width=0.8\textwidth, angle=0]{images/triangle-trig-addition-formula.png} + \caption{O triângulo de onde se extrai a fórmula da adição.} + \label{fg-trig-add-formula} +\end{figure} +% +\begin{exercise*} +Observe a desenho exibido pela Figura~\ref{fg-trig-add-formula}. Use-o +pra provar a fórmula da subtração +% + $$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta.$$ +% +\sol. A estratégia é computar $c^2$ duas vezes, a primeira pela +distância entre os pontos $A$ e $B$ e a segunda pela lei dos +cossenos. Primeiro note que o ângulo que subtende o lado $c$ é o +ângulo $\alpha - \beta$. Agora, executando a estratégia, obtemos o +quadrado da distância de $A$ pra $B$ sendo $c^2 = (\cos\alpha - +\cos\beta)^2 + (\sin\alpha - \sin\beta)^2$. Pela lei dos cossenos, +obtemos $c^2 = a^2 + b^2 - 2ab \cos(\alpha - \beta)$. Assim obtemos +% + $$(\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2 = a^2 + b^2 - 2ab \cos(\alpha - \beta).$$ +% +Daí expanda os quadrados do lado esquerdo e observe que pode-se +aplicar $\cos^2 x + \sin^2 x = 1$ pra obter a constante $2$. Note +também que, pelo desenho, $a = b = 1$. Daí a gente consegue eliminar +todas as constantes da expressão e obter +% + $$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta,$$ +% +como desejado. +\end{exercise*} + +\begin{remark*} +Teríamos trabalhando menos se tivéssemos notado mais cedo que $a = b = +1$. +\end{remark*} \begin{exercise*} -Descreva o gráfico da equação $9x^2 + 16y^2 = 144$. Note que $144 = -12^2 = (4\times 3)^2 = 16 \times 9$. Logo, a equação é equivalente a -$(1/16)x^2 + (1/9)y^2 = 1$. O gráfico é uma elipse que intercepta o -eixo $x$ em $\pm 4$ e intercepta o eixo $y$ em $\pm 3$. +Use o exercício anterior pra provar a fórmula da adição relativa ao +cosseno. \sol. Declare $\beta = -\beta$ pra obter +% + $$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta,$$ +% +observando que $\cos -\beta = \cos \beta$ uma vez que o cosseno é uma +função par e que $\sin -\beta = - \sin\beta$ porque o seno é uma +função ímpar---obtida assim a fórmula da adição relativa ao cosseno. \end{exercise*} +\begin{exercise*} +Use o exercício anterior pra provar a fórmula da subtração pro seno e +dela obtenha a fórmula da adição relativa ao seno. \sol. Começe com +% + $$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta,$$ +% +e agora substitua $\alpha$ por $\pi/2 - x$ e $\beta = y$ pra obtermos +% +\begin{align*} + \cos(\pi/2 - x + y) &= \cos(\pi/2 - x)\cos y - \sin(\pi/2 - x)\sin y\\ + \cos(\pi/2 - (x - y)) &= \sin x \cos y - \cos x \sin y\\ + \sin(x - y) &= \sin x \cos y - \cos x \sin y, +\end{align*} +% +obtendo assim a fórmula da subtração relativa ao seno. Pra obter a +fórmula da adição, terminando completamente o exercício, declare $y= +-y$ e obtenha +% +\begin{align*} + \sin(x - (-y)) &= \sin x \cos -y - \cos x \sin -y\\ + \sin(x + y) &= \sin x \cos y + \cos x \sin y, +\end{align*} +% +uma vez que o cosseno é uma função par e o seno, ímpar. +\end{exercise*} + +\begin{exercise*} +Se $\sin x = \sfrac{1}{3}$ e $\sec y = \sfrac{5}{4}$, calcule $\sin(x ++ y)$, sendo ambos $x, y$ ângulos agudos positivos. Aplicando a +fórmula da adição, obtemos $\sin(x + y) = \sin x \cos y + \cos x \sin +y$ e daí +% + $$\sin(x + y) = \frac{1}{3}\frac{4}{5} + \cos x \sin y,$$ +% +já que $\sec y = \sfrac{5}{4}$ implica $\cos y = \sfrac{4}{5}$. +Falta-nos conhecer $\cos x$ e $\sin y$. Mas se $\sin x = +\sfrac{1}{3}$, então aplicando $\sin^2 x + \cos^2 x = 1$, obtemos +% + $$\left(\frac{1}{3}\right)^2 + \cos^2 x = 1,$$ +% +que implica $\cos x = \pm \sfrac{2\sqrt{2}}{3}$. Similarmente, $\sin +y = \pm \sfrac{3}{5}.$ Precisamos decidir que sinal nos interessa. +Como $x, y$ estão restritos ao primeiro quadrante, então $\cos x = +\sfrac{2\sqrt{2}}{3}$ e $\sin y = \sfrac{3}{5}$. Logo, +% + $$\sin(x + y) = \frac{1}{3}\frac{4}{5} + \frac{2\sqrt{2}}{3} + \frac{3}{5},$$ +% +como desejado. +\end{exercise*} + +\begin{exercise*} +Encontre todos os valores de $x$ tal que $\sin(2x) = \sin x$ e $x \in +[0,2\pi]$. Usando a fórmula dos ângulos duplos, sabemos que $2\sin x +\cos = \sin x$, logo $2\sin x \cos - \sin x = 0$ e então $\sin x +(2\cos x - 1) = 0$. Daí estamos interessados em $\sin x = 0$, em que +são soluções $0, \pi, 2\pi$. Estamos também interessados em $2\cos x +- 1 = 0$ em que $\cos x = \sfrac{1}{2}$, em que uma solução é +$\sfrac{\pi}{3}$ --- e a última é $\sfrac{5\pi}{3}$ porque o ângulo de +referência relativo a $\sfrac{5\pi}{3}$ é $\sfrac{\pi}{3}$. Coletando +as soluções, obtemos $\{0, \sfrac{\pi}{3}, \pi, \sfrac{5\pi}{3}, +2\pi\}.$ +\end{exercise*} \section{Definitions, limits and continuity} @@ -2068,7 +2194,7 @@ function that describes the slope of a tangent to a curve in some point. If we consider a curve $f$ at some point $a$ and draw a tangent to $f$ at the point $a$, then you can estimate the slope by $$\fr{f(x) - f(a)}{x - a},$$ for some $a$ near $x$. That's an -estimate. We would get a closer estimate if we choose a number $x$ +estimate. We would get a closer estimate if we chose a number $x$ closer to $a$ than the previous. If we choose $x$ exactly equal to $a$, we get a division by zero and we go beyond what the definition of limit lets us do. Considering the limit of the expression as $x$ @@ -2160,8 +2286,6 @@ $ax^n$ & $nax^{n-1}$\\\bottomrule \end{table} \end{remark*} -\undef\d\newcommand{\d}{\frac{d}{\dx}} - \begin{theorem*}[regra da soma] Se $f$ e $g$ são deriváveis, então % @@ -5159,10 +5283,11 @@ podemos computar o que está correto. A verificação é feita observando que $\sin^2 x = (1/2) - (1/2)\cos 2x$. É uma boa hora de estudar as {\em half-angled formulas} porque elas se aplicam também às potências -trigonométricas. Por exemplo, $\cos^2 2x = (1/2)(1 + \cos(2 \cdot -2x)) = (1/2)(1 + \cos 4x)$, o que é usado na computação da integral de -$\sin^4 x$. É---vamos estudar a {\em half-angled formula} pra -especialmente entender o porquê do nome. +trigonométricas. Por exemplo, +% + $$\cos^2 2x = (1/2)(1 + \cos(2 \cdot 2x)) = (1/2)(1 + \cos 4x),$$ +% +o que é usado na computação da integral de $\sin^4 x$. \section{Studying functions} @@ -6177,7 +6302,13 @@ nisso---então vamos em frente. Quais coeficientes determinam o crescimento de uma variável em relação a outra nas retas? É o coeficiente líder porque a constante $b$ da reta $y = mx + b$ é nada mais que uma translação da reta no plano. -Pra mostrar isso à audiência, bastaria dar um exemplo de duas retas +Pra mostrar isso à audiência, primeiro precisamos saber calcular o +crescimento, o que nos exige uma definição de crescimento. O +crescimento pode ser medido com relação a uma variável---existe o +crescimento em $x$ e o crescimento em $y$---o que chamamos de +``delta''. + +Quanto cresce $y$ com respeito a $x$? \newpage \tableofcontents \newpage \printbibliography diff --git a/images/triangle-trig-addition-formula.png b/images/triangle-trig-addition-formula.png new file mode 100644 index 0000000000000000000000000000000000000000..675f88567db5fb4b9c8c6e8494eac5d2abbcef08 GIT binary patch literal 12914 zcma)jWmFY!`0ap{bjLv?MMR}rX+%J}L?pyRhjh2J96+U2N~BvF1Qg@|(h>@&lqg*S zq9ESq`1{|r?w7mn{lc2V%)Il?8~fSMe)h!aYG0=yMUf&1LZPmvqK_bWPvH0q2@(7y zXu?(?2nV9BqG;&%Y~!=Pnc@C#;_YEXw2O=XrQm}HnfUiI)SfTgf6~10`?d?ii5 z{K2bNS0{FJLVllqD(PD3GsF6!YH?s??6<%pVeCgbO#%B$moK~j+%zvYttZ{t*(qJ> z>+8$7;w$j3zrXC$!S|0jG>x8i=#)NY^!ne^nmginDv|d-x8!-RO@3_kt#b5NQzM~d zmm?=58$34bU0Oa0Vy9Vr_aiVkR!T$ z^d#iu<0uit?dvv;wVhpl-3x*aIT~%;i}mwF5Fyw^HnL33=&l&5Q^>*AY|;gTLPP13 zz6+w+a%fBq7X4CjEKixZ&UN>a67?&KlYxr 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