E-book C2 Versão Final (1) (2) (1).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <center>CÁLCULO DIFERENCIAL E INTEGRAL II - INTRODUCÃO AO CÁLCULO INTEGRAL E ÀS FUNCÕES DE VÁRIAS VARIÁVEIS</center>\n",
    "\n",
    "<center>Martins, J. P. S.; Oliveira, A. G. F.;</center>\n",
    "<center>1(Santos, M. D.): DCC, CI, Universidade Federal da Paraíba, Joao Pessoa, Brasil, mdantas@ci.ufpb.br.</center> \n",
    "<center>2(Oliveira, G. C. P): DCC, CI, Universidade Federal da Paraíba, Joao Pessoa, Brasil, gustavo.oliveira@ci.ufpb.br.</center>  \n",
    "<center>3(Souza, A. J.): DCC, CI, Universidade Federal da Paraíba, Joao Pessoa, Brasil, aparecidosouza@ci.ufpbr.br.</center>\n",
    "\n",
    "## Apresentação\n",
    "\n",
    "O presente material objetiva discutir e fornecer possibilidades para o desenvolvimento da teoria à prática voltadas para atividades de reforço para os estudantes matriculados na disciplina de Cálculo Diferencial e Integral II, ofertadas no Centro de Informática da Universidade Federal da Paraíba. Tais possibilidades serão apresentadas considerando as dificuldades que os estudantes do centro, bem como os de toda a Universidade encontram para aprender os assuntos contidos na ementa da referida disciplina. Por esse motivo, foi elaborado um livro digital, de modo a apresentar os conteúdos de forma bem clara por meio de implementações computacionais que contribuem para a promoção dos estudantes em melhorar sua participação no processo de construção de seus conhecimentos que necessitam de aprofundamento no tema.\n",
    "\n",
    "## Objetivos\n",
    "\n",
    "1. Apresentar algumas aplicações práticas e teóricas de conceitos vistos tanto nas disciplinas de cálculo quanto nas de programação.\n",
    "2. Desenvolver práticas pedagógicas que facilitem a otimização dos conceitos matemático na busca de melhores resultados.\n",
    "3. Propiciar atividades que envolvam a resolução de problemas utilizando a metodologia computacional.\n",
    "4. Utilizar o conhecimento computacional para auxiliar os discentes nas aulas de tutoria presencial e remota.\n",
    "5. Ofertar aos discentes aulas de reforço para a melhoria da aprendizagem.\n",
    "\n",
    "## Conteúdos abordados\n",
    "\n",
    "1. Introdução às integrais\n",
    "    - Conceitos iniciais\n",
    "    - Surgimento\n",
    "    - Notação\n",
    "    - Propriedades\n",
    "    - A antiderivada\n",
    "2. Métodos de integração\n",
    "    - Integrais fundamentais\n",
    "    - Integração por substituição\n",
    "    - Integração por partes\n",
    "    - Integração por substituição trigonométrica\n",
    "    - Cálculo de áreas entre curvas\n",
    "    - Cálculo de volumes em discos e anéis\n",
    "    - Cascas cilíndricas\n",
    "    - Frações parciais\n",
    "3. Introdução às funções de várias variáveis\n",
    "    - Conceitos iniciais\n",
    "    - Domínio de funções de várias variáveis\n",
    "    - Imagem de funções de várias variáveis\n",
    "    - Esboço de gráficos\n",
    "    - Curvas de nível\n",
    "    - Funções de três variáveis\n",
    "4. Limites e continuidade de funções de várias variáveis\n",
    "    - Conceitos iniciais\n",
    "    - Verificando a existência de limites\n",
    "    - Propriedades de limites\n",
    "    - Continuidade\n",
    "    - Diferenciabilidade \n",
    "5. Derivadas parciais\n",
    "    - Notação e conceitos iniciais\n",
    "    - Interpretação geométrica\n",
    "    - Derivadas de ordem superior\n",
    "    - Plano tangente\n",
    "    - Aproximações lineares\n",
    "    - Regra da cadeia\n",
    "    - Derivação implícita\n",
    "    - Vetor gradiente\n",
    "    - Derivada direcional\n",
    "    - Maximização da derivada direcional\n",
    "    - Reta normal à superfície\n",
    "6. Aplicação das derivadas parciais\n",
    "    - Pontos críticos\n",
    "    - Máximos e mínimos locais\n",
    "    - Máximos e mínimos globais\n",
    "    - Multiplicadores de Lagrange\n",
    "\n",
    "\n",
    "## Bibliotecas utilizadas\n",
    "\n",
    "Inicialmente, iremos importar todas as bibliotecas do Python que serão utilizadas durante desenvolvimento do material. Logo abaixo podemos ver todas as importações, bem como uma tabela explicando a funcionalidade de cada biblioteca, e como a mesma foi nomeada para uma utilização mais concisa (isso quer dizer que em alguns casos, trocamos o nome de uma biblioteca, ou de uma função da mesma por outro mais simples).\n",
    "\n",
    "|Biblioteca  |Nomeação|Função importada|Funcionamento|\n",
    "|:------------:|--------|----------------|-------------|\n",
    "|future      | *      |division        |Permite que algumas funções de versões anteriores do python possam utilizar a sua versão atualizada. Nesse caso, como utilizamos division, a importação da mesma referenciou à versão futura.|\n",
    "|sympy       |syp     | *              |Disponibiliza algumas ferramentas algébricas para computação simbólica.|\n",
    "|numpy       |np      | *              |Permite realizar operações matemáticas entre vetores e matrizes da linguagem (Att. no decorrer do material nos referenciamos à biblioteca como np).|\n",
    "|scipy       |sp      | *              |É uma biblioteca voltada para análises e cálculos na área programação científica.|\n",
    "|math        | *      | *              |É a biblioteca padrão da linguagem para realizar cálculos matemáticos.|\n",
    "|matplotlib  | *      |pyplot          |Proporciona a criação plotagem de gráficos de funções.|\n",
    "|mpl_toolkits.mplot3d| * |axes3d       |É uma extensão do matplotlib para geração de gráficos 3D. A mesma utiliza a função |axes3d para a criação dos eixos do gráfico.|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "import sympy as syp\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import pylab as pl\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import axes3d\n",
    "syp.init_printing()\n",
    "x, y, z, w = syp.symbols('x y z w')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## I. INTRODUÇÃO ÀS INTEGRAIS\n",
    "### A. Conceitos iniciais\n",
    "Nesse modulo buscamos compreender o conceito de Integral de uma funcão qualquer, bem como suas propriedades e métodos de resolução.\n",
    "\n",
    "**Definição -** Basicamente, uma integral representa o valor de uma área abaixo de uma curva determinada por uma funcão, ou seja, ao calcular uma integral, busca-se descobrir um valor do tamanho (em u.A.) de uma figura geometrica que está delimitada por uma funcão f(x).  \n",
    "\n",
    "Existem muitos métodos conhecidos para se calcular área interior à uma figura plana, mas nem em todos os casos, esses cálculos são tão simples de se resolver. Como por exemplo, observe a figura abaixo:\n",
    "\n",
    "![](Imagens/1.jpg)\n",
    "\n",
    "Veja que nenhuma das formulas utilizadas para se calcular áreas de figuras simples, como polígonos e circunferências iriam servir para definir um valor correto para a área dessa figura. E é por isso que utilizamos a integral.\n",
    "\n",
    "### B. Surgimento\n",
    "Alguns matemáticos, em busca de calcular a área embaixo dessas curvas tão complexas, perceberam que preenchendo essa curva com retângulos, o valor da área que se buscava encontrar poderia ser determinado pela soma das áreas de cada retângulo, já que, por convenção, sabemos calcular a área dos mesmos. Entao, \n",
    "\n",
    "![](Imagens/2.jpg)\n",
    "\n",
    "Porém, pela figura, percebemos que boa parte da área total da curva seria perdida (área amarela em baixo da curva), então, foi-se necessário utilizar de alguma técnica de refinamento para aperfeiçoar esse método de cálculo. \n",
    "\n",
    "Tendo isso em vista, esses estudiosos perceberam que quanto mais retângulos fossem colocados, mais próximo do valor real da área era alcançado.\n",
    "\n",
    "![](Imagens/3.jpg)\n",
    "\n",
    "Ou seja, para chegar ao valor exato da área, teríamos que ter uma soma das áreas de infinitos retângulos compondo essa curva, que pode ser representado por:\n",
    "\n",
    "$$A = \\lim_{x\\rightarrow \\infty} S_n$$\n",
    "Onde S é essa soma.\n",
    "\n",
    "Então, a integral é uma técnica que faz exatamente o mesmo que a soma explicada anteriormente.\n",
    "\n",
    "$$A = \\lim_{x\\rightarrow \\infty} S_n = \\int_a^b f(x) dx$$\n",
    "\n",
    "### C. Notação\n",
    "Vamos tomar a integral exemplificada anteriormente.\n",
    "\n",
    "$$\\int_a^b f(x) dx$$\n",
    "\n",
    "***f(x) →*** Função a ser integrada (ou integrando).  \n",
    "***a →*** Limite inferior.  \n",
    "***b →*** Limite superior.  \n",
    "***dx →*** Define que variável dentro da função está sendo integrada.\n",
    "\n",
    "### D. Propriedades\n",
    "\n",
    "**1ª propriedade:** A integral da soma de duas funções é igual à soma das integrais de cada função.  \n",
    "Ex.:\n",
    "$$\\int_a^b [f(x)\\pm g(x)]dx = \\int_a^b f(x)dx \\pm \\int_a^b g(x)dx$$\n",
    "\n",
    "**2ª propriedade:** A integral de uma função que multiplica uma constante é igual à constante vezes a integral da função.  \n",
    "Ex.:\n",
    "$$\\int_a^b c\\,f(x)dx = c\\int_a^b f(x)dx$$\n",
    "\n",
    "**3ª propriedade:** Se os limites de integração forem invertidos, o sinal da integral muda.  \n",
    "Ex.:\n",
    "$$\\int_a^b f(x)dx = -\\int_b^a f(x)dx$$\n",
    "\n",
    "**4ª propriedade:** Se os limites de integração forem iguais, o valor da integral é 0.  \n",
    "Ex.:\n",
    "$$\\int_a^a f(x)dx = 0$$\n",
    "\n",
    "**5ª propriedade:**  \n",
    "Ex.:\n",
    "$$\\int_a^b f(x)dx + \\int_b^c f(x)dx = \\int_a^c f(x)dx$$\n",
    "\n",
    "\n",
    "**6ª propriedade:** Se uma função é maior que 0 em um intervalo [a, b], a sua integral é positiva nesse intervalo. Caso contrário, a integral é negativa.\n",
    "\n",
    "**7ª propriedade:** Se uma função $f(x)$ é maior que outra $g(x)$, a integral de f também é maior que a integral de g.\n",
    "\n",
    "### E. A Antiderivada\n",
    "Por meio dos assuntos vistos até agora, é de se esperar que já saibamos como calcular a derivada de uma função qualquer (ou pelo menos grande parte das funções). Esse é o método que chamamos de derivação. Mas o que seria a antiderivação?\n",
    "Vamos lá! Se eu te mostrasse uma função, como por exemplo:\n",
    "\n",
    "$$f(x) = x² + 2x + 1$$\n",
    "\n",
    "É uma função simples de se calcular a derivada, que nesse caso seria:\n",
    "\n",
    "$$f’(x) = 2x + 2$$\n",
    "\n",
    "Agora vamos entender o conceito de antiderivação.\n",
    "Retomando a função $f’(x)$, irei te fazer um questionamento. Que função eu anteriormente derivei para me gerar $2x + 2$?\n",
    "\n",
    "Nesse caso, sabemos que a função foi $x² + 2x + 1$, porque fizemos todo o processo lá em cima, mas como eu posso descobrir isso utilizando manipulações matemáticas?\n",
    "\n",
    "Esse é o conceito de ***antiderivada***, ***primitiva***, ou melhor ainda, ***integral indefinida***.\n",
    "\n",
    "Mas até agora não falamos muito sobre integral definida ou indefinida, mas o que você precisa saber é que a definida apresenta limites de integração, e o seu cálculo geral um número como resultado, já a indefinida não contém limites de integração e o seu cálculo só gera a função integrada.\n",
    "\n",
    "Pense na integral indefinida como uma etapa da integral definida, porém, com a diferença de que os limites não são aplicados.\n",
    "\n",
    "**Obs.** Lembre-se sempre de acrescentar uma constante à integral indefinida, para fazer com que ela represente todas as primitivas da função.\n",
    "\n",
    "## II. MÉTODOS DE INTEGRAÇÃO\n",
    "\n",
    "Nesta seção iremos discutir um pouco sobre como funciona a resolução de integrais, como também alguns métodos de integração pré-definidos.\n",
    "\n",
    "Primeiro, a técnica base para cálculo de integrais é a seguinte:\n",
    "\n",
    "Suponhamos uma função $f(x) = x^2.$ Qual a integral dessa função?\n",
    "\n",
    "Para fazer isso, basta somar 1 ao expoente e dividir a função pelo mesmo. Ou seja,\n",
    "\n",
    "$$\\int x^n dx= \\frac{x^{n+1}}{n+1}$$\n",
    "\n",
    "Então, a  função $f(x)$ integrada fica.\n",
    "\n",
    "$$\\int x^2 dx= \\frac{x^3}{3}$$\n",
    "\n",
    "\n",
    "Façamos mais um exemplo:\n",
    "\n",
    "Ex. 1: Determine a integral da função $g(x) = 3x^2 + 5x - 1$?\n",
    "\n",
    "Inicialmente, vamos aplicar a primeira propriedade da resolução de integrais. \n",
    "\n",
    "\n",
    "$$\\int (3x^2 + 5x - 1)dx = \\int 3x^2 dx + \\int 5x dx - \\int 1 dx$$\n",
    "\n",
    "Agora, basta aplicarmos a técnica base vista anteriormente. Então, \n",
    "$$\\int g(x) dx = \\frac{3x^3}{3} + \\frac{5x^2}{2} - x + C$$\n",
    "\n",
    "**Obs.** Lembre-se sempre de adicionar a constante após integrar uma função (caso a integral seja indefinida).\n",
    "\n",
    "Para fazer esse cálculo em Python podemos fazer o seguinte:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{3} + \\frac{5 x^{2}}{2} - x$"
      ],
      "text/plain": [
       "        2    \n",
       " 3   5⋅x     \n",
       "x  + ──── - x\n",
       "      2      "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def g(x):\n",
    "    return (3*x**2 + 5*x -1)\n",
    "\n",
    "syp.integrate(g(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### A. Algumas integrais fundamentais\n",
    "Então, assim como na derivada, também há algumas funções cuja sua integral já é pré-definida, como por exemplo, as funções trigonométricas, logarítmicas e exponenciais.\n",
    "\n",
    "Esta subseção irá apresentar algumas dessas transformações pré-definidas que serão mais utilizadas no estudo das integrais.  \n",
    "Primeiro de tudo, vamos examinar um pouco sobre as funções trigonométricas $sen(x)$ e $cos(x)$. Quais as integrais dessas funções? Vamos lá!\n",
    "\n",
    "\n",
    "1. $$\\int sen(x)dx = -cos(x)$$\n",
    "2. $$\\int cos(x)dx = sen(x)$$\n",
    "\n",
    "As funções desta seção têm de ser decoradas para que sejam aplicadas futuramente. Mas uma dica para decorar essas integrais trigonométricas básicas pode ser a da análise do ciclo trigonométrico.\n",
    "\n",
    "![](Imagens/4.jpg)\n",
    "\n",
    "Como vocês podem ver, o ciclo já simplifica muito esse trabalho para nós. Então, analisando o mesmo, podemos perceber que o percorrendo em sentido horário, a função trigonométrica é derivada, em contrapartida, se percorermos o ciclo em sentido anti-horário, a função é integrada. Daí, podemos confirmar as equações (1) e (2), assim como definir que as integrais de $-cos(x)$ e $-sen(x)$ são:\n",
    "\n",
    "$$\\int -cos(x)dx = -sen(x)$$\n",
    "$$\\int -sen(x)dx = cos(x)$$\n",
    "\n",
    "Analisemos mais algumas integrais trigonométricas:\n",
    "\n",
    "$$\\int tg(x)dx = ln|sec(x)|$$\n",
    "$$\\int sec(x)dx = ln|sec(x)+tg(x)|$$\n",
    "$$\\int cossec(x)dx = -ln|cossec(x)+cotg(x)|$$\n",
    "$$\\int sec^2(x)dx = tg(x)$$\n",
    "$$\\int cossec^2(x)dx = -cotg(x)$$\n",
    "$$\\int sec(x)tg(x)dx = sec(x)$$\n",
    "$$\\int cossec(x)cotg(x)dx = -cossec(x)$$\n",
    "\n",
    "Agora basta guardar essas integrais com você que o resto do assunto vai ser tranquilo.\n",
    "\n",
    "Outras integrais que serão super necessárias com o decorrer do assunto são as exponenciais. Mas elas são simples de se aprender. Veja só!\n",
    "\n",
    "$$\\int e^xdx = e^x$$\n",
    "$$\\int a^xdx = \\frac{a^x}{ln|a|}$$\n",
    "\n",
    "### B. Integração por Substituição\n",
    "\n",
    "A partir desse ponto, o cálculo de integrais começa a se complicar um pouco mais. Isso porque funções mais complexas e elaboradas não podem ser integradas utilizando apenas a técnica base e as propriedades de integração.\n",
    "\n",
    "Vamos supor uma função bem simples, $f(x) = e^{x+2}$, você saberia me dizer qual a sua integral?  \n",
    "Para resolvê-la, utilizamos uma técnica chamada Integração por substituição, que obedece os seguintes passos:\n",
    " \n",
    "1. Tome uma parte da função como sendo uma variável qualquer (convencionamente chamamos de $'u'$).\n",
    "2. Derive essa parte e veja se essa derivada está na função integranda. \n",
    "3. Caso a mesma esteja, substitua a variável e a derivada na equação.\n",
    "4. Verifique se não ficou nenhum termo da variável de integração original na função. Caso isso ocorra, integre a função.\n",
    "5. Substitua a variável utilizada pelo seu valor na função original.\n",
    "\n",
    "Vamos aplicar essa técnica em $f(x)$.\n",
    " \n",
    "$$\\int e^{x+2}dx = ?$$\n",
    "\n",
    "Tomemos $u$ como sendo $x+2$. \n",
    "\n",
    "Fazendo alguns cálculos vemos que a sua derivada $du$ é igual à $dx$, que claramente está na função integranda.\n",
    "\n",
    "Agora o que resta é substituir $u$ e $du$ na função e verificar a existência de termos que não puderam ser substituídos.\n",
    "\n",
    "Como $u = x+2$ e $du = dx$, temos: $$\\int e^udu = \\,?$$\n",
    "\n",
    "Agora basta aplicar a propriedade de integração de função exponencial que vimos anteriormente, e depois substituir de volta a variável $u$. Ou seja,\n",
    "\n",
    "$$\\int e^udu = e^u = e^{x+2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle e^{x + 2}$"
      ],
      "text/plain": [
       " x + 2\n",
       "ℯ     "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (exp(x+2))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### C. Integração por Partes\n",
    "\n",
    "O método de integração por partes é uma técnica que é geralmente utilizada quando, na função a ser integrada temos uma multiplicação de funções de tipos diferentes.\n",
    "\n",
    "Poderíamos até provar esse método à partir de derivadas de um produto de funções, mas o escopo deste livro não se aprofunda tanto nesse sentido.\n",
    "\n",
    "Sem mais delongas, a integração por partes afirma o seguinte:\n",
    "\n",
    "$$\\int f(x)g(x)dx = u.v - \\int v du$$\n",
    "\n",
    "Tal que ***u*** é uma das funções e ***du*** é sua derivada, e ***dv*** é a outra função, onde ***v*** é sua integral.\n",
    "\n",
    "Aí você deve estar se perguntando, como eu sei escolher qual das funções será \n",
    "***u*** e qual será ***dv***?\n",
    "\n",
    "Então, muitos materiais que abordam o cálculo de integrais utilizam um padrão chamado ***LIATE***, que basicamente você deve seguir para definir quem será cada uma das funções.\n",
    "\n",
    "Esse padrão segue uma sequência, onde o valor de u se dá de acordo com a função que aparecer.\n",
    "\n",
    "LIATE significa:\n",
    "\n",
    "1. L - Função logarítmica\n",
    "2. I - Função inversa trigonométrica\n",
    "3. A - Função aritmética\n",
    "4. T - Função trigonométrica\n",
    "5. E - Função exponencial\n",
    "\n",
    "Então, para se definir o valor de **u**, você deve perceber qual das funções aparecem primeiro, seguindo essa sequência, e então atribuí-lo.\n",
    "\n",
    "**Obs.** Nem sempre essa sequência dá certo, então é sempre bom ficar de olho quando for utilizar esse método.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Calcule a integral de $f(x) = x.e^x$.\n",
    "\n",
    "Primeiro, notemos que f(x) é composta por um produto de duas funções, uma aritmética e outra exponencial. Então nesse caso, utilizaremos a integração por partes. Seguindo a sequência LIATE, temos que a variável u será igual à x, já que a função aritmética aparece antes da exponencial. \n",
    "\n",
    "Substituindo os valores e calculando, temos:\n",
    "\n",
    "$$\\int x.e^xdx \\rightarrow\\left\\{ \\begin{array}{c}\n",
    "u = x, du = dx\\\\\n",
    "dv = e^xdx, v = e^x\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Seguindo a fórmula, temos:\n",
    "\n",
    "$$uv - \\int vdu = xe^x - \\int e^xdx = xe^x - e^x$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x - 1\\right) e^{x}$"
      ],
      "text/plain": [
       "         x\n",
       "(x - 1)⋅ℯ "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x*exp(x))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Calcule a integral de $f(x) = \\frac{ln(x).x^3}{6}$.\n",
    "\n",
    "Vamos começar definindo u e dv.\n",
    "\n",
    "$$\\int \\frac{ln(x)x^3}{6}dx \\rightarrow\\left\\{ \\begin{array}{c}\n",
    "u = ln(x), du = \\frac{dx}{x}\\\\\n",
    "dv = x^3dx, v = \\frac{x^4}{4}\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "$$\\frac{1}{6}\\left[\\frac{x^4ln(x)}{4} - \\int \\frac{x^4dx}{4x}\\right] = \\frac{1}{6}\\left[\\frac{x^4ln(x)}{4} - \\frac{1}{4} \\int x^3dx\\right]$$ $$=\\frac{1}{6}\\left[\\frac{x^4ln(x)}{4} - \\frac{1}{4} \\frac{x^4}{4}\\right] = \\frac{1}{6}\\left[\\frac{x^4ln(x)}{4} - \\frac{x^4}{16}\\right].$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{4} \\log{\\left(x \\right)}}{24} - \\frac{x^{4}}{96}$"
      ],
      "text/plain": [
       " 4           4\n",
       "x ⋅log(x)   x \n",
       "───────── - ──\n",
       "    24      96"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return ((ln(x)*x**3)/6)\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### D. Integração por Substituição Trigonométrica\n",
    "\n",
    "Esse tipo de técnica é utilizada quando nos deparamos com funções que apresentam raízes em denominadores, nas quais as mesmas não podem ser resolvidas por substituição simples ou por partes.\n",
    "\n",
    "O método se baseia no seguinte:\n",
    "\n",
    "Quando em uma função a ser integrada aparecer funções com $\\sqrt{x^2+a^2}$, $\\sqrt{x^2-a^2}$ ou $\\sqrt{a^2-x^2}$, podemos fazer uma substituição da variável x pelas seguintes funções:\n",
    "\n",
    "\n",
    "Caso a função contenha $\\sqrt{x^2-a^2}$\n",
    "1. Caso $\\sqrt{x^2+a^2} \\rightarrow x = a.tg\\theta$\n",
    "2. Caso $\\sqrt{x^2-a^2} \\rightarrow x = a.sec\\theta$\n",
    "3. Caso $\\sqrt{a^2-x^2} \\rightarrow x = a.sen\\theta$ ou $a.cos\\theta$\n",
    " \n",
    "Após fazer essa subtituição basta calcular a integral normalmente, e daí substituir de volta as funções trigonométricas do resultado final pelo ciclo trigonométrico.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Calcule a integral de $f(x) = \\frac{dx}{\\sqrt{16+x^2}}$.\n",
    "\n",
    "Vamos fazer a substituição da variável x e da sua derivada dx.\n",
    "\n",
    "Como $x^2+4^2 \\rightarrow x = a.tg\\theta, a = 4$ \n",
    "$\\rightarrow\\left\\{ \\begin{array}{c}\n",
    "x = 4tg\\theta\\\\\n",
    "dx = 4sec^2\\theta d\\theta\\\\\n",
    "\\end{array}\n",
    "\\right.$\n",
    "\n",
    "Substituindo as variáveis temos:\n",
    "\n",
    "$$\\int \\frac{4sec^2\\theta d\\theta}{\\sqrt{16+(4tg\\theta)^2}} = 4\\int \\frac{sec^2\\theta d\\theta}{\\sqrt{16+16tg^2\\theta}} = 4\\int \\frac{sec^2\\theta d\\theta}{\\sqrt{16 (1+tg^2\\theta)}}$$\n",
    "\n",
    "E como sabemos pelas relações trigonométricas, $1+tg^2\\theta = sec^2\\theta$, nossa integral fica:\n",
    "\n",
    "$$4\\int \\frac{sec^2\\theta d\\theta}{\\sqrt{16 sec^2\\theta}} = 4\\int \\frac{sec^2\\theta d\\theta}{4 sec\\theta} = 4.\\frac{1}{4}\\int sec\\theta d\\theta = \\int sec\\theta = ln|tg\\theta+sec\\theta|$$\n",
    "\n",
    "Como $x=4tg\\theta \\rightarrow tg\\theta = \\frac{x}{4}$. E $\\theta = arctg(\\frac{x}{4})$, temos que:\n",
    "\n",
    "$\\int \\frac{dx}{\\sqrt{16+x^2}} = ln|\\frac{x}{4}+sec(arctg(\\frac{x}{4}))| = ln|\\frac{x}{4}+\\frac{\\sqrt{16+x^2}}{4}|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{asinh}{\\left(\\frac{x}{4} \\right)}$"
      ],
      "text/plain": [
       "     ⎛x⎞\n",
       "asinh⎜─⎟\n",
       "     ⎝4⎠"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (1/sqrt(16+x**2))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### E. Cálculo de áreas entre curvas\n",
    "\n",
    "Agora chegamos em um ponto desse material em que colocamos em prática os conceito de integral para calcular de fato as áreas mencionadas anteriormente.\n",
    "\n",
    "Nesse caso, iremos calcular à princípio a área delimitada por uma curva e um eixo.\n",
    "\n",
    "Dada uma curva delimitada por uma função $f(x) = x^2$, como calcular a área entre essa função e o eixo X no intervalo $[-2, 2]$?\n",
    "Para isso basta integrar a função normalmente tal que os limites de integração são substituídos por -2 e 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{16}{3}$"
      ],
      "text/plain": [
       "16/3"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x**2)\n",
    "\n",
    "syp.integrate(f(x),(x,-2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs.** Esse conceito se aplica apenas para funções positivas. Quando a mesma for negativa, deve-se inverter o sinal da integral, ou seja:\n",
    "\n",
    "\n",
    "![](Imagens/5.jpg)\n",
    "\n",
    "**Obs.** Em alguns casos é faz-se necessário dividir a função para calculá-la em intervalos menores, para facilitar esse cálculo.\n",
    "\n",
    "### F. Cálculo de volume de discos e anéis\n",
    "\n",
    "Assim como área, podemos utilizar integrais para fazer o cálculo de volume de sólidos obtidos pela rotação de uma função em um eixo.\n",
    "\n",
    "![](Imagens/6.jpg)\n",
    "\n",
    "Devemos observar que essa função ao ser rotacionada gera infinitos discos, que ao serem somados geram áreas. A soma dessas áreas resulta no cálculo do volume do sólido.\n",
    "\n",
    "![](Imagens/7.jpg)\n",
    "\n",
    "Para calcular esse volume, basta calcular a integral da função $f(x)$ na forma:\n",
    "\n",
    "$$\\int^b_a \\pi[f(x)]^2dx$$\n",
    "\n",
    "**Obs.** Isso somente para a rotação em torno do eixo X. No caso de a rotação ser em torno de uma variável qualquer A no eixo Y, deve-se subtrair da função esse valor para calcular a integral. Ou seja,\n",
    "\n",
    "$$\\int^b_a \\pi[f(x)-A]^2dx$$\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Calcule o volume gerado pela rotação de $f(x) = x^2$ em torno do eixo X, entre 0 e 3.\n",
    "\n",
    "Vamos entao analisar o gráfico da função.\n",
    "![](Imagens/8.jpg)\n",
    "\n",
    "Limitando-a entre 0 e 3 temos:\n",
    "![](Imagens/9.jpg)\n",
    "\n",
    "Agora calculando a integral pela formula temos:\n",
    "\n",
    "$$V = \\int^3_0 \\pi[x^2]^2dx = \\int^3_0 \\pi x^4dx = \\pi \\int^3_0 x^4dx$$\n",
    "$$ = \\pi \\left[\\frac{x^5}{5}\\right]^3_0 = \\pi\\frac{3^5}{5} = \\pi\\frac{243}{5}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 152.681402964464$"
      ],
      "text/plain": [
       "152.681402964464"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x**2)\n",
    "\n",
    "syp.integrate(math.pi*(f(x)**2),(x,0,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Calcule o volume gerado pela rotação de $f(x) = 6+x^3$ em torno do eixo X, entre -1 e 2.\n",
    "\n",
    "Analisando o gráfico.\n",
    "\n",
    "![](Imagens/10.jpg)\n",
    "\n",
    "Limitando entre -1 e 2.\n",
    "![](Imagens/11.jpg)\n",
    "\n",
    "Rotacionando.\n",
    "![](Imagens/12.jpg)\n",
    "\n",
    "E por fim, calculando a integral.\n",
    "$$V = \\int^2_{-1} \\pi[6+x^3]^2dx = \\pi\\int^2_{-1} x^6+12x^3+36dx $$\n",
    "$$= \\pi\\left[\\frac{x^7}{7}+3x^4+36x\\right]^2_{-1} = \\pi\\left[\\frac{2^7}{7}+3.2^4+36.2\\right]$$\n",
    "$$-\\pi\\left[\\frac{(-1)^7}{7}+3(-1)^4+36.(-1)\\right] = \\pi\\left[\\frac{128}{7}+48+72\\right]$$\n",
    "$$- \\pi\\left[\\frac{-1}{7}+3-36\\right] = \\pi\\left[\\frac{128+336+504}{7}\\right]$$\n",
    "$$-\\pi\\left[\\frac{-1+21-252}{7}\\right] = \\pi\\left[\\frac{968}{7}+\\frac{232}{7}\\right] = \\pi\\frac{1200}{7}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 538.558740615393$"
      ],
      "text/plain": [
       "538.558740615393"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (6+x**3)\n",
    "\n",
    "syp.integrate(math.pi*(f(x)**2),(x,-1,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quando temos duas funções e queremos calcular o volume do sólido de revolução entre as mesmas, podemos utilizar o mesmo princípio dos discos, subtraindo a função menor da maior. Ou seja,\n",
    "\n",
    "$$\\int^b_a (\\pi[f(x)]^2 - \\pi[g(x)]^2)dx$$\n",
    "\n",
    "Para o caso de a rotação das funções não serem em torno do eixo X, basta subtrair das duas funções o valor da posição da reta no eixo Y. Ficando assim com:\n",
    "\n",
    "$$\\int^b_a (\\pi[f(x)-A]^2 - \\pi[g(x)-A]^2)dx$$\n",
    "\n",
    "Exemplo 3.\n",
    "\n",
    "Calcule o volume do sólido obtido pela rotação das funções $f(x) = x^2+4$ e $g(x) = x+2$ em torno do eixo X, entre 0 e 5.\n",
    "\n",
    "Analisando o gráfico das funções.\n",
    "![](Imagens/13.jpg)\n",
    "\n",
    "Rotacionando.\n",
    "![](Imagens/14.jpg)\n",
    "\n",
    "E por fim, calculando a integral.\n",
    "\n",
    "$$ = \\int^9_0\\pi [x^2+4]^2-\\pi[x+2]^2dx $$\n",
    "$$= \\pi \\int^5_0 x^4+8x^2+16 - (x^2+4x+4)dx$$\n",
    "$$= \\pi \\int^5_0 x^4+7x^2+12dx = \\frac{x^5}{5} + \\frac{7x^3}{3} +12x$$\n",
    "$$= \\pi \\left[\\frac{x^5}{5} + \\frac{7x^3}{3} + 12x\\right]^5_0 $$\n",
    "$$ =  \\pi \\left[\\frac{5^5}{5} + \\frac{7.5^3}{3} + 12.5]\\right]$$\n",
    "$$= \\pi \\left[625+\\frac{875}{3} -50+60\\right] = \\pi\\left[\\frac{1875+875+30}{3}\\right]$$\n",
    "$$\\pi\\frac{2780}{3}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2911.20919232654$"
      ],
      "text/plain": [
       "2911.20919232654"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x**2+4)\n",
    "\n",
    "def g(x):\n",
    "    return (x+2)\n",
    "\n",
    "syp.integrate(math.pi*(f(x)**2-g(x)**2),(x,0,5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### G. Cálculo de volume com cascas cilíndricas\n",
    "\n",
    "É um método utilizado para calcular o volume dos sólidos de revolução que são sólidos gerados pela rotação de uma figura em torno de um eixo. Esse método é bastante semelhante aos que vimos anteriormente, porém nesse caso, calculamos utilizando a forma:\n",
    "\n",
    "$$\\int^b_a 2\\pi xf(x)dx$$\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Calcule o volume obtido pela rotação da figura delimitada por $f(x) = x^2 + 2x + 1$ em torno do eixo X entre -1 e 3.\n",
    "\n",
    "Analisando o gráfico.\n",
    "![](Imagens/15.jpg)\n",
    "\n",
    "Rotacionando.\n",
    "![](Imagens/16.jpg)\n",
    "\n",
    "E por fim, calculando a integral.\n",
    "\n",
    "$$V = \\int^3_{-1}2\\pi x(x^2+2x+1) = 2\\pi \\int^3_{-1}x^3+2x^2+xdx $$\n",
    "$$= 2\\pi\\left[\\frac{x^4}{4}+\\frac{2x^3}{3}+\\frac{x^2}{2}\\right]^3_{-1} = 2\\pi\\left[\\frac{3^4}{4}+\\frac{2.3^3}{3}+\\frac{3^2}{2}\\right]$$\n",
    "$$-2\\pi\\left[\\frac{(-1)^4}{4}+\\frac{2(-1)^3}{3}+\\frac{(-1)^2}{2}\\right] $$\n",
    "$$= 2\\pi\\left[\\frac{81}{4}+18+\\frac{9}{2}-\\frac{1}{4}+\\frac{2}{3}-\\frac{1}{2}\\right]$$\n",
    "$$=2\\pi\\left[\\frac{80}{4}+\\frac{8}{2}+\\frac{2}{3}+18\\right] $$\n",
    "$$= 2\\pi\\left[42+\\frac{2}{3}\\right]=2\\pi \\frac{126+2}{3} = 2\\pi\\frac{128}{3}=\\pi\\frac{256}{3}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 268.082573106329$"
      ],
      "text/plain": [
       "268.082573106329"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x**2+2*x+1)\n",
    "\n",
    "syp.integrate(2*math.pi*x*f(x),(x,-1,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### H. Frações Parciais\n",
    "\n",
    "Em alguns casos, as funções que iremos integrar podem ter divisões de polinômions na qual o grau do numerador é menor que o do denominador, fazendo com que o cálculo dessas integrais sejam bastante complexos. Para se resolver isso, utilizamos esse método que busca simplificar bastante as funções a serem integradas, dividindo-as assim e somas de frações com polinômios com graus mais baixos.\n",
    "\n",
    "Existem quatro casos que devemos seguir para se calcular corretamente as frações parciais.\n",
    "\n",
    "**1º Caso:** Fatores do denominador lineares e de raízes distintas.\n",
    "\n",
    "Quando ocorrerem funções desse tipo, basta simplificar o polinômio pelas suas raízes e igualar a função à soma de uma constante dividida pelo primeiro monômio com uma outra constante dividida pelo outro monômio.\n",
    "\n",
    "E por fim, basta calcular o sistema, substituir A e B na função, e calcular a integral da soma dessas funções.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Para calcular a integral de $f(x) = \\frac{x+3}{x^2-4x+3}$ devemos desmembrar a mesma em frações parciais. Para isso, façamos:\n",
    "\n",
    "$$\\frac{x+3}{x^2-4x+3} = \\frac{A}{x-3}+\\frac{B}{x-1}$$\n",
    "\n",
    "Agora vamos tirar o MMC do lado direito da equação:\n",
    "\n",
    "$$\\frac{x+3}{x^2-4x+3} = \\frac{(x-1)A+(x-3)B}{(x-3)(x-1)}$$\n",
    "\n",
    "Podemos então simplificar a equação tirando os denominadores das duas frações, já que ambos são iguais.\n",
    "\n",
    "$$x+3 = (x-1)A+(x-3)B\\rightarrow x+3 = Ax-A+Bx-3B$$\n",
    "\n",
    "Agora basta montar um sistema com essas equações, resolver e substituir tudo na equação final.\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "Ax+Bx = x\\\\\n",
    "-A-3B = 3\\\\\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "\n",
    "Pela primeira equação temos:\n",
    "$$x(A+B) = x \\rightarrow A+B=1 \\rightarrow A = 1-B$$\n",
    "\n",
    "Substituindo na segunda equação:\n",
    "\n",
    "$$-(1-B)-3B = 3 \\rightarrow -1+B-3B=3 \\rightarrow-2B = 2 \\rightarrow B = -2$$\n",
    "\n",
    "Colocando o valor de B na primeira equação chegamos a:\n",
    "\n",
    "$$A = 1-(-2) = 3$$\n",
    "\n",
    "Tendo os valores de A e de B podemos então substituir na equação que nós montamos lá em cima.\n",
    "\n",
    "$$f(x) = \\frac{A}{x-3}+\\frac{B}{x-1} \\rightarrow \\frac{3}{x-3}+\\frac{-2}{x-1}$$\n",
    "\n",
    "E enfim, podemos calcular nossa integral de maneira mais simples.\n",
    "\n",
    "$$\\int\\frac{3}{x-3}+\\frac{-2}{x-1} = 3\\int\\frac{dx}{x-3}-2\\int\\frac{dx}{x-1}$$\n",
    "\n",
    "Fazendo duas substituição simples temos:\n",
    "\n",
    "$$u = x-3, du = dx$$ e $$v = x-1, dv = dx$$\n",
    "\n",
    "$$3\\int\\frac{du}{u}-2\\int\\frac{dv}{v} = 3ln|u| -2 ln|v|+C$$\n",
    "\n",
    "E por fim, basta substituir de volta u e v.\n",
    "\n",
    "$$\\int\\frac{x+3}{x^2-4x+3} = 3ln|x-3| -2 ln|x-1|+C$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3 \\log{\\left(x - 3 \\right)} - 2 \\log{\\left(x - 1 \\right)}$"
      ],
      "text/plain": [
       "3⋅log(x - 3) - 2⋅log(x - 1)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return ((x+3)/(x**2-4*x+3))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**2º Caso:** Fatores do denominador lineares e de raízes iguais e múltiplas.\n",
    "\n",
    "Esse caso é bem semelhante ao anterior, porém ao invés de separarmos as raízes vamos repetir uma delas como denominador de A e a mesma elevada ao quadrado como denominador de B. E por fim calcular da mesma forma vista anteriormente.\n",
    "\n",
    "Exemplo 2.\n",
    "\n",
    "Para calcular a integral da função $$f(x) = \\frac{2x+5}{x^2-6x+9}$$ podemos fazer:\n",
    "\n",
    "$$\\frac{2x+5}{x^2-6x+9} = \\frac{2x+5}{(x-3)^2} = \\frac{A}{x-3} + \\frac{B}{(x-3)^2}$$\n",
    "\n",
    "$$2x+5 = (x-3)A + B$$\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "Ax = 2x \\\\\n",
    "-3A+B = 5\\\\\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "\n",
    "Resolvendo o sistema temos:\n",
    "\n",
    "$$A = 2$$ e $$-3A+B = 5 \\rightarrow -3.2+B = 5 \\rightarrow B = 11$$\n",
    "\n",
    "$$\\frac{2x+5}{x^2-6x+9} = \\frac{2}{x-3} + \\frac{11}{(x-3)^2}$$\n",
    "\n",
    "$$\\int \\frac{2x+5}{x^2-6x+9} = 2\\int \\frac{dx}{x-3} + 11\\int \\frac{dx}{(x-3)^2}$$\n",
    "\n",
    "$$u = x-3, du = dx$$\n",
    "\n",
    "$$2\\int \\frac{dx}{u} + 11\\int \\frac{dx}{u^2}$$\n",
    "\n",
    "$$\\int \\frac{2x+5}{x^2-6x+9} = 2ln|x-3|-\\frac{11}{x-3}+C$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAAfCAYAAACcRu1HAAAABHNCSVQICAgIfAhkiAAABiZJREFUeJztmnuIVUUcxz+7a5svWDNLC8xL6YahZYlBpbaoGRYSkUQFpWlUoJWBkhrVgpVrCCVlpVasUplhJEkPH0ml9rB8pJUV+SxdN80yM99tf3znsOeeO3Mfu/eec1fOBw733t/85szvzJ3zm9/8ZiAmJibmNGUg8B6wG2gARjVRJyOlTTIvJgbaA98BDwFHmqETExMK/5DZS2ajY6U5njSBXHhtM+6RD+YBvwPtIrYj3/RF/XtP1IZEjX+Qno065F3gF+SeDwKrgTEUZ2jQD7gTqAEOR2xLJqYDHwO/or49AGwAnkB9H2QdsBiYiqbNGOB+9ObuAd4ApgGvAX8Z+SKgxKefIHpPugzZ1yZCG7LlOPAl6tMa4Hnga9SHu4GuljpXmvIpIdnYVAo63fsZBAwn1WN2AXahzrrFJ08Q7SCtBP4D5kTUfq60dsifQv34oqN8C7CT4pzJPEKLSVcCS9Af72cv8LL5XpXDvW8FPkMhwxFgMzAZONOiW4JWgD8AR5FneQGoAHaYK8hoU2+ho/1lpL5YXlu1pqwmu0fJC0cd8rfNZw9H+VvABcB1ebeohZDt23nCfJ7MUv9pNHh6Am+iAVdi5EuB8oD+LOA5NCjnAAuAocBy4AxHG0OAU2gKtTERvXBTgTKffAYw0rQzKcvnKSTDzecmR/ka81lsg7Q90MdcpehF6mM+c9HJC62QF2wArvfJE9in+6uMfBcKFfz3WUJqjDXAyH4COvjk5cgTN5DqSduhF2ZzBttrSU4iTzG/FxLd9DkBqAaeBVYZe74FznHoVxidtWEYlwNVyK7gVZujTl6YYW78fkCecDQ418jvtdyrEnm/bT7ZK0b/Lov+NdgHaaWRL8tge1cUamwHxpk6H5HqycNkL8l/2IdA5wx1jph6MRYeRB25BegYKEtgH6TrjLy74547TXlFQP9Ci24ZCjV2BOSet3bFo36m0Tgg1gBts6izA7sHcF2vZ3HPIJ2Bm9EMsge4Io3ubjKHWmHYHAmt0pSNA2aixcxglNfLBm/w1TnK61BM0gEtqjz9eovuKeAPi9zbYnOtmP3s830fA/ybRZ2tuBc6NvbkoOtRj3LS64GfgflAL4duGzJvK4Zhc1ExHr1tm4FzHToJ0nvSixz1PE/qxZ/ryd2Tnm/qrHa04XEHWjzVGf2XMuhHxQZkXydLWSl6hq0h2pOLRw7jSuERU7ABe6d5JLAPUi/GHGOp053UmPRVco9JS9BW6L5gBR83oAT6JrQo+REN+IvT1ImKevScZ1nKepqyd0K1qIh5DHXIN6TGoEES2Afp1Ua+neQVaxna5msAHvXJr6VxdV/hk5cDn2IfpKAdMFfs2x9N69uA84xshNFfnOaZCkUlyc/mUUpjMn+NpRzgblM+rjCmtSxGos44idIj1ZZrlE8/gTudMN2U1aMc6DM0prFWkbq6nm3KfkNx8Aw0aNeiRcM2Urnd1BkbkPdBW6V1pIYc3jbkAMv9Csl4FFMuR/lZb8t5q7GnDrjEUXcB+k9s26YtnbFopvvbXF8AN6arUE3m2OATn36C9Dmv21DMeAgF9N8jD2pb7JQCD6Mp+RgK6mch73MI2GipU45egq98su4oVfMncKmlzhBjs2sDoFD0QhsaG4H9aNAdRC9NNe5ZqwIN7ii8fxjcBAxD/1slmlVOYP/vipYeaFAtcJRPNuWXh2ZRuDyAnq9/BG2PQA6jm082E3n/THnd5nAAuK+A928yXUjdAWqLNhAa0DkAG61RtmBJ4UyLjDZoRlkUUfslaG0y1/yegBarrjMGzaUMzb7Hgd4FaqNZ1KDBNs98r0XnLhuAD0g+IhhkIDqTebodeu6JQoFEhDYMRdPvJBQz9itAG73RCamTaC2RNiaNksFoe7AOTTGHUQpsIu4DJjHh8DkaQMPS6DxJ5vVMlaNuOYpJ+6IF5X4CmxrpPFRMzCAUSrVGXnS9Q68T6XPqoANH2ez2rUCzqi3PHhOTxGUoAzEabSQsDandlbSgcwUx0dENLdoeN797oa3Zqjy3U4Ny1gkUm04z7aQLLWJi6IhOvc0OyBeiZHs+qUVT+zGUOVhB8pnlmJiYmJiYmJiYmKLgf7Df6y9xfpNTAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 2 \\log{\\left(x - 3 \\right)} - \\frac{11}{x - 3}$"
      ],
      "text/plain": [
       "                 11 \n",
       "2⋅log(x - 3) - ─────\n",
       "               x - 3"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return ((2*x+5)/(x**2-6*x+9))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3º Caso:** Fatores do denominador lineares e quadráticos irredutíveis.\n",
    "\n",
    "Quando isso acontecer, basta colocar a constante A dividindo o menor fator e um monômio dividindo o polinômio, e então calcular a partir do sistema linear.\n",
    "\n",
    "Exemplo 3.\n",
    "\n",
    "Vamos calcular a integral de $f(x) = \\frac{-3x+7}{x^3-4x^2+5x}$.\n",
    "\n",
    "Desmembrando em frações parciais temos:\n",
    "\n",
    "$$\\frac{-3x+7}{x^3-4x^2+5x} = \\frac{-3x+7}{x(x^2-4x+5} = \\frac{A}{x} + \\frac{Bx+C}{x^2-4x+5}$$\n",
    "\n",
    "$$-3x+7 = A(x^2-4x+5)+x(Bx+C)$$\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "Ax^2+Bx^2 = 0\\\\\n",
    "-4Ax+Cx = -3x\\\\\n",
    "5A = 7\\\\\n",
    "\\end{array}\n",
    "\\right.\n",
    "$$\n",
    "\n",
    "Resolvendo o mesmo chegamos a $A = \\frac{7}{5}, B = -\\frac{7}{5}$ e $C = \\frac{13}{5}$.\n",
    "\n",
    "Substuindo tudo na equação equivalente temos:\n",
    "\n",
    "$$\\int \\frac{-3x+7}{x^3-4x^2+5x} = \\int\\frac{7dx}{5x} + \\int \\frac{-\\frac{7}{5}x + \\frac{13}{5}}{x^2+4x+5}dx$$\n",
    "\n",
    "$$= \\int\\frac{7dx}{5x} + \\int \\frac{-7x + 13}{5(x^2+4x+5)}dx$$\n",
    "\n",
    "Vamos tentar simplificar o polinômio completanto os quadrados. Somando 1 e subtraindo 1 ao mesmo, podemos fazer:\n",
    "\n",
    "$$= \\int\\frac{7dx}{5x} + \\frac{1}{5}\\int \\frac{-7x + 13}{x^2+4x+5+1-1}dx$$\n",
    "$$= \\int\\frac{7dx}{5x} + \\frac{1}{5}\\int \\frac{-7x + 13}{x^2+4x+4+1}dx$$\n",
    "\n",
    "$$= \\int\\frac{7dx}{5x} + \\frac{1}{5}\\int \\frac{-7x + 13}{(x-2)^2+1}dx$$\n",
    "\n",
    "Agora vamos integrar a primeira parte:\n",
    "\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\int \\frac{-7x + 13}{(x-2)^2+1}dx$$\n",
    "\n",
    "Fazemos então uma substituição $u = x-2$, $du = dx$ e manipulando u, encontramos x como $x = u+2$.\n",
    "\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\int \\frac{-7(u+2) + 13}{u^2+1}du$$\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\int \\frac{-7u-14 + 13}{u^2+1}du$$\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\int \\frac{-7u -1}{u^2+1}du$$\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\left[\\int \\frac{-7u}{u^2+1}du - \\int \\frac{du}{u^2+1}du \\right]$$\n",
    "\n",
    "Substituindo $v = u^2+1, dv = 2udu$ chegamos a:\n",
    "\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\left[\\int \\frac{-7u}{u^2+1}du - \\int \\frac{du}{u^2+1}du \\right]$$\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\left[-\\frac{7}{2}\\int \\frac{dv}{v} - \\int \\frac{du}{u^2+1} \\right]$$\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\left[-\\frac{7}{2}ln|v| - arctg|u| \\right]+C$$\n",
    "\n",
    "Agora basta substituirmos v e u de volta:\n",
    "\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\left[-\\frac{7}{2}ln|u^2+1| - arctg|u| \\right]+C$$\n",
    "$$= \\frac{7ln|x|}{5} + \\frac{1}{5}\\left[-\\frac{7}{2}ln|(x-2)^2+1| - arctg|(x-2)| \\right]+C$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{7 \\log{\\left(x \\right)}}{5} - \\frac{7 \\log{\\left(x^{2} - 4 x + 5 \\right)}}{10} - \\frac{\\operatorname{atan}{\\left(x - 2 \\right)}}{5}$"
      ],
      "text/plain": [
       "                ⎛ 2          ⎞              \n",
       "7⋅log(x)   7⋅log⎝x  - 4⋅x + 5⎠   atan(x - 2)\n",
       "──────── - ─────────────────── - ───────────\n",
       "   5                10                5     "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return ((-3*x+7)/(x**3-4*x**2+5*x))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**4º Caso:** Fatores do denominador lineares e quadráticos tais que os quadráticos se repetem.\n",
    "\n",
    "Esse caso é praticamente uma mistura dos casos 2 e 3. Onde colocamos o menor fator no denominado de A e colocamos cada o polinômio no denominador de um monômio aumentando as potências do mesmo em cada membro até a potência ser igual à do polinômio inicial.\n",
    "\n",
    "Como por exemplo, para se calcular a integral da função $f(x) = \\frac{-3x+7}{x(x^2-4x+5)^2}$ teríamos que fazer:\n",
    "\n",
    "$$\\frac{-3x+7}{x(x^2-4x+5)^2} = \\frac{A}{x} + \\frac{Bx + C}{x^2-4x+5} + \\frac{Dx + E}{(x^2-4x+5)^2}$$\n",
    "\n",
    "E então resolver utilizando o mesmo método utilizado até agora.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{x - 9}{10 x^{2} - 40 x + 50} + \\frac{7 \\log{\\left(x \\right)}}{25} - \\frac{7 \\log{\\left(x^{2} - 4 x + 5 \\right)}}{50} + \\frac{23 \\operatorname{atan}{\\left(x - 2 \\right)}}{50}$"
      ],
      "text/plain": [
       "                                      ⎛ 2          ⎞                 \n",
       "        x - 9         7⋅log(x)   7⋅log⎝x  - 4⋅x + 5⎠   23⋅atan(x - 2)\n",
       "- ───────────────── + ──────── - ─────────────────── + ──────────────\n",
       "      2                  25               50                 50      \n",
       "  10⋅x  - 40⋅x + 50                                                  "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return ((-3*x+7)/(x*(x**2-4*x+5)**2))\n",
    "\n",
    "syp.integrate(f(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## III. Introdução às funções de várias variáveis\n",
    "\n",
    "### A. Conceitos iniciais\n",
    "Nas disciplinas de cálculo estudadas até esse momento só trabalhamos com o cálculo de funções de uma variável, porém, o estudo de cálculo diferencial e integral se extende um pouco mais em relação ao número de variáveis trabalhadas.\n",
    "\n",
    "\n",
    "**Definição -**\n",
    "Uma função de várias variáveis é simplesmente uma associação de mais de um valor à outro em um domínio maior que o conjunto dos números reais. Ou seja, antes tinhamos por exemplo uma função $f(x) = x^2$, onde f(x) é também chamado de y, sendo determinado no eixo das Ordenadas, mas agora iremos associar uma função $f(x, y)$ à um valor representada pela associação de valores de x e y, onde a função agora é chamada de z e representada pelo eixo das Cotas. Claro que isso apenas para funções de duas variáveis.\n",
    "\n",
    "Façamos um exemplo prático:\n",
    "\n",
    "Dada a função $f(x, y) = x^2 + 3xy - y^3$, vamos calcular $f(1,2)$.\n",
    "\n",
    "\n",
    "$$f(1, 2) = 1^2 + 3.1.2 - 2^3 = 1 + 6 - 8 = -1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAASCAYAAABM8m7ZAAAABHNCSVQICAgIfAhkiAAAAJVJREFUSInt1KENAjEUh/Ef5IZgAYbAcWEBBKMwADuwwAn2QLAADkVyCoUCgwNBFaLpXUpDyH3JSyte+4n+X/kTVtjigBue2JUQH4PsjlNJcY0pRpjHxFVm8T61cZxZnMwgbr0DkVq9E/sZrjMeHc5fcokXfS/qys+8cTFyfyDLUDAJ6wxN2F+xzuwEG/EpaL8hHYjyAqjXJ2KDRzE8AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(a, b):\n",
    "    return (a**2+3*a*b-b**3)\n",
    "\n",
    "f(1, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### B. Domínio\n",
    "\n",
    "Outro ponto importante no cálculo das funções de várias variáveis é determinar e esboçar o domínio dessas funções. Mas como podemos fazer isso?\n",
    "\n",
    "Pense o seguinte, uma função de duas ou n variáveis tem o domínio em $\\mathbb{R}^2$ ou $\\mathbb{R}^n$, respectivamente, exceto em três casos:\n",
    "\n",
    "1. Dada uma função na qual apresenta uma raiz quadrada, o seu interior não pode resultar em um número negativo.\n",
    "2. Se a função contém uma fração, o seu denominador tem que ser diferente de 0.\n",
    "3. Caso apareça um logaritmo natural na função, o seu parâmetro deve ser maior ou igual a 0.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine o domínio da função $f(x, y) = ln(x - 3y)$.\n",
    "\n",
    "Para definir o domínio façamos, $$x - 3y \\geq 0 → x \\geq 3y$$ então, $$D = \\{(x, y)|x \\geq 3y\\}$$\n",
    "\n",
    "Fazendo uma analise do gráfico do domínio de f(x, y) temos:\n",
    "\n",
    "![](Imagens/17.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x/3)\n",
    "p1 = plot(f(x),(x,-5,5),show=False)\n",
    "p2 = plot_implicit(y<=f(x),(x,-5,5),(y,-5,5),show=False, line_color='yellow')\n",
    "p1.extend(p2)\n",
    "p1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Determine o domínio da função $f(x, y) = \\frac{x^2}{y + 2}$.\n",
    "\n",
    "Para definir o domínio façamos, $$y + 2 \\neq 0 → y \\neq -2$$ então, $$D = \\{(x, y)|y \\neq -2\\}$$\n",
    "\n",
    "Fazendo uma analise do gráfico do domínio de f(x, y) temos:\n",
    "\n",
    "![](Imagens/18.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (-2)\n",
    "\n",
    "p1 = syp.plot(f(x),(x,-5,5),show=False)\n",
    "p2 = syp.plot_implicit(y<f(x),(x,-5,5),(y,-5,5),show=False, line_color='yellow')\n",
    "p3 = syp.plot_implicit(y>f(x),(x,-5,5),(y,-5,5),show=False, line_color='yellow')\n",
    "p1.extend(p2)\n",
    "p1.extend(p3)\n",
    "p1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 3.\n",
    "\n",
    "Determine o domínio da função $f(x, y) = x - 2y + \\sqrt{x + y^2}$.\n",
    "\n",
    "Para definir o domínio façamos, $$x + y^2 \\geq 0 → x \\geq -y^2$$ então, $$D = \\{(x, y)|x \\geq -y^2\\}$$\n",
    "\n",
    "Fazendo uma analise do gráfico do domínio de f(x, y) temos:\n",
    "\n",
    "![](Imagens/19.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (sqrt(-x))\n",
    "\n",
    "def g(x):\n",
    "    return (-sqrt(-x))\n",
    "\n",
    "p1 = syp.plot(f(x),(x,-5,5),show=False)\n",
    "p2 = syp.plot(g(x),(x,-5,5),show=False)\n",
    "p3 = syp.plot_implicit(y>=f(x),(x,-5,5),(y,-5,5),show=False, line_color='yellow')\n",
    "p4 = syp.plot_implicit(y>=g(x),(x,-5,5),(y,-5,5),show=False, line_color='yellow')\n",
    "p1.extend(p2)\n",
    "p1.extend(p3)\n",
    "p1.extend(p4)\n",
    "p1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### C. Imagem\n",
    "\n",
    "Aqui estudamos um pouco os valores das quais a variável depende pode assumir, ou seja, em uma função f(x, y), que representamos por z, queremos saber quais os valores máximo e mínimo de z. Mas como fazer isso?\n",
    "\n",
    "Nessa situação, devemos ficar atentos ao surgimento de expoentes pares, raízes quadradas e exponenciações, isso porque em ambos a função só consegue assumir valores positivos.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine a imagem da função $f(x, y) = e^x - 10$.\n",
    "\n",
    "Primeiramente, tomemos a função $e^x$ e, após uma análise, vemos que a mesma tem como imagem $[0, +\\infty]$, ou seja, $e^x \\geq 0$.\n",
    "\n",
    "Mas por outro lado, temos um -10 associado à função exponencial, o que extende o limite de f(x, y). Então podemos subtrair -10 dos dois lados da equação, tendo assim: $$e^x - 10 \\geq -10$$ ou seja, $$I = \\{z|-10 \\leq z \\leq +\\infty\\}$$.\n",
    "\n",
    "Fazendo uma analise do gráfico da imagem de f(x, y) temos:\n",
    "\n",
    "![](Imagens/20.jpg)\n",
    "\n",
    "Exemplo 2.\n",
    "\n",
    "Determine a imagem da função $f(x, y) = \\sqrt{2+x}$.\n",
    "\n",
    "Inicialmente, devemos observar que a função raiz quadrada também deve resultar em um número estritamente positivo, assim, já limitamos o intervalo de imagem para $[0, +\\infty]$, agora, analisemos o interior da raiz. Em $2+x$, podemos perceber que qualquer valor de x dentro do domínio da função é aceitável, ou seja, não há nenhum valor de $x \\geq 0$ que possa tornar inviável a operação da raiz. Assim sendo, temos como imageM de f(x, y): $$I = \\{z|0 \\leq z \\leq +\\infty\\}$$.\n",
    "\n",
    "Fazendo uma analise do gráfico da imagem de f(x, y) temos:\n",
    "\n",
    "![](Imagens/21.jpg)\n",
    "\n",
    "### D. Esboço de gráficos\n",
    "\n",
    "Agora iremos aprender algumas técnicas para esboçar o gráfico de funções de duas variáveis, tendo em vista que é o máximo que se pode ser representado geometricamente, já que o mesmo é esboçado no $\\mathbb{R}^3$.\n",
    "\n",
    "Para fazer esses esboços, quatro pontos que devem ser seguidos, e são os seguintes:\n",
    "\n",
    "Dada uma função f(x, y),\n",
    "\n",
    "1. Caso possível, manipule essa função para transformá-la em uma outra que o gráfico seja conhecido.\n",
    "2. Faça $x=0$ e tente plotar o gráfico entre os eixos y e z.\n",
    "3. Faça $y=0$ e tente plotar o gráfico entre os eixos x e z.    \\item Faça $z=0$ e tente plotar o gráfico entre os eixos x e y.\n",
    "4. Observe a função como um todo e repare na sua fórmula, associando-a a uma figura geométrica, e após isso, interpole entre os três eixos utilizando a figura observada.\n",
    "\n",
    "Façamos alguns exemplos.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Esboce o gráfico da função $f(x, y) = 2x + 4y + 8$.\n",
    "\n",
    "Primeiro, verificamos que não há muita coisa para se manipular na função, vamos zerar o x. Então, $$z = 4y + 8$$ vamos desenhar o gráfico nos eixos y e z.\n",
    "\n",
    "![](Imagens/22.jpg)\n",
    "\n",
    "Agora vamos zerar o y. $$z = 2x + 8$$ vamos desenhar o gráfico nos eixos x e z.\n",
    "\n",
    "![](Imagens/23.jpg)\n",
    "\n",
    "Zerando o z, temos: $$2x + 4y + 8 = 0 → 2x + 4y = -8$$ $$→ 4y = -2x - 8 → y = \\frac{-2x - 8}{4}$$ $$→ y = -\\frac{2x}{4} - \\frac{8}{4} → y = -\\frac{x}{2} - 2$$\n",
    "\n",
    "Vamos desenhar o gráfico nos eixos x e y.\n",
    "\n",
    "![](Imagens/24.jpg)\n",
    "\n",
    "E por fim, vamos analisar a função como um todo. Pela equação, podemos notar que a figura geométrica representada é um plano, então basta interpolar o desenho com a figura nos três eixos.\n",
    "\n",
    "![](Imagens/25.jpg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = axes3d.Axes3D(fig)\n",
    "X = np.arange(-4, 4, 0.25)\n",
    "Y = np.arange(-4, 4, 0.25)\n",
    "X , Y = np.meshgrid(X, Y)\n",
    "Z = 2*X+4*Y+8\n",
    "ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='hot')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### E. Curvas de nível\n",
    "\n",
    "Uma outra técnica muito utilizada para representar graficamente uma função é o esboço a partir das curvas de nível, que tem como fundamentos a interseção do gráfico da função com um plano determinado por uma constante. Ou seja, basicamente, deve-se igualar a função à uma constante k. $$f(x, y) = k$$\n",
    "A partir do desenho de várias curvas de nível podemos determinar o gráfico da função fazendo uma interpolação entre as mesmas.\n",
    "\n",
    "Vejamos um exemplo para um melhor entendimento:\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Ache as curvas de nível da função $f(x, y) = 2x + 4y + 3$ para $k \\in \\{1, 3, 4, 7\\}$.\n",
    "\n",
    "Pela definição das curvas de nível devemos igualar f(x, y) às constantes dadas. Então: \n",
    "\n",
    "Para $k = 1$, temos:\n",
    "$$2x + 4y + 3 = 1 → 2x + 4y = -2 → y = -\\frac{1}{2} -\\frac{x}{2}$$\n",
    "\n",
    "Para $k = 3$, temos:\n",
    "$$2x + 4y + 3 = 3 → 2x + 4y = 0 → y = -\\frac{x}{2}$$\n",
    "\n",
    "Para $k = 4$, temos:\n",
    "$$2x + 4y + 3 = 4 → 2x + 4y = 1 → y = \\frac{1}{4} -\\frac{x}{2}$$\n",
    "\n",
    "Para $k = 7$, temos:\n",
    "$$2x + 4y + 3 = 7 → 2x + 4y = 4 → y = 1 -\\frac{x}{2}$$\n",
    "\n",
    "A partir das curvas encontradas, podemos agora plotar a sua projeção no plano xy.\n",
    "\n",
    "![](Imagens/26.jpg)\n",
    "\n",
    "Agora, a ideia é desenhar o gráfico de f(x, y) a partir dessas curvas. Para isso, podemos analisar que as constantes k se referem à altura de do gráfico no eixo z, ou seja, $k = 1$ é a mesma coisa que $z = 1$. Então, fazendo a interpolação das curvas do $\\mathbb{R}^2$ temos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f1(x):\n",
    "    return (-1/2-x/2)\n",
    "\n",
    "def f2(x):\n",
    "    return (-x/2)\n",
    "\n",
    "def f3(x):\n",
    "    return (1/4-x/2)\n",
    "\n",
    "def f4(x):\n",
    "    return (1-x/2)\n",
    "\n",
    "p1 = syp.plot(f1(x),(x,-5,5),show=False, line_color='orange')\n",
    "p2 = syp.plot(f2(x),(x,-5,5),show=False, line_color='cyan')\n",
    "p3 = syp.plot(f3(x),(x,-5,5),show=False, line_color='lawngreen')\n",
    "p4 = syp.plot(f4(x),(x,-5,5),show=False, line_color='yellow')\n",
    "p1.extend(p2)\n",
    "p1.extend(p3)\n",
    "p1.extend(p4)\n",
    "p1.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### F. Funções de três variáveis\n",
    "\n",
    "Uma função de três variáveis segue o mesmo princípio das funções vistas até então, porém agora um valor chamado $w$ é associado à outros 3 valores, representados pelas variáveis x, y e z. Ou seja, $$f(x, y, z) = w$$.\n",
    "\n",
    "Quando falamos sobre funções de duas variáveis vimos que o seu gráfico é representado por uma superfície. No caso das de três variáveis, o gráfico dessas funções é formado por um sólido, e ao invés de desenhá-los à partir de curvas de nível, utilizaremos agora superfícies de nível.\n",
    "\n",
    "As outras características como cálculo do domínio, imagem também se aplicam à essas funções.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Ache o domínio da função $f(x, y, z) = \\frac{x+2}{z-1} + ln(x-2y)$.\n",
    "\n",
    "Partindo da mesma ideia de cáculo do domínio de funções de duas variáveis, temos que:\n",
    "\n",
    "$$z-1 \\neq 0 → z \\neq 1$$\n",
    "\n",
    "$$x-2y \\geq 0 → x \\geq 2y$$\n",
    "\n",
    "Então o domínio de $f$ fica definido como:\n",
    "\n",
    "$$D = \\{(x, y, z) \\in \\mathbb{R}^3|z \\neq 1, x \\geq 2y\\}$$\n",
    "\n",
    "E pode ser representado por:\n",
    "\n",
    "![](Imagens/27.jpg)\n",
    "\n",
    "Exemplo 2.\n",
    "\n",
    "Ache as curvas de nível da função $f(x, y, z) = (x-2)^2+(y-3)^2+z^2$, para $k \\in \\{4, 16, 25\\}$.\n",
    "\n",
    "Primeiro temos que analisar a equação da função. A partir daí, podemos perceber que a função representa uma esfera centrada em $(2, 3, 0)$.\n",
    "\n",
    "Calculando $f(x, y, z)$ para cada $k$, temos:\n",
    "\n",
    "$$(x-2)^2+(y-3)^2+z^2 = 4 = 2^2$$\n",
    "\n",
    "Que é representado por:\n",
    "\n",
    "![](Imagens/28.jpg)\n",
    "\n",
    "$$(x-2)^2+(y-3)^2+z^2 = 16 = 4^2$$\n",
    "\n",
    "Que é representado por:\n",
    "\n",
    "![](Imagens/29.jpg)\n",
    "\n",
    "$$(x-2)^2+(y-3)^2+z^2 = 25 = 5^2$$\n",
    "\n",
    "Que é representado por:\n",
    "\n",
    "![](Imagens/30.jpg)\n",
    "\n",
    "E por fim, temos respectivamente as superfícies esféricas de raio 2, 4 e 5, respectivamente.\n",
    "\n",
    "Que é representado por:\n",
    "\n",
    "![](Imagens/31.jpg)\n",
    "\n",
    "## IV. Limites e continuidade de funções de várias variáveis\n",
    "\n",
    "### A. Conceitos iniciais\n",
    "Agora iremos recaptular o estudo dos limites de uma função, visto lá no Cálculo I, com a intuição de ver o comportamento de uma função e se a mesma existe em um certo ponto do plano xy.\n",
    "\n",
    "**Definição -**\n",
    "Limite de uma função é basicamente um ponto para a qual a função está se aproximando de acordo com a aproximação das variáveis independentes em algum ponto do seu domínio.\n",
    "\n",
    "Então, dada uma função $f(x, y)$, o seu limite é calculado por $$\\lim_{(x, y) \\to (a, b)} f(x, y)$$\n",
    "\n",
    "onde $(a, b)$ é um ponto no domínio de $f(x, y)$, que nesse caso pertence ao $\\mathbb{R}^2$.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine o limite de $f(x, y) = \\frac{x^2}{x+y}$.\n",
    "\n",
    "Façamos $$\\lim_{(x, y) \\to (0, 1)} \\frac{x^2}{x+y} = \\frac{0^2}{0+1} = 0$$.\n",
    "\n",
    "### B. Verificando a existência de limites\n",
    "\n",
    "Há uma certa peculiaridade em determinar se limites de funções de duas variáveis existem, fazendo com que se torne mais complexo do que o que acontece com as funções de uma variável.\n",
    "\n",
    "Vamos relembrar como o limite de uma função de uma variável era determinado.\n",
    "\n",
    "A partir do domínio da função e um ponto 'a' que utilizávamos para fazer o teste, tentamos aproximar x para valores próximos de 'a', tanto pela esquerda quanto pela direita, e caso os mesmos sejam iguais, o limite existe naquele ponto.\n",
    "\n",
    "O problema com as funções de duas variáveis é que não temos somente duas direções para testar (esquerda ou direita). Nesse caso, temos uma quantidade infinita de direções, já que aumentamos uma dimensão do domínio.\n",
    "\n",
    "Que é representado por:\n",
    "\n",
    "![](Imagens/32.jpg)\n",
    "\n",
    "Devido às infinitas soluções, é mais fácil descobrir se o limite não existe do que o contrário. Nesse caso, basta fazer o teste por algumas direções, e caso um deles seja diferente, o limite não existe.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine a existência de $\\lim_{(x, y) \\to (0, 0)} \\frac{x}{x+y}$.\n",
    "\n",
    "Vamos testar o limite por alguns eixos.\n",
    "Primeiro, aproximando pelo eixo x, façamos somente y = 0.\n",
    "\n",
    "$$\\lim_{(x, y) \\to (x, 0)} \\frac{x}{x+y} = \\frac{x}{x} = 1$$\n",
    "\n",
    "Agora, aproximando pelo eixo y, façamos somente x = 0.\n",
    "\n",
    "$$\\lim_{(x, y) \\to (0, y)} \\frac{x}{x+y} = \\frac{0}{0+y} = 0$$\n",
    "\n",
    "Então, como o limite diverge em direções diferentes, o mesmo não existe.\n",
    "\n",
    "### C. Propriedades de limites\n",
    "\n",
    "Vamos relembrar um pouco das propriedades utilizadas no cálculo dos limites de funções de uma variável que também se aplicam à funções de 'n' variáveis.\n",
    "\n",
    "$$\\lim_{(x, y) \\to (a, b)} c = c$$\n",
    "$$\\lim_{(x, y) \\to (a, b)} (c.f(x, y)) = c .\\lim_{(x, y) \\to (a, b)} f(x, y)$$\n",
    "$$\\lim_{(x, y) \\to (a, b)} (f(x, y) \\pm g(x, y)) = \\lim_{(x, y) \\to (a, b)} f(x, y) \\pm \\lim_{(x, y) \\to (a, b)} g(x, y)$$\n",
    "$$\\lim_{(x, y) \\to (a, b)} (f(x, y).g(x, y)) = \\lim_{(x, y) \\to (a, b)} f(x, y).\\lim_{(x, y) \\to (a, b)} g(x, y)$$\n",
    "$$\\lim_{(x, y) \\to (a, b)} \\frac{f(x, y)}{g(x, y))} = \\frac{\\lim_{(x, y) \\to (a, b)} f(x, y)}{\\lim_{(x, y) \\to (a, b)} g(x, y)}$$\n",
    "$$\\lim_{(x, y) \\to (a, b)} f(g(x, y)) = f(\\lim_{(x, y) \\to (a, b)} g(x, y))$$\n",
    "\n",
    "E pelo teorema do sanduíche, temos:\n",
    "\n",
    "Se $$g(x, y) \\leq f(x, y) \\leq h(x, y)$$\n",
    "e $$\\lim_{(x, y) \\to (a, b)} g(x, y) = \\lim_{(x, y) \\to (a, b)} h(x, y) = L$$\n",
    "Então: $$\\lim_{(x, y) \\to (a, b)} f(x, y) = L$$\n",
    "\n",
    "### D. Continuidade\n",
    "\n",
    "A continuidade de limites de várias variáveis também não difere muito quando se comparada à de uma variável. Isso porque para descobrirmos isso, basta verificar se a função está definida em todos os pontos e se $$\\lim_{(x, y) \\to (a, b)} f(x, y) = f(a, b)$$.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine se $f(x, y) = \\frac{1}{x+y}$ é contínua.\n",
    "\n",
    "Nesse caso, já podemos observar de cara que essa função é descontínua para $(x, y) = (0, 0)$, tendo em vista que aplicando os pontos teríamos $$\\lim_{(x, y) \\to (0, 0) \\frac{1}{x+y} = \\frac{1}{0}}$$ o que não existe.\n",
    "\n",
    "### E. Diferenciabilidade\n",
    "\n",
    "Toda função diferenciável tem duas características essenciais:\n",
    "\n",
    "1. Ter derivadas parciais nesse ponto.\n",
    "2. Ser contínua no ponto de diferenciação.\n",
    "\n",
    "\n",
    "Então, para se calcular a mesma, basta verificar se: $\\frac{\\partial f}{\\partial x}(a, b)$ e $\\frac{\\partial f}{\\partial y}(a, b)$ existem, e se o limite extendido da continuidade é igual à 0.\n",
    "\n",
    "$\\lim_{(x, y) \\to (a, b)} \\frac{f(x, y) - f(a, b) + f_x(a, b)(x-a)+f_y(a, b)(y-b)}{\\sqrt{(x-a)²+(y-b)²}} = 0$\n",
    "\n",
    "## V. Derivadas Parciais\n",
    "\n",
    "**Conceitos iniciais**  \n",
    "As funções de várias variáveis têm uma técnica especial para calcular suas derivadas, isso por que, não sabemos ao certo qual será a variável independente utilizada para realizar a operação, já que no caso de uma função $f(x, y)$ temos duas funções a se considerar.\n",
    "\n",
    "Então, como calculamos a derivada de uma função de várias variáveis?\n",
    "\n",
    "A resposta é bastante simples, basta calcular uma derivada para cada variável da função, tomando as outras subsequentes como constantes. Ou seja, uma função $f(x, y)$ tem duas derivadas, uma em função da variável x e outra de y. Daí o nome derivadas parciais.\n",
    "\n",
    "### A. Notação\n",
    "\n",
    "Dada uma função $f(x, y)$, suas derivadas são:\n",
    "\n",
    "$$f_x(x, y) = f_x = \\frac{\\partial f}{\\partial x}$$\n",
    "$$f_y(x, y) = f_y = \\frac{\\partial f}{\\partial y}$$\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine as derivadas parciais de $f(x, y) = 5x^2+2y+3$.\n",
    "\n",
    "Primeiro, vamos calcular a derivada em função de x, tratando assim, y como uma constante. Daí temos: $$f_x = (5x^2+2y+3)' = 5.2x+0+0 = 10x$$\n",
    "\n",
    "Agora calculando em função de y, temos:\n",
    "$$f_y = (5x^2+2y+3)' = 0+2+0 = 2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 10 x$"
      ],
      "text/plain": [
       "10⋅x"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (5*x**2+2*y+3)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Determine as derivadas parciais de $f(x, y, z) = cos(x) + sen(y) + 2z$.\n",
    "\n",
    "Calculando a derivada em função de x, temos: $$f_x = (cos(x) + sen(y) + 2z)' = -sen(x) + 0 + 0$$\n",
    "\n",
    "em função de y: $$f_y = (cos(x) + sen(y) + 2z)' = 0 + cos(y) + 0$$\n",
    "\n",
    "e por fim, em função de z: $$f_z = (cos(x) + sen(y) + 2z)' = 0 + 0 + 2$$\n",
    "\n",
    "### B. Interpretação geométrica das derivadas parciais\n",
    "\n",
    "No estudo das derivadas de funções de uma variável vimos que a derivada representava a inclinação da reta tangente à curva em um determinado ponto determinado por $(x, f(x))$. As derivadas parciais têm o mesmo princípio, ou seja, a derivada parcial de uma função de duas variáveis em relação à x nos retorna a inclinação da reta tangente à superfície que representa $f(x, y)$ paralela ao eixo x. Mas e com relação ao y, o que fazemos?\n",
    "\n",
    "Dado que y virou uma constante, a sua posição vai nos informar em que lugar do eixo a reta tangente ao gráfico vai ficar, ou \n",
    "seja, a projeção da mesma fica paralela ao eixo x e posicionada na direção de y.\n",
    "\n",
    "Representação da derivada parcial na direção de x:\n",
    "\n",
    "![](Imagens/33.jpg)\n",
    "\n",
    "Representação da derivada parcial na direção de y:\n",
    "\n",
    "![](Imagens/34.jpg)\n",
    "\n",
    "### C. Derivadas de ordem superior\n",
    "\n",
    "Da mesma forma que nas derivadas convencionais podemos aplicar a derivação mais de uma vez, nas parciais também temos essa liberdade, porém, com a diferença de que podemos fazer uma mistura com as derivações em função de x e de y.\n",
    "As mesmas são representadas da seguinte forma:\n",
    "\n",
    "$$f_{xx}(x, y) = f_{xx} = \\frac{\\partial^2 f}{\\partial^2 x}$$\n",
    "$$f_{yy}(x, y) = f_{yy} = \\frac{\\partial^2 f}{\\partial^2 y}$$\n",
    "$$f_{xy}(x, y) = f_{xy} = \\frac{\\partial^2 f}{\\partial x \\partial y}$$\n",
    "$$f_{yx}(x, y) = f_{yx} = \\frac{\\partial^2 f}{\\partial y \\partial x}$$\n",
    "\n",
    "**$f_{xx} →$** A derivada segunda de $f(x, y)$ em função de x.\n",
    "\n",
    "**$f_{yy} →$** A derivada segunda de $f(x, y)$ em função de y.\n",
    "\n",
    "**$f_{xy} →$** A derivada segunda de $f(x, y)$ primeiro em função de x e depois em função de y.\n",
    "\n",
    "**$f_{yx} →$** A derivada segunda de $f(x, y)$ primeiro em função de y e depois em função de x.\n",
    "\n",
    "E por fim, essa notação se mantém para 'n' derivações de f(x, y).\n",
    "\n",
    "**Obs.** Uma técnica para saber se o seu cálculo das derivadas parciais está correto é verificar se as derivadas parciais mistas estão iguais, ou seja, $$\\frac{\\partial^2 f}{\\partial x \\partial y} = \\frac{\\partial^2 f}{\\partial y \\partial x}$$\n",
    "\n",
    "Isso sempre irá acontecer, exceto se a função for descontínua em algum ponto.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine $f_{xy}$ e $f_{yx}$ de $f(x, y) = 3x^6+5xy-4y^2$.\n",
    "\n",
    "Derivando f(x, y) em x e y respectivamente, temos:\n",
    "\n",
    "$$f_x = 3.6x^5+5y-0 = 18x^5+5y$$\n",
    "$$f_{xy} = 0 + 5 = 5$$\n",
    "\n",
    "\\noindent Agora em y e x:\n",
    "\n",
    "$$f_y = 0+5x-4.2y = 5x-8y$$\n",
    "$$f_{yx} = 5 - 0 = 5$$\n",
    "\n",
    "E podemos comprovar a igualdade das derivadas mistas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAaCAYAAACjFuKcAAAABHNCSVQICAgIfAhkiAAAA/1JREFUaIHt2FmoVVUYB/CfmhUWmM0DZYPRSFoRFqSeBoLqIUvrQUIOJD0VlYGERPmUGRZGUVovNwiyLBUxKA3CwtKC6sEGk/TSpI2aipqZt4dv2d1u9z2zntLzh80561vrG9Ze6/vWf2066KAOTENP7tnQzoD2Fw5rdwA1YjVKmfbfbYpjv+L/shi7HKTZkEX/XHs8nsH72CxKwstVbNyMJfge27EW83BVC+M8Gz9iHeamdjvQbd+S2bLSmc+MhzEcW8XLPb+K/gxMwW9YiF8xDLdgHCaqvpjVsBJlfIUTU4wf4KLk90DjD8wqkG9ttaNrcC76iRpdKTNOFrV7g3hJeTs9IktajaPxMyY3oFsWcZUa9N2dnv2CfJl6F2tEwNUwNOmvFC8nb2cLTijQW5Lsj8vJ+6Er9T1ewe9WfC42zX8Zk8VcHuyj/zz8ifdqMVZSOTOOTcbW4/hc3+iku6BAb7jIqC8wICN/MunMqRLXkcnnI1XGFaGs+cxYjzsxFfeJKjCgYOzI5GteH7aWCGJySS2OS6of4Pdjt8iMFzAdr2FHcpYvX3vQlWyXU3tqar9q32ydiTE4S0xwsSAXQ2uZRA5lzS9G0eG9NsWYxUBsw7cFdm5Pek/X6rikNjY1Fr/ngluDCRV0ThfMax3uSTpv4fCCsXMFk9qJH/AGLqxxDnmUNbcYj+JanIRBuBizxYbcJrI+i2XJ3ykZ2VH4Dj9hcK2OS6ovxhSRak8JujkIl+HtpPtEBd3pehdvedJtJboV7+K+nq4mfM1UXJYfS/LbMrIZ9q4K/6KZS18pGV5gb2bzCW7F1+Lwmq2YVf2S+X+X2FmtxCwck5ONELT7Jfuyos+a8DVbzHV0Tr48/Y7EfHFVeAAfphhqRknlzNizG+7to3++YtZElLDd4jDswfP1BNYEyporU31hcLK7IycfIua5LLXfEeTl0iIj+cOyHhyRfovoa1a+Mye/SZSEVYJJrMYkQfX+r7gy/eYrwEZ8icvFBrxOsMVP63VQUjkz7tD7GeC0XN+NYkdsx3EZ+dWiHK3Ve6iNT3YW1htgAyhrPDMuEIdvHmfqvZtNLeifk/q2iNI8pFaHY8Wu7RLspgffZGQzM2P7Y2kas1nUwBlYJBaiR/DwPRiBTaI0nZPz+3EaP6rWQBtEWeOLMU280DfxnJjr62LD9SR5ERucqJckTKrXYSXG0Z0bP1DcNVaIBdkl7hyLcUNm3DCRQRsVX3KuT/ZX1BNsAyhrfDHG4BXxjWwT/hI7fal44f360BuVfH5UYUwHBwiLxKF9RbsDOdQxQWTFs+0O5FDFGXgIL4pvd6u0/kLbQY24W2TDRvGd7tT2htNBBx100MHBiH8AEhEqG5FMJ2YAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 18 x^{5} + 5 y$"
      ],
      "text/plain": [
       "    5      \n",
       "18⋅x  + 5⋅y"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (3*x**6+5*x*y-4*y**2)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 5 x - 8 y$"
      ],
      "text/plain": [
       "5⋅x - 8⋅y"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAMNJREFUKJHF0LtKQ1EQheEvEhLQwtJOLNIIgm26gM8hPkgabYUUqcQ3SBNIdSq7PIKVIIRTBkQwJF5Acmm2zWGfGwhZsNkwMz9rzfAPSrHNefO/oWYEXGAYqa+KnNKyOAdlAzHF4rVxjVN84hlTrIvixY4wQy8PusUVTnCICzxigy9c1lljEBwndaBOgN7rQMcB+qH6ybvhn2Ub5ziKAGd4DU79bPMOSyR4wD3G+A5AglYW6mGEF3zgF294wg0aFVfZt3bMkzEsK/o3tgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 5$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fxy = syp.diff(fx, y)\n",
    "fyx = syp.diff(fy, x)\n",
    "fxy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 5$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fyx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Determine $f_{xxxx}$ da função $f(x, y) = x^2y^2 + cos(x) - 5y^3$.\n",
    "\n",
    "Derivando f(x, y) quatro vezes em função de x, temos:\n",
    "\n",
    "$$f_x = 2y^2x - sen(x) - 0$$\n",
    "$$f_{xx} = 2y^2 - cos(x)$$\n",
    "$$f_{xxx} = 0 + sen(x)$$\n",
    "$$f_{xxxx} = cos(x)$$\n",
    "\n",
    "### D. Plano Tangente\n",
    "\n",
    "Quando nós calculamos as derivadas parciais de uma função em um ponto (a, b) e fomos analisar a representação geométrica dessas primitivas, podemos notar que ambas resultaram em retas que faziam uma intersecção com o gráfico da função, onde sua projeção ficava paralela ao eixo da variável utilizada como dependente e o seu deslocamento no outro eixo variava de acordo com a posição dada no mesmo.\n",
    "\n",
    "O plano tangente utiliza dos mesmos princípios, só que nesse caso, utilizaremos as duas retas encontradas nas derivadas parciais, tal que as duas retas estão sobre esse plano.\n",
    "\n",
    "![](Imagens/35.jpg)\n",
    "\n",
    "Então, para determinar uma equação para esse plano tangente teremos que fazer uma associação da inclinação das retas encontradas como derivadas parciais, do ponto de intersecção entre essas retas e de uma equação de plano.\n",
    "\n",
    "Como todo plano em um certo ponto $(a, b, c)$ é formado pela equação, $$A(x-a) +B(x-b) +C(z-c) = 0$$ Basta fazermos algumas manipulações para colocar na equação do plano os requisitos mensionados antes.\n",
    "Então, dividindo essa equação por $-C$ ficamos com $- A/C(x-a) - B/C(x-b) + z-c = 0$. E por fim, $z-c=A/C(x-a) +B/C(x-b)$. E fazendo uma pequena observação, notamos que $A/C$ é a inclinação da reta tangente ao gráfico no eixo xz, assim como $B/C$ é a inclinação no eixo yz. Como essas inclinações podem ser calculadas a partir das derivadas parciais, ficamos então com $z-c = fx(a, b) (x-a) + fy(a, b) (y-b)$.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine a equação do plano tangente à função $f(x, y) = 6-3x^2-y^2$ no ponto $P_0(2, 1, 4)$.\n",
    "\n",
    "Para isso, basta aplicar cada parte da equação que nós encontramos separadamente. Então achando $$f_x = - 6x$$ e $$f_y=-2y$$\n",
    "\n",
    "Aplicando os pontos:\n",
    "$$f_x(2, 1) = - 6.2 = -12$$ e $$f_y(2, 1)=-2.1 = -2$$\n",
    "\n",
    "E agora basta aplicar isso na equação do plano.\n",
    "\n",
    "$$z-4 = -12(x-2) -2(y-1) \\rightarrow z-4 = -12x +24 -2y +2$$\n",
    "$$12x + 2y + z = 24+2+4 \\rightarrow 12x + 2y + z = 30$$\n",
    "\n",
    "\n",
    "### E. Aproximações lineares\n",
    "\n",
    "A partir dessa equação do plano temos uma função que podemos utilizar como uma aproximação de $f(x, y)$ no ponto utilizado.\n",
    "\n",
    "Para isso, basta fazermos: $$z-c = fx(a, b) (x-a) + fy(a, b) (y-b)$$\n",
    "Como sendo, $$z=c + fx(a, b) (x-a) + fy(a, b) (y-b)$$\n",
    "E como $c = f(a, b)$, temos:\n",
    "$$z= f(a, b) + fx(a, b) (x-a) + fy(a, b) (y-b)$$.\n",
    "\n",
    "E a partir daí, podemos fazer:\n",
    "$$L(x, y) = f(a, b) + fx(a, b) (x-a) + fy(a, b) (y-b)$$ na qual L é chamada de ***função linear*** e $f(x, y) \\approx L(x, y)$ é a ***aproximação linear*** de $f(x, y)$ em um ponto. \n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Dada a função $f(x, y) = 6-3x^2-y^2$ no ponto $P_0(2, 1, 4)$, determine a aproximação linear dessa função nos pontos $(1, 0)$, $(2, 3)$ e $(4, 2)$.\n",
    "\n",
    "Primeiro vamos calcular f(2, 1).\n",
    "\n",
    "$$f(2, 1) = 6-3.2^2-1^2 = 6-12-1 = -7$$\n",
    "\n",
    "Agora calcular $f_x$ e $f_y$ e aplicar nos pontos.\n",
    "\n",
    "$$f_x = -6x \\rightarrow f_x(2, 1) = -12$$\n",
    "$$f_y = -2y \\rightarrow f_y(2, 1) = -2$$\n",
    "\n",
    "E por fim, basta aplicar na equação da linearização, o que nos dá:\n",
    "\n",
    "$$L(x, y) = -7 -12(x-2) -2(y -1) = -7 -12x +24 -2y +2$$\n",
    "Então a linearização é: $L(x, y) = -12x -2y +19$.\n",
    "\n",
    "Para encontrar a aproximação linear de cada ponto basta aplicar cada par ordenado na função de linearização. Sendo assim, a aproximação de cada ponto fica:\n",
    "\n",
    "$$L(1, 0) = -12.1 -2.0 +19 = 7$$\n",
    "$$L(2, 3) = -12.2 -2.3+19 = -24-6+19 = -11$$\n",
    "$$L(4, 2) = -12.4 -2.2+19 = -48-4+19 = -33$$\n",
    "\n",
    "### F. Regra da Cadeia\n",
    "\n",
    "Como já sabemos, a regra da cadeia é uma técnica utilizada para se calcular as derivadas de funções compostas de funções de uma variável. No caso de funções de mais variáveis essa técnica também é utilizada, porém com uma pequena diferença, já que nessas funções as derivadas são calculadas parcialmente.\n",
    "\n",
    "Enfim, a fórmula utilizada para calcular a regra da cadeia é a seguinte:\n",
    "\n",
    "**1º CASO -** Funções compostas com apenas um parâmetro.\n",
    "\n",
    "Esse caso se refere à funções do tipo $z = f(g(t), h(t))$, e a derivada dessa função em relação à 't' é calculada por:\n",
    "\n",
    "$$\\frac{dz}{dt} = \\frac{\\partial f}{\\partial x}.\\frac{dx}{dt} + \\frac{\\partial f}{\\partial y}.\\frac{dy}{dt}$$\n",
    "\n",
    "**2º CASO -** Funções compostas com dois parâmetros.\n",
    "\n",
    "Esse caso se refere à funções do tipo $z = f(g(t, u), h(t, u))$, e a derivada dessa função em relação à 't' e 'u' é calculada por:\n",
    "\n",
    "$$\\frac{\\partial z}{\\partial t} = \\frac{\\partial f}{\\partial x}.\\frac{\\partial  x}{\\partial t} + \\frac{\\partial f}{\\partial y}.\\frac{\\partial y}{\\partial t}$$\n",
    "\n",
    "e\n",
    "$$\\frac{\\partial z}{\\partial u} = \\frac{\\partial f}{\\partial x}.\\frac{\\partial  x}{\\partial u} + \\frac{\\partial f}{\\partial y}.\\frac{\\partial y}{\\partial u}$$\n",
    "\n",
    "Aqui aparecem derivadas parciais devido às funções internas às compostas apresentarem duas variáveis.\n",
    "\n",
    "**CASO GENERALIZADO -** Funções compostas com N parâmetros.\n",
    "\n",
    "Observando os dois casos anteriores, podemos perceber uma certa semelhança entre as derivadas das duas funções dadas, por esse motivo podemos definir uma fórmula para se calcular as derivadas de qualquer função de composta de várias variáveis. \n",
    "\n",
    "Tais funções são calculadas da seguinte forma:\n",
    "\n",
    "Sendo $a = \\{a_1, a_2, ..., a_n\\}$ o conjunto com as variáveis da função dada e $t = \\{t_1, t_2, ..., t_v\\}$ o conjunto com os parâmetros de cada função.\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial t_i} = \\frac{\\partial f}{\\partial a_1}.\\frac{\\partial  a_1}{\\partial t_i} + \\frac{\\partial f}{\\partial a_2}.\\frac{\\partial  a_2}{\\partial t_i} + ... +\\frac{\\partial f}{\\partial a_n}.\\frac{\\partial a_n}{\\partial t_i}$$\n",
    "\n",
    "Para cada $i \\in [0, v]$.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Qual a derivada de $f(x, y) = 2x^3y+3xy^2$ tal que $x = x(t) = 2t^2, y = y(t) = t^3+2t$ e $t = 1$?\n",
    "\n",
    "Resolução:\n",
    "\n",
    "Primeiro vamos calcular x e y aplicando os pontos.\n",
    "\n",
    "$x(1) = 2$ e $y(1) = 1+2 = 3$\n",
    "\n",
    "Agora vamos calcular cada parte da equação separada e depois aplicar esse ponto.\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x} = 6x^2y+3y^2$$\n",
    "$$\\frac{\\partial f}{\\partial y} = 2x^3+6xy$$\n",
    "$$\\frac{dx}{dt} = 4t$$\n",
    "$$\\frac{dy}{dt}= 3t^2+2$$\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x}(2, 3) = 6.2^2.3+3.3^2=72+27=99$$\n",
    "$$\\frac{\\partial f}{\\partial y}(2, 3) = 2.2^3+6.2.3=16 + 48 = 64$$\n",
    "$$\\frac{dx}{dt}(1) = 4$$\n",
    "$$\\frac{dy}{dt}(1) = 3.1^2+2 = 5$$\n",
    "\n",
    "E por fim, aplicando tudo na equação final, temos:\n",
    "\n",
    "$$\\frac{df}{dt}(2, 3) = 99.4+64.5 = 716$$\n",
    "\n",
    "\n",
    "Exemplo 2.\n",
    "\n",
    "Dada a função $f(x, y) = x^3+2y^2-xy$, tal que $x = x(s, t) = s^2+t$ e $y = y(s, t) = 2t-s$, encontre as derivadas parciais $\\frac{\\partial f}{\\partial s}$ e $\\frac{\\partial f}{\\partial t}$.\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x} = 3x^2-y$$\n",
    "$$\\frac{\\partial f}{\\partial y} = 4y-x$$\n",
    "$$\\frac{\\partial x}{\\partial s} = 2s$$\n",
    "$$\\frac{\\partial y}{\\partial s} = -1$$\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x}(x, y) = 3(s^2+t)^2-(2t-s)$$\n",
    "$$\\frac{\\partial f}{\\partial y}(x, y) = 4(2t-s)-(s^2+t)$$\n",
    "$$\\frac{\\partial f}{\\partial s} = [3(s^2+t)^2-(2t-s)].2s+[4(2t-s)-(s^2+t)](-1)$$\n",
    "$$\\frac{\\partial f}{\\partial s} = 6s(s^2+t)^2-2s(2t-s)-4(2t-s)+(s^2+t)$$\n",
    "\n",
    "Agora para $\\frac{\\partial f}{\\partial t}$.\n",
    "\n",
    "$$\\frac{\\partial x}{\\partial t} = 1$$.\n",
    "$$\\frac{\\partial y}{\\partial t} = 2$$.\n",
    "$$\\frac{\\partial f}{\\partial t} = [3(s^2+t)^2-(2t-s)].1+[4(2t-s)-(s^2+t)]2$$\n",
    "$$\\frac{\\partial f}{\\partial t} = 3(s^2+t)^2-(2t-s)+8(2t-s)-2(s^2+t)$$\n",
    "\n",
    "### G. Derivação Implícita\n",
    "\n",
    "Quando uma função y está também em função de uma variável x, e queremos encontrar a derivada dessa função y, precisamos utilizar de uma técnica chamada de derivação implícita para fazer isso. Utilizando uma função $F(x, f(x)) = 0$, à partir de uma outra dada e a partir daí, aplicaremos a fórmula da regra da cadeia. $$\\frac{\\partial F}{\\partial x}.\\frac{dx}{dx}+\\frac{\\partial F}{\\partial y}.\\frac{dy}{dx} = 0$$\n",
    "\n",
    "Como $\\frac{dx}{dx} = 1$, para descobrirmos $\\frac{dy}{dx}$ basta fazer,  $$\\frac{dy}{dx} = -\\frac{\\frac{\\partial F}{\\partial x}}{\\frac{\\partial F}{\\partial y}} = - \\frac{F_x}{F_y}$$\n",
    "\n",
    "No caso de uma função de três variáveis, com $F(x, y, f(x, y)) = 0$, fazemos algumas manipulações e como derivadas implícitas, temos:\n",
    "$$\\frac{dz}{dx} = -\\frac{F_x}{F_z}$$\n",
    "e\n",
    "$$\\frac{dz}{dy} = -\\frac{F_y}{F_z}$$\n",
    "\n",
    "Agora, basta aplicar as fórmulas na função dada.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine $\\frac{\\partial y}{\\partial x}$ para $6x^2+12y+3xy-x=4$.\n",
    "\n",
    "A resolução é bastante simples. Para isso basta encontrar $F_x$ e $F_y$ e aplicar na fórmula.\n",
    "\n",
    "$$F_x = 12x+3y-1$$\n",
    "$$F_y = 12+3x$$\n",
    "\n",
    "$$\\frac{\\partial y}{\\partial x} = -\\frac{F_x}{F_y} = - \\frac{12x+3y-1}{12+3x}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAWCAYAAAACcfiCAAAABHNCSVQICAgIfAhkiAAAA8JJREFUaIHt2VmIXEUUgOEvMYqooHFXiA9ijA9BoiAqGGw3hEBwiy++2BANPgRxRRHEQdEkEFBJcAUZRR8UUXFBEbeYDAYNJiZq3B0UY4ZEx6i46/hwapyezu3ue29PTxvSP1xud9U5VafuqXvqVF167NYswAqsxo8YwaMNZA/CZXgan+FX7MAaLMTUThs7QSzDq/hajOF7rMctYoy7AkX8No4NSfgnbG6heEWq34LHsAQP4YdU/iSmlB3BJPIH1grbl4oH944YwzeY0T3TclPEb+M4AzOFoyotFM/EfDu/yYfjq6R7UTG7S1NN/VVK6O7doPz21OY95UyaVHL7rd5Zr+PTpNCK1/Ac/qkr34r70u9Kht7LsifDFPSnuqU5+p8ofmtQ/kS6z6wrv0bYeG0DvVn4HW+2b1pucvutU+vsn+n+V0bd9WKS3IY9asqX41I8gBs7ZFcR5qf7xrrygXQ/pYHeCjGuxZ0wqpNUFFgLapiGTUn33AYy/am+mv7flP4/rtwkrCof0ke5Dn24UyQ/I3gPh9TJ7YlfxLJVz8VJ7+427GiXinJ+K624POm90ERmhsiIvxRvwghewl5FjUxUte/wramN0etFHNZAdlWSOaKmbF+R6Q9h/zbsaJeKSXT4lUlnMw5sIbvE2MMdwD45+xg03jGtrv4C9hNOvgAfix3IiRkyd6S2L6wpW2Z81JrIMRTxQaWZzrQCDbVisQhlH+IssZ9txraa3wtFmMzDXTigrmwOzsPD4mHWsiFnu6MMibOFd/EJHsHsOpnRdfxkPIXjcDXeSja04nONk8UsthSQLU1F/tl1VZLdhENzyF8iErdvk9695Uz8j6r2Q3oW61O7B9eVTxf2r0r/X8HfOGGC+y9DRYdD+g1Jbr2dH0wW88Rhx0aREH0ksvpZRQ2soaozDh9K7U7PqPsAP4vJ+3/ar1d00OE3J5l1Wq/ZcJoI3V8YS3gWpDaeKWpgDVXlHH6s7ARrqrGDl4GMerjf2OnWNtmTohtUFFjDz08XcWIGpxpLfLaL7QuxZ75VhLLVImGrZ7BGdw6eF+ft54hwThzBrhNr8NzU1mQxTySPa8SO4TuRtJ2Oo0XmfnkD3QEswn5i/R7utLFNKOK3cfRpni0OFpAdwRtJ9hjx8IZxfEa/Zyf5tXlGl0FVuTd8NlaKxG67OCjaIc7S+zSPWnNTn2/r/jeDPvn91qMkz4rodlK3DenReUYTtZXdNqRH5zhKnPE/KD6OvC//YVGPXZBF4q0eFl/SjuyuOT169OjRYzfnX0foM3mtrsT/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 12 x + 3 y - 1$"
      ],
      "text/plain": [
       "12⋅x + 3⋅y - 1"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (6*x**2+12*y+3*x*y-x-4)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 x + 12$"
      ],
      "text/plain": [
       "3⋅x + 12"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{12 x + 3 y - 1}{3 x + 12}$"
      ],
      "text/plain": [
       "-(12⋅x + 3⋅y - 1) \n",
       "──────────────────\n",
       "     3⋅x + 12     "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "implicita = -(fx/fy)\n",
    "implicita"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Determine $\\frac{\\partial z}{\\partial x}$ e $\\frac{\\partial z}{\\partial y}$ para $3x^3+6y^4+4z+xyz = 2$\n",
    "\n",
    "$$F_x = 9x^2+yz$$\n",
    "$$F_y = 24y^3+xz$$\n",
    "$$F_z = 4+xy$$\n",
    "\n",
    "$$\\frac{\\partial z}{\\partial x} = -\\frac{F_x}{F_z} = -\\frac{9x^2+yz}{4+xy}$$\n",
    "$$\\frac{\\partial z}{\\partial y} = -\\frac{F_y}{F_z} = -\\frac{24y^3+xz}{4+xy}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 9 x^{2} + y z$"
      ],
      "text/plain": [
       "   2      \n",
       "9⋅x  + y⋅z"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y, z):\n",
    "    return (3*x**3+6*y**4+4*z+x*y*z-2)\n",
    "\n",
    "fx = syp.diff(f(x, y, z), x)\n",
    "fy = syp.diff(f(x, y, z), y)\n",
    "fz = syp.diff(f(x, y, z), z)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x z + 24 y^{3}$"
      ],
      "text/plain": [
       "          3\n",
       "x⋅z + 24⋅y "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x y + 4$"
      ],
      "text/plain": [
       "x⋅y + 4"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{9 x^{2} + y z}{x y + 4}$"
      ],
      "text/plain": [
       " ⎛   2      ⎞ \n",
       "-⎝9⋅x  + y⋅z⎠ \n",
       "──────────────\n",
       "   x⋅y + 4    "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "parcial_implicita_x = -(fx/fz)\n",
    "parcial_implicita_y = -(fy/fz)\n",
    "parcial_implicita_x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{x z + 24 y^{3}}{x y + 4}$"
      ],
      "text/plain": [
       " ⎛          3⎞ \n",
       "-⎝x⋅z + 24⋅y ⎠ \n",
       "───────────────\n",
       "    x⋅y + 4    "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "parcial_implicita_y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### H. Vetor Gradiente\n",
    "\n",
    "É um vetor calculado a partir das derivadas parciais de uma função de 'n' variáveis. A notação do mesmo é denotada como: $$\\nabla f(x_1, x_2, ..., x_n) = (f_{x1}, f_{x2}, ..., f_{xn})$$\n",
    "\n",
    "Uma característica muito relevante é o fato de que em funções de duas variáveis, o vetor gradiente é ortogonal às curvas de nível em um determinado ponto e no caso de três variáveis é ortogonal às superfícies de nível.\n",
    "\n",
    "![](Imagens/36.jpg)\n",
    "\n",
    "O termo gradiente refere-se à variação do crescimento da função, onde o mesmo sempre aponta para a direção onde a função cresce mais rápido.\n",
    "\n",
    "\n",
    "### I. Derivada direcional\n",
    "\n",
    "Quando nós calculamos as derivadas parciais $f_x$ e $f_y$ de uma função de duas variáveis, estamos calculando a derivada dessa função na direção dos eixos x e y, respectivamente, o que também pode ser entendido como o cálculo dessas derivadas na direção dos vetores $(1, 0)$ e $(0, 1)$.\n",
    "\n",
    "A questão é a seguinte, como eu poderia calcular a derivada de uma função de duas variáveis em uma direção qualquer? \n",
    "\n",
    "É para isso que a derivada direcional é utilizada. Para determinar a derivada dessas funções na dureção de um dado vetor.\n",
    "\n",
    "Há duas formas dde se calcular essa derivada. Primeiramente, como queremos somente a direção do nosso vetor dado, basta utilizar a sua projeção no plano xy.\n",
    "\n",
    "\n",
    "A partir daí, podemos notar que a direção desse vetor é dada pelo $cos(\\theta)$ e pelo $sen(\\theta)$.\n",
    "\n",
    "E então, podemos utilizar a fórmula: $$D_u(a, b) = f_x(a, b)cos(\\theta) + f_y(a, b)sen(\\theta)$$\n",
    "\n",
    "Onde $D_u$ é a derivada direcional da função no ponto $(a, b)$.\n",
    "\n",
    "A segunda forma de se calcular essa derivada é dada pela multiplicação do vetor gradiente com a normalização do vetor dado, ou seja: $$D_u(a, b) = \\nabla f(a, b).\\frac{v}{||v||}$$\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Calcule a derivada de $f(x, y) = 6x^2 +12y +3xy-x$ no ponto (2, 3) na direção de (4, 3).\n",
    "\n",
    "Primeiro vamos achar o vetor gradiente de f e aplicar no ponto.\n",
    "\n",
    "$$\\nabla f(x, y) = (12x+3y-1, 12+3x)$$\n",
    "$$\\nabla f(2, 3) = (12.2+3.3-1, 12+3.2) = (24+9-1, 12+6) = (32, 18)$$\n",
    "\n",
    "Agora devemos encontrar o módulo do vetor que usamos como direção.\n",
    "\n",
    "$$||v|| = \\sqrt{4^2+3^2} = \\sqrt{16+9} = \\sqrt{25} = 5$$\n",
    "\n",
    "Substituindo tudo na equação da $D_u$ temos:\n",
    "\n",
    "$$D_u = (32, 18).\\frac{(4, 3)}{5} = (32, 18).(\\frac{4}{5}, \\frac{3}{5}) = 32.\\frac{4}{5}+18.\\frac{3}{5} $$\n",
    "$$D_u = \\frac{128}{5}+\\frac{54}{5} = \\frac{182}{5}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{182}{5}$"
      ],
      "text/plain": [
       "182/5"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (6*x**2+12*y+3*x*y-x-4)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "gradiente = [fx, fy]\n",
    "gradiente = [32, 18]\n",
    "v = [4, 3]\n",
    "v_mod = syp.sqrt(4**2+3**2)\n",
    "normalizacao = (1/v_mod)*np.array(v)\n",
    "Du = np.dot(gradiente, normalizacao)\n",
    "Du"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### J. Maximização da derivada direcional\n",
    "\n",
    "A maximização dessa derivada é basicamente a própria na direção em que a função cresce mais rápido. Mas como eu posso descobrir isso?\n",
    "\n",
    "Na verdade, é bastante simples, retomando as características do vetor gradiente, nós podemos notar que a função cresce mais rápido na direção do mesmo, então para maximizar a derivada direcional, basta calcular a mesma na direção do vetor gradiente. Ou seja: $$D_u(a, b) = \\nabla f(a, b).\\frac{\\nabla f(a, b)}{||\\nabla f(a, b)||}$$\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Maximize a derivada de $f(x, y) = 2x^6+3y^2-2xy$ no ponto (1, 2).\n",
    "\n",
    "Calculando o vetor gradiente e aplicando o ponto (1, 2).\n",
    "\n",
    "$$\\nabla f(x, y) = (12x^5-2y, 6y-2x)$$\n",
    "$$\\nabla f(1, 2) = (12.1^5-2.2, 6.2-2.1) = (12-4, 12-2) = (8, 10)$$\n",
    "\n",
    "Agora para maximizar basta calcular o módulo do vetor gradiente.\n",
    "\n",
    "$$||\\nabla f(1, 2)|| = \\sqrt{8^2+10^2} = \\sqrt{64+100} = \\sqrt{164}\\approx 12,8$$\n",
    "\n",
    "Substituindo tudo na equação chegamos a:\n",
    "\n",
    "$$D_u(1, 2) = (8, 10). \\frac{(8, 10)}{12,8} = (8, 10).(0.625, 0.781)$$\n",
    "$$D_u \\approx 8.0,625+10.0,781 \\approx 5+7,81 \\approx12,81$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 12.806248474865697$"
      ],
      "text/plain": [
       "12.806248474865697"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (2*x**6+3*y**2-2*x*y)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "gradiente = [fx, fy]\n",
    "gradiente = [8, 10]\n",
    "v = [8, 10]\n",
    "v_mod = math.sqrt(8**2+10**2)\n",
    "normalizacao = (1/v_mod)*np.array(v)\n",
    "Du = np.dot(gradiente, normalizacao)\n",
    "Du"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### K. Reta normal à uma superfície\n",
    "\n",
    "Quando calculamos o vetor gradiente de uma função, temos a ciência de que esse vetor é normal à uma superfície gerada pela mesma. Como poderíamos achar uma função que calculasse uma reta também normal à essa superfície?\n",
    "\n",
    "Para isso, basta utilizar dos conhecimentos pré-dispostos do cálculo vetorial, e associarmos uma equação da reta com um ponto dado e um vetor que seja normal também, para podermos direcionar a nossa reta.\n",
    "\n",
    "Sabemos que uma reta pode ser descrita como: $$r(t) = (a, b, c) + \\vec{v}t$$.\n",
    "\n",
    "Onde $(a, b, c)$ é o ponto dado e $\\vec{v}$ é o nosso vetor normal.\n",
    "\n",
    "Nós sabemos também, que uma das características do vetor gradiente é que o mesmo é normal à uma superfície de uma função de três variáveis, então como vetor $\\vec{v}$, utilizaremos o gradiente aplicado no ponto, então, a equação da nossa reta fica como sendo:\n",
    "$$r(t) = (a, b, c) + \\nabla f(a, b, c)t$$\n",
    "\n",
    "E a partir daí, podemos calcular a fórmula da nossa reta.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Encontre a reta normal ao gráfico da função $f(x, y) = 4+3y-2x^2$ no ponto (2, 1, 0).\n",
    "\n",
    "Resolução:\n",
    "\n",
    "Para fazer isso vamos passar o f(x, y) como z para o outro lado da equação.\n",
    "\n",
    "$$F(x, y, z) = 2x^2-3y+z-4$$\n",
    "\n",
    "Agora devemos calcular o vetor gradiente de F e aplicar no ponto.\n",
    "\n",
    "$$\\nabla F(x, y, z) = (4x, -3, 1) \\rightarrow F(2, 1, 0) = (4.2, -3, 1) = (8, -3, 1)$$\n",
    "\n",
    "Assim encontramos o nosso vetor $\\vec{n}$.\n",
    "\n",
    "Montando a equação da reta temos:\n",
    "\n",
    "$$r(t) = (2, 1, 0) + \\vec{n}t = (2, 1, 0) + (8, -3, 1)t = (2+8t, 1-3t, t)$$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8*w + 2, 1 - 3*w, w], dtype=object)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def F(x, y, z):\n",
    "    return (2*x**2-3*y+z-4)\n",
    "\n",
    "fx = syp.diff(F(x, y, z), x)\n",
    "fy = syp.diff(F(x, y, z), y)\n",
    "fz = syp.diff(F(x, y, z), z)\n",
    "gradiente = [fx, fy, fz]\n",
    "gradiente = [8, -3, 1]\n",
    "p = [2, 1, 0]\n",
    "w_gradiente = np.array(gradiente)*w\n",
    "r = np.add(p, w_gradiente)\n",
    "r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## VI. Aplicações das Derivadas Parciais\n",
    "\n",
    "Este capítulo do livro aborda o conceito visto até então para uma única variável, a funcionalidade de se calcular uma derivada de uma função, porém agora, extendendo os nossos conceitos para um domínio em um espaço de dimensões maiores.\n",
    "\n",
    "### A. Pontos críticos de funções de várias variáveis\n",
    "\n",
    "**Definição -**  Quando estudamos os pontos críticos em funções de uma variável nós vimos que os mesmos eram pontos no Plano Cartesiano onde a derivada da função não existia ou se igualava à zero. Os pontos críticos de funções de várias variáveis são calculados da mesma forma, porém, nesse caso utilizamos as derivadas parciais da função, montando um sistema linear em seguida, para então descobrir os pontos.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Dada uma função $f(x) = x^2 + 2x + 3y^2$, calcule os pontos críticos dessa função.\n",
    "\n",
    "Resolução:\n",
    "\n",
    "Como foi explicado anteriormente, para se calcular os pontos críticos dessa função vamos inicialmente calcular as suas derivadas parciais. Então:\n",
    "\n",
    "$$f_x = 2x+2$$ e $$f_y = 6y$$\n",
    "\n",
    "Agora, vamos igualar as duas à 0 e montar um sistema.\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "2x+2 = 0\\\\\n",
    "6y = 0\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Resolvendo o sistema chegamos a:\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "2x+2 = 0\\\\\n",
    "6y = 0\\\\\n",
    "\\end{array}\n",
    "\\right.\\rightarrow\n",
    "\\left\\{ \\begin{array}{c}\n",
    "x = -\\frac{2}{2}\\\\\n",
    "y = \\frac{0}{6}\\\\\n",
    "\\end{array}\n",
    "\\right.\\rightarrow\n",
    "\\left\\{ \\begin{array}{c}\n",
    "x = -1\\\\\n",
    "y = 0\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Então, nesse caso, o ponto crítico encontrado para a função $f(x) = x^2 + 2x + 3y^2$ foi $(-1, 0)$.\n",
    "\n",
    "Para confirmar, analisemos o gráfico da função:\n",
    "\n",
    "![](Imagens/37.jpg)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAb4AAAEuCAYAAADx63eqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy8QZhcZAAAgAElEQVR4nOy9eZAc2X3f+Xkvs+7q+wS6ATSAbtzAYA5wMBxKpkxRpqSVaDkUpEOKMINrhbT+i7JiV+JKQdu7G1KMvbuKZXAdtldBS7Palc0Jei2aou7hDMfDuTgzJDCYC+gLjW70fdddmfn2j54sVFW+6q6jq4FG5ycCASCrXr6s633z93u/Qyil8PHx8fHxOSjI+30BPj4+Pj4+e4kvfD4+Pj4+Bwpf+Hx8fHx8DhS+8Pn4+Pj4HCh84fPx8fHxOVD4wufj4+Pjc6Awd3jcz3Xw8fHx8dmPiEoP+Bafj4+Pj8+Bwhc+Hx8fH58DhS98Pj4+Pj4HCl/4fHx8fHwOFL7w+fj4+PgcKHzh8/Hx8fE5UPjC5+Pj4+NzoPCFz8fHx8fnQOELn4+Pj4/PgcIXPh8fHx+fA4UvfD4+Pj4+Bwpf+Hx8fHx8DhS+8Pn4+Pj4HCh84fPx8fHxOVD4wufj4+Pjc6Dwhc/Hx8fH50DhC5+Pj4+Pz4Fipw7sPj77GsdxsCyLTCaDaZoYhoFhGEgpEUIgRMUmzT4+Pg8pvvD5PJS4gmfbNo7jFP5WSpWInSuEviD6+BwcfOHzeahwHIfl5WXC4TCmaSKEQEpZ+FOMUqogiuVYlkUgECASiRTG+oLo4/Nw4Aufz75HKYVSinw+j+M4TExMcOLECYLBYOFxHZWETCnFzMwM0WiUnp6ekudLKTEMA9M0fUH08dmn+MLns28pFzy4J07lYleLMBVbiYZhlMznWoi5XM7zfF8QfXz2B77w+ew7XAGyLKtE8FyREUJUtPKqRXeO7SzE7QTRDarxBdHH58HAFz6ffUO54LkCUi4iOouvVmoRz50EMZvNlhzf3NwkGo0SjUZ9QfTxuQ/4wufzwKOUwrbtQmRmJcFz0YlWeTRntfM2QqVrXFhYoLu7u8SN6j6/PMJUF5Tj4+PTGL7w+TywuIJnWVZBuKqxioQQBRdo8bFaaKbl5b6G8v1DoPB6y59fvn9oGIZvHfr41IkvfD4PHK7gLSwsANDZ2VmT1VNu8dUjELuxT1jrfMV/u1QSRKVUSUBNucvUx8enMr7w+TwwlFt4yWQSIQRdXV01nWe3RGsvha8S1QqiaxEnEgkA2tvbfUH08amAL3w+9x2dS9NdrHXJ5TuxW1GdzWS3rq/8OhOJBEopWlpaSt5PoGAhlu8h+oLoc9Dwhc/nvqGUKpQVKxY8l3oFTErp2eOrFd0+4W7RbKEpfx/hntBalkU+ny95zBdEn4OGL3w+e44reO6elW6hhvoF7GFyddaKu/dXzk4uU18QfQ4SvvD57Bk6wdtuEa03H28/uDofFGoRRLeU29GjR31B9NnX+MLn03RqFTyXet2NzarccpDQCaJSioWFBY4ePepbiD77Gl/4fJqGW8ZrenoawzDo7e2taRGs19W515VbHiTqSdSv9dy6G5diCzGXy5U87guiz4OGL3w+u44reK6Fl8/nsW27riTyel2duxGY0kzhe9hEtdhCLN5jLC4k7guiz4OCL3w+u0Zx81corVBST1pCI8EtuxHV2Sz268JejzW5XR3TckHc2NjAcRw6OzsLVWp8QfRpBr7w+TRMJcFzacRyu5/BLQ+bVfYgnVsniKlUqiCIuVzOM1+5deiWbfMF0adWfOHzqYtKvfB0i1Aje3V+cMuDQzNF1T1/paLcxa2fthNEVxR9QfTZDl/4fGqiFsFz2esgleJxSimWlpa4e/cukUiEWCxGPB4nHA4/lAtjs/clmy182+0h7tQL0RdEn2rxhc+nKnZq/rode71XJ4TAtm0WFxcZHx8nGo3S399PLpdjc3OTubk50uk0UkpisVhBDGOxGMFgsPC69qvFtx9cnTocx6m5BVM1guiyvLxMOBymtbXVF8QDji98PttSLHg3btxgeHiYUCj0wKYlKKXY3NxkZmaGTCbDxYsXiUQinohC2CrwnEwmSSaTrKysMDU1RS6XwzRNTNPEtm3W1taIxWIEAoGar3+n62wG+9niq0f4KqETso2NDQzD8AiiiyuExe2ffEF8OPGFz0eLrtu5696sdSHYC4vPdWmOj48jpaSnp4fz588XHtNds2EYtLa20traWnI8n88zOzvL8vIy8/PzJJNJLMsiGAwWLMN4PF7ooF7P62om+9Xi2wthdS083dyuIOZyucJxNz3DF8SHC1/4fErYrvnrXgepVDOuWPCi0SgXL14klUqxvLxc83wugUCAlpYWstksIyMjhXlyuRzJZJJEIsH09DTJZBLHcUr2DmOxGJFI5KHsmv4wCF+lz6Ual6kviA8PvvD5AJVbAxVzP/LxKrnuXMEbGxsjHo9z8eJFotEoAJlMZlfy+Mqb2YZCIUKhEJ2dnSXXkclkSCQSJJNJFhcXSaVSAESj0RJB3IuAGt/Vubvnr0UQNzY2EEJU7IXoC+KDgy98B5xqBM/lQbD4ygXv0qVLBcFz0QlmrYt2tcEtQggikQiRSISenp7CccdxSKVSJJNJNjY2uHv3LplMpvCastksUkri8TiBQGBXF8VmujqbSaXOErtFs/cQE4lEwX2ezWY9z9dVqXkYPQP7AV/4Dii1CJ7L/RC+8rSE7QTP5UGIyHRFLR6Plxy3bZubN29iGAbLy8slATXF+4exWAzTfPB+ns22+PazRWnbNqFQqGIvxOIyfi46QXQT832ax4P3y/JpKjs1f92O+1V6bHFxsSrBKx73oCawG4ZBKBSipaWlxELM5/OF/cO5ublCQE0oFCpJudgpoGa/VG65H+e/X67U4jqmxVQjiMX7h74g7h6+8B0QXMFbW1tjcXGREydO1LwI7KXF51p4qVSK+fn5qgSveD6daNWysDbTatRdQyAQoL29nfb29sIxN6DG3T+83wE1D6sw7Ra2bdcU5VurINq2zerqKv39/do9RJ/q8YXvIae8F55SilQqVdcPZS+Er9ylGYlEuHDhQk3z6QJTHgT3Z60UB9R0dXUVjiulSKfThRzEhYUF0uk0sLW4ZzIZgF0PqPEtvu2pVfgqUUkQLcvi7t279PT0lGxRuM91LURfEHfGF76HlErNX93E7Hpwk39rpdq0BLfSSrFL85VXXql5vnorvpSf40EVSiEE0WiUaDTqCaj58MMPCQQCrK+vlwTUFFuH8XicYDBY87z7XZj2+x6i4zgF12cx5RZi8efkt37S4wvfQ8ZO3c7r3aeDrR9RPWO3+5FVErxGeJD3+JqJlJJgMEh7e3uJhWhZFqlUikQiwfLyMrdv3yafz9ccULPfLT5obnDObll8tZ5/J5epZVnk8/mSxw66IPrC95BQ3vy1Ut5Qve5Kd2z5D6hemiF4LrslWvtN+EB/zaZpVqxQ4+4fzs3NkUgkCpGJxWLoBtTsd4uv2Yu6WxmmWViWVVOkbz2CmM1mCQQChc/8YRVEX/j2OTv1wiunUYuvURdiMwXPZbcsvmbSTFGt9toDgQAdHR10dHQUjpUH1KysrJBKpQqLuhCChYWFpgTU7IXF10xs297Xe4hKKebn54lGo57X4VqIr7/+OpcvXy4p4rAf8YVvn1Kr4Lk0Il6NiKZSioWFhaYKnstuCPReR3U+KGwXUDMzM8Pa2ponoCYajZa4TGstYu7SbIuv2ewX4auEEFtdTQKBQMk8xRbi1772NX73d3/XFz6fvcVxHPL5PLdu3eLkyZM1l0JqZNGtNy3BLeO1uLjII488QiQSqfsaqsF3de4+QohCke6hoaHCcbdCTSKRKAmoMQxD2/Jpp2t/kG8KdqLZlWeaLXygd6cWW4ibm5slHoL9ii98+wBd89fFxcVCAeW9ota0BNel2dLSQiQSKXRLqJVmlRvb6Rz7lb1MYK9UoaY4oGZpaYnJyUny+TyBQKBEDMsDavZrubW9YK+Eb7s5NjY2SnJN9yu+8D3A1NPtvJlUk85QLniuhVdPWgLcS0ZvVPhqFcP9GNXZbGr5HCoF1LgdLpLJJLOzsySTyUJATTabZW5uriCIu2k97YUbtdm/S8uymrY94GLb9rYBNNlslnA43NRr2At84XsA0fXCexCqu2+XzlBJ8MqfU+trcHPyalm0ymt8rqysMD4+XnC/uRbKdsnd+1X4HvSSZcFgkGAw6AmoyWazvP3222Sz2ZKAGrfDhfu5RSKRA7l/CPfP1emyH38PlfCF7wFiLwWvnkWsUrcEt5Zma2trxT08d2ytP9x6A1VcwRsdHSUUCjEyMoJSimQyyebmJrOzsyV7UcXBGW639Wa3+NlvNEtUhRCEw2FM0+TYsWMl86XT6UKEaXFATfn+4U4BNc0Wvr34PPdC+Kp5n+73Dfhu4AvfA8B2zV8rUY8l5FKvCBW7OssF7/Lly9sGrdQ7Zz2W1/r6OslkkqmpKc6dO0c8Hi+4i2OxGL29vYXn2rZdKA69uLjIxMQElmURCARIpVLMzs4WFtfdbGnTTPbrPln5dRdXqCnGcZyCu3RtbY3p6Wmy2aznJqY4oGa/V4WBvRE+qPz9yeVyhZvC/Y4vfPeReloDubipBfX8mN2x9VhflmUxPz/P+Ph4VYJXPLbZdT7X19cZHR0tWBCXL1/ecYzbP614L8qtZ3rjxg3y+Tx37twhlUqhlCISiZRYh3vRXLYW9lqc7gdSSlpaWmhpaSk5bllWQRAXFxdLAmrcPcT19fWmtHxqdvI67J3wVWJ9fd2zZ7tf8YXvPuCWFVtaWqK9vb0mwXNxxaueO7BGuiVsbGywtLRUteA1Mqc7bqfFfHNzk1u3bqGUYnh4mLa2trqDaeBeLptpmhw9erRw3HGcQnHoat2lDxPNDtdvVLRN06StrY22traS47lcrpBSUxxQEw6HPS2f6n1993v/bTfY6f1/WCI6wRe+PaXYwnMchw8++ICPf/zjdZ2rkWTyWsYWJ563trbWnZbQaE8+HYlEgtHRUfL5PMPDw7uaX1Sp3Ju7UBa7S11Lo9xd6ua9Fbve9nProP2aZ+d+Di0tLZw+fRq4F1BTXqHGterLWz7t9Lr3InjmftUCddnY2PDcVOxXfOHbA3TNXxv9AjcqfNWkJbiC19bWVrDwGklL2C2LL5lMMjo6SjabZXh4uO4qEtJ5D4SJI055Hqtlb1FnaVQq/eWmqEQikYKl+KC5Syuxn0W1XJhcd3g4HKa7u7vkea5V7zYFzmQyhf3G4puY4oCaByXwpBF2sijX1tZ84fPZmUa6ne9Es7osVBK8RtkNiy+VSjE2NkYqlSoIXrWLpW5+03mOoPOvyRlfJie/BKI0kboR11ul0l+O4zA+Po5t22xsbHgqnTTqLm12JOrDInyVKLbqi7Ftm1QqRTKZZHV1tSSgJh6PF/a/3T3FZtDs92gn4fMtPp9t0bUG2u07td12dTZL8FzqFRIpJel0munpaTY3Nzl58iTd3d01LwDargXqBQRZQvb/hOl8i4zxr3HkxZqvsRbc1kHBYJD+/v7C8Uru0uJOCdXuQ+1HcXrQe+UZhrFtQI2bauEGRAUCgZKbmGg02tT9ud1gJ6t1fX3dFz4fLzv1wqs0pp4f/G4JX7ngPfroo02pzFCPxZfJZFhdXWV5eZlTp05x7ty53Vsc1TJSXSv811A/Imp9kpz8dXLGbyFE7Y1aG2End2kikWBlZYVkMglQiC51F9e9cJc22+JrdrpBM87vfm65XA7TNDl+/DhAiZv77t27TQmo2W12svjW19c5fPjwHl5R8/CFbxeoR/Dg3l5bPXsDu+HqdNMSmil4xXNWK3zZbJaJiQlWVlaIRCIcOXKkpNN4vRQv3KbzPQSl1yPIE3L+V2TmJvnQrwL3tzTTdu5SN7G7vDB0LpcjGAxi2zaxWGxX3W6+8FWmPLUoGAzS2dlZsv+slCKTyRRSLpaXl0mlUgA7BtTsRWBRNa7OM2fONPUa9gpf+Bqg2uavlTAMY8eisDuNrRU3R216epqenp66BG+3qr6Uk8vlmJiYYHl5maGhIU6fPs3o6GhN81SLoV7QHlcKjNzLBLLf4lzPTyKcf4eSjYvublK8D9XX11c4blkWN27cKPTMSyQSdbtLdRxkV2c159/pdyyEIBKJEIlEKgbUbG5uMjc3RzqdRkpZCKhxLfpmfgbVCN/D0JkBfOGrC1fw3C/o0aNH99xdaZom2Wy26ue7TSYnJiYwDIPBwUFOnjxZ87z1FI12x1USvnw+z+TkJAsLCxw7doyRkZHCIrVdOkMjmI5e+Bw1gqFuAdDf8reotcfJRv8Z+dB/C2J3m67uNqZpEgwG6e3tLXRLKHeXFlsZbh1MVxB3KvvVbOFrtjA1c49tp+LO21EpTcYNqEkkEqyurpJOp/nBD35QCKgpdpmWWvZ5AoFZ8vmj3sm2wb1JqoQf3HJAKW/+CltuuXoXA9M0m56LVyx4rktzeXmZfD5f17yum7TWRUonfJZlcfv2bebm5jh69ChPPfWUtvNzvSJRaaEWagzJlH6M3QvcKnruGuHkbxDI/D9kIr+PE3qsrmspmb+Jlk35a67HXVoclFHsLt3PUZ22be/YD/BBO39xQE1rayv5fJ4LFy5sEwgVYGjI5tChdTKZLmx7oCZv0k7ivb6+7lt8B4lK3c4DgUBd7kaXet2V7tjthE8neK5L0zAMMplMXfPuRukxy7KYmpri7t27HDlyRCt4LvVafNu5hSpZewDCSuiP27NEZ38CK/Z5sh3/E8o8VPM1PUhs5y51F9X5+XmSyWTBEsjlckgpCQQCux6U0Wzh24s9xL1KLvcGQikMY5ZQaBTD2Ppdz8w4zMz8EMdxCgE17o1Mpc/OT2fwqaoXnmmaDQlfI+MrCd92grfT2GrnrVf4LMticnKS6elpBgcHeeqpp3ZcLBoV2mKXqYtRQfiUCiOtD7SPOWoIk1kCyf+Imfo2ubYvkWv9EsjmdpPfaypFl2azWT788MOClV6vu7QSe+Hq3MvglmacXydKUi4RDo9iGBuFY0pJDh16hEOHjJKAGrcpcHFATbF1n8/ntxW+zc1NX/geVmpp/tqIqxJ2NxevGsFz2S6BfSfqGes4TiEt4dixY1y9erXq/ZB68/8qjlM2pnqpwnWexuCa9jGse69ZqCShtd8jsPl/k+34X7Bi/wAekMorzbCciquc9Pf3FxY/x3EKSd3r6+vMzMyUJHUXC+JOn/d+j+rcS4sPwDBmCQbfALoxjGTZczuBrefWElDjpsu0tLSU7B8Gg8GC5+V+FsneTXzh+4jyXniwc5Tmg2Dx1SJ4xWPrDRipxQJzHIeZmRmmpqaIxWIMDg5y4sSJmuerR6QrCZ9UbyFY145Rdov+uBLI3C3PcWlPE1j5twTnv0q2759hx3+y5uvcT5SLqpSyIGzl7lI3h81NmXHdpeVJ3a4Y7UVU5/2sc7lb55dyjlDoB5jmNEoJwPudtaxu7wnK0AXUvP3225w/f74QDLW6usqdO3f46le/yq1bt0gmk/ybf/NvuHjxIhcuXKh5v29tbY1f+ZVfKUQe//t//+85ffo0n//855mcnGRoaIjnnntuT/YRD7zwNdL8tZE9Ond8I5ZXKpXitddeo729vaa0hGa7Oh3HYXZ2lsnJSXp7e/nYxz7GysoKm5ubNc/nukhrpVz48vk82WyWYPYtnGg3kiXvGGtOey6HUxjOh/qJHDAyPyJ6+x9gxX6cbO8/x4le2fH6mlVa7EEoWWaaJu3t7SWV/MuLQpdHl7qfcyaTqdtduh3NdkU226I0jCWGhj4gFlsomrMfw/D+NiyrvvQbN0AnFAqVVKj5wz/8QxYWFvjsZz+LYRh84xvf4Ctf+Qq/+Zu/yc/8zM9Uff4vfelLfOYzn+Gb3/wmuVyOVCrF7/3e7/GpT32KL3/5yzzzzDM888wz/Mt/+S/ruv5aOLDC5wre4uIibW1tdXU7byTiELYWiFwuV9MY18IbHx8nm83y1FNP1ZyH16jFt12dT1fwurq6uHLlSiHSrRndGXYa5xYWmJycZG5ujkAgwCPpP8KUWVInHyHeds+t6TjdSHuswuvqAbzCpxTI7Hjh/2byJcyJT5Fv+W/I9vwzVESf7LufG9HWe+7tikK77YJSqRQffvhhibu02EJsJB1hL/b4mmHxGcY4weCLDA4qAoHStUKpLiBddh0xlKp/37nS5xuNRmltbeXXfu3X6jrv+vo6L730En/0R38EUCjb961vfYsXX3wRgC984Qt88pOf9IWvGZQ3f33//fd56qmn7kt1/FosxmKXZnt7O5cvX+batWt1VVtpdI+vXIiKxbijo4PHH3/ckw/UzH58OoQQTE1NsbCwwNGjR3nyySdxMou0vPkhwnYI3LxGqvs85uA8wcASGxt9tBteKxBA5De0xx3jJIZGLAObf4ZILaDMAXKH/wec6IWar/9BpBn7h667tL29vaTkl+sudaNLx8bGsG17W3fpduwv4VMYxvuEQi9iGLex7QFM01uBR/dR1Gvt7USjnRkmJibo6enhi1/8IteuXePxxx/nq1/9KvPz8xw6tBUh3d/fz/z8/G5d8rYcGOHbrtv5/WoJU01wTLmoFLs067U2G3WxugKmlGJxcZGxsTHa2tp47LHHtg2o2QuLz91XXFlZIRqNFiJHbdtGrj1fUqYsuvQuaj2OdeYp4mEFmtRG2w4hsu+D5iuiOAR4hU8piZH6AGG/gbn6p1jtP0Pu8G/ixB6t+nXUy4Pg6qyHcmHayV2qS8YvthDL3aX7I2rUwTSvEQy+iGHcc7sr1Qusls3XgmGUWnsAtr3z/l49NJrKYFkWb7/9Nl/72td48skn+dKXvsQzzzxT8pxaPW6N8NAL33aCB/cW1ka+tI0Umq5k8ZUL3naiUs+8jezxWZZVELx4PF5VJ4d6XZbVCqbrZp2YmKC3t5euri4GBwdL7sLNtb/xXlc+gfnOq1jtT+IcOoFU4yWPO+IsAfEj7Zy55LL2B+QETmOk3986P4rA2ncIrH0Hq+3TZPt+ExjY8fU0woPo6tyJaoJbdnKXJhIJ1tbWCi2DTNMsWIa5XK4pFYBcGotKTRAMPo+U7xEIrHkeFcJ7M+M4h5CyvMOKiW3X1yHdtu1t3/9GOzMMDg4yODjIk08+CcAv/uIv8swzz9DX18fs7CyHDh1idna2pHJNM3mohc8tHL1dLzw3srLeqguu1VbP/oPO4lNKMTc3x8TExK4LnksjvebS6TR37tyho6ODS5cuEY1GqxrXiMtyu3HFVmdHR0dhX/H69esl4wQKc+157Tns4AjmnddRMybWmU9gxN5GsGVJCBXTjlEqRNgZ1z6WycXRjTI2Xyc6+9MMBc+y2vlFUF8EsX/Cwx/UItXF0aXF5PP5Qv5aNpvl3XffLemQsFNCd7OR8g6BwF8QCLyMEDks6ypQKnyOE0dKbxSyEGGgNI1hK5qzvtdRTdWWRoSvv7+fI0eO8OGHH3L69Gmef/55zp07x7lz53j22Wf58pe/zLPPPstnP/vZuueohYda+Fy2+1K71VcaEb6dKh5Uotji2wvBc6ln8VpdXWV0dBTLsjh06BAjIyM1jd+Nii/lLC8vMzo6SiwW80S1lgut2HwLaS1rz6NkH3AL4ViY772M03II59Q5DPkmMlch0tM4UzHnz7D06RJZcZywukYk+w6R2d/AWfk/yA3+d+QP/SMwW7VjamW/lhVrxrkDgUDBXTo7O8tjj22Vmyt2lxYndBe7S4vz13YXC8P4IcHgX2Oa75Q8IoTXC+M4RzHNVMkxpQykzJQdA8vqol6qaUnUaPL61772NX75l3+ZXC7HiRMn+MM//EMcx+Fzn/scX//61zl27BjPPfdcQ3NUy0MtfNX4jE3TrLtupTu+3pQGd6zroqtH8Jpd6mltbY3R0VEMw+DMmTNsbm7WHIkKuxvcsra2xq1btwgGg1y4cMHTLRu8rlVj5a8qziEypfsncnMW3pplpfMi7f1e1xOAUhVy/kQLIVtvCVr5UutOZqcIj/02cub/xWn9OPbRX8GJn614nfebZrs6m92o1b32ndylxR3Wi92lroWou86dvBlC3CUQ+DaBwJ/jOOcxzTKXutOKlIuakREgVfbcAQzjnkjadgj4EZb19PZvwDZUU66seL+1Hi5fvsybb77pOf7883pPTDN5qIWvGu5XErq7h7exscHa2lpdFp67V9eMBWN9fZ3R0VGEEJw6dYrW1i2LxG2oWSu7EdyyubnJrVtbieRnzpzxdMMuH1e8GBkrf619npKtyA19mTKxrBDTc1jnnsaIXEeoe3mIooIlaJunMNVb2seizoL2eCYfpHXmD2DmD1gPXGKt4/NYfX+feGuXpy9bNewnq2wvzl0N1bhL5+bmSCQSWnepXvgsTPMlAoFvYxhvFvbqlPIGpTjOSUxzpeSYUhLD8Oa+KtUGpLDtMHATw/gRljUCWgd7dVQjfLV6eR5kHmrhq+aHtNeFpstdmtFolLNn67vLb4bwbW5uMjo6iuM4DA8Pe9wbzXBZ7jQun89z7do1crkcIyMjVd15lghfbhGx+bb2eXbwNKb6gX5uy0bYecx3vo8T68Y+ewFDvI6iE5nT5/yh9G1dHHMAmZzWPhYL2PCR56otf522hetYS/8bM5HPM8HjpIMnS6wObxuaoukfkqjOB4Vid6lLef3LxcXFQsugWCxGd/ckPT0vEg6nCAZLA6McpxvDmNXM5A0Os+2jmKbXuyIEWNYKhvH9gpha1qWGXudO6Rjr6+sNW3wPEg+18FXDXrk6ywXPzXV75ZVX6p67kehM95rchSyRSDA6Oko+n2d4eLhi2aBGilTXOi6dTjM6Osrq6iqXL18uaauzE8XCJ1f+xtNtvYCt/7ErDGLZ24X/y+QS8s0l7N4RnGPHCaC3IGXmjva4YxxDMuOdR0SRCa/FaTqrDDpvcmz132K1XiQR/QcsOZ9mcTFd1IYmVCKGO0XWNsrDbPHVQnn9y63izpNcvPgWpvmnmObWDU4mc94zNp8/TihUKnxblt2K57nQCRftRnUAACAASURBVNw77jgRbNvCNP8TQpSuWZZ1saHXdJA6M8BDLnzVlldqpvAVC15nZ6c2ubtediMfL5PJMDY2RiaTYXh4mM7Ozh3HNdvVmcvlGBsbY21tjcHBQRzHqUn0oNRFKlb0Fp1SIDe9NTgBnMhZzM0bnuPGwi1UshM7dhn6kxjWvfGOMYBM6YVPWPp9USd6GmPth95rQyI3bwJgbrxD+8Y7tInfxe75CXKD/wir55NknVAhSMO1OlKpFLdu3aK1tbWkyPBu0Exr8kG1+LZDiFsEjP9CMPg+V66U7lPZ9nHCYe/+cD6vKP/553JHCYW87k8pt9YVx4mwvj5Pe/tfIsSTHtGz7cMo1Vj+XjXBLb7F9xBhmibptPdLV8t4nXDWIniN5AE2YvHduHGDTCbDyZMn6erqquoaGtmr22nhLO7Efvz4cc6cOUM2m2VhQb83ttN1KqXAsTBe+A/YJ68iYjeRVtEddOgURuKmdrwS+rtbpQTG6i3E3ApqDKwzH0N2zSKtOzjGUb1Vh4FMVppHnw7iRE5jrL5fckwoC2P+b4ks/ADsNNHenyA+8Fmswz8NwSFgq9Dw4cOHyWQyLC8vc/v2bfL5PMFgsMQ6rDeEfz9afLsp2FL8ENP8NqbxbQz5IUoFUNLbm1Gp45SXuVPKJBbTiWEnoVBpxHEu14phZMnlHAKBP6WjY8sXrov8bNTNuXUOa9sYg90IbnmQOPDCFwgEGrb4ioWzuF5lNRZeI/t09Qifa+ElEgkGBwcZGBiouT5pvcJXCdu2uX37NrOzs55O7I0KrZh+CZFdw3jvNZx4J9aFK5jZLQtQyV5AL0gyWSGNIX4aY27LNSkA84M3UIaJdf7jFVOonOBpjOR7+nky+n0/ZegtXCd+GmNta35z7i8x5/4SJUzs3p/A6vt7hPOHiMcvlriqlVKFivvlFU/K9w63sw736x5fQ9etNjDVSxjO32LErmMYpVGJjrqIIW97hulTE85iGKue4+UeatuOkc0KYrFvEQxmC8fz+TZM03sT2Kibc+scO1t8D0v3dXjIha9aV+duRHXWKnjl45stfNlslvHxcdbW1jhx4gS5XI6Ojo6aF4RGrcxiHMfhzp07TE9PMzAwwNWrVz0b7I0mvsux79w7V2IF+doK1qknkG23EWl9bU4n0I9cqVCwWiNIwrYwrr+OEu1Ypz6OjI4jizo9KKFfMJzAIWTKu2gCiGyFnMOA16UllIVIzxL+4X/Pk4C9fAL70Kex+n8Su/fHEUaEUChEKBQqcRm7Ifybm5usrKwwNTVFLpcjEAiUiGEsFmt6eb8HRlSVg1TvYKq/2RI79QYCC0f0IKT3+6LoBko/Q6XiSDnpfa7qxFt6rAvDWP7o3+04zjqG8Z+IRp9GymzJc237FIFA6fhsNsboqEUsNl/Y563nBsKyrG2DW7LZbNP3kPeSh1r4YGcXW6PCZxgGGxsbvPrqq3Xt4e1mM1oduVyO8fFxVlZWCu5DIQQLCwt7mpZQjFKKu3fvMjk5SV9fH08++WRF4W+k1JlSCmPszzyPmTffxGk7ijraiUIgKP1+OMHjSPQWn0jqcq3AaTuLMX8Dee0VlBHAOv9xZPw2Mj+DyOoCF8AJDSFT3gg/ZbQiK7hgRdZrMQAo854bykiOY4z+O4Kj/w6r6ylAYvc8vfWn+0kwt9yrlUL4i63DO3fukExuVQhJp9Pcvn27KQnezbT4tj23yiOttzGtVzDy30dabyHiWQSJ0nMEhzGF97PX5d7Z9gVM07vXK6W32LnjHEcpB6WmMYxvY5rqI+Gc8jw3EIhSLpy2fYmWllaSyWRhnxcoSbWo5rPazuvkNufeL8FH1fDQC99ONJKHNzs7y9jYlmXwsY99rK6gld1oRqsjl8sxOTnJ0tISQ0NDnD59uuSLW6/gNiJ87r7n+Pi4p23RdvPVa/GZq+8hNvQWlRMYwHzlFexjZ2Egj5EevTc2XyEQJdCNXNEHw6jAPeERdh7z+isoaWCd/zGkeVd/kY7+c7cjpzGT3oAcZbQgN/V9AUWmwj6oAnPp+5iL39/6rwzgtD9CfuDnUPEh7M7LqPjxkiHBYJDOzs6SQCfHcXjjjTcIhUKeBO9y67CeLgV7ZfGJ/CRG7ofI7I8wzBsY+ZcRRa197OBFJO94ziEC3s/KcQaRAa9A6ZrDOs7hEpFUSuLYh1HqFqb5UslzbfsiZtl3xrZDSKnrXPAoPT099PT0FD3XLknGv3PnDrlcruSzcpvQumJXjdfJF759xG5bfOUuzUceeYRbt27VHam52xZfcYDIsWPHuHr1qvZut14BqyedQSlVKA21srJSU7J+vXVFhRBE73iLUhcez23t6xq330dNSazHnsYQPwI7h1x7XzvGiQxjahrYAsgNb1CLcGy4m0XcGcM+/Rj0WBiZ68BHQTIVAl5Q+hw9O3YaM+OtfOEEuypGp8pEaYUQ4eQxVt6EfAZjbStqVQU7sDsvY3c+it31BKrlOE7LCBj3bkqklBiGQX9/f8n58vl8wTqcmZkhmUziOI6n/NdOzWV33eJzUsjsTWT2QyK59zgffZn41CjC2bKY7OBxjNCEZ5gKeIOaFAbS8N5wOJxCasrWSen1FjjOcaS8jVIxbKsVyQtI8TzSPKy5+LjnSDp9kni8vB9fGNs+5XmuYRi0tLR4CjwUf1azs7OFYhThcJhEIsHKygotLS0ed+lObtD9yEMvfDtRrUVRaQ/PLYRdL9W0JqqEYRiF8mGWZXH79m3m5uY4cuRISYBIpbH1Wny1jFtdXeXWrVuEw2EikQjnzp2rec56EEIQvasvhaSMMHL+3XvPVQ7mW9/Hae/DPv8IgU19jh6W/nU7kcPIVe8iujWZRADGh2/Dh2APniJ7KIY0E4TTFcQqOVnhRemtYyc2jJl+3XPcjp3A2PSWT1NmC3Lj3kIucquYcy9gzH0PAi2I/DpKGFsWYetpnLazOK3DdKQ2EZsdqOgRMLZu9AKBAB0dHZ5gGtfiWF9f5+7du2QymYLFEYvFaGlpKbEO67L47E1k9g4iN4XM3UGoWYzMNWT2JiI/hUBt7aMGZ7dWuqL7NRUaALyfmZTeGxjbPIspvKktQnq/D7Z9WpuTp1QAKx/HEN/BlFuuY8t5GlNMlj3PxDC81wCtUHTTpZQkn79MLUt4pc8qk8nwwx/+kHQ6XbhBFUIQi8X43ve+R3t7Oy0tLQ1Z5UNDQ7S0tGAYBqZp8uabb7KyssLnP/95JicnGRoa4rnnntuzAJoDL3w7sVPQSqPBHrVWfikfm8/nmZiYYGZmhiNHjmgDRCqNree6q/3ib2xscOvWLaSUnDt3jng8ziuvvLJnewVmZp7QqnexAkhHRoiued1Zcm0e52YS27gIfRmM5D1hUhgYK3o3oxMbQq563ZlKgVwuFTdj+ibRaUj0P4V1uA8pbyLz99yUdugYxrrePSuT+uPaZoGACh8CjfDZLacxlzSWY+tpjI2P2ikpG7E5htwcw974ECM1xqMAfw4KgQr3YHddAaFQoW5UsGvr71AnTrifVjNMSzACPVFUfxsYh8g5Jqn0liDOTc+TTCRwHIdIJEwusURiPokRNwmbDsJOgcoic3OI/NK9P7klCOUxUtcR9r3UACfQhwx4XYFO5DjS9u6jCrlBeU0DxziM1IghOitQhZCGrhfjIPDBR/8OY+eOQXoaM/5/IURpYWk0KTOOc8ET+amUJBy+t+do2/0I8TK2/VPea60RNxk/EAgUmgBvzbHlLg0Ggzz//POMjo7y2GOP0dHRwcWLF/nt3/5t+vr6aprrhRdeKKmP+swzz/CpT32KL3/5yzzzzDM888wze9J9HQ6A8NW7yBYLXldXV8WglUYX8XotPtu2WVxcZG5ujhMnThQarlbLbgSp6HArwFiWxcjISEm1h0KKwR4IX/zudz1BKwVU5X1FsTCGsTmHuiXYOH2JeNs0Mr+CHdMntAOIXEZ73Gk/gzGrrwNqri1gTo5t7QOeu4LoziLT76CCA5RHCQI44SPITW/AxFYS/qjnOICwEtrjSP3rV+Eu0DSbV9F+SN1b5AUKkVnAya9jLr3seb7V+zTm8vc1x5+ic/VVz/F850UCqXeg6B5BAVaog4BTFgUZHkDmvBaREzuBzHmFT8gclP28lIgjHa872wkOIfHewEjTK5yOuoghvCIpRBbbOorKgJF+CVO9gRV6wiN6SkltAIuu6Ww2e5JwOINt9wI3MIzvoFQYy/q7nvG7hesu/cf/+B8XtiaeffZZlpeXuXHjhicgqh6+9a1v8eKLLwLwhS98gU9+8pO+8O0lrvvOMIyqBW+3cK22anEch+np6UJPvJ6enpI7tVrm3a20BLhXXiyVSjEyMqKtAOOKbbMrdCilCE7puzEoBOH1Se1jmegxwnNboiOUovWD69jhGNnLV5GiQsSbDCFX9HuCKqjPxbOCHYTWtiwx4diYN7YCWZy+o6jhNpxgLzJXGqxSSfic+AjGhtdlqmQYua63UGWiUgqFfv9SWN5CyQAir1FJ2LLKdPOmJz3HlBHFTHuv044OE8h7BX2Tbto0RQKE9HpNlAggLe9nY0dOY6IpJG5mPYcc0Y80NO5i0UGxq9SyOlGZk8jcKIb1bumTg97vjqMuYWgiQqX0prFYVhvp9BSRyHeKjv04UF0vzJ3YaatnY2OjUKS+q6uLv/N3/k7Ncwgh+Kmf+imEEPzar/0av/qrv8r8/DyHDm0l//f39zM/rwveaQ4PvfBVm8uXy+VYXV3dM8ErnjuT0VsMxTiOw8zMDFNTU4UUADcZvR4MwyCb9f7QayWbzTI2Nsb6+jrDw8N0d3dXfM+bZWW6uEE0E+/9iEtzSaxIG2a+tDee03YWY1qfTG62HIG5UlEwMkmM115jvf0Mqv88bfb7JXU/nbazGLP6Du0ioReAXOw40Q1v0WyxtIA5twhOFnvkEVRfCCP7DsJOIyz9d0SFtnoJluO0nsFY8V6XEz2K3NRYGoE25IY32EYZUW33CmXEkJve406gQxtoY8eOY2S8FpLTdgYjpSkgHukHjfBFY6JQ0LtwDiQq4/1Mneg5DKXpmRgIedyciiAGOitQn9oi5RxKxbHTI+QW5ohk3sOOd2IES0VPITECmveVw0Cp8Nn2MIZx7zvj2P04+TTB4F8TDJY+17J+2vu66mQv6nS+/PLLDAwMsLCwwKc//WnOnDlT8ng1LeR2k4de+HbCcRzy+Txvvvkmvb29PPHEEzXXNmzEktlpj684562np4ePfexjhcr8jUaENiJCuVyOiYkJlpeXOXHiBGfPnt3xi9tM4XODaCKRCI+paWLvX8OKt+FceRK5dC/wQ5mVN8/Fmv6O02k9QtvdD+Au5AdPYB+NEU5u7RFuJG10Z3RC3cjlCrl4Fd4Dp+csxp2tup3Gh9fgQ1CRONbZj4NloYSJUKXflUpWlzL0LWqcyBGt8NmtpzGX3/A+v/0MxppXmJy20xWOj2Cues+jYochp4ui1Ef3Ctubr6hEEDOnia6MncVU73qOJ3KCNs1PWSqvBeeEzmobC4tAeSRlHDv5KGRtjNWbmOp2YREVQe/nagfPYgrvtUnDa9kpNQTcKgieYf1nhDxDMLxY9rwglvUp7wurk72o0zkwMABAb28vv/ALv8Abb7xBX18fs7OzHDp0iNnZWXp7exuaoxb2V1XYXcRxHO7evctrr72G4zicOXOG06dP11XQtxm5eK7L9dVXXyWRSHDlyhVOnTpV0o6m0SLV9Yy1LItsNltowXL16lX6+/ubWudzOxKJBG+//TYTExOcPXuWixcvEr625RIyE+vIF17HlldQka0flVzTRcyBEz2EsVihYHXb0cK/A9PjhF95Byt1Abv1HG2OvlHthujX7jEqYRBcrVQVxlsZQ6QTiJkk5nffgPdiWM5V7PijKBlESb01BiBT+txBYaW0x5EVulSYemFSRoV0lArfA2Hr9xtlxut2VWYrMqmxJlvPIhxNXd2QflGORbyfTcoZRCqvBadM756VIoA03sdxurEST2FPPQpvZ2HOwVx5BaGyRc8NItF4EoLe2yLHOYGUmlJ1ysDKtCHS/xnT/guEUChj0PM02/4EulzBeml2S6JkMsnm5mbh33/913/NhQsX+Pmf/3meffZZAJ599lk++9nP1j1HrTz0Fl/5guw4DrOzs9y+fZuuri6eeOIJJiYmGjKzXattN0TTbVA7MTFBe3v7ti7XZld9Kca27UJ5MSHEttVWKrGbwufuKabTaUZGRu6FQac3kTdK8/eMaz9AxVvJP/ETBCZf0J7PaTmOnNP1SQORWPccM2/dwF4aQrV3Yfe2YKyVLnrxkF4YkpHjxFf0wShyRb/3pgLxwnWYb722dSzaSv7SU8joKkb2JiJfFOEY6kcmNdaVCCLXKwilJvoTqNhtQldqTQEy4X1tytC3XnKiR5FZjfUZr5DAH2wBzf2lcDTRnIHDmPak53ig7SjgFR3LGsf86OerlMBRJ3HSJ5DL08iV90vyN4Vc9wTMWOEzBLjuOa8uwd1hBMm9QCnbOomzGcYM/BFCOCVBulJTmzOf/4znWCNU4+o8efJk3eefn5/nF37hFwpz/dIv/RKf+cxnuHLlCp/73Of4+te/zrFjx3juuefqnqNWHnrhc9EJnitU96sLO9wTIKUUi4uLjI2N0dbWxqOPPrpjkncjQlKtq9O1jG/fvs2hQ4e4evUqb72l7zC+E41WfRFCFEqwra6ucvLkSXp6ekpuWuSPvoOwvHuXIrGBGM9j2ReR3QnkeqkwiFRSO68dakfOVgheaRnAvLEVvWifvwR9DsbaDZQwMCq4OYPxHtAIXyY6QHhdb43KVa/4iNQGcnYTc+INlJTYQ2dRhzvJ5u4Qig8i05ok6razGMted54dP4GR0Lj/ooNa4XOiR5BpzfGWUxgp7+t22k5jbHpbLznxQeSGpvKJoV+WpKWJ5gz0YeS876cTGULamghNualJYxggGMiQST6OvWoRWLpNMD/KRjRCR7D0hkYZbUhb830IeQNNbOOYvoC1zG6Ja/4Mam0VM/USquXjW6JXPF6MYBilIq2UiWV92jt/A1QjfI1YfCdOnODaNe/3rquri+ef1+faNpuHXviUUszMzGgFz2U3OjQ0InypVIrXX3+deDzO5cuXqy4G24iVupOrs7itUnd3t2dv8X50Yb9z5w5zc3McP37cU4Kt8Nw3/7+K5xFL8xh3b211U/ixT2Ak3kJYaVSgFTmnD3ixOkYIrer7+YmNe4nKxrvX4V2wz17AOdpL4PZ3tWPMRIXSYq2DoBG+XOwwwVVdWSyQy25kqIMx/j6Mb/2greGjWOEr0BFEMI9MjiFQKEMfgq6ih0EnfPGjyKzXOnJiR7QNd1WkFzTCp0z991kofTswmfa6gp3QYaRmj7BSGgMyq0ljaCmkMTiiH8c6BgmJysQxV/+GMKX7bm3RZY+Fuc4g7Xj37Axd5GfoCOWpKY7ThbIjOGsxjMzfFo6LoHevVhlDFDeiBbDtq8Dutgc6aL344AAIn2VZJJPJbYNWTNNsKMKxXuFbXl5mdHSUTCbDU089RTS6O+HJ1bDd3qJreVZyte5FM1oXx3HI5XK88cYbDA4Obl+RJpNAvqOvuuK0D2JMbu3hCdvCfPFlnP4BnAt9KGlqAzsARF7/uTqxbuRdr/vOeP8GKvMJ7Mwp1Il2jNW3ER/V5HSihyp2fQjk9XtvdnwANMnxubZjBFc1+X7CxJj4ISJ3z4JVsVaskyOodBS75RIyPVGSolAxLcHRR5Jq99kAkdcX0NaKpAwhk5pUg+gwhiaa04kPITPe96FSGoNRlMbgqBAWR0GdRC6vIhenkJuzhWhNu+8RTwkAOzaMYXmvo6U96hHUTXuQFo37VASLu6d342yeQW2YBPiL0tcmu5HCe+MlTW8FmN2M5rx3zu1Lkj1svfjgAAhfMBjk1ClvPbtiTNMsVKCvh1qFb3V1ldHRUYLBIOfPn+f69et7Knqgt9pcIY7FYttano1YbtWOK7Y4lVI89thjO75H8tqfI/IVwv7bhijf25FzMzA3Q/6pz+C0HUGuly7Qtgxizun3xJzOEcyFCvlq87eRS3dgDJy+Q9gXT2AkruO0HkeuaLoxhFqRC3p3atDWF8zORXq0wpdoOUZrWfCMSG4gJ+4if7TlolYCnP6jOIf6oSWIsFNbUahFeXxKBvRpDDKoP262agtoV3SLtp3FSHvTLVSkT5vGgPTemCphIMuiPB3ZhW2eR1gWJB3E8gJy6TZBbmENdGFulJZ2U2YLMu0VHRXrgzIXqhIBbeK7Feii/LuVs1sJyA9Ibg7hLLcRW30XU72M1fso5TFPTvQUpnil9BhDSGOy5JhtHcfK/z3P/I3i1uushC98+5Bq3IF75epcW1tjdHQUwzA4c+aMp4jsXlJsta2vr3Pz5k0CgQAXLlwgFtOHwhePbabwLS0tMTo6SmtrK48//jg3btyoqiqNfPM/V3xMrOhFSgXCmC+/BHYe6xNPY2xcQ+S2IhDTbcPEZ/QuUJHVC6zdfQJj6p7bS87PIudnUbEW1JNtOK3HkGUdI+yuM5gJTQpAIIac1wtipILVlZf6mxWr4yjB9S3RFQrE7BRydgrryOMYd7bSElS8DfvQAHS04bS2YLAMzjoyu4CwtqxCp+0sxrpmn7DtFOaapgxa7AhyTWPxBaOgeQnC1rj8RAAjc09sldGGY/TgiCMIOw9JC7GxgVi+i0wt4xyzMVdKhUQhMNJeN6zdfhYz633vhaYYuRM/j6G8Yh0Ml7pIFRFQV3Empoit3btuixjSueGpMKcC3gIBjjmCZOtzyaS7MNN9kAyg2vVFERphL/L4HjQeeuGD5vfk2ykJ3a1bKYTg1KlThSoIu4Hbr67WHEK3wPXbb7+NUorTp09XfV3NamnkCnAoFOLSpUsFC6+qnnyZFPJ6hWot7QPISb3lZg9ewHx3a8E2X/w+Tmc39uMXMeZeR1UobaaCMeQdvSCqlkOAJkLScjC/+wJYGTJnzuH0mEQ239tygyr9Z2f3nMWc9IqJCrchF/SvJ5rW9wvMJdbRvRpl3EuPEYl1zFtbEazWyacxJu4FMTnRFjLxNkzzMIooBA0wFUgLQRYlOnAiQ+Bktlyk9tbfFffxsjMoJMgwSkbIWwIz2A422OHHUCoEtgR7K3dOLs/DxjpyfQGRXcdgHXW0H3PBWzJNpr2WcL7tNMG85j0zgTJj0gn2IHOavcqg183pmIeIhbesPVudRq13Ydy5gdG9hJErm6/9AlKUWpyWasEwNBanXMFxOnA2jxFcfAkpbpJp+T+9178L7CR8lmWVpFE9DBwI4duJZkV1bm5uMjo6im3bDA8PN8Vd4IpQLcKXSqUKqQDnz5+vuSL6blt8yWSSmzdv4jiOVoCr6aAhv/dfUOYZiMwi1kvdiU7bCQxNmautB0vfN7myhPybJezhUxDVv0a75xxmpYCXNb1laQ+cw/xwa0z4/ffgfXC6e7EvnUKk9GOgQnuinjOYt73dGJz2o0SWNVVZzDAxTeFrBThzFcqalVlpMrVJNLWJ40hkWbCNEiBaepFlgTsqGEVGJ1BWGBAfWToCp20A+e5thOUAKQQpQoB19Djmgvd9tU48jTHntbR0Ead22wmMtCbQJNIOZbsZShgYGY17tnUY09KUE9MlvgdPsr7YQ2sihbG6dS5lxJF5bwCMCNoeNyctF5CitH5pLn+ITCJMdPM9TDEKAhwVYD7xCSKkCYfDu1rlZDvhq6cl2H7gQAjfThbfbrs63ULN+Xye4eHhHYWlXqsN7glfNXdkbomzzc1NhoeH2dzcrKsNSL3CV/45ZDIZRkdHSSaTFet7uuN2mk/+5X9Evvk2KhbH/vGnkVOvID6aS6xWcHOaIYwJ7wIFQC5My99eJ/vIBcyeHMZikQVQwUJzWvuRdyv02NPUipBLC6ib3cixD7AvXkQNxDFWryNyya2cuKXaytE57UeQGuFz+s9iTGnSCXpPEVrwXm8m3kdYE0majvUR0Rx3ek5hLGnSGPpOY8x751WxXsSqplhABXe2rrao03JEe1y1HYZVr0AZljcwxmk9i2Fr2g0ZWU80px0ZwXC2rlkRweYiYtlCLGzSkSrN37Pbz2GqUvepElGkprqMCBYlwasW7OQjyKRJq3qxxCWaFj/G2iZMz94qtHhyu6u7rZ5qzastXO8O3dfh4WpCCwdE+Hai/kTwZSBaSGAvtqSGh4fp6qrOH1+P1VY8didr1c19W1lZ4eTJk5w7dw4hBDdvVlqkd56zEYsvn88zPj7O8vIyJ0+e5Pz589v+sHa0+FYXka9vhYaLZALjL76PdekCdniJoJTIKf0+mX3kYsHNWY6bNB66dgMlwHrqY8jgHGJtBmOmQsBL1wnMOU1VEGlgTOvHqEgnQoFx/R24DiocwXrsSVR3CHPS2+FACYmxUKkLu76YdMXqK/Ee0Aif2TcME94UgVSkm0jGe1zFe0AjfCpYIY0h771OhcBY1ezBdRzHSGjSGNqPIhc1+Y22NxIyE+wnrEmUV5E2rxUoI8icxvUY7sdSrbBuYEzdwMy/gRNoR7R6K9KIoIKymCS79Twmpdaski1IeQOlWrGTlzCmr2PmX8buG/FYhrL1lznRe6Lwf8uytE1lI5FISQPgaqzD7aI6M5lM1elV+wlf+GjkbqaFQOA3kPIXWV7ectm5glfLOd3WRPX40bdra2RZFhMTEywsLGyb+1Yr9aYzACwuLjI1NcWxY8c4depUVdezk8Un//Y/IexS8Tev30CEQtif/imM6fmtbujlOPobDSUEcvLeIiwUmK+8gTIMrJ/8FMbSdQTeai6VkuCdwfMYY96qHgByvnRBFpk05iuvY517GjXVhn3pNCKcQi69i3As7L6zmHe9loMKtiDn9AIvl/RNckWyQjeGrF5AQ+Wr+Udklu9oeoaD1JRmVN79GgAAIABJREFUU6FW5KL3OpOtQ8Q1RaxV+wDMa9r/WN4gGCfcqY04zYb7CGvKlMm8pmpM2zlM5Ua/RnHkedSGRCwsYCTKIki7zmLmSt2USkaQeU37qpDwiJkdPA9JE+POdUzr5Y9ew9GCZVl4nhPGCv1MyTHTNGlvby/ZPlFKkU6nSSaTWz0P5+ZIp9MYhlGwCl1BLLbwtitZtr6+vqsxCQ8KB0L4mmWmZzIOS0snGRj4n7lw4Qm6ur6CELVHajbajLZchGzbZmpqipmZGY4ePbpjN/ZaqdXV6XaWGB8fp6WlpepmudXOZ/zlf9Afz2ZxXr+JCpxEDZjIoghNZYYwJvUBKvbgBcz3vI1qhW0jbicQ15exfvzjSGsSubblQlOhVuRUhYCXkD5x3O4ZwrgzqX1MLM0jV1eQ39taWFVbG9als6hoByowjsiXBo3YfWcwJ737Y3b3SYwFTUJ4rBupsfZUqFUbSeqEWoiuafa4Yj3EEl7LK9lyRHvc7juDOeuNorTCbR4rCfS5gSrUilzTXGPXKcyN1zzHDZn0BKXYsSGM3KT33IEWLOspxHIWeeddDOsHOPEBpKlphWRmvZZdx3lMVepFUCKIoe59NxzRi5MagdUcZrb0ep2Wo0hKBXk19zQhsX2kNWytc9FolGg0Sk9PT+G4m8ucSCSYn59nbGyskMIQj8fJ5/Ok02kikYhnrXwYUxnggAhftVTbJDWbzTI+Ps7a2honT/59lHqegYE3UepXsKzfwHGerGneepvRQqnwub36pqamGBgYqKo5bT2NYat1Dbt1R8fHx+nu7mZkZIRMJlOT6MEOe7TTY8gb3kAPgHTnYSLjHwUc3BJsXHmEaGYMM5Mgf/gcwQ+8+08AhPQ3L0oI5PhNRD6P+fwrKNPE+rGrSDWN03pY25UAQM7pLS7VMQga4XO6BjBmSnPIxPo65n99Dad3CFYc7IuXUb1RZPrOVv5hpX6BrYdAJ3y9w5gTXovP7j+NeUcTYNJ3iuCMt1Sd0zuMedsbCBLqOwpTXuFLpJLauiOhvPccTqQDuapJcu86i7ni/cy1yeyBVqK2xpKMD0BqEiVCOMHTqGwrYn4RY+YdZLY0PcHpPI7cKBW+rfw/jWUXNDxi6LRcwOBtHHECZ70X485bGMY70OGNBJdyxlNSbSX39zjknalqTNOkra2tJCVBKUUmkyGRSHD37l3Gx8dJp9MIIYjFYrS0tPD+++8jhGjY4rNtmyeeeIKBgQH+7M/+jImJCf7hP/yHLC8v8/jjj/PHf/zHddU5boQD0Z2hmoW9msU8l8vx4Ycf8tZbb9He3s7Vq1fp7T2Obf/8R/MsEgj8jxjG/055F+Wd5m7E4rMsi5mZGV599VWy2SxXr17l+PHjOwpMs9ISYCsZ/vXXX2d5eZnHHnus0Flit/P/jL/6RsVx2fi9NifCUbS+fg0xFSB5+DKZjP793nJzVujScOwCsihQRlgW5guvIf7rLE6uC7tv2DPG7h/RJq0DiFV96oHTO6Q9bvcdQ85OIrJZjDd/hPnnryBfuIO9fAw2gtiHrpALluZbiYR3zwvwuIYLVPIMVOy6oHd/irSm7Y6UtGa9llM63E0kranD2XMaoTSfu6HpemEEkZtei9vuOossMveUMLEjp1GZOPbGZXhHYLx2HfOHL4Md8IgegHC8x+yuswhV1rJIBDAsTV9A2Y29/Cji+jjm7dcQTh6745xnvB05iXRKRdqhlYTzCc85G0UIQSQSobu7m1AoxIULF7hy5QqPPvooAwMDKKV47rnn+MpXvsK3v/1tfu7nfo7f+Z3f4Rvf+EbNv+GvfvWrnD17tvD/3/qt3+Kf/tN/yujoKB0dHXz961/f7Ze3I77F9xFuZKYuuimXyzE5OcnS0hJDQ0OevSnb/kWE+CaGsbWYGMZ3kPI1bPuXcJzPstP9Rb0Wn1KKVCrF9PQ0fX19XLlypaY7p2Ykom9sbHDz5k1M0+TixYslyfCNzFfJ4pN/9R+1xxUQuutdTI3VVaKvpLGHz2EfHsG4Wypyia7jtEzoOxWocAU3thEg8FffQ6RT2FcegR4w7mwleavWXnSNYvOxLszpCv36NN0gAFTPINzVRDKKGObffG/rUgD7xBBq6DDE0BaBVjKg3Q9UQmhbMykhMJd1XRdC2oozTrwbuajpqt57DnPZayUZfcMw77U+E6l1j3WopImh6TDhdJ/HSGkseMNkgxGigT7E8gbyzigisIxp3PKIqoq1eZvcRvu0nSVEQEFZILjTfgHD2boGJSLY4jLi7irm4muexHwRyHgsQxU/BJRa50l+FqlpV7VblEeTG4ZBa2srra2t/MEf/AHf/OY3uXPnDl/84he5fv06H3zwQU3bJtPT03znO9/hd37nd/j93/99lFJ897vf5U/+5E8A+MIXvsC/+Bf/gn/yT/7Jrr+27fCF7yPclIbi0j35fJ7bt28zPz/PsWPHuHr1aoUPvZO5uccYGNhydQkBSnUQCPxzHOdPsayvoJTXGnCp1eJTShXKiwkhGBwcrKttSL0Wny6qM5VKcfPmTSzLYmRkRFvpoZE0CN048f7byNt68Uj0HqdlTO9itE9ewnztDZQUWH/3KWTiFnJja+GNtOsT0BUC52aFPcGTFzF/uOUeNH6wJXj26WHUqa6KeX3ZzmMElr2WhBPvQE7qg1TEut56I16aBmKMT8L4JNb5q8h3pnE6O1HDx1CdMYSRQgVDmLdf9ZzG6T+Dodvf6z+LsaCxZA6dw7irS5MYxpzWvO5oG3hfMljeoCBlBGjLe12lm5EhWjXdGFQ4gsq34YSPoewWSFiIpRWMO+/Qmt2g+ObDHryMuVhe2QVkSuMS7jqJuVkayaqMCDLjFXAVDuPYR3CyRzHG38HMvorV/7hH9JxgJzLnHS/lRMHN6YghnLUeNoyfrnlroBaqLVB95MgRjhw5ws/+7M/WdP5f//Vf51/9q39V6Me3vLxMe3t7Yc7BwUFmZirk2DaRAyF81bg6i3PxLMvi9u3bzM3NceTIkaqCQ6amnubw4TcLrUWEmESpTqT8IYHA57DtL2Dbvwp4795qESC303g4HObSpUusra3VXWC7EVenOy6bzTI2NsbGxgYjIyPbpnDstsUn//hPcM78GOLWqx7XXbTzMFQQPvGRm1M46v9n782j5MruOs/Pve/FHpH7vu9KKSWlpNRaUtnlNtjY2NiAsQvo6WraA83gpj3Mac7MNObMgYbjdmPDYWYOMO7D4jHQM9Me2hgPdrugKFNSSaV9l3JVpnLfMzIiY33v3vkjUpkZ8V5orbLB8u+cOqV4GW+LeHG/9/e739/3i/k3Z9GRMNZLpzDuXyhe5mzfi/emk/ACkEqkHKxGY3AEtZaCZArr2ClkfAy5Q2xaptwVTVTjLsxVJ0FDlVYhJ4p46c26+/ix2ZsqV1bg/DZoWvtfQE2Wo1qaoCoCfhAqhgpXugKfDrn3ehZtV9DukzgZc/bSaV8Ez6pLBlezx1UaLVhZi4quYZnVZO0AdkKj1zcwFicJr0Uxdnji2a0HEGkn+1NoJ9Cq6l6MpEtmh5Phalf2YVrbBBaNge0/CHMZxPQkJjsA2284WiZU+S5MlT/xsEO9GOpuzjUi1o5x/y2kzyLechjzHewhfxy5svb29qc69te//nVqamoYGBjg9ddff8orfGfiuQC+xwnTNMlkMty7d4+ZmRkaGxufiH2YzdZg2+/CNF8HQAgL296FYZxFCAvT/EOk/Dts++dQ6gPs7E59HHeInbJnu3fv3tL5jMViz0SMeVogsiyL4eFhFhcX6ejoYPfu3Y+cYLytGV8qhfHnf4ZYWyW7u5NMVZrQUk46Shsmskh2piJlyALGpojFMf/qNNkjxxAlGrG2uNX8/iB0McKL109wwr3RPFleR2jiIvKvTqOlwD5xCCo1LIzin3EHWFFESEE19jj0JwHs+g6MKTdLnCDGmEvbAyCnx5Crq8jV/HVo0dSBXg2g6mqgsgJd4gefRFt+MlV7UakoXp1EJFcRykauuJBGTC/SLTssa0KuOd9v1/ZuqbVoaaJ9ZeCJoHx16HI/KA+kFWIjCWsx5PIMcmkJL0tbEmx2Qy+Gix5ozLIcpVLlL0WuuHwukUpHNqp8Fci4y3PkM8AC5W0ntVGF5/4IsszCSOery2gjgOFCgBFmzEmACdShU5UY4xcwVa71IlP7cWyln7ox/XHindTpPHPmDF/72tf467/+a1KpFOvr63z6059mbW1t67xTU1M0NjY+7eU/dTwXwPeoAdm27S1204OS5pOWF0zTJJn8cSKR17e2STmJ1iGE2Nh8PQb8AYbxJ1jWr6H17q19Ewl3W5qNjQ2Gh4fJZrN0d3c7qMXP4sL+NP14tm0zPz/P3NwcPT09Dyn/up/v7VpTlH/1VcRabuD23BnF9JjYH3gROfYWqn0/xiX3xnTVuhtzxlnmAxApA/O1N7H374IWP8bY5jrdJpvTLXaWOQvDXNkeSYXSGGdyosPRgcPIak0oO4dc3i7zaG8Aea+IKHbK/fnQVfXgAnx2+x7M2y6i0a27MCZcZLrqmpEzueMY9ybgXi6LVDVNyOUCM1TTxP7Z30BM3sSevoj2+RBeE+31oEuqEHYUvGwTYgQoXxmSZrBssG2EZaGzFnohgjVdj1iPYiQTCJbQYgmjLomM5vfe2fVdyHUXsC2thLWCbVJS4qLWEg03UZ52Ap9MuSnS7MKMF/TpGRHI+rBXdmPM3CFI7nqshh6H5qddsw/Tymf6Kl8tMrN9fiXLUXYfxuhtZCZf8s2q/SnsRdthC/Z2xsN6+ODZ2hk++9nP8tnPfhaA119/nc9//vP82Z/9GT/xEz/BV77yFV5++WW+9KUv8ZGPfOSpjv8s8VwAX7F4QP+fnJwkEAjQ1tZGW1vbUx0rlzG2YiXfjeG/ihBRhEih1D6E2Fm6Ukh5Bo/nfSj132BZ/8Z1jS+ZTDI6OprXFO8WzwJ8T7LvTkPfiooKKisraW5ufqLzvV1SZxsbGxi/97+xcx4qshbG195A9XaAt/gMVawUIY54vBg3c7Nz4/ogXIfkod3I8gyGL4h5y73MWUg939pc24Rvwr3UKlcTRM7dRguI7e5EtEQILA1jN/TgHXKR+ApEigPiijszFOk+mOmyKnAFvhbkgot9UENbPvC96yOIT30Oo3UXvPqf4Bv55CKr/wXM286Jhd2zF+N+fvajDRPCk4hEwRpY0y6MZec16spaWHeu78moC2g19GGsO7+zSIkfCj6yRLCR4Ibz3pGpzesMYAf2waoNGwbm5N/nn0t6MFwYpcKnHNJnqrwL055HGZWobC/G8FVERRyp8kHPjhxChXqwZke+62t8b7czw+c+9zlefvllPvOZz3Dw4EE++clPvq3Hf5x4LoFPKcXMzAwTExPU1tZy9OhRlpaWSCbd110eJx6sESr9YYzVL6IiH0R6RhFiHq09CLG53iLvo9QepLyNYXwJKf+ScPhTzM29ABT2CHZSXV390Iz1WTO+RwHRA2PakZERKisrOXLkCNlsluFh91Lds56v2H62bW+tJ2YG73L06mX3N8+uYA3OoF46Cfev4kluL7Co6kaMwSIElZ79mOfyM6TA5TtoAdkf+QCqKYGcyi9paq8fY8hd61PVtyEnnOakOhgmdC93HKEhcnsUboMK+ImXhDBqewkvj2FY27Uwu7kX844zq1SV9Rj3XZrQpYFRxI1CLjqzIAARc2+/EbHNtcG+o/CvfgsOvmv7j6d+BPxB2MxGtRTIKRdQLa10dbNQrX1b7Ne86y+vAhfgkzFnW4hd04mx5qIQE45AwfKe9oYwXMqc3ro2WM4nWGRliI31FCLWR3hmDDOby9ysnqOO/dNVuwlY+co82lOCTLhMlowUVuYkxtBlTCsnSacDXsc6YLb2J3P39xAdzbcjHqfU+TR6voXx0ksv8dJLLwHQ0dHB+fPuPa/fqXiu+vi01szMzHDu3DkSiQRHjhyhq6sLj8fztjk0aOMQSp5Arn8ZFfcDJlr3571X6+2GUCHWiER+k87OX2Bu7n/n0qWLO3oEax5Zpn0nM76VlRXOnz/PwsIChw4dYteuXXi93u+IEW1hLC4ucvHiRcrLyxm4dd2xBvcg7N69iGQK/Y0zZGZN7L3Ht/6m6oov0guryOcsTcxvnUO8NopVfxi7uXv7XJ37EEn3EqRYcGpaAtg9fciMcz1XWDblb16n9G/uIoc9pCsHSDTvR3l8RU2Si92P6uhzbYmw61qRMy66l+XVrkxSVVaFzCThN/4v+MO38kEPIBCCF7ZZfnbrHuS6C1O1pce1H08H3I2F5apzwqCqW5BLLmuZFXXObQJk1KX8W9+HsF2axtO5kqr2lGKVHsPiCCJ7kLKbdyiduIWxqZJjG15YcQEzr3MYtWv2IPT296y8TWTN92LcvIU5eAZh5Y6pzQhGMv+YWnjI1n4sd5xHlCKfNR6X1fm9Fs9FxvfAzXtsbIzKykoOHz7s6Hd7Ox0abO9nMJJfR6ZfQ2dK0eEfRXsEQuQGa8MYQqk2pBwHckshlqVpbfk3NDcNYNm/idaPp9XwTgBfLBZjaGgIKSV9fX2Ew+HH2u9R8Vi+ejvigdTZ6OgokUgkx67VGuM/fbnoPnpmG3A8K1Hsr53DPt6P6Ykip5wDKoAOlyBvuGtpJrt7CV7NlejM1y/mBKvfNYDwxR9a5jTGipFX3CdXqncfxuVcFiviG/hez6mkqPIyQpESrK7jiMVRjOh2nS65NI8b5UYH3CXSdG0zzLk4HbR0Y94oqP/VNsAv/zvErj6oe4gS0Xs/Aa/959y/S8sKjchz9+MCNloI5JyzbGnXtWMsu4BzTTNy3FnSlBtODU5V24ux4ea955ws2eV9aFmGTgWQt+9gqpwijL3rgPOam/sx1/Mzb0v68CZdJg0yNyGy/b3oWATjzkVEVyvCyv8s7Lq9mOn80rBV+X7w5JY2vhPA98D30i0SicQjjan/McZzAXxCCDKZDAMDA0UXip8149sJBto4gG38CIb9NYSOYsT+BNv4KLJkDSFnAIHWjcD41v7l5TNoXYmUl/DKH0Kp92HZv4HW+x963me57kJW5wN3iXQ67UqkeRDvRCP6zthZXq2qqqKnp4dkMomUEvmtbyKK9P3Y3b1ww2XAO3eNVF8foqEM//ISYiNfUd/u6MOcdSe85FrCt0NoML99CRUKkTm0G3r2Yw7lg+bDypzyjntpVBvuwgOqpRPzb3JrxFqAvbcb3VkDYoOwS5amNdhjg64/bBEtIkq9kzhTVQM/8yn46CeQ3s3fyvAV6D7oui8nPgjBCCRiyLlx5/UEIsj7zntWLbsxZlxcEGoawYXA4qYEoyqakcsuZc6ySihIxLXpw1i9jfJXooKdkDCQExNoyjCv57tgqGAZctGFjel1Prupih7CKj9js8wS0imb9HwHkeXc86gBXHwCheFsl7A2y5zwzgPfw46vtUZr/bbq/P5DiecC+ADa2toeOli/3Wa0uazva1uvRfYK3J/Hrn4/IjCCEINks5V4PLkftJQWtt2PIV7bfP0tPPwdduanUeIX0aLP9bzPusaXzWbJZDKMjo6ytrZGV1cXVVVVj7QJeqdKnWtrawwNDREIBDh06BB+v5+FhYWt/TKvvooZCiFcyn92WfEewmxpKdarZ0jWVBE6uA/vtbcQm8cUUae1DIAdDOG/475Wltm7F/XGW6SBTF83nrYKzLuXEJZVvMzZvQfzonNtQ3u8GHfd1x6R2z9RocG4MQw3hrGOnoBBD/aB3eiaECK9jJwZYaOigfC8MzNKhcvxuZBadLAEee8WlFfCKz8PP/5PwV9gYxSfBYoAny8Ap34E+855jDlnlmt37MYcdbnnkgpwWW4UMednp8rqkC5WTKqmBemiByp3iGMrfyUbRh3+UDNyYRQ5MYypd4Bo2GnZpJp2Yy4UsDm9YeSqizanx95ic2pPCbZvHzrtJ3L7b/Pelq7chT+dfw9pXy3exPakQAsvduhULuPbjIdZBr0d8ahSpxDie86LD54j4HtUeDyeZwa+nb142tiPbXwUw/4qAFJOkBKH8C/+JWnZjFm3H2l4gW2GmJSjaB1EiNx0VcgsQo7gUYdR+sexxf/sAMBnfSgXFxeZmZmhvb2d3t7exzre42ZuhfGwYz9QfrFtO69P8cH5lFLY9++z9nt/wEZdLRVdPZjXthmQKhBAX72O2xlUMIh1PZeV6YUl4t9awuzbRajMRK4sI2+5DGhApquXwHmnMDOAvfO7vjVM5hZkmuowjvThu1HEnd1yB33Vu3erzLkztM+Hcds9QxSJJCIWx3hju3dMR8IkBurxdzUjSCIWJ5DR3CBvtO9G3HD2AcZ276fk3fth30uw/4jruehoAsuCYgPkez+BXpgAF+BDFJGaWxx3bFOVjRjzzmOohg7MCWdJU6Tys0BteLBr94HPh/LXI+9PIWemibCMdSCEUSBOYNd1YKy4ZGGFTBPAbuzDjOYLY2tvCcHMMFagE7I1GHeuYmbO5CoPBWFWVDhaLlRlKyI7j0YStXfjn5tnraKetdnFLfugpzWoftx4GPAppb4nQQ++D3xb8Swec+CeMdreX8FIfnXrtfDMoTMCn5qEmUns0h9Al1YjZG59RYg4Sh1FiNe3r8u8DpkABl9B6v+3KAA+SSilmJycZHx8nGAw+LbbFj1JZDIZRkZGHqr88qCdIfXFL4Jtk52eYX56hpJTJwgP3UasR7H7DiBOu5crrQMH4M38Qd+6NUgUMD78ASLZmxhTLqamcXeWr11Vhb5xwwmyU3OkWjvZmMniP34Cf2IZ416OdamDYUfj/IPQhrsPo927F/OiE3h1aakrWItYnJLb9zAXtqnxqqUB1d0IlGJ3HUJE5xFLM4jmRnjlJ4l86P05QLvqrk8KQGkJ3LkMu52MRgCOvR/5J/+L8zq9PowpF+Hohi6MRef6nmpoQ4662P+4GNeqsibAxKp7ARIaMb+EnBgHGcQcPJ1/Po8XY9alab2mHu7n37eKVLmXOc0CQWojQLbyONmZEULzozzQ2LQrWzHW8qsE2vRjxJ3H9Mk57OBhmF6mbDl3fbrvk2itmZ2dJR6Pk0gkuHXr1hYQRiIRvF7v2wZIDwO+WCyWNwH9XornBvge9aA864PkBnzL0Ubkxg9SW/4qAD7fDEocQ6RzM0cj+jfY6y8hKlsgeA0pLYScRWsDIXIgLOQ6SpxC6NMIdA4A1V9gZ38a5fkZtPHCY1/jgx/UvXv3qKurY8+ePSwtLX1XQM+2bcbHx5mbm3uk8ouUEpVMkvrjP87bvn76LIm6WioOdaOLlCsBsutO2aoHsXH1JrG5WUqPv0B4bAi5nFsHU1U1+O4WKXP29CDedGZPAPbUFCSSpF47Swow9/YSaChDmgbmORdHdY8X406RMqdwL3HZu/ZgnnXplevuwT+c394g789ALImIXUFYFhzoRv/qr6M/fBIh5RZ4611NkE4jijVLu7gWbIXHi+w+ACP5PYiqvQ9jwiWTrawFF+ATqfyUSPtLUKUNIPxY9SchqWBlHTkzh/K0Yl52fp5yxVnmTTb3EF5x0cZcd+tb7MGcL9Dx9JdulTnt0t3oTDnGnZsY2Tm8C/nAqWubYDGfQGQ37MdM5pd7sxUnkMurGHPbLTRW3QuEWw7nyd+99dZbtLe3E4/HiUajTE9Pk06n8Xg8eU7roVDoqX7HjzKhfbt7+P6hxHMDfO907AS+aDTK8PAwpmnS0/TrqLUUouxyroRizqHTArFpxyz1JZiQpEUAb1c/0nsNpY4hxPaPT3hG0RkDsWmvIoRCiHt4k/8EJY9TVfJB0MdBFHEU15qlpSVGRkYoKyvbcnGIRqPPlOU+TWitmZqaYmJiYssz8FE/WCklvm9+E+0i6mzNzbMQDOGprqa0thYxn79GZLe1oW66lzLVvn2oG7ksLPrGm6wHg5QdP0Xo5jVUezfm1IL7fkvuJBF71y4YzF/HsW7eJXYT9MkX8Pe/gD+xijm8TUqxe/dhXnbJ6h5W5twoouJSXQ3Dzr4+1duLWZ6Aj78IP/Ux13KwCHjh3DU47p7V6Z52yGYQHicJx54ewtjfCN8ovIfce7XpBY8v93/TC5aN3bgXDB9aekglswjpw5fU2J5+WIki5xcRG+voI3sxzzgnGbrO2XdoN3e7ukvgcd6x3diDseb8rIS95thmNexDAGJ8CuN27rtT5fXIhYI2BEBuuJROvRnYLB7YJYdgLoEwwVjOn1hld73i2FdKSSgUIhQKUVtbu7U9k8kQj8eJx+NMTk5utbzsdFkPh8OPdGt5GHnle9V9HZ4j4HvcjO5pjFkhB3ypVIorV66glMpzKNBDexCzw6hdPUh5BeU7jth0XhbEsIOn8MdPw+AcqvIwujqY12Ep5GwuU9TbawzSvILORpDqHPvazqESf4rt+TTK/EmQ2/TknWSRAwcOEAhsCws/rVbn08QDpubGxgYbGxscPXoUj8e9xFcYQgiC/3dx3z2rvp7YmTOshcPUnzqF9+xZxCagZ5ubYXzcfb8CGrdOJFh9/TTrFeWE/UHCgQCyQNQg29oKQ0Xky4qo6+jycqy3zhO3LOKAbGvGbKqhbHEKDPefoN27D/Oii+RYeTmyCCDKifH8DRE/vPIixi/9MDRvMnTvTUN7EW1EWVzAQQT92DdvYezdQXK5fw9m38A4FoL6/ej/qQo9uwqGQJseWBhGr7LpO5dBAHZLK+bZfBHuMJA8eMS1SV9knJm8qq5HTrowWqtqYTkf+JQ/RHDZpcm/sho28rersnrk0iaw+ctRJbsRC3HkeBRjJh/kVGMn5kx+Q71q3odR0LSuQlXI9ZvYJYdheh1j8HJO91XnZ4XaW4bV/tH8fR/y2/R6vVRUVFBRUZH3/gdO68vLy0xMTJDNZvH5fHlgGAwGH2uM+37G95zEg6ztcQfkB5FIJBiugV1wAAAgAElEQVQeHiYWizEwMOBQOrDa/i3ec3+GvHQF1XkCytbysz5xDUuHMMUGcvkietnAqn8/RsUlhNjMLoyVPPkjITZQnhcR2Tdyx9BDyMynUMk/R5nHiNk/zd2RHAGjkCzyIJ51XfNxIxqNMjg4SCAQIBgMsmvXric7wMWLeIqUHYlESF3NETxUPM706dOEdu2iWkr00BDZO+4WPyocxrrmVA0BsKtrmHn1VYyyMsr27KHqzh3EppZqtqkJJlx64QwDNehkHgKwe3feGqManyQzPsl8KIinNEvoyAv4RgeRO7Q9i2XvqrvXvczZ04PxAJA7a+BTPwivnIJIID/Dm7xfHPj6G9HRGKLUfV1HJjefxeUluPtf4agXWjZ7vKRE/MS7EL/zF2BDuns3viHnmmaquobQgvPzk0mXdbyKKuSYM1tXbZ2Yd50qLnLRmW2lW3cRmM8vt2qBq2C2qusG1YpYSiGv3cC030RVNiBwOVfCpV0lEsrz89PCwK44hJwfxbi7PYmx2/ZhRvOz2GzXx8HMd7t4UtUWKSWRSCTvt661Jp1Ob2WHi4uLJBIJpJSEw2EymQxra2uEw2HHub5Xm9fh+8CXF08KfKlUirGxMaLRKF1dXcTjcXd5H28tdsv/gHnv15GjZ9HBGlTbuzBUzjxUEGNDHKSUK5uvbeTCfRhPYHe+iCy9jDSHUVY/ku3BWhhD6KyJ2IGIigRm8rcp07/DwZYfQJb8ItrjbMaFZ2uFgEdnx25MzTeLrI09LOyHODSr/n706Xwyw8bgIBtCoI8fp/rOHdfSXqKnB+nCpASwNmfR9toay5cuES0vp/bQIcJXr6LGx92Zo/v3wxUXI1TAXinigr6nj+T5C7kqmCEJ7t9PsDyMb3ICowjT1K2NA0DX18IBH/zMYTDL4d1F+j/bijcr4/Og79xDHHDfV3RWw7f/Hzik4aSzDYCX3w2/8xcAGGH3JnrPirNdIVVW4dpqobp6MG85nxcRd5ag7bbdGC6muEI6RSlU026M9c3MrqQBFWpHLMURt2cdLRmquR3zfn7fhd24C2Mt/3q1J4CxdnPz3yHsyAHkyDiGPewAWeFiSpjd9c+d9/Q29PAJIfD7/fj9fqqqqvKOHYvFWFlZYX5+ntHRUWzbJhAIYJomV65cYWVl5alLnalUine9612k02ksy+JjH/sYv/Zrv8a9e/d4+eWXWV5eZmBggC9/+ctPZJ79dsVzA3yPk9o/bkvDA/ui5eXlPGLGw/Qr7eZPY0x/EZGZQyQWkMMGdtkxZNU4Qs8TCg6ikhGkzs18pX0H5T2AMfgG2l+B6jyA9qVgB04JOY8yTyCs7QzAlFewdBOmmMLPq7D+Kkp0YPv/O5T/x8Fo2Hrvs6q+KKVcf5gP+gKj0aiDqflAveVxF+Izk5Msjo8TqqrCV7C2poH0pIu4MIDWRBcW2LAs6o4eJXD5co6SvxlmPO4qvGJ7vSSvXcvT8rNWV5k+fRrz0CFKgkGCqRSiYL1RFZks6eZmdJFsVe387G1F4sp1EoA4ehTTW0K4uhL/4jzmaO65UhUVyFsFZc49NfAzhzH+2QCUb4LRsLsQNwAtpXBzBPYWMUYOuqgXJZLYdy9gtE6BJw2RIqDa1QiHu9HXxjGGneVYu60D76wzK0s1NuMfdU4O7PVlxwCl6powXJzrdXkFFOChDpXgXShgWAqJKm9GByoQE/MYt0eQzGA3d2LEnM3wMu6S2VVUwXQ+8Nmt+5CZcezgAYxb1zE3zmC19GGu5X8Odl03RrRg3+oBVOVex3neyeZ1wzAIhUIEAoGtCozWmmQyyczMDJcvX+bChQssLCzwjW98g/7+fk6ePMknPvGJxzq+z+fjtddeIxwOk81mOXXqFB/4wAf47d/+bX7pl36Jl19+mZ//+Z/nD//wD7/j7uvwnGh1Pm6YpvlQ2TLLshgZGeHChQuEw2FOnDhBXV3d460JGiGs9l/deimys4iYB67EUNlTGDKF5St4+M3ctYjUCvLWGeTQLCr9Inrn12YUzKAFSH9L3iapxzDSX8O70IW58lFk8i9Ap59JO9OtTGrbNqOjo1y4cIHS0lKOHTvmaE940nMu/O7vkrh0iYVEgvTJk+R1hR08iOVSdgRI19djjI6i43Fmz59nvqUFdSCX+arublSRdTp58CCyiEXUWibDzOnTjMZirBw5gmrJfc4qHN4iyRSGbmlx3W6VlmJfd5dJU5ZF4tYdFl4/zf1bw0yW1rB2+ATJgaPocBjKffDPd8PrPwdXPw2fPoko35GBdZfCsLM8t3VN0eIMTdFVAQubE4xkCt74JjrxVYxDc1BpQkmm6L4AvPwSqm8fIuZk0uqGBpcdwJdyUS+prMHrAnDJGqeUn5YC6do8vwdpZ8mUNGI1v4AVOYyeLsU4exnz9TMY97aZpbrWqflpt/QiVwrW4gwTYyU/s7QrdqHTEcSNVczzpxEbm/de7iwZ66rq7X8jsCqOkun+lON98J3X6RRCEAwG6erq4vOf/zwf/OAH+cIXvsBXv/pVfuzHfuyJloCEEFtSh9lslmw2ixCC1157jY99LKdD+sorr/DVr371YYd5x+L7Gd+OKKbeYts2k5OTTE9PP7Yju1uouldQk/8rMpGb8YnsLdAG8uppUqWtyI4YWpYgVO6HI+1bqGA/MpErb4r4DOpeE6hmVEc90nceaYwRS/YQCWwPEoKLaFGB0NuzaKFuAx6M9Dcx0t9EiwpU4BOEzL2bjNAnI/TsBLCdlkWPYmo+CfBZKyusbLYwiESCxTNnCPb3U7a8jJyaIvuQ78DT0UFmdnvw3xgbYwOoOHSIQFkZFMnOs0VKiUQieEZH0YDOZFi+cIElIfDv2oW/ooISl3U3AHXfSa8HyHZ0YLqVRsvLsQpANDu/wGpshfAvnIKffYHg+1sQXokaChafuc7OQ3cRvde95TlfPNM5qAop4c5NGExDzzK8aCJ2yrbtLYf7C9BS437sHz2J/i9FyDdzLvZBdY0Epl16CDu7EbecJU2PS5kz2di9RWDRQqCr2lCROkiHyCxW4709DeT6A+29BzGm8j93LSVywfk86MoKKODW2G39mKuX0KYfu/IAqZEZgitxTPFannC6DpRgLOavIWtPACOa+26t8gHEdBQ5N4r14Q8575/vvmrLAy++8vLyLWeFJwnbthkYGGBkZIRPfepTdHZ2UlZWtnXOpqYmpovID77T8f2Mb0cUAt+DRu9z586hlOL48eO0tLQUHdQfKcIsTayOf4+WOUKAsFdRFTnnBn90As+Vm6jMCbTcsaBs5gOxkTqPvaExrpwje6MOO3UMw59/PUJk0J4ChReW0d6B7dd6BSPx+3SX/T7emV6M1V9BZK7kxB4fIx6USRcXFzl37twWU7Otre2hk4InAb6l3/s9VAEQJa5dY25lheR737tFaikMUVJCssjfVgYHGbt8meQLLyCqq/P/2NxMtkjrg9y3D13A8BRakx4cZDUaZaq1lfWDB7F3yH2lOzvRxUqxUfdypNi9GzarDsInCfxgA9Vffi+tCz9D9W/tJvThNsSmG0B6vnhWR3cIrdy/S1HqhavO8qveSMEbfwvBb8OLUagtMigWybIBKAtjtDh7Ae3ObuSMC/C1ujtMiLgzK7Wb2vAW9MipYCmqrIFY7QAxcxfWZAh5+h7q3CDG3/8d3sV88W0ddGYtqnsfcj0fULXpQS66EKNCEazKk3Dfh/n35wjP3Ee1tDvcQuyOPkQ2/3mx2/pRoXbs7G7Mi5cwZkfIHn4FPC7rpfzDsCR6FlanYRhcvXqVqakpzp8/z91iBLXvQjw3Gd/jxAOHhp2N3jU1NY9NvX8AnA9brNVVP4TKfggRGkSmriJTF9C+WkR6HgHIsTfQuhrd04cQ53JZX2A/MrlZFhOQNkrxWODdmIErM5ihBlINx/EFL24RXYS4hcaHyLOFdg6UPu8iwp7HjH0BYl9AmZ2o4I+jAj+K9h4omglalsX169cJhUKONomHxeM6NKhEgsXf//2if1tLpUh3dRFMpwkUtCuI/ftRBYSXB2H295N4801m33wTIxjMtT9cuwaxGHZLCxQBqqxLDyGA0dxM+vZtskBqYgIjEqHm0CEC09PoIj5m2fp6zDF3pRSVXSP0k02EPtpA8AO1WLMW3p421/f6DgRRKQvpd/kZ1wexhxYxe4pkZtkdwLu4ih68APtX4EUBBGBwHnbVuu/b5UMrlcsOXUJ85Dh85e/ytunaWph2ZlVizZnBqbpGjImCtTkpUU0dUFmN1l7EWhJxbwYxvERw9hpyPb+vz+7sxjuYLzFmBYLIey7+f2E/FKwY2F39mAs5Jqb2BLBr9kMMjG+/idjhlaiFRK663JeVvx5tV/WhY2CObpe3tTTIHi9uwvrdtiR6Fvf1nVFWVsZ73vMezp49y9ra2tZ5p6amaGwswjB+h+O5yfgep9RpGAZra2ucO3eOaDTK4cOH6e7ufuza9mMJXQuBPfCriPO3UNYJkF50Wcf2n1UCwk3Ii2fQ4y0oox9b5h8zJG+gfW1br70bMzCWhOuVqMSLaEIIsYL2DOTtJ/U9lOdQ3raAdx7l3d4mrVHM9f+AufLLeCc7MZd+AbnxdVC5zCuRSHD16lXi8Titra3s37//sUEPHl/nc/mP/gi7CNiIykpWL14kMTjI0v37JE6eRG9SuDWQLJZlAamZbYaenUgwdfo0k4aBdeoU2SKZjOjqIlukVUEUrOHZsRizb77J2OwsK1pjnTgBBQzHTH1+CdJo9xH6b1uoev29NJ/pofbPjxL+eBMy4sHbEyAz6G4SKyMeMkPujFGA7ELxMpI+WIuamENdfBUd+Rbi1CqiZMdvZMG9eR+A+hCMugtx63QWAnGoy38m5LRL+0BDM8bEdole+wKoumbs9j6snhNYHaewqgawrTYYNjHODGN8/QLm/3cG48xl5Mwc9r5+B+gB+DacIgPWrj5kNt8L0fYGkPedYCh8Aru2D6viONwzMF97CzDzQA8g3tyDLBDWtht3YazkwNCu3o9l7IPZLJ7R/P5Fa/cH0WXNjnNvHec7AHwPO/6ztDMsLi6ytpYTA0gmk7z66qvs3r2b97znPXzlK18B4Etf+hIf+chHnur4zxrPVcb3QPPRLZaXlxkdHUUIwcDAAP5ChfrHCNM0H4slqcu7sAd+CfP8f0CX1sKeDFlfC550rhQkYufQoWbk4j1YhI263VC7D0/2xuZ9gArWItLjW8f0W9dQdCCvvoH2RbC7X0REXKS6pEu2JZ3XLNQ42LMYsT/CiP0RGh8x+zALseO0NXyAGW/1EwHe1qkex/Xdslj43d8t+vd4YyM8IIYoxfKZM3hqaqjeswdPNkumSJuCp7+fhEvvXnZtjfl0mkQmQ8PJk3DpEqR2NGTV1MCIU2JLS0mqyFqh98ABVs6fZwUwAgFqjh0jkkwibt7Em1wg8NEK/O8rw/eDpXi6AqRet/G/272spNadxrVbYa8C7lmd70AZKpFGBvNLj+nLw+jsEDqTIPCiO/lGHwhBxkZ4i1jWpJYRbAO4thVcvgrt03BKo372IPqPpsD0YFdWI60kdlUDWnpy5CylIVCKaVSSnl3CvxpDrkQRTMKCF3kvn2Fp79mN4dK0LnwuJrDtXRhTzu/Lo50N+lbPXnxT243ziUg9aX8DwRvT+Obz2xjkuouOaNDYUmXZ+iyqKrGtg7CQwngr95xaAyccXoXZE//Scby8+7Dtd5Tq/6hS6rNkfLOzs7zyyivYto1Sio9//ON86EMfYs+ePbz88st85jOf4eDBg3zyk8Uz3ncynivgc4u1tTWGh4fxer10dXWxtLT0VKAHT2ZtZB/7HzHu/DkiOoU4O0+68wTCiGKqKAKbuFlGmFzmEpm7g9rYg11zEhm8g7BXEKnzaF8zIr2Z3QggUA2JMUQ6hnHzDbQ0sTvfj4iMI8llLNK+ijK6kPb2wCCtayizC2ltbxP2FMp3GJG+uHn4NCXGGUI1YMS+QIWnguz6KaR8HyrwHvB0PBZB5nGAb+pP/5RUWxvG/DxkCsSBg0HXNabswgIzCwtE3vc+zI4OtEsp0X5I5p5OJkktLDC2sICvqorGgQG4dAll21i33bU0PQcOFAVZa4d7g/ClSVfcxHfCT8UfHyXcLxBG/mdldhfPgs0ujbYUwnQO8r79Aaz5OGats29Ohr0kz9wjcLIXtZEmfeMOZtUYvkO5ZzR9vTigiogkdWEa/xF3YBRdHlQ0gSwNkjl3HVlzD/PI9j3pH6ln7dNnwdZ4TtZhnHFOOAKtrciJCXZ2Fto93Rj3XIgm1eVQkMjrkhLkkJMZq+trYSkf+FRtg6svoCnTqKpWVLAZMTpN8MI9zONteAtAL97YQbigQd4urSK4ul2S1dLAbjiMmFjCmNzOZFW4HGMun1Bj1+7B7ixwtC8Iy7KeanL5uPEoE9pMJlPUv/RRsX//fq64kLc6Ojo4f95pU/Wdjuem1An55c5YLMbly5cZGxujt7eX/v5+IpHI2+rJ99DwhLDe/VtbLwMjZ0lOt7MhcpT7sH0DVbptbyJjtxEzKbiURukXQQbQwab8+0u8hQ60br9WFmJ6HPnWIGpiN8o+jhZeMAtIHQLnNiDrooovM1fRIoxHrBDMfg3P4r/Cd78Pc+ZHMac/jrHyeUTi26DcRaMfBXx2Os3Qb/wGc2+8wWpjY071ZEd4BgawixBDvD09zHzrW0zev0/m1CnUDgULo6GBeJEGc09XFxs7SC3ppSXGzpxhKhQievw4ush3qtyAXkDwhQYi/ffo/oMaDt9o5eRqF/v+upHWX60Ej3KAHoDZaJK+7F62NMsl6cvupUdhSDJ3HlLSzK6S+PZraOtrBI4P4enavhfffh/ZseI9f7KkePVCBEwyp98ie+Ov8B4fx+zIvyejOYTnh5vAMJAupAbR24t0mcDoemf2qr0ejFEXl4e9exGZfPDWhkS6tEGo9nwCit3QTbb7JRjbQJ6ewPzWaYzRXDnWjDktkHwNzt9HtKYBqWxsT4hU3THUeg0kjDzQA1A9fQ7n9eyJn3Mcz3F/30Vyy9NKN/5jiecu49vY2GBkZIRMJuNwGX+7zWgfFar7o6jW9yIn/hYEGKlpAlfXsFv7EPVJEB7YOS6JFcgkkeffQEeq0H0S7W1EZHIDX64EWo9Ibg8oMjGIKj2InLsCc6CDFegOE+VvQ6rx7UNbF9CyCqG210Z8DKLMbqS1PQMXbKD8pxDJfPKIyI4iM8OwkTPf1Ui0tw8V/iEwalG+PrR37yOB7/5//I+kpnI1ocS9eySkpPL4cbyXLyO1Jlak/w7YIpNoy2L+9GlkSQnVx4/jOX8e2dkJMzPuO1ZXu5YyM8vLJGdmQEoaT57EHB5Gba59yepqUrevEew3CB40CR3c/H+/gbIUZmmtK8Dp7CrgHEQBLFNSdH5tu/cWApgF/ACVskhfHESExwmcimEvVmOUulcxrMlFPB3uJVZPN9gzUYyG/L9nb8+iUzcwOuN4eovLz/l/vgc1VY1wyYrNHRqTD0IbEjnq/H7t/v2Yd51C3iLhFJRWe/ZjjDsZvWJ9Hrv9EDrrQ94cw7g7jH5XLcZIfknV6u3DnMvPDHW4BHMyP7PUAoJei/XSg4TuDuHfJNIkwvmDqpYSub6jX9BTgl11nOzBlx3X6Ljv74D7+qOA9XsV/J4r4BsbG2Nubq6o79sDVufThmEYTwZ8WjPZ+8vUT1/Ca60RtBdRbS9ijL2Bvm+gBl5AVRxCruQGDpm4h6o/gZg5i4gtYZxbwm4+iV3RgEddQqAQifNofxMitWNBQewouyVWMG6+gV1yCuUvYaNEEy4ZRogUGdmNdwfw5TLBSrDyS0/CvodmW2sUQFrDKN9eZPrm5q4KkbkBGxKZ3i5z7ZGV2NE9SN6DMErRno7N/9qwEhYjn/tcwYekWD53DllbS+OhQ2S/UWABsBmepiZWC0Sd1fo68+fOEd6zh1CRtVdZUsJ6kdYHT2cnqdFRfHUQN84Q+VGD+hfrCdZG8XXFSY0blL7kLDEaZFm/kKTkiLOMFDkA6Zk0vgYnxPm7E1jrIcwSZ0nWNxDAXk1jlDv383aHyAwugLLJxifx90wQOPVgciHIDK5g1rs3j3v3muisjfC49fQJskPTW8CXuTEJ5gTePQ9IRwbZmyt49jpBDMDz/gbML6agAPe0aWK6aKiqA/sxbrmAltc5+KrGJtcsUG+uZ2ohUE1dxHUQv78S79k3EcntrEtLgZxxTnaojEB+B0TOF3FsWzrNbt6H8tfju/C3+Hc8V3ZvP8Hl/JJurLmXkthtbOEj4e8jODFCsrcPZfofWW77bpJbMpnME2sW/2OK5wr4mpqaaG1tfajv29M4iz+IQhf2YqG1Zn5+nrGxMaqqGqjt+WWsub/EXDmPWLmADtch4nMY599A1XSjmo4i13N1cZEZQxs+hJ07j7x/Bj3XRkbW4Nnbjsi8hQ615AGf3LiNKtmHXN9ujJbxy7DqJXJ/DeUNE6vvR5Sv4vHkt0CIzEVHJijsaZJiPwFdOAt2rkfI9DW0px2RzZWRPCzjsd5AxVaQ1vbMWiNgI8zefx3DDEFqEVJLkF7K/Tu5ssj88hUC7zuMuDaKPZ/P5JOtregpp7wUABUVjJw+TcX+/ZSm0+ihQcySnKRl8N3t+Cau4SkDTw0EWiDQCP5G8HdM4auElStQ/RLk9OK2W0LMaokd1xhhl+dJpQAn8Akp2BiKuQKfEZCsXlym/EWniojwSjK3EgQKgM+aipEZGcdOLhD5gB83KoSny0IrjZDO6zQqBam35vEfcwdGs0OSuT4J2bt4B5xZp15bBtyBT0iB92SGbIE4h9Hfj7zkYsUUcH4muqwUece5jqc6WpFXt79vLQSqoR3wYtUfQd4Yxrg7TCmQffexPNCDnMarcS8fZHW4BMOl3UEkFtCBEuz6fYiRaYxv34Djni0HkO39TQplOENlXqzsSeTIIJH1yyhPgMHu97F+6RJCCEKhEJFIZMs5YWcG9t1sZ/hedmaA5wz4fD7fO+pGYJrmli9WsVheXmZ4eJhIJMKhQ4dyRJqOf434d/+ZmLmLcPUaqrwFI55bZ5ALwyhOoYz9ULOB3BhF1Z9CTOVKjUKADpXjm7sCZ+ZQ1c3QDdpXj0jv6Nsr6LkSKoFVehBz6QwyE6dk4hpMgN30T6AsgwjNINUYQljYnl6MVH5pU4oMFMwRZPoi2mhA2PklRW3UbwHf1nszt1C+HmQ2V9oSaHyhGHXvBm8ZyILJZnTSpLRtDsh9LiorsOKa2CAEWiS29SYqC1qBtiGzCr4akH6QgbMIAwzvdZC59Slf2YPMfHugW7kMFad2njU3AfC7VyYxgorl85rKY05AiQwoMosW3mrnTyzQXvwZNMqKS4JZvihQTnYtS3p8Hk9kHF/nDGZTjtypUvVIvzOP8DRJ0tfj+PYXcdOW7uuxmQtTYI6ANvAOuIsVe46AWssiy9yzA88/ayP7K5cgs13eNl36/3Q4jHHLCXD2vj7Mqy5C1WuLWD0H0J4weimOvjuGqK/F+1/eyHufFQhgDrrIyQWc34u9dy/mcP65rF2Hcw/j2DTmrZzxrWpoRBaApqppxLi/vYasDQ9WywmM4RGMle3fg/XSP2XXkZO589k2GxsbxGKxPKHoYDBIOBwmmUxi2/Y7tt72feB7TuKdrlc/bI1vfX2doaEhTNNk3759hEKhHTt6sD75RUK/9gLMBxCHJaq+Hzm7KVW2eh2SJgytoo6cRGQm0N4SRCbXruBZvUIi1EFwYwy5OAmLk9gtL0JlBzJ7EaHSyPg1VGQPMrZdHtKxK1gigLmD5i2X7sD4AkLbqOoOaGlAyDVHM7yPu6Rpxcf2eqIQGtvbiZHMBz6RupInxfYgbF3mKPf4K2D5OlQezN9e2pwhOgalmy2P0qPxlkPlcUiueojUODPttSEo6YY8ZW9g7oJF3YuOt2OGnNsAIp0QvQGl+5x/M4IxwDlACFOQuJXA+5ITMAKtBrGrMSIHnEBUss9L4l6cYHt+CTU9lyS7OMvi3w5R9VIUz4H8WYdRDhu3TEJ97uunamMVcAc+3yEv1twGZl0IbSnSN+YxQrfwHslleOm3DMAd+IRPkL2zgfeAO+1dZxXGh0qw/yK3HqdLSjBdNErt/r2Yl845touNNVRNHXZNA7YngJXMooUX7+tnEDuUaQRguPTzxXo7KS+wNlKVVchBl5JqMlfjVGU1qJoexNA0bHgwb+bL0amOdszBfEKRam/HHJvOSZnVDCBHRhGZNHIH6GlpkPnhX9x6bRgGJSUleQ4ID4SiY7EY2WyW4eFhMpkMXq93KyuMRCKP7an3sHiYWPz6+vr3rAktPGfA9zjxpO4BO8Otj++BV18mk6Gnp6f4LKplPxMHfpr2y19CfPscavdBdKQREZtGZNdRjSeQw2cR58+gw6WoAyeQC3+D0DYIUAXfpLx/BmabQYVRfUcR3EAb+QQHj0gQDw0Qjm+XnURqFlV5ArF0Nudvtknhttt+AErSiNAyQt1BCI1FRR7wAcjMNbQIIHaAqSCJ8g4gCrJGI30ZZZYhdT5JwSjSTZJc3ga+nZFY9hNwAT6rSPIddq/qUbILNoYh1O38W7ZIn3jZXovYsEWk2/lT8rUXJ/FY8STFgCg1sUawPczGcJTU9Czeylkie1bxVcPyaT/CcEdoZa4A7gDkPyKxV22Mcpe1PEOQvTVDdjCLp2sC/8H8dW7vEQtrKo3Z5E69MaqXHOdVaxnUtbsYx6fw/FuNnXMrwuzrQ7jpmmoLq6sbu6QMy+vDylpYhom4cQO9GoVb20zLwPHjeaAHoHu6MIac7FFvytnLqvb0Yt7Ifxbtjh50oAy78gDy0nVMeyFn+pvIVzvSgSBGIdHFF0AujWLVn0QO3sW8dwZV04Axnl/OtY59FF3T5rz3HfFAKDoYDDIxMcGBTWH1B5OZz8sAACAASURBVJ56sViMpaWlPE+9naXSJy2NFgPPtbW171kvPnjOgO9JhKqfpnF0Z8aXTqcZHR1lfX29KJmmMCYO/BSti5eRkzeQd65gt78LUdWOWDmHnDuLqu1Fzt9FxKPIN19D1/ejG03k6nnC6TFU7SHkpummQKHK6pATbyHOvYHy+om1ZfFFOglY20w2vxpBST9S7VgDycw4ySvzt2FsDqEVOlRBuqEDwlm0vxahtpUrhF7H9p/ESJ7JuzeRHXMeU1go314oAMSyHojfl4Rb8oGjag+k18FXMBGN1EXJJsFTsMRYsRdSK7kscmeEG2FtyE9ZT/66D0B6yUOo20lwKtsH2XXwuEyCEzNpV+ALtNrEhiHiAqQlh8CK2ZiR7YHKWreI31pHJdZITY0Q6lghVAD0pQcs7KTGCLiUV3dtkBgLEOxwApQwYe3aApUv5avGZGYtsrNzmJ338TaHEC7jppACayJWHPgawbowj3mkFr2RxbowjHFkCvPduc/RGDDR/2IX1vUIVkkJmVOnyNo22WSSTCxGPJMhcu4yFLB9A6dOwWpBu0VJCYaLCIGsr4X7+WQVu6eb0Ew+MUsLkJt6n9rrw27fC2kTpIn5zfxnVu3vw7xZAJD9BzBHdxBdguWonmOYN89ixrf3V51tmCP5lY/Mh/97x3U/LHaOVz6fD5/PlzeO2La9BYazs7PE43GUUoRCobzs8GnGsrdLruwfajxXwPc48azAl8lkGB4eZnFxMc+r73FCGx6sf/FFPL/5boSVQY6fRs93or3N6M4gqO1sUqgsSguM18+jOnrZqMgQUtE8cJELb2GVd2GujiAzKcqGbmDXHMQuOYyotpCJq5gqSjJylEB0u6lUJidQlUcRy9vbRHIGVXkcsXQOsbGCb3gFH2BXv4jwVKJrKhGBdYQeRFguChf2DNFsL6We/Fm5vX4TDJAFA258WRAu6J02/bB0EeqO5W/3hmHmMjQUbJceWLsKdS84P+tMke6AUEcWOy0xfPmDsBmEpQuSqhedWVzZ3jR2JoTh8shszK0S6XZqdhpBSfSyjZAbWEtrmKEo4Y44ZT25725tWOJ3MVcwwxYrZ9NUnHJPi5P3Y67AB+Bv3Qb02O0NVPw2pQOLeDdb59YuSMqOuzdMG3vTqLRCuiilAKSNOMk3U/i7h/C85FyntD86w/3/WoK3gHkLQH+/UyPVNBEuzE/v3r2IAjNj7ffjueNcx9ON1TCfD3x23z5EyIMVaMC4eAfz7iV0eSkECoQSDImcc7I+ZSI3ybOrOkhkSwgN38Hw3EbEtwFaR0oxJrdLqRpB9thPoNr6nff+DGEYBqWlpXlVJKXUVql0dXWV+/fvk81m8fl8ednho0Q6nlWg+h96fB/4CuJpWxps22Z6epq1tTXq6uo4fvz401kXtR7A+vDnMM7+H8i5u+ggiLH7iHEbdfIEquMkciw3szSmL6FaDyHHLhMZA3vvIah/ATG7OfMUkBQyr6BmzF9BpfuQN2+h6ltJtVRj2rNo4UHoHfdtu+hDZpxN1HL1GigbOZVbO9SGB5pKsKvfj/AsgBpFkis3hYNeKPhoPeYa8zegtmBMqOqxSccFvnB+SStcxGkn4K4HTaQ5R3gRBV9FZWeK5BIEqvK3+yIwfUHReApH+Ou9gDNL9FVolq8ZVB5wklYqDymycYUnLEmv2kQHk9hraQIRhSHmiOxaBze5xkzxMqlZkgbcB67I/uIA5SlRzH/tLpF9c0R2OZFfeuOAO/B5Si2iZ9YoPZmfPqusJv7mPL66aagI4Kl2L7WVfthL8P+sxPpK/qRIS4nfRVvVe/AgXLjg2O5x0RAVh/oRl/MFqXXAjzGcW9tT5ZXYLV1YSYWZlJh/n/9e+9A+zCsFmd2hQ5hj+SBt9+0HKbBLDmBcuUoESB88iK/A5sju24c5ehoN2I1HkDNLZH/gZx3X/bB4Wna5lJJQKJTHIdBa55VK5+fnSSQSpFIphoaGtrLDUCi0VSpdW1ujqamp2Gn+0cdzq9xSLJ60CV1rzdTUFOfOnUNrTSgUeqh10cPigau5+tC/RCeasRtOIdbuo/eeAECeOYu4NIRqPIUWIrein1nKgQ1g3LwMl0aIZw+S9OYWsiLJIVTtDsdswdZ0R85OEHzrIuaFJZT/vajQtkqK3BhGVeQLWsuNMVRlvvC1sNbRFdtMFGFnkRPX0UPTyNNXkG+uYw+1oKInIBMmzT6Uzk+NpEumZPphbcq5BhZuhCUX56Dydlh20ZEO1cHKbedgbHhgxcV1BsBXhOQSbkoRLbKPLpgoZOOalatZ1m54mPrqLNE3x/FFR6mpm6G+d5myxlUiDeusumtfU9YHyQn3v5X0WcT/f/beO0iy9Lru/H3vvfS+vPdd1d3V3k/3YAZuSEWQC8bO7AI0sSAIYldYRnCXCwqK0e5SIEUGApKWFEgGQ2JQ5BL8Q6TAUEBckQIEYDCDwfS097a891mVWZX2+f3jlUnzqrunewZDYnAiKjo6X5qXmS+/8917zz13yP0a9dbA+tUdxx3btElf2iD12hRK4QHe4DjBDvdwN3pYIz+6+6ZPju+oPy3NInsjjz51neiJcXxtKupEtRdmKeo/UZ0JUAYH8ay5FFAL1c8lDwwguRgNyMXyYq4tBPrp02i9B1mLdpEfXkP79iWs4WmUG+V1N1sSSAvVBtpC2qkZ2/4IRt9ZbDOIfPFWWa+hRMVr+/xISw8wm49gefpRrlzBau3FGqxIRzwGT6szcIMQAr/fT11dHd3d3Rw8eJDDhw8Ti8VoaGjY3rTfuHGDv/7rv+ZTn/oUV65cYXp6mjW37+YxmJmZ4SMf+Qj79+9ncHCQ39/03V1bW+Oll15iz549vPTSS6RS7ubrPwx8oIjvSfCkxGfbNsvLy1y4cIFcLsfJkyfp6el5JqXVdgO8JGH883+PdGMUe70JjA3sJmdumUitwLyOXejDajiAtD6N3Xdm+zmkzBJywUPg4jyWdBor1gNS+WImrd7DaimZyKAVkO5cR5yfwFroxgo8j+2pxS26cXMQERtD2BXspazfRQ87xS05OY107wLyhbfhgQ9xQce438rSmyEW3/ZhZWHlPlgVH3sgsoHtEvjYu3w96i7uW5rhvnuOtuH6/HUD4DJpBoB8trw3zyhAdsqHldGZ/Uae5f+SIfN6Cs/EGjXyBvWhReKJHLEW9zaFgvuQA+fY1O4JGXXR5bvZhBQqUJhUSV/Mow49JN49SeLgOpLXJrYftOQjPEvndm/HCQ/aZG9lSL+xiDZznXD/TXzNOwQRPqSir+4eqUT+Rxu5ofz3EXQrKTQ2Irv5o0ZcJpp3diA9uIfZ0Unx1FnSh0+y6I+hTi1j/Lfz+EYmt4UwysAeRMVv2zx5DGmxnJDNnh7k0TuYzXswOs7CnImYmke+VR4pZnt78VTYo5mHzmKbTcjXbm5PnlA/+3+5fyCPwA+jh8/j8RCPx2lvb2ffvn2cOHGCT3ziE/z6r/86ALdv3+aVV17hyJEj/PZv//YTP7eiKPzu7/4u9+/f5+LFi/zRH/0R9+/f5ytf+Qof+9jHGBkZ4WMf+xhf+cpX3qu39/hzfN9e+X3Ak0Z8j0t1plIphoeHCYVCO7147wK2hrsCUNuI/lt/hudXfwqmbcwPfxg5tYRQ80ijl7A6jiN97xrW6VMY6UlsXwzf5srvX76JXdOKdPUStgD7+GmsljNI8yVycavch1PklrE6PoQ0/gOYm8CWJOz93Zi1LyBtXEJYmw3zmQdYiUNIqR1lm9BWMGvPIq+U1F4ESIF41QRrb+omhqcGT3puZ86pCgv3gRRkClDQwQoF0KwCyYcQaQV/DAI1oPigbhByKxKh+nLWahyEYgr8FWnPpn0WmUWIVPSFR5ph+RY0VLROAOTmIbEH9DwUlkHL+SmuFNE38kzNCEJhP0FFIxgyCaMS9sNCskDDmernSnTA6kOo3Vt9rGGfQ9g+l3JKsMPA1HCtHcYOaxhZG6WkeV5dssjeVPFIGpa6TNxFASspkLsL3herjwFEj+bR16N4YuV7YmPDInsxh5VdoOYn3dOwkh8KDzx4at13JpJfEP9ffKz+jkPaUl0dlotQJdDXh1gq3xHYgQC+TTK0ZJlCaytqbS1mPIZUKGANT8GwEyIrAwNIFfVB2+vFM+wy3aFi0LPlD2J1DsCoF/nyQ8DZAdldTYg7k+XPGVbYzOJjdhyFDRNp6D7S6o4C1XjuH2HtLc+SPAneL7syn8/HqVOnCIVCfPGLX+To0aPYtk0mU+3buxuam5tp3hy9FYlE2LdvH3Nzc/zN3/wNb7zxBgC/+Iu/yIc//GH+ZaVT0w8JHyjiexI8qsaXyWQYHh5GkiQGBwcJh6vtquDpDV4r2yHskx/F/Mw/Rfl//yXK629gnvgIwruBNHkNsT6DFQgjXbqM4vWxduAgXus+Qs8jTA2zpQV5bQ5hg7h6CTvRhtl8EhHPI63dQ0qPYLWfQpopEbAs38H2hhFaFmFZyHevYjUMwqoHa/9RiGmI7O3qghmgp0eQkBDsLIrS6hXscCciW9Lrh0F6I0B9RSmpoQvUgkwkZG7WJJ1U19I0zn3XgAkoaqAKL0tLBonWCHou40h5ZLBlsH1RkDdAbJ7m5r+5Nci2etA1AyFkvL4A5noWNW0zNw+y7MHSDSTbRhZOpSvzTYjEwYmPiuAD6mH6gU3d8epUXE0XGEUnTVv9+ezynfth6Sa0uvQV+uOwcBmaXeqNSsQmfUsh1GuTuZxHtlSi7Qa1mx7l6ckwdLs3pofadMwCyC7lPDlks/5WnpoXnWtbW5XIX04R7sgR77exTVDnwLfL/FBfaxIrF0MKlV//xoqFdiVP4hMqqd939l2BvXuhcmCwEIixEv9MSYK2NuyBAfKFAoW1NfLj41jT07C8TMjjgYpFWXKZOKAcP4p0taK219uN/OAWFmD2HcIkiJiexfvd1xHqTkRtNTYgPyz3XTN7egkvDKN3nkBaWke+eAPj3FnkimkR6i/9n+4f1GPwfg+hLZ3FJ4R46p6+yclJbty4wenTp1laWtomxKamJpaWHpHueI/xgSO+R83kA4d8ChX1hXw+z+joKMVikf7+/kfKfLeitqdxVXfz+jQ/9xuI6Qnk176OdPV17NaD6PFj2PokxZZeomO3kDSVumtXMQbOItoE0vxF5OkrWF2HkSa3muBnEYkupO9ewdo7CJ0BUJewhez0AgJCTWO2O16hW5CW72E1HEXabC62wzHY68eIPYe0fhVpU63iN1ew6k4jkjuLixBghVrKiA8gKmYpFsBfsvDKMixPWHRWeB43dsDqPNRu9t75veBHw1cDwshURUq5bI5AtEIlakFeAa9qoCg2YICegSDYAchtQDhRvdmZHXG8NSsRcXfowheB5YkQDfuq04X1/U4fYsClqyWyizMMgMcl2svOwfpdsNZWiPpsalzEMZGWLFoKvC6iH28cUpchsUvUF+jIkhvyoU1kifXliZcMyBAyFB4IfK3uvyElapK5XCTyEefLNdct1Lte/PElgpvfbc0vQfIPQExN7TS3CIFoaMAeGEAzDLTubgqpFLmpKazpaeJ+P2aFQXno2DGkCoWnSCQI3KseP8TqStVN5p5ejGwrDI0hzt9GAsRHzyGulFvfWQf7UW7tELQtK1jte9Bu5wlc2JzS7vMhJcsNr/VzP4XV73IBPQHez8kM8O60M2SzWV555RW++tWvVhGnEOJ9NcD+wBHf41Ca6tQ0jbGxMdLpNH19fdTV1T32y9qqET7NRetaX1QUjH/6+7CYRSzfwlifRcxaKLZO+IUwVt9hpNFbTmpxaQxxL4Pd1Ia9vwEKSWxJRmy2QYi5G9jROqSH9+AhWO2dqP0v4F06j2Q5NShp8Tp2IIEolBSeS+p6IruOuHqRfGIA/4aEOXAMud6HUIdcu7zF6hVsfwOiuKPG8/lhcgi6KkiuqdVG0yS83vJUWtGl7OQPwsQIdFekD0Nhk9lRaKt47mAYJodtuvZXnJ+A1JqfcF11zSyyi5o70QgLD6B5X/UxWXKvkckeSN2GwLnqY9FWWLkJ9S5rZF0/ZKb96BtF8hMQkiFRD+EaoMZmZQjq91c/TvbCypCXpjPutUWvywbeNmHjIoiUiWUsk9glQxftB2MNlN02AG1FzKyf/PkigcYCwaZykkx8TiI19QLFpI5aX8/60hLm6irW0hKJ1lbUimkOgf37MV1qfvLSUqVrHoHBQURFFGkP9OMfd0gz09BIur4Ju1Cg4TtvIZUMHLbDITxD5copOxpFHnU2j3YohtlyEFIbKK9/B0+pQfWp4ygPHRK2JRmz7zTqL/9z9w/oCfC46ejPisetUdlslohLTfVJoes6r7zyCr/wC7/Ayy+/DEBjYyMLCws0NzezsLBAQ4P7AOUfBj5w4pbHEZfH40HTNEZHR7ly5QqxWIwzZ85QX1//RDuUsjrdO8Ruj9VDYYZ/9h+jPVSxm/qR9vQg5fJI3zwPcwWsjkEApPUl7MGjSJNTSP/1Ciz7sQZ2tvVCzWG3923/X5qZwnvxFvpkDLPmQ9iRNoSew2o+UPb60uoQVsfJstvCa0MUAy147l5Hev0CXFiDaQUz9pNYkUHszU5oYRtY8erRNfXNYFQEWb4AzIxX14+auyHjIi6rravqeQYgEHZfMCLumWkaWlSKLkLHWAMkZ90lnvYuZeDaNliZdO+ji7W6i2kAREUbQW4R5r8PC38nSP9XlZoCtDU5pFeKXcYeAhCp1TB3Oc9QB6zecn7+2oaH1TfDqG97iAUg2gKe4O4DUCWvTWHM/cO0dIE6GyLzmk6kL48SqY4M5aBF8Pkh5s6fJ3n9OvrcHFaxiK+tDdVlUobfpaTgO3AAe6w8wkKW8YxUq5KkjjbSZ55nuauP3NwSnpu3iNbUlpEeQG5fH2KjXCFlnjiEHUpgdJ+FKQ3le29BXaDMoNr2+5EWR7CFwOg7jS23YLUMYPe47IyeED+MGt+jnt+27ad+fdu2+eVf/mX27dvHF77whe3bP/GJT/C1r30NgK997Wv8zM/8zFM9/7sB8Zh+kacfVfD3FLqu7zoPzrIsxsbGmJqaYs+ePbS3t79jSfHdu3dpb29/qubPqakpZFne7p8xTZPp6Wnm5+fp6uqibXwUz3//U9iH+7G7GpEvvO6c9/5T5ESBoFhETq9gNQ4ijTnpHtsfxDpwDBFYQ1q4DzZYtXuRZncayddbjxCbvIktwDpyDFEnI9anEdmdHLwWbsezNlvmvJKJ9hJJly8+Vv0A0uIQdiCE1TeAqA2CnMJOTyDb5ewyPQIdFa4mGxsQijqpz1LMjMu091VvCqYnoKOv6maS6yHq6qqjr7lZaHW5/9hD6HXx4lychKbB6tsB0usQd+krXJgO0eyS7gRYXgjT0F/NVsW0THLERBQg7IVYSZbJ0ED3QsAl82RbjiAoukvNbWEYml0ENwDTr4PXJ6hvtJFdNv9ry4KaQfclQF0HucVp7AcwVcHamzbhAARikFuBQL8jpnGDkYNLL4JRUvtMnDuHer7cPUWpryeQTkNF3T168iR2RZ+f79QpApcvYykK6v795IJB8uk0ibGx8scHAiQCAUSJVN/2KCgddcjLmyboQLpzD7KsEH3wcHuIrdnThVScRpSsIcaHzoGuIc0nkaYnsEMRct+8iV37iBz2Y7CwsIBhGLS3uzV5PjvGx8eJRqPU1dVVHbNtmxdeeIEbN248VTryrbfe4kMf+hAHDx7cXj+//OUvc/r0aT75yU8yPT1NZ2cnX//616lxmcv4LmLXk//ApTrdvkjbtllYWGBiYoK6ujqi0SidnZ0uj348nmWY7VbEV3o+zc3NnDlzBlmWsVtbMX7vD/H8b/8r1t0RzJ/4GNKt7yPdv4zoOoIYzWB9/HnILWB7fAhdRRTzsJ5H+v59rJOHoVaHis1OZP42dkMnYnkK+YaTZjIPnXMKU7mHyPkVvNkZzM7TyFM7NbzIxhhW62GkuR1lnrQyhNV5AmnqKvKdnZTVWFLGGwbNB7441LVAvNmLbWllWploFCbGobuCEBvbLYpFCb+/fNPi22Vya24th8tvutKvehs1m9Fj5T6nqQvSSQ/xuurQKT3rTnwNLTkyK+61OyObxTJhbUqgpkIoeQhRJOw3EKPQ6qIwVbywOAxtp6qPCQkyU7sTX2W3QH4J0rchLEFHArKWF1lxH6VlZXbf9/pisHQZak/B6lsQj0F9iWo2VA9rd73UHHFPtSohaP8cTPw/zv+lYBD9TrX7Sri/H7OSDJuasG+UN41bDQ3o8Ti5kycpDg1hbxphe48dqyLNwNGjVe4v4swJ5JsXMRO1qN17McemCCbq8V8sv1+mIUh80rkGbUDdcwzP3Ary+E79Ufuff/2ZSA/eX3HLVjD0tDW4559/flcdxWuvvfZUz/lu4wNHfKWwbZtkMsno6CjxeJyTJ0+iKAqXLl16/IN3wbMQn6IopFIpLl68uH0+ldZp1qd/iWKhgO9ffRm+fxWj7yhyuEBw6gFWbSPKf3kLu7MV68SLyJe/DYA8ch3r6BmkK45AxTp5BGvvC0gP3wRAsi3MeC3y8o4IRb51nnykg8DqCsbhw0gtfkRxuaxmCIDpsrAVqhtTW6ImmBCwgZTzl9U0hoWXaEsQ00yTaIBQ2LVdC6/HZnjYpr+iptfYAivJAPV15YKkzm5YW4GaivWntRNWF6C2grASdZsRqMtA8eS0TtyFRJu7QSuAtyIrKCuQfOAQn16E1DQUUzJm1sJn2SxNQnO7DWTLDFjqOyQ0Fby+6oxETRMYqtPOUYmGHlAzjrimErVdsHIbMAPYiwXq6qClRGBTWBCEd1mj63ohPQpxlwhZ2wBzDrSr0NgMbsmhUBQszd2gwNIg3OuIb7QURI4cQa8gIxQFu0LQAk67g6Vp6L29qB4Phbk5PIEAnm9/u+x+IhIh8LDCuFqS8E2Vi61sIbB9XrL7T2HfuAnT5yEQwGuWp0LN/h5i0w+xZRmt5yjG6Bwm9nb9EEBrbGXxv/s5Irr+TINcDcMg6KJOfbfwKOLL5XK7KtZ/VPCBJb50Os3w8DB+v58jR44QCOysXs86jPZpiG9jY4Px8XFM0+T48eOPvOjFP/4VNnJFAt/8/5AXF9HvraEd3o9XKSIv+xBTc0hTc5gvfhRhLiJN3UfMP8SO1SDW15Cu3MSOJjD7TiJqLKSpa8jj17F6jyKNbe6kBXhjIUQSlJu34CbY0RjWcy8h9DmkZWd3Li0/wGo/jjSz44ghrY5hdZ1CmtxplQh4YWwOeksIp84LImeSWE4jCWANsjoULbg3DIEo+OIywahNNGbR1OTUBZWK9SSfLoALMaVciA8guyaobXapPe2S1e7ogkIWAhVrgS8IU/eh8zhkVyG3KlFM2Vg5G3RY+TbUxqBBAiRze7LP5CSuNmVer8XYA+h1EbkEozB/D1qOVR/z+GH2DrSVeJJaBiTvgbEEQvfSPFiAxurH1jYVyS1DaBedgboElBBfbh6yd6G2FlqaYXkMgifcH+uLaKxeg9rndm6zLVi/AH4Zalqh+3+CoT8AXKTtkePHsS5dwgbsri7MpiY0RaG4soKaSpVZmkVPnKDyV+ffvx+pYhPrP3EC6bJzXeqtreS6usDrJfDd18vu5zlxDOlieaRpdzVg6k1Iw1P4fnAVu7GO4HR5b+Dq536d1UyWyXknVen3+4lEItt/Xq/3iSKp9zPiS6fTP9IjieADSHy5XI77mwqxffv2PZNyyQ3vlPgKhQIjIyOoqkpHRweZTOaJdnr+L3yBbDaL9J3/Rqi5Be/5S1ixMMZHXsTz5rcRgHT3HqSLmEdOIEkr2G1NiGvOQiA2UghNQvrbK+Q7WvAd7kBfn8cnJMSmAkOZe4C17xTSg83p7xvrSG9fhqKOHWkk3dFALGE4IxMqkV3GpjzJ3tYAeRWCJVFLbchkbAl6NxflsAfCgCcFjQFgxYQVMG3QDbifdSIJUwFPxIc/6kMRG8xNQqwW/P4dYuzoFuSyNqEKwmrpsCkUFAKB8u+ptQuWZ6Fh06LQNKCQA7UAi/MQioEkghRSeWQTfIqCT/hQ38ptnvdmpLaph5mchXqXdoL2NthIQdTlWF0NmGZ1jRMgVOMF3FOHiSYwdcHaQxt9ARJhaPCzuSHQWJ+HmMs4JkmGjZHdia+hD7KzTk3OXPCQiOuESjYv0VqBqdrIu6ScQ7VgqiD7nBaKAArxxM7n3vRxWLrbgfY9p1ZsC4Foa8NqbKQYiVA8fJj8xATm5CRMTlJ79izqULnPm6+jA6NCCYrPh9dF6OIpFMg89xz5TAb93j2Ym6Out7f8TsEg3pIRR1YkinrsBL6bd5FKvEK1nkb8E0nnPg1tGCc/SvjnP8dWlt62bYrFIplMhvX1debm5lBVdXu23hYZBgKBKjL8YRDfbs//o25QDR9A4jNNk+7ubhKJXVyNnxGKoqCq7jWTUui6zvj4OGtra9utEluO6k+K4G/8BnnDIPnv/h3ec2eJvPkDrP/0bfI/9RP4k1NI40OY+48jv3EVWxLwEy1YB04g3XV6j6Q7l7AGjxK8dwOm57Ea6zHPfRx58iIi75CZSE1je/0IzUn7iMwa5uCHkG/+gMTmLt1ubcE8+o8Q6ixi5T7CtpBSk2WEBuDzwETaQ7evQqgQAMvGifo20ZiAxVyQppAjiJEF1HqctFlU3ozOcqrzB0yvBwgnnHSnYYFqO39z61AbB0s4r8HmvwVvgICUIxAIYBeKYJpIQCoDoXkFj23glR0SDgNKFkIKKEq+xMfZAAzGZ6Gnu/r7iYfd64ayAskJd+KLJWDyAXQdcDkW11icDNLUtSMS0jVBaiaEnRKkb+q0dhVdo9/CkjvxAdR3OtnpSqNvy4DkLSguQsdxnPpwBfxhm5VbUO9SfwTHRWfp285mxVGkVm8KO/+HNLcLZyiupchPT2PNzBCLSL2E+wAAIABJREFUxdj47nfL7yhJ2BVpSoBQWxvm9HTZbeFjx7bn/lnBIPqBA9iJBPk33yzzAQ2dPImoEMl4TxxFunAevb2dYnM75o2bhFdXykjPOLCX6MQ9rNpmrJou5Ft30P79/172PEIIAoEAgUCgTLqvqiqZTIZsNsvy8jKFQgFZlreJMBwOv+ftDI/qE/wx8f0IIh6PP9aSTJKkp95xPS7isyyLqakp5ufn6ezspL+/f3u35zbI9lEQQhD4zd+ElhbyX/oS+bNnqZ+bQ/rWa2wMDuI9eJbA7BDW8+eQ3jqP+NYF7LpazNMfQZq5jVhfRVueRvH6UbQinqUV7G++hR1twN53AFhGWhzF2vc8oqSBV3p4AbuhHbHsuOqLuXkkFVhLQihKqj5OKj1JKASmVZ5CbEno5HWZoGfnfdZHqSJJAEWuTkcmQjC6BH0Vi3hrpEBa9RD36SiSc2GHsPGFnejJX1FnyhYzKD7wmzkoOVbnh/WMQajidx8Lw8xaiPaGarVmYJdfUTwO0+PuqtPWFlBVGZ+v+vsOBSTAXXksihYbSUiOgicHTRGbBiULEqxa7kQL0NAO2TWFcE31tal4YXkIApvqz/wKrN+HmB8agmC1QTapEK5zv65jDaDnwFPR+bE+7EPOQI1fxw4LdlMWxds28CsXSZYEcn6vl8oOk8SJExiXL5fd5qmrw7xWbjyNEMjJJJmjRymYJubICPbly9QMDJSbXwtBIJksf2w4jK3rrB885ghoxmfwnDyBcnNnUoMtwKwPkQ8cIXL3PsrDBdRX/2/sXpfBiy7Ymq1XqqjUdX17esLMzAzJZJJMJkM0Gi0jxHeTDB81hPbHxPcBxBZ5PS3xuZHXbkrNUrg5tzwOkiQR+vznSSUSiK9+lZWODqLt7fiHh8mnfBTTKqEeCU/fHqTREURyFWtyFe5nyZ09Snhjko2ePcQfOjU7UchjdyaQvuUIDazjg2DY2IlGRMqJ8IRpYNY1Ii/vjJMRyXmsA88j3XqLRDpNAjCX4Z7uWG+FE9BUAz4FxjY89MbLP6NooHrRrvMWmFiV6K4tJ4GacDWhyhIsLunEK9SNIR8MLcJAxWy/sB8ezMO+CvGuJGC9GCQRq27sC0uqK7E018P0nKDDxc0kFAkB1WTp88HkpEzXQPW1Ut9gMTe603ZRzMHyBKAGCBYKFEahp4UywgaorYGpEarcb8A555VJw2l8d0Fth2B5LIRIQa0nS7Cm/LG5Ve+uxOcNwcKYn+ZDTlYgORxEXsmTSKjb0fHSmJfGA7tv6g58BhYuOVqpcE8P+QrVJoCUSlVRZ2RgYFv1abe0oHd2IkWjLL7xBnZJ5iV87BhWRTo0dPIkYpNIrWiUwoEDeAIBzFLloUfBv7pTfzTqGzGPHsFz/vv4NvsAzb370H71/9j1vT0JPB4PiURiOxN18+ZNBgYG0HW9bNCsbdsEg8FtIoxEIs8konHDj/oQWvgx8bliy73Ft5tW/hFwI6/V1VVGRkaIxWKuSs3S133a5vfAxz/OciBA7M/+jPViEcvvp7arC+1Gmuy3foDc10Pg6Al8N67iuXcb7fhzRF+7gB0KYJ8Aa89+pBGn9ik9vIF14gzS1YtI1+7BNTBPnIaufqTCDGJpEnn4KtmWPYRLJlwb9y6i6hDZ/B3KAnoEKGnwZ6AwDcmQF1VRmTGgtWaHROqjm5FcRdTnFS4KxxAMLcFAhTKzrwFSOScqLEVrHDQdvBXrQ1sCNAO8Fb+C1lCejYJENFD+2omgwfg09HRVf/626i6Iqg3mmJyELpfH1IY1DAMqM07FPGzMOinGkAUNIeiQAKkAIZherH6uLQQfsVfraIONVYiWqDrTy5AahVrFxrAkWjpc6rVAY02e5DTUdbgeJhEtMnsJIpJMXSgPFWnThnqNjTn3toviGuTHYOAluP93EG5oIDc+Xnaf6OHD6BWG1iIcRldViufOkVtYoDg+DgsLNHZ2lpEegDeTKY+hJYnA0hLa3r0UEgmKN28ij4/jXy9vYPedOY188TxGWzvF5jbs0TFC169sN7/bQlD83T+Ad5l8TNPE6/USCATKhCaWZZHP58lkMqyurjI5OflMIho3rK+v/zji+1HDk1wMHo/nmVoSth6byWQYGhpCURQOHTr0WNHKVor1aSBJEnZfH3X/9t8y/8Uv4gHm3niD2NmzeEdGMEfHWUAi0LOXWp+M59YVrP49SMMjJL5/B7OzHe3AC3iWRxHL84jpIex4ApF2ao7ylUtYB44jbk5iHR2ErhhiYwlbkrabeb22wWRxh/jAEatMqH66KRIA2nOOOOPhDGghWFRAj3kJxGX8SqGq1tcah7Fl6K0QX9QGq6M+RYZVPUiiIkkW9sPYmpfexnJhSCQAD+dgb0XU51FgeN5i0GW6QXCXX0xnC8wvQktT9TF7l5JvJAozCxGCvgLrcwbkoCboIy5U9kmwlIVGFyVmRxMsL0KDy2vV17kbA8BObdEfFizcs/HmoTkO8c11NaBvkN2A8C6CPiMFVBCfVoDle5uC1XWIHXK/foXk2JxRQnx6QSF926AmBnV1UPsRWJ2vIV+ZugR8tk1RkhB79mDW11PIZgnEYsx9//tl96s9dQqjQskZPnECq2Tyux2Nopw+zerkJEZJu0P00CEoeaxIxJEzG2SPncK+eg0mZvB/aKctSIslsD73eayT72zW3pNgt3l8kiRtD47dfj+PENGURoZbIhrLsh65Dm5sbNDSsktB+EcEHzjiexI8ay+eqqrcuXOHQqHAwMDAE++ensW0deucvc3NdPzFX5D6D/8BLRgkff06hUSCmu5uIjduYJ06xcL580ROHSfskRGzQUQ+jzw1g9HSRv76IsqLp/EYaxCrRVzeHGUkQCzNYgdDSDfuwQ3w+32kjh0hO3eP1oCGjE1fFJYK0FjSn9amFNnAQ7Rk/Hp/GJaK0CWANc2ZvgDcn/LiVTQ0n1MzikXB5xLF1IVhdE2hryL91hnKu0Z9NT4NwxIoUnlk1hQDw3RIsxS9dZAtQLiiR6+pBqYXZDqaqxf4wi72Yd0tsLDg1PxWlsCyIhSXM4QskPQMiWaoldhUg+6wpJrFtQUBoLAOuBAfOEMkKmHoMD8GPi1I9nKezgRVqVKPB+anIOziYAPQ1AJLY9DYC/kNmbV7JnUBaNvcz0UCkFqAhEtTPzi9iMv3IdEFq/clakMG9SXRp5Dg6M8UuHBHd7oCg0Gk3l6UhgbWUikyoRDm0BAMDaFEo9W/F0lCmZ+vks94UylMjwfr8GFUoDAyQs3t21glLRT+gQHkkvqhfugQnro61Ndf3zZ8kI8exnPlImY8Tr6rHytfRPq1L/Be4J1MeHmnIppgMIhpmmQyGUKhUBXB/jji+xHEezGFfQtbSs1MJvPEptbvFkp9PoUQWD/5kyxEowQiEeLz82iaxmJrK43FIt5z58icP09GEkRffBH/nbv4kkm8Fy5gnT6N8b1LGIB8JoRy4hzKVaeGIpJLWCeeR1xwhC5yUSXw1nVkHQoKLNf58McFakXjr0eGybxFtGRFliQouOg3WgyNIBJew3JKY8tQtOHWKMQaQmS0HLIPgkFQZKMqQvTIMJ+qJr5EEO7O2Ryo6J+Lh2BkWWFPc/n37ffCvXlcoz5Ld49qultgYckxwDXsMBvLWYQOPhMKeWhu3QqYMjuk44ehGRhwMQrqaIBkJkBdpHoEUmeTQ6L1LsTYWA8Lsz7qm1QWxsBMQZMPOjyAlGdyCWp2ETW3N0E6iWvDPkBhHhY3AtR7CrRVrI1CgJkGdiE+vQj6NKhFaIq7i3eC0QLtn27kxutBCtPT2Hfu0Hr0KNmKml/toUPkK8yo606dwri4M3PSBjzPPUdeCPJra5ibUV/NuXNYFW4wYUXB9njQjh+nuLyMYpr4SkgPvx+vXkQ99Txcv4nn1h2m/+N/pPkpyiFPgndj3dhNRLMlnJmZmSGXyyGEIBQKUSwWSafTZSOJ3ik++9nP8rd/+7c0NDRw965j+L22tsanPvUpJicn6erq4utf//p7pqp/Usi/+Zu/+ajjjzz4DxW7eXVuIZvNYlnWE+96tpSaDx8+pL6+nnw+z969e5/q4p2ZmXkqfz7LslhcXCQcDm9Hm4MnT9L6yitkMxnWLl4k1N3N0ttvI9XU4Gtqwl5YRJ2cZHFgAKmrC79loczNYdTXQyaDPbuIkS5SaOlF6u9HSicR02Nk65rx5ZzwxiNg1nKCj0TOJLJq4CkqPMhZFGyI+ASScGpI03mIlaRBE14Yzzr/bsEvwYOMTUPJeqII8MleGgpFmnSoL0B8A+LrcHcU5LQgk40wP6ezuAqmDisZSGVhowB5TaBZErJt4/U6RFlKlrKs4FVMKr8uvwya5CGbt0hnYDUNK2ug5WF6DjKZICvTFsVVP/q8wJ61WJtw6nHRnEY9UCdBQnFmCi5knaioErYF/lD5OW1hftlgNzvDDS1KJFKeRy0WYHkpSGakSCQPtUDcWx7RRgOQykPAJfMuJFhZcJSaW9A1mH8I2hS0eiCf9xKJuCujg0FYmIVICXFqeVi5C/51SAQgnfIRqtk9pZ+oyzF5IU0+bePv6sKqcF9RolHkxUVsbSd1LRSFsGliZbOIwUH0PXtYMwy86+uo9+9jb9bkPI2N+CYnoWRjGzxzBjMYJJvJoA0PY2ezjpfnZmuRFY0iP/884v4DxPAIQtNIfulLqB/60HvW6D0/P09r6y4+dM8AWZaxbRtN09i3bx+tra00Njbi9/uZnJzkz//8z3n77bf51re+xYULF5iYmCAajT7xJIVEIsFnP/tZvvGNb/Arv/IrAHzpS19icHCQr3/968zPz/Pd736Xl1566V1/by74rd0OfOBMqsEZN/So9z0/P4+qqnR3uzRnlaBUqdnU1ERXVxeyLPP2229z9uzZRz52N7z99ts899xz75g0c7kcly5dIhwOu6ZX07dvc+8zn8FvGPhWVrBVldrnn8dz/TpmMolx5AjG3bs0nDpF0DQxrl51OqkB6/hxtGvXMH1eVnw2itDplRxCcj4IGFOht2RxXUciYAlMy2Q6DFbUGZXTXxGJLarQoJQv+kXTcW+JV+gF7mVhsGKdSWkQ8gu8ovz7vJ+F/S77lrtZOLC52dRtMCWBYcOCN0Kb2EC2nfe1dT731mHQJfqZNfy0RatHGVk2pL0+arzVhb3xlKBnlzl202aEjoT7lOu5ArTusu4sqKDIgo2Un6AKjVZh+9xndT9t9dXnCDCzAu0u6s8tJFXweAXrEzb1HgiUfBf5Isj1zngpNxSKEkojmKZgfSFMQs3glco3m8sqNLhEucUspMfBr8Hf/Seo33uYbIWoJXjoEPLtkoGvikL8hRcwi0Uyo6Pom/120WPH8FQoOevOnMG6eNFZ2A4dQg+HCY6PYy/uKIZizz+P9623sOrqUAcGkDSNyNWr20bVxssvM/Ev/gUej2d7sOq7Cdu2uXr1KidPnnz8nZ8Ca2tr2/3Dbnj55Zf5kz/5EzKZDDdu3KCtrY2PfvSjT/z8k5OT/PRP//R2xDcwMMAbb7yxPY7owx/+MEMVJgTvEX5sUv1OoCgKuZy7u/4WnlSp+U7xTgfZGobBxMQEy8vLyLLMyZMnXUkzfugQZ69d4+4/+2cs/MVf4Pf5MN98E6ulhWBDA4mZGeyODpbefhuhKESfe47A8DCe5WWka9eYBLpUjabN9fyqAnEf9HqdqK/WC0UD/JsvHcPingGDAgYyQMYhhWte8IUhGIH2sJOCG9H87FF2Fmi/DGP5auJrD0DBhEAJwSa8cLcgcyBYnqrcG95cXCsyUW0+h1j9snPens3FLJ7ZQImAp0JP0B1ybNTClYpQpcjoGvRVRGOSgOVVjRqX9bAnYTO35qhZKxFSs1VinS0UMkAJ8W3kIJ0Nk5/PYuThQIdN/ebE+tKful8rutYvAdrrYSUdpD5eLgTSDcHSUhC76KMutEbMJSoM+mF2JURbu/tvRCtaTLwJ/a2CRmnddfhZyHIccXybEbCuyyQfmtR5oSkEhODjL4c5/1flpOdJJPCMjUFdHVZXF3lVpTg3h379OqTTZfdTKhbXwMAA1sgI5vPPk5+eRr99m4Zz58pIz9PVhTwzQ/7sWbTr15EePKBOURC2jdHVhX7iBOIP/xBrZeU9azB/v0cSbWxs0NDQQFdXFwcP7lLwfQf4+zR5fQsfuHl88GQz+Xar8WUyGa5du8bMzAyHDh1i3759VaS3pZx6GjxpfdG2bWZnZ7l06RJer5fnnnsOj8fzyPcmhODgV77CyddeQ2trY0GWWZ+aYnF0lJHVVWZmZ9EUBdswSL/1FldWV7kfCjEP9AClrb49BuRzMJ+GmznIC5mFCm+wfQrMlHwMkoA9KnStQM8E2HdhdBQyySLzVgC9JBjaG4Ylo/zHGZVhRlSb53YrBhsVH5kEbFSqN3DIdMRFsV/vg/suM/+CCkymq28H8OySrdsbtZlddz+mm+6y99qgzciM6yEaQnDnHjy4CYvXIDoOHctZ9ipwIArTSffH1YVgdNr9GEAxuUN6MzMwegPMBzZt6zna1TXGdzkfgBZvjmTFUPOlBZi+CZEF2OuDVGr3xTUUgNWxzTTqkII9ZtIcdBS1W2isybJvM+gRfj++Q4cInzlDrqWFpdVVVq5eJXfnDk3795eRHkCwowOxuXm1JQlzcBC7tpZUJsP6W2+hT08T2rcPe9PdBYC+PuT2djJzc2hvvw3FIonOTkyfj9zp0xQMA/E7v4MIhd5Tcno/fTrBEcWUehe/m3i/J69v4ccRnwtKp7BvoVgsMjIyQqFQoL+//5HF361+vHc6yw+ebJBtMplkZGSEmpoaTp069Y4bWOP79vGxK1e493u/x+S/+TdYq6tEfT4UVWUGCIfD1GezNJsmK7kcHmBFkpAsixjgAWQgjhPFNamAaqL5C9yNRGktZkhgI0mOF6RpOT19AFEJRv0B+goFvDb05YAc3F4qUC/DYiLAqijgjShoCjRWXKGdVpaUJUgoOywZkuFWFg5XpDb7FI3xHPRUpFd7/ZC1JMIV6bduP+SN6paF/hhsGDJRpfx76QzBUBIGXFKhuSzgkmrtCulMpSQ6E9Ubo3ovFDVYSUEmDWE5RHA9R50MkTx0teB88BUQ1dqXbbQFoahL+D3Vrxfyebl5UaPDD+1equSgdbJzPpWuN+CIk7QUWLUwPy2hrFs0+aF0nq43Z6BH3VvcdA2sFCxdgLa23Td6+07Ciq+dB5eWCK6uknn4EKukthdobqZQ0qoAEOrpQbl7F72jA9HSQnZkhJDPR7F0+oMs48/nsW0b+9gxVF0nHIth/uAHO8/z4otomoY5Pw9rayjf+hZiU+a/W7vBu4H3c/r6sxj074a/T5PXt/CBJL7H7ThKoy5d15mYmCCZTNLX1/dEk9i3Hv80jgqPivgymQzDw8PIsszhw4efeWzJ4Be+QM9nPsN3f/ZnSV25ghyNksjnSWWzrEsSsWCQSDZLHghuRrA3cXq2unHW9blIhEDGqU3FiiZrxQ1WgIVIAE0r0KPAAwtKrSf71AITQqK7ZBz5oIB5C9rXCo5Kf9X5DG5JEPBDwQ9yAOIBWDNsEhWkMhh0T226Bd5BBR4UPewLltfhogrcTMGRiokOXgH3U2bV7eD4eFYqSwEG4jCVwmkbqICWtyDhCEwyRoj11SI+20PccAbr9ge30r65baLrCsLYCvS6nEN7BIbnod+l9Srsg4lVP91NTnS3vAZrS5DQoNGjIVQI7XIZRf0wMgN7equP5QtQSMJcLkZ7cL1svNIWYgGYniq3bFOLkJx2RDdtCugCNtYF0Zj7grs0D/u9MySD4G1uZmNurux4TXs7uYWF7f/Lra0E+vrICUFhbAymp/E1NOAbHS1rYI+dPk1R01CLRcT160g9PYhN83qrowNlYAD7+9/H1DTsUAjPn/0Z0uHD24//hx7x+f27FGg38W5GZVuT11999dX3ffL6Fj6QxPc4bEV8k5OTzM3N0dHRwZkzZ554h/c01mOlj62M+FRVZWRkhFwux8DAwLtqJxSoqeHAH/8xqfPnmf7yl5leWiIWCiGbJplcjpzPB5KEt1hEsW2agMVYjPGNDfK2TWcmw1woROtmWqkGGAG6Mk4YklKhKMMtn5cuYREzDRAQkMAwBcpWj5RwSMoS5fn3diETzJn4S0pRTcBND3h9UPSBJ+whFvGyms9T67W3o0uAviDcz8D+iiEcfYpKyvKQkMoj+71hwbpmE6uIcg5EIakr1HkqJjoE4f4K7HfZxGoFyAQhY/pJrRYxVJA1CBlwdxoORCCxbWfmfOcBA3I6hFz2TBHTmVIhu6xJDV4Phqm71vM8mTx3k1BvQqNns1y4+fwJHwwvQP8uQuKemEOWDZt1yYUlUJPQJkGvBOrGOhs2REPuj++IwcK8Y76dmpOpx6S1ZNXxSFBYtyn6wV+yaVleBGUDGn1AAH76rMRrb5VHdrG9e8ldvozS3g4dHeSXlwklEiQr5vLVdnaibhpRS319KF1dFC9dwspkEIAci1Gby2H09FAMBGB1lcSFCxhCkD9+HP2znyXw3HOESyKl95r4nrTG/7TPv9u5P2u0+XM/93O88cYbJJNJ2tra+K3f+i1effVVPvnJT/Knf/qn25PX32/8mPgqYNs2KysrrK+vU1dXx+nTp9/xRfisDfBbxGeaJpOTkywtLdHT08Pg4OBjd2LvpPF1C7IsU//iixz5+Z/n7S98gdn//J+xFxfxx+OE0mlkYNznQ1ZVOoH69XVWQyFiuRxrwHouR1EIum0bCWd82whOVCgBjSbMqybLpsmwAFOGZsnirgVHSk61U4I7kszBkkG3NabJUDjMQG6nO1wBYip0aiBlgVUd0OkAZiNBInqBtGSTU0DzQFGCG3nHrDoUCWBpBXwyLGg6+xICryyQbQuPBH5h82AdjpZEVoYFhg1LBRnJNCgYoJqO3ZlhguULM5JWMDIFyKn4DQha0G7D8lKADqlAZTA2q7lHihEFho0w/Z7qbvgGP4xkQ+yJVItK4orOgznY1wGqDrPLoG9Aq5Bps0yKha2BsdXoDcDKhmMbVwlZAr3gY3xMJVKAZg9lq4ZPhrVVCAfd2zEymg815cHIqbTK7i0QUR8szkFjNyRXBdK6RINilqVevZLFR87AaxcgnYFAfz/+9nYyus7a2BjMzOCJxwmurpY9d+L4cdR795DOnKGYTlOcmqLRMDAyOwra+OnTqEtL6LduIW/6ZVqHD2NMTeH9tV9DO3mSpaUlxsbGME2TUCi0bSjt9Xrfda/M97PGt7Gx8UwtGn/5l3/pevvfl8nrW/hAEt9uxLCl1IxGowSDQXor53Q9IZ6F+GRZRtd15ubmmJycpLW19YmjzXeqCN3CllWaJEk8/9Wvknv1VV7//OdZe+MNMoqCX1GoKRbZ8PuZ0DRsy6IxlyOrKIQNgziwZtvcAPw+Hy2WRbuuk/F6iWzWY1pMk8lIhNZMBowdn/7XBdRI0Ov1EDZ0ekyTNaCm5CvqzWaZA1pLbuuW4IYFRyvNO/IaYWynvFaSybwBHPIACzsFsQ5gbNamt6ReaAL9wPQ0tGy2bCg4f4OojEvQU1XzynLNgOMuQ6u9RR0zUB2ltXnhdgoOuSg8e6ys437joi9oLObI+asjwrkNgZn3MDwi0a0X2b5ybeeT7gvAVNZDZ7iafGQJiutsD8oFJ/qeXXEa0ttRmRUe6hPuxNUcgKl0iM7EDiHPrTgWZ+2ySkSoLKUEwWZQdrmMJQ3GLkNfkw2Ke43b54EjxyNceiiQamtZ+M53ys9j716yJQ3sUk8PIhhkQ5axNm9vPnMG4+JFbI8H+dgxguEwhc0I0fZ4iJ48iXbzJtatW4T+6q9QXnih9GPZ9sq8d+8e6XSa+fn57WnppTP2nkXl/X4S3wfBtQU+oMRXidLa2cGDBwmFQrxdWgh/h3gW4lNVlZmZGRobG9+xcGUrWnynxCfLMlqJYED1eql79VViL7/MzB/9EcWJCea8XnzZLJplEQaWAdUwaAIiOCnOeUkipKrOsUAAVdfZy075pz2TYdHrpWHztaJAAUHQguWCzkMBcixKLp/jqGUS2iQLRYDpDzjd2SXoE5CyIVFCKi2mwVXgRMW6MQCkhUTcLi/6CRMseSdakXGcwxY9ARTLRTWiU2X1BTAgQcpwmtVL0WQbDBOhn+oevU4BOcOZ81cKRcBqxp34oh64tQx1Eaeh3qsK2gybVtumFY27Gnh20Q4ECzpqwN0Crj0Eo0kPYa/B2gq0aLZjjL2JFlN3vENdyB2gRc2xnJHJZ20CGctJZ5YqNL02k3PQVZFSXUyCyDgipgYZ5pckWhqrC7OLq+DPQ6uc4RNNcHniGqUxcXxwkOzFi9j19RitrVipFPW1tayXiFVqT53CGB5GnDtHfmiIqGVhfe97jjXasWOEsln0H/wA5Sd+gsCXvoTi0ue25ZWpKAp79uzZbggvFArb8zSnp6fRdb3KOPpJTe/fT3HLs7i2/EPCB5L4tiK+xyk1nyZtCE83ZSGbzTI0NISqqjQ3N9Pf3/+OX/dpTa63IsV8Ps/Q0BC2bTsbgDNn4NOf5sK//tc8+OM/Jl0oEPB4UD0earNZgsC8z4eQJBKWRYuqMgq0Ab5CAR9wNRjEl88TAXoBv6ZRZIcMG22b+8B+oMYG0htEgSvhMMFcFlVAbTRCNJvhug3HShbjiID7skLCLN9k7LchaUNdyVcXBCbwEKdc0NIjwX3Fz/4Km7VetcAd4GDFvqNHhpsFOFJBSmEJbhQgUVFLBGjKZtgIOOKZUsQUuJWGwy6q0P1hGFuH3hhkdFgzg6STefwqtOtObfOQDyo9Jg54YSwDvS7nUe+BMTVKb7C8n2MxCxktjLycJRCB/V6qGp28ElhZMILVUVu6AEtJiEkemoNFfLusKl1+mFyEriZYWvcipzSaZMoe7ViPAAAgAElEQVRWoRbJYn4jREvUiR5X1RBiOefcb5MLFAnOtmgkZLg+ByIaxQ6FyHR1YU5NwcoKiaNHyZUMmA3u24fs8ZDJ5bDPnyc6OIhvehpx7hzq2BiJbBYRDCJ9+tN4fuM3sAMBdF1HkiSEEFUZl1JVpxCCYDBIMBikcdNVvNI4enZ2Fk3T8Pl8VWToNn393eoLdsPjiO+9cqP5+4QPJPHpus7w8PAjlZpPmzYEh/hKI6hHQdM0RkdHyWQy9Pf3o2ka2ewubsePwZO0QuyGZDJJMpmkv7+f2trasmNn/sk/4div/iqvffGLTH7jG3hSKWZ9PiRVpVnTyCgKa7pOEgj7/SwDDZsWUc35PHObKc+HQAHwBoMM5PPbF99eYArYMvIQQG02iyZJNFkWpDMUcEazfsdyVJYB8f+z9+ZBktzlnfcnM+u++u6eo7vn6Onu6Rkdc0ogywIThrUxa4Ux9i7ErrGB1xFGQjKgDQlkDJLDQhwWGORgF+Fd2xBgE6/BB/gVxsharGA0YnSNZvo+pq/p+6j7yOv9I+uXk1Vd1XfXCKa/ERUBmq7KzKrM3/N7nuf7/T7QKEG7rjEeCtOSuJZRBSQY0KG+6Kc7nssy5PfQphX+Ni3ZLDHZklo40WBADglPUXA5YELSgGDR39/qgpEsHCra2EdkeDEOp0swPI97YcH0UidZAXkuAwtpi/2opmFq2rK+DJOyvx8kS+Bf0o0aCOVANVaK8QEOZGNM5ROqhUWoTcM+E/bk86eh5UK7Mif2eqBnBrr2Wv3JsUUwY3DAgGoJIMNYCpqbSvf7NAPMRZjUI+w3YiWlGWBlj/1XoD7goU5Llvy7pCrTWB3kV90qrwYamHIYTIcPHULp7sb0+zHa21HTabzRKLGeHgDcLS0E6+pIj4wgXbxIwxvegBmN4vvgB/HkGYe6rltSB9PEMAz7uTJN087yVtsYlzKONk3TNo6Ox+O2Q5SYouCcvr5TOjpxHuVaJzfCLD64QQOfuLFW652JcuVmA18qtXKQqRO6rjM6OsrU1BSHDx+mq6sLSZKYn5/flpFI64EQwY+MjODz+Va4voiH3jAMZEXhbV/4AplPfpKnP/xhZp59FimTYSSZxKuqREIhlESCXCaDBlypqiKcSFCn6zTlciwoCrW6bpEJUyn+HUsOEcAiwVS7XKQ0DcGsDwKLLhcRxwaiCiuT25ePQyowJMHVpTgDEkQiYdyaii+ToVWCbgOOFf287lQOze2wWwPCpsmLKpwuCiR7ZDifMbm9iPldI0Ofv4bO7FLBf5cl0HKUDEi3umE8Ay0+SOgwl4VYztKzpSeztAY81CVzNJjgVCxcxste90r7szYXDJlh2qSVJdQmBS4uwi2OTFIzYNYIsDidQk5DZwj2lliz2xRLyH+sjD/oPpfCxRGdFiM/WQMK3GJaFRics2YjCqQNhYVskNqlJId0HTUdY9IP+0tkpZNRa7J8BzCbATVUGMDTkod40kV9IkUwf+2/FB9irM6SoqS9fmr270fbu5foq69idHez79gxMgMDuDo6MOvq8A0MkDh/Hm66iYhpkrnrLoLvfz/ecNi+/53rgjCjEAFwcnISSZLQdd0OkIqilHyvE5Ik4fP58Pl8NDRc+5WdwXBmZobl5WW8Xi/Ly8t2QAwEAhURfu/2+H6OEQgE1jSC3ipBpdx7nf6e+/bt441vfGPBg7KVYbQbyfgWFhbo7++ntraWm266yX6YxTmKgAeFbgv+mhp+46/+iqXhYf7pgx8kcfEisqqylEzirqoilE7jyeXQYjGm/X4WdR3ZNAnnciSwdG9gZXczWPNKJ4CYpiF5PByQJBryQ0RbcjlewyqDCrQBPUCXuGYTWoDLJhyMXgsCS0DG7eK5rIYpgYYVmFwS9JmwT4HqgA85kyGiyJyUDUYMiUNFY4tulWFGt4KJE23RJcaU/MQDICPJpHQIyPB8HIIY5HQwNZA18Ghg+gKE0ymqjWvfg0CPIV3L6Bxoz+aYlmBPiSd1TyJBNCBRpazUwHXJMByDdAaUJBxQYZ+Rstmll2JwU5n17YgOi6aPWsnK2jM6jC6DLwkHVJ3DJuQilOx1gtV7HcsGCEhZlud1Dmg6zdK18qpbgsYUTEqwP2Rl8pPLEEgVjOujMZsjZnrIVYFk6CRUP3XRBI1SriDYxlUIGAr/2aMTDbq5cu45JnSLqNR8xx2g6yhHjyLlclTNzKAcOYI/l8P3pjeRe9/7iAPTIyOkUincbjeRSIRwOGyT3MTzmU6n6e3tJRQKcfr0aRRFsZ8T8czAtWxRlmX7vauR04qnKPT19dmTXeLxOPPz86RSKRRFKSiTOs9tuxCNRu1y7c8zbsjAtx643e4V7i3rRbmgubi4SH9//6r+ntutASxGMpmkr68PWZZtEXwymbQfWtM07Qd3NXuhmsOHee/TTzP12mt87777WLp0iVA6zWIuh9vjoUqS8KdSRLEytRgQx8rkmrFuvFqXi4SuEzJNqgFyOV71emmUZQy/H5JJSzdI4ei5NmDJ66XGMWX7ODCIJaUQqFY1lsJhDsULs6IskPN6MRYyGMAihrX4Siaax41pmmimiarrGBIsANOStdbKJnjdbnRVZc4ErwkBE2QMvFjJnsvlImBKhIt/i2SKl1xwqlQQi2eJBVeWWz2myVIK9pRouwQxuRiDW/Il1LipMC8F0ZJQHc3iMyQOuLIoJbzmj5kwbnhokVeW5D0yaHGYCATQkjL7FpN0Gtc+IyTBdEoh7dLxF52vZsJIAgLLBilT54ifklbBbhlqkjCMn7pYmnLbUCmVY3YO/LURGrVYQQk1JbnJaF5qlhOE8zzhcDzGPgnaZAnZ5Sb905+yoKp4QyHcXi9qNou7s5PIl76EkndhcbZYc7kc8XicWCzG3NycHXDA4gS0t7fT1NRUNrsrDoROaRJQ0C8sF7Q0TcPr9RIKhah1jOZQVdWWUYyOjpJMJpFluSAYlpqv58RazizxeJz29hJTjH/OcEMGvp2cyVfqvSLYSJJks0bLYSt9utXeq6oqg4ODRKNROjo6Ch4oQYrRdb1klrca9t58M//Pv/87V195hX9+4AG0y5eRs1mms1n8bjdej4dYMkkEKwDGfT4GMhl0oErTCHo86JqGkj9uczZrSRfygvhlrMAzjBU09/t8lrQimyVCYfunCmuEn/PbPRSPMyjLHHFYuHiB6WyWKsf7Zaze3WtZlVPF1wh0A7Zvh25tiBqAS8CZor8PaRqvAW8o8X11aDCnQEPRV1sDvJyBkyVcVLokeDkJJ4tumykVjAw8m4TDOWhVdcIUEldedMHpErebDITTJgkfhPJfgm5avUOZAI2zKeZccLxEORJgj64zsGyZdEvAfM4aBdWcAmvZzJAzYQxoLWpXxTWYjcOeJBw000zvi1BlFJ73sgrxBOzLwCEJmIyxrIBaJRH0uUibQWpmlwnkhxsbSMzrJn5cVGsahgTLuRyyolDt9ZIFjBMnqP2Lv7AE72Xg8Xioq6uz+9zxeJzu7m6CwSD19fXMzs5y5coVO/sS2aEIOM4sT8AZCMsFQ/ESz2KpFovb7aYmrzMU0DTNDobF8/Wc5yaC91rtm2g0et1n5VUCN2TgA+tmW233sx2BL5fLMTQ0VDLY7MRxSwU+wzAYHx9nYmKCgwcPrpgTKEoysViMS5cuUVVVZT8w6+0pGIaBVlvLLX/yJ/gSCZ579FEyg4PWkhSLYcoyMb+fKk0jnMmQCIfxJRKopslsLscw0Kgo7DUMFNOkHivgiRZ7HTAfDhOIx1nKZJjByh6HsVxImiSJetOkFhgJBDhS1F+tNQziWLILgQPAReBk0bUco5BoI3BIUYjJMpGiKsBhVmakADdj2budKPrvIWDaG6Qht1KEflKH7lyeVVmEGtUyAwcvrniWvRmoM6zvJgooPolSU8ROajCky7QpKyUC1arKaznLS1NNwsEUdJgA1vd3XIXXUnBzGUuzg6bESzMmtQYc0gozJ7BISHvi14LfnK6QyXrZt5C6pjME9l2NMeqBPdWQdvuIz2VozuYJM45bUJE9LM/n0H0KWnYZn2JZgy74AvgyGtVqDgON5UgE0mkwTVL19UgnT1L92c/iPlxiqnAZGIbB8PAwS0tLHD9+nFCosDitqqrdl7ty5UpB9hWJRIhEIgXBECiQKDiDobNMqmka2WzWrryUYpQ64XK5qK6uLiCk6LpOMpkkFosxOTlJMpnENE2CwSA+n896XssEwFgsttvju5FRyqh6vZBlmUQiwU9/+lMOHTq0oaG0W+3xOc95bm6OgYEBGhoaVjjQFBBXZJk77riDZDJpU68TiQSSJNkPclVVFcFgcMV1LCwsMDg4SH19PbfddhuKonDyrW9l4rXX+M5997HY14dbVfEmk8yFQizqOmY8joIVtDxYGsAxXScFqC4XNX4/ZjxOjmttpPp4nDGvl9ZsFg/Wgg9WkPKYpqUd9HqRZZkXg0HCySR1+b+rBS5jlUOdOApMuVzsdWw0FKzekIrt6gWAX9fpM80Cz1GwyDkzLhd7SmxWaoGkLBMsMgw9EkvyohtOl2ArBkwPg8ByViOXM/DrUKdB2IQ54FZWEl2qgHHZS7WeWcGmlIGahEEsfK2MOq/DnOJHXU6zPwejEpwqw7C8OQOjkQAHtGubiSs5SGYlmjMmNxswUMLgWkCRJaCKS/MJbkpriKBaDCkD41E/vnSaFpmCgLdgQkyHOjVn/e5Jq0Q94wsBJmY8TcwwkBQFIxJBTSYxIhE8b30rTY8/jquhhMHpKlheXqa3t5e9e/dy5syZks+u2+2mtra2YDOraZodDEdHR0kkErbuTwTDUChUNhgmEgl6e3uprq7G7XYX9AxhZWZYDoqi2McTMAyDZDLJ3NwcuVyOV155BcMwCAaDhMNhDMMgEolsm47v6aef5v7770fXdT7wgQ/w0EMPbfkztxM35CBaWHsY7dTUFOl0msMb2CWapsn09DTDw8PkcjnuuuuuDQtRTdPk3LlzmxpkOz09TTKZpKmpib6+PlwuFx0dHSuo0evt4+m6TiwWs1/JZNJ+qHw+H/Pz82WPITA7OMh3H3yQiRdewJVOk0unqcG6sRJYgaNJklBMk0mssiL5f89VVaHG41QZBnux1sIJLMcVgQyger2EHf2+DJB2u6lRVVSspdYTDDKfL7n6sDIvURo9yMod4KusLGECvAYrSqEAvS4XJ0oEvx63mzP5zUgWq2wbBTKhILFMEtO0+oZ+E2pMK36MBAIcL8MKHlAUTpXZGL3mgtMltrJZEy5I4JWhOgt7i95uAOPKSgasQAYYDUJOhYY01JYw/h6vj9CWvVaunDNgOgeNumXfBjAuw1H5mtQhZ8KoAR6jcJDFottF2NQwvT7SqYyl78xDk2UWAwFIJPBgbVKSwSCoKiag1dUR+LVfY99jjyGv0lIoBU3TGBwcJJlM0tXVtWUTeLCeIdEzjMfjtlRJ9OREZnj16lWuXr3K0aNHCwJPucxQYL3BUGB5eZnZ2Vk6OjpsF5p4PM6//du/8b/+1/9iZmaG22+/nTvuuINTp05x9uzZDQdCXdfp6Ojghz/8Ic3NzZw9e5ZvfetbHDt2bO03by/KZhs3bOBTVXXVmXlzc3MsLS2tW0i+tLREf38/4XCYI0eOcOHChS1NYd/MewVbVFGUkmbWq7E114t0Os3AwADLy8v4/X67ES92mCIoFiM6M8P/+9BDDD39NArgNU28iQRZrxdT1/H6fAQBXZapil4bZjetKNTrOgZWAJOxMqm9XNNYLyoKAV0vGBAwj9Xrc4bjLFaJ1MlZM4DpUIhwIoEGKPmsMZvJoJsmQUWxMulcDglrofVgZYYmEAyFiOffG/T58Hs8GLpOLp3G6/WiptPEsAg5xVuDEazSaqnlqo+VZViw2KkLksSREs+tDsxX+WnKZRh3eYhKClImR21OwwVcKROcAXKyRMbnYq+jYjDj9rDkDyCnMrhzOVyKi6ZV9KlDIQ+ylsNUYX+ZosWMBNUBLwtZlWoTvHrhM6gDk4Di8WDkclZlIC8VmAM8moYbKwDGAwGUZBLV58M4cIDQ3Xez7+MfR96EBEnYFba0tLBv374dlQ7ouk4ikSAWi7G4uMjCwgKKolBXV0dVVZUdFMttmot7hs41XLQuygXD+fl5YrFY2Q39XXfdxde//nUuXrzISy+9xNve9jbe9ra3bej6zp07x6c+9Sl+8IMfAPDpT38agI997GMb+pxtQNkfcbfUWQbr7bU53U5K9QIqAcMwGBsbY3R0lGAwyOnTp8vq8WBzAc80Ta5evcrY2Bitra3cfPPN9mdks1mi0SixWIyJiQl7kKUokUYiEaqamnj///k/ZNNpvvPHf8xr3/0uS6aJmUhQHQiQymTQVBUNiAaDhEyTcCrFHl1nAiHithB1uViSZdRcDhmo0nXSjgkRYPWbpvx+/OlrtmNeIOd2k1ZVOwjJwL5EgkEs7RiOzDHhcuEzTQJFi/0kVg/QBZBI2NKEyUyG+kzmWok0f+yILJMC/EUbrUPAS5TOLFuxstvmov/uAmTTLOhZXs2/koAZTbMANJBdMSnooKbR7fNxLJOhGB7DJJFS6VEgakLYsEqL1alr157Uc0zKMvuLruMKkA6FcCcSLFLIrBUwgalgkKyikI3HSZomTpuEdP4a/OSlHvnvPCFJ9LvdKJkMpiShRCJIikJyaQlcLqQTJ+Cd70R94xtJuN2khoftDZjf71/zPhdmFqqqcuLEiTXH9WwHRNVkeXmZTCbD6dOnCYfDBcL2RCKBYRgFZVIRDDfSM3RKKyRJQlXVVcktYh276aabeM973rOp65ucnCyQizU3N3P+/PlNfdZO4YYNfOuZwr5aj09VVYaGhuyssNjtBDZvebZeiEkSg4ODNDU1cfPNN3P16tV16fE2gqWlJQYGBqiurubMmTMr/EO9Xi+NjY0FDhWZTIZoNMrCwgIjIyOoqmozzX7l4x/nXY89xjNPPcUL3/wmi5cvo6sqoUAAfzaLkkwyjrWwK34/ZDLMBQI05AObT9NIut24sUqlBpBJJvkxUOfxWJmXYRBKJOjF6uUJhFWVUVnmcNHi3YKlK3RmgyFN4wrWQu7cN+8HrpQg0ezH6iUWE1p8hsE4Vjmv+IHrBPrJB10H/FhaxCzX2mcaVnBYwCq5NodCeBIJglhZsOg2Lfp8ZDKZUiPyaMxk6MsfF6wsetzrZSnPcs3pFsO0FK8vmD+f8fw5TeTPMwyE8yU8HzAly1R5PNRmMiRcLsawMrVQMmlvODz597vCYdLxOFVFx5zPHytomvgyGTJ+PylNw5VMotfWEvz1X6ftU58i2NlpvyeXy9klxZmZmQJdXqlgODMzw/DwMIcOHSqQKOw0kskk3d3d1NTUcPbsWTuQVVVVFRBLDMOwM8OpqSn6+/vtYOhklLpcrnUFQ1VVmZubo66uDlVVV5RJd2II7esVN2zgWwvlMj6RXU1OTnLw4EE6OztLPjCyLGMYxqbMZiVJWnPCczwep7e3F5/Px6lTp/D5fCQSiQJrpfX08VaDKGvqus7x48dXlWEUn7+wa9qzZ499PoJpNjMzw8DAAFWnTvHON72JhYEB/uOrX2X+tdeIyzJuwC9JZHM53Pk5gJlkkm6XC5emEQYaVJWUy4Vmmrh0HQkrO1r0+fDHYuQHuyPJMv83b6ztwgqUDS4XfaZJp2NjI4JLmsKS5EEsycItRdfYnEpxJRDgYFHw68r/fTEJpgUYiURojxXS9mWgSlFIyDKh/PksyzILhkEMmKmuhuVlPFwLnFX513AicU1i4UBtJsMAFpmn+A6SsDSV/5G/5kYgkM3i7GYtY31v1UWbg6jHw3g+E5OwMtZS8BkGw7kc3cABTaOYy6wCM4qC4vXijcfRsJifEb+fXDaLZBj4AUWSiCoKat4EQW5vp+Y3foPDH/0oSgnDZ4/HQ319vS0Eh9LBUJDAPB4PXV1dVFVVVSTomabJ6OgoMzMzdHV1remJKctyWZKKeI4GBwftUUnOYOh2uwuC4dLSEn19fTQ3N7Nnz54COzawMkPhLbrV72L//v2Mj4/b/39iYoL9+/ev8o7K44bt8Wmatip7UtM0XnzxRW6//XbAumnFTK49e/Zw8ODBVYPahQsXuPnmm9ftyO7ECy+8wMmTJ0tOZhBDaVOpFJ2dnQU7xHQ6TXd3NydOnNhSlifmAM7NzXHkyJGChWQ7IXa0okw6NzXFj7/6VWZfeQV1bg5XLoc7T1wI5nJ4gHQwiDuVAtMki3WD7vP5CGezyPl7uViOYADx/PxA+xqxSpY1+f8tAT63G1cwSFU2i55O4wf7Nc7KrCyNRfwoHnOXwSLuOB/1rCyTMgwmsIJWGiuj0QB/IEA0laIOS8JRrGaYDoc5HF9pTQZWmfHmkv9iCfpvAa4qCouShJYfIeXFyvRcHg8NZXp2CVkmqCgYssy8x4OiqgSKSqTTLhftmmZrISexysPhfB8OrM2H7PGwP5cj5vUS93pxx+P2AGKwyD5aKIScTCKZJllJgnAYPZuFmhqqbr+djo9+lNrTp8tc6fogXJOuXLnCnj17kCSJWCy2IjPcCYuwRCJBT08PtbW1HDp0aFsdVwRJRZDQ4vE4uq7bo5Li8TiqqnL8+PGSJDTDMHj55Ze5//77uemmm/ibv/mbLZ2Ppml0dHTwox/9iP3793P27Fm++c1vcvx4Ma96x7FLbimGruur9vCc7Mrl5WX6+voIBoO0t7evK5i98sortLe3rztLcuKll17i2LFjBf0GwzC4cuUKU1NTtLW1rSjNmKZJLpfj3LlzVFdX22WTcDi87odMBPeRkRH2799Pc3PztlsilYPolcqyzJ49e3jxBz/gP/7mb1js60OKxzF9PhRVxSvLeGQZTzaLnM+QYlhkEwnw+/2EZZlUOk2LI2PRJYklRaHe8ZvrWOOVivtoI1CgMwNrAScYxJ1MYmA9GAaQw5I9VHm9ZLNZJMDtcmG4XHgBX77nJ36pXP58i48pjlucKTr/rdSyoQOzskxH/lqjLhcLbjdpWQbDQHW5VjjXCGQlCcPtZo8j+GmSxKhpEsPKFj2s1DQ6MYb1XURYacMGVk910eUim+931mKVRlVg2e1GUhRcGUueEAcknw9JknAfOcK+3/otuu67D3kbBr2m02l6enoIBAIcOXJkRZ/LmRluZzA0DIPR0VHm5ubo6uoiHC7jCLDNEM/ywMAAfr/fciPKzw2MRCLouk4oFKKuro7Pfe5zPPPMMzz11FPbFpz+5V/+hT/8wz9E13Xe97738fDDD2/L524Qu4GvGGsFPoDnnnuOUChk03M3ctNeunSJlpaWTYlBX331Vdra2giFQgWZ5t69ezlw4EBBplncx3OWFKPRKPF43BbWCqJJKT1eLBajv7+fYDBIW1vbjo5FcULXdUZGRlhcXKS9vX2Fa0Q2nebvPvMZLvzTPxGdmEBJp3Ebhq21i0gS1ZKEJklkdN0eGmoCuepqfLpOLpNByRNaFApNoDUg4fFQV5T5lAp+WZcL0zSJFFUKon4/VfkxTE5MA/slCXfRM5aWJDTTpJS6bIiVZVVxnnOOcxL/fwkrewtXVRGKRim1zRqlfFaYxeqnqYA3P0LKGRJMrBLnoTyRSFxXNhhETyYJ5P8mEw5Tn8ngy0tIpgDD5SKoafb7TCAVCKApClo2C4aBz+9HM010rxff3r003nUXx/7H/yDQWGZExAZhmibj4+NcvXqVzs7ODbmSbDUYJhIJuru7qaur2/YsbzUYhmEbZxw7dsyWZZimacsXfvSjH/EXf/EXTE1NUVtby7vf/W7Onj3LG97whp8n55bdwFcMwzDKkldUVWV4eJjR0VFOnTq1qVJfb28vDQ0NJUkva+Hy5cs0N1s5QV9fH4FAoGSmud4+nhDWipJiMpnE7XZTVVWF3+9nYWEBVVXp7Oys+I50ZGSE5uZmmpub19xNz46N8e0/+zN6/+M/SFy9iltV0WQZUikkWbb6ipJEIF8yg5WuKhmszMKH5bkZ9nrREgmqvF6qs4Xi8FLBbxkIShLBoudmAqsfWFz8Fv+9uJme8Hjw6TqhEuX2qx4P7bkcBlZgi2IFN3ckQjwWo4Y88cfxnlz+ug6u+DQLY1yTTiy53UznA1QAq/QZDwRoLqMd1IFZnw+v2403k8FT4rnRJYlZ00TPf54zqKe8XhKKgpLJoBgGKqAGg9ZmLRRizy//MqcfeICao0dXfO5WIMqL1dXVHD58eFuGu64nGPp8PkZHR5mfn69olgfWBranp4c9e/bQ2tpa8pnSdZ0nn3yS73znO3zlK1+hpqaGF198kZdeeom3vOUtG5YvvI6xG/iKUSrwCXuv8fFxDh48yOjoKHfcccemav2Dg4NEIhGb6bgRXLp0iUwmg2madHZ2rmiCO7M8cW4bPcdMJsPQ0BDz8/N4vV5M08Tn8xVIEHYq64vH43bpeLPZ5WhPD9/58z/n4jPPkF5YwK8oyKkUkmkSBSIeD16XC7JZNF2nkWuBwgCy4TB+RwkwyzWfT8XjwevzocgyS8vLBLAClyf/ygB7JAm/aRaQRxbCYfaVKCuOYJFetPx78XiI53IsYpUHc/l/k1wu3IqCms2ieTy0OLIsARVQPZ6SejoNWKQwWGfBtnhTgX0eD/5yfb1QiNpMBm+eNRvz+UilUgR0HRd5R5tQiEh+CLEKzMsykteLkk4XBOKM10s6m0XHKj97TBPD40F1u3FVVxO6+Waa3vMezLq6slZfm4VoC8zPz3P06NEdH6zqDIaLi4v2WKGGhga73bDTY4UMw7ArJ8eOHSvbYhkeHuaee+7h9ttv59FHH62IfOM6YjfwFUP0xMT/np2dZWhoiMbGRg4ePIjL5VqVZLIWrly5gsfjYd++fWv/cR6CVDI2Nsb+/ftpb2/fET3e3Nwcw8PDNDU10draag/WzGQydok0Fqh0vvYAACAASURBVIvZEgQRCCORyJZ2zcIoO5lMbmt2eXVkhO8++SQvP/MM6bk5XKaJkc2iZzIE8ve34fUiud2Y6TRuXScEpF0uGhzlbg2IKgoNRVlYPBSitmg4cEyW8efZh4bjFQWa3G7I+yxKpolpGMxh9cqKl/NFyvfHxrkmO3BCw8oEiyXISWBRknDV1KDGYii6TtA0C57+BFYQLx48k8QqeaaxWIDNeaZsMZL5a3T5fPhkmWAqZf9dHDAjETKxmOWqoigYgQCqpuGpr6ft13+d2z/8YUJ7C+lAoiJR7BBUHAzXc69Ho1F6e3tpbGzkwIEDFS0visDT1dWFx+PZkZ5hKYiSakNDQ9lrNgyDv/zLv+Sv/uqv+PKXv8ydd9655eP+DGA38BVDBL5oNEpfXx9+v5/29vaCHdBLL71EV1fXpqYhj4+PY5omra2ta/6t0+ps//79GIaBz+ezg+Z26fESiQT9/f14vV6OHDmyJklH9AtFIIzH45imSSgUsskz69mdi4G3wihbMOp2Asl4nH/86lf5j7//e2YGBvDIMqauQzqNDzACATz5xVrDWujdQDgQwC/LuFWVFFBfVPacxiKkOM8643JhahrFXVxRXi3eLsWDQWpTKZSiZy6pKLhNk0gJJ6FoOMxeRxapYQUeMxi0ZiBiBVM/FOj2ZoF9lLbQFKSeqkCARL78WJwfJLFIOrW6TjIQQHO70VMpFEe2KAK9CbgCAXweD0Y2S0bTcNfWUtXayqFf/mXO3HMP/g32jYQJtAiGQobg1OQ5A4eu6wwNDRGLxejq6toUqWyziMVidrBtbW0t+zxsN4HGNE2bfb1aSXVycpJ77rmH9vZ2PvvZz1b0u7nO2A18xVBVlVdeeYVcLlc2+7h48SKHDh3aVGayXq9PwRgVVmcej4exsTEkSaKlpWVb9HhCbB+Px+no6NiS+7qwWxLBMJFIrLogCfF7TU0Nhw4d2tRE+81AlLv+7z//M/3PPcfMyAhLo6PIhoHicuGVJNyahpROY3g8pFSVcP5ZMIGULBP0enFJEqZh4HW7ieaF1m6ulT1NQPb5CBVR/eexpAnFW6Y5rP5XcYEpLsvIeVG+5nKhyTKmy0U2l0M1TTyahieXWxHI5rBkE6UCXAKQJYmwJLFgGJa9tSzjM03cponq8aB4PFQ5sllDkoi7XMQ1DdM08bhcmJKEV1WJkLeYrq62LP/yPUEzGEQFpEiEcEcHb/qDP+D4r/3atm9uVFUtCByiV+3xeIhGo+zbt6/iJBIxwaGrq2tTrk3O+X8bCYZCBL+aPMIwDP72b/+WL3/5y3zuc5/jrW99a8VE+q8T7Aa+YghyRXV1ddmboaenh6ampnWNEyrGWl6f6XTatko6evRowUNz9epVstksra2tW+rjGYbB5OTkjmdaYncugmEqlbKnW8iyvMJ4d6cxPz9vu9k4Sz+JWIz/7xvf4Cff/z6TfX1k02mMTAYjH7RCwaAVkLJZ3JqG7najAQFHL1hzuzEd/83EyqDEwF0Bt8eDKx9YzFTKsmPLlw91QJFlqmUZQ7N8NEUGukRpqUMCcMsytSWywoyi4PN48GYyJBSFjKKgmqa1yUinSUkS1SWyOrB6jotAVSRCLhbDRyFpRnO70Xw+NNPE1HV0TUMHfH4/wT17qG1vp/7WWwmdPk3XyZM0NjZWbHFVVZXe3l5SqRQ1NTWk02lSqRQej6dgivp6rMs2ClFSFffYdn7+asEwHA6TSCRYWFjg2LFjZfuXs7Oz3H///dTU1PDFL36xos/f6wi7ga8U1prQMDAwQFVV1aYIKktLS0xNTa1wJNc0jZGREebm5mhvb6ehaGSK6MENDAzQ1NRklxQ32lsT44IElXo7GG3rgdAtTU1N2d9bLBYjm83aGiLx2kzvdDWIzYQkSXR0dKzZuJ+fmeHpv/s7up9/nsm+PuJzc2ipFJlcDreu4/F4rHOUJLR0GlnX8QB+WUb3+wkmkwVP1jIW29J5VB0rmInpEgImoEYihGOxFU/nApawvjhbzOY/S1iHiQBk5HKQH+tUDSUDnA6kPR6Cskwsk0GSZVxgDwAGq3Sp+v14FIWcqqJls3glCcnnw11Vha++nuoDB7jpV36F07/1W5guFz09Pfh8Ptrb27f991wNwqqv1IZOlBTFK51O28HQaaS+mWDllApUsqSay+Xs3rwYa+TMDIVRvCzL/NM//ROf/vSnefTRR7n77rtvtCzPid3AVwprBb6RkRG8Xu+GCCoC8XickZERbrnFUmUJk+crV67Q3NxMS0tLQXmi2G09nU4XaPFM0yyYjRcKhUre0KlUiv7+fmRZpr29fVP9yc1ibm6OoaGhFZmWuD6nu0QsFrNFtM5r2kyAFsF2ZmaG9vb2TUlIwCrj/vj73+f8j37E1PAw85OTJBcWIJfDVFWQZavHlR+Bo+ZfXq6J5+V8QHFJEpgmhq6j5nJkDYOGQABD18lks/aDlSM/JikYRDcMsvmp4bIso7jdeDQNSdchPx0CwPB4ULxe/PH4iifblCSMmhqUbBZdUaxMTdMs0+c8aScLmLJMxOcjmZ9C4VIU0HVkvx/Z78ddU0Nk/34iBw/SevvttNx6a8FMuZmZGaampjasjdsqcrmcbQp/9OjRdTOCs9lswb2XyWRWTBXxer2rBgmR5a0mFdgJiB755ORkQfXEmRn+67/+K08++SSSJKEoCh/96Ef5pV/6Jdra2ipW+n0dYjfwlcJagW9iYgJd1zlwYDXvitJIp9P09vZy8uRJ2yevqqqKI0eOrNgZr6eP5+ytRaNRkskkLperYEjs1atXWV5eLikE30kI1xW3282RI0fWTZEWvoNO8sx6ht86IcqaYjHa7ofcNE1eff55fvKv/8pQTw8Lk5PEFxZQUymMbBZT11Hy/qJSNoupacj5fpqKlYGJszfJk2t0HdlBnjGBrNuNV9PwF92PwtasnmtOMYrfTzqXI5OXGZiAR1FwKwqGI0DmgGBVFXIuhyFJmIoCkoSey6EbBkoggK+6moaDB9nb1saBEye4+Zd/mcYiQpbTdHx+fp7Z2VkURbEdgpz+kDsFQQC7cuUKbW1tm6rCFEOwmEXwyGQytqTHGQx1XWd4eHiFILwSyGQydHd3EwwGOXLkSMmNoWma/Nu//Ruf+MQn+OAHP8jBgwd56aWXePHFF3n729/O+9///oqd7+sMu4GvFNaayScGu7a1FcuY1/fZL774Ij6fD8Mw6OzsXFEW2aoeT1VVe2L64uIiLpfLdmgRC9JOLkaibCt6mdvRR3AOvxUBXpR0xDX58tMHNlLW3E4IQpKpqsyPjjI6MMDklSvMTkyQS6dJLS+TWlpCz+WQDAO314ssSRiqipn/rRVFIZfJoGsaEtivYCiE2+VCTaVA3BeGAbKMy+/HVFXkTMaSSSgKssdjZXmShKTrVhD0+5ElCU1Vcfl8BOvrqd23j+b2dpo7O2k7eZKDt966If2kYE06S3ylMngxgWM75C8CmUyGnp4evF7vjpZUTdNckRmmUilyuRzV1dW0tLTYwXCnISpE4+Pjq2bV8Xichx9+mKmpKZ566qlNVad+jrEb+EphrcA3Pz/PwsICnZ2l1FTloWkaQ0NDjI2NcerUqRWlt+2SJywvL9Pf3091dbXNmHRq8aLR6LaVE4vPX+y+W1pa2L9//46WfZw9m+XlZduEt7GxkT179uyo2L74PAYGBmwm8Hp2/nMzM4z09jIzMcHC1BTzU1PEFheJLiwQW17GNAxMXUfNZDDyTEpZknC53bhdLmRJQlYUNFW1ynM+H8GqKoKhEMFwGK/XSyAUorG1labWVpo7Otjf1oZ/G3tPYkir8G8t91s7JweITMowjAI93kbuP2eJr6OjY1Mks81C13UGBweJx+McPnzYZpTGYjFyuZw9b1K8tvP+y2azdHd3273TUkxo0zR57rnnePDBB7nnnnt4//vffyOXNMthN/CVwloTGqLRKOPj49x0Uznr4EKYpsnk5CSjo6O0trYyNjbGL/zCLxT8+3YEvEwmw8DAgO2CvlqDvVw5UTywVVVVG9IOCdeVUChEW1tbRQkNTrZmU1OTXaKKRqMF8/5EZrhdhB7xu46Pj3P48OEdYy46F1cnm0+4DHV0dFBfX1+x3lIul6O/vx9d1+ns7NxUVu2cKSfuP2BFMCxetJPJJD09PbbMp1LkLMBuTZQL9M4evHipqrpl8pZzQ1luxidYbZRHHnmEy5cv87WvfY1Dh8oNiNp+6LrOmTNn2L9/P9/73vcqdtxNYjfwlcJagS+ZTDIwMMCJE8WjRVdiYWGB/v5+amtrOXz4MG63m5/85CfccccdwPbMx9uucUGinOj07hSsN1EmLS7n5HI5hoaGtt11ZT1Ip9P25IZyZc1S5twi2xCBsNQCuxbE3MOqqioOHz5cMR0iWPdUX18f1dXVthNINpvF5/MVuOlsd7a7E/00J3RdL6DrJxIJe/acoOsLbdxWNKebOa/BwUESiQTHjh3bEDGsmLwlRgE5S7+r9UFzuRw9PT24XC46OjrK/t2FCxf48Ic/zH//7/+dD33oQxXdEAA88cQTXLhwgVgsthv4flaxVuDLZrO89tprnDlzpuzfJJNJ+vr6kCRpRfnrJz/5CW984xu37KtZiXFBorchgqFTfpDNZllcXOTw4cMVnVSt6zqjo6PMzs5uqtRlGEZBVuhcYEXgKJftinJ1PB6veKAXzEXRG3YGeqe13GrZxmYDdCqVore3F7/fX5KItVPQNM2eiK4oCpIk2eStUsYI243FxUX6+/tpbm7ettJ9OSZz8W+1sLDA8PAwR44cWSFvEsjlcjz++OM899xzPPXUU3R1dW35/DaKiYkJ3vve9/Lwww/zxBNP/EwHvht6AvtaN7cQYZeCcENZWlqis7NzxaIsNhTT09ProkqXg3Nc0OnTp3eslyVMdcWDJ3b9Q0NDuN1uXC4XV65cYWFhoaBfuFN9BSGN2LNnD7fddtumjiPLsp3BtrS0ANYCKwLh7OysLXh2ZlDLy8uMjIxw4MABOjo6KkpbFyXVcpmWc7p9U1OT/b5UKmVf09DQ0IZ7u4ZhMDY2xszMDJ2dnRUVPAuXnaWlJU6ePGmbOThLv87fajv0eAKapjE4OEgqleLWW2/dVvmPJEkEg0GCwSB78/6kzsrE9PQ0Fy9exDRNamtrSaVSLC0tEQ6HCzYuly5d4t577+Xuu+/m2WefrWjVwYk//MM/5LOf/axdrv5Zxg0d+NaCMG92wjAMJiYmGB8f58CBA3R2dpY1ku7o6GB+fp7x8fEVhs/hcHjVhSibzTI4OEgmk6l4xiF6iLquc+rUKTuLdU5MHxsbW2FXJsYcbWUhEmVNRVE4ceLEtrM1XS4XtbW1BRsVke3Oz8/T3d2NaZpUV1fbme5Os2PB8lHt7e0lHA5z9uzZDS1uzgVWsPqcvbXJyUm7tyt8Vp1TEITXZF1dHWfPnq0oSWJ5eZne3l727t3LmTNnCu4dt9tNXV1dQa9LEJ2i0ShXr14tqcdb7z0jsryWlpYVz/FOQfwG6XTaZsg2NDQUBMOBgQG+853vMDQ0hNfrpbe3l7/8y7/kjW98446fXzl873vfo7GxkdOnT/Pss89et/PYLtzQpc7VZvIJOPt08/Pz9Pf3U19fX7LfI/p4hmEgy/KKgChIJqIHVYpkYpomY2Nj9qT1hoaGimUcTiH4enuITruyaDRKOp3G6/Xai2tVVdW6slTRv5yfn6e9vb3iDD7hrC+8TMXCVEpsv56Ny0aOPTw8zPLyMkePHt3RDU6p3pqYUNLS0kJjY+OOj88REJlWMpmkq6tr09q4UhKEbDa7KutS0zQGBgbIZDJ0dXVVVAqjqir9/f1ommZPcSiFy5cv8+CDDxKJRGhoaOC1115D0zSeeOIJ7rrrroqdr8DHPvYxvv71rxcwx9/5znfyjW98o+LnsgHs9vhKYb2B75ZbbqGvr89uPBeXQzarx3OSTJyjgCKRCC0tLdTU1FRsErqztLhVIXixpMKZ7Yr5ZM6gIY69d+/eFY42Ow3BFN23b9+qvdNS7FigYOOy3tE5xcdeSyawExDH3rt3L6FQqIBJ6iwnCqLTdp6bkEe0tLSwb9++bb/u4j6ouAcDgQAul4vFxUUOHjxY8e9cEOBW883VdZ2vfe1rfP3rX+fJJ5+0N91gVSYMw6ioG1MpPPvss3z+85/f7fH9rGKtmz6Xy5HJZLh06VJJEelW5QmKolBTU4Pb7WZxcZGamhpaW1ttl4yJiYkNl0g3imQySX9/P263m5MnT26LONfn8+Hz+ewelbOvMTU1RX9/P6ZpEggESCQSeL3ebe+vrIVMJmOTktZz3WJYqjMjExlUNBplZGTEniPnzHZL9aCy2Sx9fX0AO1LOXQ1O4ozz2MXlRBHgRTlxO4YUi2xHVdUdve5SfVDBmozFYtTW1jI9Pc3k5OQK1uVO9M+cGebJkyfLXvf4+Dgf/OAHOX78OM8999yKLLgSwvkbBTd0xuccRuuEaPRPTk5imiZnz54tuOm2S4+3nnFB6y2RbvT4O+G6sl6I8t7s7Cy1tbVomrbCgm0nMg249ttOT09vydezHAQhQ/xexTZYiUSCmZkZu4xdKZimydTUFKOjoxuWKGwHk1QwNndSB1kO8/PzDAwMrMi0DMOwWZdOCYyzD7pVwwehCVwtuzUMg2984xt85Stf4YknnuAtb3lLRb6fTCbDXXfdRTabRdM03vWud/HII4/s+HEriN1SZykUBz4xGWFwcJDGxkYOHTrExYsXbZmC+K7K9fHWi62OCyoukaZSKbvBL4JGuR15pV1XirFaWdNJXBDeiaJXsx0WbEtLS/T399PQ0MDBgwcrUlIVPaiZmRlGR0dtmr5TbL9TmYZAKpWip6fH9nvcjmM5qfoiaJTqg2qaRm9vL4qi0NHRUbHSPRRmmF1dXevKmDYruC/GejWB09PT3HfffTQ1NfHEE09UVLcoNtWhUAhVVbnzzjv58z//c97whjdU7Bx2GLuBrxyyebNg4Uji8XgKRNKXLl2yPfq2qseDwnFBBw8e3LYFL5vNFvQKc7ncir6aKGteD9cVp5F1e3v7uhYhpzmyuK5imn44HF5zERIOJKqqrttqbLvg9LgU5BWn/EAssE5rr+2Sigiy0uzsbEUkCsV90IWFBXK5HDU1NTQ2Ntp90EpsOMQG9tChQ1vWnpYaviz0oOLl7O9Go1F6enpW7d2apsl3vvMdPve5z/HYY4/xazswuHcjSKVS3HnnnXzlK1/h9ttvv27nsc3YDXzlEI/HGRgYIJFIcPTo0RU7rp6eHurr6+1FY7NlzUqPC3KWSJeWlpifn8cwDOrr66mvr990iXSjcLI1Ozo6tjw1wrm4ClF6udJvsS6ukgxZuLb4riezdortRaaxXrF9KUSjUfr6+qivr69YdiuQTqfp6ekhEAhw6NChAnuvYgnMdgvTVVWlr68PXdc5evTojvXFNE0r+L1Ef9cwDNvirba2tuR1LSws8NGPfhRFUXjyySe3vdy+Eei6zunTpxkcHOSee+7hM5/5zHU7lx3AbuArBdM0OX/+PHv27FlRbhR9vOnpaXsun3PqwXpLNs5eWqXHBTlLqocOHaK+vr5AerCREulmIBb+ffv27ShbUyxCzuuSZZlsNmuPgqrUwFC4RpwRFmubXXyF2F6UE52MS+ekiuL3iL5xJQelgvXMjI+Pc/Xq1VUnCggJjLiudDpdMFR1s/1dId4XDkOVRCwWo7u72yY0FTNkX375ZW655RaGh4d55JFHePjhh/kv/+W/vG6GxC4vL/Mbv/EbfPnLX163N/HPAHYDXzkUT2go18crpuhrmlbQBC8uuTnJBNejl7a4uMjAwMCaE9iLS6SCtFBOerAebKasuV0QhKFYLMaePXtshmIul7Ova6u2XuXgXPh3gjgDhQNVo9GorVmrqqqyre2ux/2WSCTo6emhurqaw4cPb/iecTJJS83GW21TttnhtNsBwzBsDeixY8dWbDRE3/ozn/kMzz33HGNjY5w4cYI777yTM2fO8I53vKPifpvl8OijjxIIBHjggQeu96lsF3YDXzk4A99G9HjFJbd4PG6XcNxuN7Ozs9TU1NiG1ZWCmFMnnGM22s8qxyJ1zvkrV5oSQvCFhYVtKWtu9LwFaefAgQPs3bt3RQbvJGM4+2riurbSfxLuJ7W1tatuNLYbpmkSjUbtHqaY5rCTEhgnhN3Y/Pw8R48eJRKJbMvnlmKSltq8CJ/L65HlJRIJuru7aWho4MCBAyXvHdM0+fGPf8xDDz3E/fffz+/+7u8yPT3NhQsXePXVV/mjP/qj65b1zc3N4Xa7qa6uJp1O87a3vY0HH3yQd7zjHdflfHYAu4GvHFRVtbO7rcoThO1UNpvF6/Wiqqq9GxcP606x95yGzluZ3FDus0tNc3BeVzQaZWhoaMdMtFdDMpmkt7eXQCCwIWPlUibWYvMirm0tL0hnafHo0aO2z2QlIIaVjo2NFRgcO+fiic0LsGJSxVYX3Gg0Sm9vL42NjWUX/u2Ec/OytLTE7OwspmlSV1dHdXX1jgd553mIKSldXV1l3XZSqRSf/OQn6e/v52tf+xoHDhzY0fPaKC5evMh73/tee/377d/+bf74j//4ep/WdmI38JXDT37yEw4fPmzv9rdzXJCY2+UsJZqmuSLL2MoCJCQYw8PD2+K6sl6IEunCwgIzMzOYpklNTQ01NTWbLpFuFE6rsc7Ozm2hgjuNkUX/qZR4W3zvQ0NDtLa27ogDyWoQwX69EgWnXZmYbO8M8pFIZN0+q4KpGovFKt5HhGuawLa2Nurr60s66jgZstvJJE0mk3R3d9uZfbnPfeGFF/jIRz7C7/3e73HPPfdU5JkcHx/nd37nd5iZmUGSJH7/93+f+++/f8eP+zrGbuArBdM0ue+++3j++ecBy0XjzJkznD17lo6OjjVv1s2MCxJZhgiGyWQSt9ttB8KNEEzESCSPx1PxXlpxWbO6unrTJdLNQAQdQZzZqaAjdHhiYRX9Qk3T8Hq9HD58mNra2oqVNp0ShVIs5I3AKbaPxWLr8lkVxs7Xw2ZNuK8oikJnZ2fZzL6UJ6miKAUZ70bvRdG/nZqaoqurq2xJN5vN8thjj3H+/HmeeuopOjs7N3Wtm8HU1BRTU1OcOnWKeDzO6dOn+Yd/+AeOHTtWsXN4nWE38K0G0de6cOEC58+f5/z58wwMDNiO8bfddhtnzpyhpqbGflimp6eZmJggGAzS1ta2pYa6aOyLV7EGLxKJFARUTdNsY+NKu644M521gv1aJdLNsEjT6bQtC9kKY3IzEK4vU1NTtLa2AhSUErfi27keVKK06CRxOftqws/TMIwND2ndKpwbzNVm1q2G1Riya404EgYAkUhkVeLOxYsX+dCHPsRv/uZv8sADD1y38UECd999N/feey9vfetbr+t5XEfsBr6NQkxJeP7553n++ef56U9/SjKZ5PDhw8zPz+N2u/nrv/7rHSFwrGZTZhgGS0tLHDhwoOLMve1ga5bKnlYzsBZwWo1tZijtViF0ccJ4oPgcnb6dxZn8Vs2enZMMjh49WnGJwsTEBCMjIwSDQbsf5HQyWY+JwGaRzWbp7e1dczL5ZrAWkzQSiTA7O8vk5CRHjx4tu8FUVZUvfOELPP3003z1q1/llltu2bZz3CyuXLnCXXfdxaVLl7aNcPQziN3At1XkcjmeeOIJ/vf//t+86U1vIplMcvnyZSKRCGfPnuXs2bPcdtttG7YfWy+Wlpbo6enB5XLhdrvth9SZFe4Ue9RZ1txuBxCngbVYhJwl0kgkQi6XY2BgoGIkCieEPGIzQcdpwVYsPRAL61q/mdBCXo8+YjmZgHMuY7HYfiv+sU44Wbrt7e3bStZa7ZhiY7a4uMjMzAwA1dXVBeQZ52/W19fHvffey1ve8hY+8YlPVFRKUQ6JRII3velNPPzww7zzne+83qdzPbEb+LaKK1eu8K1vfYsPf/jDtmjYNE3m5+ftrPD8+fPMzMzQ0dFhl0hPnDixpbJQLpdjcHCQdDpNZ2enzRp00r2Xl5dtO6/toueLY6y3rLmdECXSxcVFrl69iqqqhMNhmzgjJtrvJEzTZHZ2luHh4ZLyiM1+pnAxKbZgK9aDOic4dHZ2VrSk6ww66zW0dpoIOMvamxGlZzIZent77d51JeVAQn87NjZmS3IEk1S8vvvd79LX10coFOLixYs89dRT12VGXimoqso73vEO/tN/+k985CMfud6nc72xG/gqBV3X6enp4dy5c5w/f55XXnkFRVE4efKknRWuxgYTEJPeJycn1+1o79yJC3q+y+VaMSZnPRDEGa/Xy5EjRyq+8E5MTDAxMWEvvJstkW4G6XS6YOHdyV280xRZlLU1TUPTNPbu3Utzc3PFhsOCFXR6enrwer1bDjqlSomrmY47TR86OjoqbuWVzWbp7u7G6/XS0dFRtkc3ODjIAw88gMfjoampiUuXLmEYBo888ghvf/vbK3rOTpimyXvf+15qa2v54he/eN3O43WE3cB3vWCaJvF4nJ/+9Kd2MBwZGaG1tdUOhKdPnyYcDtuLm3D/ED6LW1nQS5XbnM4skUik4POLp5FXkjgDlhC8r69vTQeQUiVSoGBR3SjBxDmBfjXLrZ2CU4/opOkXW8vtRMYrNhuTk5M71kN1mo6LYKhpGsFgkEAgwMLCAqFQaNWgsxNwZrirBVzDMPjrv/5rnnrqKb74xS/y5je/2f63dDptD5G+Xnjuuef4xV/8RW6++WZ7Y/3YY49d12B8nbEb+F5PEDZHokR64cIFstksbW1tjI+P09raypNPPrkjYmjnZIDigCFJEgsLC7S0tOyoRKAUnL00Z0l3IyhFMBEsUme5rRSWl5fp6+ur6MgiAeF+Mjc3V1aiUDxRXGS82zHaKJlM2qzFtra2ilpoiWdhcnKSNHZ+mAAAGW9JREFUcDiMqqq21nUndHjFEBKJtcgzU1NT3HvvvbS2tvL5z3++rGh9F68r7Aa+1zMymQyPP/44f/d3f8eb3/xm5ufn6e/vp66ujtOnT3Pbbbdx9uzZHZsuEIvF6OnpAcDj8ZDJZDZtyr1RrGU1tlUIb8tiqYhYVP1+P8PDw2QyGY4ePVrRkUVwLeBulLhTarTRRgOGYMrOzMxsWRO4GTinODhF+OV0eJsR268GYWq9mkTCNE2+/e1v84UvfIHHH3+cX/3VX63YhvB973sf3/ve92hsbOTSpUsVOebPGXYD3+sZr732Gj/84Q/50Ic+ZO84RUB4/vnn7RLp0tISXV1dNov0lltu2VLJS0xCLzWFfaOm3JvBZq3GtgKnVGR6eprl5WW8Xi+1tbXb5qazHjglCl1dXdsScJ3mCM6AUdzjlSSJeDxOT0+PbWJeyQxXjIuamJhYd1m1lKPOZsu/qqrS29u7pqn13NwcH/nIR/D7/XzpS1+quITmxz/+MaFQiN/5nd/ZDXybw27g+3mApmlcunSJc+fO8fzzz3Px4kV8Pp/NIL3tttvYv3//uhxnBGOxubl5XQ4cq5lyr9fXUmAnrMY2glQqRW9vLz6fj/b2dmRZ3nSJdDMQmcZOZLjFcLqziIBhGAamadpDWivJmnRmee3t7Vsqq5Yq/641gUPIQ1YztTZNk+9///v8yZ/8CZ/85Cf5zd/8zetmJH3lyhXe8Y537Aa+zWE38P08wjRNlpaWeOGFFzh37hwvvPAC4+PjHD582M4KT506VZDBLCwsMDo6arP2tlLCLF5UBWtvNVPuYnlEJRcUZy9tLT3iWiXSYlLQeiDE2LIs09nZWXHN1/LyMj09PTQ0NNhOLOWy+e3u8znJMztFHCqewBGPx225SDgcZnFxEYBjx46t2ut98MEHSSQS/M//+T8rPvGhGLuBb0vYDXw3CgzDYGBgwM4KX3rpJUzT5Pjx4ywsLLC4uMi3v/3tHVt4yply+/1+FhYWbKp4JeURYBkA9PX10dTUtCkRfDEpKB6Pr9tw3Fna2+7JGevBWmXVUtMcik0EtlL+FZZfoVCII0eOVJw8Mzk5ycjICH6/3x455rQpq6qqQpZl/v3f/52Pf/zjPPDAA/y3//bfKlr+LYfdwLcl7Aa+GxWGYfDNb36TT33qU5w4ccIOjKv5kG4nVFVlcHCQubk5AoEAqqpu2a9zIxCuL7lcjqNHj26rx2Q5Fqmz/Ktp2nVjTALMz88zMDCwYecXYSLgnOaw1vT3YjgH865m+bVT0HWdgYEB0uk0XV1d9vk6xfbPPPMMX/ziF5FlGVVVeeihh3j7299e8WpEOewGvi2h7A94fV1Ud7HjWFxctLM/57gk4UP64x//mM9//vMkEgmOHz9uawuPHz++5d6PmALf2NjInXfeae+gnWXEsbExu4wobKGKTbk3A6cYer0GABuFoii2nZWAEG0vLy/T399PNpuluroaj8dDLBbbVIl0M1BVlb6+PjRN4+TJk+s2LhBQFMUeMyXgFKRPTEzYmlBnMBSl7VQqRXd3t23pV+mALzL8lpYWOjs7C357l8tlX1tXVxehUIh3v/vd3HTTTVy4cIE/+IM/4NSpUzz66KMVPeddVA67Gd8uAGtRe/XVV20G6eXLlwmHw5w9e5YzZ85w++23r9uHNJvNMjAwgKZpdHZ2rpllrWbKLTKnjdDXnbPq2traKkregGsShaamJlpaWmzR9kZLpJuFmFe3UwFfwFnaFtmhrutIkkQ2m+XIkSPs2bOnoiVDXdcZHBwkkUisOkUik8nwp3/6p1y4cIGvfe1rtLe3V+wc14t3v/vdPPvss8zPz9PU1MQjjzzC+9///ut9Wj9L2C11bhZ/9md/xgMPPMDc3FzFezPXE8KH9Pz583YwXMuHVNd1rl69yuTkJG1tbZsaHyMgxsg42YhrmXKLgcDCTLvSbFFR1k2n06tqAotLpKlUCrfbXRDoNzv5ore3F0VR6OjoqDh5Rgxp9fv9hMNhEomEbWDtZMhuhwavFKLRKD09PWsSp15++WXuu+8+/ut//a985CMfqVg2+vTTT3P//fej6zof+MAHeOihhypy3BsYu4FvMxgfH+cDH/gAvb29vPjiizdU4CsF4UMqDLlffvllZFnm1KlTNDY28vd///c8/vjjvPnNb94RVqAzcyo25Qbr9xKDaStNTNiqRKHUTMbVrOWcME2Tq1evMjY2VrFJBsXHF1ZvpYTwzk1MLBZbYcG21T6vYRgMDQ0RjUY5duxY2Q2Hqqp87nOf45lnnuGrX/0qN91006aPuVHouk5HRwc//OEPaW5u5uzZs3zrW9+6kYfEVgK7gW8zeNe73sUnPvEJ7r77bi5cuHDDB75iCPLChz70IS5fvsyxY8cYHh62fUjPnj3L6dOnbTu07YaYTTg4OEg2m8XlchUQZ9ZDwNgqMpkMfX19255lrZdFKkyli91PKgWR5dXU1HD48OF1bzhKDbzdjFwkHo/T3d1ts3XL3Wfd3d3ce++9/Mqv/AoPP/xwxcvf586d41Of+hQ/+MEPAPj0pz8NwMc+9rGKnscNhl1yy0bxj//4j+zfv59bb731ep/K6xaSJPGlL32J3/7t3+Y973kPkiTZWrlz587xgx/8gD/90z8lm81yyy232MSZo0ePbjkjdI6PEWVVSZIKTLmdBIz1ZE4bPb7QpbW3t2/7JAFJkggGgwSDQfbt2wcUlkiHh4dZXl5G0zQaGhqora1F07SKBT5h6D07O0tXV9eGzZl9Ph8+n88eeeQ0HZ+enmZgYGCFBVsoFLIDm/D4XFxc5Pjx42W9XXVd58knn/z/27v3mKbONw7g30JFcVwUN4a0IuDGKHVYB1UXddlkZhnRqeMPt3hJ5pjLVLyQKdmYm7LI4oX8JuqUeZ1Ol2w6A0OCOLzOCXjNYKWAQyKoDIot5Vbanvb3x3JOikBBenqBPp+/Vpec8zZGHt7L831x6tQpZGVlISYmxrYvPkAPHjzAuHHjuM9isRjFxcVOGQtx88L35ptvor6+vtufb9myBenp6SgoKHDCqAaXHTt2dPns4eGB8PBwhIeHY9GiRQD+++3+5s2bKC4uxvbt26FUKjFmzBhur/Bpc0hbW1uhVCq5wzeWP+y9vLzw7LPPdjnBys6c6uvrUVlZ2aVHbSCXprLvd/SJRfYUqVAoRENDA4KDgyESibirqOxZ6C21traivLwcAQEBkMvlvCwrCwQC+Pj4wMfHp1uh12q1qKmpQVtbGzw9PeHt7Y3m5mY899xzeOWVV3r9fvfu3cOKFSswZcoU/PHHH3af/ZPBg5Y6e1BaWoq4uDhur6Curg7BwcEoKSlBUFCQzc/fuHEjsrOz4eHhgcDAQBw5coT7x+4OnswhLSkpgVqtRmRkJFcMe8ohtYw6i4yMHPAVMGyPmuXhEstQbn9//x6XwthZRlNTk03vHyh2Nq1SqSCRSHq8IcBaeLWtp0jZWV5jY2Ov77cns9mM6upq1NfXIyAgADqdDjqdjjv0ZDab4efnhzFjxuDQoUM4fPgwMjMzMXPmTIeOsye01OkUtMdni9DQUF73+Nh+LgDIzMyEQqHAvn37eHn2YNVXDmlLSwvOnDmDjIwMu1yZ9OTBGcsYL39/fxgMBq7x3xmHZ5qbm6FUKp/6Fgeg50b7YcOGPdVeKBtqzd4R6ejv39teouWhp8LCQuzatQsqlQq+vr746KOPMHPmTMhkMocnBT3JaDQiIiIChYWFEIlEkMvlOHHiBKRSqVPHNcTRHp8rsZwptLW1uURChLMJhULIZDLIZDJ88sknMJvN0Gg0yM/PR3p6Otrb2zF69Ghs2LCh1xxSW7B7Tmw2Ixvj1dTUhNLSUnR2dnKHSRobG7liYe+/O4Zh8M8//0Cr1WLixIl45plnnvoZ1hrte1oitQx47s8s057YA1SPHj3q8cSoQCCAt7c3hg8fzv1dHDhwACKRCNevX8fBgwcRHh7u9NYBoVCI3bt346233gLDMFi2bBkVPSeiGZ+TpKam4ujRo/D398eFCxds6nkbqlpaWjBr1ix8+eWXmDt3brcc0tu3b8NkMkEmk3H3Ftqa+M+yvMEiNDQUQUFB3XoL+xPKbYvHjx+jsrLSIYHePS2RGo1G6PV6jB49GmFhYfD19XXoL2kdHR1QKBTw9fW1GvfW0NCANWvWwN/fHzt37rRLDi0ZlGip09GsHZyZN28e9/mbb76BTqfD5s2bHTm8QYNhGKv9a+3t7bhx4wbXW8hHDqlOp4NSqcSwYcOs3mBhLZTblv00dlm1s7OT93zR/mD3MlUqFUJCQmAwGNDc3IzW1tanXiIdCMtQb2sZn2azGTk5OUhPT0daWhrmz5/v8NWTX375BZs2bUJ5eTlKSkoQGxvr0PcTq6jwuar79+8jPj6eQmh5wi6NFRUVoaioCCUlJf3OIeWjRaG34Or+hnKz98Wxs0xH/yDXarUoLy/H888/j5CQkG57eZZ5nc3Nzb0ukQ6UTqeDQqHo874+tVqN9evXQ6/XY+/evU5bMSkvL4eHhwc+/vhj7Nixgwqfa6HC50qqqqq4bMBdu3bh0qVLOHnyJC/PXr9+PX777Td4eXlhwoQJOHz4sMNT8V1NTzmkfn5+iImJgVwux9SpU1FbW4vc3FwsWbIE4eHhvLYAdHZ2dpkV6vX6LgdnfH19YTQaoVQqAcDqreD2YjKZUF1dDbVajaioqH7vJfa0RGoymbgi+GT/nbXnsH2Z1m5lN5vN+P3337Fx40akpKRw/aPO9vrrr1Phcz1U+FxJQkICKioq4OHhgfHjx2Pfvn0QiUS8PLugoACzZs2CUChESkoKAGDr1q28PHuosMwhvXLlCn7++Wfo9XruFnu5XI7JkyfbbYmRbdbWaDRobm7G48ePodfrERAQgKCgILvmWfaEPTHaV/pJf1n23z25RMoWQ8sl0s7OTpSXl8PLywsRERG9zhhbWlqQmpqKhw8fYv/+/bz9m+EDFT6XRIXPHZ0+fRonT57E8ePHnT0Ul1RXV4d58+ZhyZIlWLFiBXdw5skcUnaJNCwsjNdj/Gzc2PDhwxEWFtZlv7A/ody2YhiGS4B5mlneQPS0ROrt7Q1PT09oNBpERERYve38ypUr2LBhA1auXInExESHtlP0Z7+eCp9LosLnjubOnYuFCxdi8eLFzh6KSzKZTHj48CHEYnG3/2c2m9HS0oLr169zTfb37t2DWCzmZoUDzSG13EvsbVmvr1Bu9uDMQAuARqOBUqnE2LFjERIS4vDlws7OTpSVlYFhGPj4+KC1tbVLRJler+cO1qSlpeGvv/7iWhNcERU+l0R9fENJf34D3bJlC4RCIRcbRrrz8PDosegB4O4DjIuLQ1xcHAB0yyFNT0+HTqfjckjlcjkiIyOtHu5oa2vjbmS3FnfG9qd5e3tzaUEmk4k7OMNGeAmFwi4HZ/pq1LbsC4yOju71JgN7Ym+ymDBhApfVyY6NXSL97rvvcPbsWbS3t0MikWDFihUO3/ckQxfN+IagI0eOICsrC4WFhbz/YKPj213pdDrcunWLO0XaWw6pwWBAXl4exo4d22Mj9kBZNqJrtVqu0d7Pzw+jRo2Cr68vV1zZWR57dZOjZ3nsrfAmk8nqAR69Xo+tW7fi8uXL2L59O5qbm1FcXIzi4mIkJSUhPj7eoeO25vTp00hKSkJjYyNGjRoFmUzGxZIRp6OlTneRn5+P5ORkXLp0yS5HvOn4tnWWOaRsb+HDhw/R2dmJ2NhYJCUlYdKkSXaL0HryOiOtVgvgv9mi2WzGSy+9hICAAIcXPZVKhaqqKoSFhVnNu/3777+xatUqzJ07FykpKQ6/PogMKVT43MULL7yAzs5Orgdt2rRpdskBpT2Nvun1eqSlpeHixYtYu3YtGhsbUVRUhNLSUowYMYJLm5HL5RCLxXY5sKFWq6FUKhEQEAAvL68uF8H2FcrNB6PRiIqKChgMBkgkkl4LvtFoRGZmJnJycpCVlYXJkyfbZTx9oXagIYUKH+EXFb6+GY1GHD9+HIsXL+6yl8fmkJaUlHCnSOvq6hAWFsZbDinDMKiqqkJbWxuioqK6tWZYHpxpbm7mDpmwhdDHx8fmQtzU1ITKyso+b6WvqqrCqlWrMH36dGzevNmpgdLUDjSkUOEj/UfHtx2PzSFll0hv3boFhmEgk8m4/cL+5pCyGZ9isRgikahfxdNkMnH3+rG9d56ent0OzvTnWWzRbW9vR1RUVK+xZiaTCfv378fRo0exe/duTJ8+vc9nOxK1Aw16VPgIv/gsfPn5+VizZg0YhkFiYqLTk/RdwUBySNva2nD//n10dHRAIpHY3IBvMBig1Wqh0Wig1Wqh0+kwcuRIrgm9p3gytVqNioqKPotubW0tVq5cCYlEgq1btzrldGlfqB1o0KPCR/jFV+FjGAYRERE4d+4cxGIx5HI5fvrpJ0RFRfE00qHDWg7pmDFjcObMGRw+fBgxMTF2ObxiLZTb19eXu7FCKpX2WnRNJhN+/PFH7N27FxkZGYiLi3P4QZv+tgPduHEDv/76q0tEopEBocJH+MH38W26mdo2KpUKy5cvR2VlJaKjo7lrfGJjY7kcUnuGXTMMg0ePHqG6uhpeXl4wm829hnLX19dj9erVCAwMxP/+9z/eWjr4Zs92IOJQ1MBO+LFgwQIsWLCAt+c9ePAA48aN4z6LxWIUFxfz9vyhbs+ePZg/fz6WLFkCgUDQJYf02rVrOHToEBoaGvDiiy9yxZCvHFL2+iK1Wo2YmBgu8owN5dZoNLh48SK+/vprBAUFobq6GuvWrUNSUpJdrjPiQ35+PrZt24ZLly5R0RvCaMZHnOrkyZPIz8/HgQMHAADHjh1DcXExdu/e7eSRDR0Mw0CpVPKaQ9rS0gKFQtFnsHVTUxOSk5PBMAxmzpwJhUKBO3fuICQkBKdOneLrK/LGUe1AxCFoxkdck0gkQm1tLfe5rq6O99T9ZcuWITc3F4GBgW5576GnpyekUimkUikSExO5HNIbN27g2rVrSE1NRXV1NcaNG9dnDikb26ZSqSCVSuHj49PjO81mM86ePYtNmzbh888/x8KFC7s8S6/X2/U7D9Tdu3edPQTiADTjI05lNBoRERGBwsJCiEQiyOVynDhxAlKplLd3XL58GT4+Pli6dKlbFr7+sMwhLSoqws2bN7kcUvYUaUdHB3744QckJycjNDS01xmiVqvFZ599BpVKhe+//x5jx4518LcBNm7ciOzsbHh4eCAwMBBHjhxBcHCww8dBnIoOtxDXlZeXh7Vr14JhGCxbtgypqam8v6OmpgZz5syhwvcU2BzSP//8E8eOHUN9fT2io6Mhk8m65JCyMzmz2YwrV64gJSUFq1evxgcffODQ64MsabVa+Pn5AQAyMzOhUChoydL90FIncV3x8fEuFTxM/jNixAjExsYiNTUV77zzDr744guo1WoUFRXh2rVr2LNnDx4/fozIyEjIZDKUlZWhoaEBOTk5GD9+vFPHzhY94L/+RmpJIJZoxkfcAs34Bq62trbLyVtLRqMRZWVlyMvLQ3V1NbKysvqVLuMIqampOHr0KPz9/XHhwgW7hLYTl0ZLncS9UeEbevrTiA781xuq0+mwefNmRw6POB8VPuLe7FX4amtrsXTpUvz7778QCARYvnw51qxZw+s7iG3u37+P+Ph4+qXH/fRa+Jyz80yIA73//vt49dVXuQzJgwcP8vZsoVCIjIwMKBQKFBUVYc+ePVAoFLw9nwxMVVUV99/Z2dmIjIx04miIq6EZHyE8mjdvHlatWoXZs2c7eyhuLSEhARUVFfDw8MD48eOxb98+3vtDicujpU5C7K2mpgavvfYaysrKupwqJIQ4BS11EmJPra2tSEhIwLfffktFjxAXR4WPEBsZDAYkJCRg0aJFePfdd3l/vk6nw5QpUzBp0iRIpVJ89dVXvL9jMMjIyIBAIIBKpXL2UMggRw3shNjAbDbjww8/hEQiQXJysl3eMXz4cJw/fx4+Pj4wGAyYMWMG3n77bUybNs0u73NFtbW1KCgoQEhIiLOHQoYAmvERYoOrV6/i2LFjOH/+PGQyGWQyGfLy8nh9h0Ag4MKgDQYDDAaD2yWRrFu3Dtu2bXO7703sg2Z8hNhgxowZ6OOAGC8YhkFMTAzu3r2LlStXYurUqXZ/p6vIzs6GSCTCpEmTnD0UMkRQ4SNkEPD09MSdO3eg0WiwYMEClJWVYeLEic4eFm+spbCkp6ejoKDACaMiQxW1MxAyyKSlpWHkyJH49NNPeX0uwzCIjY2FSCRCbm4ur88eqNLSUsTFxXG3odfV1SE4OBglJSUICgpy8uiIi6N2BkIGq8bGRmg0GgBAR0cHzp07Z5ckkp07d0IikfD+XFu8/PLLaGhoQE1NDWpqaiAWi3Hr1i0qesQmVPgIcXGPHj3CG2+8gejoaMjlcsyePRtz5szh9R11dXU4c+YMEhMTeX0uIa6I9vgIcXHR0dG4ffu2Xd+xdu1abNu2DS0tLXZ9j61qamqcPQQyBNCMjxA3l5ubi8DAQMTExDh7KIQ4BBU+Qtzc1atXkZOTg9DQULz33ns4f/48Fi9e7OxhEWI3fZ3qJIS4EYFA8DqAT81mM7+biIS4EJrxEUIIcSs04yOEEOJWaMZHCCHErVDhI4QQ4lao8BFCCHErVPgIIYS4lf8DOWubsUY2zBgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = axes3d.Axes3D(fig)\n",
    "X = np.arange(-4, 4, 0.25)\n",
    "Y = np.arange(-4, 4, 0.25)\n",
    "X, Y = np.meshgrid(X, Y)\n",
    "Z = X**2+2*X+3*Y**2 \n",
    "ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='hot')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x + 2$"
      ],
      "text/plain": [
       "2⋅x + 2"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return (x**2+2*x+3*y**2)\n",
    "\n",
    "fx = syp.diff(f(x), x)\n",
    "fy = syp.diff(f(x), y)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 6 y$"
      ],
      "text/plain": [
       "6⋅y"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ x : -1, \\  y : 0\\right\\}$"
      ],
      "text/plain": [
       "{x: -1, y: 0}"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = syp.Eq(fx, 0)\n",
    "eq2 = syp.Eq(fy, 0)\n",
    "syp.solve([eq1,eq2],(x,y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Calcule os pontos críticos da função $f(x) = y^3+2x^2+2xy$.\n",
    "\n",
    "Resolução:\n",
    "\n",
    "Calculando as derivadas parciais da função temos:\n",
    "\n",
    "$$f_x = 4x+2y$$ e $$f_y = 3y^2+2x$$\n",
    "\n",
    "Agora, montando e resolvendo o sistema:\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "4x+2y = 0\\\\\n",
    "3y^2+2x = 0\\\\\n",
    "\\end{array}\n",
    "\\right.\\rightarrow\n",
    "\\left\\{ \\begin{array}{c}\n",
    "4x +2y= 0\\\\\n",
    "3y^2 = -2x\\\\\n",
    "\\end{array}\n",
    "\\right.\\rightarrow\n",
    "\\left\\{ \\begin{array}{c}\n",
    "4x+2y = 0\\\\\n",
    "x = -\\frac{3y^2}{2}\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Onde nós encontramos o valor de x em função de y.\n",
    "\n",
    "Agora substituindo esse valor de x na primeira equação temos:\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "4\\left(-\\frac{3y^2}{2}\\right)+2y = 0\\\\\n",
    "x = -\\frac{3y^2}{2}\\\\\n",
    "\\end{array}\n",
    "\\right.\\rightarrow\n",
    "\\left\\{ \\begin{array}{c}\n",
    "-6y^2+2y = 0\\\\\n",
    "x = -\\frac{3y^2}{2}\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Resolvendo a primeira fórmula por Bháskara:\n",
    "\n",
    "$$-6y^2+2y = 0 \\rightarrow \\Delta = b^2-4ac = 2^2 - 4.(-6).0 = 4$$\n",
    "\n",
    "Calculando y' e y'':\n",
    "\n",
    "$$y' = \\frac{-b+\\sqrt{\\Delta}}{2a} = \\frac{-2+\\sqrt{4}}{2.(-6)} = \\frac{-2+2}{-12} = 0.$$\n",
    "e\n",
    "$$y'' = \\frac{-2-2}{-12} = \\frac{-4}{-12} = \\frac{1}{3}$$\n",
    "\n",
    "E agora, como achamos dois valores para y, vamos substituí-los na equação que encontramos para o x anteriormente para encontrar os pontos críticos.\n",
    "\n",
    "Para $y = 0$, temos: $$x = -\\frac{3.0²}{2} = 0$$\n",
    "E para $y = \\frac{1}{3}$, temos:\n",
    "$$x = -\\frac{3}{2}\\left(\\frac{1}{3}\\right)^2 = -\\frac{3}{2}.\\frac{1}{9} = -\\frac{3}{18} = -\\frac{1}{6}$$\n",
    "\n",
    "Então os pontos críticos encontrados para a função $f(x) = y^3+2x^2+2xy$ foram $(0, 0)$ e $(-\\frac{1}{6}, \\frac{1}{3})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4 x + 2 y$"
      ],
      "text/plain": [
       "4⋅x + 2⋅y"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (y**3+2*x**2+2*x*y)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x + 3 y^{2}$"
      ],
      "text/plain": [
       "         2\n",
       "2⋅x + 3⋅y "
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( - \\frac{1}{6}, \\  \\frac{1}{3}\\right), \\  \\left( 0, \\  0\\right)\\right]$"
      ],
      "text/plain": [
       "[(-1/6, 1/3), (0, 0)]"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = syp.Eq(fx, 0)\n",
    "eq2 = syp.Eq(fy, 0)\n",
    "syp.solve([eq1,eq2],(x,y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### B. Máximos e mínimos locais de funções de duas variáveis\n",
    "\n",
    "Agora que sabemos calcular os pontos críticos de uma função, como podemos definir se os mesmos são pontos de máximo, mínimo ou inflexão?\n",
    "\n",
    "A resposta é simples. Calculando-se a derivada segunda da função.\n",
    "\n",
    "Porém você deve estar se perguntando como iremos fazer isso, sendo que, no caso de uma função de duas variáveis, por exemplo, temos quatro derivadas segundas parciais. Mas para fazermos isso também não há muita dificuldade, nós devemos calcular todas as derivadas segundas parciais no ponto crítico encontrado, colocá-las em uma matriz e calcular o seu determinante, tal que a matriz chamada Hessiana tenha a forma:\n",
    "\n",
    "$$\\left[\n",
    "\\begin{array}{c c}\n",
    "f_{xx} & f_{xy}\\\\\n",
    "f_{yx} & f_{yy}\\\\\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "**Obs.** Lembre-se sempre de que o cálculo do determinante deve ser feito aplicando o ponto crítico nas derivadas.\n",
    "\n",
    "Agora, para definir de fato qual é o tipo do ponto crítico, temos as seguintes classificações:\n",
    "\n",
    "Para A(p) sendo o determinante da Matriz Hessiana no ponto, temos:\n",
    "\n",
    "Caso $A(p) > 0$:\n",
    "1. Se $f_{xx}(p) > 0$ o ponto crítico é de máximo. \n",
    "2. Se $f_{xx}(p) < 0$ o ponto crítico é de mínimo. \n",
    "\n",
    "Caso $A(p) < 0$:\n",
    "\n",
    "A função não tem máximo nem mínimo naquele ponto.\n",
    "\n",
    "Caso $A(p) = 0$:\n",
    "\n",
    "Não se pode afirmar nada sobre o ponto crítico.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Utilizando os pontos críticos encontrados na seção anterior $(0, 0)$ e $(-\\frac{1}{6}, \\frac{1}{3})$ para a função $f(x) = y^3+2x^2+2xy$, classifique os mesmos como máximo ou mínimo.\n",
    "\n",
    "Primeiro vamos calcular as derivadas parciais segundas de cada função:\n",
    "\n",
    "Como já havíamos calculado anteriormente, as derivadas de $1^a$ ordem são:\n",
    "\n",
    "$$f_x = 4x+2y$$ e $$f_y = 3y^2+2x$$\n",
    "\n",
    "E então, as derivadas segundas são:\n",
    "\n",
    "$$f_{xx} = 4$$ $$f_{yy} = 6y$$ $$f_{xy} = 2$$ $$f_{yx} = 2$$\n",
    "\n",
    "Vamos aplicar agora essas derivadas no ponto (0, 0).\n",
    "\n",
    "$$f_{xx}(0, 0) = 4$$ $$f_{yy}(0, 0) = 0$$ $$f_{xy}(0, 0) = 2$$ $$f_{yx}(0, 0) = 2$$\n",
    "\n",
    "Montando a Matriz Hessiana e calculando o seu determinante chegamos a:\n",
    "\n",
    "$$\\left[\n",
    "\\begin{array}{c c}\n",
    "f_{xx} & f_{xy}\\\\\n",
    "f_{yx} & f_{yy}\\\\\n",
    "\\end{array}\\right]\\rightarrow\n",
    "\\left[\\begin{array}{c c}\n",
    "4 & 2\\\\\n",
    "2 & 0\\\\\n",
    "\\end{array}\\right]\\\n",
    "$$\n",
    "\n",
    "$$A(0, 0) = 4.0-2.2 = 0-4 = -4.$$\n",
    "\n",
    "Como o valor de A deu menor que 0, o ponto não é de máximo nem de mínimo.\n",
    "\n",
    "Fazendo o teste para o ponto $(-\\frac{1}{6}, \\frac{1}{3})$.\n",
    "\n",
    "$$f_{xx}\\left(-\\frac{1}{6}, \\frac{1}{3}\\right) = 4$$ $$f_{yy}\\left(-\\frac{1}{6}, \\frac{1}{3}\\right) = 2$$ $$f_{xy}\\left(-\\frac{1}{6}, \\frac{1}{3}\\right) = 2$$ $$f_{yx}\\left(-\\frac{1}{6}, \\frac{1}{3}\\right) = 2$$\n",
    "\n",
    "Montando a Matriz Hessiana e calculando o seu determinante chegamos a:\n",
    "\n",
    "$$\\left[\n",
    "\\begin{array}{c c}\n",
    "f_{xx} & f_{xy}\\\\\n",
    "f_{yx} & f_{yy}\\\\\n",
    "\\end{array}\\right]\\rightarrow\n",
    "\\left[\\begin{array}{c c}\n",
    "4 & 2\\\\\n",
    "2 & 2\\\\\n",
    "\\end{array}\\right]\\\n",
    "$$\n",
    "\n",
    "$$A(0, 0) = 4.2-2.2 = 8-4 = 4.$$\n",
    "\n",
    "Nesse caso o valor de a foi maior que 0, então já sabemos que é ponto de máximo ou de mínimo. Para definir qual dos dois é basta vermos se $f_{xx}\\left(-\\frac{1}{6}, \\frac{1}{3}\\right)$ é maior ou menor que 0. Como $f_{xx}\\left(-\\frac{1}{6}, \\frac{1}{3}\\right) = 0$ temos que esse ponto é um ponto de mínimo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [],
   "source": [
    "fxx = syp.diff(fx, x)\n",
    "fyy = syp.diff(fy, y)\n",
    "fxy = syp.diff(fx, y)\n",
    "fyx = syp.diff(fy, x)\n",
    "\n",
    "fyy = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 2],\n",
       "       [2, 0]])"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matriz = np.array(((fxx, fxy), (fyx, fyy)), dtype='int')\n",
    "matriz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -4.0$"
      ],
      "text/plain": [
       "-4.0"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.det(matriz)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4, 2],\n",
       "       [2, 2]])"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fyy = 2\n",
    "matriz = np.array(((fxx, fxy), (fyx, fyy)), dtype='int')\n",
    "matriz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4.0$"
      ],
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.det(matriz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### C. Máximos e mínimos de funções de três variáveis\n",
    "\n",
    "Essa seção foi separada da anterior devido à o cálculo da mesma não ser tão simples quanto para duas variáveis, isso por que, para definirmos se os pontos críticos são de máximo ou de mínimo, nós teremos que usar um conceito visto lá na Álgebra Linear. Os chamados Autovalores.\n",
    "\n",
    "Basicamente, os princípios são os mesmos, ou seja, dada uma função de três variáveis, calcula-se as derivadas das mesmas, monta-se um sistema e descobre os valores de x, y e z, formando então os pontos críticos. Após isso, para fazer a classificação dos pontos, monta-se a Matriz Hessiana e calcular o determinante da subtração da mesma pela matriz identidade de autovalores igualada à 0. Caso todos os autovalores dessa operação sejam positivos, o ponto crítico é de mínimo, caso todos sejam negativos, de máximo, e caso apareçam tanto positivos quanto negativos é o chamado ponto de sela.\n",
    "\n",
    "Nesse caso, a Matriz Hessiana é:\n",
    "\n",
    "$$A = \\left[\n",
    "\\begin{array}{c c c}\n",
    "f_{xx} & f_{xy} & f_{xz}\\\\\n",
    "f_{yx} & f_{yy} & f_{yz}\\\\\n",
    "f_{zx} & f_{zy} & f_{zz}\\\\\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "A matriz de autovalores é:\n",
    "\n",
    "$$\\left[\n",
    "\\begin{array}{c c c}\n",
    "\\lambda & 0 & 0\\\\\n",
    "0 & \\lambda & 0\\\\\n",
    "0 & 0 & \\lambda\\\\\n",
    "\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Determine os pontos críticos da função $f(x, y, z) = 2x^2+y^3-4z^2+3x-12y$ e classifique cada um como máximo, mínimo ou sela.\n",
    "\n",
    "Resolução:\n",
    "\n",
    "Primeiro, vamos calcular as derivadas parciais da função.\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "f_x = 4x+3\\\\\n",
    "f_y = 3y^2-12\\\\\n",
    "f_z = -8z\\\\\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Resolvendo esse sistema chegamos a:\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "4x+3 = 0\\\\\n",
    "3y^2-12 = 0\\\\\n",
    "-8z = 0\\\\\n",
    "\\end{array}\n",
    "\\right. \\rightarrow \n",
    "\\left\\{ \\begin{array}{c}\n",
    "x = -\\frac{3}{4}\\\\\n",
    "y^2 = \\frac{12}{3} \\rightarrow y = \\sqrt{4} = 2\\\\\n",
    "z = 0\\\\\n",
    "\\end{array}\n",
    "\\right. \n",
    "$$\n",
    "\n",
    "Assim encontramos dois pontos críticos $p_0 = (-\\frac{3}{4}, -2, 0)$ e $p_1 = (-\\frac{3}{4}, 2, 0)$.\n",
    "\n",
    "Calculando as derivadas parciais segundas no ponto $p_0$ temos:\n",
    "\n",
    "$$\\begin{array}{c c c}\n",
    "f_{xx}(-\\frac{3}{4}, -2, 0) = 4 &  f_{xy}(-\\frac{3}{4}, -2, 0) = 0 & f_{xz}(-\\frac{3}{4}, -2, 0) = 0\\\\\n",
    "f_{yx}(-\\frac{3}{4}, -2, 0) = 0 &  f_{yy}(-\\frac{3}{4}, -2, 0) = 6.(-2) & f_{yz}(-\\frac{3}{4}, -2, 0) = 0\\\\\n",
    "f_{zx}(-\\frac{3}{4}, -2, 0) = 0 &  f_{zy}(-\\frac{3}{4}, -2, 0) = 0 & f_{zz}(-\\frac{3}{4}, -2, 0) = -8\\end{array}\n",
    "$$\n",
    "\n",
    "$$A(p) = \\left[\\begin{array}{c c c}\n",
    "4 &  0 & 0\\\\\n",
    "0 &  -12 & 0\\\\\n",
    "0 &  0 & -8\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "Agora devemos calcular o determinante dessa matriz A subtraindo-a da matriz identidade de autovalores.\n",
    "\n",
    "$$det \\left(\\left[\\begin{array}{c c c}\n",
    "4 &  0 & 0\\\\\n",
    "0 &  -12 & 0\\\\\n",
    "0 &  0 & -8\\end{array}\\right] -\n",
    "\\left[\\begin{array}{c c c}\n",
    "\\lambda &  0 & 0\\\\\n",
    "0 &  \\lambda & 0\\\\\n",
    "0 &  0 & \\lambda\\end{array}\\right]\n",
    "\\right) = 0\n",
    "$$\n",
    "$$det \\left(\\left[\\begin{array}{c c c}\n",
    "4-\\lambda &  0 & 0\\\\\n",
    "0 &  -12-\\lambda & 0\\\\\n",
    "0 &  0 & -8-\\lambda \\end{array}\\right]\\right) = 0\n",
    "$$\n",
    "\n",
    "$$(4-\\lambda)(-12-\\lambda)(-8-\\lambda) = 0$$\n",
    "\n",
    "Nesse caso, para a expressão ser satisfeita, teríamos que ter $\\lambda = 4$, $\\lambda = -12$ ou $\\lambda = -8$.\n",
    "\n",
    "Então como tivemos autovalores positivos e negativos o ponto $p_0$ é ponto de sela.\n",
    "\n",
    "Calculando as derivadas parciais segundas no ponto $p_1$ temos:\n",
    "\n",
    "$$\\begin{array}{c c c}\n",
    "f_{xx} = 4 &  f_{xy} = 0 & f_{xz} = 0\\\\\n",
    "f_{yx} = 0 &  f_{yy} = 6y & f_{yz} = 0\\\\\n",
    "f_{zx} = 0 &  f_{zy} = 0 & f_{zz} = -8\\end{array}\n",
    "$$\n",
    "\n",
    "$$\\begin{array}{c c c}\n",
    "f_{xx}(-\\frac{3}{4}, 2, 0) = 4 &  f_{xy}(-\\frac{3}{4}, 2, 0) = 0 & f_{xz}(-\\frac{3}{4}, 2, 0) = 0\\\\\n",
    "f_{yx}(-\\frac{3}{4}, 2, 0) = 0 &  f_{yy}(-\\frac{3}{4}, 2, 0) = 6.2 & f_{yz}(-\\frac{3}{4}, 2, 0) = 0\\\\\n",
    "f_{zx}(-\\frac{3}{4}, 2, 0) = 0 &  f_{zy}(-\\frac{3}{4}, 2, 0) = 0 & f_{zz}(-\\frac{3}{4}, 2, 0) = -8\\end{array}\n",
    "$$\n",
    "\n",
    "$$A(p) = \\left[\\begin{array}{c c c}\n",
    "4 &  0 & 0\\\\\n",
    "0 &  12 & 0\\\\\n",
    "0 &  0 & -8\\end{array}\\right]\n",
    "$$\n",
    "\n",
    "Agora devemos calcular o determinante dessa matriz A subtraindo-a da matriz identidade de autovalores.\n",
    "\n",
    "$$det \\left(\\left[\\begin{array}{c c c}\n",
    "4 &  0 & 0\\\\\n",
    "0 &  12 & 0\\\\\n",
    "0 &  0 & -8\\end{array}\\right] -\n",
    "\\left[\\begin{array}{c c c}\n",
    "\\lambda &  0 & 0\\\\\n",
    "0 &  \\lambda & 0\\\\\n",
    "0 &  0 & \\lambda\\end{array}\\right]\n",
    "\\right) = 0\n",
    "$$\n",
    "$$det \\left(\\left[\\begin{array}{c c c}\n",
    "4-\\lambda &  0 & 0\\\\\n",
    "0 &  12-\\lambda & 0\\\\\n",
    "0 &  0 & -8-\\lambda \\end{array}\\right]\\right) = 0\n",
    "$$\n",
    "\n",
    "$$(4-\\lambda)(12-\\lambda)(-8-\\lambda) = 0$$\n",
    "\n",
    "Nesse caso, para a expressão ser satisfeita, teríamos que ter $\\lambda = 4$, $\\lambda = 12$ ou $\\lambda = -8$.\n",
    "\n",
    "Então como tivemos autovalores positivos e negativos o ponto $p_1$ é ponto de sela."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 4 x + 3$"
      ],
      "text/plain": [
       "4⋅x + 3"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y, z):\n",
    "    return (2*x**2+y**3-4*z**2+3*x-12*y)\n",
    "\n",
    "fx = syp.diff(f(x, y, z), x)\n",
    "fy = syp.diff(f(x, y, z), y)\n",
    "fz = syp.diff(f(x, y, z), z)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x$"
      ],
      "text/plain": [
       "2⋅x"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 y^{2} - 12$"
      ],
      "text/plain": [
       "   2     \n",
       "3⋅y  - 12"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 8 z$"
      ],
      "text/plain": [
       "-8⋅z"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( - \\frac{3}{4}, \\  -2, \\  0\\right), \\  \\left( - \\frac{3}{4}, \\  2, \\  0\\right)\\right]$"
      ],
      "text/plain": [
       "[(-3/4, -2, 0), (-3/4, 2, 0)]"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = syp.Eq(fx, 0)\n",
    "eq2 = syp.Eq(fy, 0)\n",
    "eq3 = syp.Eq(fz, 0)\n",
    "syp.solve([eq1,eq2, eq3],(x,y,z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  4,   0,   0],\n",
       "       [  0, -12,   0],\n",
       "       [  0,   0,  -8]])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fxx = syp.diff(fx, x)\n",
    "fxy = syp.diff(fx, y)\n",
    "fxz = syp.diff(fx, z)\n",
    "fyx = syp.diff(fy, x)\n",
    "fyy = syp.diff(fy, y)\n",
    "fyz = syp.diff(fy, z)\n",
    "fzx = syp.diff(fz, x)\n",
    "fzy = syp.diff(fz, y)\n",
    "fzz = syp.diff(fz, z)\n",
    "\n",
    "fyy = -12\n",
    "\n",
    "matriz = np.array(((fxx, fxy, fxz), (fyx, fyy, fyz), (fzx, fzy, fzz)), dtype='int')\n",
    "matriz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[w, 0, 0],\n",
       "       [0, w, 0],\n",
       "       [0, 0, w]], dtype=object)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matriz2 = np.array(((w, 0, 0), (0, w, 0), (0, 0, w)))\n",
    "matriz2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4 - w, 0, 0],\n",
       "       [0, -w - 12, 0],\n",
       "       [0, 0, -w - 8]], dtype=object)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrizFinal = (np.subtract(matriz, matriz2))\n",
    "matrizFinal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 4,  0,  0],\n",
       "       [ 0, 12,  0],\n",
       "       [ 0,  0, -8]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fxx = syp.diff(fx, x)\n",
    "fxy = syp.diff(fx, y)\n",
    "fxz = syp.diff(fx, z)\n",
    "fyx = syp.diff(fy, x)\n",
    "fyy = syp.diff(fy, y)\n",
    "fyz = syp.diff(fy, z)\n",
    "fzx = syp.diff(fz, x)\n",
    "fzy = syp.diff(fz, y)\n",
    "fzz = syp.diff(fz, z)\n",
    "\n",
    "fyy = 12\n",
    "\n",
    "matriz = np.array(((fxx, fxy, fxz), (fyx, fyy, fyz), (fzx, fzy, fzz)), dtype='int')\n",
    "matriz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[w, 0, 0],\n",
       "       [0, w, 0],\n",
       "       [0, 0, w]], dtype=object)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matriz2 = np.array(((w, 0, 0), (0, w, 0), (0, 0, w)))\n",
    "matriz2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4 - w, 0, 0],\n",
       "       [0, 12 - w, 0],\n",
       "       [0, 0, -w - 8]], dtype=object)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrizFinal = (np.subtract(matriz, matriz2))\n",
    "matrizFinal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### D. Máximos e mínimos absolutos\n",
    "\n",
    "Após estudar os pontos críticos das funções de várias variáveis, vamos agora tentar estudar o comportamento dessas funções a fim de descobrir quais pontos críticos são menor e maior. Esse não é um conceito totalmente novo, tendo em vista que também fazemos isso para funções de uma variável.\n",
    "\n",
    "Encontrar os pontos de máximo e de mínimo de uma função utiliza basicamente os mesmos métodos vistos para cálculo de pontos críticos, com a exceção de que para os mesmos encontrados, devemos testar quais entre eles são extremos e se também são para todos os pontos encontrados no domínio da função.\n",
    "\n",
    "#### 1) Máximos e mínimos em uma região aberta\n",
    "\n",
    "Primeiro, vamos estudar os pontos absolutos das funções em todo o domínio da mesma, sem nenhuma restrição.\n",
    "\n",
    "Para este conteúdo é bem interessante rever um pouco sobre a imagem de funções de várias variáveis. Isso porque para termos a certeza que de um ponto crítico é de máximo absoluto ou de mínimo absoluto devemos saber qual o maior e o menor valor a função pode atingir.\n",
    "\n",
    "Como já sabemos calcular os pontos críticos de uma função vamos fazer alguns exemplos para um melhor entendimento e só então definí-los como sendo de máximo ou de mínimo.\n",
    "\n",
    "Então, vamos definir os pontos de máximo e de mínimo, caso existam, da função $$f(x, y) = x^2+y^2$$.\n",
    "\n",
    "Calculando as derivadas parciais e igualando à 0 encontramos:\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x} = 2x = 0 \\rightarrow x = 0$$\n",
    "$$\\frac{\\partial f}{\\partial y} = 2y = 0 \\rightarrow y = 0$$\n",
    "\n",
    "O ponto crítico encontrado foi P = (0, 0).\n",
    "\n",
    "Analisando a função f(x, y) agora podemos ver que para qualquer valor de x, seja ele positivo ou negativo, o menor valor que influenciará na função será 0, já que mesmo que colocássemos valores negativos, a potência quadrada o tornaria positivo.\n",
    "\n",
    "Isso também serve para o y, ou seja, o menor valor que a função pode assumir é para x = 0 e y = 0.\n",
    "\n",
    "Como o nosso ponto encontrado foi P = (0, 0) e ambas as coordenadas atenderam à análise que acabamos de fazer, P é um ponto de mínimo absoluto."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x$"
      ],
      "text/plain": [
       "2⋅x"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (x**2+y**2)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 y$"
      ],
      "text/plain": [
       "2⋅y"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy = syp.diff(f(x, y), y)\n",
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ x : 0, \\  y : 0\\right\\}$"
      ],
      "text/plain": [
       "{x: 0, y: 0}"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = syp.Eq(fx, 0)\n",
    "eq2 = syp.Eq(fy, 0)\n",
    "syp.solve([eq1,eq2],(x,y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exemplo 2.\n",
    "\n",
    "Calcule os pontos de máximo e de mínimo, caso existam, da função $$f(x, y) = sen(x+3)+sen(y)$$.\n",
    "\n",
    "Calculando as derivadas parciais temos:\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x} = cos(x+3) = 0 \\rightarrow x + 3 = arccos(0) \\rightarrow x = \\frac{\\pi}{2} - 3$$\n",
    "$$\\frac{\\partial f}{\\partial y} = sen(y) = 0 \\rightarrow y = arccos(0) \\rightarrow y  = \\frac{\\pi}{2}$$\n",
    "\n",
    "O nosso ponto crítico encontrado foi $P = (\\frac{\\pi}{2} - 3, \\frac{\\pi}{2})$\n",
    "\n",
    "Analisando a função vemos que o seu valor mínimo é de -2 e o máximo é de 2 já que cada seno tem valores extremos de -1 e 1, respectivamente.\n",
    "\n",
    "Aplicando o ponto encontrado na função, temos:\n",
    "\n",
    "$$f\\left(\\frac{\\pi}{2} - 3, \\frac{\\pi}{2}\\right) = sen\\left(\\frac{\\pi}{2} - 3+3\\right)+sen\\left(\\frac{\\pi}{2}\\right) = 1 + 1 = 2$$\n",
    "\n",
    "Vemos que a substituição do ponto resultou no valor 2 que é o máximo da função, ou seja, o ponto P é um ponto de máximo.\n",
    "\n",
    "#### 2) Máximos e mínimos em uma região fechada\n",
    "\n",
    "Agora vamos estudar os pontos de máximo e mínimo absolutos em regiões que não sejam todo o domínio da função.\n",
    "\n",
    "Para isso, todos os cálculos são efetuados da mesma maneira que nós fizemos até agora, porém, temos que testar vários pontos que iremos considerar como críticos.\n",
    "\n",
    "Os possíveis pontos que devemos encontrar são:\n",
    "\n",
    "1. Pontos críticos de todo o domínio (ou seja, aqueles pontos que nós calculamos normalmente zerando as derivadas e igualando à 0, porém, nesse caso o ponto deve ser testado para saber se está contido na região fechada).\n",
    "2. Pontos críticos na fronteira da região fechada.\n",
    "3. Vértices da região fechada, caso a mesma tenha.\n",
    "\n",
    "Após descobrir todos esses pontos, deve-se aplicar todos na função e então descobrir quais são o maior e o menor para definir os pontos de máximo e mínimo.\n",
    "\n",
    "Como exemplo, tomemos a função $f(x, y) = x^2 + 3y^2 +2x - y$. Vamos então seguir essa sequência para encontrar todos os pontos críticos dentro da região tal que $x \\leq |3|$ e $-3 \\leq y \\leq 5$.\n",
    "\n",
    "Para uma melhor visualização, podemos observar o gráfico da região fechada:\n",
    "\n",
    "![](Imagens/38.jpg)\n",
    "\n",
    "Para resolvermos isso, vamos primeiro calcular os pontos críticos em todo o domínio de f e verificar se os mesmos estão na região fechada. Então,\n",
    "\n",
    "$$f_x = 2x+2 = 0 \\rightarrow x = -1$$\n",
    "$$f_y = 6y-1 = 0 \\rightarrow y = \\frac{1}{6}$$\n",
    "\n",
    "Assim, encontramos o ponto $P_0 = (-1, \\frac{1}{6})$.\n",
    "\n",
    "Como $-3 \\leq -1 \\leq 3$ e $-3 \\leq \\frac{1}{6} \\leq 5$, significa que o ponto $P_0$ está na região fechada.\n",
    "\n",
    "Agora vamos encontrar os pontos críticos nas fronteiras da região.\n",
    "\n",
    "Primeiro, vamos analisar o gráfico dessa região.\n",
    "\n",
    "A partir disso podemos perceber que a mesma possui três lado denotando toda a sua fronteira, então, para testar os pontos críticos da fronteira devemos separar cada lado da mesma, aplicar em f(x, y) e então tirar os pontos críticos da função resultante.\n",
    "\n",
    "Para o lado em que $x=-3$ temos:\n",
    "\n",
    "$$f(-3, y) = (-3)^2+3y^2+2(-3)-y = 9 +3y^2-6-y = 3y^2-y+3$$\n",
    "\n",
    "$$f'(-3, y) = 6y-1 \\rightarrow 6y-1 = 0 \\rightarrow y = \\frac{1}{6}$$\n",
    "\n",
    "Então, o nosso primeiro ponto crítico encontrado na fronteira da nossa região fechada foi $P_1 = (-3, \\frac{1}{6})$\n",
    "\n",
    "Para o lado em que $x=3$ temos:\n",
    "\n",
    "$$f(3, y) = 3^2+3y^2+2.3-y = 3y^2-y+15$$\n",
    "\n",
    "$$f'(3, y) = 6y-1 \\rightarrow 6y-1 = 0 \\rightarrow y = \\frac{1}{6}$$\n",
    "\n",
    "Então, o nosso segundo ponto crítico encontrado na fronteira da nossa região fechada foi $P_2 = (3, \\frac{1}{6})$\n",
    "\n",
    "Para o lado em que $y=-3$ temos:\n",
    "\n",
    "$$f(x, -3) = x^2+3(-3)^2+2x-(-3) = x^2+27+2x+3$$\n",
    "$$= x^2+2x+30$$\n",
    "\n",
    "$$f'(x, -3) = 2x+2 \\rightarrow 2x+2 = 0 \\rightarrow x = -1$$\n",
    "\n",
    "Então, o nosso terceiro ponto crítico encontrado na fronteira da nossa região fechada foi $P_3 = (-1, -3)$\n",
    "\n",
    "Para o lado em que $y=5$ temos:\n",
    "\n",
    "$$f(x, 5) = x^2+3.5^2+2x-5 = x^2+75+2x-5 = x^2+2x+70$$\n",
    "\n",
    "$$f'(x, 5) = 2x+2 \\rightarrow 2x+2 = 0 \\rightarrow x = -1$$\n",
    "\n",
    "Então, o nosso quarto ponto crítico encontrado na fronteira da nossa região fechada foi $P_4 = (-1, 5)$\n",
    "\n",
    "Agora que encontramos os pontos da fronteira devemos encontrar os últimos possíveis pontos críticos, que são os pontos do vértice.\n",
    "\n",
    "Pela figura, percebemos que os mesmos são:\n",
    "\n",
    "$$P_5 = (-3, 5)$$\n",
    "$$P_6 = (3, 5)$$\n",
    "$$P_7 = (-3, -3)$$\n",
    "$$P_8 = (3, -3)$$\n",
    "\n",
    "Agora vamos substituir todos os pontos encontrados na função para saber quem é o maior e o menor.\n",
    "\n",
    "Testanto para $P_0$ temos:\n",
    "\n",
    "$$f\\left(-1, \\frac{1}{6}\\right) = (-1)^2 + 3\\left(\\frac{1}{6}\\right)^2+2(-1)-\\frac{1}{6} = 1+\\frac{3}{36} -2 -\\frac{1}{6} $$\n",
    "$$= \\frac{36+3-72-6}{36} = \\frac{-39}{36} = -1,083$$\n",
    "\n",
    "Testanto para $P_1$ temos:\n",
    "\n",
    "$$f\\left(-3, \\frac{1}{6}\\right) = (-3)^2 + 3\\left(\\frac{1}{6}\\right)^2+2.(-3)-\\frac{1}{6} $$\n",
    "$$= 9+\\frac{3}{36}-6-\\frac{1}{6} = 3 - \\frac{1}{12} = \\frac{36-1}{12} = \\frac{35}{12} \\approx 2,91$$\n",
    "\n",
    "Testanto para $P_2$ temos:\n",
    "\n",
    "$$f\\left(3, \\frac{1}{6}\\right) = (3)^2 + 3\\left(\\frac{1}{6}\\right)^2+2.3-\\frac{1}{6} = 9+\\frac{3}{36}+6-\\frac{1}{6}$$\n",
    "$$= 15 - \\frac{1}{12} = \\frac{180-1}{12} = \\frac{179}{12} \\approx 14,91$$\n",
    "\n",
    "Testanto para $P_3$ temos:\n",
    "\n",
    "$$f(-1, -3) = (-1)^2+3(-3)^2+2(-1)-(-3)=1+3.9-2+3=1+27-2+3=29$$\n",
    "\n",
    "Testanto para $P_4$ temos:\n",
    "\n",
    "$$f(-1, 5) = (-1)^2+3.5^2+2(-1)-5=1+3.25-2-5=1+75-7=75-6=69$$\n",
    "\n",
    "Testanto para $P_5$ temos:\n",
    "\n",
    "$$f(-3, 5) = (-3)^2+3.5^2+2(-3)-5=9+3.25-6-5=9+75-11=73$$\n",
    "\n",
    "Testanto para $P_6$ temos:\n",
    "\n",
    "$$f(3, 5) = 3^2+3.5^2+2.3-5=9+75+6-5=85$$\n",
    "\n",
    "Testanto para $P_7$ temos:\n",
    "\n",
    "$$f(-3, -3) = (-3)^2+3.(-3)^2+2(-3)-(-3)=9+3.9-6+3=9+27-3=33$$\n",
    "\n",
    "Testanto para $P_8$ temos:\n",
    "\n",
    "$$f(3, -3) = 3^2+3.(-3)^2+2.3-(-3)=9+27+6+3=45$$\n",
    "\n",
    "Analisando todos os pontos, vemos então que o ponto $p_0$ é o menor quando aplicado à função, ou seja, é o nosso ponto de mínimo. Vemos também que o ponto $p_6$ é o maior entre eles aplicados à função, ou seja, o mesmo é o ponto de máximo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x + 2$"
      ],
      "text/plain": [
       "2⋅x + 2"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (x**2+3*y**2+2*x-y)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 6 y - 1$"
      ],
      "text/plain": [
       "6⋅y - 1"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fy = syp.diff(f(x, y), y)\n",
    "fy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ x : -1, \\  y : \\frac{1}{6}\\right\\}$"
      ],
      "text/plain": [
       "{x: -1, y: 1/6}"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = syp.Eq(fx, 0)\n",
    "eq2 = syp.Eq(fy, 0)\n",
    "syp.solve([eq1,eq2],(x,y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1.0833333333333335$"
      ],
      "text/plain": [
       "-1.0833333333333335"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(a, b):\n",
    "    return a**2+3*b**2+2*a-b\n",
    "\n",
    "f(-1, 1/6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.9166666666666674$"
      ],
      "text/plain": [
       "2.9166666666666674"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(-3, 1/6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 14.916666666666668$"
      ],
      "text/plain": [
       "14.916666666666668"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(3, 1/6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 29$"
      ],
      "text/plain": [
       "29"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(-1, -3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 69$"
      ],
      "text/plain": [
       "69"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(-1, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 73$"
      ],
      "text/plain": [
       "73"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(-3, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 85$"
      ],
      "text/plain": [
       "85"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(3, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 33$"
      ],
      "text/plain": [
       "33"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(-3, -3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 45$"
      ],
      "text/plain": [
       "45"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(3, -3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### E. Multiplicadores de Lagrange\n",
    "\n",
    "Enfim chegamos ao último assunto deste material, que são os famosos Multiplicadores de Lagrange. Mas não se preocupe, pois não é nada de muito diferente do que nós vimos até agora.\n",
    "\n",
    "Esses operadores servem para calcular os máximos e os mínimos de uma função em cima de uma área restrita, tal que essa área seja uma outra função. Mas como assim?\n",
    "\n",
    "A partir de uma função f(x, y) dada, podemos descobrir os pontos críticos da mesma em uma única área, ao invés de seu domínio completo, tal que essa nova área é representada por uma outra função g(x, y).\n",
    "\n",
    "Para fazer esse cálculo g(x) deve ser igualada à 0 e $\\nabla f(x, y) = \\lambda\\nabla g(x, y)$, tal que $\\lambda$ é o chamado ***multiplicador de Lagrange***.\n",
    "Após isso, deve-se montar um sistema composto pelos vetores resultantes dessa igualdade como também a função g(x, y).\n",
    "\n",
    "A partir da resolução do sistema os pontos são encontrados, e então basta substituí-los em f(x, y) para achar quem é o maior, no caso do máximo, ou o menor, no caso do mínimo.\n",
    "\n",
    "Exemplo 1.\n",
    "\n",
    "Ache os pontos de máximo e de mínimo da função $f(x, y) = 3x^2+2y^2+6x+4y$ sobre a curva $x^2+3y^2+2x+6y$.\n",
    "\n",
    "Resolução:\n",
    "\n",
    "Então, vamos começar calculando o vetor gradiente de f e g.\n",
    "\n",
    "$$\\nabla f(x, y) = (6x+6, 4y+4)$$\n",
    "$$\\nabla g(x, y) = (2x+2, 6y+6)$$\n",
    "\n",
    "Agora, montando o sistea com os gradientes e g(x, y) temos:\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "6x+6 = \\lambda(2x+2)\\\\\n",
    "4y+4 = \\lambda(6y+6)\\\\\n",
    "x^2+3y^2+2x+6y = 0\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "Resolvendo o mesmo chegamos a:\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "3(2x+2)-\\lambda(2x+2) = 0\\\\\n",
    "2(2y+2)-3\\lambda(2y+2) = 0\\\\\n",
    "x^2+3y^2+2x+6y = 0\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "$$\\left\\{ \\begin{array}{c}\n",
    "(2x+2)(3-\\lambda) = 0\\\\\n",
    "(2y+2)(2-3\\lambda) = 0\\\\\n",
    "x^2+3y^2+2x+6y = 0\n",
    "\\end{array}\n",
    "\\right.$$\n",
    "\n",
    "No caso da última equação basta completar os quadrados para fazer os produtos notáveis. Para isso, se somarmos 1 dos dois lados da equação, do lado esquerdo fazemos o produto $(x+1)^2$ bem como se somarmos 3 dos dois lados também podemos fazer o outro quadrado $(\\sqrt{3}y + \\sqrt{3})^2$, ficando com:\n",
    "\n",
    "$$x^2+3y^2+2x+6y = 0 \\rightarrow x^2+2x+1+3y^2+6y+3= 1+3$$\n",
    "$$\\rightarrow (x+1)^2 + (\\sqrt{3}y + \\sqrt{3})^2 = 4$$\n",
    "\n",
    "Agora vamos calcular os pontos a partir do sistema.\n",
    "\n",
    "Aqui temos dois casos a considerar. Primeiro:\n",
    "\n",
    "Pela primeira equação temos $(2x+2)(3-\\lambda) = 0$, o que nos dá $(2x+2) = 0$ ou $(3-\\lambda) = 0$. Então vamos encontrar $\\lambda$.\n",
    "$$(3-\\lambda) = 0 \\rightarrow \\lambda = 3$$\n",
    "\n",
    "Agora analisando a segunda equação vemos que $(2y+2)(2-3\\lambda) = 0$ e $(2y+2) = 0$ ou $(2-3\\lambda) = 0$, só que como $\\lambda = 3$, $(2-3\\lambda) \\neq 0$ , o que nos dá $(2y+2) = 0 \\rightarrow 2y = -2 \\rightarrow y = -\\frac{2}{2} = -1$.\n",
    "\n",
    "Substituindo y na terceira equação:\n",
    "\n",
    "$$(x+1)^2 + (\\sqrt{3}y + \\sqrt{3})^2 = 4 \\rightarrow (x+1)^2 + (\\sqrt{3}(-1) + \\sqrt{3})^2 = 4$$\n",
    "$$\\rightarrow(x+1)^2 + (-\\sqrt{3} + \\sqrt{3})^2 = 4 \\rightarrow (x+1)^2 = 4$$\n",
    "$$\\rightarrow x+1 = \\pm\\sqrt{4} \\rightarrow x+1 = \\pm 2$$\n",
    "\n",
    "Então ficamos com $x = -3$ e $x = 1$. O que nos dá dois pontos $(1, -1)$ e $(-3, -1)$.\n",
    "\n",
    "Olhando para o segundo caso, o que $2x+2=0 \\rightarrow 2(x+1) = 0 \\rightarrow x+1 = 0 \\rightarrow x = -1$.\n",
    "\n",
    "Vamos descobrir y pela última equação utilizando esse x. Então, $$(x+1)^2 + (\\sqrt{3}y + \\sqrt{3})^2 = 4 \\rightarrow (-1+1)^2 + (\\sqrt{3}y + \\sqrt{3})^2 = 4$$\n",
    "$$(\\sqrt{3}y + \\sqrt{3})^2 = 4 \\rightarrow \\sqrt{3}y + \\sqrt{3} = \\pm2$$\n",
    "$$\\sqrt{3}y = \\pm 2 -\\sqrt{3} \\rightarrow y = \\frac{\\pm 2 -\\sqrt{3}}{\\sqrt{3}}$$\n",
    "\n",
    "Deu um número bastante complexo, mas calculando os dois chegamos a: $$y' \\approx 0,15$$ e $$y'' \\approx -2,15$$. Encontrando assim mais dois pontos $(-1, 0.15)$ e $(-1, -2.15)$.\n",
    "\n",
    "Como já achamos todos os pontos críticos da função na restrição, basta substituir os mesmos em f(x) para vermos quem é máximo e mínimo.\n",
    "\n",
    "$$f(1, -1) = 3(1)^2 + 2(-1)^2+6(1)+4(-1) = 7$$\n",
    "$$f(-3, -1) = 3(-3)^2 + 2(-1)^2+6(-3)+4(-1) = 7$$\n",
    "$$f(-1, 0.15) = 3(-1)^2 + 2(0.15)^2+6(-1)+4(0.15) = -2.3$$\n",
    "$$f(-1, -2.15) = 3(-1)^2 + 2(-2.15)^2+6(-1)+4(-2.15) = -2.3$$\n",
    "\n",
    "Como $f(P_1) = f(P_2) > f(P_3) = f(P_4)$, $P_1$ e $P_2$ são pontos de máximo e $P_3$ e $P_4$ são pontos de mínimo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwoAAAAeCAYAAACMl+kxAAAABHNCSVQICAgIfAhkiAAACxJJREFUeJztnW2sHkUVx3+9VGhTQ0GU3mA0VWkRJQGpXDHBsoUWlLRIkW+KXF9Toza+RYNRuVFRNBpfaozyAS9EE0EMmNa2NgoGiiIIJUAsIsQHQWyxVqrRGi3WD2c29/qwbzM7s7N3nvNLbp7m2d0zZ6bnP8/ZnZcFRVEURVEURVGUBkwADwHbgZuBDXHdUUaAo2I7oCgjxpGRy1fNK0q3xNa80n82Inn/DuQ+4NSyEzPgMHBmJ24po8464NzYTii95LDnP2WGCeD1kcpWzcdnEaqtUSOm5pW5Rcas+4CxqK4oo87JwHrgZ7EdUXrHCuA9wDyPf8oMdwFrqHhiFAjVfHxWAJei2ho1YmleSZAMHVFQwjMG7ATGYzui9JJPAS+K7UTiLAbuAI7oqDzVfD9QbY0uXWtemZtk6IiC0gMuA54A9sR2ROklLwYej+1E4hwAHgQmOypPNd8PVFujS9eaVxIlQ0cUlLDMAx4GzovtiNJLlgCfje3EiDCBJO8LApejmu8Hqi2lK80rc5cMHVFQIpMhT7VuieyH0k/eAGyN7cSIcBewELgwcDkZqvk+oNpSutK8kgh6o6DE4BLgAeBQbEeUXvIa4M7YTowQuxBNhkQ13w9UWwp0o3klEVxvFK4FnkK2WFPSYAUy1PTODso6G+moFGWY5wDPAP+N7cgIcS+iyZCo5uOj2lJyutA8aK7YR6xzPZcbhTOQrdWuAv7hcH0VX0C2zXscOAjsR35crgCO81xWl1wCbAJuB/6G/Cd9t4NybdrzHuRFG58BnhvQpyORLRIHAcuAeG0emr7V6zikw7kJeASJswPI7jbvwL6POctcW4Sv/mEJkjBt8mzXhrbt5rMOjwHHG5shUM2741NfVdoCPzoYjktfdn0TI1ZGSfNQnSumqFWIUy/bOPCS62VUL2beATyNzHHzzb+RYdFrkODaBNxt/Pkjc3dLt/uQOvwd2E13orBtzwlz7OMBfVpmynhvwDIgXpuHpm/12mB8eBL4HvB5JN6eNt/fiN0+61cBx5Yc89U/vNtcs8qzXRvatpvPOryFsBtYqObd8amvKm2BHx0Mx6Uvu76JESujpHmozhVT1CrEqZdLHNTlehk18VF1wnJk2PLqBs67ULYK/0rj0zcDlRuaVciP5Txm2rcLUbi0527kaUOo9St5gF4ayH5OrDYPTYh6TRo7mcO15yBv2h2Ol3HgD8bumwquGwPmF3z/7YqyfPUP24B9zOwlHqPfcW23HJ91uNCcE+qtrap5d1zixEVb4EcHw3Hpy65vYsTKKGm+LldMUasQp16ucVCV62W02PXo7UgDXG95XVP+VfL9DeZzWaByQ3Mr8Du6f9W9S3t+H9mdZE0Qj+Ao8/nPQPZzYrV5aPpWr1uAzTx73vMe4Fvm39nQsWORnVfePPT9S4FHK8ry0T8cjfxgb0aG8X3ZtcWl3XJ81yHXYqjtElXz7tjGiau2oL0OiuLSh90QxIiVUdJ8Xa6YolYhTr1c46Bxrmd7o7AaCdKud01YZz7v77jcVKlqzzvMZ6gbhYPmM+Q6CKUf/Md8Du9080pgL7B26Pu1wI8dyrHpH9Yic+Zv8mzXJ2XtluO7DrkWD1ac0wbVfBiK4sS3tqC5Dmzi0sbuKJCa5mPlisoMdXHQONcrGp4sYxFwGjJc4XsR8zAfQQJ5MfBqZBHW/cj8K8Uem/a823yuDOTLfvN5dCD7Sj+YD7zV/Hv70LGdSDLza2QnlvxH8hXA1xvYbtM/rEf6rx2e7fqiqt1yfNch1+JfbJ1tiGreP2Vx0lZb4K6DqrhsYzd1UtN8l7miMoNtHLTK9TKK1ygsN9+XdQI+2WPKyv+2EXZ1fpdkdD8fz7Y9D5prQjAf+fH6RCD7RWSkMwdyNhnx1yiU8SVjs+op5m+Bc82/FwFfbmjbtX9YgCwy+4Fnuz6pa7cQdXi/ObdqoWsbVPP+qYsTV22BWwzVxaWr3S7IiBsrqWneNlfMSFOrGd3WyyUOynK9DMc1Cvk2S38tOT4YcrLur6rxxpH5bePAxcj8yl3A6Rb+ujLAXz36gm177geeH8iXQ8CDwIkV5wxI4/9gQP/qUeTTd8yxWwuOTTuUsRH4MPAQ1QtYtzAzRWI18NOG9l37hzXIE5ey4fuY/Q40a7cQdTgRiYuyvr0tqnm/9WgSJ67aArcYqotLV7s5A9KIj2FS1HxdrtgXBqQVUy5x0CjXs5l6lM9lK1v88ijliyqKeLLBOXsRcdwLPAxcB5xiUYYLIerRF5q250LCzV0ESUhXVxxP5f+gj/X4KnDM0HenAW9EXo4zGDp2n6X99wFfA36DPNHcX3HuFmQnlg8iTzAutyzLtn+4GNlKrm6udox+p2m7hajD6YgmQ6KaL8a2Hk3jpK22wC6Gmsalrd2cVOJjNqlqvi5X7AspxhTYxYFzrpcxa8hhFieY76te2BKSXab8UE+6uyKjH3enZe05huzKULdDRhvORKYihHgXRxEZ/Whz32T4qdeksZO1tPMBY+cB5GU+dcxH9tk+ifZbJNb1D0cgWwtu82zXB03bLUQdxpAXA51vadMW1Xx7bPTlU1tQHUOucVlntysyuo+VlDVvmytmpKdV6Ee96uKgLNfLcJx69Cfgz0jHE4MTzOczlWcpTSlrz5OQ4SvbJ8k23Im8HfCCgGUo3fIx4CtI3KwCnmpwzSHgJ8jc9bbxVtc/rESGxJvuyNLUblts2i1EHc4xZdpMTXFBNd8OW3351BZUx5BrXNbZTZXUNR87V1RmqIqDxrmezY3CYeA25M6kaq6pK8uR1drDjCEvjjge+AXPnvc2bXybDOBTTKZpVy/X9sxHkoqGJdv6NJuPAm/zYMcX06QZR13wSWRnhXuQIfR9FtduQfZ831pznms856xHnp78yJPdadrHi227+a4DyH7nV1KeqE2jmo+Nq76aagvaxVBZXLa12zemUc03IUSuOM3c0Kot08TJ86A61/s/bNYoAPwQeXPg+cAjltfWcQHySvOdwO+RbbuWAGcjizL2AO8quC6/2Snbf7gPXGT+QBaaALyWmYWi+5CtrWbTtl6u7Xke0oEUdfo+2/o2ZPh1GfKCEt/YtvlciCNwi6WQXAZ8GomZ25HFecMMKF8UvRUZHn2iphzXeAZ5anIR8Etk/qYPu23jxbbdQtThhcgTp2sr/FTNx6WNvppqC9xjqCou29gNTYzf5FHSfF2umKJWoft6tdFXVa5XS0bxGgWQF37sBX7lYriGU4BvIMMg+5CGO4Ds9ToFPK/kul3IfLtQW/v5YIrq1fSDgmva1sulPRcjC1tuLrHpu61fAFyDdIi+mcKuzedCHIFbLNUxifsahTp/DgM/r7GxtEE5rv0DwBnGjw95tNs2Xqawa7cQdbgOeFmNn6r5uEzRTl9LG5bjGkNVcdnGbmim6P43ua7MlDRflytOkZ5Woft6ucZBXa6XUX4f0OiEy83xV5X73hnHIHdEX4ztiGdi1SvfW/msgmOhfHo54d4C3ZRU40gRPofE9Us82YsRL77r8DpkfnQVqnmlDt9x2VdU8/b4yhVT1WrMelXleuDhRmEB8Biw2dVDj6xDtrcarztxjhGjXguR7b9uLDke0qcQTxdtSDWOFGE3fhfnx4gX33VoojnVvFKH77jsK6p5e3zliqlqNVa96nI9aHCjMIG8/GM7MiyxoeCclcAVyBsflTQ4GRmqWhrXDUVRFEVREkBzxf5RlettRPL+Hch9wKmdeaUoiqIoiqIoiqIoiqIoiqIoiqIoiqIoiqIoSmr8D6dq+BmR4af+AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -3, \\  -1, \\  3\\right), \\  \\left( -1, \\  -1 + \\frac{2 \\sqrt{3}}{3}, \\  \\frac{2}{3}\\right), \\  \\left( -1, \\  - \\frac{2 \\sqrt{3}}{3} - 1, \\  \\frac{2}{3}\\right), \\  \\left( 1, \\  -1, \\  3\\right)\\right]$"
      ],
      "text/plain": [
       "⎡             ⎛         2⋅√3     ⎞  ⎛      2⋅√3         ⎞            ⎤\n",
       "⎢(-3, -1, 3), ⎜-1, -1 + ────, 2/3⎟, ⎜-1, - ──── - 1, 2/3⎟, (1, -1, 3)⎥\n",
       "⎣             ⎝          3       ⎠  ⎝       3           ⎠            ⎦"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y):\n",
    "    return (3*x**2+2*y**2+6*x+4*y)\n",
    "\n",
    "def g(x, y):\n",
    "    return (x**2+3*y**2+2*x+6*y)\n",
    "\n",
    "fx = syp.diff(f(x, y), x)\n",
    "fy = syp.diff(f(x, y), y)\n",
    "\n",
    "gx = syp.diff(g(x, y), x)\n",
    "gy = syp.diff(g(x, y), y)\n",
    "\n",
    "eq1 = syp.Eq(fx, w*gx)\n",
    "eq2 = syp.Eq(fy, w*gy)\n",
    "eq3 = syp.Eq(g(x, y), 0)\n",
    "syp.solve([eq1,eq2, eq3],(x,y,w))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Obs.** Para o caso de se querer calcular com mais restrições basta seguir o mesmo princípio de uma restrição, porém dessa vez a fórmula utilizada é: \n",
    "\n",
    "$$\\nabla f(x, y, z) = \\lambda \\nabla g(x, y, z) + \\mu \\nabla h(x, y, z)$$"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Raw Cell Format",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}