materiais/notebooks/Introdução_à_Computação_Simbólica.ipynb

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      "name": "Introdução à Computação Simbólica.ipynb",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "88xVBbXkpTLA"
      },
      "source": [
        "# **INTRODUÇÃO À COMPUTAÇÃO SIMBÓLICA**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eByvUBsCcXYv"
      },
      "source": [
        "_Este Notebook acessará conteúdos abordados no **e-book** [TAIL — Introdução à Matemática para Inteligência Artificial](https://www.overleaf.com/read/gtspgphgvstz), seja livre para revisitá-lo quando necessário._"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Wpi_UivqXjY5"
      },
      "source": [
        "A partir dos conhecimentos já adiquiridos ao longo das nossas aulas, você já é capaz de fazer várias operaçôes matemáticas e resolver diversos problemas usando algumas bibliotecas do [Python](https://www.python.org). Agora iremos dar um passo adiante e introduzir um assunto que, talvez, você não tenha visto antes, a Computação Simbólica, em especial, aplicada às funções.\n",
        "\n",
        "<center><img alt=\"Gráfico_3d\" width = \"30%\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Sympy_plot_screenshot.png/637px-Sympy_plot_screenshot.png\"></center>\n",
        "\n",
        "Contudo, antes de realmente abordarmos o que é Computação Simbólica, devemos compreender o porquê devemos utilizá-la e quais as suas vantagens a fim de facilitar a resolução de diversos problemas matemáticos.\n",
        "\n",
        "Por exemplo, imagine o seguinte:\n",
        "\n",
        "Em uma determinada situação, surge a tarefa de você fazer o cálculo do volume de um tanque cujo formato é o de um cilindro. Para isso, como nós sabemos, esse cálculo pode ser feito multiplicando a área da base do cilindro pela sua altura, ou seja, $ V = \\frac{1}{4} \\pi D^2 h $. Contudo, o valor de $ \\pi $ que obtemos é um número finito, o que não chega a ser um problema, pois realizamos os cálculos do mundo real com bastante facilidade e que nos fornece excelentes resultados. Mas, podemos ir além...\n",
        "\n",
        "Geralmente, os cálculos com o número $ \\pi $ são feitos adotando dupla precisão, ou seja, 15 casas decimais, como podemos verificar abaixo."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sGxodJwcfB2M",
        "outputId": "b65b1bda-4a62-4571-ea5a-79f1518d2001",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "import math as mt\n",
        "\n",
        "print(mt.pi)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "3.141592653589793\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CSjafo8YfOTV"
      },
      "source": [
        "> De acordo com um artigo publicado pela NASA [[1]](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/), para o cálculo do raio do universo observável, cerca de **46 bilhões de anos-luz**, seria suficiente uma aproximação de $ \\pi $ com apenas **40 casas decimais** para garantir precisão do tamanho do **diâmetro de um átomo de hidrogênio**.\n",
        "\n",
        "\n",
        "Porém, mesmo essa precisão sendo realmente impressionante, pense que você poderia realizar o cálculo do volume do tanque de maneira \"exata\". Sabemos, no entanto, que matematicamente isso é impossível, pois o $ \\pi $ é um número irracional. Mas, é nesse ponto que a Computação Simbólica entra. Ela nos permite realizar esse tipo de cálculo com precisão infinita, admitindo o $ \\pi $ apenas como um símbolo, ampliando totalmente as capacidades matemáticas, nos permitindo trabalhar com modelos abstratos e também com diversas aplicações reais."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "OzqCc6sSswpK"
      },
      "source": [
        "### **Mas, afinal, o que é Computação Simbólica?**\n",
        "\n",
        "A Computação Simbólica (CS), também conhecida como computação algébrica, é uma subárea da matemática e da ciência da computação relativamente moderna, tendo surgido na segunda metade do século XX. Essa área procura resolver problemas usando objetos simbólicos representáveis em um computador. \n",
        "\n",
        "Existem diversas possibilidades de aplicação da CS, resolvendo problemas que surgem em pesquisa básica, na indústria e no desenvolvimento de softwares para computação avançada, denominados de *sistemas de computação algébrica* (SCAs).\n",
        "\n",
        "Os sistemas de CA permitem, dentre diversos exemplos, a implementação de regras de álgebra, trigonometria e cálculo, derivaração, integração, obter a solução de equações diferenciais, simplificar expressões, representar graficamente campos escalares, vetoriais e curvas paramétricas [[2]](https://www.scielo.br/scielo.php?pid=S0102-47442002000200017&script=sci_arttext&tlng=pt). Em casos mais avançados, a CS é útil para solucionar equações da macroeconomia, manipular números para a ﬁnalidade de criptograﬁa e criar modelos probabilísticos [[3]](https://kevinkotze.github.io/mm-tut1-symbolic/),[[4]](https://www.ukma.edu.ua/~yubod/teach/compalgebra/%5BJoel_S._Cohen%5D_Computer_algebra_and_symbolic_comp(BookFi.org).pdf).\n",
        "\n",
        "Existem diversos SCAs, dentre os que estão disponíveis com licença comercial, os mais populares são o [Maple](https://www.maplesoft.com/), [Mathematica](https://www.wolfram.com/mathematica/) e [MATLAB](https://www.mathworks.com/products/matlab.html), que apesar de possuírem versões para estudantes, os custos das liceças ainda são altos. Contudo, também existe uma certa varieadade desses softwares com licenças gratuitas, como o [Maxima](http://maxima.sourceforge.net/), [Scilab](https://www.scilab.org/), [Octave](https://www.gnu.org/software/octave/index), [Sympy](https://www.sympy.org/pt/index.html) e [Scipy](https://www.scipy.org/), estes dois últimos são bibliotecas do Python, que inclusive, serão abordados com mais ênfase ao longo deste *notebook*."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RUBuhXqtZE6x"
      },
      "source": [
        "## **BIBLIOTECA SYMPY**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "goQlIV86pZUx"
      },
      "source": [
        "<center><img alt = \"Logo_Sympy\" width=\"15%\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/5/54/Sympy_logo.svg/1200px-Sympy_logo.svg.png\"></center>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ESFRB_RspeQZ"
      },
      "source": [
        "O SymPy é um sistema de álgebra computacional completo (CAS) escrito na linguagem de programação Python. Atualmente, na sua versão 1.6.2, o projeto do SymPy começou a ser desenvolvido em 2005 pelos pesquisadores Ondřej Čertík e Aaron Meurer e, desde então, cresceu exponencialmente, contando com mais de 500 colaboradores nos dias atuais [[5]](https://peerj.com/articles/cs-103/?utm_source=TrendMD&utm_campaign=PeerJ_TrendMD_1&utm_medium=TrendMD). \n",
        "\n",
        "Ao contrário de muitos CASs, o SymPy não inventa sua própria linguagem de programação. O próprio Python é usado tanto para a implementação interna quanto para a interação do usuário final.\n",
        "\n",
        "De acordo com os próprios desenvolvedores da biblioteca, as principais vantagens do SymPy são [[6]](https://www.sympy.org/pt/index.html):\n",
        "\n",
        "* É um software livre e de código aberto, licenciado sob a licença BSD;\n",
        "\n",
        "* É escrito inteiramente em Python;\n",
        "\n",
        "* Além de ser usada como uma ferramenta interativa, o SymPy pode ser incorporado em outros aplicativos e estendido com funções personalizadas.\n",
        "\n",
        "* Leve: SymPy depende apenas de mpmath , uma biblioteca Python pura para aritmética de ponto flutuante arbitrário, tornando-o fácil de usar.\n",
        "\n",
        "Você pode conferir todos os detalhes da biblioteca SymPy acessando o [site oficial](https://www.sympy.org/pt/index.html) e/ou  através de artigos dos próprios autores da ferramenta clicando [neste link](https://peerj.com/articles/cs-103/?utm_source=TrendMDutm_campaign=PeerJ_TrendMD_1&utm_medium=TrendMD)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ROr3assB6Lo0"
      },
      "source": [
        "### **Primeiros Passos com o SymPy**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_K-cElPv6RIB"
      },
      "source": [
        "O primeiro passo que devemos fazer, antes de começar a escrever códigos, é fazer a instalação ou importação da biblioteca SymPy. No nosso caso, como estamos trabalhando no ambiente do Google Colab, podemos importar o SymPy digitando e executando o seguinte comando:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-jiqvAvoXkEB"
      },
      "source": [
        "import sympy as sy\n",
        "\n",
        "from sympy import *"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LMPXkbmp7-GL"
      },
      "source": [
        "Na primeira linha de código, estamos importando o SymPy e atrabuindo o \"apelido\" '**sy**' para facilitar na hora de chamar essa biblioteca posteriormente.\n",
        "\n",
        "E ao digitar o **asterisco (*)**, estamos informando que queremos importar todos os módulos dessa biblioteca."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RDPEk3YX8zOw"
      },
      "source": [
        "**Inicialmente, podemos verificar a diferença entre objetvos numéricos e objetos simbólicos no Python.**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5jTVB-0y78Nx"
      },
      "source": [
        "# MELHORAR A IMPRESSÃO DOS SÍMBOLOS DA BIBLIOTECA SYMPY\n",
        "\n",
        "sy.init_printing(pretty_print=True)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "j_-MNWls9Mup",
        "outputId": "85f4179a-e614-43d5-d0bc-e07fb21fa06f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "# OBJETO NUMÉRICO\n",
        "\n",
        "mt.pi"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$3.141592653589793$$",
            "text/plain": [
              "3.141592653589793"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 4
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WoyMFTny9SHj",
        "outputId": "036a302a-c542-48bf-9343-00ad65e5fe30",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# OBJETO SIMBÓLICO\n",
        "\n",
        "sy.pi"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPCAYAAAA71pVKAAAABHNCSVQICAgIfAhkiAAAAL5JREFUKJHV0D1qgkEUheFHEYRAIFgHrISAFuo2bOxcQCp7cQXuwU1Y2Nu6AH9IY5UFpBEhTchPM8J49RMLmxwYLrxz7j13hjuph98bTwcqWfMeUwzxhll294IBllhgfSm9nyZPAh8n/prDcjC1U10F3i3gJ5qnhEbgO3yheq35HQeUMvaIH2yjOV/7CXVsUvpR7TTsbOVyMHH+k4XvvdQcTa1Ur65dlFxLtYnnOOCoFb7xEPgAH/jEqKj5H+kPZCwqsiZWNEcAAAAASUVORK5CYII=\n",
            "text/latex": "$$\\pi$$",
            "text/plain": [
              "π"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6_VgFP6U9qT7"
      },
      "source": [
        "Já podemos perceber claramente a diferença entre a maneira de como cada biblioteca trabalha a tipologia das variáveis.\n",
        "\n",
        "Agora, podemos verificar o tipo desses objetivos."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JI6rebwc9etb",
        "outputId": "c7b365b7-56e9-436e-a33a-50ce49595599",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "type(mt.pi)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "float"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Ga2xPzjx-FP8",
        "outputId": "4eac733d-52fd-4b99-b0f0-8f6349a0df8e",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "type(sy.pi)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "sympy.core.numbers.Pi"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gQFEb9eC-OLx"
      },
      "source": [
        "Enquanto no primeiro exemplo foi verificado que o objetivo é do tipo *float* (numérico), o segundo é do tipo 'sympy.core' (objeto simbólico do SymPy)."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Tv1LYoeu_NDx",
        "outputId": "cfa20775-7b99-4925-be54-3aa11fd1c12c",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "mt.sqrt(5)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$2.23606797749979$$",
            "text/plain": [
              "2.23606797749979"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "NBTOV605_bKV",
        "outputId": "65688356-f11d-4d6a-daf0-7129c5c779d3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "sy.sqrt(5)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\sqrt{5}$$",
            "text/plain": [
              "√5"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1lHexvnB-67F"
      },
      "source": [
        "**Mas então, como podemos escrever equações usando o Sympy?**\n",
        "\n",
        "Isso é bem simples de ser feito. Podemos atribuir símbolos às variáveis de uma equação usando a função 'symbols'.\n",
        "\n",
        "Por exemplo, queremos escrever uma função de apenas uma variável, no caso, 'x'."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ANPDy_dX-HKg",
        "outputId": "269aedd5-0489-4ad5-bbca-e854690675f6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "x = sy.symbols('x')\n",
        "\n",
        "x"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPCAYAAAA71pVKAAAABHNCSVQICAgIfAhkiAAAAMFJREFUKJHVziFPwmEUxeFH2Qh2goEEG035ANJMVEkmNiqfgC8gxexoVJqR2d3cbARHwqgbQZJuMIRy2Rj740ugeNN7z3l/9xyOOE9Y4WZHP0E/vO4++BJLvCG3pd8H2EulbxKasXdiH+A0BRfxg3e0AxwinwI3cxfQCs84y/q0r8Z0693C96Gpt/jFRyQ/HArWMccIBYyxQCUFXkW9Cc5Da0T6419gFbOoWtrxXuNALQss4xNfuMjwrwN+SVX/J7MGliUqwx10gTUAAAAASUVORK5CYII=\n",
            "text/latex": "$$x$$",
            "text/plain": [
              "x"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "R5v-Z7BHA2zG"
      },
      "source": [
        "Perceba que a variável foi impressa como no formato semelhante ao do LaTeX. Essa é justamente a forma visual que SymPy utiliza. Ao verificar o tipo dessa variável, iremos perceber que está no formato de símbolo. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dxeUqYb5Ax2U",
        "outputId": "693c985a-afa3-45d4-c0c0-23fccc663b6c",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "type(x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "sympy.core.symbol.Symbol"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dFLtokkBBLa1",
        "outputId": "55603dfa-9144-4398-a01c-c7a950aa72e6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "x**2 + 2*x - 3"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$x^{2} + 2 x - 3$$",
            "text/plain": [
              " 2          \n",
              "x  + 2⋅x - 3"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LlRoUidQByFT"
      },
      "source": [
        "**E se quisermos declarar mais de uma variável?**\n",
        "\n",
        "Bom, neste caso, existem duas formas. Pode ser feito da maneira da situação anterior, ou seja, uma por uma. Ou também podemos declarar todos ao mesmo tempo."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "drKuLTIMBhzU"
      },
      "source": [
        "# DO JEITO ANTERIOR\n",
        "\n",
        "y = sy.symbols('y')\n",
        "z = sy.symbols('z')\n",
        "\n",
        "# TODAS AO MESMO TEMPO\n",
        "\n",
        "x, y, z = sy.symbols('x y z')\n",
        "\n",
        "### ESSAS DUAS MANEIRAS SÃO EQUIVALENTES ###"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5vMWdoOuCPpQ",
        "outputId": "9b435cc2-e590-48d5-f086-f75681c777dc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "x + 2*y/3 - 5*sqrt(z)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$x + \\frac{2 y}{3} - 5 \\sqrt{z}$$",
            "text/plain": [
              "    2⋅y       \n",
              "x + ─── - 5⋅√z\n",
              "     3        "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l0N2iA3QFqCF"
      },
      "source": [
        "**Alfabeto em formato de Símbolos** \n",
        "\n",
        "Como mencionado anteriormente, o SymPy possui vários submódulos em sua biblioteca, uma delas o 'sympy.abc', um módulo que permite importar e trabalhar símbolos de letras gregas ou latinas, por exemplo. Vejamos a seguir."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cIpcOGrAGcwu"
      },
      "source": [
        "from sympy.abc import alpha, beta, gamma, theta, a, b, c"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OcxDxUDKC8Ff",
        "outputId": "562eac9e-7693-43ef-99d6-15032297f18e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "alpha + beta + gamma + theta"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\alpha + \\beta + \\gamma + \\theta$$",
            "text/plain": [
              "α + β + γ + θ"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0_jjUOvyG2Ti",
        "outputId": "19ecd419-1e8f-406f-dca9-f9f6278f84ac",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "(2*a - b + c)*(alpha/2 - 3*beta)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left(\\frac{\\alpha}{2} - 3 \\beta\\right) \\left(2 a - b + c\\right)$$",
            "text/plain": [
              "⎛α      ⎞              \n",
              "⎜─ - 3⋅β⎟⋅(2⋅a - b + c)\n",
              "⎝2      ⎠              "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vgYYgqyyH5fh"
      },
      "source": [
        "Devemos nos atentar ao fato de que o SymPy já utiliza algumas letras com símbolos especiais, como por exemplo o '*$I$*' que é o número complexo *$i$*."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "6awq77vlHHyI",
        "outputId": "7a539128-3708-49d5-d155-ecf08212ef88",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "sy.I"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAATCAYAAACtHkzTAAAABHNCSVQICAgIfAhkiAAAAIJJREFUKJG1zj0KAkEUA+DPPyz0DNYWgiB7gb2Gx/B6lnYiq52lh7AQBNFmBpZhxl0E04SXhJfwA3Z4YwvDTKAKfCp9WGCJQa++UXLXuGGOfW7DJnAThc5AiivupYEzvHBoi+2KdbibUiD2n7sCxYEXPDDOmVM8cUyNWLHC5Nv7P+IDVSkSNjuW8sEAAAAASUVORK5CYII=\n",
            "text/latex": "$$i$$",
            "text/plain": [
              "ⅈ"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DxiJ6P-bIXvS",
        "outputId": "d5ede022-a02f-4ba9-8a86-c4fb22630b4d",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "type(I)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "sympy.core.numbers.ImaginaryUnit"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 19
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ksTHKeLe4brY"
      },
      "source": [
        "Trabalhar com valores infinitos é tarefa simples com SymPy."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8SHDQtHx4Zol",
        "outputId": "a1f7049e-6f88-466b-c4b9-382df1363328",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "# infinito negativo\n",
        "-sy.oo"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAALCAYAAAAa5F88AAAABHNCSVQICAgIfAhkiAAAAO1JREFUOI3d0z9LglEYBfBffyaXJmkTGkMkB+cigz6CS30CoT5GU4uLNAWBW0tbtAniYLW0ZyVEBAktYQWRDe8ryEWjhkvQgTucc5/7PId7z+UfYxs9vKKBTKqv4xLPOEUupokC+qhgE12cIY8BDrCGGi4wE8vIIXbHeB7veMRxUNtBeVqjOwx/sRrB+XOsBFodH1gK9H3sjMh8sNnF2zSXE/AwQRsG/AZzyOI2qIv2NEeojvEF3OMJzWBwBxuxjBTToRWsooW2JMQDSYZK2MOViDcCW5Lv+4ITLKZ6Gdf4lGRpOaaJn2D2rw18iy+QgDkVbgq5XQAAAABJRU5ErkJggg==\n",
            "text/latex": "$$-\\infty$$",
            "text/plain": [
              "-∞"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 20
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SPxiQiQcFs42"
      },
      "source": [
        "**Simplificando Expressões**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VDEL4K3eFyoD",
        "outputId": "0cc5eeaa-c6fd-4d95-dfa5-0b2336c060e7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        }
      },
      "source": [
        "simplify(x*x*x*x * sqrt(x))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$x^{\\frac{9}{2}}$$",
            "text/plain": [
              " 9/2\n",
              "x   "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sz5WOsU-JL2k"
      },
      "source": [
        "**Substituição de Variáveis**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3YL8wG0_cvkR"
      },
      "source": [
        "Com uma expressão simbólica definida, podemos fazer uso do método `subs()` para avaliar o comportamento desta expressão sob os valores desejados."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "U5xGNNkjIZjl",
        "outputId": "de93564d-0a75-4c3b-9a65-c2e47923966f",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "expr = (x**2 - x)/2\n",
        "expr.subs(x, 10)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$45$$",
            "text/plain": [
              "45"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sVVT0UMhYEd0"
      },
      "source": [
        " **Avaliando Expressões** \n",
        "\n",
        " Adentraremos agora em uma série de características da biblioteca, de modo que sem o conhecimento destas seríamos incapazes de prosseguir para aplicações mais práticas do SymPy na resolução de problemas do mundo real. \n",
        "\n",
        " Para definir uma expressão podemos atribuir a esta qualquer nome, dentro do bom senso científico.\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yHUia49yYr6Q",
        "outputId": "0f8f3383-14fc-4216-9e73-4bb53b1bc189",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "expr = sqrt(x*x*x - x**2 + x)\n",
        "expr"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\sqrt{x^{3} - x^{2} + x}$$",
            "text/plain": [
              "   _____________\n",
              "  ╱  3    2     \n",
              "╲╱  x  - x  + x "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 23
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mMdJx_dSaz-o"
      },
      "source": [
        "É importante ressaltar que, uma vez que alteramos o valor simbólico de um objeto SymPy, as expressões já definidas com a versão anterior deste objeto mantém o seu valor e estrutura originais, não mais dependendo do objeto utilizado; vamos observar um exemplo:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "U3HlrhuSbgN1",
        "outputId": "aec77310-7e91-4b43-f18b-fbf15b1a0d4d",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "z = sy.symbols('100')\n",
        "e = z*(x**2) + z\n",
        " \n",
        "# experimente modificar z para um objeto SymPy ou para um valor definido no \n",
        "# ecossistema Python puro.\n",
        "z = 0\n",
        " \n",
        "e"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$100 x^{2} + 100$$",
            "text/plain": [
              "     2      \n",
              "100⋅x  + 100"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kc96E8sKe8uD"
      },
      "source": [
        "Para avaliar igualdades em SymPy devemos estar atentos ao uso de `=` e `==`. O primeiro faz parte do sistema Python nativo, não sendo extendido nem alterado pela biblioteca SymPy. Seu significado permanece sendo o de atribuição de valores a espaços de memória em Python. O segundo comporta-se como um avaliador estrutural. **Atenção!** Para este separador, temos que `(a+b)**2` é diferente de `a**2 + 2*a*b + b**2`, e de fato, estruturalmente falando são expressões distintas. Cientes disto, podemos prosseguir para a avaliação de igualdades de uma maneira geral."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2wDrZskXgCvS"
      },
      "source": [
        "É definido que `a` e `b` são iguais se e somente se `a - b = 0`; nos ateremos a este fato para avaliar igualdades com SymPy, como segue:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "tvnq-SDwdKCD",
        "outputId": "765f8d31-c6f0-4b11-b8f5-58ff781dbdd2",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "ex1 = x**2 + 2*x\n",
        "ex2 = x*(2 + x)\n",
        " \n",
        "# ex1 e ex2 só são iguais se a linha seguinte nos retorna 0\n",
        "simplify(ex1 - ex2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAMRJREFUOI3N0r1OgkEQheEnlhqDxlYrA9wCLRfkdWhlLK30FmixsOMe+CnQRmOnUQstxGa+8GVgCRgLpjnZs/PuZHcP/1jHuMETvvCAKxyugk7xghl6uMB9rIc4KoH9aDpL/mX416VpM0yxk/b28YFP7EkN3dA7/CTwHQPsopPBdui4cI1JaCuDjdC3Alj5BxncqOpgdWJjWWPNf83gKLRVAJuhC2+w0Xfk+lMAqqn1yJ2bR25kReTgBLd4xjcerRHyLa5fD28wqzSkqLgAAAAASUVORK5CYII=\n",
            "text/latex": "$$0$$",
            "text/plain": [
              "0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 25
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GQpid7Z9g7xE"
      },
      "source": [
        "**Funções e Determinação de Raízes**\n",
        "\n",
        "Finalmente podemos aplicar os conceitos até então apresentados para lidar com funções em SymPy. \n",
        "Apresentamos, a seguir, a função modular:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ERhGoI6Gy4xr",
        "outputId": "4fc21a94-5f1f-45e9-f727-72d37bd58d02",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "x = Symbol('x', real=True)\n",
        "expr = 1 - x**2\n",
        "sy.Abs(expr)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left|{x^{2} - 1}\\right|$$",
            "text/plain": [
              "│ 2    │\n",
              "│x  - 1│"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 26
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "liXz_tOx-wL4"
      },
      "source": [
        "Além disso, podemos pedir para o SymPy resolver equações algébricas, assim como essa acima."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dZyK63gu-4aF",
        "outputId": "9d52ec8e-9367-4f3d-d1c5-71f387c68e79",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "solve(expr, x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left [ -1, \\quad 1\\right ]$$",
            "text/plain": [
              "[-1, 1]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 27
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tX72ByLo1Li6"
      },
      "source": [
        "Essa sintaxe significa `resolva(equação, em relação a variável)`. Ou seja, encontre as raízes da equação.\n",
        "\n",
        "Quando se trata de métodos numéricos, ou seja, aproximados, esse tipo de cálculo é feito usando o **Método de Newton-Raphson** para estimar as raízes."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "x73lFL4iziXh"
      },
      "source": [
        "Contudo, perceba que a função do exemplo acima foi escrita considerando que ela está igualada a zero. Isso pode não parecer tão óbvio à primeira vista.\n",
        "\n",
        "Mas, e se quisermos que a igualdade seja diferente, como fazemos?\n",
        "\n",
        "Bom, existe duas formas. Vejamos a seguinte equação: $x^2 + 2x + 1 = 5 $\n",
        "\n",
        "Para resolvermos de acordo com o exemplos acima, podemos escrever a equação já passando o **5** para o lado esquedo, igualando a zero: $x^2 + 2x + 1 -5 = 0 $. Daí, pedimos para o Sympy resolver."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VAFQr05Q06EW",
        "outputId": "f71e98bc-5827-4ce7-f677-6e697f82e630",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "eq = x**2 + 2*x + 1 - 5\n",
        "eq"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$x^{2} + 2 x - 4$$",
            "text/plain": [
              " 2          \n",
              "x  + 2⋅x - 4"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 65
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0_nn3fg21Cao",
        "outputId": "cde41a33-4c5d-4298-c51e-15f109ceae44",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "solve(eq,x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left [ -1 + \\sqrt{5}, \\quad - \\sqrt{5} - 1\\right ]$$",
            "text/plain": [
              "[-1 + √5, -√5 - 1]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 66
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MQNeItWT2A-y"
      },
      "source": [
        "A segunda forma de fazer é usando o método `Eq` do SymPy. Nele, podemos declarar a função e a sua respectiva igualdade. Vejamos."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Bi7ZSSZP2NTM",
        "outputId": "2d6a1f3b-b147-4f82-8a01-f90ef05a57a8",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "eq2 = sy.Eq(x**2 + 2*x + 1, 5)\n",
        "eq2"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$x^{2} + 2 x + 1 = 5$$",
            "text/plain": [
              " 2              \n",
              "x  + 2⋅x + 1 = 5"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 67
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "L8q0Ds0Y2UsP",
        "outputId": "5feeb9d1-61e6-42d8-b565-08fc79416e18",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# Resolvendo a Equação\n",
        "\n",
        "solve(eq2,x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left [ -1 + \\sqrt{5}, \\quad - \\sqrt{5} - 1\\right ]$$",
            "text/plain": [
              "[-1 + √5, -√5 - 1]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 68
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gj0VBFct2a2B"
      },
      "source": [
        "Agora você como declarar funções de uma variável das duas maneiras e achar suas raízes.\n",
        "\n",
        "E se quisermos fazer o mesmo processo para funções de várias variáveis e resolver um sistema?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JSJnIuDg9TJ4",
        "outputId": "29824d56-64aa-400d-c220-9ea181eadb0e",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "# Declarando as Funções\n",
        "\n",
        "f1 = sy.Eq(3*x + 2*y, -3)\n",
        "f2 = sy.Eq(x - 3*y, 2)\n",
        "\n",
        "pprint(f1)\n",
        "\n",
        "pprint(f2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "3⋅x + 2⋅y = -3\n",
            "x - 3⋅y = 2\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "UCZZ9lwX-Bxn",
        "outputId": "246e5227-09da-4327-ec9b-2aef19c328d2",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# Resolvendo o Sistema de Equações Algébricas\n",
        "\n",
        "sy.solve([f1, f2], [x,y])"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left \\{ x : - \\frac{5}{11}, \\quad y : - \\frac{9}{11}\\right \\}$$",
            "text/plain": [
              "{x: -5/11, y: -9/11}"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 91
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ao6YrmtLzjWS"
      },
      "source": [
        "**Funções Trigonométricas**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DaySeJGKztV9",
        "outputId": "9a501148-e4e3-4378-ad46-471876e6e2e5",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "# seno\n",
        "expr = sy.sin(sqrt(x))\n",
        "expr"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\sin{\\left (\\sqrt{x} \\right )}$$",
            "text/plain": [
              "sin(√x)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ozmDZKcG0BVV",
        "outputId": "60b7cc68-951f-408d-b4d8-bddb2dc23669",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "# cosseno\n",
        "expr = sy.cos(expr)\n",
        "expr\n",
        "\n",
        "# observe que utilizamos o valor anterior de expr para gerar uma função composta\n",
        "# no espaço de memória utilizado pela função anterior\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\cos{\\left (\\sin{\\left (\\sqrt{x} \\right )} \\right )}$$",
            "text/plain": [
              "cos(sin(√x))"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 29
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KqHDUlIo0-l_",
        "outputId": "7050cf81-7af7-49b1-b485-f5bc9a761b2b",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# tangente\n",
        "expr = tan(x)\n",
        "expr"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\tan{\\left (x \\right )}$$",
            "text/plain": [
              "tan(x)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 62
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "X1D_v46FzM3s",
        "outputId": "59713fdd-623a-460d-d5e6-41c2fe431521",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# Identidade Trigonométrica Fundamental\n",
        "\n",
        "sy.Eq(cos(x)**2 + sin(x)**2,1)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\sin^{2}{\\left (x \\right )} + \\cos^{2}{\\left (x \\right )} = 1$$",
            "text/plain": [
              "   2         2       \n",
              "sin (x) + cos (x) = 1"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 64
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gqAskwW48wn-"
      },
      "source": [
        "A função `expand_trig` pode ser usada para verificar e expandir as identidades trigonométricas, aquelas que geralmente esquecemos nas horas essenciais. rs"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ocYV5-lm9Js0",
        "outputId": "7e8828c7-e465-40a2-e417-3bee8cb9adac",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "sy.expand_trig(sy.cos(a + b))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$- \\sin{\\left (a \\right )} \\sin{\\left (b \\right )} + \\cos{\\left (a \\right )} \\cos{\\left (b \\right )}$$",
            "text/plain": [
              "-sin(a)⋅sin(b) + cos(a)⋅cos(b)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IIWSk8vn1JUn"
      },
      "source": [
        "Muitos fenômenos da natureza podem ser estudados por meio de expressões envolvendo o número de Euler, `e`. Tratamos agora de funções exponenciais."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "RjeFAfAV1kMX",
        "outputId": "1f0c8463-7af6-4e1a-c3a9-bc17ce8b5ee1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "sy.exp(x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAUCAYAAACXtf2DAAAABHNCSVQICAgIfAhkiAAAATZJREFUOI3t1D8oRWEYBvDf9afMUgbFZKBbim4Wg5CyoswMNptBBtmslJJMJinKIimLAYsFgySrJCmK8vcazqdOt3uuf7cYPHV6z/e+z/s839d5v8MfRz8eUBfLzeAM1cUwSGEfC2E9ikvUF0P8Hd14whhukSmm+Dt28Yyezzb0YQNXeMQpxlGah9uBO7yg+SPhUiwhG0TnMI2TkFvM4TfhBkNYxeZHBrNBaAplsXw5dkKtMeTqcI6JsE7jFe1J4q2BsJZQHw4Gg6jEMeZzOMvYiyfiuxwRjd09JvMYpEMswTUa8nAGknZPNL/ZTzxdhUSSUBGat7/TXAglIaZCrCq2QRwHolP0JtTb5L8HBZGKvXdjXfTht3AoOmENWkSjWvtVg1xksIIL0f/lCkeicez8qfg/fgdvnuZCZjwZckUAAAAASUVORK5CYII=\n",
            "text/latex": "$$e^{x}$$",
            "text/plain": [
              " x\n",
              "ℯ "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 32
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3IQn5UqU1qb_",
        "outputId": "cc6996d6-664d-4617-b962-fd9ad706a75d",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 42
        }
      },
      "source": [
        "# exemplos mais complexos podem ser alcançados por meio de intrincadas \n",
        "# expressões em python\n",
        "sy.exp(x**2 + sqrt(sy.sin(1 - x))) - x**sy.exp(1)\n",
        "\n",
        "# isso mostra a versatilidade da biblioteca SymPy, fator decisivo no sucesso de \n",
        "# estudos e pesquisas científicas"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$- x^{e} + e^{x^{2} + \\sqrt{- \\sin{\\left (x - 1 \\right )}}}$$",
            "text/plain": [
              "         2     _____________\n",
              "   ℯ    x  + ╲╱ -sin(x - 1) \n",
              "- x  + ℯ                    "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 33
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "H9EVvC7y-CZi"
      },
      "source": [
        "**Polinômios**\n",
        "\n",
        "O Sympy também nos permite trabalhar com polinômios de maneira simples, podemos realizar expansões, simplificações e fatoração. \n",
        "\n",
        "Vejamos um exemplo para um polinômio de 2° grau."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Gg0w1FvpAdT7",
        "outputId": "966151c0-c07b-4dfd-a70c-5915103b5039",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "a0, a1, a2 = sy.symbols('a0 a1 a2')\n",
        "P = a0 + a1*x + a2*x**2\n",
        "P"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$a_{0} + a_{1} x + a_{2} x^{2}$$",
            "text/plain": [
              "                2\n",
              "a₀ + a₁⋅x + a₂⋅x "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 34
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gYqF-AGyA5xb",
        "outputId": "c05a2291-e656-4699-a9d4-5e79471e1235",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "b0, b1, b2 = sy.symbols('b0 b1 b2')\n",
        "Q = b0 + b1*x + b2*x**2\n",
        "Q"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$b_{0} + b_{1} x + b_{2} x^{2}$$",
            "text/plain": [
              "                2\n",
              "b₀ + b₁⋅x + b₂⋅x "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fbo4EhdKA_7v",
        "outputId": "c5b65fea-0a24-49c2-89f7-abbfd573988b",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# PRODUTO ENTRE POLINÔMIOS\n",
        "\n",
        "P*Q"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUgAAAAZCAYAAABTsbfVAAAABHNCSVQICAgIfAhkiAAABtFJREFUeJztm3uoFUUcxz+WeqWuaYmUhmllZKB5rT8MMrlIQkgR9pKEbCvFsoigIhQyE8owSi0uYWXdntrTQoKeVESlmZpRUhpmFBVpGGYPX93++M3hruPMnt3Z3bO7nvnA5dw7szPz/Z353t3ZeYDH4/F4PJ6GMBtYC+wCtgOrgJGFKvI0M7n58SKgXxYVeZqKt4BrEBOOAlYCvwLHha65EOifoE7vRY8rcfwICT05FpiZkUBPc9MKHEBucjX6AB3AkTHK27x4A9AFXJtWYAFUWXsSyhinyY9g8eQRhgp6IMPSx/NQ52k6+iI+2xlK+xf4AJhWp2yUF89Wn+tS6iuCKmtPQhnjNPkR4nuSK4H5mcvyNCsvAhs4dLTYotJbIspGeXED8A/QM63AAqiy9iSUMU6bHyGeJ9lI+SbVA2So3l6sjNwJOLzifBD4GTjFkv8kcHVEeZsXW4C9wJpU6twISNdHRWpPSoB7rGWMs54fQfOk/oo9AhgObLIUngo8B2wG/kSGqWuRSdAyk4futxHzXKql9wA6Vd59Kep3oUxxLkJGgBOArZa61wFXWPKivDgK6KXKtwGvAL8Dfym9ZXvAh8lLe9n8mGcfucQax48Q7UmuBz615PUF9qv8ZcAC9blDCbojotG0BLg/yfLSPRqZ7N3EwcP1B1S9Sx3qDDg84lyCrBSeUafuc5HtF6bXnSgvzlRtv4H8w70MLAQ+Uum/kWyVPAkB6UaQeWkvmx/z7KOkscb1I0R7kg4kEBOtwAmG9EHIaOWbGI27EuDeUXnq7kR0BervOervFzAvgNUjoPpxdiAGm6D01H5aDfWerOoZZsiL8uJSVW4HMEbLe1rlzbGUTUtAuhtknto7KY8f8+6jTuLFmsSPEO1JXgeecBD7LfJEyIuAfObm0uoegkxCfw/chGh8E+jtWF9A9ePssvzMM1w7QOWNM+RFefFz7NtHxqq85ZayaQlI10d5ai+TH/Puo7ixJvEjRHuSd4HFloLHAncCq5G5rQNag+vrhhSPbdiDMv101qkvb90LQnV9DBwVs9w2miPOKHqp+i4w5Nm82AvYg3x/ptXRoarOlRno20a2fdQI7WXwY6P6KHdP6uIPYH7anIlMjB4PfAasQCZc9yND0mnIimOYWcDtyCve18AtyPxDPRZz6NxEG3Ax8BTypYf5IqIuF93jgduQPVyDkQWOzog2tod+vw74O+LaMEXHORu4BDgdMfNqlfaVpQ3XOKOoeW2fIc/mxZEqfRUSn85Q9fmDlu7ixyz7CNy0V9GPLnEm9SM03pOsAJ41pG9ENlK2G/LmI3fcG0NpU1QDM5CJ0YeB3cBJLopxH+on1Q0wCbgXuAz5woOI+qcC/wG/qLoeSahPJ6BxccY9ggXZx1ljkKrvLEOezYvTVZlbLXUuUvmTQmlZ+jHA/bXTRXsV/egSZxI/QjGe5B7krh9mCN3v9zr9gR9V/jmh9DXAY9q1W5AhsQsByTvKRbfObuyGnITs8/oSGIgsguxDnoCuBBQTJ9iPYOURZ40RSptpNdPkRZB/hC6VrzMIuYls5uC3oyz9GOB+g3TRHqYKfoT0cYLdj1CcJ5kMfKelDVQFNiPv5zUGIHf9LiWuj0rvjQyrL9fq6QA+dBQdkLyjkuo2YTPkOKSTtyIdDvKE7wJeS6BRJ6CYOKH7yRmenM4rzhqTkQUkW57uRZCbXZfSdHQovRV4B/mnCs9pZu3HAPcbZFLtOlXwI6SPE8x+hGI9ST9kZegYLf09uif6FwLPIMv3LyHBhue3Bqtrx2t1zI1quA4Bbh2VRLcJkyHbgD+Qof2pWt5a1d55CXXWCCgmTjj0CFaecda4G9mjZsLkxZ4qbT0yatgC3A88BPyEvHLdrNWTtR8D3PrIRbtOFfyYRZxgPhJYtCcBOYmhD2sHInuXtiN76j5BJv7HKFHLQteW6QaZRLcJ3ZDDkXmRncjCiM75qt7VCXXWCCgmTv0IVt5x1ngfMb0N3YujVbuPAicCryJ73HYho+V2Qx1luUG6aNepgh+ziNN0JLAsnuQ04PkUDeTxil0UUXM+hwuLkCfyiAa3OwwZJUSR1ovg/Vg1ivIjGDxp2l2/BRE41JAXh73IecaJWvpEZGTjKQ9L6D6fmudJKBMzgLvqXJPWi+D9WCWK9CPE8yQgE/rzUjQ0BTHmdGRbxRLk6ZfG6I2iFRlityGTwXPV765blMpK0iNYWTIKuCrmtWm9CN6PVaBIP0IyTwIiLs2B/1nI5tI9yBNcnwMqK+24nZKoGrYTEvMa0HbS16e0XgTvx7JTpB+hmFd6j8fj8Xg8Ho/H4/F4moX/ASIliFB5e6QmAAAAAElFTkSuQmCC\n",
            "text/latex": "$$\\left(a_{0} + a_{1} x + a_{2} x^{2}\\right) \\left(b_{0} + b_{1} x + b_{2} x^{2}\\right)$$",
            "text/plain": [
              "⎛                2⎞ ⎛                2⎞\n",
              "⎝a₀ + a₁⋅x + a₂⋅x ⎠⋅⎝b₀ + b₁⋅x + b₂⋅x ⎠"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 36
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qT6bE0HwBJMS",
        "outputId": "cb5f8f75-1e55-41ab-f5b2-a512787a173e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# EXPANDIR EXPRESSÕES\n",
        "R = P*Q\n",
        "sy.expand(R)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAu0AAAAaCAYAAAADtmLvAAAABHNCSVQICAgIfAhkiAAACS1JREFUeJztnX2oZVUZhx8zNepKkzXUGDZ9SCY4OuIfGqjdJKkmIijDjz9kZYo0RhQVMUISgY0UNU0lojFw+v6yFCQwTb0SqdM0ppJhU41XiozGNCbLRsduf7zrMPvuu/c9a6+9373XOud94DL37nP2u9d9zm+tOeuetfcGwzAMwzAMwzAMw5hStgC7gP3APuAW4KRBW2QY1VhW++EK4CHE837gXuBdg7ZoerFM94Nlehi2AEvA14ZuiGFMCz8DPoD8R7EBuAn4G3DMkI0yjAosq/3wHuCdwPHAG4GrgeeAk4ds1JRime4Hy3T/nAE8CjzIQG/aP4TMGC4Z4uCJYk7akaK/OeB54N1DN6RDUvScGyk6tKz2x5PA5UM3omNSdD2NmYY0XVum9Xgp8CfgrcACpTftL+ipEaf5f3f3dLwcMCftSNHf0UifemrohnRIip5zI0WHllV9DgcuQN5M3jNwW7omNdcwnZmGtFxbpvW5AbgRuGvIRvwGeAZ44ZCNSAxz0o4U/f0QadfhQzekQ1L0nBspOrSs6rEBeBo4CPyT6Vz/m4rrItOYaUjDtWW6Hy5DJg1H+J8XGGB5zFHAs8DOvg8MOOTjjvkBjr0aQzppiiM9hyn6+xLwV+D1kfs7zHMbHOn5gzQdWlZ1ORJZ/3sasBV4gvgTJB3mOoRpzDSk49oyrc8JyAnVJxS2LRCwPOYi4DvAHuBfyEdNu5ATPmLYgMwadgMbgR8D/wD+DdxG+md7d+0D9JzchnSG95W2HwaM/GPXRNaOZRb8bQMuBM4B9kYeuwty6ruWVcvqtGb1WeCPvi1bgAeAj7U4fhss0/0xza4t03E08fxm4BXAw8gnGgeBtwCb/fdHVR3gaP/gvcAOZEa1A5lVLQGfimj05X7fnyICbgQ+D/zCb/87sCaibgiOdjM6DR+g5+QU5ESc37H8I8Iv+rrXR9R0xDucBX/bkSsWnBhxvCKO9LKq2Xe7zqojPX9gWa1i1rJ6J/DtyGM7bPyd1UxDWq6LWKbDaOJ5DTI5KH7tAr7rvz+s6gBzwKsqtq9DZjSPRDT6et+4J4BTS4990z92ZUTdEBztOqKGD9B1MvL7O//zlf7nHxB34rEj3uG0+7sWuW7tOcjvOf6aiziuI72savfdEd1l1ZGeP7CsVjHNWb0GOAt4LfLXu63A/5BL5sXgsPF3VjMNabi2TA/3nmyBFmvaf4/MNprya+ovo3O6f+x7sY2agENv7VSsD9B1chxyMsWjwId9rVuRNWkxOHQcToO/pZqvz0Qc15FeVrX7bpdZdaTnDyyrTck9qyPgMeAA8nv8HHh7i+M6bPyd1UxDGq5HWKaHek+2wIQ37S8DPg3ch6wTep7lneH+ho09AnmhF6k+I3e9r3tTw7pVLFLfiau+RgE1u/YB/TjZyqE2/hJ4ceB+i3TrcNb8hbJIt54h374b43qR9P2BZbWOWcpqExbp1rVluppFuvUM5rqORbp1PfOeiw08GVk0/0rgV8D3kYX4B4HXARcjd2cqshn4JPLRxMPAR5E1QGNOQmYTt/g6Zdb7fx9rWLeKL7NyzdFG5G5e30BekCIPTKin4QOaOzkb+ARy1vaxyMkWowlt31f4/oPAfyY8f0yXDlPxB3LizHuRs7IPIB1+C/DbmrbH+gtl6KyGZKqvvhvjOgd/kHZfDyUH132MCZCX65jxN8RJH+NCTp6huevQ7Nn4sZyUMp3Ee4oHgf9S/ZHFZ5EZwhWFbecjt7K9DDkJ5KvIdTxfU3jOpX6/j9ccc5t/fFPDuqE4X38+Yl8NH9DcySbgc8B5yAvtJrT7ImS92eO+znUTnj8JR5zDVPxBs9tdd+0vFEd/WQ3JVB99t0vXjrT8QX59PRRHWq61xwTIz3VTzxDmRHtcyM0zNHcdmj0bP5aTUqYHf09xnC92a8Vja4A/+8fPKGzfCXy99Nw/IB8DjLnO73d1Rd11SLD2sPwv/iF1Q3HEhUPLB8Q5GfM0q3fETcilmR4C1iInZTzH8ut+NsXR3GGq/sbU3e5aw18ojv6yWqQuU9p9t2vXjrT8QX59PRRHWq41xwTIz3Vbz2OqnGiOC7l5hm5c12XPxo9DpJbpkLqg6Hkt0ug9HLoTE8DLkRnFkj/Qi/z2I5GPEd5fqnMtcHfh551+373ASwrb54DbkV/yHYXtoXVDccR1RC0f0NxJkdU64plIuPYiQQOZcS8BN9fsE4KjucNU/Y1Z52ucWdim5S8URz9ZLVOXKc2+q+HakZY/yK+vh+JIy7XWmAB5um7reUyVE61xIUfP0I3ruuzZ+HGIlDIdWlfd8x2+2P3I9Sm/hVz+5ke+0cW1Qsf6555dqnEVcgYvyKzkGV/vEWRm/QXgK8BfkI8LPlLaP6RuExzxH3l17QPinBSp64gbkVsLPw68ofTYLt+2s1apuxqOOIcp+htTvt21pr9QHP1ktUxVpjT7rpZrRzr+IM++HoojHddaYwLk7bqN5zFlJ1rjQs6eob3rquzZ+LGSVDIdUrcXz2uR61HuQ653eQ+ysP9Uf4AdheeGdMRT/HNuAF4N/AS5pup+ZGY0X9GGlN60d+0D4pwUqeqIxyNrqZ5CTtQo8zZ/zPsm1K7DEf8XiNT8wcrbXWv7C8XRT1bLVGVKq+9qunak4w/y7OuhONJxrTEmQP6u23iGaica40LunqGd6yrPYONHFalkelLdVDwvo+tlLNp1temr3ZPWqeVKX/62ITPfN3VYM3e6ylSufbctWn1yWvt6GzSc2Jiwki6dzOq4EIJm9mz8WI6W697Hj5g7ZIIsrN8NnFvafi4y+4lFq642ubY7Ffrwtx24ELl7Xuxd04x6rA8YuWFjwkq6dmLjQjWWvf7Qcp3da3g+0iEvRS7jtB2Z3a1PtK42Wu2eQ9ZHbUROarjKfx9zCcyU0Xzdu7zd9TSglalc+25TtPzNSl9vgpYTGxNWouVkVsaFULQ82/ixEi3X2Y4fm5GL4x9AZtPldWup1dVGo93zyPqn8teog9qpofW6V/lbIu5219PAPHqZyrXvNmEeHX9adXNmHh0nNiasRNPJLIwLoWh5nq+pO2pZN2e0XNv4YRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYRiGYcwc/wfDhIbWOFAd8gAAAABJRU5ErkJggg==\n",
            "text/latex": "$$a_{0} b_{0} + a_{0} b_{1} x + a_{0} b_{2} x^{2} + a_{1} b_{0} x + a_{1} b_{1} x^{2} + a_{1} b_{2} x^{3} + a_{2} b_{0} x^{2} + a_{2} b_{1} x^{3} + a_{2} b_{2} x^{4}$$",
            "text/plain": [
              "                         2                    2          3          2         \n",
              "a₀⋅b₀ + a₀⋅b₁⋅x + a₀⋅b₂⋅x  + a₁⋅b₀⋅x + a₁⋅b₁⋅x  + a₁⋅b₂⋅x  + a₂⋅b₀⋅x  + a₂⋅b₁⋅\n",
              "\n",
              " 3          4\n",
              "x  + a₂⋅b₂⋅x "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 37
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FEf0gfriBUWM",
        "outputId": "2da0c1c8-4453-451c-f745-bf453c2c5fc9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# FATORAR POLINÔMIOS\n",
        "\n",
        "sy.factor(R)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\left(a_{0} + a_{1} x + a_{2} x^{2}\\right) \\left(b_{0} + b_{1} x + b_{2} x^{2}\\right)$$",
            "text/plain": [
              "⎛                2⎞ ⎛                2⎞\n",
              "⎝a₀ + a₁⋅x + a₂⋅x ⎠⋅⎝b₀ + b₁⋅x + b₂⋅x ⎠"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 38
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3O1X4tZP2olw"
      },
      "source": [
        "**Não poderiam faltar as funções logarítmicas no arsenal do SymPy.**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mfPdIs4T2nGY",
        "outputId": "2e508789-e1af-4a6a-c4f3-8641adcf3844",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "# log\n",
        "pot, base = 8, 2\n",
        "sy.log(pot, base)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAOhJREFUOI3N0r1KQ0EUBODPYCEkIGggkEZIYSUIYmEhpLESLHwI38DeNEHsxVexEDutBBWRQCqLQBD/wEqUSCx2F8Mlyr2QIgPLLmfPcGaGw4RwhHP08IE33OAAi6ONMxniF67RwRPK2MA6+vHdGzdx7g8lbQxxUtTCaiSepUIpJ3En3nepkPWYsI8K5gV/m5G0hef/JjxGaemcopZTHbF5F10h1bUiZFjCJ+6LEgmLMESV/KlCPd7f2Y9lIcUsSn4X4DIVZ0catnGICzzgVQiniYaQ9N44KSs4xi1eMMA7rtDCQk5L04YfBeYtAc2H2hEAAAAASUVORK5CYII=\n",
            "text/latex": "$$3$$",
            "text/plain": [
              "3"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 39
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "drS4J0ue3A1a",
        "outputId": "de20dfd3-f40a-4a27-dd0f-f41c2f6248df",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "# ln\n",
        "sy.ln(sy.exp(x))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPCAYAAAA71pVKAAAABHNCSVQICAgIfAhkiAAAAMFJREFUKJHVziFPwmEUxeFH2Qh2goEEG035ANJMVEkmNiqfgC8gxexoVJqR2d3cbARHwqgbQZJuMIRy2Rj740ugeNN7z3l/9xyOOE9Y4WZHP0E/vO4++BJLvCG3pd8H2EulbxKasXdiH+A0BRfxg3e0AxwinwI3cxfQCs84y/q0r8Z0693C96Gpt/jFRyQ/HArWMccIBYyxQCUFXkW9Cc5Da0T6419gFbOoWtrxXuNALQss4xNfuMjwrwN+SVX/J7MGliUqwx10gTUAAAAASUVORK5CYII=\n",
            "text/latex": "$$x$$",
            "text/plain": [
              "x"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 40
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eKuTlduX75A7"
      },
      "source": [
        "Propriedades de Logaritmo\n",
        "\n",
        "A função `expand_log` nos permite explorar as propriedades dos logaritmos no SymPy."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BxpWxLYu7PQK",
        "outputId": "f975a517-04a5-47cc-81b2-45e558a9cd98",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "a, b = sy.symbols('a b', positive = True)\n",
        "\n",
        "sy.expand_log(sy.log(a*b))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\log{\\left (a \\right )} + \\log{\\left (b \\right )}$$",
            "text/plain": [
              "log(a) + log(b)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 69
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HDrbcYgx3Typ",
        "outputId": "610f15f6-7598-4d82-f2cb-3ae7a507aa67",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "sy.expand_log(sy.log(a/b))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\log{\\left (a \\right )} - \\log{\\left (b \\right )}$$",
            "text/plain": [
              "log(a) - log(b)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 70
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "u6AiQZSH3X8q",
        "outputId": "2f8a293d-d383-4f68-a189-eeaedd8420f1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "sy.expand_log(sy.log(a**b))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$b \\log{\\left (a \\right )}$$",
            "text/plain": [
              "b⋅log(a)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 73
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TmCDhpLV1ELe"
      },
      "source": [
        "Há também suporte nativo à biblioteca MatplotLib, o que nos permite desenhar gráficos de funções quaisquer."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "aRxsV6zn1MEB",
        "outputId": "c8d972bb-2939-4dfb-ff25-1def24276544",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        }
      },
      "source": [
        "# considere a função a seguir\n",
        "sigmoid = 1/(1 + exp(-x))\n",
        "sigmoid"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAADQAAAAgCAYAAABdP1tmAAAABHNCSVQICAgIfAhkiAAAAa1JREFUWIXt1zFrFEEUwPFfYlJIrpKAYCAiphQ8UbsoKcz38AuItgYCVoqtIKQRDu0UG7sUsRPSKApJJ5gIF2yMIREDuSIWsyeb5Y67XTNslPnDY3dn5s17b2d4M4943MQbtHGI2xFt/WE04twNrOEu9iPaqYWf/oMVqoUU0EknBXTSGYs4dwMz2fsoptHENr5GtBuNOeFALUqrPpcSicQgRrLnYa1eJPpzqm4HKvIO60Kt9QwXscq/G9AW7mEKk1joNaiWCrMPH4XisCjncmPeYwXjecX81adbYT7PpAwtbOBBSb1+NAf0X8cZbKIzzIRlK8yWcsFMCT/tO3bwGmdL6H7CBXzApXxnHbftriNtzAp3vkksDaF7Gq9wB1/wCIvDGI25Qst4WGi7hd0S9kozKKCFbExXOjgotN3ooXdeSDi/CmP3he3311Sth5bwMvf9WNhCT3Jt7R56l4WVuNqj76CiL0eoGtB2Jl32su/PA/Q6mMA3YWWOnXxSaAjpsulohTl9jPZW8QMvcEU44efxVIQENad6hdkyfFK4hrdCut4TUvD9Mo4mEol4/AbdzGcquVF0VwAAAABJRU5ErkJggg==\n",
            "text/latex": "$$\\frac{1}{1 + e^{- x}}$$",
            "text/plain": [
              "   1   \n",
              "───────\n",
              "     -x\n",
              "1 + ℯ  "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 41
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VcOqtTxv1tgq",
        "outputId": "9830d50c-3acc-45f6-ef9c-629f6843cba7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 407
        }
      },
      "source": [
        "from sympy.plotting import plot\n",
        "\n",
        "x = symbols('x')\n",
        "\n",
        "plot(sigmoid)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/sympy/plotting/plot.py:890: MatplotlibDeprecationWarning: \n",
            "The set_smart_bounds function was deprecated in Matplotlib 3.2 and will be removed two minor releases later.\n",
            "  self.ax.spines['left'].set_smart_bounds(True)\n",
            "/usr/local/lib/python3.6/dist-packages/sympy/plotting/plot.py:891: MatplotlibDeprecationWarning: \n",
            "The set_smart_bounds function was deprecated in Matplotlib 3.2 and will be removed two minor releases later.\n",
            "  self.ax.spines['bottom'].set_smart_bounds(False)\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<sympy.plotting.plot.Plot at 0x7f8250e45860>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 78
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WBTSyjYnIhpl"
      },
      "source": [
        "Uma aplicação interessante que envolve tanto a função Sigmóide quanto a composição de funções é o de Regressão Logística, que nada mais é do que a função composta `sigmoid(linear(x))` onde `linear(x)` é uma função linear em seus argumentos.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YqdFSPllNVMc"
      },
      "source": [
        "Vamos definir as seguintes funções $f$ e $g$, tal que:\n",
        "\n",
        "$$\n",
        "f(x) = \\frac{1}{1 + e^{-x}}, \\text{ com } x \\in (-\\infty, \\infty) \\text{ e } f(x)\\in (0,1)\n",
        "$$ e,\n",
        "$$\n",
        "g(X) = X\\cdot \\beta, \\text{ com } x \\in (-\\infty, \\infty) \\text{ e } g(X)\\in (-\\infty, \\infty)\n",
        "$$\n",
        "\n",
        "Perceba que $h(x) = f\\circ g (x)$ é dado da seguinte forma:\n",
        "\n",
        "$$\n",
        "h(X) = f\\circ g (x)= f(g(x))= \\frac{1}{1 + e^{-g(x)}} = \\frac{1}{1 + e^{-X\\cdot \\beta}}\n",
        "$$\n",
        "\n",
        "com $\\{ x \\in \\mathbb{R}: 0 < h(x) < 1 \\}$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9Dhtjq54UCyn"
      },
      "source": [
        "Como visto no **Ebook**, **subseção [4.7](https://www.overleaf.com/read/gtspgphgvstz)**, temos a seguinte expressão final para cada observação de $h(X)$:\n",
        "\n",
        "$$\n",
        "h(y_i | \\beta_0, \\beta_1, \\dots, \\beta_p) = \\frac{1}{1 + e^{-(\\beta_0 + \\beta_1 \\cdot x_{i1} + \\beta_2\\cdot x_{i2} + \\dots + \\beta_p \\cdot x_{ip})}}\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qDpwhcyDOyFy"
      },
      "source": [
        "Podemos definir `h(x)`, compondo `g` sobre `f`:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "NTgJZtWjJ8PB",
        "outputId": "83bed1cd-fc40-4171-976a-8043c882e2e1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 40
        }
      },
      "source": [
        "# Definindo a função f\n",
        "f = sigmoid\n",
        "\n",
        "# Definindo a expressão de g\n",
        "g = x * beta\n",
        "\n",
        "# Definindo a função g\n",
        "g = Function('g')(x)\n",
        "\n",
        "# Definindo a função composta h(x) = f(g(x))\n",
        "h = 1/(1 + exp(-g))\n",
        "h"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\frac{1}{1 + e^{- g{\\left (x \\right )}}}$$",
            "text/plain": [
              "    1     \n",
              "──────────\n",
              "     -g(x)\n",
              "1 + ℯ     "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 43
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8T-qQaUb8nTp"
      },
      "source": [
        "É importante definir 'positive = True' para que o SymPy entenda que se tratam de números positivos, já que os logaritmos possuem restrição quanto a isso."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Xe-zuwSj5vL9"
      },
      "source": [
        "**Fatorial**\n",
        "\n",
        "O cálculo do fatorial de um número pode ser feito usando a função `factorial(n)`. Vejamos um exemplo."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3OG6dJ8W6Iei",
        "outputId": "2da63be2-ded5-452e-c660-8e76eb06830e",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "sy.factorial(x)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAASCAYAAABb0P4QAAAABHNCSVQICAgIfAhkiAAAAOtJREFUOI3l0S9Lg1EUx/GPCga7wWBS0DR9AbOZrC6ZBKuvwDcwi1nWVm2LYhcEm0FM29MmGDQpKG6GncnzjOfPFAXBH1y459xzvvd37uUPKokFZn+a/uvACwyxO5GfQTvOjr9ywQbecYu5VP4kYK2cnkTqDfM0drIf8VHEZ/KfqBK4jBf0cBiwc8wX1FcCoRmgIS6xUFKbARb98kNqf4DnKgdl2sMA/XB4WlGfKBl5B6+4wSLu8Ia17zirG43WxVLkGuGyU9K3HiujTTwZjbkycXYd0K0C4PjzPrWKezyiltOwHQ1X0wL/mT4AQjw5lVVu+sYAAAAASUVORK5CYII=\n",
            "text/latex": "$$x!$$",
            "text/plain": [
              "x!"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 45
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "432nt9Ly6SFt",
        "outputId": "1ffbbdee-cb46-4028-d4b1-59fc29527850",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 37
        }
      },
      "source": [
        "# 8!\n",
        "\n",
        "sy.factorial(x).subs(x,8)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$40320$$",
            "text/plain": [
              "40320"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 46
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UslqqY5MB6PS"
      },
      "source": [
        "**Matrizes**\n",
        "\n",
        "As matrizes no SymPy seguem uma sintaxe semelhante ao do Python. Assim, podemos definir uma matriz da seguinte forma:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MmiyljtvCHwn",
        "outputId": "0ca8e292-c5f3-43ff-fd3c-b2d7637b11d1",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        }
      },
      "source": [
        "A = sy.Matrix([[1,2,3],[4,5,6],[7,8,9]])\n",
        "A"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/latex": "$$\\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$$",
            "text/plain": [
              "⎡1  2  3⎤\n",
              "⎢       ⎥\n",
              "⎢4  5  6⎥\n",
              "⎢       ⎥\n",
              "⎣7  8  9⎦"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 47
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "x7LydJ2t6ygy"
      },
      "source": [
        "O SymPy nos permite executar vários cálculos envolvendo matrizes... autovalores e autovetores, diagonalização, transposta, inversa, entre outros."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Vv9SFleJ76Ed",
        "outputId": "c97739a9-270c-4f99-fdba-28f9c5cfa202",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        }
      },
      "source": [
        "# Transporta de uma Matriz\n",
        "\n",
        "A.T"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/latex": "$$\\left[\\begin{matrix}1 & 4 & 7\\\\2 & 5 & 8\\\\3 & 6 & 9\\end{matrix}\\right]$$",
            "text/plain": [
              "⎡1  4  7⎤\n",
              "⎢       ⎥\n",
              "⎢2  5  8⎥\n",
              "⎢       ⎥\n",
              "⎣3  6  9⎦"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 81
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "k87ejMXj8CQt",
        "outputId": "a1b3fb99-51c4-4581-df55-760416fa001a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        }
      },
      "source": [
        "# Matriz de zeros\n",
        "\n",
        "zeros(3,3)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/latex": "$$\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$",
            "text/plain": [
              "⎡0  0  0⎤\n",
              "⎢       ⎥\n",
              "⎢0  0  0⎥\n",
              "⎢       ⎥\n",
              "⎣0  0  0⎦"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 82
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gyD5Rg568MOJ",
        "outputId": "a5d320e0-d2fd-4ee2-83da-4fe3ffa3148d",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        }
      },
      "source": [
        "# Matriz de uns\n",
        "\n",
        "ones(3,3)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/latex": "$$\\left[\\begin{matrix}1 & 1 & 1\\\\1 & 1 & 1\\\\1 & 1 & 1\\end{matrix}\\right]$$",
            "text/plain": [
              "⎡1  1  1⎤\n",
              "⎢       ⎥\n",
              "⎢1  1  1⎥\n",
              "⎢       ⎥\n",
              "⎣1  1  1⎦"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 83
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "obDpg3Nf8Uck",
        "outputId": "e65b0bdb-942f-4947-ad92-2e23676dcc01",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 78
        }
      },
      "source": [
        "# Matriz Identidade\n",
        "\n",
        "eye(3)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/latex": "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$",
            "text/plain": [
              "⎡1  0  0⎤\n",
              "⎢       ⎥\n",
              "⎢0  1  0⎥\n",
              "⎢       ⎥\n",
              "⎣0  0  1⎦"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 86
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zVS3p_ZMr0zc"
      },
      "source": [
        "**Exercícios**\n",
        "\n",
        "(1) Um vendedor tem seu salário definido como um valor fixo, de `1045` reais, acrescido de um bônus por cada artigo negociado com sucesso, cada um somando `55` reais ao salário do vendedor. \n",
        "\n",
        "  "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-65BIhzUtWjo"
      },
      "source": [
        "a) Determinar a expressão matemática que corresponde ao salário final do vendedor."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xwIaq9YjtYkq"
      },
      "source": [
        "# R:\n",
        "expr = 0\n",
        "\n",
        "# PS: defina expr = \"sua resposta\""
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JXfs2J_Itfcb"
      },
      "source": [
        "b) Qual será o salário deste profissional, tendo ele vendido 7 artigos?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "QMKK9lUItq6b"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vaUnohlqv4kk"
      },
      "source": [
        "c) Quantos artigos ele precisa vender (no mínimo) para bater a meta de `3600` reais?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Bh28WpSVwFh9"
      },
      "source": [
        "i = 1\n",
        "while(expr.subs(x, i) <= 3600):\n",
        "  i += 1\n",
        "print(i)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "um1BG2Imwq21"
      },
      "source": [
        "(2) Um projétil é lançado ao ar e sua trajetória obedece à parábola determinada por"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Xsyn_5lYzAZE"
      },
      "source": [
        "expr = -x**2 + 5*x - 1\n",
        "expr"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "McWAHyjHyodn"
      },
      "source": [
        "a) Determine o ponto vertical ao solo que coincide com a altura máxima alcançada pelo projétil."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YHoLApb_yY7S"
      },
      "source": [
        "# x do vértice de uma parábola: -b/2a\n",
        "xv = -5/(2*(-1))\n",
        "\n",
        "# altura máxima: f(xv)\n",
        "expr.subs(x, xv)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ca4hZ5tOyYNR"
      },
      "source": [
        "b) Determine o ponto de onde foi lançado o projétil."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "spjUW__8w6iq"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hTMqYo2AxlIn"
      },
      "source": [
        "c) Determine o ponto onde irá cair o projétil."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WCdrh2sbznzI"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cnscWqu69xNQ"
      },
      "source": [
        "(3) Dada $f(x) = \\sqrt{x} - 2x +1$, calculo a $f$ no ponto x = 4 e plote o gráfico a mesma função."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wgX9muYm-Ahd"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "K6vcxHuc975p"
      },
      "source": [
        "(4) Seja $f(x) = -x^2 + 2x$, simplifique as expressões:"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UwA6_II7-Cq5"
      },
      "source": [
        "a) $ \\frac{f(x) - f(1)}{x - 1} $"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "TlHZLDY3-BoR",
        "outputId": "497666d5-e79b-40b9-d085-f61d1f1bcb83",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        }
      },
      "source": [
        "# R:\n",
        "# armazenando a função f\n",
        "f = -x**2 + 2*x\n",
        "\n",
        "# calculando a expressão do item (A)\n",
        "g = (f - f.subs(x, 1))/(x - 1)\n",
        "g"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$\\frac{1}{x - 1} \\left(- x^{2} + 2 x - 1\\right)$$",
            "text/plain": [
              "   2          \n",
              "- x  + 2⋅x - 1\n",
              "──────────────\n",
              "    x - 1     "
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 79
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dgj1OU3WZlsj",
        "outputId": "7430e1d6-e710-4163-99db-6e5cf42a421a",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 38
        }
      },
      "source": [
        "\n",
        "# simplificando a expressão g\n",
        "simplify(g)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "image/png": "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\n",
            "text/latex": "$$- x + 1$$",
            "text/plain": [
              "-x + 1"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 80
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uoIe2IVk-MSp"
      },
      "source": [
        "b) $ \\frac{f(x + h) - f(x)}{h} $"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Qujwxk7h-Mt0"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "soFfI6n2-fzu"
      },
      "source": [
        "(5) Resolva o sistema de equações algébricas formado pelas equações abaixo:\n",
        "\n",
        "\n",
        "$ 2x+y-z = 0 $\n",
        "\n",
        "$ 4x-y-z = 5$\n",
        "\n",
        "$ y+3z = 10 $"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-BKepr-0_o39"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8CxA4pYl_q2k"
      },
      "source": [
        "(6) Dado o sistema $Ax = b$, sendo as matrizes $A = \\begin{pmatrix} 2 & 4 & 1 \\\\ 5 & -7 & 2 \\\\ -1 & 3 & 6 \\end{pmatrix}$ e $b = \\begin{pmatrix} 1 & 3 & 7 \\end{pmatrix}$, determine:"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ak8sfMMCBMMx"
      },
      "source": [
        "a) O determinante da matriz $A$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "SwreVUkvBHA1"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DGJBdnp8BQu8"
      },
      "source": [
        "b) A matriz inversa de $A$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "07XfU1dSBXtR"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gzY-TdQQBYFk"
      },
      "source": [
        "c) Os valores de $x$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "p5d_lDc7Bbb8"
      },
      "source": [
        "# R:"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uplpL_CN5eaz"
      },
      "source": [
        "Todo o aparato apresentado pode ser muito útil quando lidamos com problemas envolvendo funções, mas toda referência é bem-vinda. Por isso recomendamos que, desejando-se obter um conhecimento mais aprofundado da biblioteca, o leitor visite a documentação oficial do SymPy. \n",
        "\n",
        "https://docs.sympy.org/"
      ]
    }
  ]
}