APS1_SN_2022_2_Q01.ipynb

{
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  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyOo9y33YMcw4zwEwyukm15s",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/rodrigosilvaluz/SistemasNumericos/blob/master/APS1_SN_2022_2_Q01.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Questão 01"
      ],
      "metadata": {
        "id": "EaHS3iZ2-I9V"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Feito a importação das blibiotecas necessárias para rodar o código"
      ],
      "metadata": {
        "id": "YTC6vS0bz_BL"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 68,
      "metadata": {
        "id": "O8kJVZxb2DW0"
      },
      "outputs": [],
      "source": [
        "import math\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Input com a função alvo"
      ],
      "metadata": {
        "id": "OVEOFkVg0KOE"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def f(x):\n",
        "  return(x**4 - x**3 - x**2 + x + 8)\n"
      ],
      "metadata": {
        "id": "uEem34diYcvZ"
      },
      "execution_count": 69,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "a = float(input(\"Valor 'a' para o intervalo [a;b]: \"))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "NwKoHrLeYxEo",
        "outputId": "424c0f3f-1bc1-40b9-d52a-fe2ffbd084fc"
      },
      "execution_count": 70,
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor 'a' para o intervalo [a;b]: -2.5\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "b = float(input(\"Valor 'b' para o intervalo [a;b]: \"))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AFNGZNqsZIUG",
        "outputId": "c340e4e0-8dd6-499d-b183-c8c4620144eb"
      },
      "execution_count": 71,
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Valor 'b' para o intervalo [a;b]: 2.5\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "print('Dada a equação: x**4 - x**3 - x**2 + x + 8')\n",
        "\n",
        "print('Intervalo \\033[31m(-2, 5) e (2, 5)\\033[m')\n",
        "\n",
        "print('Raizes negativas: \\033[32m-2, -1, 0\\033[m')\n",
        "\n",
        "print('Raizes positivas: \\033[33m0, 1, 2, 3, 4, 5\\033[m')\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "G7_07LcvLMco",
        "outputId": "1f9049f9-d996-4a40-afb4-a51960df2e84"
      },
      "execution_count": 72,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Dada a equação: x**4 - x**3 - x**2 + x + 8\n",
            "Intervalo \u001b[31m(-2, 5) e (2, 5)\u001b[m\n",
            "Raizes negativas: \u001b[32m-2, -1, 0\u001b[m\n",
            "Raizes positivas: \u001b[33m0, 1, 2, 3, 4, 5\u001b[m\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "l = 0.001\n",
        "t = 1e-6\n",
        "c = b - a\n",
        "cont = 0\n",
        "Ni = 100"
      ],
      "metadata": {
        "id": "e-x2zgiqZMTL"
      },
      "execution_count": 73,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "x0 = (a + b)/2.0"
      ],
      "metadata": {
        "id": "a60SqSOnZXHp"
      },
      "execution_count": 74,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "while(c > l or math.fabs(f(x0)) > t):\n",
        "  if(f(a)*f(x0) < 0.0):\n",
        "    b = x0\n",
        "  if(f(a)*f(x0) > 0.0):\n",
        "    a =x0\n",
        "\n",
        "  c = b - a\n",
        "  x0 = (a + b)/2.0  \n",
        "  cont = cont + 1  \n",
        "\n",
        "  if(cont >= Ni):\n",
        "    break\n",
        "\n",
        "  print(\"\\n\\n\\nRaiz: %f\\nIteracoes: %i\\nf(%f) = %f\" %(x0,cont,x0,f(x0)))\n",
        "  \n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "O8cDOEAKaeNu",
        "outputId": "0b5fde89-97a1-464e-8d75-4852d03bc76e"
      },
      "execution_count": 75,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\n",
            "\n",
            "\n",
            "Raiz: 1.250000\n",
            "Iteracoes: 1\n",
            "f(1.250000) = 8.175781\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 1.875000\n",
            "Iteracoes: 2\n",
            "f(1.875000) = 12.127197\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.187500\n",
            "Iteracoes: 3\n",
            "f(2.187500) = 17.832535\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.343750\n",
            "Iteracoes: 4\n",
            "f(2.343750) = 22.150834\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.421875\n",
            "Iteracoes: 5\n",
            "f(2.421875) = 24.754779\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.460938\n",
            "Iteracoes: 6\n",
            "f(2.460938) = 26.178482\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.480469\n",
            "Iteracoes: 7\n",
            "f(2.480469) = 26.922129\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.490234\n",
            "Iteracoes: 8\n",
            "f(2.490234) = 27.302074\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.495117\n",
            "Iteracoes: 9\n",
            "f(2.495117) = 27.494099\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.497559\n",
            "Iteracoes: 10\n",
            "f(2.497559) = 27.590627\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.498779\n",
            "Iteracoes: 11\n",
            "f(2.498779) = 27.639020\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499390\n",
            "Iteracoes: 12\n",
            "f(2.499390) = 27.663249\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499695\n",
            "Iteracoes: 13\n",
            "f(2.499695) = 27.675372\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499847\n",
            "Iteracoes: 14\n",
            "f(2.499847) = 27.681435\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499924\n",
            "Iteracoes: 15\n",
            "f(2.499924) = 27.684467\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499962\n",
            "Iteracoes: 16\n",
            "f(2.499962) = 27.685984\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499981\n",
            "Iteracoes: 17\n",
            "f(2.499981) = 27.686742\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499990\n",
            "Iteracoes: 18\n",
            "f(2.499990) = 27.687121\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499995\n",
            "Iteracoes: 19\n",
            "f(2.499995) = 27.687310\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499998\n",
            "Iteracoes: 20\n",
            "f(2.499998) = 27.687405\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499999\n",
            "Iteracoes: 21\n",
            "f(2.499999) = 27.687453\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.499999\n",
            "Iteracoes: 22\n",
            "f(2.499999) = 27.687476\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 23\n",
            "f(2.500000) = 27.687488\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 24\n",
            "f(2.500000) = 27.687494\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 25\n",
            "f(2.500000) = 27.687497\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 26\n",
            "f(2.500000) = 27.687499\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 27\n",
            "f(2.500000) = 27.687499\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 28\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 29\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 30\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 31\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 32\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 33\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 34\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 35\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 36\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 37\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 38\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 39\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 40\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 41\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 42\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 43\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 44\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 45\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 46\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 47\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 48\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 49\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 50\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 51\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 52\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 53\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 54\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 55\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 56\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 57\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 58\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 59\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 60\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 61\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 62\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 63\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 64\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 65\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 66\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 67\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 68\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 69\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 70\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 71\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 72\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 73\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 74\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 75\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 76\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 77\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 78\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 79\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 80\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 81\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 82\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 83\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 84\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 85\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 86\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 87\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 88\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 89\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 90\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 91\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 92\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 93\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 94\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 95\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 96\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 97\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 98\n",
            "f(2.500000) = 27.687500\n",
            "\n",
            "\n",
            "\n",
            "Raiz: 2.500000\n",
            "Iteracoes: 99\n",
            "f(2.500000) = 27.687500\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "a = -2.5\n",
        "b = 2.5"
      ],
      "metadata": {
        "id": "ISFtFhPdqDWK"
      },
      "execution_count": 76,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def f(x): # definindo a função a ser aproxima sua raíz (intruduza a função objetivo)\n",
        "  f = x**4 - x**3 - x**2 + x + 8\n",
        "  return(f)"
      ],
      "metadata": {
        "id": "QFr9p-G7qRqc"
      },
      "execution_count": 77,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"f(x) = x**4 - x**3 - x**2 + x + 8 -> x = a \", f(a))\n",
        "print(\"f(x) = x**4 - x**3 - x**2 + x + 8 -> x = b \", f(b))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "DEepChfEqZ6t",
        "outputId": "7053e590-b6c6-4d14-b6f9-cad38466e323"
      },
      "execution_count": 78,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "f(x) = x**4 - x**3 - x**2 + x + 8 -> x = a  53.9375\n",
            "f(x) = x**4 - x**3 - x**2 + x + 8 -> x = b  27.6875\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "k = (math.log(b - a, 10) - math.log(0.000001, 10)) / math.log(2, 10)\n",
        "print(f'k ->: \\033[36m{k}\\033[m >> arredondado para ->: \\033[33m{math.ceil(k)}\\033[m') # referente ao [iii. abs(b-a) <= 1E-06] - APS"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "tSp192gQqkn4",
        "outputId": "6bb07e0d-964a-4d62-8795-05df070b1194"
      },
      "execution_count": 79,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "k ->: \u001b[36m22.253496664211536\u001b[m >> arredondado para ->: \u001b[33m23\u001b[m\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "x = np.linspace(-2.5, 2.5) \n",
        "y = f(x)\n",
        "plt.figure(dpi=70, figsize=(8, 6))\n",
        "plt.title(\"f(x) = x**4 - x**3 - x**2 + x + 8 \")\n",
        "plt.plot(x,y)\n",
        "plt.hlines(y=0, xmin=-5, xmax=5) \n",
        "plt.vlines(x=0, ymin=-5, ymax=20) \n",
        "plt.ylabel(\"f(x)\") \n",
        "plt.show() "
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 380
        },
        "id": "bf7lEQ8IqsgW",
        "outputId": "b062519e-25eb-41f4-c60e-57648aed87da"
      },
      "execution_count": 80,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 560x420 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}