Cours_codekodo_1G_cours_exponentielle.ipynb

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  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.3"
    },
    "colab": {
      "name": "1G-cours-exponentielle.ipynb",
      "provenance": []
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Denq-NUXzg96",
        "colab_type": "text"
      },
      "source": [
        "# La fonction exponentielle\n",
        "\n",
        "Il existe de nombreuses situations dans lesquelles la situation est modélisée par une équation de la forme $f ′ = k \\cdot f$ avec $k\\in\\mathbb{R}$. Ce type d’équation dont l’inconnue est une fonction et qui fait intervenir sa dérivée est appelée *équation différentielle*."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WnpmVCOczg-C",
        "colab_type": "text"
      },
      "source": [
        "## La fonction exponentielle\n",
        "\n",
        "L’unique fonction $f$ définie et dérivable sur $\\mathbb{R}$ vérifiant $f ′ = f$ et $f (0) = 1$ est appelée *fonction exponentielle* et est notée exp.\n",
        "\n",
        "$$ f : x \\mapsto \\exp(x) $$\n",
        "\n",
        "Ainsi,  $\\forall x\\in \\mathbb{R}\\,,\\exp′(x) = \\exp(x)$ et $\\exp(0) = 1$.\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TDp_LcE9zg-H",
        "colab_type": "text"
      },
      "source": [
        "Par la méthode d'Euler, on peut obtenir une approximation de $\\exp(a)$ pour tout réel $a$\n",
        "\n",
        "Rappel : on $f(a+h) = f'(a) \\times h + f(a)$\n",
        "\n",
        "D'où, comme ici $f'(a) = f(a)$, on a $$\\forall a,h\\in \\mathbb{R}\\,, f(a+h) = f(a)\\cdot (1+h)$$\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OQrorEz1zg-T",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Approximation de exp(a)\n",
        "def exp(a):\n",
        "    h = .0001\n",
        "    x= 0\n",
        "    f = 1\n",
        "    if a>0 : \n",
        "        while x < a :\n",
        "            f = f*(1+h)\n",
        "            x += h\n",
        "    else :\n",
        "        while x > a :\n",
        "            f = f*(1-h)\n",
        "            x = x-h\n",
        "    return f"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mr-9gf8Pzg-1",
        "colab_type": "text"
      },
      "source": [
        "### Approximation de $\\exp(1)$ et conjectures sur les règles opératoires"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gNno0Pg8zg-4",
        "colab_type": "code",
        "colab": {},
        "outputId": "e44667f9-6bc1-4979-f512-7470bcb40d9f"
      },
      "source": [
        "exp(1)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.7184177414175807"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1nAA1Zo9zg_R",
        "colab_type": "code",
        "colab": {},
        "outputId": "edf5b564-656d-42bc-ef15-4b94fc37b7d3"
      },
      "source": [
        "print(exp(2+3),exp(2)*exp(3))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "148.37606292299085 148.39090052927986\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ScZvWX9dzg_j",
        "colab_type": "text"
      },
      "source": [
        "### Représentation graphique (méthode d'Euler)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vWhdUtwCzg_n",
        "colab_type": "code",
        "colab": {},
        "outputId": "22b99f56-86b3-4764-c088-fd655dbcf2c9"
      },
      "source": [
        "# Courbe de exp sur [a;b]\n",
        "\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def courbe_approx(a,b):\n",
        "    x = a\n",
        "    liste_x=[]\n",
        "    liste_y=[]\n",
        "    # courbe approchée de f\n",
        "    while x<b :\n",
        "        liste_x.append(x)\n",
        "        liste_y.append(exp(x))\n",
        "        x += .01\n",
        "    plt.plot(liste_x,liste_y)\n",
        "\n",
        "courbe_approx(-5,5)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "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"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "BW4oO9nOzg_5",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}