notebooks/3-calculus-i.ipynb

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      "source": [
        "# Calculus I: Limits & Derivatives"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yqUB9FTRAxd-"
      },
      "source": [
        "This topic, *Calculus I: Limits & Derivatives*, introduces the mathematical field of calculus -- the study of rates of change -- from the ground up. It is essential because computing derivatives via differentiation is the basis of optimizing most machine learning algorithms, including those used in deep learning such as backpropagation and stochastic gradient descent. \n",
        "\n",
        "Through the measured exposition of theory paired with interactive examples, you’ll develop a working understanding of how calculus is used to compute limits and differentiate functions. You’ll also learn how to apply automatic differentiation within the popular TensorFlow 2 and PyTorch machine learning libraries. The content covered in this class is itself foundational for several other topics in the *Machine Learning Foundations* series, especially *Calculus II* and *Optimization*."
      ]
    },
    {
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      "source": [
        "Over the course of studying this topic, you'll: \n",
        "\n",
        "* Develop an understanding of what’s going on beneath the hood of machine learning algorithms, including those used for deep learning. \n",
        "* Be able to more intimately grasp the details of machine learning papers as well as many of the other subjects that underlie ML, including partial-derivative calculus, statistics and optimization algorithms. \n",
        "* Compute the derivatives of functions, including by using AutoDiff in the popular TensorFlow 2 and PyTorch libraries."
      ]
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      "metadata": {
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      "source": [
        "**Note that this Jupyter notebook is not intended to stand alone. It is the companion code to a lecture or to videos from Jon Krohn's [Machine Learning Foundations](https://github.com/jonkrohn/ML-foundations) series, which offer detail on the following:**\n",
        "\n",
        "*Segment 1: Limits*\n",
        "\n",
        "* What Calculus Is\n",
        "* A Brief History of Calculus\n",
        "* The Method of Exhaustion \n",
        "* Calculating Limits \n",
        "\n",
        "*Segment 2: Computing Derivatives with Differentiation*\n",
        "* The Delta Method\n",
        "* The Differentiation Equation\n",
        "* Derivative Notation\n",
        "* The Power Rule\n",
        "* The Constant Multiple Rule\n",
        "* The Sum Rule\n",
        "* The Product Rule\n",
        "* The Quotient Rule\n",
        "* The Chain Rule\n",
        "\n",
        "*Segment 3: Automatic Differentiation*\n",
        "* AutoDiff with PyTorch\n",
        "* AutoDiff with TensorFlow 2\n",
        "* Machine Learning via Differentiation \n",
        "* Cost (or Loss) Functions\n",
        "* The Future: Differentiable Programming \n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
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      },
      "source": [
        "## Segment 1: Limits"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ab2YePNtcxYx"
      },
      "source": [
        "### The Calculus of Infinitesimals"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Bzug184ZcxYx"
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Kgjg4q4kcxY1"
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      "source": [
        "x = np.linspace(-10, 10, 10000) # start, finish, n points"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MHWrqwBacxY4"
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      "source": [
        "If $y = x^2 + 2x + 2$: "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "y4oTgUY_cxY5"
      },
      "source": [
        "y = x**2 + 2*x + 2"
      ],
      "execution_count": 3,
      "outputs": []
    },
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      "source": [
        "fig, ax = plt.subplots()\n",
        "_ = ax.plot(x,y)"
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5IjlYM5LcxY_"
      },
      "source": [
        "* There are no straight lines on the curve. \n",
        "* If we zoom in _infinitely_ close, however, we observe curves that _approach_ lines. \n",
        "* This enables us to find a slope $m$ (tangent) anywhere on the curve, including to identify where $m = 0$: "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "nJ1n043-cxZA",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "ca8a8659-68d2-41cd-e7c3-14770e02123d"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.set_xlim([-2, -0])\n",
        "ax.set_ylim([0, 2])\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 5,
      "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": "tgrXcqfScxZD",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "2f9bac4e-cba5-4511-b33d-492d98c35fd8"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.set_xlim([-1.5, -0.5])\n",
        "ax.set_ylim([0.5, 1.5])\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAD4CAYAAAD8Zh1EAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+WH4yJAAAbpElEQVR4nO3deXzV9Z3v8dcneyAhCVlJQhLCYsoqEiuKFtzFBW1rF69d7LT1dl8evTO1985Ub28fD9vpY9remevUOuqotdU6M1apFcFWKoigBAUl7IQEsq9k38/3/nEOEiGQYE7OSX68n49HHuf3O+d7fr8PX5L3+Z3vbzPnHCIiMvlFhLsAEREJDgW6iIhHKNBFRDxCgS4i4hEKdBERj4gK14rT0tJcQUFBuFYvIjIp7dixo9E5lz7ca2EL9IKCAkpKSsK1ehGRScnMKs70moZcREQ8QoEuIuIRCnQREY9QoIuIeIQCXUTEIxToIiIeoUAXEfEIBbqIiEco0EVEPEKBLiLiEQp0ERGPUKCLiHiEAl1ExCMU6CIiHqFAFxHxCAW6iIhHKNBFRDxixEA3s0fNrN7Mdo/Q7mIzGzCz24NXnoiIjNZottAfA244WwMziwR+CmwIQk0iIvIBjBjozrlNQPMIzb4J/BdQH4yiRETk3I15DN3McoCPAr8aRdu7zazEzEoaGhrGumoRERkiGDtFfwl83znnG6mhc+4h51yxc644PT09CKsWEZETooKwjGLgaTMDSANuNLMB59xzQVi2iIiM0pgD3Tk368S0mT0GvKAwFxEJvRED3cyeAlYBaWZWCdwLRAM45x4c1+pERGTURgx059wdo12Yc+6uMVUjIiIfmM4UFRHxCAW6iIhHKNBFRDxCgS4i4hEKdBERj1Cgi4h4hAJdRMQjFOgiIh6hQBcR8QgFuoiIRyjQRUQ8QoEuIuIRCnQREY9QoIuIeIQCXUTEIxToIiIeoUAXEfEIBbqIiEco0EVEPEKBLiLiEQp0ERGPUKCLiHiEAl1ExCMU6CIiHqFAFxHxiLAF+qDPhWvVIiKTUu/A4FlfD1ugH6hr5/mdVTinYBcRGclbR1u4+Z9fO2ubsAV6dGQE3356J3f9+3aONXeFqwwRkQmtvaef+9aW8vFfvU5H78BZ24Yt0OdkJHDvLfPZXt7Mdb/YxMObyxgY9IWrHBGRCWd9aS3X/nwTj28t57PL89nw3Y+ctb2Fa8ijuLjYlZSUUHW8m394bjev7KtnUU4S939sEQtzksJSk4jIRFB9vJt715by8p46irISuf9ji1ialwKAme1wzhUP976wBzqAc44/vVvDfWv30NLVx5cun8V3rplHfExkWGoTEQmHQZ/j8dfL+acN+xl0ju9eM4+/uXwW0ZEnB1POFuhRIav0LMyMmxdnc8WcdO5ft5dfbyrjhXdquG/NAq6dnxnu8kRExt3uqlb+5x/e5Z3KVlbOS+fHty1k5vQp57SMCbGFfqptZU388PndHKjr4OqiDO69ZQF5qef2DxMRmQzaevr55csHeXxrOSlTYrj3lvncvHgGZjZs+wk/5DKc/kEfj20p55d/PsCAz/G1VXP47ysLiYvWMIyITH4+n+PZt6v4ybq9NHX2ceclefzt9UUkxUef9X1jCnQzexS4Gah3zi0c5vU7ge8DBrQDX3XO7RrpHzNSoJ9Q29rDj/+0hxfeqSE/dQr3rVnAlRdkjPg+EZGJqrS6lR8+X8qOihaW5iXzf25dOOqDQcYa6B8BOoAnzhDolwF7nXMtZrYauM85d8lIRY020E947WAjP1y7m7KGTq5fkMk/3Dyf3BQNw4jI5HG8q49/2nCA375RQcqUGL6/uojbL8olImL44ZXhjHnIxcwKgBeGC/RT2qUAu51zOSMt81wDHaBvwMfDr5XxL385hM857v5IIV9ZOZupsRNi366IyLB8PsczJcf4x/X7Od7Vx+cuLeC7184bcXhlOKE8yuWLwLogL/M9MVERfG3VHG67MIefrNvHv7xyiGdKjvF31xfx0aU55/QpJyISCm+UNfHjP+3l3apWLi5I4X+vuYT52dPGZV1B20I3syuBfwUud841naHN3cDdAHl5ecsqKio+QMkn7aho5kd/3MOuylaWzEzmhzfPZ1l+ypiWKSISDBVNndz/4j5eKq1lRlIc37+hiFsvzD7j0SujNe5DLma2GPgDsNo5d2A0RX2QIZfh+HyOP7xdxU9f2kd9ey9rlmRzz+oispPjx7xsEZFz1drdzwMbD/HYlnKiIo2vrpzNl64oDNqJkuM65GJmecCzwGdHG+bBFBFhfHxZLjcszOLBVw/z0KYyNuyp5ctXFHL3RwpJjDv3MSoRkXM1MOjjqTeP8os/H6Slq49PLMvle9ddQOa0uJDVMJqjXJ4CVgFpQB1wLxAN4Jx70MweBj4OnBg/GTjTp8dQwdpCP1VlSxc/WbePF96pIXVqDN+8ag7/7ZJ8YqJ0Lw8RCT7nHK/sq+f+dfs4VN/B8sLp/P1N88ftmlST8sSisdp17Dj3r9vLtrJm8lOn8D+uu4CbFs3QjlMRCZqS8mZ++tI+tpe3MCttKj9YXcS18zPHPE5+NudloIP/k/OvBxr46bp97KttZ3FuEvesLuKy2Wnjul4R8bb9te38bP0+/ry3nvTEWL599Vw+dfHM911Ea7yct4F+wmBgx+nPN+ynurWHlfPS+f4NReN26JCIeFNlSxe/ePkgz75dSUJsFF9ZOZsvrChgSkzozoU57wP9hJ7+QZ7YWs4DGw/T2t3PjYuy+M4185iXmRjSOkRkcmnq6OWBjYd5clsFGNx1WQFfXTmblKkxIa9FgX6K1q5+HnmtjEe3lNPZN8CaJdl86+q5zE5PCEs9IjIxNXf28dCmMp7YWk5P/yC3L8vlO9fMC+th0Qr0M2jp7OOhzWU8tqWc3oFBPro0l29dPYf81KlhrUtEwqu5s49/21zG46+X090/yC2L/Rt9czLCv9GnQB9BY0cvD/71ML/ZVsGgz3H7sly+cdUcXfxL5DzTMiTIu/oHuXlxNt++eg5zMibOsKwCfZTq2nr41V8P87s3juJzjtuW5vDVVbM1FCPicce7+nh48xEee90/DHvTohl86+q5E3L/mgL9HNW0dvPrV8t4evtRegd83LhwBl+7cjYLsnXzahEvqW3t4eHNZfzuzaN09Q1y0+IZfOuquVyQNfGC/AQF+gfU2NHLI68d4TdbK+joHeCqogy+fuUcXQBMZJIra+jg16+W8ezblfgcrFmSzVdWzp7QQX6CAn2MWrv7eeL1ch7dcoSWrn6WF07na6vmcMXctHE9I0xEguvdylZ+9eoh1u2uJSYygk9dPJMvX1F4zjdjDicFepB09g7w1JtHeWhTGfXtvRRlJfKlKwpZsyRb14oRmaCcc7x+uIkHXz3M5oONJMZG8dlL8/nCilmkJ8aGu7xzpkAPst6BQZ7fWc0jm4+wv66djMRYPn9ZAXdekkfylNCfaCAip+vpH2Ttzmoe3XKEfbXtpCXE8sXLZ3Hn8jymTeKrsCrQx4lzjk0HG3l4cxmbDzYSHx3JJ4pz+eLls3Qsu0iYNLT38uS2Cn77RgWNHX0UZSXyN5fPYs2SbOKig3NN8nBSoIfAvto2Ht58hOd3VjHgc1z7oUw+d2kBK+akapxdJAT21rTxyGtHWLuzmr5BH1cVZfDFy2dx2Wxv/Q0q0EOovq2HJ7ZW8Ls3j9Lc2Udh2lQ+szyfjy/L/UA3hBWRM+sb8LFhTy1PbqtgW1kz8dGR3L4sly+sKKDQo+ePKNDDoHdgkBffreGJrRW8ffQ48dGR3LY0h88uz9dVHkXGqLKli6fePMrvt1fS2NFLbko8d16Szx0fnun5/VgK9DDbXdXKb7ZW8PyuKnr6fRTnp/CZ5fncsDDLE2N6IqEw6HO8eqCeJ7cdZeP+egy4qiiDO5fn85G56USeJzevUaBPEK1d/fzHjmM8ua2C8qYupsVFcdvSHD5ZPHPcblclMtlVH+/m2bcqeerNY1Qd7yYtIZZPXzyTOy7JI+c8vBm8An2C8fkc28qa+H3JMdbtrqVvwMf8GdP41MUzue3CHJKmaKxdzm89/YOsL63lP3dU8tqhRpyDSwtT+czyfK5bkBmSOwNNVAr0Cay1q5+1u6r4fckxdle1ERMVwfULsvhkcS6XzU47b75Gijjn2HnsOP+xo5I/7qqmvWeAnOR4bl+Wy8cvyiUvdfKczTmeFOiTRGl1K89sP8ZzO6tp7e4nPTGWWxZnc9vSbBblJHnq0CuRE441d/HHd6p59q0qDtV3EBcdweqFM/jEslyWF6bqxu6nUKBPMj39g2zcV89zO6vYuK+BvkEfhWlTufXCHG69MJuCNJ20JJNbQ3svf3qnmrW7qnnr6HEAluWn8Illudy0eAaJk/hMzvGmQJ/EWrv6Wbe7hud2VvHGkWacgyUzk7l1STY3LMwK662wRM5Fa1c/L5XWsHZXNVsPN+FzUJSVyJoLs7llcfakukBWOCnQPaKmtZu1O6t5bmc1e2vaAH+4r16YxeqFWbrcgEw4TR29/GVvPetLa9l0sIH+QUdB6hTWLMnmliXZzJ2AN5CY6BToHlTW0MG63bW8tLuWd6taAf/WzuqFM1i9KIu5GQkac5ewONbcxYY9dawvraWkvBmfg5zkeFYvzGLNhdofNFYKdI+rbOnipUC47zjagnNQmDaVK4syuKoog4sLpuvyvjJunHPsrWnn5T11bNhTS2m1/9tjUVYi183P5LoFWSzInqYQDxIF+nmkvq2H9aW1vLy3nm2Hm+gb9JEQG8Xlc9K4qiiDVRekkzEtLtxlyiTX2t3PlkON/HV/Pa8eaKCurRczuCgvhesXZHLd/CztvB8nCvTzVFffAFsONfHKvnr+ur+emtYeABbmTOPKCzJYMSeNpXnJxEbp8gNyds45SqvbePVAA6/ub2DH0RYGfY7EuCiumJvGqnkZrCpKJyNRGwvjTYEuOOfYV9vOK/vq2bivnreOtuBzEBcdwcUF07lsdhor5qSyIDtJJzMJzjmONHaytayJrYeb2FbWTGNHLwALsqex6oJ0Vl2QwdKZyUSdx2dthoMCXU7T1tPPG2XNbDnUyOuHGzlQ1wFAUnw0ywv9AV9ckEJR1jQF/HnAOcex5m62ljWy9XATW8uaqGvzB3jmtFguLUxlxZw0Vl6grfBwO1ugR4W6GJkYpsVFc+38TK6dnwlAfXsPWw83seVQI1sONbG+tA6AhNgoluYlsyw/heL86SzNS2ZqrH5tJrvegUF2V7XxVkULOypaeOtoC/Xt/gBPS4jhksJULpudyqWFqcxKm6odmpOEttDlNM45Klu62VHRQklFMyXlLeyva8c5iIwwPjQjkSW5ySzJTWZRbhJzMxL0tXsCc85xtLmL3VVt7Ko8zo6KFt6tbKVv0AfAzOnxLMtL4aL8FJYXpuqQ1wlOQy4yZq3d/bx91L81V1Lewu6qVtp7BwCIjYpgQfY0FuUksSg3mQXZ05idnqBDJcPA53OUN3XyblUrpdVtvFvZyu7qVtp7/P9XMZERLMpNYll+ChflJXNRXoqOeppkFOgSdEOD453K1veCo6tvEICoCKMwfSrzMhMpykoMPE4jNyVeF1sKAucctW097K9t52BdBwfq2jlQ38HBuvb3/g9iIiMompHIwpwkFmYnsSgniXlZCTqqaZJToEtIDPocZQ0d7Klp40BdO/tr29lf186x5u732sRHR5KfOoXC9KkUpE6lIG0qhWn+x9SpMfqqf4rWrn7KmzqpaO6iorGT8qYujjR2cLC+472tboC0hFjmZSYwLzOR+TOmsSBnGvMyE8/r64Z71Zh2iprZo8DNQL1zbuEwrxvwf4EbgS7gLufcW2MrWSajyAhjbmbiadfn6Ogd4GAg4A/UdVDe1Mm+mnY2lNYx4Du5QZEYF0VuyhRykuPITo5/7+fEfEZinKeOuBn0ORo7eqk+3k1taw/VrT3UtnZT3dpDZUs3FU2dHO/qf997ZiTFkZ86hdsuzGFeZgJzM/3ffqZP9fZ9NGV0RnO4wmPA/wOeOMPrq4G5gZ9LgF8FHkWAE0fKpLA0L+V9z/cP+qhq6eZIUydHGjopb+qkqqWbypZu3jzSTNuQLVAAM5g+JYbUhBjSEmJJS4gdMh3DtLhoEuOimRYf5X+M8z+GYiy/b8BHZ+8AHb0DtPf4H9u6+2nu7KOps4+mjt73ppsD8/Xtve/7QAP//gj/h1g8Ny2aQUHqVPJTp1CQNpW86VN0D1o5qxED3Tm3ycwKztLkVuAJ5x+72WZmyWY2wzlXE6QaxaOiIyMoCAy3XHnB6a+39/RT09pD1fFuao77t14bA2HY2NHHrsrjNHX00dE7cPqbh4iNimBKTCSxUZHERUcQGxVJbHQEsVH+6ahIw+C94R7/9Il3G/2DPvoGfP7HwPSJx+6+Qdp7B+gb8J21hvjoSKZPPfFhFMO8zESykmLJSoonOymOrKQ4spPiSZ4SrWEn+cCCcUBxDnBsyHxl4LnTAt3M7gbuBsjLywvCqsXLEgNb3PNGuMRqd98gTZ29tPf4t4rbewZo6+k/Od87QHffIL0Dg/QO+Ojt97033d0/SH+PjxO7khzu5LQDB8REGjFREcRERZAQF0VMZATRURHERkYQHxNJQlwUibFRJMRGkRAXTUJsJAmx0STGRb0X4lNidOy+jL+Q/pY55x4CHgL/TtFQrlu8Kz4mktwY3RxBJBiDi1XAzCHzuYHnREQkhIIR6GuBz5nfcqBV4+ciIqE3msMWnwJWAWlmVgncC0QDOOceBF7Ef8jiIfyHLX5hvIoVEZEzG81RLneM8LoDvh60ikRE5APRaWQiIh6hQBcR8QgFuoiIRyjQRUQ8QoEuIuIRCnQREY9QoIuIeIQCXUTEIxToIiIeoUAXEfEIBbqIiEco0EVEPEKBLiLiEQp0ERGPUKCLiHiEAl1ExCMU6CIiHqFAFxHxCAW6iIhHKNBFRDxCgS4i4hEKdBERj1Cgi4h4hAJdRMQjFOgiIh6hQBcR8QgFuoiIRyjQRUQ8QoEuIuIRCnQREY9QoIuIeIQCXUTEIxToIiIeoUAXEfGIUQW6md1gZvvN7JCZ3TPM63lmttHM3jazd8zsxuCXKiIiZzNioJtZJPAAsBqYD9xhZvNPafb3wDPOuaXAp4F/DXahIiJydqPZQv8wcMg5V+ac6wOeBm49pY0DpgWmk4Dq4JUoIiKjETWKNjnAsSHzlcAlp7S5D9hgZt8EpgLXBKU6EREZtWDtFL0DeMw5lwvcCPzGzE5btpndbWYlZlbS0NAQpFWLiAiMLtCrgJlD5nMDzw31ReAZAOfcViAOSDt1Qc65h5xzxc654vT09A9WsYiIDGs0gb4dmGtms8wsBv9Oz7WntDkKXA1gZh/CH+jaBBcRCaERA905NwB8A1gP7MV/NEupmf3IzNYEmn0P+LKZ7QKeAu5yzrnxKlpERE43mp2iOOdeBF485bkfDpneA6wIbmkiInIudKaoiIhHKNBFRDxCgS4i4hEKdBERj1Cgi4h4hAJdRMQjFOgiIh6hQBcR8QgFuoiIRyjQRUQ8QoEuIuIRCnQREY9QoIuIeIQCXUTEIxToIiIeoUAXEfEIBbqIiEco0EVEPEKBLiLiEQp0ERGPUKCLiHiEAl1ExCMU6CIiHqFAFxHxCAW6iIhHKNBFRDxCgS4i4hEKdBERj1Cgi4h4hAJdRMQjFOgiIh6hQBcR8QgFuoiIRyjQRUQ8QoEuIuIRowp0M7vBzPab2SEzu+cMbT5pZnvMrNTMfhfcMkVEZCRRIzUws0jgAeBaoBLYbmZrnXN7hrSZC/wAWOGcazGzjPEqWEREhjeaLfQPA4ecc2XOuT7gaeDWU9p8GXjAOdcC4JyrD26ZIiIyktEEeg5wbMh8ZeC5oeYB88xsi5ltM7MbhluQmd1tZiVmVtLQ0PDBKhYRkWEFa6doFDAXWAXcAfybmSWf2sg595Bzrtg5V5yenh6kVYuICIwu0KuAmUPmcwPPDVUJrHXO9TvnjgAH8Ae8iIiEyGgCfTsw18xmmVkM8Glg7SltnsO/dY6ZpeEfgikLYp0iIjKCEQPdOTcAfANYD+wFnnHOlZrZj8xsTaDZeqDJzPYAG4G/dc41jVfRIiJyOnPOhWXFxcXFrqSkJCzrFhGZrMxsh3OueLjXdKaoiIhHKNBFRDxCgS4i4hEKdBERj1Cgi4h4hAJdRMQjFOgiIh6hQBcR8QgFuoiIRyjQRUQ8QoEuIuIRCnQREY9QoIuIeIQCXUTEIxToIiIeoUAXEfGIsN3gwswagIqwrPz90oDGcBcxQagvTlJfnKS+OGki9EW+cy59uBfCFugThZmVnOnuH+cb9cVJ6ouT1BcnTfS+0JCLiIhHKNBFRDxCgQ4PhbuACUR9cZL64iT1xUkTui/O+zF0ERGv0Ba6iIhHKNBFRDzivAt0M/uEmZWamc/Mznr4kZlFmtnbZvZCqOoLpdH0hZnNNLONZrYn0Pbboa4zFEb7e2FmN5jZfjM7ZGb3hLLGUDGz6Wb2spkdDDymnKHdPwb6bK+Z/bOZWahrHW/n0Bd5ZrYh0Bd7zKwgtJX6nXeBDuwGPgZsGkXbbwN7x7ecsBpNXwwA33POzQeWA183s/mhKC7ERuwLM4sEHgBWA/OBOzzaF/cAf3HOzQX+Eph/HzO7DFgBLAYWAhcDK0NZZIiM2BcBTwA/c859CPgwUB+i+t7nvAt059xe59z+kdqZWS5wE/Dw+FcVHqPpC+dcjXPurcB0O/4PuJxQ1BdKo/y9+DBwyDlX5pzrA54Gbh3/6kLuVuDxwPTjwG3DtHFAHBADxALRQF1IqgutEfsi8KEe5Zx7GcA51+Gc6wpdiSedd4F+Dn4J/B3gC3chE0Xga+RS4I3wVhI2OcCxIfOVePDDDch0ztUEpmuBzFMbOOe2AhuBmsDPeuecF7/NjtgXwDzguJk9Gxii/Vng21zIRYVjpePNzP4MZA3z0v9yzj0/ivffDNQ753aY2apg1xdKY+2LIctJAP4L+I5zri1Y9YVSsPrCC87WF0NnnHPOzE47ttnM5gAfAnIDT71sZlc45zYHvdhxNta+wJ+jV+Df2DkK/B64C3gkuJWOzJOB7py7ZoyLWAGsMbMb8X+tnGZmTzrnPjP26kIrCH2BmUXjD/PfOueeHXtV4RGEvqgCZg6Zzw08N+mcrS/MrM7MZjjnasxsBsOPB38U2Oac6wi8Zx1wKTDpAj0IfVEJ7HTOlQXe8xz+/U0hD3QNuQzDOfcD51yuc64A+DTwymQM82AIHLnwCLDXOffzcNcTZtuBuWY2y8xi8P9urA1zTeNhLfD5wPTngeG+vRwFVppZVOADfyXePIBgNH2xHUg2sxNXQLwK2BOC2k7nnDuvfvBvWVQCvfh34qwPPJ8NvDhM+1XAC+GuO1x9AVyOfwfYO8DOwM+N4a49XL8XwI3AAeAw/qGasNc+Dn2Riv+IjoPAn4HpgeeLgYcD05HAr/GH+B7g5+GuO1x9EZi/NvA38i7wGBATjnp16r+IiEdoyEVExCMU6CIiHqFAFxHxCAW6iIhHKNBFRDxCgS4i4hEKdBERj/j/B+OpHPUONyYAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8osTRq2AcxZG",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "e3bf3dcf-9bfd-4db9-d6c9-e3e36b090121"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.set_xlim([-1.1, -0.9])\n",
        "ax.set_ylim([0.9, 1.1])\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 7,
      "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": "tpcDMwpDcxZJ",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "c8ae08bf-c360-454d-901c-ff931b747cdf"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.set_xlim([-1.01, -0.99])\n",
        "ax.set_ylim([0.99, 1.01])\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZwAAAD8CAYAAABDwhLXAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+WH4yJAAAda0lEQVR4nO3df5BdZZ3n8fenuwMOLEhCWmUTflnGZWNNDPEK6I5FwBESGI0io6FUAoubWZW1piymhKJWtuJQyOgUsywOTBYzkCkn4DAzGp0wIQKpUAtBOkhiUIEGdUlgJBIIg7gwnf7uH+e53U/f9I+b5PZzb+PnVXXqnvN9fpznnO6c7z0/+kQRgZmZ2WTravcAzMzst4MTjpmZFeGEY2ZmRTjhmJlZEU44ZmZWhBOOmZkV0VTCkbRK0nOSto9RLknXS+qXtE3SgqzsnyW9KOl7DW1OlPRganO7pENS/NC03J/KTzjwzTMzs07R7BnOLcCiccoXA3PStBy4MSv7KvCpUdpcC1wXEW8DXgAuSfFLgBdS/LpUz8zMprimEk5EbAJ2j1NlCbA6KpuBoyQdk9reDfxrXlmSgDOBO1LoVuDDWV+3pvk7gPen+mZmNoX1tKifWcDT2fKOFHt2jPpHAy9GxEBD/RF9RcSApD2p/q/yDiQtpzqb4vDDD3/XSSed1ILNMDP77bFly5ZfRURvqfW1KuEUFxErgZUAtVot+vr62jwiM7OpRdIvSq6vVU+p7QSOzZZnp9hYnqe67NYzSv2hvlL5G1N9MzObwlqVcNYCF6an1U4D9kTEWJfTiOqNofcC56fQMuA7WV/L0vz5wD3hN4yamU15TV1Sk7QGWAjMlLQDuAqYBhARNwHrgHOAfuAV4OKs7X3AScC/S20viYj1wBeB2yT9KfBD4BupyTeAv5HUT/WgwtKD3EYzM+sAej2cPPgejpnZ/pO0JSJqpdbnNw2YmVkRTjhmZlaEE46ZmRXhhGNmZkU44ZiZWRFOOGZmVoQTjpmZFeGEY2ZmRTjhmJlZEU44ZmZWhBOOmZkV4YRjZmZFOOGYmVkRTjhmZlaEE46ZmRXhhGNmZkU44ZiZWRFOOGZmVsSECUfSKknPSdo+RrkkXS+pX9I2SQuysmWSnkjTshQ7QtIj2fQrSX+Ryi6StCsr+3SrNtTMzNqrp4k6twA3AKvHKF8MzEnTqcCNwKmSZgBXATUggC2S1kbEC8D8emNJW4B/yPq7PSIu3c/tMDOzDjfhGU5EbAJ2j1NlCbA6KpuBoyQdA5wNbIiI3SnJbAAW5Q0lvR14E3DfgW6AmZlNDa24hzMLeDpb3pFiY8VzS6nOaCKLfTRdmrtD0rEtGJ+ZmXWAdj80sBRYky1/FzghIuZRnRHdOlZDScsl9Unq27Vr1yQP08zMDlYrEs5OID8TmZ1iY8UBkPROoCcittRjEfF8RLyaFm8G3jXWSiNiZUTUIqLW29t78FthZmaTqhUJZy1wYXpa7TRgT0Q8C6wHzpI0XdJ04KwUq7uAkWc3pHs/dR8CftKC8ZmZWQeY8Ck1SWuAhcBMSTuonjybBhARNwHrgHOAfuAV4OJUtlvSl4GHUlcrIiJ/+OBjqV3u85I+BAxQPahw0QFtlZmZdRyNvF8/NdVqtejr62v3MMzMphRJWyKiVmp97X5owMzMfks44ZiZWRFOOGZmVoQTjpmZFeGEY2ZmRTjhmJlZEU44ZmZWhBOOmZkV4YRjZmZFOOGYmVkRTjhmZlaEE46ZmRXhhGNmZkU44ZiZWRFOOGZmVoQTjpmZFeGEY2ZmRTjhmJlZEU44ZmZWRFMJR9IqSc9J2j5GuSRdL6lf0jZJC7KyZZKeSNOyLL5R0mOSHknTm1L8UEm3p74elHTCwW2imZl1gmbPcG4BFo1TvhiYk6blwI0AkmYAVwGnAqcAV0manrX7RETMT9NzKXYJ8EJEvA24Dri2yTGamVkHayrhRMQmYPc4VZYAq6OyGThK0jHA2cCGiNgdES8AGxg/cdX7ujXN3wG8X5KaGaeZmXWuVt3DmQU8nS3vSLGx4nV/nS6n/fcsqQy1iYgBYA9wdOMKJS2X1Cepb9euXS3aDDMzmyztfGjgExHxu8D70vSp/WkcESsjohYRtd7e3kkZoJmZtU6rEs5O4NhseXaKjRUnIuqf/wr8LdU9nhF9SeoB3gg836JxmplZm7Qq4awFLkxPq50G7ImIZ4H1wFmSpqeHBc4C1kvqkTQTQNI04A+A7Vlf9afZzgfuiYho0TjNzKxNepqpJGkNsBCYKWkH1ZNn0wAi4iZgHXAO0A+8AlycynZL+jLwUOpqRYodTpV4pgHdwPeB/53qfAP4G0n9VA8qLD3YjTQzs/bT6+HkoVarRV9fX7uHYWY2pUjaEhG1UuvzmwbMzKwIJxwzMyvCCcfMzIpwwjEzsyKccMzMrAgnHDMzK8IJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7MinHDMzKwIJxwzMyvCCcfMzIpwwjEzsyKccMzMrAgnHDMzK2LChCNplaTnJG0fo1ySrpfUL2mbpAVZ2TJJT6RpWYodJumfJP1U0qOSvpLVv0jSLkmPpOnTrdhIMzNrv2bOcG4BFo1TvhiYk6blwI0AkmYAVwGnAqcAV0mantp8LSJOAk4G/pOkxVl/t0fE/DTdvD8bY2ZmnWvChBMRm4Dd41RZAqyOymbgKEnHAGcDGyJid0S8AGwAFkXEKxFxb+r7NeBhYPbBboiZmXW2VtzDmQU8nS3vSLGx4kMkHQV8ELg7C380XZq7Q9KxY61U0nJJfZL6du3adbDbYGZmk6xtDw1I6gHWANdHxFMp/F3ghIiYR3VGdOtY7SNiZUTUIqLW29s7+QM2M7OD0oqEsxPIz0Rmp9hY8bqVwBMR8Rf1QEQ8HxGvpsWbgXe1YHxmZtYBWpFw1gIXpqfVTgP2RMSzwHrgLEnT08MCZ6UYkv4UeCPwx3lH6d5P3YeAn7RgfGZm1gF6JqogaQ2wEJgpaQfVk2fTACLiJmAdcA7QD7wCXJzKdkv6MvBQ6mpFis0GrgR+CjwsCeCG9ETa5yV9CBigelDhotZsppmZtZsiot1jOGi1Wi36+vraPQwzsylF0paIqJVan980YGZmRTjhmJlZEU44ZmZWhBOOmZkV4YRjZmZFOOGYmVkRTjhmZlaEE46ZmRXhhGNmZkU44ZiZWRFOOGZmVoQTjpmZFeGEY2ZmRTjhmJlZEU44ZmZWhBOOmZkV4YRjZmZFOOGYmVkRTSUcSaskPSdp+xjlknS9pH5J2yQtyMqWSXoiTcuy+Lsk/Si1uV6SUnyGpA2p/gZJ0w92I83MrP16mqx3C3ADsHqM8sXAnDSdCtwInCppBnAVUAMC2CJpbUS8kOr8F+BBYB2wCLgTuBy4OyK+IunytPzF8Qb35K6X+dhfPUBPl+juEl0SPV2iq0t0S3R3p89U3q2qrF5fYqi8q0t0peWu1Fe9zy6x73y9v1S/u4tUXm/HyD6G6pPqD/dXb6f6/FBfII2sp6x+va+RdUfpL2ub8rvZ60ZEEAGDEQymz+HlKhZZ2WBj/cFqOcjbZ/UHh9vtHRzuc+9gMDiY5uvrGqzXoaF+sLfeT95mcPR+I9XfG/X5GKpf72dgcLCaHwwGsn6Gpmx5MIKBvcN9lNZUwomITZJOGKfKEmB1RASwWdJRko4BFgIbImI3gKQNwCJJG4EjI2Jziq8GPkyVcJakdgC3AhuZIOF0SQj4t72D/Obf0g8j7dgROz/90gz9gCIY2DtIpB/63vQLV59/PcuTUJ6gRLUsMTSf14F6AgMxnMwAurpGxlL1ofl6soM88Y2+XurL2XjFcKFSLI1oeL5eLyvP+9KI9tqnL/K+DmoPt04AEfkS6cCYIhHZfN5m+Hd4OB7D86PFhldBZOsaTOuor7d+cA+qf1P5+gaHyqp5sjaDMby+iOGx77OOofnqc3BwZL3REki2ua9LQ18iuxq+QGdfsqsvwdDT1TX0pbi7q4vuLka06ZI4pKf8HZVmz3AmMgt4OlvekWLjxXeMEgd4c0Q8m+b/BXjzaCuUtBxYDnDcccdx+x+95yA3YV/5N4pI3zj2RhCDw0lpxDeelMSGv7Gwb52Gb0VDiS6tb/hb1ATfxmL4288+38SG2jZ80xsc2bax/t7B/KASI/7hD68fYHgsjQeW/KBRnycfwxgHlcEYedCqj2FovnE5fVYHs8ER8frPLj8IR32GkQfwxgNw3r6TNCbf+meeXIdz5r6JdKwEqyyDq17WVY/v+wUhnx/5JWKMLyKNX1zq25LVy/upn33XxyiGv/Dse6aen8k3XAXo2s/6aSDjXUWotx//akXjFY4qAYy4YpL6rh/8lV1hkYavwkgjr6hMxlWJNctb3uW4WpVwJkVEhKRR/+VHxEpgJUCtVpuUo4MkerrV2TvJzGyKaNU51U7g2Gx5doqNF589Shzgl+lyHOnzuRaN0czM2qhVCWctcGF6Wu00YE+6LLYeOEvS9PS02VnA+lT2kqTT0tNpFwLfyfqqP822LIubmdkU1tTVIklrqG7kz5S0g+rJs2kAEXET1VNm5wD9wCvAxalst6QvAw+lrlbUHyAAPkv19NvvUD0scGeKfwX4lqRLgF8AHzvwzTMzs06hTrs5eiBqtVr09fW1exhmZlOKpC0RUSu1Pr9pwMzMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7MinHDMzKwIJxwzMyvCCcfMzIpwwjEzsyKccMzMrAgnHDMzK8IJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7Mimko4khZJekxSv6TLRyk/XtLdkrZJ2ihpdlZ2raTtafp4Fr9P0iNpekbSt1N8oaQ9WdmXWrGhZmbWXj0TVZDUDXwd+ACwA3hI0tqI+HFW7WvA6oi4VdKZwDXApySdCywA5gOHAhsl3RkRL0XE+7J1/D3wnay/+yLiDw5248zMrHM0c4ZzCtAfEU9FxGvAbcCShjpzgXvS/L1Z+VxgU0QMRMSvgW3AoryhpCOBM4FvH9gmmJnZVNBMwpkFPJ0t70ix3FbgvDT/EeAISUen+CJJh0maCZwBHNvQ9sPA3RHxUhZ7j6Stku6U9I7RBiVpuaQ+SX27du1qYjPMzKydWvXQwGXA6ZJ+CJwO7AT2RsRdwDrgfmAN8ACwt6HtBams7mHg+Ih4J/C/GOPMJyJWRkQtImq9vb0t2gwzM5sszSScnYw8K5mdYkMi4pmIOC8iTgauTLEX0+fVETE/Ij4ACHi83i6d9ZwC/FPW10sR8XKaXwdMS/XMzGwKaybhPATMkXSipEOApcDavIKkmZLqfV0BrErx7nRpDUnzgHnAXVnT84HvRcT/y/p6iySl+VPSGJ8/kI0zM7POMeFTahExIOlSYD3QDayKiEclrQD6ImItsBC4RlIAm4DPpebTgPtS/ngJ+GREDGTdLwW+0rDK84HPSBoAfgMsjYg40A00M7POoNfDsbxWq0VfX1+7h2FmNqVI2hIRtVLr85sGzMysCCccMzMrwgnHzMyKcMIxM7MinHDMzKwIJxwzMyvCCcfMzIpwwjEzsyKccMzMrAgnHDMzK8IJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7MinHDMzKwIJxwzMyuiqYQjaZGkxyT1S7p8lPLjJd0taZukjZJmZ2XXStqepo9n8Vsk/UzSI2man+KSdH1a1zZJC1qxoWZm1l4TJhxJ3cDXgcXAXOACSXMbqn0NWB0R84AVwDWp7bnAAmA+cCpwmaQjs3Z/EhHz0/RIii0G5qRpOXDjgW6cmZl1jmbOcE4B+iPiqYh4DbgNWNJQZy5wT5q/NyufC2yKiIGI+DWwDVg0wfqWUCWviIjNwFGSjmlinGZm1sGaSTizgKez5R0pltsKnJfmPwIcIenoFF8k6TBJM4EzgGOzdleny2bXSTp0P9aHpOWS+iT17dq1q4nNMDOzdmrVQwOXAadL+iFwOrAT2BsRdwHrgPuBNcADwN7U5grgJODdwAzgi/uzwohYGRG1iKj19va2ZivMzGzSNJNwdjLyrGR2ig2JiGci4ryIOBm4MsVeTJ9Xp3s0HwAEPJ7iz6bLZq8Cf0116a6p9ZmZ2dTTTMJ5CJgj6URJhwBLgbV5BUkzJdX7ugJYleLd6dIakuYB84C70vIx6VPAh4Htqf1a4ML0tNppwJ6IePYgttHMzDpAz0QVImJA0qXAeqAbWBURj0paAfRFxFpgIXCNpAA2AZ9LzacB91U5hZeAT0bEQCr7pqReqrOeR4D/muLrgHOAfuAV4OKD3kozM2s7RUS7x3DQarVa9PX1tXsYZmZTiqQtEVErtT6/acDMzIpwwjEzsyKccMzMrAgnHDMzK8IJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7MinHDMzKwIJxwzMyvCCcfMzIpwwjEzsyKccMzMrAgnHDMzK8IJx8zMinDCMTOzIppKOJIWSXpMUr+ky0cpP17S3ZK2SdooaXZWdq2k7Wn6eBb/Zupzu6RVkqal+EJJeyQ9kqYvtWJDzcysvSZMOJK6ga8Di4G5wAWS5jZU+xqwOiLmASuAa1Lbc4EFwHzgVOAySUemNt8ETgJ+F/gd4NNZf/dFxPw0rTjQjTMzs87RzBnOKUB/RDwVEa8BtwFLGurMBe5J8/dm5XOBTRExEBG/BrYBiwAiYl0kwA+A2ZiZ2etWMwlnFvB0trwjxXJbgfPS/EeAIyQdneKLJB0maSZwBnBs3jBdSvsU8M9Z+D2Stkq6U9I7mt4aMzPrWD0t6ucy4AZJFwGbgJ3A3oi4S9K7gfuBXcADwN6Gtn9JdRZ0X1p+GDg+Il6WdA7wbWBO4wolLQeWAxx33HEt2gwzM5sszZzh7GTkWcnsFBsSEc9ExHkRcTJwZYq9mD6vTvdiPgAIeLzeTtJVQC/whayvlyLi5TS/DpiWzo5GiIiVEVGLiFpvb29zW2tmZm3TTMJ5CJgj6URJhwBLgbV5BUkzJdX7ugJYleLd6dIakuYB84C70vKngbOBCyJiMOvrLZKU5k9JY3z+wDfRzMw6wYSX1CJiQNKlwHqgG1gVEY9KWgH0RcRaYCFwjaSguqT2udR8GnBfyh8vAZ+MiIFUdhPwC+CBVP4P6Ym084HPSBoAfgMsTQ8WmJnZFKbXw7G8VqtFX19fu4dhZjalSNoSEbVS6/ObBszMrAgnHDMzK8IJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7MinHDMzKwIJxwzMyvCCcfMzIpwwjEzsyKccMzMrAgnHDMzK8IJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKaCrhSFok6TFJ/ZIuH6X8eEl3S9omaaOk2VnZtZK2p+njWfxESQ+mPm+XdEiKH5qW+1P5CQe/mWZm1m4TJhxJ3cDXgcXAXOACSXMbqn0NWB0R84AVwDWp7bnAAmA+cCpwmaQjU5trgesi4m3AC8AlKX4J8EKKX5fqmZnZFNfMGc4pQH9EPBURrwG3AUsa6swF7knz92blc4FNETEQEb8GtgGLJAk4E7gj1bsV+HCaX5KWSeXvT/XNzGwK62mizizg6Wx5B9XZSm4rcB7wP4GPAEdIOjrFr5L058BhwBnAj4GjgRcjYiDrc1bj+iJiQNKeVP9X+QolLQeWp8VXJW1vYlvabSYN29GhPM7WmgrjnApjBI+z1f5DyZU1k3CacRlwg6SLgE3ATmBvRNwl6d3A/cAu4AFgbytWGBErgZUAkvoiotaKfieTx9laHmfrTIUxgsfZapL6Sq6vmUtqO4Fjs+XZKTYkIp6JiPMi4mTgyhR7MX1eHRHzI+IDgIDHgeeBoyT1jNLn0PpS+RtTfTMzm8KaSTgPAXPSU2WHAEuBtXkFSTMl1fu6AliV4t3p0hqS5gHzgLsiIqju9Zyf2iwDvpPm16ZlUvk9qb6ZmU1hEyacdJ/lUmA98BPgWxHxqKQVkj6Uqi0EHpP0OPBm4OoUnwbcJ+nHVJe/Ppndt/ki8AVJ/VT3aL6R4t8Ajk7xLwD7PIY9ipVN1OkEHmdreZytMxXGCB5nqxUdp3zyYGZmJfhNA2ZmVoQTjpmZFdFxCUfSH0p6VNKgpDEfKxzrdTuSLk2xkDQzi0vS9alsm6QFWdkySU+kaVnjuiZpnGO92uc6SY+k6XFJL2Zt9mZla0dbX6ExXiRpVzaWT2dtOmlffkHSj9PP+25Jx2dt9mtfTvI4x3ydk6QrUvwxSWc3Oc4Zkjakn8EGSdPHqDfWa6fOlPRwit+q9DSppD/J9tn2tA9npLKfS/pRKmvqUdtJHOdCSXuysX4pazPua7oKj/MT6XfzR5Lul/TOrE0n7U+pVcfOiOioCfiPVH+MtBGojVGnG3gSeCtwCNUfmM5NZScDJwA/B2Zmbc4B7qR6NPs04MEUnwE8lT6np/npBcb5LWBpmr8J+Mwo7f8bsCpbfrnwvhx1jMBFwA2j9NVR+5LqD40PS/OfAW4/0H05yeP8LHBTml9aHyfVmzq2AocCJ6Z+u5sY558Bl6f5y4FrR6lzLrCB6m/xDqd6GvVIqi+hTwNvT/VWAJeM0v6DVE+Q1pd/Tvbvrcn9OSnjpHqI6Xv787Np0zjfS/r3QfXqsAc7dH+27NjZcWc4EfGTiHhsgmpjvm4nIn4YET8fpc0Sqve9RURspvo7oGOAs4ENEbE7Il6g+mEsmsxxSuO+2id3AbBmorG0eYy5jtqXEXFvRLyS4pup/t7rgE3i/hzrdU5LgNsi4tWI+BnQn/qfSN7fWD+3UV87RfXE6GsR8XiqtwH46CjtD+p3s+A4c828pqvYOCPi/vTvBFrw+zlZ46SFx86OSzhNGu11O7PGqDtRmwPpq1lj9T3eq32A6g3cVN9q78nCb5DUJ2mzpIkO/pM9xo+m0+s7JNX/MLgj92VyCdW3tLrJ2JcHOs4Rr3MC6q9zOtD9+eaIeDbN/wvVnyo02kr1XsPDVF16PoPqD65/BfRo+JLh+Yz8w28kHUZ1YPn7LBzAXZK2qHrtVDMmc5zvkbRV0p2S3pFiHbk/k8bfz07any07drbq1Tb7RdL3gbeMUnRlRHxnlHhbtHmcS4E7IiJ/FdDxEbFT0luBeyT9CPirNozxu8CaiHhV0h9RfZs6c7wG7dyXkj4J1IDTs/A++zIinnw9/G7mCxERkvb524cY47VTqf5S4DpJhwJ3se/rqD4I/J+I2J3Ffi/tzzcBGyT9NCI2tWmcD1P9fF+WdA7wbWDOKGMY0s79KekMqoTze1m4k/Zny7Ql4UTE7x9kFxO+bmc/2uykuuabxzdO8jiHXu2TvtGONv6lwOfyQETsTJ9PSdoInNyOMUZE/qqhm6muHdf7WtjQ18bUpi37UtLvU/1jPD0iXq3HR9uXwJNtGme9zQ6NfJ3TmL/n441T0i8lHRMRz6ZLH8+NVi8irib9kbakv6V67RQR8QDwvhQ/C3h7Q9OlNFxOy/bnc5L+kery1aZ2jDMiXsrarpP0l+nbfMftT1VvYLkZWJz/u+qk/ckBHDvHFPt547TUxPg3ZnuoblCdyPDNv3c01Pk5Ix8aOJeRN75+EMM3vn5GddNrepqfMdnjBP6OkTeQP5u1OymNX1lsOnBomp8JPEETNzwnY4zAMVn7jwCbO3FfkpIIMKehrwPel5M0zs8x8qGBb6X5dzDyoYGnaO6hga8y8ubxn41Spxs4Os3PA7YDPWn5TenzUOBu4Mys3RuB3cDhWexw4Ihs/n5gUbvGSXUGUP+j9lOA/0v1737C40bhcR5HdV/uvQ19ddr+bNmx86CSwmRMVAewHcCrwC+B9Sn+74F1Wb1zqDLzk1SXO+rxz6f2A8AzwM0pLqr/SO5J4EdkBwzgP6cffD9wcaFxvhX4QVrn35EOgKnsfwBfaVjfe9O4t6bPfZ4cKjVGqv9g79E0lnuBkzpxXwLfT/09kqa1B7ovJ3mcb0jL/an8rVmbK1M/j1F9C25mnEdTHTCeSPtgRorXGP738Aaq/yrkx1Q3rOdn7b9K9Rqrx4A/buj7IqoHGWjYrq1pejTf5naMk+pVXPXfz81kB/SxfjZtGufNVP/5ZP33s69D92fLjp1+tY2ZmRUxVZ9SMzOzKcYJx8zMinDCMTOzIpxwzMysCCccMzMrwgnHzMyKcMIxM7Mi/j+iOISgTRDs6gAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BJ0B4YNPcxZL"
      },
      "source": [
        "**Return to slides here.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "38-6wbQocxZL"
      },
      "source": [
        "### Limits"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "S7GSdz-ucxZM",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "3686163d-44c8-4698-da51-be594c100f3a"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.xlim(-5, 10)\n",
        "plt.ylim(-10, 80)\n",
        "plt.axvline(x=5, color='purple', linestyle='--')\n",
        "plt.axhline(y=37, color='purple', linestyle='--')\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 9,
      "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": "markdown",
      "metadata": {
        "id": "AaRfRUYtcxZO"
      },
      "source": [
        "$$\\lim_{x \\to 1} \\frac{x^2 - 1}{x - 1}$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZdDhpdiPcxZO"
      },
      "source": [
        "def my_fxn(my_x):\n",
        "    my_y = (my_x**2 - 1)/(my_x - 1)\n",
        "    return my_y"
      ],
      "execution_count": 10,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5xKbepf445LM",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8c8fac8a-9026-45d6-9476-906cd80111ad"
      },
      "source": [
        "my_fxn(2)"
      ],
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3.0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "umrgvf3BcxZQ"
      },
      "source": [
        "# Uncommenting the following line results in a 'division by zero' error:\n",
        "# my_fxn(1)"
      ],
      "execution_count": 12,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yeGqX8oscxZS",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "041d14a7-d67c-4302-f05d-3527ade11ab2"
      },
      "source": [
        "my_fxn(0.9)"
      ],
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.9"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4unTKbu8cxZU",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "58b57aa7-7550-471e-bb3f-9827307db589"
      },
      "source": [
        "my_fxn(0.999)"
      ],
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.9989999999999712"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YuVfXKa5cxZW",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "4a63cee9-c029-477d-a7cb-fd805a08653b"
      },
      "source": [
        "my_fxn(1.1)"
      ],
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.1"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 15
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "x4eY1qWEcxZY",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "1c1a6d44-0137-44f4-e7cf-cb5a701f5006"
      },
      "source": [
        "my_fxn(1.001)"
      ],
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.0009999999999177"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pjUtPQwAcxZb"
      },
      "source": [
        "y = my_fxn(x)"
      ],
      "execution_count": 17,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Tg4Ow8k3cxZe",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "3a13c87a-7eb0-47da-d959-a1b4bdedb65c"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.xlim(-1, 5)\n",
        "plt.ylim(-1, 5)\n",
        "plt.axvline(x=1, color='purple', linestyle='--')\n",
        "plt.axhline(y=2, color='purple', linestyle='--')\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 18,
      "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": "markdown",
      "metadata": {
        "id": "VQWYjKjVcxZf"
      },
      "source": [
        "**Return to slides here.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hc3PDahDcxZg"
      },
      "source": [
        "$$\\lim_{x \\to 0} \\frac{\\text{sin } x}{x}$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XDLZLagKcxZg"
      },
      "source": [
        "def sin_fxn(my_x):\n",
        "    my_y = np.sin(my_x)/my_x\n",
        "    return my_y"
      ],
      "execution_count": 19,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HU-B5ebBcxZi"
      },
      "source": [
        "# Uncommenting the following line results in a 'division by zero' error:\n",
        "# y = sin_fxn(0)"
      ],
      "execution_count": 20,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YjIqAWnwcxZk",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "f9760170-6248-444d-c8a5-e2a817100298"
      },
      "source": [
        "sin_fxn(0.1)"
      ],
      "execution_count": 21,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.9983341664682815"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "mmTAnugUcxZm",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "b73c423a-cd63-4fdd-aa72-8e0a922e9e28"
      },
      "source": [
        "sin_fxn(0.001)"
      ],
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.9999998333333416"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "UFM_c1t_cxZp",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5f082cc3-a1e1-4f3f-926f-b3a4a18b4490"
      },
      "source": [
        "sin_fxn(-0.1)"
      ],
      "execution_count": 23,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.9983341664682815"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 23
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xrSXdv0ScxZr",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8c66240a-ec06-400f-ddf7-02e0d1c8fc48"
      },
      "source": [
        "sin_fxn(-0.001)"
      ],
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.9999998333333416"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5nhhRmJTcxZs"
      },
      "source": [
        "y = sin_fxn(x)"
      ],
      "execution_count": 25,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "bolWRmXmcxZu",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "f06aec8a-ffcd-4110-da53-41b517ab8b95"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.xlim(-10, 10)\n",
        "plt.ylim(-1, 2)\n",
        "plt.axvline(x=0, color='purple', linestyle='--')\n",
        "plt.axhline(y=1, color='purple', linestyle='--')\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 26,
      "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": "markdown",
      "metadata": {
        "id": "2OI8JVY6cxZw"
      },
      "source": [
        "**Return to slides here.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ly5b30kjcxZx"
      },
      "source": [
        "$$ \\lim_{x \\to \\infty} \\frac{25}{x} $$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-WbP-XFncxZy"
      },
      "source": [
        "def inf_fxn(my_x):\n",
        "    my_y = 25/my_x\n",
        "    return my_y"
      ],
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "C-35M67YcxZ3",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "c6f52cd5-d3a4-4fa5-b15e-79d6aa4e981a"
      },
      "source": [
        "inf_fxn(1e3)"
      ],
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.025"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jZ88FSTfcxZ5",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "e7313985-aa06-4730-eab9-382cc307fd14"
      },
      "source": [
        "inf_fxn(1e6)"
      ],
      "execution_count": 29,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.5e-05"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 29
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PwTJXJWWeDZT"
      },
      "source": [
        "y = inf_fxn(x)"
      ],
      "execution_count": 30,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "AwwKi3RMeBjV",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "e364db18-17e4-4fc6-ea04-b667a5f9965e"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.xlim(-10, 10)\n",
        "plt.ylim(-300, 300)\n",
        "_ = ax.plot(x, y)"
      ],
      "execution_count": 31,
      "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": "2Rjoo4OKbmnB"
      },
      "source": [
        "left_x = x[x<0]\n",
        "right_x = x[x>0]"
      ],
      "execution_count": 32,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "sb9QiH6vcxZ0"
      },
      "source": [
        "left_y = inf_fxn(left_x)\n",
        "right_y = inf_fxn(right_x)"
      ],
      "execution_count": 33,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vleskD0VcxZ7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "outputId": "a1e13c1c-378e-40de-d0b1-9c32569c78ee"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.xlim(-10, 10)\n",
        "plt.ylim(-300, 300)\n",
        "ax.plot(left_x, left_y, c='C0')\n",
        "_ = ax.plot(right_x, right_y, c='C0')"
      ],
      "execution_count": 34,
      "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": "markdown",
      "metadata": {
        "id": "bA2xg33LcxZ8"
      },
      "source": [
        "**Exercises:**\n",
        "\n",
        "Evaluate the limits below using techniques from the slides or above.\n",
        "\n",
        "1. $$ \\lim_{x \\to 0} \\frac{x^2-1}{x-1} $$\n",
        "2. $$ \\lim_{x \\to -5} \\frac{x^2-25}{x+5} $$\n",
        "3. $$ \\lim_{x \\to 4} \\frac{x^2 -2x -8}{x-4} $$\n",
        "4. $$ \\lim_{x \\to -\\infty} \\frac{25}{x} $$\n",
        "5. $$ \\lim_{x \\to 0} \\frac{25}{x} $$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zIhHIcmzLTQQ"
      },
      "source": [
        "FYI: While not necessary for ML nor for this *ML Foundations* curriculum, the `SymPy` [symbolic mathematics library](https://www.sympy.org/en/index.html) includes a `limits()` method. You can read about applying it to evaluate limits of expressions [here](https://www.geeksforgeeks.org/python-sympy-limit-method/)."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AbAn1ChdcxZ9"
      },
      "source": [
        "**Return to slides here.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "e7cnhfOYcxZ9"
      },
      "source": [
        "## Segment 2: Computing Derivatives with Differentiation"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CDn8BeCDcxZ9"
      },
      "source": [
        "Let's bring back our ol' buddy $y = x^2 + 2x + 2$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Dk2EXlqccxZ-"
      },
      "source": [
        "def f(my_x):\n",
        "    my_y = my_x**2 + 2*my_x + 2\n",
        "    return my_y"
      ],
      "execution_count": 35,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "SLmyyYdEcxaA"
      },
      "source": [
        "y = f(x)"
      ],
      "execution_count": 36,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eNIlXWeQcxaC",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "0fef978e-f7ed-40cc-e353-2759a0297b73"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 37,
      "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": "markdown",
      "metadata": {
        "id": "h7C7nPEhcxaD"
      },
      "source": [
        "Let's identify the slope where, say, $x = 2$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DeVa3RrtcxaE"
      },
      "source": [
        "First, let's determine what $y$ is: "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hz5oolNEcxaE",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8dd78728-b4ed-49d6-fc05-f5a65bbc7bed"
      },
      "source": [
        "f(2)"
      ],
      "execution_count": 38,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "10"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 38
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7fkzBRR5cxaF"
      },
      "source": [
        "Cool. Let's call this point $P$, which is located at (2, 10):"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "IOI2oxj2cxaG",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "f60d568f-c926-4a70-d5ca-2783fc50082c"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.scatter(2, 10) # new\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 39,
      "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": "markdown",
      "metadata": {
        "id": "PPCiF6CacxaI"
      },
      "source": [
        "The _delta method_ uses the difference between two points to calculate slope. To illustrate this, let's define another point, $Q$ where, say, $x = 5$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "YAP0LJnxcxaI",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "7d340508-1c97-4d88-bcb9-cf48bdcac524"
      },
      "source": [
        "f(5)"
      ],
      "execution_count": 40,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "37"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 40
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "usfMMyTvcxaK",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "a11a27e7-704c-4819-b920-cd0813f8b86a"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.scatter(2, 10)\n",
        "plt.scatter(5, 37, c = 'orange', zorder=3) # new\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 41,
      "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": "markdown",
      "metadata": {
        "id": "kdony9sucxaM"
      },
      "source": [
        "To find the slope $m$ between points $P$ and $Q$: \n",
        "$$m = \\frac{\\text{change in }y}{\\text{change in }x} = \\frac{\\Delta y}{\\Delta x} = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{37-10}{5-2} = \\frac{27}{3} = 9$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kgoU-_wzcxaM",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "a44c6af7-fe97-4a90-fafa-7228c5b44982"
      },
      "source": [
        "m = (37-10)/(5-2)\n",
        "m"
      ],
      "execution_count": 42,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "9.0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 42
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZWtSUfLVcxaP"
      },
      "source": [
        "To plot the line that passes through $P$ and $Q$, we can rearrange the equation of a line $y = mx + b$ to solve for $b$: \n",
        "$$b = y - mx$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "UuYXrltscxaP",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "4aa81301-d991-4288-d314-2ea828c8f53a"
      },
      "source": [
        "b = 37-m*5\n",
        "b"
      ],
      "execution_count": 43,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-8.0"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 43
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "AwawB1LLcxaR"
      },
      "source": [
        "line_y = m*x + b"
      ],
      "execution_count": 44,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Lkg2Bl7TcxaT",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "89e9c6a9-5e5b-45b7-8091-98a09f1ae21f"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.scatter(2, 10)\n",
        "plt.scatter(5, 37, c='orange', zorder=3)\n",
        "plt.ylim(-5, 150) # new\n",
        "plt.plot(x, line_y, c='orange') # new\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 45,
      "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": "markdown",
      "metadata": {
        "id": "RGEaUT_OcxaV"
      },
      "source": [
        "The closer $Q$ becomes to $P$, the closer the slope $m$ comes to being the true tangent of the point $P$. Let's demonstrate this with another point $Q$ at $x = 2.1$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xWAJUiEbcxaV"
      },
      "source": [
        "Previously, our $\\Delta x$ between $Q$ and $P$ was equal to 3. Now it is much smaller: $$\\Delta x = x_2 - x_1 = 2.1 - 2 = 0.1 $$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-XDEG_o7cxaV",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "6c6c7666-257b-405c-eaac-0cb721957e53"
      },
      "source": [
        "f(2.1)"
      ],
      "execution_count": 46,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "10.61"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 46
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "85n3ApTmcxaX",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "9ac2ffd7-8eaf-40c3-e815-d87ae6b783c3"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.scatter(2, 10)\n",
        "plt.scatter(2.1, 10.61, c = 'orange', zorder=3)\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 47,
      "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": "YlWhoumBcxaY",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5edba389-0ca7-4d46-dd51-076c0a3e2622"
      },
      "source": [
        "m = (10.61-10)/(2.1-2)\n",
        "m"
      ],
      "execution_count": 48,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "6.099999999999989"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 48
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KHMZNdjQcxaa",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "a49f1b35-05c4-402f-fcf2-bb666a3c5a9c"
      },
      "source": [
        "b = 10.61-m*2.1\n",
        "b"
      ],
      "execution_count": 49,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-2.199999999999978"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 49
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "J1-_0DBJcxab"
      },
      "source": [
        "line_y = m*x + b"
      ],
      "execution_count": 50,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eDtjdRYQcxad",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "aab8a88c-df28-4d2e-e5f9-79bcf70942c9"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.scatter(2, 10)\n",
        "plt.scatter(2.1, 10.61, c='orange', zorder=3)\n",
        "plt.ylim(-5, 150)\n",
        "plt.plot(x, line_y, c='orange', zorder=3)\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 51,
      "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": "markdown",
      "metadata": {
        "id": "elxPRnoTcxae"
      },
      "source": [
        "The closer $Q$ becomes to $P$ (i.e., $\\Delta x$ approaches 0), the clearer it becomes that the slope $m$ at point $P$ = (2, 10) is equal to 6."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "srMjU62xcxaf"
      },
      "source": [
        "Let's make $\\Delta x$ extremely small, 0.000001, to illustrate this:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "c_B0wY47cxaf",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "88d748a1-8ae1-4913-f995-82c03f61ab87"
      },
      "source": [
        "delta_x = 0.000001\n",
        "delta_x"
      ],
      "execution_count": 52,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1e-06"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 52
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "g8LSgemacxai"
      },
      "source": [
        "x1 = 2\n",
        "y1 = 10"
      ],
      "execution_count": 53,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gsa2I2Jxcxaj"
      },
      "source": [
        "Rearranging $\\Delta x = x_2 - x_1$, we can calculate $x_2$ for our point $Q$, which is now extremely close to $P$: \n",
        "$$x_2 = x_1 + \\Delta x$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4PgqVCZVcxak",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "9efbfa02-c14b-416a-da25-d7e07c44e1ed"
      },
      "source": [
        "x2 = x1 + delta_x\n",
        "x2"
      ],
      "execution_count": 54,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.000001"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 54
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FoBqhu1zcxan"
      },
      "source": [
        "$y_2$ for our point $Q$ can be obtained with the usual function $f(x)$: \n",
        "$$y_2 = f(x_2)$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9_2mjm1Ycxan",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "8d908d59-7e94-4648-8867-9950ea7b3661"
      },
      "source": [
        "y2 = f(x2)\n",
        "y2"
      ],
      "execution_count": 55,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "10.000006000001001"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 55
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Eekls7AIcxap"
      },
      "source": [
        "To find the slope $m$, we continue to use $$m = \\frac{\\Delta y}{\\Delta x} = \\frac{y_2 - y_1}{x_2 - x_1}$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yQAwKXPUcxap",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "c07ba5e1-2cec-489f-aa71-3a40db79f0b7"
      },
      "source": [
        "m = (y2 - y1)/(x2 - x1)\n",
        "m"
      ],
      "execution_count": 56,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "6.000001000088901"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 56
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IEy7jMVpcxar"
      },
      "source": [
        "Boom! Using the delta method, we've shown that at point $P$, the slope of the curve is 6. "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GJ7bRkBvcxar"
      },
      "source": [
        "**Exercise**: Using the delta method, find the slope of the tangent where $x = -1$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pV8Vpgvdcxar"
      },
      "source": [
        "**Spoiler alert! The solution's below.**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "65FeJSRocxaz"
      },
      "source": [
        "x1 = -1"
      ],
      "execution_count": 57,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fAIGLk8Mcxa0",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "85a2864a-dc7a-45a1-fce9-1fc164ab90fe"
      },
      "source": [
        "y1 = f(x1)\n",
        "y1"
      ],
      "execution_count": 58,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 58
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SZowMdIQcxa1"
      },
      "source": [
        "Point $P$ is located at (-1, 1)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-CPHl783cxa1",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "e74a4347-652e-4e40-a7e1-83f87a51f776"
      },
      "source": [
        "delta_x"
      ],
      "execution_count": 59,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1e-06"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 59
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jzw0J1n9cxa2",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "53c6056e-caec-4c33-fe0b-8369ec1cc0bd"
      },
      "source": [
        "x2 = x1 + delta_x\n",
        "x2"
      ],
      "execution_count": 60,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "-0.999999"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 60
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WaynVKiwcxa3",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "f5c682ad-93bb-49b2-f171-952f521e45c9"
      },
      "source": [
        "y2 = f(x2)\n",
        "y2"
      ],
      "execution_count": 61,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.000000000001"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 61
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rApig2ipcxa4"
      },
      "source": [
        "Quick aside: Pertinent to defining differentiation as an equation, an alternative way to calculate $y_2$ is $f(x + \\Delta x)$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yCW283sZcxa4",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "87008c5f-0184-4262-edc4-b571d533f2c4"
      },
      "source": [
        "y2 = f(x1 + delta_x)\n",
        "y2"
      ],
      "execution_count": 62,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.000000000001"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 62
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "S1Dr3Ankcxa7"
      },
      "source": [
        "Point $Q$ is at (-0.999999, 1.000000000001), extremely close to $P$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZtokXlMncxa7",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "676f0135-60cc-44b1-f1c0-adaf6def3395"
      },
      "source": [
        "m = (y2-y1)/(x2-x1)\n",
        "m"
      ],
      "execution_count": 63,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.0000889005535828e-06"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 63
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "91slg8F4cxa8"
      },
      "source": [
        "Therefore, as $x_2$ becomes infinitely close to $x_1$, it becomes clear that the slope $m$ at $x_1 = -1$ is equal to zero. Let's plot it out: "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "2WToJBOhcxa8",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "722eab24-e3da-4180-9e51-ac003be91f41"
      },
      "source": [
        "b = y2-m*x2\n",
        "b"
      ],
      "execution_count": 64,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1.0000010000889006"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 64
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ah0cXRhrcxa9"
      },
      "source": [
        "line_y = m*x + b"
      ],
      "execution_count": 65,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MjLttDg9cxa-",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "e3827009-0d8f-46a1-81cd-c2425c2b308b"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "plt.axvline(x=0, color='lightgray')\n",
        "plt.axhline(y=0, color='lightgray')\n",
        "plt.scatter(x1, y1)\n",
        "plt.scatter(x2, y2, c='orange', zorder=3)\n",
        "plt.ylim(-5, 150)\n",
        "plt.plot(x, line_y, c='orange', zorder=3)\n",
        "_ = ax.plot(x,y)"
      ],
      "execution_count": 66,
      "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": "markdown",
      "metadata": {
        "id": "hqmMtmpGcxbA"
      },
      "source": [
        "As $Q$ becomes infinitely close to $P$:\n",
        "* $x_2$ - $x_1$ approaches 0\n",
        "* In other words, $\\Delta x$ approaches 0\n",
        "* This can be denoted as $\\Delta x \\to 0$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nsh_7o7kcxbA"
      },
      "source": [
        "Using the delta method, we've derived the definition of differentiation from first principles. The derivative of $y$ (denoted $dy$) with respect to $x$ (denoted $dx$) can be represented as: \n",
        "$$\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UxuWTEKlcxbA"
      },
      "source": [
        "Expanding $\\Delta y$ out to $y_2 - y_1$: \n",
        "$$\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{y_2 - y_1}{\\Delta x}$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "b0H4kFkYcxbB"
      },
      "source": [
        "Finally, replacing $y_1$ with $f(x)$ and replacing $y_2$ with $f(x + \\Delta x)$, we obtain a common representation of differentiation:\n",
        "$$\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) - f(x)}{\\Delta x}$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SuYWQ9e1cxbB"
      },
      "source": [
        "Let's observe the differentiation equation in action: "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Do6EHZfHcxbB"
      },
      "source": [
        "def diff_demo(my_f, my_x, my_delta):\n",
        "    return (my_f(my_x + my_delta) - my_f(my_x)) / my_delta"
      ],
      "execution_count": 67,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hXbAgWvrcxbC"
      },
      "source": [
        "deltas = [1, 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001]"
      ],
      "execution_count": 68,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "iYtlA3S5cxbD",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "ea82a37a-e15a-4343-ccd1-73e459070f08"
      },
      "source": [
        "for delta in deltas:\n",
        "    print(diff_demo(f, 2, delta))"
      ],
      "execution_count": 69,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "7.0\n",
            "6.099999999999994\n",
            "6.009999999999849\n",
            "6.000999999999479\n",
            "6.000100000012054\n",
            "6.000009999951316\n",
            "6.000001000927568\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "CMW45DQmcxbF",
        "scrolled": true,
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "307410ee-4154-4b91-f9a3-ff6263557a38"
      },
      "source": [
        "for delta in deltas:\n",
        "    print(diff_demo(f, -1, delta))"
      ],
      "execution_count": 70,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "1.0\n",
            "0.10000000000000009\n",
            "0.009999999999998899\n",
            "0.001000000000139778\n",
            "9.99999993922529e-05\n",
            "1.000000082740371e-05\n",
            "1.000088900582341e-06\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kh8vawMMcxbG"
      },
      "source": [
        "**Return to slides here.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Ez8pKj_lcxbG"
      },
      "source": [
        "## Segment 3: Automatic Differentiation"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "laWM6cgvcxbG"
      },
      "source": [
        "**TensorFlow** and **PyTorch** are the two most popular automatic differentiation libraries."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RLi7jwLEzaDB"
      },
      "source": [
        "Let's use them to calculate $dy/dx$ at $x = 5$ where: "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qmgeCDWycxbL"
      },
      "source": [
        "$$y = x^2$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MhMxT9oQcxbL"
      },
      "source": [
        "$$ \\frac{dy}{dx} = 2x = 2(5) = 10 $$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Zg-PepdncxbN"
      },
      "source": [
        "### Autodiff with PyTorch"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "n8FJ1OkWcxbN"
      },
      "source": [
        "import torch"
      ],
      "execution_count": 71,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DTYVufujcxbP"
      },
      "source": [
        "x = torch.tensor(5.0)"
      ],
      "execution_count": 72,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "13j9vFZgcxbP",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "982fb7b9-cd4a-4d6d-dcfb-397626d30ca5"
      },
      "source": [
        "x"
      ],
      "execution_count": 73,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(5.)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 73
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PBLvpsCWcxbQ",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "6f50d93d-6e9d-40b2-f66a-2190b2b1e445"
      },
      "source": [
        "x.requires_grad_() # contagiously track gradients through forward pass"
      ],
      "execution_count": 74,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(5., requires_grad=True)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 74
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zper8WoVcxbR"
      },
      "source": [
        "y = x**2"
      ],
      "execution_count": 75,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1Pbdg7zVcxbS"
      },
      "source": [
        "y.backward() # use autodiff"
      ],
      "execution_count": 76,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "UbV3BecacxbT",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "9d3f5270-a22c-40cc-dede-e51d75209c6a"
      },
      "source": [
        "x.grad"
      ],
      "execution_count": 77,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor(10.)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 77
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dyB3FghkcxbG"
      },
      "source": [
        "### Autodiff with TensorFlow"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "4D12um9kcxbH"
      },
      "source": [
        "import tensorflow as tf"
      ],
      "execution_count": 78,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "f7mQuelZcxbJ"
      },
      "source": [
        "x = tf.Variable(5.0)"
      ],
      "execution_count": 79,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "M-WeFuBfcxbK"
      },
      "source": [
        "with tf.GradientTape() as t:\n",
        "    t.watch(x) # track forward pass\n",
        "    y = x**2"
      ],
      "execution_count": 80,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Txjp5NBNcxbL",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "00862329-4eae-4129-f6e7-971c718039f5"
      },
      "source": [
        "t.gradient(y, x) # use autodiff"
      ],
      "execution_count": 81,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<tf.Tensor: shape=(), dtype=float32, numpy=10.0>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 81
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dS--vwVWzaDD"
      },
      "source": [
        "**Return to slides here.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "l599s7vacxbV"
      },
      "source": [
        "As usual, PyTorch feels more intuitive and pythonic than TensorFlow. See the standalone [*Regression in PyTorch*](https://github.com/jonkrohn/ML-foundations/blob/master/notebooks/regression-in-pytorch.ipynb) notebook for an example of autodiff paired with gradient descent in order to fit a simple regression line."
      ]
    }
  ]
}