0023 Calculus - Functions and Limits.ipynb

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   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import sys\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "import seaborn as sns\n",
      "\n",
      "sns.set_palette(['#00A99D', '#F5CA0C', '#B6129F', '#76620C', '#095C57'])\n",
      "sys.version"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "'3.4.1 |Anaconda 2.1.0 (64-bit)| (default, Sep 24 2014, 18:32:42) [MSC v.1600 64 bit (AMD64)]'"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Functions\n",
      "We often express/encode relationships between different real-world quantities as a function. A function assigns in each number in its domain another number. The function 'transformes' the input variable into an output variable by using a rule. The domain consists of all numbers that can be plugged into the function. In the example below, the domain of $f$ is all real numbers $\\mathbb R$ except $0$.\n",
      "\n",
      "$$\n",
      "f(x)=\\frac{1}{x}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Function f(x) as a lambda expression\n",
      "f = lambda x: 1/x\n",
      "\n",
      "f(2), f(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "(0.5, 0.2)"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A more interesting example of a function with piece wise notation for selecting between subrules below. Here al values greater than $4$ are handled differently compared to values less or equal to $4$.\n",
      "\n",
      "$$\n",
      "g(x)=\\begin{cases}\n",
      "    x^2 & x \\gt 4\\\\\n",
      "    2x & x \\leq 4\n",
      "\\end{cases} \n",
      "$$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Function definition g(x)\n",
      "def g(x):\n",
      "    if x > 4:\n",
      "        return x**2\n",
      "    elif x <= 4:\n",
      "        return 2*x\n",
      "\n",
      "g(3), g(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "(6, 25)"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In fact, functions can describe rules in plain English, although it tends to become more ambiguous. In the example below $c(1325)=3$ and $c(1325.5)=100$.\n",
      "\n",
      "$$\n",
      "c(x)=\\begin{cases}\n",
      "    the\\ number\\ of\\ digits \\lt 4 & x=\\mathbb{W}\\\\\n",
      "    100 & otherwise\n",
      "\\end{cases}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Function definition c(x)\n",
      "def c(x):\n",
      "    result = 0\n",
      "    \n",
      "    if float(x).is_integer() == False:\n",
      "        return 100\n",
      "\n",
      "    digits = str(int(x))  # can be a .0 float\n",
      "\n",
      "    for i in range(len(digits)):\n",
      "        if int(digits[i]) < 4:\n",
      "            result += 1\n",
      "\n",
      "    return result\n",
      "\n",
      "c(5678), c(1634508995), c(1634508995.031)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "(0, 3, 100)"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can also combine functions to get completely new aggregated function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Combine functions a, b, and c\n",
      "a = lambda x: 1/(abs(x)+1)\n",
      "b = lambda x: a(x)**2\n",
      "c = lambda x: b(x)+a(x)\n",
      "\n",
      "c(-2), c(0), c(5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "(0.4444444444444444, 2.0, 0.19444444444444442)"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Plot the function c(x) from -2 to 5\n",
      "x_values = np.linspace(-2, 5, 250)\n",
      "y_values = c(x_values)\n",
      "\n",
      "plt.plot(x_values, y_values)\n",
      "plt.fill_between(x_values, y_values, alpha=.5)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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       "text": [
        "<matplotlib.figure.Figure at 0xbc69be0>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Finding the Domain of a Function\n",
      "A function is supposed to be unambiguous, or [well-defined](http://en.wikipedia.org/wiki/Well-defined). If we take the square root function $f(x)=\\sqrt{x}$ we cannot plug in any number x and get an unambiguous result, since $f(25)=5$, but also possible $f(25)=-5$. So the domain of $f$ is from zero (included) to infinity (excluded) $[0, \\infty)$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Examples:\n",
      "$$\n",
      "g(x)=\\sqrt{2x+4}\\\\\n",
      "2x+4 \\ge 0\\\\\n",
      "2x \\ge -4\\\\\n",
      "x \\ge -2\n",
      "$$\n",
      "\n",
      "So the domain of $g$ is $[-2,\\infty)$\n",
      "\n",
      "$$\n",
      "c(x)=\\sqrt{1-x}+\\sqrt{1+x}\\\\\n",
      "1-x\\ge0\\ \\&\\ 1+x\\ge0\\\\\n",
      "1\\ge x\\ \\&\\ x\\ge-1\\\\\n",
      "x\\le1\\ \\&\\ x\\ge-1\n",
      "$$\n",
      "\n",
      "So the domain of $c$ is $[-1, 1]$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Limits\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}