{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Sympy \n",
    "\n",
    "[Sympy](http://sympy.org) je Python biblioteka za simboličku matematiku. Prednost Sympy-ja je što je potpuno napisan u Pythonu (što je katkad i mana). Mi ćemo u nastavku kolegiju obraditi i puno moćniji Sage, koji je [CAS](http://en.wikipedia.org/wiki/Computer_algebra_system) u klasi Mathematice i Maplea. No Sage nije biblioteka u Pythonu, već CAS koji koristi Python kao programski jezik.\n",
    "\n",
    "Korištenje Sympy-ja počinje kao i kod ostalih biblioteka, s importiranjem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Da bi dobili lijepi $\\LaTeX$ izlaz:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from sympy import init_printing\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Koristit ćemo i interaktivne widgete, pa ih ovdje učitavamo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from IPython.display import display\n",
    "from ipywidgets import interact, fixed, interact_manual\n",
    "import ipywidgets as widgets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Simboličke varijable\n",
    "\n",
    "Kako je Sympy samo Python paket, trebamo deklarirati koje simbole ćemo koristiti kao simboličke vatrijable. To možemo napraviti na više načina:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "# ili x,y,z = symbols('x,y,z')\n",
    "# ili from sympy.abc import x,y,z\n",
    "# ili var(x:z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAYCAYAAABdlmuNAAAABHNCSVQICAgIfAhkiAAAAxVJREFU\nWIXt2FuIVWUUwPGfGuakYEkPGdFDeCksnVIiYYQQMygiiiB8O14e6yVEUFAUgkIq86GCnqZ60TBM\n7KHmJegCXYZSvBIZBtmUCmlEQ2TZw/pOs91z9p59Lo2HOH/Y7LPXXutb61t7fbdDjx5Nshlf4lec\nw0HceVUj6lI+wFqRnLuwHz9hTpnRmziLmf91dJPIUlzG+or6s/AXHilSWIa/8UzboXUf+zEikjAR\nc0ViB4oUhnABfR0Jrbu4V3R+SwXdt/E1pjV6uUBU0+sdC62cmgj8/knyByfwvYIEJF7Cj7itLpia\nU1iHKdjbwHhIdOrxnHwKBtO755uJuE0eTj6rXP0Zuz24FasK2t2FNViJ74qcD+OSxpP4EjG5HXfl\n13gxBdNKFda0XlEDeDXZH8X2zLUnyT/GNlcWxKr07oUGbe7Gz7ijzPFMkaQjJTqDyUktPW9Jz3uN\nr84q1LQ39B5L9s/m5JsUr3Cz07svcvJXxB5qJW7KXOMm/gWpgaGSwG7BKE7jqaT/PqaX2JRR016i\ndiT7J3LyekUtLbAbFXukLEXDdnveeLmx6ijjuUwjn+K6CfTrnC4JptE1WKHNA0l3fk7+Df7EtQV2\nZ8Toqcw1md+j6T5jAptzmd/r8XtFXy/j+pysH4/iDZHILIcqtNmP3/BtRjYL83AMfxTY9Rnrb9Pc\nLL7OJyU6a8T2YSTpvtaqs0RN60PvBmNVnWVFkr9VYDdV9OFUM86yE/CIqJaFBboPiS9/DItxEhtw\nezMOO0h9yT+ck9+T7kUVuVBsaapU7L9kE3UZH+FGUbpZBrAPP2C1SOhWMXQnc++UpZ6ofIfrJ/6i\n1fu+dP+wGWf5Jf2ddH8wI1uC93ARD4jKIxI3LOaYFc047RBFFVU/7S8Sq3Se1WI/eKAd59PFsvl5\nep6Xnn8Rwy1PffP2WYv+alqfow6JDudX3SdxXiwyG3PvZotJ/N0W/I1jswj+7k401mU8LfrWkREw\nQxwaD3aisS6iTxx097Vi3OgEfQlfic3asNi4/R+YL/ZVO8XfSD169Oh+/gEOGMhezE4VYQAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "(x + π) "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(pi + x)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "a, b, c = symbols(\"stranica_a, stranica_b, stranica_c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAXCAYAAACLbliwAAAABHNCSVQICAgIfAhkiAAABA9JREFU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\nBhpooIETGH8DzT1MpScBoTIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$stranica_{a}$$"
      ],
      "text/plain": [
       "stranicaₐ"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHcAAAAaCAYAAACNU8MOAAAABHNCSVQICAgIfAhkiAAAA75JREFU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\nRK5CYII=\n",
      "text/latex": [
       "$$\\alpha^{2} + \\beta^{2} + \\gamma^{2}$$"
      ],
      "text/plain": [
       " 2    2    2\n",
       "α  + β  + γ "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a, b, c = symbols(\"alpha, beta, gamma\")\n",
    "a**2+b**2+c**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPIAAAAXCAYAAAAx4RKVAAAABHNCSVQICAgIfAhkiAAABHtJREFU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\nFP8CxCmyi1aQdhoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( x_{0}, \\quad x_{1}, \\quad x_{2}, \\quad x_{3}, \\quad x_{4}\\right )$$"
      ],
      "text/plain": [
       "(x₀, x₁, x₂, x₃, x₄)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "symbols(\"x:5\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Možemo navoditi i dodatne pretpostavke:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "x = Symbol('x', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.is_imaginary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "x = Symbol('x', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAASCAYAAADLw4ffAAAABHNCSVQICAgIfAhkiAAAAa1JREFU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\n",
      "text/latex": [
       "$$\\mathrm{True}$$"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x > 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Možemo kreirati i apstraktne funkcije:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAAVCAYAAAAuJkyQAAAABHNCSVQICAgIfAhkiAAAAkJJREFU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\nb4O2X2FYkxpX3bGKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$f{\\left (0 \\right )}$$"
      ],
      "text/plain": [
       "f(0)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Function('f')\n",
    "f(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAAAhCAYAAAD9JwTTAAAABHNCSVQICAgIfAhkiAAABdlJREFU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\njIyMjIyMjP8f/wKmhIwuLjLcxQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left ( \\frac{d}{d x} g{\\left (x \\right )}, \\quad 0\\right )$$"
      ],
      "text/plain": [
       "⎛d          ⎞\n",
       "⎜──(g(x)), 0⎟\n",
       "⎝dx         ⎠"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = Function('g')(x)\n",
    "g.diff(x), g.diff(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Kompleksni brojevi\n",
    "\n",
    "Imaginarna jedinica se označava s `I`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAAATCAYAAAAEaoRHAAAABHNCSVQICAgIfAhkiAAAARVJREFU\nSInt1LFKA0EQxvGfoljYWFpZCIKIYBAbu8sTaGFjZR7ASnyKVIIvoOB7WFgbU2lho72VWlgoWuwm\nHpvI5dDEC+QPw8A3u3PfwcwyYWgc4RP7MJ0U93CKKzzHgxejdPcDWzFf9yu2BaMvuFMd00tYxVS/\nYh0rsZj5G9ON2Cf7ZZ8u6Xhc4j5+pCrUBT/NjpCariKbMd90hHEy3eoI42L6VRjbQjLlF/Eh3hk0\nzgr6zeNDeIK7zJQwNAgnWEi0GnZwLvxUnnZBvw1hGloF57pk/v/JO4x3G3mx6jPds4SMh+k33ObF\ndKZ3Y8BizNu+F+YJx8Px18Mc1oS5f88XUtM1HCTacgx4NDrT65hVYgknTMAXBic96mSZPnEAAAAA\nSUVORK5CYII=\n",
      "text/latex": [
       "$$1 + i$$"
      ],
      "text/plain": [
       "1 + ⅈ"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+1*I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAASCAYAAABM8m7ZAAAABHNCSVQICAgIfAhkiAAAAJVJREFU\nSInt1KENAjEUh/Ef5IZgAYbAcWEBBKMwADuwwAn2QLAADkVyCoUCgwNBFaLpXUpDyH3JSyte+4n+\nX/kTVtjigBue2JUQH4PsjlNJcY0pRpjHxFVm8T61cZxZnMwgbr0DkVq9E/sZrjMeHc5fcokXfS/q\nys+8cTFyfyDLUDAJ6wxN2F+xzuwEG/EpaL8hHYjyAqjXJ2KDRzE8AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$-1$$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAYCAYAAABtGnqsAAAABHNCSVQICAgIfAhkiAAAAyZJREFU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\nYbD4D1Vq1+0+9QbRAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(i x + 1\\right)^{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "(ⅈ⋅x + 1) "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x * I + 1)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Razlomci\n",
    "\n",
    "Postoje tri numerička tipa: `Real`, `Rational`, `Integer`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "r1 = Rational(4,5)\n",
    "r2 = Rational(5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAfCAYAAAAx6zerAAAABHNCSVQICAgIfAhkiAAAAQZJREFU\nOI3V0z8rxVEcx/HX1S0GYjIpEREPgEWsHoRSuk/gjjKYTEaTxaTuwGb2IESRwXSR8iex6PIz/M5P\nx+l3L4PCp771Pee8z/l+zj9+QKvIsBV3dpWAs6jhKB1I4X7sYgX3X8Hb2MNhmbdqlNcwhqUyMIYn\nsIE5vLSDCy3Ld9+KIsNbyLuhEuABDCUL7OA8VDxBVth4CBHrGXc4LjrKzvmvqziN7FddfNK63E4c\n1zFQTSacYSFqv3aCW+lqsdIbHEUTF2iE9ocqUb6IPpxiEGuYxDRu21Ur1Isb1NvZiPUkf5rj34F7\ngo2rssFNzGMEMzjAI4bL4AYu5X+wiX1Mdaj8L/QO2go2Bq7+2awAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{4}{5}$$"
      ],
      "text/plain": [
       "4/5"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAfCAYAAADnTu3OAAAABHNCSVQICAgIfAhkiAAAAaBJREFU\nSInt1b1rFUEUBfBfYojNk4BISBWbgAQUBBsREzR96tgIEUyTxlorK+2Uh9poEf+AgKKVKUI6CSqI\nWBn8QBAMigZR/IzPYu7CZNgHSwQDwQOXnb337JnDzJ1Z/hHOoYOrWW4cd/AmatN1H/bW5A5jBk+K\nfAtPcQZfmzobwHNMYKlwmONzU4fXMY/Fpg5K9GXjGYzg5GbFcsF9uIAx/PgbwQrT0s79yqKD3zHe\nWfC7rmHl8DYeFrU5rEjOG7uuBNcicnzBB6lVSG0zEuNeDONgcF43mWzJxrY5Ji1DGTebut+m6Iln\nZ0tdbBnO4gE+4R3uYn/BmcVLfMMj6ah2xT2cCpEDuIW32B31KfyULpFRXJGO4HBTxy2sYzLel3Gj\n4KzgYp6ou7Er7Ir6R/TjEBYKzgKONBVs4zHuYw92YLXgrGIoT/SpxyUcjVjP8mW/9pS5OsHLOIHj\neBG59yE8VHAHa1xvQDsIozW1Zemfk+OZYlNyXJN6cCKcVNGK+pR00Z6OCdtS2+ztJlh313VwPuPM\n4hW+S4093k3sPzaPPz8bY9qNF6+bAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{41}{20}$$"
      ],
      "text/plain": [
       "41\n",
       "──\n",
       "20"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1+r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAfCAYAAADnTu3OAAAABHNCSVQICAgIfAhkiAAAAaBJREFU\nSInt1TFIVVEcBvBftjTcIGqppSiiLCpyiKChRFrkEbTonHNa4CaCNOkSRUO11lZBSzoGtrwgGkoQ\nxRZp0MqGKCRFjddwzoPD671373sQQvTBH+455zvf+c495/z//EVcxAssoYJrdTgH8BhfsY45XEoJ\nHcl3hlncxFodsT0oYwdKOIEhrBRxu1rH4XgUbIqOPEKCq3iDp9HVewwKjttyuB5jAl0YiLzBdgU3\n8LqmbxzzaUcrW/4knGqKeRxsV7CM4zV9x/Cx0YQMZ2P8xFj8rjo4h02M4ij68B3XGwl2Cxe6Nh4l\nnBJmhMP5gBsKnvI/jOr+K9vqYtswgrf4IeS7SZxKxm/580p9rhVJX0o3HuACerCFl9ibcBaEJFuN\n0604zvALVxKHs3mTmr3l3XH8W9J3RCgRi3gS24XxDO+wM7Z70Y8zuIxXwj/cV0TsDpZzHGRC5h7O\nE7sr5L7OAgtP42Ezwj18ESpaHnbFhccaEe4Ld7AH+5PI4vhtoQYfxnlMRf6hRoL1cmFFuC6EU10W\nassSnuNkgZ38R4v4DUy4YlG3c1b5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{16}{25}$$"
      ],
      "text/plain": [
       "16\n",
       "──\n",
       "25"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1/r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAMNJREFU\nKJHF0LtKQ1EQheEvEhJQsLXTJo0gWKcK+Bzig6TRVkiRSnyDNIFUp7LLI1gJQjhlQARD4gUkl2bb\nHPa5gZAFmw0z87PWDP+gFNucN/8bakbABYaR+qrIKS2Lc1A2EFMsXhvXOMUnnjHFuihe7Agz9PKg\nW1zhBIe4wCM2+MJlnTUGwXFSB+oE6L0OdBygH6qfvBv+WbZxjqMIcIbX4NTPNu+wRIIH3GOM7wAk\naGWhHkZ4wQd+8YYn3KBRcZV9awfLRDErZz+D5wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denom(r1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Numerička evaluacija\n",
    "\n",
    "SymPy može računati u proizvoljnoj točnosti te ima predefinirane matematičke konstante kao: `pi`, `e` te `oo` za beskonačnost.\n",
    "\n",
    "Funkcija `evalf` ili metoda `N` s ulaznom varijablom `n` računaju izraz na `n` decimala."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAoMAAAASCAYAAAAgy9RBAAAABHNCSVQICAgIfAhkiAAAC/VJREFU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\n",
      "text/latex": [
       "$$3.1415926535897932384626433832795028841971693993751$$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi.evalf(n=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "y = (x + pi)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIQAAAAYCAYAAAA74FWfAAAABHNCSVQICAgIfAhkiAAABQpJREFU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\n",
      "text/latex": [
       "$$\\left(x + 3.1416\\right)^{2}$$"
      ],
      "text/plain": [
       "            2\n",
       "(x + 3.1416) "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(y, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ukoliko želimo zamijeniti varijablu s konkretnim brojem, to možemo učiniti koristeći funkciju `subs`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF4AAAAYCAYAAABz00ofAAAABHNCSVQICAgIfAhkiAAAA4RJREFU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\nrkJggg==\n",
      "text/latex": [
       "$$\\left(1.5 + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "(1.5 + π) "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMYAAAASCAYAAAAANKFkAAAABHNCSVQICAgIfAhkiAAABiVJREFU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\n",
      "text/latex": [
       "$$21.5443823618587$$"
      ],
      "text/plain": [
       "21.5443823618587"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(y.subs(x, 1.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "No `subs` možemo korisiti i općenitije:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAYCAYAAAB+zTpYAAAABHNCSVQICAgIfAhkiAAAA9dJREFU\naIHt2FuIlVUUB/BfZuWoYE0SFfQi3qALliUFSWWaD92wAntroh6SSiSiUIgsoiKqSSKKouhC0MUw\nscCmK6XQvSkppQtMpWkWlRFqZGMPax/m85tvz5w5njGb5g8f+7Bue521915r7c0whrGfYzE+wO/4\nCatw3L/q0RDDK7hcBPV4rMAWtPal9CS2Ysxge/cfwHTsxhV1yo/F3zg/J3AyunHdXrs2+DgcV4pd\n8zV2YBvWiICMaNI8K7BZBK8/HCUW5PScQAd+Q0tTXBtcXCX+zA94GnfgMeH/bizHAU2YZ0ayt6QO\n2efwCQ6sYk4Wu/fhJjhVL9qE82c2oDtLHMXyTj0S3yW7F++Fb0Wsx7cygUu4Vyz2hJzAncmpszP8\ng7AIneI4bkQ7DsZo/Ch20kDQpvEA94Ulye79Jfq5iV7PN62gd3Oizc3M1y7SyNQicWRJaLZI0O9W\nGGjFapyCl0T1PE8EfJPY+a3Jkf0Bf6VxV4m+DQ9iAT4XaaSGqZgvcvir+KzAW5vGOeK/F7EMl4pN\nsiHn0JjkzLoMv0Os4MICbTx24nWxeo/kjPeBNs3fwSPF/8jtuHmJd1uJfoN8xzAu8d4v0R8QPfAs\nkZpqX6+CODkZ6KgwPjvx3ta7aGxIvJ04pkK3P7RpfoDvTjZfzvBvSfxLSvRnEn16Rm+H6HGLyKWX\npWXl0xLj2QrDTyTeRRW8zsRblnGqiK4+HKr6Hq/DZhkLk+56+WZ/ZZKZVKJ/KVLLIRm9TXqnnD5R\nzME70jiqQu6MNPHqjJ3tuL2O+e7DoSXaNFwoFrGrxOusw2YRV4uF/kIU6l8yctPwh+idaxiLiSIv\n/5nRa9ETpwHjaLGqayqMduOrCp0JIjW80eikmpciFiU763BEH3KHJbm1JfrMRH8qozdCxOGbgThV\n7B83i8eKKSWZFpF3uyv028VxGtCxGQTcmHzpxFnimp9DrfX6tEQ/KY25UzNFxGFAp6oY4FoRGy+O\nSg2/iuM0EScU6AtwQfpdPvb7EjeJ/v0jkRZ+7ke+FuByoGovYLku6tQ0vjkQ58p98Avi5jNXT36q\nFZtr8BqeF23IPFEsxonj/RAeFU93+wqX4VbRu79jzxayhi57FsvcDq4VxGNFDt9Y4p+T5lnZsLfi\nRrYF75Xoo0Tr870odltxj7jZzdDTqs1pYM42jefgpfrvRN4q6XSKQI0u0eeL3b8d15d440Rxe7EB\nH3thcXLsxGYYGyK4VsRkZjOMjRKPGquaYWwIoEU84CzvT7AKVS9Du/Cx6A4+1HOn/79ikuiL7xLP\noMMYxjCG0Sz8A6jNB2QCdfE7AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(\\alpha + 2 \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "(α + 2⋅π) "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, a+pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Sympy i Numpy se mogu simultano koristiti:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "import numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "x_vec = numpy.arange(0, 10, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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f47HAgnLtY3yzbwYB+d8O8YhIYCi7KVk6d0xfSdP6kSx4YLBCPsSdtUdvZrOBIUAzM8sA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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f779b45d278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.pyplot import subplots\n",
    "%matplotlib inline\n",
    "fig, ax = subplots()\n",
    "ax.plot(x_vec, y_vec);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Efikasniji kod se postiže funkcijom `lambdify` koja kompajlira Sympy izraz u funkciju:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# prvi argument je lista varijabli funkcije f, u ovom slučaju funckcija je x -> f(x)\n",
    "f = lambdify([x], (x + pi)**2, 'numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "y_vec = f(x_vec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Razlika u brzini izvođenja:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 loops, best of 3: 22.6 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 19.02 times longer than the fastest. This could mean that an intermediate result is being cached.\n",
      "1000000 loops, best of 3: 1.73 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = f(x_vec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Ovdje smo mogli koristiti i [theano](http://deeplearning.net/software/theano/) ili uFuncify."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Pretvaranje stringa u Sympy izraz:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKwAAAAfCAYAAAB6byYDAAAABHNCSVQICAgIfAhkiAAAAzNJREFU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\nKPS0Q3cikUgkEolEIlEZ/wLzMldrj8OttQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x + 1 + \\frac{1}{x + 1} + \\frac{1}{x - 1}$$"
      ],
      "text/plain": [
       "          1       1  \n",
       "x + 1 + ───── + ─────\n",
       "        x + 1   x - 1"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "string = '1/(x-1) + 1/(x+1) + x + 1'\n",
    "izraz = sympify(string)\n",
    "izraz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Jedan interaktivan primjer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "def factorit(n):\n",
    "    return display(Eq(x ** n - 1, factor(x ** n - 1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "`Eq` kreira matematičke jednakosti, tj. jednadžbe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAArIAAAAYCAYAAAAGYR4hAAAABHNCSVQICAgIfAhkiAAACIVJREFU\neJztnVuoJMUZx3/rirpRUYJCDImst0WD6wWNRPDSXlBUvKKw+yCU+iDkRsxTdsVwQLOKdxERfJpV\nUVcURX3QRRQ0XqKiqwu7oFEnGt0km2ii4m52dY8PXx/PnDndM9XVXV1V098PmnNOT52ar/7/b2qq\na6prQFEURVEURVE6yEnAE8AnwDRgCsosBK4FPgS25j+vA3ZuJ0RF6RwrgNeBL4DNwJPA4UEjUmxR\n79JnP2A14t9WYANwctCI7PgV8A6Se18ArwDnBI0oblL0WT0u4GxgFXAx8DXFA9mVwGfAucBi4Dzg\nc+CaViJUlO7xDHAZMgBaCjwG/AP4YcigFCvUu7TZG/gAuBc4DjgAOA04LGRQlpwPnAUcDCwB/gRs\nB44IGVSkpOqzejyGrygeyD6FXLUMsjo/r8xyL/AvYPfQgTTMMchs/RUWZbuuga/27wF8i1xMKtUI\nnb/qnTshvFsFvFSzjpj4DLgydBBjUJ/rkYLHUM1nZ8oGsn8APgIOzf/+GfAx8EufwZRwMXAn8CIy\nrT4N3B8gjmGOBXYAvw8diAUuGj4GbELelMvougY+278fEuMJHupOjdTyV72bS1X/2vZuA3ArsAYZ\nMK0Dfg0saKDuNlkILAO2IZ8MtI367J/UPAY7n2tRNpBdgExf70CmsKeRNbIhWJc//5fARuIZyK4F\n/gssCh2IBS4aHpeXWzmiTNc18Nn+h4G3kI6r66SWv+rdXKr617Z3W/PjeuBoZJnIV8ggJwWWIvF+\ng2gSav2k+uyPVD0GO59rUTaQXYbMwC5DBLwUmcr2Oj1cwinAIcjgOiOOgewSZJB/T+A4bHHVcCPw\nN4rfkNvWwCBxZ47/37QGPtt/K/ApcKCHukNhcPcvpfxtwjtDvVxvGkP7r702vdsGvDx0blUeQwrs\ngqyfPBYZpP0bt5sNDeqzLYZ2X6NNeQyR9MU7DRVYm1d20dD5BUAvf+yGisHeBNwMPASsB+5DOugV\nFetpgueB95B2xMLliL5rSh734UkdXDV8CNgfOL3gsVEaxNZ+aF4DXzlwG7AcOBW5MSEEsfnXdv6C\nmwbqXTEu/rXp3SbkY+dBNubP3zYu/m0D/gq8gbxHrwOu8hplMeqzPVVjT9ljGO0zAEciNxZsYO5V\nzS35k426mimbkf0P86fbVxCuc54hI44Z2TeQKf6yxed1PPFNhr2Gp+dlby54bJQGPtpvaO4KOKO+\nBj5y4A7gn4S/izZm/zL85y9U16BJ7wzuWsXsHdj716Z3DyBr/ga5lvmDnjZowr/ncHuPNKjPthjq\naVXXZ1ePIb6++Ht6zN0TdmX+9xrmz+DuARyVH18Df8x/H7wq6QF/R9ZhLAYuRPZdu8UiYJ9kVB/I\n/g6YqnBcMKa+3ZEX1/ox5XrYe9ImGfYa7pWXfW3ovI0GPZptvyHMQLZIAx85cBeyeP5U4EcDh7cF\n8mPoEad/Ge3kL9hr0LR3hnpa9YjTO7D3ry3vAH6O3AtyNfLx7SXA/5D9O0PQwz72G4ATkffppcjH\nzjuQ7ZqqYlCfbTHU16qHXexNegxx9cVz+AmwBegjM6nTwNPIuoqyJx8+egNl9gRuR9Y0bEFmYlcB\nu1kE7JOM6gPZPsXtLTt6Y+pbkpdbO6ZcFU/aJKOahluQPTEHsdGg6fYbwgxkYb4GPnKgLB+nLGNs\nmlj9y2gnf8Feg6a9M9TTKlbvoJp/bXg3wznA28jNQO8CvyXc3exVYu8h79P/R+7EfxY40/F5Deqz\nLYb6WtnG3qM5jyGuvnge1zPbgb4E/MAxuDr0Ke/Ui46qU+OZ4/81yfHMXjWNw8WTPnFp+AlyZTyI\nrQauOdmnmgY9y3pnyKinge8cqEIff/kSo38Z7eUvxOdfz7LeGL2Dav5NmndtvPZs6RfEpz4X08ef\nVinHnlGjLy77mtjNA79fgSwbaJv3kSsdWz71FYhHtuQ/bWanXTyJTcNFzLZ5BlsNXHPyduRbWAY5\nCvmGk9XIi3OQdZb1ujKsge8cqILPfJkE/+rkL/j3z5dW6l03X3u2qM/2+NQq5dirUuTzHJYjayY2\nISPkuz0GE5KM6le2Ta+R/XEew5/HlIvVkwx7DXdC2vD+0HkbDZpuv8nryWrWA/U1SD0HbIjVv4x2\n8hfC+Weop1Ws3oG9f6l61wSp5t0gGZPts6G+VinHDs30xd9zNrI1w3pgX2Q7ie3MfivXJJFRfSDb\np9mp9QXIOpXNI8rE7EmGvYaH5WUfHTo/TgMf7TeEGcgWaZB6DowjZv8y/OcvhPXP4K5VzN6BvX+p\neleXVPNumIzJ9tlQT6uUY58ho0ZfPHg32wnAI8gOA2cgpl+DLD9oe6/AWFmMvChsDzOmvmngBWAf\n5M7HYSbJk1/kP58fOj9Kg0lqPxRrMMk5kHLsw7jkL6SrQapxF9E17yDt2F1Rn9OK3ZUynzkS+aqy\nTcBBQ4+9jiTCiV5Da4cLkFnSHnJH3zQyPT1zbuS+ZJ5YnscxvG1HrJ64avggsjj7pwWPFWngs/2G\neleRTWuQWg7YEKt/beUvxOGfobpWsXoHbv6l6p0rMcRuUJ9tMbhplXLs0HBffDCyjcHnwBEF/zSz\n+eyrDoHGxhSjlwL0A8S0C6L/XwbOxezJFNU13AtZmP14SZ3DGvhuv6FeJztFsxqklgPjiNm/Kfzn\nL8Tjn6GaVjF7B9X9S9k7F2KJ3aA+22KorlXKsc8wRfN9sRKQFYhxR4cOxBO/YfzVYdc1mPT2p4zm\nb7qod91Afe4GNj4rgdgN2aT4ydCBeGARslXMI2PKdV2DSW5/ymj+pot61w3U525Q6vPC+WWVAHwD\nvAnsinwn9Paw4TTKIci3iNyIrOcpo+saTHL7U0bzN13Uu26gPncDW58VRVEURVEURVEURVEURVEU\nRVEURVEURVEG+Q5gQW7oskzB4QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{18} - 1 = \\left(x - 1\\right) \\left(x + 1\\right) \\left(x^{2} - x + 1\\right) \\left(x^{2} + x + 1\\right) \\left(x^{6} - x^{3} + 1\\right) \\left(x^{6} + x^{3} + 1\\right)$$"
      ],
      "text/plain": [
       " 18                       ⎛ 2        ⎞ ⎛ 2        ⎞ ⎛ 6    3    ⎞ ⎛ 6    3    \n",
       "x   - 1 = (x - 1)⋅(x + 1)⋅⎝x  - x + 1⎠⋅⎝x  + x + 1⎠⋅⎝x  - x  + 1⎠⋅⎝x  + x  + 1\n",
       "\n",
       "⎞\n",
       "⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "factorit(18)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x^{17} - 1 = \\left(x - 1\\right) \\left(x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1\\right)$$"
      ],
      "text/plain": [
       " 17               ⎛ 16    15    14    13    12    11    10    9    8    7    6\n",
       "x   - 1 = (x - 1)⋅⎝x   + x   + x   + x   + x   + x   + x   + x  + x  + x  + x \n",
       "\n",
       "    5    4    3    2        ⎞\n",
       " + x  + x  + x  + x  + x + 1⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(factorit,n=(2,20));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x^{11} - 1 = \\left(x - 1\\right) \\left(x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1\\right)$$"
      ],
      "text/plain": [
       " 11               ⎛ 10    9    8    7    6    5    4    3    2        ⎞\n",
       "x   - 1 = (x - 1)⋅⎝x   + x  + x  + x  + x  + x  + x  + x  + x  + x + 1⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(factorit,n=(1,20,2));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x^{9} - 1 = \\left(x - 1\\right) \\left(x^{2} + x + 1\\right) \\left(x^{6} + x^{3} + 1\\right)$$"
      ],
      "text/plain": [
       " 9               ⎛ 2        ⎞ ⎛ 6    3    ⎞\n",
       "x  - 1 = (x - 1)⋅⎝x  + x + 1⎠⋅⎝x  + x  + 1⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "interact(factorit,n=widgets.widget_int.IntSlider(min=2,max=20,step=1,value=2));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Algebarske manipulacije"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcYAAAAgCAYAAACICNgCAAAABHNCSVQICAgIfAhkiAAAB8RJREFU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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\left(x - 1\\right) \\left(x + 1\\right)} \\left(x \\left(x - 1\\right) \\left(x + 1\\right) + 2 x + \\left(x - 1\\right) \\left(x + 1\\right)\\right)$$"
      ],
      "text/plain": [
       "x⋅(x - 1)⋅(x + 1) + 2⋅x + (x - 1)⋅(x + 1)\n",
       "─────────────────────────────────────────\n",
       "             (x - 1)⋅(x + 1)             "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(izraz)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAhCAYAAAB3JF/XAAAABHNCSVQICAgIfAhkiAAABQRJREFU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\n",
      "text/latex": [
       "$$\\frac{1}{x^{2} - 1} \\left(x^{3} + x^{2} + x - 1\\right)$$"
      ],
      "text/plain": [
       " 3    2        \n",
       "x  + x  + x - 1\n",
       "───────────────\n",
       "      2        \n",
       "     x  - 1    "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cancel(together(izraz))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAVCAYAAAD7GFqYAAAABHNCSVQICAgIfAhkiAAABKdJREFU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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x+1)*(x+2)*(x+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALcAAAAWCAYAAACCLg3UAAAABHNCSVQICAgIfAhkiAAAA9BJREFU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\nE/8DExTYlykimo8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x^{3} + 6 x^{2} + 11 x + 6$$"
      ],
      "text/plain": [
       " 3      2           \n",
       "x  + 6⋅x  + 11⋅x + 6"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "`expand` prima dodatne argumente. Npr. `trig=True`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAXCAYAAABtR5P0AAAABHNCSVQICAgIfAhkiAAABFRJREFU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\nDI8Lah5AH7BX7aVsAAMYwABOFfwHDSIilmEDLAwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (\\alpha + \\beta \\right )}$$"
      ],
      "text/plain": [
       "sin(α + β)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQgAAAAXCAYAAAD3AdEFAAAABHNCSVQICAgIfAhkiAAABrlJREFU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\nZMiQIX34H6Jn1trRAXrgAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sin{\\left (\\alpha \\right )} \\cos{\\left (\\beta \\right )} + \\sin{\\left (\\beta \\right )} \\cos{\\left (\\alpha \\right )}$$"
      ],
      "text/plain": [
       "sin(α)⋅cos(β) + sin(β)⋅cos(α)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(sin(a+b), trig=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFU\nKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSx\noPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA\n6XoX52TB9a4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(sin(a)**2 + cos(a)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC0AAAAgCAYAAACGhPFEAAAABHNCSVQICAgIfAhkiAAAAmJJREFU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\n",
      "text/latex": [
       "$$\\frac{1}{\\tan{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "  1   \n",
       "──────\n",
       "tan(x)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(cos(x)/sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "f1 = 1/((a+1)*(a+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAfCAYAAADZa4KAAAAABHNCSVQICAgIfAhkiAAAAndJREFU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\n8BkZhQ42rQIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{1}{\\alpha + 2} + \\frac{1}{\\alpha + 1}$$"
      ],
      "text/plain": [
       "    1       1  \n",
       "- ───── + ─────\n",
       "  α + 2   α + 1"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apart(f1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "f2 = 1/(a+2) + 1/(a+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAgCAYAAAB0OKThAAAABHNCSVQICAgIfAhkiAAAA3NJREFU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\nLX6pOnG5eMG9r0tenXXTX3tDQ0NDVfwHLFs4OCxJMwUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{2 \\alpha + 5}{\\left(\\alpha + 2\\right) \\left(\\alpha + 3\\right)}$$"
      ],
      "text/plain": [
       "    2⋅α + 5    \n",
       "───────────────\n",
       "(α + 2)⋅(α + 3)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Analiza"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Deriviranje"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAYCAYAAABdlmuNAAAABHNCSVQICAgIfAhkiAAAAxVJREFU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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "(x + π) "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFcAAAAYCAYAAACPxmHVAAAABHNCSVQICAgIfAhkiAAAA59JREFU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\n/E5so4022mgV/wCCn/ML2LDqqgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$4 \\left(x + \\pi\\right)^{3}$$"
      ],
      "text/plain": [
       "         3\n",
       "4⋅(x + π) "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Više derivacije:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAYCAYAAAAMAljuAAAABHNCSVQICAgIfAhkiAAABBVJREFU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\n55JHxd8LQxjCEIbwf8a/qTUhO8pgme8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$12 \\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAYCAYAAAAMAljuAAAABHNCSVQICAgIfAhkiAAABBVJREFU\naIHt2WuIVVUUB/CfJj5SsBdREX0IH0Evy4gCByYbFYroJZifnNIPRQ8ieikIFoIR0oMIoyCmF2gY\nJhaUQUYq9G5KzOjFFJaiUlmhhOb0Ye3LXE/3nHvuzHUu5vzhcM5de629/2evvdda+1yGMIRjGAvx\nMf7AbqzDeS1ldIzjbdwsnHA+1mAnTmolqQpexC6MbTWRJmIqejG/pP44/INr8hRm4ylsFNuqFy/n\n6J6MBcLL32E/9mJTIjS8gMglOIR7ShI/mrAGO8Rk18PpYo6n5Sl0J4U/sU2xQ25N7b/gFSzD8/g9\nyVdjWI7t+qQ3pgTpow2XivdfVEL3VXyO4/IUrsBEMZHtih0yXWy17E44DT8l2xtr2E0Su+PZEoSb\ngc7EpX2QxiMW848KJhqPicV8drUwO5kb8K14gXp4V1QJhzLynXgmPbfXsLtFOHxVTr/r0/g3ZOTD\n0JXaHinBrxm4Oo1X5ppSZbcSZ6Ejp9/HMVcs6h+qG0Y0j/thOJDuB2u0dYhE9kGO7X34DEuxNunC\ncszDc3iwaUyLsRcrcBu2ijBcwTmYI3LmO/iyqm1zus8QlVU1nsRNYrF+3QiZdsUhKw8jsCXZzsq0\njRVO2lKnj65k35l+L0q/VykuFmqh08BC1vXJfmlGfr/8imp8avsoI39aFEvTRWivXGUKgH47ZHmy\ne7NG26TUtr5OH2eKqq0HdySbtzCyQS4M3CEPJfvZGfnKJJ+aY7dfhO9q5IW7JWWItGvcIXclm21q\nH3Yu17fS62GZPsKbcXwJmx75L13r6irR59qkOzEj/0aE5lE5dj+rHbIL0cwccruIj1/hSvxaQ2d/\nuo8u0d/uquf52FfC5gmckJFNwbV4QTisGt0l+pyCv8RZq4JxmCDyyt85dmP0vW9T0K78Drk76W7B\nqQV6ZyS9TXX6myuqtx1Jf0UJDnno1P+QdaK+HVqNtiR/KcduuOD/faMDNpoga+EBUcZ1i3PMrgLd\nHWLlTy7QuUqs5q24QFQiC0RVM9iolLJfZOQXp3veDpssyvQyO/AwDNQhi8WZ4FMRpvbU0e/F+zhF\nbPksponScjtmCuctFqF1sM4e1ag4JDuxlS+0edXiZem+odEBsznkunQR5RiRiLvS8x7cm57n4WFx\nTtgoEnoWPf6bOF8TJ/hZDo/LF+INUfvPELuJcNAnIg+0pbEGC3k7pFKwnCty5vZM+0wxL2sHSmCJ\n4qqkpwHdXrxXY4yRohz8sEo2Icl+E2Eqi47UX95hsgid+p9DusXEZiu8OWJx7tO3QCsYL5L56/0Y\nr2VYKCbpolYTOQK4U7xbW6uJNILR4uPbulYTaTLGiA+Gq+sp5qHoa+SRxEHxvWqUyA8HitWPGkwU\n55JHxd8LQxjCEIbwf8a/qTUhO8pgme8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$12 \\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from sympy.abc import x,y,z\n",
    "# ili npr. symbols ('x:z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "f = sin(x*y) + cos(y*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Želimo izračunati $$\\frac{\\partial^3f}{\\partial  x \\partial y^2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPcAAAAXCAYAAADXyNnRAAAABHNCSVQICAgIfAhkiAAABnNJREFU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\ns1MAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- x \\left(x y \\cos{\\left (x y \\right )} + 2 \\sin{\\left (x y \\right )}\\right)$$"
      ],
      "text/plain": [
       "-x⋅(x⋅y⋅cos(x⋅y) + 2⋅sin(x⋅y))"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x, 1, y, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$- \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def deriv(f):\n",
    "    display(diff(f,x))\n",
    "interact_manual(deriv, f='x');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integracija"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAXCAYAAABqKY+kAAAABHNCSVQICAgIfAhkiAAABe5JREFU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\n",
      "text/latex": [
       "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$x \\cos{\\left (y z \\right )} + \\begin{cases} 0 & \\text{for}\\: y = 0 \\\\- \\frac{1}{y} \\cos{\\left (x y \\right )} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "             ⎛⎧    0       for y = 0⎞\n",
       "             ⎜⎪                     ⎟\n",
       "x⋅cos(y⋅z) + ⎜⎨-cos(x⋅y)            ⎟\n",
       "             ⎜⎪──────────  otherwise⎟\n",
       "             ⎝⎩    y                ⎠"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definitni integrali:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAXCAYAAAB6ZQM9AAAABHNCSVQICAgIfAhkiAAABGJJREFU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\n",
      "text/latex": [
       "$$2 \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "2⋅cos(y⋅z)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, (x, -1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nepravi integrali:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAWCAYAAAChWZ5EAAAABHNCSVQICAgIfAhkiAAAAYhJREFU\nSInt1b9LV1EYBvCPkoopoo7VUmAELeGsjiIEUYsSgUsg5BaYf0FbNDuIg/QLpIa2KGipVrGipUFo\nCiKKfliWqA3nXPl6uJfOrS8I4gN3uM85z/s+7/ueey4HqIcx/MJ2E59auIvp/yzin9GDz+hvZtDW\nGnvH8RSfmmmgDp7h3F4lP4EPaKtYPyv/0J1pFB5KAnUJs36f8JO4h40KA18whyt4g/sNa6cwged4\ngldlAU7GBGu4k6y1YBWDFckLXBAqvJ7ws5G/XCYqOvAVN3EU59GN73FtGD+w/BcDRWtXEn6wgi/F\nVHQ72cAt4FqG9mHUDiT8W2F0HTkGerGOx/G9U/j2j2Ro3+GbMLIC3djC65zkBR5gMya9hEcZmj6h\n+hcJPxz5W1XCsovoduQvCqNYzDBQzP9lwteaf4F24bZbFdrfmaG5KlQ6lfDzkR+tEpZ14DeWcFwY\nx88MA1UdKP4bp3EsI84OhgTnI5n7V4RzczjhJ/BR+Ixn6hhowQ27T/QB9if+ACoTZalac/bRAAAA\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\pi}$$"
      ],
      "text/plain": [
       "√π"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Sume i produkti"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n = Symbol(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAA7CAYAAAAw23kDAAAABHNCSVQICAgIfAhkiAAAA7NJREFU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\n",
      "text/latex": [
       "$$\\sum_{n=1}^{10} \\frac{1}{n^{2}}$$"
      ],
      "text/plain": [
       "  10    \n",
       " ____   \n",
       " ╲      \n",
       "  ╲   1 \n",
       "   ╲  ──\n",
       "   ╱   2\n",
       "  ╱   n \n",
       " ╱      \n",
       " ‾‾‾‾   \n",
       "n = 1   "
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n, 1, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMcAAAASCAYAAADv9spaAAAABHNCSVQICAgIfAhkiAAABaBJREFU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\nMcYYY4wxxhiN8X+SPoDo8RLHZQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.54976773116654$$"
      ],
      "text/plain": [
       "1.54976773116654"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n,1, 10)).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMYAAAASCAYAAAAANKFkAAAABHNCSVQICAgIfAhkiAAABl9JREFU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\n",
      "text/latex": [
       "$$1.64493406684823$$"
      ],
      "text/plain": [
       "1.64493406684823"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n, 1, oo)).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAA6CAYAAAAZW7HfAAAABHNCSVQICAgIfAhkiAAAAs9JREFU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\n",
      "text/latex": [
       "$$\\prod_{n=1}^{10} n$$"
      ],
      "text/plain": [
       "  10   \n",
       "┬───┬  \n",
       "│   │ n\n",
       "│   │  \n",
       "n = 1  "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Product(n, (n, 1, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Limesi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFU\nKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSx\noPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA\n6XoX52TB9a4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAXCAYAAABqKY+kAAAABHNCSVQICAgIfAhkiAAABe5JREFU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\n",
      "text/latex": [
       "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAXCAYAAACRUrg+AAAABHNCSVQICAgIfAhkiAAABA1JREFU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\nH/3ox7HAP8idRXuXL4K/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$y \\cos{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$ \\frac{\\partial f(x,y)}{\\partial x} = \\lim_{h\\to 0}\\frac{f(x+h,y)-f(x,y)}{h}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "h = Symbol(\"h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAXCAYAAACRUrg+AAAABHNCSVQICAgIfAhkiAAABA1JREFU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\nH/3ox7HAP8idRXuXL4K/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$y \\cos{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((f.subs(x, x+h) - f)/h, h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAALCAYAAACZIGYHAAAABHNCSVQICAgIfAhkiAAAANNJREFU\nKJG10TFLglEUh/GfurU0SZvQGCI1OBcZ+BFc6hMI+jGaXFzEKQjaWtqiLRChpKU9hcBBMGgRRRBt\neN9BLu8LDXq289x7Hs79X/ZUN/jGAg84iPklPvCLZxTSBCX8oIYqhnhBEXN0cYF2LMwkSe7Q3OqL\nWGKCx+DuGypJkgFOA9bBCscBb6EB2eAgg03ARsghH/CNlOfco77VH2KMKV6DoXdcJUnO4oEaztFD\nXxT4XJRZGbf4TNsErkVfPMMTjmJewRfWouxO0gT/qTDH3dQfdTgpAZFH3n4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x, x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAALCAYAAAAa5F88AAAABHNCSVQICAgIfAhkiAAAAO5JREFU\nOI3d069Lw0EcxvHX1GQxDZtglCEalhUn+CesuL9A0D/DZLGIaTBYs9jEJoigYrFvuoGIoGCR6UDc\nwveEL4cTDYfgAxee53583tx9jn+sGrp4RROTIV/BFZ5xhJmUEPN4QhVraOMYJfSwj2XsBqhCKpA6\ntnK+hD4ecBCtPUdl1EEdDH4xmtH+SyxE2R7eMRvlO9j8NBPRZBtvoyi/0H3kCwEwrxuMo4jbXD6Q\n8Gka2Mj5KdzhESdR4QuspgJZDEWrWMIpzmRN3JP1UBnbuJbwRmBd9n1fcIjpkFfQwoesl+ZSQvxE\nY38N8K2Gk+85FrKxPB4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x, x, 0, dir=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### (Taylorovi) redovi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAa4AAAAYCAYAAACiERlGAAAABHNCSVQICAgIfAhkiAAAClNJREFU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\nBwcHBwcHBwcHBwcHh/6H/wF/rrMiV/JWpgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "         2    3    4     5        \n",
       "        x    x    x     x     ⎛ 6⎞\n",
       "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n",
       "        2    6    24   120        "
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Rastav oko $x=1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\mathcal{O}\\left(\\left(x - 1\\right)^{6}; x\\rightarrow1\\right)$$"
      ],
      "text/plain": [
       "                         2            3            4            5             \n",
       "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       6\n",
       "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + O⎝(x - 1) \n",
       "                    2            6            24          120                 \n",
       "\n",
       "        \n",
       "       ⎞\n",
       "; x → 1⎠\n",
       "        "
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\frac{e}{720} \\left(x - 1\\right)^{6} + \\frac{e}{5040} \\left(x - 1\\right)^{7} + \\frac{e}{40320} \\left(x - 1\\right)^{8} + \\frac{e}{362880} \\left(x - 1\\right)^{9} + \\mathcal{O}\\left(\\left(x - 1\\right)^{10}; x\\rightarrow1\\right)$$"
      ],
      "text/plain": [
       "                         2            3            4            5            6\n",
       "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1) \n",
       "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + ──────────\n",
       "                    2            6            24          120          720    \n",
       "\n",
       "            7            8            9                      \n",
       "   ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       10       ⎞\n",
       " + ────────── + ────────── + ────────── + O⎝(x - 1)  ; x → 1⎠\n",
       "      5040        40320        362880                        "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 1, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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zUds7HS1GS2Izwr/M10n3cKe3JR7zgVeJdj8y/Ufrf0fghznlT+Tz6ME52pd2JnLGzxvX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\n",
      "text/latex": [
       "$$- \\frac{1}{x - \\frac{\\pi}{2}} - \\frac{\\pi}{6} + \\frac{1}{45} \\left(x - \\frac{\\pi}{2}\\right)^{3} + \\frac{2}{945} \\left(x - \\frac{\\pi}{2}\\right)^{5} + \\frac{x}{3} + \\mathcal{O}\\left(\\left(x - \\frac{\\pi}{2}\\right)^{6}; x\\rightarrow\\frac{\\pi}{2}\\right)$$"
      ],
      "text/plain": [
       "                     3            5                         \n",
       "              ⎛    π⎞      ⎛    π⎞                          \n",
       "              ⎜x - ─⎟    2⋅⎜x - ─⎟         ⎛       6       ⎞\n",
       "    1     π   ⎝    2⎠      ⎝    2⎠    x    ⎜⎛    π⎞       π⎟\n",
       "- ───── - ─ + ──────── + ────────── + ─ + O⎜⎜x - ─⎟ ; x → ─⎟\n",
       "      π   6      45         945       3    ⎝⎝    2⎠       2⎠\n",
       "  x - ─                                                     \n",
       "      2                                                     "
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tan(x).series(x,pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOwAAAAYCAYAAAD50BEbAAAABHNCSVQICAgIfAhkiAAABqhJREFU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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{5}\\right)$$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 5⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = cos(x).series(x, 0, 5)\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAYCAYAAAB6OOplAAAABHNCSVQICAgIfAhkiAAAA4dJREFU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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2 = sin(x).series(x, 0, 2)\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAYCAYAAAB6OOplAAAABHNCSVQICAgIfAhkiAAAA4dJREFU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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(s1 * s2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "S metodom `removeO` se možemo riješiti $\\mathcal{O}$ dijela:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAYCAYAAAB9VvY1AAAABHNCSVQICAgIfAhkiAAABA5JREFU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\nSkpKSqYk/wH5zLKUlmvXWwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$$"
      ],
      "text/plain": [
       " 5    3    \n",
       "x    x     \n",
       "── - ── + x\n",
       "24   2     "
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(s1.removeO() * s2.removeO())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Ali oprezno s time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAYCAYAAADzqehgAAAABHNCSVQICAgIfAhkiAAAB8dJREFU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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "       3      5        \n",
       "    2⋅x    2⋅x     ⎛ 6⎞\n",
       "x - ──── + ──── + O⎝x ⎠\n",
       "     3      15         "
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cos(x)*sin(x)).series(x, 0, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Reziduumi:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAMJJREFU\nKJHF0j9LQlEYB+DH6BNYhJ/HxcmpD9CgQxCImxAIbi5u7eLSJji5NzQGDRItDtHQ3J0aSnS4LxJy\ntBMN/eBw4XCe+77nD3/MCdqYYYkPFLhHC0cpdIk13nCLIcZ4j/kpKruojmbijzW8Bjz/TevXgW7s\n6zORz/h+5VY5xiIqNXLRKMA8F3QCPKOaA64CPClP8Md0AyxwlgN6AR5xmgP6AR4c2MP3Z3GBCVbK\nSywS619izTaDqHJo3OW0+8/ZALBUMPfsVQVVAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "residue(2/sin(x), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Linearna algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Matrice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n",
    "b1, b2 = symbols(\"b1, b2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡m₁₁  m₁₂⎤\n",
       "⎢        ⎥\n",
       "⎣m₂₁  m₂₂⎦"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[m11, m12],[m21, m22]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⎤\n",
       "⎢  ⎥\n",
       "⎣b₂⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix([[b1], [b2]])\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2                               ⎤\n",
       "⎢ m₁₁  + m₁₂⋅m₂₁    m₁₁⋅m₁₂ + m₁₂⋅m₂₂⎥\n",
       "⎢                                    ⎥\n",
       "⎢                                 2  ⎥\n",
       "⎣m₁₁⋅m₂₁ + m₂₁⋅m₂₂   m₁₂⋅m₂₁ + m₂₂   ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⋅m₁₁ + b₂⋅m₁₂⎤\n",
       "⎢               ⎥\n",
       "⎣b₁⋅m₂₁ + b₂⋅m₂₂⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALEAAAATCAYAAADf/ewgAAAABHNCSVQICAgIfAhkiAAAAsNJREFU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\naE7MZEJfNTU1NTU1NTU1NUuK9718w9lp8jSlAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$m_{11} m_{22} - m_{12} m_{21}$$"
      ],
      "text/plain": [
       "m₁₁⋅m₂₂ - m₁₂⋅m₂₁"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def funkcija(A,f):\n",
    "    return display(getattr(A,f)())\n",
    "interact(funkcija,A = fixed(A), f=['det','inv','adjoint','charpoly']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Rješavanje jednadžbi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAVCAYAAADCUymGAAAABHNCSVQICAgIfAhkiAAAAVtJREFU\nWIXtmLFKw0AYx3+Ig4ui4NDNzVVXJwPOQgdHh66Ogg8QV5/CQZ/ABzA+gd3qUsjk5KQgDkod7opp\ne6T9knx3bbwfHAn00u/Hn+Ny+SDijQQYFcZLUJvVYpfJ7EbjH9Ydk5+ADHhTEDkDjoFD4ADYBO6B\nc4VadZG4fgLX9r4H7Ln+MMEknzaqOUnf1vgABvb+TrFeHaq6ZhRW8JqGWQmXwD6wBVx4ri2lEVfX\nFqHJo+d6dWjE1fcK/nfEgJWJASsjCThn9qxXNpb1dOAVyUtuCHwJ5r8KXVqJJOATNYsWE/dgZVYh\n4FvMnt4Lq1EN3x8aXTsAOvZ6hAkRTP/jauqZ8SL4VjWbpYprKQn6vYiU8pNH7njmGXgHdhS9XKTI\nXWGqF1EkQT9gKdvAD3ATWkRAxpyAl6kffIo5GnbmTQzMQv3gnL+eJuj0g6U8ABuhJRag2A+ORFrE\nLzLCY61VgewcAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ -1, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAnoAAAAsCAYAAADy4hXPAAAABHNCSVQICAgIfAhkiAAACiJJREFU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\n",
      "text/latex": [
       "$$\\left [ - i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad - \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}\\right ]$$"
      ],
      "text/plain": [
       "⎡       __________        __________       ________      ________⎤\n",
       "⎢      ╱   1   √5        ╱   1   √5       ╱ 1   √5      ╱ 1   √5 ⎥\n",
       "⎢-ⅈ⋅  ╱  - ─ + ── , ⅈ⋅  ╱  - ─ + ── , -  ╱  ─ + ── ,   ╱  ─ + ── ⎥\n",
       "⎣   ╲╱     2   2      ╲╱     2   2     ╲╱   2   2    ╲╱   2   2  ⎦"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**4 - x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAM8AAAAWCAYAAABnsMi4AAAABHNCSVQICAgIfAhkiAAABUBJREFU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\n",
      "text/latex": [
       "$$x^{3} + 2 x^{2} + 4 x + 8 = 0$$"
      ],
      "text/plain": [
       " 3      2              \n",
       "x  + 2⋅x  + 4⋅x + 8 = 0"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = Eq(x**3 + 2*x**2 + 4*x + 8, 0)\n",
    "eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAVCAYAAADM1pWsAAAABHNCSVQICAgIfAhkiAAAAoxJREFU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\nCw+BrdIiEmbc/cnXY3Z0dHR0NGYdXOgk5nfMkUIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ -2, \\quad - 2 i, \\quad 2 i\\right ]$$"
      ],
      "text/plain": [
       "[-2, -2⋅ⅈ, 2⋅ⅈ]"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(eq, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sustavi jednadžbi:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAAXCAYAAAAldIx0AAAABHNCSVQICAgIfAhkiAAABCFJREFU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\n",
      "text/latex": [
       "$$\\left \\{ x : 1, \\quad y : 0\\right \\}$$"
      ],
      "text/plain": [
       "{x: 1, y: 0}"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + y - 1, x - y - 1], [x,y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR0AAAAXCAYAAAA2hpupAAAABHNCSVQICAgIfAhkiAAABstJREFU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\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{\\alpha}{2} + \\frac{\\gamma}{2}, \\quad y : \\frac{\\alpha}{2} - \\frac{\\gamma}{2}\\right \\}$$"
      ],
      "text/plain": [
       "⎧   α   γ     α   γ⎫\n",
       "⎨x: ─ + ─, y: ─ - ─⎬\n",
       "⎩   2   2     2   2⎭"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + y - a, x - y - c], [x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Više o interaktivnim widgetima možete naučiti preko primjera koji se nalaze [ovdje](https://github.com/jupyter-widgets/ipywidgets/tree/master/docs/source/examples). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table><tr><td><strong>Python verzija</strong></td><td>3.5.3</td></tr><tr><td><strong>kompajler</strong></td><td>GCC 4.8.2 20140120 (Red Hat 4.8.2-15)</td></tr><tr><td><strong>sustav</strong></td><td>Linux</td></tr><tr><td><strong>broj CPU-a</strong></td><td>8</td></tr><tr><td><strong>interpreter</strong></td><td>64bit</td></tr></table><table><tr><td><strong>sympy</strong> verzija</td><td>1.0</td></tr><tr><td><strong>matplotlib</strong> verzija</td><td>2.0.0</td></tr><tr><td><strong>IPython</strong> verzija</td><td>5.3.0</td></tr><tr><td><strong>numpy</strong> verzija</td><td>1.11.3</td></tr><tr><td><strong> ipywidgets</strong> verzija</td><td>6.0.0</td></tr></table>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from verzije import *\n",
    "from IPython.display import HTML\n",
    "HTML(print_sysinfo()+info_packages('sympy,matplotlib,IPython,numpy, ipywidgets'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Zadaci za vježbanje\n",
    "\n",
    "1) Napišite funkciju koja prima listu izraza, varijablu i točku, a ispisuje vrijednost svakog izraza s liste u toj točki. Funkcija treba izgledati ovako\n",
    "\n",
    "```\n",
    "def evaluiraj(izrazi, x, x0):\n",
    "\"\"\"\n",
    "Za svaki izraz iz izrazi funkcija ispisuje vrijednost izraza za x = x0\n",
    "\"\"\"\n",
    "```\n",
    "\n",
    "2) Izračunajte \n",
    "$\\frac{d}{dx}\\sin(x)e^x,\\,\n",
    "\\frac{\\partial}{\\partial x}\\sin(xy)e^x,\\,\n",
    "\\frac{\\partial^2}{\\partial x\\partial y}\\sin(xy)e^x.$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "3) Izraz `(x**2 + 3*x + 1)/(x**3 + 2*x**2 + x)` pojednostavite do izraza `1/(x**2 + 2*x + 1) + 1/x`\n",
    "\n",
    "4) Izraz `(a**b)**c)` pojednostavite do izraza `a**(b*c)`\n",
    "\n",
    "5) Napišite funkciju koja rješava (egzaktno) kvadratnu jednadžbu.\n",
    "\n",
    "6) Napišite funkciju koja za ulazne parametre prima funkciju (danu simboličkim izrazom) te točku (tj. broj) a crta danu funkciju i njenu tangentu u danoj točki. Učinite funkciju interaktivnom na način da se može birati funkcija (upisivanjem u polje) te točka (klizačem)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "7) Izračunajte vektorski produkt vektora $(a,b,b)$ i $(b,a,a)$\n",
    "\n",
    "8) Riješite Cauchyjev problem\n",
    "$$\\begin{align}\n",
    "u'(t)&=-au(t)+b,\\\\\n",
    "u(0)&=I\n",
    "\\end{align}$$\n",
    "i pojednostavite dobijeno rješenje. Ovdje su $a$, $b$ i $I$ nepoznate konstante."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "9)  Riješite diferencijalnu jednadžbu\n",
    "$$ y(t) + \\frac{d^2}{dt^2}y(t) = 0$$\n",
    "i nacrtajte partikularna rješenja koristeći sympy modul za crtanje. "
   ]
  }
 ],
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