Prvi kolokvij - rješenja.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matematički softver\n",
    "## Prvi kolokvij\n",
    "## Rješenja"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Prvi zadatak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Napišite funkciju `usporedi` koja prima dvije liste, u svakoj od kojih se nalaze samo cijeli brojevi (`int`) i stringovi (`str`). Ako nisu jednake duljine, `usporedi` vraća `False`. Ako jesu, `usporedi` treba ustanoviti jesu li elementi na odgovarajućim pozicijama \"slični\".\n",
    "\n",
    "* dva cijela broja su slični ako je jedan od njih (bilo koji) djeljiv onim drugim\n",
    "* dva stringa su slični ako su jednaki do na veličinu slova (_case insensitive_)\n",
    "* string i cijeli broj nisu nikada slični\n",
    "\n",
    "Primjerice, `usporedi([a, b, c], [d, e, f])` treba vratiti `True` ako su `a` i `d` slični, `b` i `e` slični, te `c` i `f` slični."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def slični(x, y):\n",
    "    if type(x) is type(y) is int: return not (x and y and x % y and y % x)\n",
    "    elif type(x) is type(y) is str: return x.casefold() == y.casefold()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def usporedi(prva, druga):\n",
    "    return len(prva) == len(druga) and all(map(slični, prva, druga))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all([\n",
    "    usporedi([24, 'v'], [12, 'V']) == True,\n",
    "    usporedi([1, 2], [1]) == False,\n",
    "    usporedi([1, 't'], ['t', 1]) == False,\n",
    "    usporedi(['', 0, 'Wäßrig'], ['', 52, 'WÄSSRIG']) == True\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Drugi zadatak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "U varijabli `kocke` nalazi se tri milijuna rezultata bacanja igraće kocke."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "kocke = 1 + np.random.randint(0, 6, 3_000_000, dtype=np.int8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### a)\n",
    "\n",
    "Interpretirajte `kocke` kao niz pokusa bacanja po četiri kocke (ne kopirajte podatke, samo im promijenite oblik).\n",
    "\n",
    "Izračunajte, i izrazite u postotcima na dvije decimale, _a posteriori_ vjerojatnost da se u bacanju četiri kocke pojavi bar jedna šestica. Isplati li se kladiti (1:1) na taj događaj? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def vjerojatnost(događaj):\n",
    "    vj = događaj.any(axis=1).mean()\n",
    "    print(format(vj, '.2%'))\n",
    "    return vj > .5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "51.83%\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pokus1 = kocke.reshape(-1, 4)\n",
    "vjerojatnost(pokus1 == 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sada reinterpretirajte `kocke` kao niz pokusa, svaki od kojih se sastoji od 24 bacanja 2 kocke (trodimenzionalni tenzor).\n",
    "\n",
    "Odgovorite na ista pitanja kao u **a)**, za događaj da se u 24 bacanja 2 kocke bar jednom pojavi dvostruka šestica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "49.13%\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pokus2 = kocke.reshape(-1, 24, 2)\n",
    "vjerojatnost(pokus2.min(axis=2) == 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Treći zadatak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "U datoteci `bodovi.csv` nalaze se bodovi studenata nekog izmišljenog kolegija. Stupci su redom ime i prezime studenta, bodovi s prvog i drugog kolokvija, te bodovi s tri zadaće.\n",
    "\n",
    "* Učitajte datoteku u DataFrame.\n",
    "* Dodajte stupac `Kolokviji` koji predstavlja zbroj bodova na oba kolokvija.\n",
    "* Dodajte stupac `Zadaće` koji predstavlja ukupan broj bodova na zadaćama. Najslabije napisana zadaća se ne računa.\n",
    "* Odredite je li bolje (u prosjeku) bio napisan prvi ili drugi kolokvij.\n",
    "* Ispišite _top 10_ studenata po ukupnim bodovima (kolokviji plus zadaće).\n",
    "\n",
    "Sve radite u jednom DataFrameu. Smijete dodavati i pomoćne stupce (osim dva gore zadana), ali ih morate obrisati kasnije."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = pd.read_csv('bodovi.csv', delimiter='\\t', names='ime prezime k1 k2 z1 z2 z3'.split())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "df['Kolokviji'] = df['k1'] + df['k2']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "zadaće = df[['z1', 'z2', 'z3']]\n",
    "df['Zadaće'] = zadaće.sum(axis=1) - zadaće.min(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'k2'"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df[['k1', 'k2']].mean().idxmax()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>ime</th>\n",
       "      <th>prezime</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>410</th>\n",
       "      <td>Eliza</td>\n",
       "      <td>Mora</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>599</th>\n",
       "      <td>Mercedes</td>\n",
       "      <td>Boone</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>170</th>\n",
       "      <td>Leslie</td>\n",
       "      <td>Glass</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>Lea</td>\n",
       "      <td>Henry</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>250</th>\n",
       "      <td>Madilyn</td>\n",
       "      <td>Martinez</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>380</th>\n",
       "      <td>Brittany</td>\n",
       "      <td>Kaufman</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>204</th>\n",
       "      <td>Makaila</td>\n",
       "      <td>Schaefer</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>603</th>\n",
       "      <td>Joy</td>\n",
       "      <td>Dorsey</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>892</th>\n",
       "      <td>Braylon</td>\n",
       "      <td>Sherman</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>909</th>\n",
       "      <td>Hillary</td>\n",
       "      <td>Gentry</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          ime   prezime\n",
       "410     Eliza      Mora\n",
       "599  Mercedes     Boone\n",
       "170    Leslie     Glass\n",
       "50        Lea     Henry\n",
       "250   Madilyn  Martinez\n",
       "380  Brittany   Kaufman\n",
       "204   Makaila  Schaefer\n",
       "603       Joy    Dorsey\n",
       "892   Braylon   Sherman\n",
       "909   Hillary    Gentry"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df['Ukupno'] = df['Kolokviji'] + df['Zadaće']\n",
    "df.sort_values('Ukupno', ascending=False).head(10)[['ime', 'prezime']]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "del df['Ukupno']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Četvrti zadatak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nacrtajte, jedan pored drugog sa zajedničkom osi ordinata, grafove funkcijâ $f(x)=1/x$ i $g(x)=1/x^2$ na segmentu $[-1,1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x257ba992d30>]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x257ba8f32e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, (ax1, ax2) = plt.subplots(1, 2, sharey=True)\n",
    "xneg = np.linspace(-1, 0)[:-1]\n",
    "xpoz = np.linspace(0, 1)[1:]\n",
    "ax2.set_ylim(-50, 50)\n",
    "ax1.plot(xpoz, 1/xpoz, 'r')\n",
    "ax1.plot(xneg, 1/xneg, 'r')\n",
    "ax2.plot(xpoz, 1/xpoz**2, 'g')\n",
    "ax2.plot(xneg, 1/xneg**2, 'g')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Peti zadatak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Riješite egzaktno diferencijalnu jednadžbu $xy'+y=x\\sin x$ uz početni uvjet $y\\left(\\frac\\pi2\\right)=-\\frac12$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = sp.Symbol('x'), sp.Function('y')\n",
    "jedn = sp.Eq(x * sp.Derivative(y(x), x) + y(x), x * sp.sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = - \\cos{\\left (x \\right )} + \\frac{1}{x} \\sin{\\left (x \\right )} + \\frac{1}{x} \\left(-1 - \\frac{\\pi}{4}\\right)$$"
      ],
      "text/plain": [
       "                               π\n",
       "                          -1 - ─\n",
       "                 sin(x)        4\n",
       "y(x) = -cos(x) + ────── + ──────\n",
       "                   x        x   "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "opće = sp.dsolve(jedn)\n",
    "konstanta, = sp.solve(opće.subs({x: sp.pi / 2, y(x): -sp.S(1)/2}), dict=True)\n",
    "opće.subs(konstanta)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}