docs/_static/notebooks/assign-full-example-v1.ipynb

{
 "cells": [
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# ASSIGNMENT CONFIG\n",
    "generate: true\n",
    "export_cell:\n",
    "    run_tests: true\n",
    "    pdf: false\n",
    "seed:\n",
    "    variable: rng_seed\n",
    "    autograder_value: 42\n",
    "    student_value: 90"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "%matplotlib inline\n",
    "rng_seed = 42"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN QUESTION\n",
    "name: q1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 1.** Write a function called `sieve` that takes in a positive integer `n` and returns a `set` of the prime numbers less than or equal to `n`. Use the Sieve of Eratosthenes to find the primes."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sieve(n):\n",
    "    \"\"\"\n",
    "    Generate a set of prime numbers less than or equal to a positive integer.\n",
    "    \"\"\"\n",
    "    # BEGIN SOLUTION\n",
    "    is_prime = [True for _ in range(n + 1)]\n",
    "    p = 2\n",
    "    while p ** 2 <= n:\n",
    "        if is_prime[p]:\n",
    "            for i in range(p ** 2, n + 1, p):\n",
    "                is_prime[i] = False\n",
    "        p += 1\n",
    "\n",
    "    is_prime[0]= False\n",
    "    is_prime[1]= False\n",
    "\n",
    "    return set(i for i in range(n + 1) if is_prime[i])\n",
    "    # END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN TESTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_low_primes(sieve):\n",
    "    assert sieve(1) == set()\n",
    "    assert sieve(2) == {2}\n",
    "    assert sieve(3) == {2, 3}\n",
    "\n",
    "test_low_primes(sieve)  # IGNORE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\" # BEGIN TEST CONFIG\n",
    "points: 2\n",
    "hidden: true\n",
    "\"\"\" # END TEST CONFIG\n",
    "def test_higher_primes(sieve):\n",
    "    assert sieve(20) == {2, 3, 5, 7, 11, 13, 17, 19}\n",
    "    assert sieve(100) == {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}\n",
    "\n",
    "test_higher_primes(sieve)  # IGNORE"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END TESTS"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END QUESTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN QUESTION\n",
    "name: q2\n",
    "manual: true"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 2.** Evaluate the integral:\n",
    "\n",
    "$$\n",
    "\\int x e^x \\, \\mathrm dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN PROMPT"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\int x e^x \\, \\mathrm dx &= \\texttt{# YOUR MATH HERE} \\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END PROMPT"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "tags": []
   },
   "source": [
    "# BEGIN SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**SOLUTION:**\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\int x e^x \\, \\mathrm dx &= x e^x - \\int e^x \\, \\mathrm dx \\\\\n",
    "&= x e^x - e^x + C \\\\\n",
    "&= e^x (x - 1) + C \\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END QUESTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN QUESTION\n",
    "name: q3\n",
    "manual: true"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 3.** Graph the function $f(x) = \\arctan \\left ( e^x \\right )$."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "tags": []
   },
   "source": [
    "# BEGIN SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# BEGIN SOLUTION\n",
    "f = lambda x: np.arctan(np.exp(x))\n",
    "xs = np.linspace(-10, 10, 100)\n",
    "ys = f(xs)\n",
    "\n",
    "plt.plot(xs, ys)\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.ylabel(\"$f(x)$\")\n",
    "plt.title(r\"$f(x) = \\arctan \\left ( e^x \\right )$\");\n",
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END QUESTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN QUESTION\n",
    "name: q4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 4.** Write a function `hailstone` that returns the hailstone sequence for a positive integer ``n`` as a ``list``."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hailstone(n):\n",
    "    \"\"\"\n",
    "    Generate the hailstone sequence of a positive integer.\n",
    "    \"\"\"\n",
    "    # BEGIN SOLUTION\n",
    "    if n == 1:\n",
    "        return [n]\n",
    "    elif n % 2 == 0:\n",
    "        return [n] + hailstone(n // 2)\n",
    "    else:\n",
    "        return [n] + hailstone(3 * n + 1)\n",
    "    # END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN TESTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_low_hailstones(hailstone):\n",
    "    assert hailstone(1) == [1]\n",
    "    assert hailstone(2)  == [2, 1]\n",
    "    assert hailstone(4) == [4, 2, 1]\n",
    "\n",
    "test_low_hailstones(hailstone)  # IGNORE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\" # BEGIN TEST CONFIG\n",
    "points: 3\n",
    "hidden: true\n",
    "\"\"\" # END TEST CONFIG\n",
    "def test_high_hailstones(hailstone):\n",
    "    assert hailstone(20) == [20, 10, 5, 16, 8, 4, 2, 1]\n",
    "    assert hailstone(9) == [9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]\n",
    "\n",
    "test_high_hailstones(hailstone)  # IGNORE"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END TESTS"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END QUESTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN QUESTION\n",
    "name: q5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 5.** Write a function ``gcd`` that takes in two positive integers ``a`` and ``b`` and returns their greatest common divisor."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gcd(a, b):\n",
    "    \"\"\"\n",
    "    Find the GCD of two positive integers.\n",
    "    \"\"\"\n",
    "    # BEGIN SOLUTION\n",
    "    if a == 0:\n",
    "        return b\n",
    "    return gcd(b % a, a)\n",
    "    # END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN TESTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_simple_gcds(gcd):\n",
    "    assert gcd(1, 1) == 1\n",
    "    assert gcd(2, 1) == 1\n",
    "    assert gcd(5, 5) == 5\n",
    "    assert gcd(10, 4) == 2\n",
    "\n",
    "test_simple_gcds(gcd)  # IGNORE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\" # BEGIN TEST CONFIG\n",
    "points: 1\n",
    "hidden: true\n",
    "\"\"\" # END TEST CONFIG\n",
    "def test_larger_gcds(gcd):\n",
    "    assert gcd(121, 11) == 11\n",
    "    assert gcd(807306, 622896) == 4098\n",
    "\n",
    "test_larger_gcds(gcd)  # IGNORE"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END TESTS"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END QUESTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN QUESTION\n",
    "name: q6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 6.** Write a function `box_muller` that takes in two $U(0,1)$ random variables and returns two standard normal random variables using the Box-Muller algorithm. Use this function to generate 1000 standard normal random variables. Make sure to use the provided RNG to generate your $U(0,1)$ RVs."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN SOLUTION"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = np.random.default_rng(rng_seed)\n",
    "\n",
    "def box_muller(u1, u2):\n",
    "    \"\"\"\n",
    "    Generate two standard normal random variables from two U(0,1) random variables.\n",
    "    \"\"\"\n",
    "    # BEGIN SOLUTION\n",
    "    r = np.sqrt(-2 * np.log(u1))\n",
    "    theta = 2 * np.pi * u2\n",
    "    return r * np.cos(theta), r * np.sin(theta)\n",
    "    # END SOLUTION\n",
    "\n",
    "rvs = np.array([box_muller(rng.random(), rng.random()) for _ in range(500)]).flatten() # SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END SOLUTION"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# BEGIN TESTS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_validity(np, rvs, box_muller):\n",
    "    assert rvs.shape == (1000,)\n",
    "    assert np.allclose(box_muller(0.5, 0.5), (-1.1774100225154747, 1.4419114153575892e-16))\n",
    "\n",
    "test_simple_gcds(gcd)  # IGNORE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\" # BEGIN TEST CONFIG\n",
    "points: 1\n",
    "hidden: true\n",
    "\"\"\" # END TEST CONFIG\n",
    "def test_values(np, rvs):\n",
    "    test_rng = np.random.default_rng(1048)\n",
    "    idx = test_rng.choice(np.arange(len(rvs)), 10)\n",
    "    assert np.allclose(rvs[idx], [2.06150769, -1.24834522, -1.03226524, -0.90264717, -0.71005571, -1.340234  ,  1.77047974, -0.31023336,  1.51417285,  0.64113867])\n",
    "\n",
    "test_values(np, rvs)  # IGNORE"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END TESTS"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "# END QUESTION"
   ]
  }
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