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   "metadata": {},
   "source": [
    "Submit your answers to gradescope before, or at 3:40pm."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f2fabe0-0a28-4eb8-a899-2f68684b378b",
   "metadata": {},
   "source": [
    "# Question 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb3fa653-a074-44e7-90f2-333c5f4a253a",
   "metadata": {},
   "source": [
    "Let $N = \\cos(\\pi/5)$.\n",
    "\n",
    "1. Compute a nearby rational approximation for $N$,\n",
    "   with the denominator bounded by 999.\n",
    "\n",
    "2. What is the accuracy of your nearby rational approximation?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d751818-8407-453d-9651-5543027bafce",
   "metadata": {},
   "source": [
    "# Question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2638f774-369f-4733-b6cf-ed95d2ca1657",
   "metadata": {},
   "source": [
    "Compute the square roots of $c = 5 - 4 I$.\n",
    "\n",
    "Verify that the square of your computed roots equals $c$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6749831f-9e1c-4432-9004-fcaa25ed4965",
   "metadata": {},
   "source": [
    "# Question 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3b0a674-0660-4a0e-9df4-554cbd5a6e78",
   "metadata": {},
   "source": [
    "Let $p = x^3 + x + 4$ be a polynomial with coefficients in\n",
    "a finite field with 11 elements.\n",
    "\n",
    "Show that $p$ is irreducible.\n",
    "\n",
    "Declare $\\alpha$ as a formal root of $p$ and show that $p$\n",
    "factors over the field extended with $\\alpha$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "95918f38-50c5-4267-942d-502358b35636",
   "metadata": {},
   "source": [
    "# Question 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23770e4e-0b21-4442-b919-b1d119aea512",
   "metadata": {},
   "source": [
    "Define an equation `eqn` that shows `cos(pi/2) == 0` when printed.\n",
    "\n",
    "Without retyping `eqn`, change `eqn` so `print(eqn)` shows `0 == 0`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6533c95a-42e9-4a57-839b-012986faf2e6",
   "metadata": {},
   "source": [
    "# Question 5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac095ddb-a9cd-430e-a0fd-3b8afa7329e1",
   "metadata": {},
   "source": [
    "Consider the evaluation of \n",
    "$p = x^8 - 2 x^7 + x^6 + 3 x^5 - x^4 + 4 x^3 + x + 6$.\n",
    "\n",
    "What is the fastest way to evaluate $p$ at hardware floats? \n",
    "\n",
    "Justify your answer."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "996c0e0b-45d6-4b4c-851c-816603637f1a",
   "metadata": {},
   "source": [
    "# Question 6"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78c7e392-5f6e-4372-bd2d-150b758d6e04",
   "metadata": {},
   "source": [
    "The\n",
    "``f = lambda n: float(sum([(1+k/n)**2*ln(1+k/n) for k in range(1,n)])/n)``\n",
    "\n",
    "computes the right hand side of\n",
    "\n",
    "$$\n",
    "    \\qquad \\int_1^2 x^2 \\ln(x) dx \\approx \\frac{1}{n} \n",
    "    \\sum_{k=1}^{n-1} \\left( 1 + \\frac{k}{n} \\right)\n",
    "                     \\ln \\left( 1 + \\frac{k}{n} \\right).\n",
    "$$\n",
    "\n",
    "1. Time the execution of `f` for $n = 10000$.\n",
    "      \n",
    "   Explain why `f` is inefficient.\n",
    "\n",
    "2. Apply vectorization to improve the efficiency.  Verify the correctness.\n",
    "\n",
    "   Time the execution of the vectorized function for $n = 10000$,\n",
    "   compare with timings of `f`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d185c24-858b-42e0-84ae-7ba418889d56",
   "metadata": {},
   "source": [
    "# Question 7"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74d5778b-44e9-4b6f-8b78-306b1ea43ab9",
   "metadata": {},
   "source": [
    "Consider the rational expression \n",
    "$\\displaystyle \\frac{u^{1000} - v^{1000}}{u - v}$. \n",
    "\n",
    "Explain why it is bad to simplify this expression automatically."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4861d1b0-ed55-4d80-bc0b-a4a9d3fd4370",
   "metadata": {},
   "source": [
    "# Question 8"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "987503b6-e566-41bb-9760-5fbe3c155b61",
   "metadata": {},
   "source": [
    "Transform\n",
    "$q = \\displaystyle (x-y)^2 + \\frac{(x+y)^7}{(x-y)^2}$\n",
    "into $\\displaystyle r = \\frac{(x-y)^4 + (x+y)^7}{(x-y)^2}$,\n",
    "without typing $r$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c2990b6-d622-480b-9714-7d25de67d68b",
   "metadata": {},
   "source": [
    "# Question 9"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65238de0-a5fc-459b-b129-51cc74e356b5",
   "metadata": {},
   "source": [
    "Are the expressions $\\displaystyle p = \\frac{x^2 - 6 x + 9}{x - 3}$\n",
    "and $q = x - 3$ the same?\n",
    "\n",
    "Justify your answer by appropriate *symbolic* computations.\n",
    "\n",
    "Demonstrate the application of a *numerical* \n",
    "probability-one equality test."
   ]
  }
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