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   "source": [
    "In this review sheet we consider problems\n",
    "to prepare for the second midterm exam of mcs 320."
   ]
  },
  {
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   "metadata": {},
   "source": [
    "# Question 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Count the number of points with integer coordinates $(x,y)$\n",
    "in the region defined by the inequalities $0 \\leq x < 20$, $0 \\leq y < 20$,\n",
    "$y \\geq x/2$, $y \\leq 3 x$.\n",
    "\n",
    "Do the following:\n",
    "\n",
    "1. Generate a list L of integer points $(i,j)$ for $i$ and $j$ ranging between 0 and 19.\n",
    "\n",
    "2. Select from the list L those points in the cone $y \\geq x/2$ and by $y \\leq 3 x$.\n",
    "\n",
    "3. Count the number of points in the cone.  Write also the number below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For some parameter $t$, consider the sequence recursively defined as:\n",
    "\n",
    "$F(n) = (1-t) F(n-1) + t F(n-2)$, for $n > 1$, where $F(0) = a$ and $F(1) = b$.\n",
    "   \n",
    "Using the recursive definition write an efficient SageMath function $F$\n",
    "to compute $F(n)$ as $F(a,b,t,n)$.\n",
    "What is the result of $F(a,b,0.3,100)$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the function $f(t)$ as the integral of $(1 - e^x)$ for $x$ from 0 to $t$, for $t \\geq 0$.  \n",
    "\n",
    "1. Define this function in Sage.\n",
    "   \n",
    "2. What is $f'(1)$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function $g(x,t) = (1-t^2)/(1 - 2xt + t^2)$ is a\n",
    "generating function for the Chebyshev polynomials.\n",
    "\n",
    "1. Compute a Taylor series approximation for $g(x,t)$\n",
    "   around $t = 0$ of order 10.  \n",
    "       \n",
    "2. Select the coefficient of $t^8$ and compare\n",
    "   with the output of ``chebyshev_T(8,x)``.\n",
    "       \n",
    "   Is there a difference between the two?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the point $(1,1)$ on the curve $f(x,y) = x^2 - y^3 - x + y = 0$.\n",
    "\n",
    "1. Give the Sage command(s) to compute a Taylor series about the point $(1,1)$\n",
    "   where the term of the error is of second order.\n",
    "\n",
    "2. Compute the slope of the tangent line of the curve at the point $(1,1)$\n",
    "   and use the slope to determine the tangent line.\n",
    "   Write the equation of the tangent line.\n",
    "   \n",
    "   Verify that the tangent line equals the first-order Taylor series at $(1,1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the curve $x^4 - 3 x y + y^4 = 0$.  Do the following.\n",
    "\n",
    "1. Make a plot for $x$ and $y$ both ranging between $-2$ and $+2$;\n",
    "\n",
    "2. Convert the curve into polar coordinates; and\n",
    "\n",
    "3. Plot the curve in polar coordinates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $a$ and $b$ be positive numbers.\n",
    "Consider $f = x^2/a + y/b$ and the unit circle $x^2 + y^2 = 1$.\n",
    "    \n",
    "Determine the number of candidate extremal values of $f$ on the unit circle.\n",
    "   \n",
    "Use a lexicographic Groebner basis to compute a triangular form of the equations for this problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do the following.\n",
    "\n",
    "1. Create a 5-by-5 matrix $A$ over the rationals where \n",
    "   the $(i,j)$-the element is $1/(i+j)$ (for $i$ and $j$ both from 1 to 5).\n",
    "\n",
    "2. Define $b$ as a vector of length 5 of ones.\n",
    "   Solve the system defined by $A x = b$.\n",
    "   \n",
    "3. Verify that $b - A x$ equals zero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 9"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the intial value problem $dy/dt = 2 - 6y$, $y(0) = -1$.\n",
    "\n",
    "1. Solve this problem and plot the solution trajectory for $t \\in [0,2]$.\n",
    "    \n",
    "2. Plot the slope field for $t \\in [0,2]$, $y \\in [-1,0.5]$.\n",
    "  \n",
    "   Place also the particular solution computed the first part of this question on the plot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Minimize $x+3y$ \n",
    "\n",
    "subject to $x \\geq 2$, $y \\geq 1$, $x + 2y \\leq 8$, $x+y \\leq 6$.\n",
    "    \n",
    "Formulate the linear programming problem and solve it."
   ]
  }
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