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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*For this exercise you should have received this text in .ipynb format. Complete the exercises by modifying this file, and submit the modified version*\n",
    "\n",
    "**Deadline:** Sunday, June 6."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 1 (6 points)**\n",
    "\n",
    "Use Sage to find the intersection points *in the real plane* (that is, only those points such that *both* coordinates are real numbers) of the following pairs of geometric objects:\n",
    "\n",
    "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2\\right)^2 + (2y)^2 = 4$.\n",
    "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2-2\\right)^2 + (2y)^2 = 4$.\n",
    "* The curve of equation $y^2 = x^3 -x +1$ and the horizontal line $y=10$.\n",
    "* The $x$-axis and the graph of the function $f(x)=\\log(x) - e^{-x}$. *Hint: $f(x)$ has only one real zero.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2 (6 points)**\n",
    "\n",
    "(a) Use Sage to compute\n",
    "* the derivative\n",
    "* a primite (i.e. integral)\n",
    "* the power series expansion around $0$ up to order $4$\n",
    "\n",
    "of the following functions:\n",
    "* $f(x) = e^x$\n",
    "* $f(x) = \\sin(x)$\n",
    "* $f(x) = \\cos(x)$\n",
    "* $f(x) = \\tan(x)$\n",
    "* $f(x) = \\log(1+x)$\n",
    "* $f(x) = \\sqrt[3]{1+x}$\n",
    "\n",
    "(b) Use Sage to get the Latex code that represents the objects you computed above.\n",
    "\n",
    "(c) Arrange the results of the previous points in a table in Latex. The table should have 4 columns (function, derivative, integral, series) and one row for each of the functions above. *Note: when including Latex in a Markdown cell in Jupyter you will not receive any warning if you make mistakes; instead the Latex will simply not be rendered and it will appear as plain text. If you have troubles making this work you can send me a separate .tex (and .pdf) file.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute derivatives etc..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute Latex code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(Write your table here)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3 (4 points)**\n",
    "\n",
    "The equation\n",
    "\\begin{align*}\n",
    "y^2+x^{16}=1\n",
    "\\end{align*}\n",
    "determines a closed curve in $\\mathbb R^2$ that looks like a rounded square. Determine the area of that shape, giving both an exact value (which might depend on some functions that Sage knows, but you don't) and an approximate value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 4 (12 points)**\n",
    "\n",
    "A team of biologists is monitoring the population of river shrimps in the Alzette. At first they thought that the size $P(t)$ of their population on day $t$ would satisfy the differential equation $P'(t)=P(t)/10$. However this does not work well with the data they have collected, so they now believe that the population of shrimps follows the formula $P'(t)=P(t)/10-b$ for some value of $b$ between 1 and 100. They need your help here.\n",
    "\n",
    "(a) Using Sage, find a solution for the differential equation with initial conditions\n",
    "\\begin{align*}\n",
    "\\begin{cases}\n",
    "P'(t)&=\\frac{P(t)}{10}-b\\\\\n",
    "P(1)&=1000\n",
    "\\end{cases}\n",
    "\\end{align*}\n",
    "where $b$ is a generic constant.\n",
    "\n",
    "(b) The list `data` in the cell below contains the actual number of shrimps that was measured every day from day $1$ (the $0$ at the beginning is meaningless, but it will help to keep it there). Plot in one single picture, possibly using different colors for each:\n",
    "* The data as a bar chart.\n",
    "* A curve that interpolates the data, using one of the methods shown in class.\n",
    "* The solution of the differential equation for $b=0$.\n",
    "* The solution of the differential equation for a value of $b$ of your choice ($1\\leq b\\leq 100$) that fits the data better than $b=0$. *(For this last point there is no right or wrong choice, just pick one that looks good)*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = [0, 1000, 1123, 1223, 1190, 1432, 1553, 1709, 1826, 1980, 2146, 2172, 2383, 2588, 2822, 3401, 3330, 4157, 3994, 4995, 5392, 5910, 6468, 7128, 7325, 7984, 9634, 10473, 11761, 12777]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Grading**\n",
    "\n",
    "This homework assignment is worth $28$ ($24+4$) points, distributed as described above.\n",
    "\n",
    "Your final grade for the course will be the total of points you obtained (notice that the maximum is $20+20+16+28=84$) divided by $4$, rounded to the nearest integer. More precisely\n",
    "\n",
    "\\begin{align*}\n",
    "\\operatorname{grade} = \\operatorname{min}\\left(20, \\left\\lfloor \\frac{\\operatorname{total}}{4} + 0.5\\right\\rfloor\\right)\n",
    "\\end{align*}"
   ]
  }
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