mathsoftware

A course about LaTeX and SageMath
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Homework4.ipynb (6155B)


      1 {
      2  "cells": [
      3   {
      4    "cell_type": "markdown",
      5    "metadata": {},
      6    "source": [
      7     "*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",
      8     "\n",
      9     "**Deadline:** Sunday, June 6."
     10    ]
     11   },
     12   {
     13    "cell_type": "markdown",
     14    "metadata": {},
     15    "source": [
     16     "**Exercise 1 (6 points)**\n",
     17     "\n",
     18     "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",
     19     "\n",
     20     "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2\\right)^2 + (2y)^2 = 4$.\n",
     21     "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2-2\\right)^2 + (2y)^2 = 4$.\n",
     22     "* The curve of equation $y^2 = x^3 -x +1$ and the horizontal line $y=10$.\n",
     23     "* The $x$-axis and the graph of the function $f(x)=\\log(x) - e^{-x}$. *Hint: $f(x)$ has only one real zero.*"
     24    ]
     25   },
     26   {
     27    "cell_type": "code",
     28    "execution_count": null,
     29    "metadata": {},
     30    "outputs": [],
     31    "source": []
     32   },
     33   {
     34    "cell_type": "markdown",
     35    "metadata": {},
     36    "source": [
     37     "**Exercise 2 (6 points)**\n",
     38     "\n",
     39     "(a) Use Sage to compute\n",
     40     "* the derivative\n",
     41     "* a primite (i.e. integral)\n",
     42     "* the power series expansion around $0$ up to order $4$\n",
     43     "\n",
     44     "of the following functions:\n",
     45     "* $f(x) = e^x$\n",
     46     "* $f(x) = \\sin(x)$\n",
     47     "* $f(x) = \\cos(x)$\n",
     48     "* $f(x) = \\tan(x)$\n",
     49     "* $f(x) = \\log(1+x)$\n",
     50     "* $f(x) = \\sqrt[3]{1+x}$\n",
     51     "\n",
     52     "(b) Use Sage to get the Latex code that represents the objects you computed above.\n",
     53     "\n",
     54     "(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.*"
     55    ]
     56   },
     57   {
     58    "cell_type": "code",
     59    "execution_count": null,
     60    "metadata": {},
     61    "outputs": [],
     62    "source": [
     63     "# Compute derivatives etc..."
     64    ]
     65   },
     66   {
     67    "cell_type": "code",
     68    "execution_count": null,
     69    "metadata": {},
     70    "outputs": [],
     71    "source": [
     72     "# Compute Latex code"
     73    ]
     74   },
     75   {
     76    "cell_type": "markdown",
     77    "metadata": {},
     78    "source": [
     79     "(Write your table here)"
     80    ]
     81   },
     82   {
     83    "cell_type": "markdown",
     84    "metadata": {},
     85    "source": [
     86     "**Exercise 3 (4 points)**\n",
     87     "\n",
     88     "The equation\n",
     89     "\\begin{align*}\n",
     90     "y^2+x^{16}=1\n",
     91     "\\end{align*}\n",
     92     "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."
     93    ]
     94   },
     95   {
     96    "cell_type": "code",
     97    "execution_count": null,
     98    "metadata": {},
     99    "outputs": [],
    100    "source": []
    101   },
    102   {
    103    "cell_type": "markdown",
    104    "metadata": {},
    105    "source": [
    106     "**Exercise 4 (12 points)**\n",
    107     "\n",
    108     "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",
    109     "\n",
    110     "(a) Using Sage, find a solution for the differential equation with initial conditions\n",
    111     "\\begin{align*}\n",
    112     "\\begin{cases}\n",
    113     "P'(t)&=\\frac{P(t)}{10}-b\\\\\n",
    114     "P(1)&=1000\n",
    115     "\\end{cases}\n",
    116     "\\end{align*}\n",
    117     "where $b$ is a generic constant.\n",
    118     "\n",
    119     "(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",
    120     "* The data as a bar chart.\n",
    121     "* A curve that interpolates the data, using one of the methods shown in class.\n",
    122     "* The solution of the differential equation for $b=0$.\n",
    123     "* 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)*"
    124    ]
    125   },
    126   {
    127    "cell_type": "code",
    128    "execution_count": null,
    129    "metadata": {},
    130    "outputs": [],
    131    "source": [
    132     "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"
    133    ]
    134   },
    135   {
    136    "cell_type": "markdown",
    137    "metadata": {},
    138    "source": [
    139     "**Grading**\n",
    140     "\n",
    141     "This homework assignment is worth $28$ ($24+4$) points, distributed as described above.\n",
    142     "\n",
    143     "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",
    144     "\n",
    145     "\\begin{align*}\n",
    146     "\\operatorname{grade} = \\operatorname{min}\\left(20, \\left\\lfloor \\frac{\\operatorname{total}}{4} + 0.5\\right\\rfloor\\right)\n",
    147     "\\end{align*}"
    148    ]
    149   }
    150  ],
    151  "metadata": {
    152   "kernelspec": {
    153    "display_name": "SageMath 9.2",
    154    "language": "sage",
    155    "name": "sagemath"
    156   },
    157   "language_info": {
    158    "codemirror_mode": {
    159     "name": "ipython",
    160     "version": 3
    161    },
    162    "file_extension": ".py",
    163    "mimetype": "text/x-python",
    164    "name": "python",
    165    "nbconvert_exporter": "python",
    166    "pygments_lexer": "ipython3",
    167    "version": "3.8.5"
    168   }
    169  },
    170  "nbformat": 4,
    171  "nbformat_minor": 4
    172 }