Homework4-checkpoint.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 }