From d6c61d988bfa4255baf9cdae42db59ebee38363f Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Tue, 25 May 2021 17:10:49 +0200 Subject: Added files --- .../.ipynb_checkpoints/Homework4-checkpoint.ipynb | 172 ++++ .../Homework4-solutions-checkpoint.ipynb | 61 ++ .../.ipynb_checkpoints/Untitled-checkpoint.ipynb | 6 + src/Homework4/Homework4.aux | 20 + src/Homework4/Homework4.ipynb | 172 ++++ src/Homework4/Homework4.log | 953 +++++++++++++++++++++ src/Homework4/Homework4.out | 0 src/Homework4/Homework4.pdf | Bin 0 -> 116266 bytes src/Homework4/Homework4.tex | 530 ++++++++++++ 9 files changed, 1914 insertions(+) create mode 100644 src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb create mode 100644 src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb create mode 100644 src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb create mode 100644 src/Homework4/Homework4.aux create mode 100644 src/Homework4/Homework4.ipynb create mode 100644 src/Homework4/Homework4.log create mode 100644 src/Homework4/Homework4.out create mode 100644 src/Homework4/Homework4.pdf create mode 100644 src/Homework4/Homework4.tex (limited to 'src/Homework4') diff --git a/src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb b/src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb new file mode 100644 index 0000000..47f63cf --- /dev/null +++ b/src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb @@ -0,0 +1,172 @@ +{ + "cells": [ + { + "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*}" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 9.2", + "language": "sage", + "name": "sagemath" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.5" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb b/src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb new file mode 100644 index 0000000..f8617fd --- /dev/null +++ b/src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb @@ -0,0 +1,61 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 40, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f(x) \t Derivative \t Integral \t Series expansion\n", + "e^x \t e^x \t e^x \t 1 + 1*x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + Order(x^5)\n", + "sin(x) \t cos(x) \t -cos(x) \t 1*x + (-1/6)*x^3 + Order(x^5)\n", + "cos(x) \t -sin(x) \t sin(x) \t 1 + (-1/2)*x^2 + 1/24*x^4 + Order(x^5)\n", + "tan(x) \t tan(x)^2 + 1 \t log(sec(x)) \t 1*x + 1/3*x^3 + Order(x^5)\n", + "log(x + 1) \t 1/(x + 1) \t (x + 1)*log(x + 1) - x - 1 \t 1*x + (-1/2)*x^2 + 1/3*x^3 + (-1/4)*x^4 + Order(x^5)\n", + "(x + 1)^a \t a*(x + 1)^(a - 1) \t (x + 1)^(a + 1)/(a + 1) \t 1 + (a)*x + (1/2*a^2 - 1/2*a)*x^2 + (1/6*a^3 - 1/2*a^2 + 1/3*a)*x^3 + (1/24*a^4 - 1/4*a^3 + 11/24*a^2 - 1/4*a)*x^4 + Order(x^5)\n" + ] + } + ], + "source": [ + "print(\"f(x)\", \"\\t\", \"Derivative\", \"\\t\", \"Integral\", \"\\t\", \"Series expansion\")\n", + "var('a')\n", + "f(x) = (1+x)^a\n", + "assume(a>0)\n", + "functions = [e^x, sin, cos, tan, log(1+x), f]\n", + "for g in functions:\n", + " print(g(x), \"\\t\", derivative(g(x), x), \"\\t\", integral(g(x),x), \"\\t\", g(x).series(x==0,5))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 9.2", + "language": "sage", + "name": "sagemath" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.5" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb b/src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb new file mode 100644 index 0000000..7fec515 --- /dev/null +++ b/src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/src/Homework4/Homework4.aux b/src/Homework4/Homework4.aux new file mode 100644 index 0000000..a74227c --- /dev/null +++ b/src/Homework4/Homework4.aux @@ -0,0 +1,20 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\providecommand \oddpage@label [2]{} +\gdef \@abspage@last{2} diff --git a/src/Homework4/Homework4.ipynb b/src/Homework4/Homework4.ipynb new file mode 100644 index 0000000..99481b9 --- /dev/null +++ b/src/Homework4/Homework4.ipynb @@ -0,0 +1,172 @@ +{ + "cells": [ + { + "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*}" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 9.0", + "language": "sage", + "name": "sagemath" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.5" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/src/Homework4/Homework4.log b/src/Homework4/Homework4.log new file mode 100644 index 0000000..576cde7 --- /dev/null +++ b/src/Homework4/Homework4.log @@ -0,0 +1,953 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/VoidLinux) (preloaded format=pdflatex 2021.4.20) 12 MAY 2021 14:24 +entering extended mode + \write18 enabled. + %&-line 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b/src/Homework4/Homework4.out new file mode 100644 index 0000000..e69de29 diff --git a/src/Homework4/Homework4.pdf b/src/Homework4/Homework4.pdf new file mode 100644 index 0000000..ff8ecd2 Binary files /dev/null and b/src/Homework4/Homework4.pdf differ diff --git a/src/Homework4/Homework4.tex b/src/Homework4/Homework4.tex new file mode 100644 index 0000000..ada5be7 --- /dev/null +++ b/src/Homework4/Homework4.tex @@ -0,0 +1,530 @@ +\documentclass[11pt]{article} + + \usepackage[breakable]{tcolorbox} + \usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour) + + \usepackage{iftex} + \ifPDFTeX + \usepackage[T1]{fontenc} + \usepackage{mathpazo} + \else + \usepackage{fontspec} + \fi + + % Basic figure setup, for now with no caption control since it's done + % automatically by Pandoc (which extracts ![](path) syntax from Markdown). + \usepackage{graphicx} + % Maintain compatibility with old templates. Remove in nbconvert 6.0 + \let\Oldincludegraphics\includegraphics + % Ensure that by default, figures have no caption (until we provide a + % proper Figure object with a Caption API and a way to capture that + % in the conversion process - todo). + \usepackage{caption} + \DeclareCaptionFormat{nocaption}{} + \captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt} + + \usepackage[Export]{adjustbox} % Used to constrain images to a maximum size + \adjustboxset{max size={0.9\linewidth}{0.9\paperheight}} + \usepackage{float} + \floatplacement{figure}{H} % forces figures to be placed at the correct location + \usepackage{xcolor} % Allow colors to be defined + \usepackage{enumerate} % Needed for markdown enumerations to work + \usepackage{geometry} % Used to adjust the document margins + \usepackage{amsmath} % Equations + \usepackage{amssymb} % Equations + \usepackage{textcomp} % defines textquotesingle + % Hack from http://tex.stackexchange.com/a/47451/13684: + \AtBeginDocument{% + \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code + } + \usepackage{upquote} % Upright quotes for verbatim code + \usepackage{eurosym} % defines \euro + \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support + \usepackage{fancyvrb} % verbatim replacement that allows latex + \usepackage{grffile} % extends the file name processing of package graphics + % to support a larger range + \makeatletter % fix for grffile with XeLaTeX + \def\Gread@@xetex#1{% + \IfFileExists{"\Gin@base".bb}% + {\Gread@eps{\Gin@base.bb}}% + {\Gread@@xetex@aux#1}% + } + \makeatother + + % The hyperref package gives us a pdf with properly built + % internal navigation ('pdf bookmarks' for the table of contents, + % internal cross-reference links, web links for URLs, etc.) + \usepackage{hyperref} + % The default LaTeX title has an obnoxious amount of whitespace. By default, + % titling removes some of it. It also provides customization options. + \usepackage{titling} + \usepackage{longtable} % longtable support required by pandoc >1.10 + \usepackage{booktabs} % table support for pandoc > 1.12.2 + \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment) + \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout) + % normalem makes italics be italics, not underlines + \usepackage{mathrsfs} + + + + % Colors for the hyperref package + \definecolor{urlcolor}{rgb}{0,.145,.698} + \definecolor{linkcolor}{rgb}{.71,0.21,0.01} + \definecolor{citecolor}{rgb}{.12,.54,.11} + + % ANSI colors + \definecolor{ansi-black}{HTML}{3E424D} + \definecolor{ansi-black-intense}{HTML}{282C36} + \definecolor{ansi-red}{HTML}{E75C58} + \definecolor{ansi-red-intense}{HTML}{B22B31} + \definecolor{ansi-green}{HTML}{00A250} + \definecolor{ansi-green-intense}{HTML}{007427} + \definecolor{ansi-yellow}{HTML}{DDB62B} + \definecolor{ansi-yellow-intense}{HTML}{B27D12} + 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PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}} +\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}} +\expandafter\def\csname PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@kp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@kt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}} +\expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@ow\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\expandafter\def\csname PY@tok@nb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@nf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@nc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@nn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@ne\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.82,0.25,0.23}{##1}}} +\expandafter\def\csname PY@tok@nv\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@no\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@nl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.63,0.00}{##1}}} +\expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}} +\expandafter\def\csname PY@tok@na\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.49,0.56,0.16}{##1}}} +\expandafter\def\csname PY@tok@nt\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@nd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\expandafter\def\csname PY@tok@s\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}} +\expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}} +\expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}} +\expandafter\def\csname PY@tok@ss\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@sx\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@m\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@gh\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@gd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@gi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.63,0.00}{##1}}} +\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}} +\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit} +\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf} +\expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}} +\expandafter\def\csname PY@tok@gt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}} +\expandafter\def\csname PY@tok@err\endcsname{\def\PY@bc##1{\setlength{\fboxsep}{0pt}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}} +\expandafter\def\csname PY@tok@kc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@kd\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@kn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@kr\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@bp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\expandafter\def\csname PY@tok@fm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\expandafter\def\csname PY@tok@vc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@vg\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@vm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\expandafter\def\csname PY@tok@sa\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@dl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@s2\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@sh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\expandafter\def\csname PY@tok@mb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\expandafter\def\csname PY@tok@ch\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@cpf\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} +\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}} + +\def\PYZbs{\char`\\} +\def\PYZus{\char`\_} +\def\PYZob{\char`\{} +\def\PYZcb{\char`\}} +\def\PYZca{\char`\^} +\def\PYZam{\char`\&} +\def\PYZlt{\char`\<} +\def\PYZgt{\char`\>} +\def\PYZsh{\char`\#} +\def\PYZpc{\char`\%} +\def\PYZdl{\char`\$} +\def\PYZhy{\char`\-} +\def\PYZsq{\char`\'} +\def\PYZdq{\char`\"} +\def\PYZti{\char`\~} +% for compatibility with earlier versions +\def\PYZat{@} +\def\PYZlb{[} +\def\PYZrb{]} +\makeatother + + + % For linebreaks inside Verbatim environment from package fancyvrb. + \makeatletter + \newbox\Wrappedcontinuationbox + \newbox\Wrappedvisiblespacebox + \newcommand*\Wrappedvisiblespace {\textcolor{red}{\textvisiblespace}} + \newcommand*\Wrappedcontinuationsymbol {\textcolor{red}{\llap{\tiny$\m@th\hookrightarrow$}}} + \newcommand*\Wrappedcontinuationindent {3ex } + \newcommand*\Wrappedafterbreak {\kern\Wrappedcontinuationindent\copy\Wrappedcontinuationbox} + % Take advantage of the already applied Pygments mark-up to insert + % potential linebreaks for TeX processing. + % {, <, #, %, $, ' and ": go to next line. + % _, }, ^, &, >, - and ~: stay at end of broken line. + % Use of \textquotesingle for straight quote. + \newcommand*\Wrappedbreaksatspecials {% + \def\PYGZus{\discretionary{\char`\_}{\Wrappedafterbreak}{\char`\_}}% + \def\PYGZob{\discretionary{}{\Wrappedafterbreak\char`\{}{\char`\{}}% + \def\PYGZcb{\discretionary{\char`\}}{\Wrappedafterbreak}{\char`\}}}% + \def\PYGZca{\discretionary{\char`\^}{\Wrappedafterbreak}{\char`\^}}% + \def\PYGZam{\discretionary{\char`\&}{\Wrappedafterbreak}{\char`\&}}% + \def\PYGZlt{\discretionary{}{\Wrappedafterbreak\char`\<}{\char`\<}}% + \def\PYGZgt{\discretionary{\char`\>}{\Wrappedafterbreak}{\char`\>}}% + \def\PYGZsh{\discretionary{}{\Wrappedafterbreak\char`\#}{\char`\#}}% + \def\PYGZpc{\discretionary{}{\Wrappedafterbreak\char`\%}{\char`\%}}% + \def\PYGZdl{\discretionary{}{\Wrappedafterbreak\char`\$}{\char`\$}}% + \def\PYGZhy{\discretionary{\char`\-}{\Wrappedafterbreak}{\char`\-}}% + \def\PYGZsq{\discretionary{}{\Wrappedafterbreak\textquotesingle}{\textquotesingle}}% + \def\PYGZdq{\discretionary{}{\Wrappedafterbreak\char`\"}{\char`\"}}% + \def\PYGZti{\discretionary{\char`\~}{\Wrappedafterbreak}{\char`\~}}% + } + % Some characters . , ; ? ! / are not pygmentized. + % This macro makes them "active" and they will insert potential linebreaks + \newcommand*\Wrappedbreaksatpunct {% + \lccode`\~`\.\lowercase{\def~}{\discretionary{\hbox{\char`\.}}{\Wrappedafterbreak}{\hbox{\char`\.}}}% + \lccode`\~`\,\lowercase{\def~}{\discretionary{\hbox{\char`\,}}{\Wrappedafterbreak}{\hbox{\char`\,}}}% + \lccode`\~`\;\lowercase{\def~}{\discretionary{\hbox{\char`\;}}{\Wrappedafterbreak}{\hbox{\char`\;}}}% + \lccode`\~`\:\lowercase{\def~}{\discretionary{\hbox{\char`\:}}{\Wrappedafterbreak}{\hbox{\char`\:}}}% + \lccode`\~`\?\lowercase{\def~}{\discretionary{\hbox{\char`\?}}{\Wrappedafterbreak}{\hbox{\char`\?}}}% + \lccode`\~`\!\lowercase{\def~}{\discretionary{\hbox{\char`\!}}{\Wrappedafterbreak}{\hbox{\char`\!}}}% + \lccode`\~`\/\lowercase{\def~}{\discretionary{\hbox{\char`\/}}{\Wrappedafterbreak}{\hbox{\char`\/}}}% + \catcode`\.\active + \catcode`\,\active + \catcode`\;\active + \catcode`\:\active + \catcode`\?\active + \catcode`\!\active + \catcode`\/\active + \lccode`\~`\~ + } + \makeatother + + \let\OriginalVerbatim=\Verbatim + \makeatletter + \renewcommand{\Verbatim}[1][1]{% + %\parskip\z@skip + \sbox\Wrappedcontinuationbox {\Wrappedcontinuationsymbol}% + \sbox\Wrappedvisiblespacebox {\FV@SetupFont\Wrappedvisiblespace}% + \def\FancyVerbFormatLine ##1{\hsize\linewidth + \vtop{\raggedright\hyphenpenalty\z@\exhyphenpenalty\z@ + \doublehyphendemerits\z@\finalhyphendemerits\z@ + \strut ##1\strut}% + }% + % If the linebreak is at a space, the latter will be displayed as visible + % space at end of first line, and a continuation symbol starts next line. + % Stretch/shrink are however usually zero for typewriter font. + \def\FV@Space {% + \nobreak\hskip\z@ plus\fontdimen3\font minus\fontdimen4\font + \discretionary{\copy\Wrappedvisiblespacebox}{\Wrappedafterbreak} + {\kern\fontdimen2\font}% + }% + + % Allow breaks at special characters using \PYG... macros. + \Wrappedbreaksatspecials + % Breaks at punctuation characters . , ; ? ! and / need catcode=\active + \OriginalVerbatim[#1,codes*=\Wrappedbreaksatpunct]% + } + \makeatother + + % Exact colors from NB + \definecolor{incolor}{HTML}{303F9F} + \definecolor{outcolor}{HTML}{D84315} + \definecolor{cellborder}{HTML}{CFCFCF} + \definecolor{cellbackground}{HTML}{F7F7F7} + + % prompt + \makeatletter + \newcommand{\boxspacing}{\kern\kvtcb@left@rule\kern\kvtcb@boxsep} + \makeatother + \newcommand{\prompt}[4]{ + \ttfamily\llap{{\color{#2}[#3]:\hspace{3pt}#4}}\vspace{-\baselineskip} + } + + + + % Prevent overflowing lines due to hard-to-break entities + \sloppy + % Setup hyperref package + \hypersetup{ + breaklinks=true, % so long urls are correctly broken across lines + colorlinks=true, + urlcolor=urlcolor, + linkcolor=linkcolor, + citecolor=citecolor, + } + % Slightly bigger margins than the latex defaults + + \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} + + + +\begin{document} + + \maketitle + + + + + \emph{For this exercise you should have received this text in .ipynb +format. Complete the exercises by modifying this file, and submit the +modified version} + + \textbf{Exercise 1 (6 points)} + +Use Sage to find the intersection points \emph{in the real plane} (that +is, only those points such that \emph{both} coordinates are real +numbers) of the following pairs of geometric objects: + +\begin{itemize} +\tightlist +\item + The circle of equation \(x^2 + y^2 = 4\) and the ellipse of equation + \(\left(\frac x2\right)^2 + (2y)^2 = 4\). +\item + The circle of equation \(x^2 + y^2 = 4\) and the ellipse of equation + \(\left(\frac x2-2\right)^2 + (2y)^2 = 4\). +\item + The curve of equation \(y^2 = x^3 -x +1\) and the horizontal line + \(y=10\). +\item + The \(x\)-axis and the graph of the function + \(f(x)=\log(x) - e^{-x}\). \emph{Hint: \(f(x)\) has only one real + zero.} +\end{itemize} + + +\vspace{0.5cm} + \textbf{Exercise 2 (6 points)} + +\begin{enumerate} +\def\labelenumi{(\alph{enumi})} +\tightlist +\item + Use Sage to compute +\end{enumerate} + +\begin{itemize} +\tightlist +\item + the derivative +\item + a primite (i.e.~integral) +\item + the power series expansion around \(0\) up to order \(4\) +\end{itemize} + +of the following functions: +\begin{itemize} +\item \(f(x) = e^x\) +\item \(f(x) = \sin(x)\) +\item \(f(x) = \cos(x)\) +\item \(f(x) = \tan(x)\) +\item \(f(x) = \log(1+x)\) +\item \(f(x) = \sqrt[3]{1+x}\) +\end{itemize} + +\begin{enumerate} +\def\labelenumi{(\alph{enumi})} +\setcounter{enumi}{1} +\item + Use Sage to get the Latex code that represents the objects you + computed above. +\item + 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. \emph{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.} +\end{enumerate} + + +\vspace{0.5cm} + \textbf{Exercise 3 (4 points)} + +The equation \begin{align*} +y^2+x^{16}=1 +\end{align*} 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. + + +\vspace{0.5cm} + \textbf{Exercise 4 (12 points)} + +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. + +\begin{enumerate} +\def\labelenumi{(\alph{enumi})} +\item + Using Sage, find a solution for the differential equation with initial + conditions \begin{align*} + \begin{cases} + P'(t)&=\frac{P(t)}{10}-b\\ + P(1)&=1000 + \end{cases} + \end{align*} where \(b\) is a generic constant. +\item + The list \texttt{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: +\end{enumerate} + +\begin{itemize} +\tightlist +\item + The data as a bar chart. +\item + A curve that interpolates the data, using one of the methods shown in + class. +\item + The solution of the differential equation for \(b=0\). +\item + 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 \(b=0\). + \emph{(For this last point there is no right or wrong choice, just + pick one that looks good)} +\end{itemize} + + \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder] +\prompt{In}{incolor}{ }{\boxspacing} +\begin{Verbatim}[commandchars=\\\{\}] +\PY{n}{data} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1000}\PY{p}{,} \PY{l+m+mi}{1123}\PY{p}{,} \PY{l+m+mi}{1223}\PY{p}{,} \PY{l+m+mi}{1190}\PY{p}{,} \PY{l+m+mi}{1432}\PY{p}{,} \PY{l+m+mi}{1553}\PY{p}{,} \PY{l+m+mi}{1709}\PY{p}{,} \PY{l+m+mi}{1826}\PY{p}{,} \PY{l+m+mi}{1980}\PY{p}{,} \PY{l+m+mi}{2146}\PY{p}{,} \PY{l+m+mi}{2172}\PY{p}{,} \PY{l+m+mi}{2383}\PY{p}{,} \PY{l+m+mi}{2588}\PY{p}{,} \PY{l+m+mi}{2822}\PY{p}{,} \PY{l+m+mi}{3401}\PY{p}{,} \PY{l+m+mi}{3330}\PY{p}{,} \PY{l+m+mi}{4157}\PY{p}{,} \PY{l+m+mi}{3994}\PY{p}{,} \PY{l+m+mi}{4995}\PY{p}{,} \PY{l+m+mi}{5392}\PY{p}{,} \PY{l+m+mi}{5910}\PY{p}{,} \PY{l+m+mi}{6468}\PY{p}{,} \PY{l+m+mi}{7128}\PY{p}{,} \PY{l+m+mi}{7325}\PY{p}{,} \PY{l+m+mi}{7984}\PY{p}{,} \PY{l+m+mi}{9634}\PY{p}{,} \PY{l+m+mi}{10473}\PY{p}{,} \PY{l+m+mi}{11761}\PY{p}{,} \PY{l+m+mi}{12777}\PY{p}{]} +\end{Verbatim} +\end{tcolorbox} + +\vspace{0.5cm} + \textbf{Grading} + +This homework assignment is worth \(28\) (\(24+4\)) points, distributed +as described above. + +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 + +\begin{align*} +\operatorname{grade} = \operatorname{min}\left(20, \left\lfloor \frac{\operatorname{total}}{4} + 0.5\right\rfloor\right) +\end{align*} + + + % Add a bibliography block to the postdoc + + + +\end{document} -- cgit v1.3