mathsoftware

A course about LaTeX and SageMath
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Homework4.tex (26027B)


      1 \documentclass[11pt]{article}
      2 
      3     \usepackage[breakable]{tcolorbox}
      4     \usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
      5     
      6     \usepackage{iftex}
      7     \ifPDFTeX
      8     	\usepackage[T1]{fontenc}
      9     	\usepackage{mathpazo}
     10     \else
     11     	\usepackage{fontspec}
     12     \fi
     13 
     14     % Basic figure setup, for now with no caption control since it's done
     15     % automatically by Pandoc (which extracts ![](path) syntax from Markdown).
     16     \usepackage{graphicx}
     17     % Maintain compatibility with old templates. Remove in nbconvert 6.0
     18     \let\Oldincludegraphics\includegraphics
     19     % Ensure that by default, figures have no caption (until we provide a
     20     % proper Figure object with a Caption API and a way to capture that
     21     % in the conversion process - todo).
     22     \usepackage{caption}
     23     \DeclareCaptionFormat{nocaption}{}
     24     \captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
     25 
     26     \usepackage[Export]{adjustbox} % Used to constrain images to a maximum size
     27     \adjustboxset{max size={0.9\linewidth}{0.9\paperheight}}
     28     \usepackage{float}
     29     \floatplacement{figure}{H} % forces figures to be placed at the correct location
     30     \usepackage{xcolor} % Allow colors to be defined
     31     \usepackage{enumerate} % Needed for markdown enumerations to work
     32     \usepackage{geometry} % Used to adjust the document margins
     33     \usepackage{amsmath} % Equations
     34     \usepackage{amssymb} % Equations
     35     \usepackage{textcomp} % defines textquotesingle
     36     % Hack from http://tex.stackexchange.com/a/47451/13684:
     37     \AtBeginDocument{%
     38         \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
     39     }
     40     \usepackage{upquote} % Upright quotes for verbatim code
     41     \usepackage{eurosym} % defines \euro
     42     \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
     43     \usepackage{fancyvrb} % verbatim replacement that allows latex
     44     \usepackage{grffile} % extends the file name processing of package graphics 
     45                          % to support a larger range
     46     \makeatletter % fix for grffile with XeLaTeX
     47     \def\Gread@@xetex#1{%
     48       \IfFileExists{"\Gin@base".bb}%
     49       {\Gread@eps{\Gin@base.bb}}%
     50       {\Gread@@xetex@aux#1}%
     51     }
     52     \makeatother
     53 
     54     % The hyperref package gives us a pdf with properly built
     55     % internal navigation ('pdf bookmarks' for the table of contents,
     56     % internal cross-reference links, web links for URLs, etc.)
     57     \usepackage{hyperref}
     58     % The default LaTeX title has an obnoxious amount of whitespace. By default,
     59     % titling removes some of it. It also provides customization options.
     60     \usepackage{titling}
     61     \usepackage{longtable} % longtable support required by pandoc >1.10
     62     \usepackage{booktabs}  % table support for pandoc > 1.12.2
     63     \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
     64     \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
     65                                 % normalem makes italics be italics, not underlines
     66     \usepackage{mathrsfs}
     67     
     68 
     69     
     70     % Colors for the hyperref package
     71     \definecolor{urlcolor}{rgb}{0,.145,.698}
     72     \definecolor{linkcolor}{rgb}{.71,0.21,0.01}
     73     \definecolor{citecolor}{rgb}{.12,.54,.11}
     74 
     75     % ANSI colors
     76     \definecolor{ansi-black}{HTML}{3E424D}
     77     \definecolor{ansi-black-intense}{HTML}{282C36}
     78     \definecolor{ansi-red}{HTML}{E75C58}
     79     \definecolor{ansi-red-intense}{HTML}{B22B31}
     80     \definecolor{ansi-green}{HTML}{00A250}
     81     \definecolor{ansi-green-intense}{HTML}{007427}
     82     \definecolor{ansi-yellow}{HTML}{DDB62B}
     83     \definecolor{ansi-yellow-intense}{HTML}{B27D12}
     84     \definecolor{ansi-blue}{HTML}{208FFB}
     85     \definecolor{ansi-blue-intense}{HTML}{0065CA}
     86     \definecolor{ansi-magenta}{HTML}{D160C4}
     87     \definecolor{ansi-magenta-intense}{HTML}{A03196}
     88     \definecolor{ansi-cyan}{HTML}{60C6C8}
     89     \definecolor{ansi-cyan-intense}{HTML}{258F8F}
     90     \definecolor{ansi-white}{HTML}{C5C1B4}
     91     \definecolor{ansi-white-intense}{HTML}{A1A6B2}
     92     \definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF}
     93     \definecolor{ansi-default-inverse-bg}{HTML}{000000}
     94 
     95     % commands and environments needed by pandoc snippets
     96     % extracted from the output of `pandoc -s`
     97     \providecommand{\tightlist}{%
     98       \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
     99     \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
    100     % Add ',fontsize=\small' for more characters per line
    101     \newenvironment{Shaded}{}{}
    102     \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
    103     \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
    104     \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
    105     \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
    106     \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
    107     \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
    108     \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
    109     \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
    110     \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
    111     \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
    112     \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
    113     \newcommand{\RegionMarkerTok}[1]{{#1}}
    114     \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
    115     \newcommand{\NormalTok}[1]{{#1}}
    116     
    117     % Additional commands for more recent versions of Pandoc
    118     \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
    119     \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
    120     \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
    121     \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
    122     \newcommand{\ImportTok}[1]{{#1}}
    123     \newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
    124     \newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
    125     \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
    126     \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
    127     \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
    128     \newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
    129     \newcommand{\BuiltInTok}[1]{{#1}}
    130     \newcommand{\ExtensionTok}[1]{{#1}}
    131     \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
    132     \newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
    133     \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
    134     \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
    135     
    136     
    137     % Define a nice break command that doesn't care if a line doesn't already
    138     % exist.
    139     \def\br{\hspace*{\fill} \\* }
    140     % Math Jax compatibility definitions
    141     \def\gt{>}
    142     \def\lt{<}
    143     \let\Oldtex\TeX
    144     \let\Oldlatex\LaTeX
    145     \renewcommand{\TeX}{\textrm{\Oldtex}}
    146     \renewcommand{\LaTeX}{\textrm{\Oldlatex}}
    147     % Document parameters
    148     % Document title
    149     \title{Mathematical Software - Homework 4}
    150     \date{\textbf{Deadline:} Sunday, June 6}
    151     
    152     
    153     
    154     
    155     
    156 % Pygments definitions
    157 \makeatletter
    158 \def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
    159     \let\PY@ul=\relax \let\PY@tc=\relax%
    160     \let\PY@bc=\relax \let\PY@ff=\relax}
    161 \def\PY@tok#1{\csname PY@tok@#1\endcsname}
    162 \def\PY@toks#1+{\ifx\relax#1\empty\else%
    163     \PY@tok{#1}\expandafter\PY@toks\fi}
    164 \def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
    165     \PY@it{\PY@bf{\PY@ff{#1}}}}}}}
    166 \def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
    167 
    168 \expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
    169 \expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
    170 \expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}}
    171 \expandafter\def\csname PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    172 \expandafter\def\csname PY@tok@kp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    173 \expandafter\def\csname PY@tok@kt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}}
    174 \expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    175 \expandafter\def\csname PY@tok@ow\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
    176 \expandafter\def\csname PY@tok@nb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    177 \expandafter\def\csname PY@tok@nf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
    178 \expandafter\def\csname PY@tok@nc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
    179 \expandafter\def\csname PY@tok@nn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
    180 \expandafter\def\csname PY@tok@ne\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.82,0.25,0.23}{##1}}}
    181 \expandafter\def\csname PY@tok@nv\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
    182 \expandafter\def\csname PY@tok@no\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}}
    183 \expandafter\def\csname PY@tok@nl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.63,0.00}{##1}}}
    184 \expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}}
    185 \expandafter\def\csname PY@tok@na\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.49,0.56,0.16}{##1}}}
    186 \expandafter\def\csname PY@tok@nt\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    187 \expandafter\def\csname PY@tok@nd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
    188 \expandafter\def\csname PY@tok@s\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    189 \expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    190 \expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
    191 \expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}}
    192 \expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
    193 \expandafter\def\csname PY@tok@ss\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
    194 \expandafter\def\csname PY@tok@sx\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    195 \expandafter\def\csname PY@tok@m\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    196 \expandafter\def\csname PY@tok@gh\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
    197 \expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}}
    198 \expandafter\def\csname PY@tok@gd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}}
    199 \expandafter\def\csname PY@tok@gi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.63,0.00}{##1}}}
    200 \expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}}
    201 \expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit}
    202 \expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf}
    203 \expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
    204 \expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}}
    205 \expandafter\def\csname PY@tok@gt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}}
    206 \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}}}
    207 \expandafter\def\csname PY@tok@kc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    208 \expandafter\def\csname PY@tok@kd\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    209 \expandafter\def\csname PY@tok@kn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    210 \expandafter\def\csname PY@tok@kr\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    211 \expandafter\def\csname PY@tok@bp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
    212 \expandafter\def\csname PY@tok@fm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
    213 \expandafter\def\csname PY@tok@vc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
    214 \expandafter\def\csname PY@tok@vg\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
    215 \expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
    216 \expandafter\def\csname PY@tok@vm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
    217 \expandafter\def\csname PY@tok@sa\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    218 \expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    219 \expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    220 \expandafter\def\csname PY@tok@dl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    221 \expandafter\def\csname PY@tok@s2\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    222 \expandafter\def\csname PY@tok@sh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    223 \expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
    224 \expandafter\def\csname PY@tok@mb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    225 \expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    226 \expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    227 \expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    228 \expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    229 \expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
    230 \expandafter\def\csname PY@tok@ch\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
    231 \expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
    232 \expandafter\def\csname PY@tok@cpf\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
    233 \expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
    234 \expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
    235 
    236 \def\PYZbs{\char`\\}
    237 \def\PYZus{\char`\_}
    238 \def\PYZob{\char`\{}
    239 \def\PYZcb{\char`\}}
    240 \def\PYZca{\char`\^}
    241 \def\PYZam{\char`\&}
    242 \def\PYZlt{\char`\<}
    243 \def\PYZgt{\char`\>}
    244 \def\PYZsh{\char`\#}
    245 \def\PYZpc{\char`\%}
    246 \def\PYZdl{\char`\$}
    247 \def\PYZhy{\char`\-}
    248 \def\PYZsq{\char`\'}
    249 \def\PYZdq{\char`\"}
    250 \def\PYZti{\char`\~}
    251 % for compatibility with earlier versions
    252 \def\PYZat{@}
    253 \def\PYZlb{[}
    254 \def\PYZrb{]}
    255 \makeatother
    256 
    257 
    258     % For linebreaks inside Verbatim environment from package fancyvrb. 
    259     \makeatletter
    260         \newbox\Wrappedcontinuationbox 
    261         \newbox\Wrappedvisiblespacebox 
    262         \newcommand*\Wrappedvisiblespace {\textcolor{red}{\textvisiblespace}} 
    263         \newcommand*\Wrappedcontinuationsymbol {\textcolor{red}{\llap{\tiny$\m@th\hookrightarrow$}}} 
    264         \newcommand*\Wrappedcontinuationindent {3ex } 
    265         \newcommand*\Wrappedafterbreak {\kern\Wrappedcontinuationindent\copy\Wrappedcontinuationbox} 
    266         % Take advantage of the already applied Pygments mark-up to insert 
    267         % potential linebreaks for TeX processing. 
    268         %        {, <, #, %, $, ' and ": go to next line. 
    269         %        _, }, ^, &, >, - and ~: stay at end of broken line. 
    270         % Use of \textquotesingle for straight quote. 
    271         \newcommand*\Wrappedbreaksatspecials {% 
    272             \def\PYGZus{\discretionary{\char`\_}{\Wrappedafterbreak}{\char`\_}}% 
    273             \def\PYGZob{\discretionary{}{\Wrappedafterbreak\char`\{}{\char`\{}}% 
    274             \def\PYGZcb{\discretionary{\char`\}}{\Wrappedafterbreak}{\char`\}}}% 
    275             \def\PYGZca{\discretionary{\char`\^}{\Wrappedafterbreak}{\char`\^}}% 
    276             \def\PYGZam{\discretionary{\char`\&}{\Wrappedafterbreak}{\char`\&}}% 
    277             \def\PYGZlt{\discretionary{}{\Wrappedafterbreak\char`\<}{\char`\<}}% 
    278             \def\PYGZgt{\discretionary{\char`\>}{\Wrappedafterbreak}{\char`\>}}% 
    279             \def\PYGZsh{\discretionary{}{\Wrappedafterbreak\char`\#}{\char`\#}}% 
    280             \def\PYGZpc{\discretionary{}{\Wrappedafterbreak\char`\%}{\char`\%}}% 
    281             \def\PYGZdl{\discretionary{}{\Wrappedafterbreak\char`\$}{\char`\$}}% 
    282             \def\PYGZhy{\discretionary{\char`\-}{\Wrappedafterbreak}{\char`\-}}% 
    283             \def\PYGZsq{\discretionary{}{\Wrappedafterbreak\textquotesingle}{\textquotesingle}}% 
    284             \def\PYGZdq{\discretionary{}{\Wrappedafterbreak\char`\"}{\char`\"}}% 
    285             \def\PYGZti{\discretionary{\char`\~}{\Wrappedafterbreak}{\char`\~}}% 
    286         } 
    287         % Some characters . , ; ? ! / are not pygmentized. 
    288         % This macro makes them "active" and they will insert potential linebreaks 
    289         \newcommand*\Wrappedbreaksatpunct {% 
    290             \lccode`\~`\.\lowercase{\def~}{\discretionary{\hbox{\char`\.}}{\Wrappedafterbreak}{\hbox{\char`\.}}}% 
    291             \lccode`\~`\,\lowercase{\def~}{\discretionary{\hbox{\char`\,}}{\Wrappedafterbreak}{\hbox{\char`\,}}}% 
    292             \lccode`\~`\;\lowercase{\def~}{\discretionary{\hbox{\char`\;}}{\Wrappedafterbreak}{\hbox{\char`\;}}}% 
    293             \lccode`\~`\:\lowercase{\def~}{\discretionary{\hbox{\char`\:}}{\Wrappedafterbreak}{\hbox{\char`\:}}}% 
    294             \lccode`\~`\?\lowercase{\def~}{\discretionary{\hbox{\char`\?}}{\Wrappedafterbreak}{\hbox{\char`\?}}}% 
    295             \lccode`\~`\!\lowercase{\def~}{\discretionary{\hbox{\char`\!}}{\Wrappedafterbreak}{\hbox{\char`\!}}}% 
    296             \lccode`\~`\/\lowercase{\def~}{\discretionary{\hbox{\char`\/}}{\Wrappedafterbreak}{\hbox{\char`\/}}}% 
    297             \catcode`\.\active
    298             \catcode`\,\active 
    299             \catcode`\;\active
    300             \catcode`\:\active
    301             \catcode`\?\active
    302             \catcode`\!\active
    303             \catcode`\/\active 
    304             \lccode`\~`\~ 	
    305         }
    306     \makeatother
    307 
    308     \let\OriginalVerbatim=\Verbatim
    309     \makeatletter
    310     \renewcommand{\Verbatim}[1][1]{%
    311         %\parskip\z@skip
    312         \sbox\Wrappedcontinuationbox {\Wrappedcontinuationsymbol}%
    313         \sbox\Wrappedvisiblespacebox {\FV@SetupFont\Wrappedvisiblespace}%
    314         \def\FancyVerbFormatLine ##1{\hsize\linewidth
    315             \vtop{\raggedright\hyphenpenalty\z@\exhyphenpenalty\z@
    316                 \doublehyphendemerits\z@\finalhyphendemerits\z@
    317                 \strut ##1\strut}%
    318         }%
    319         % If the linebreak is at a space, the latter will be displayed as visible
    320         % space at end of first line, and a continuation symbol starts next line.
    321         % Stretch/shrink are however usually zero for typewriter font.
    322         \def\FV@Space {%
    323             \nobreak\hskip\z@ plus\fontdimen3\font minus\fontdimen4\font
    324             \discretionary{\copy\Wrappedvisiblespacebox}{\Wrappedafterbreak}
    325             {\kern\fontdimen2\font}%
    326         }%
    327         
    328         % Allow breaks at special characters using \PYG... macros.
    329         \Wrappedbreaksatspecials
    330         % Breaks at punctuation characters . , ; ? ! and / need catcode=\active 	
    331         \OriginalVerbatim[#1,codes*=\Wrappedbreaksatpunct]%
    332     }
    333     \makeatother
    334 
    335     % Exact colors from NB
    336     \definecolor{incolor}{HTML}{303F9F}
    337     \definecolor{outcolor}{HTML}{D84315}
    338     \definecolor{cellborder}{HTML}{CFCFCF}
    339     \definecolor{cellbackground}{HTML}{F7F7F7}
    340     
    341     % prompt
    342     \makeatletter
    343     \newcommand{\boxspacing}{\kern\kvtcb@left@rule\kern\kvtcb@boxsep}
    344     \makeatother
    345     \newcommand{\prompt}[4]{
    346         \ttfamily\llap{{\color{#2}[#3]:\hspace{3pt}#4}}\vspace{-\baselineskip}
    347     }
    348     
    349 
    350     
    351     % Prevent overflowing lines due to hard-to-break entities
    352     \sloppy 
    353     % Setup hyperref package
    354     \hypersetup{
    355       breaklinks=true,  % so long urls are correctly broken across lines
    356       colorlinks=true,
    357       urlcolor=urlcolor,
    358       linkcolor=linkcolor,
    359       citecolor=citecolor,
    360       }
    361     % Slightly bigger margins than the latex defaults
    362     
    363     \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
    364     
    365     
    366 
    367 \begin{document}
    368     
    369     \maketitle
    370     
    371     
    372 
    373     
    374     \emph{For this exercise you should have received this text in .ipynb
    375 format. Complete the exercises by modifying this file, and submit the
    376 modified version}
    377 
    378     \textbf{Exercise 1 (6 points)}
    379 
    380 Use Sage to find the intersection points \emph{in the real plane} (that
    381 is, only those points such that \emph{both} coordinates are real
    382 numbers) of the following pairs of geometric objects:
    383 
    384 \begin{itemize}
    385 \tightlist
    386 \item
    387   The circle of equation \(x^2 + y^2 = 4\) and the ellipse of equation
    388   \(\left(\frac x2\right)^2 + (2y)^2 = 4\).
    389 \item
    390   The circle of equation \(x^2 + y^2 = 4\) and the ellipse of equation
    391   \(\left(\frac x2-2\right)^2 + (2y)^2 = 4\).
    392 \item
    393   The curve of equation \(y^2 = x^3 -x +1\) and the horizontal line
    394   \(y=10\).
    395 \item
    396   The \(x\)-axis and the graph of the function
    397   \(f(x)=\log(x) - e^{-x}\). \emph{Hint: \(f(x)\) has only one real
    398   zero.}
    399 \end{itemize}
    400 
    401 
    402 \vspace{0.5cm}
    403     \textbf{Exercise 2 (6 points)}
    404 
    405 \begin{enumerate}
    406 \def\labelenumi{(\alph{enumi})}
    407 \tightlist
    408 \item
    409   Use Sage to compute
    410 \end{enumerate}
    411 
    412 \begin{itemize}
    413 \tightlist
    414 \item
    415   the derivative
    416 \item
    417   a primite (i.e.~integral)
    418 \item
    419   the power series expansion around \(0\) up to order \(4\)
    420 \end{itemize}
    421 
    422 of the following functions: 
    423 \begin{itemize}
    424 \item \(f(x) = e^x\) 
    425 \item \(f(x) = \sin(x)\)
    426 \item \(f(x) = \cos(x)\) 
    427 \item \(f(x) = \tan(x)\)
    428 \item \(f(x) = \log(1+x)\)
    429 \item \(f(x) = \sqrt[3]{1+x}\)
    430 \end{itemize}
    431 
    432 \begin{enumerate}
    433 \def\labelenumi{(\alph{enumi})}
    434 \setcounter{enumi}{1}
    435 \item
    436   Use Sage to get the Latex code that represents the objects you
    437   computed above.
    438 \item
    439   Arrange the results of the previous points in a table in Latex. The
    440   table should have 4 columns (function, derivative, integral, series)
    441   and one row for each of the functions above. \emph{Note: when
    442   including Latex in a Markdown cell in Jupyter you will not receive any
    443   warning if you make mistakes; instead the Latex will simply not be
    444   rendered and it will appear as plain text. If you have troubles making
    445   this work you can send me a separate .tex (and .pdf) file.}
    446 \end{enumerate}
    447 
    448 
    449 \vspace{0.5cm}
    450     \textbf{Exercise 3 (4 points)}
    451 
    452 The equation \begin{align*}
    453 y^2+x^{16}=1
    454 \end{align*} determines a closed curve in \(\mathbb R^2\) that looks
    455 like a rounded square. Determine the area of that shape, giving both an
    456 exact value (which might depend on some functions that Sage knows, but
    457 you don't) and an approximate value.
    458 
    459 
    460 \vspace{0.5cm}
    461     \textbf{Exercise 4 (12 points)}
    462 
    463 A team of biologists is monitoring the population of river shrimps in
    464 the Alzette. At first they thought that the size \(P(t)\) of their
    465 population on day \(t\) would satisfy the differential equation
    466 \(P'(t)=P(t)/10\). However this does not work well with the data they
    467 have collected, so they now believe that the population of shrimps
    468 follows the formula \(P'(t)=P(t)/10-b\) for some value of \(b\) between
    469 1 and 100. They need your help here.
    470 
    471 \begin{enumerate}
    472 \def\labelenumi{(\alph{enumi})}
    473 \item
    474   Using Sage, find a solution for the differential equation with initial
    475   conditions \begin{align*}
    476   \begin{cases}
    477   P'(t)&=\frac{P(t)}{10}-b\\
    478   P(1)&=1000
    479   \end{cases}
    480   \end{align*} where \(b\) is a generic constant.
    481 \item
    482   The list \texttt{data} in the cell below contains the actual number of
    483   shrimps that was measured every day from day \(1\) (the \(0\) at the
    484   beginning is meaningless, but it will help to keep it there). Plot in
    485   one single picture, possibly using different colors for each:
    486 \end{enumerate}
    487 
    488 \begin{itemize}
    489 \tightlist
    490 \item
    491   The data as a bar chart.
    492 \item
    493   A curve that interpolates the data, using one of the methods shown in
    494   class.
    495 \item
    496   The solution of the differential equation for \(b=0\).
    497 \item
    498   The solution of the differential equation for a value of \(b\) of your
    499   choice (\(1\leq b\leq 100\)) that fits the data better than \(b=0\).
    500   \emph{(For this last point there is no right or wrong choice, just
    501   pick one that looks good)}
    502 \end{itemize}
    503 
    504     \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
    505 \prompt{In}{incolor}{ }{\boxspacing}
    506 \begin{Verbatim}[commandchars=\\\{\}]
    507 \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}{]}
    508 \end{Verbatim}
    509 \end{tcolorbox}
    510 
    511 \vspace{0.5cm}
    512     \textbf{Grading}
    513 
    514 This homework assignment is worth \(28\) (\(24+4\)) points, distributed
    515 as described above.
    516 
    517 Your final grade for the course will be the total of points you obtained
    518 (notice that the maximum is \(20+20+16+28=84\)) divided by \(4\),
    519 rounded to the nearest integer. More precisely
    520 
    521 \begin{align*}
    522 \operatorname{grade} = \operatorname{min}\left(20, \left\lfloor \frac{\operatorname{total}}{4} + 0.5\right\rfloor\right)
    523 \end{align*}
    524 
    525 
    526     % Add a bibliography block to the postdoc
    527     
    528     
    529     
    530 \end{document}