\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} \definecolor{ansi-blue}{HTML}{208FFB} 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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}