path: root/src/Homework4/Homework4.tex

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    % Document parameters
    % Document title
    \title{Mathematical Software - Homework 4}
    \date{\textbf{Deadline:} Sunday, June 6}
    
    
    
    
    
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        \ttfamily\llap{{\color{#2}[#3]:\hspace{3pt}#4}}\vspace{-\baselineskip}
    }
    

    
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      colorlinks=true,
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    \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}