From 462dca8585cb64bcb032ad8b1b8f3b2a6b02e172 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Wed, 2 Feb 2022 10:23:57 +0100 Subject: Changed email address --- src/Retake/Retake1.tex | 208 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 208 insertions(+) create mode 100644 src/Retake/Retake1.tex (limited to 'src/Retake/Retake1.tex') diff --git a/src/Retake/Retake1.tex b/src/Retake/Retake1.tex new file mode 100644 index 0000000..bbff371 --- /dev/null +++ b/src/Retake/Retake1.tex @@ -0,0 +1,208 @@ +\documentclass[12pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{tikz,tikz-cd} +\usepackage{enumitem} +\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} + +\title{Mathematical software - homework 1} +\author{Sebastiano Tronto} + +\newtheorem*{thm}{Theorem} +\newtheorem{prop}{Proposition} + +\theoremstyle{definition} +\newtheorem{ex}{Exercise} + +\theoremstyle{definition} +\newtheorem{remark}{Remark} + +\begin{document} + +\noindent\hrulefill + +\begin{center} +\Huge{\textbf{Mathematical Software - Retake}} +\end{center} + +\noindent\hrulefill +\begin{center} +\begin{tabular}{lcr} +\texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad & +\textbf{Deadline: Wednesday, February 9th} +\end{tabular} +\end{center} + +\vspace{0.3cm} + +\begin{center} + \emph{\large For exercises 1 and 2 submit a .tex and a .pdf file. + For exercises 3 and 4 submit your Sage code either in text format (.txt or + .sage) or as a Jupyter Notebook file (.ipynb). + } +\end{center} + + +\section*{Exercise 1} + Write a short Latex document that contains the following theorem-like + environments using the \texttt{\textbackslash newtheorem} command of the + \texttt{amsthm} package (the box around the text is not needed): + \begin{center} + \fbox{\parbox{0.95\textwidth}{ + \begin{prop}[Fundamental Theorem of Algebra] + \label{prop:fta} + Let \(p(x)\) be a non-constant polynomial with coefficients in + $\mathbb C$. Then there is \(z\in\mathbb C\) such that $p(z)=0$. + \end{prop} + + \begin{remark} + Proposition \ref{prop:fta} is not true for polynomials with + coefficients in $\mathbb R$. For example + \begin{align} + p(x) = x^2+1 + \end{align} + does not have real roots. + \end{remark} + + \begin{thm} + If $X$ and $Y$ are $\sigma$-finite measure spaces and $f:X\times Y\to + \mathbb R$ is measurable and such that + \begin{align*} + \int_{X\times Y}|f(x,y)|\mathrm d(x,y) < \infty + \end{align*} + then + \begin{align} + \label{eq:fubini} + \int_X\left(\int_Yf(x,y)\mathrm d y\right)\mathrm d x = + \int_Y\left(\int_Xf(x,y)\mathrm d x\right)\mathrm d y = + \int_{X\times Y} f(x,y)\mathrm d(x,y)\,. + \end{align} + \end{thm} + + \begin{remark} + In practice, equation \eqref{eq:fubini} means that we can switch the + order of integration in a double integral. + \end{remark} + }} + \end{center} + Notice that Propositions, Remarks and some of the equations are numbered, + and some of them are referred to in the Remarks. This numbering should change + accordingly if more numbered Theorems and equations are added before this + part of the text. + +\newpage + +\section*{Exercise 2} +Create a Latex document containing the following pictures: +\begin{enumerate} +\item[(a)] The following commutative diagram: + \begin{center} + \begin{tikzcd} + M \ar[r, "f"] \ar[d,swap,"i",hook] & A \\ + N \ar[ur, swap, "\tilde f", dashed] + \end{tikzcd} + \end{center} +\item[(b)] A triangle with verteces on a grid, as below. + Moreover, the position of the vertex $C$ below must be easy to + change at will: you should use the + \texttt{\textbackslash pgfmathsetmacro} command to set a value + for its coordinates at the beginning, so that changing only + those numbers makes the whole picture change accordingly (sides, + dots, labels). + \begin{center} + \begin{tikzpicture}[scale=1] + \pgfmathsetmacro{\ax}{2} + \pgfmathsetmacro{\ay}{1} + \pgfmathsetmacro{\bx}{8} + \pgfmathsetmacro{\by}{1} + \pgfmathsetmacro{\cx}{9} + \pgfmathsetmacro{\cy}{7} + + \draw[lightgray!30,thin] (0,0) grid (10,10); + \draw[-] (\ax, \ay) -- (\bx, \by) -- (\cx, \cy) -- cycle; + \filldraw[blue] (\ax, \ay) circle[radius=0.1] node[below left] {$A$}; + \filldraw[blue] (\bx, \by) circle[radius=0.1] node[below right] {$B$}; + \filldraw[red] (\cx, \cy) circle[radius=0.1] node[above] {$C$}; + + \end{tikzpicture} + \end{center} + +\end{enumerate} + +\newpage +\section*{Exercise 3} +Use SageMath to solve the following problems: +\begin{enumerate}[label=(\arabic*)] + \item Find the roots of the following polynomial over $\mathbb Q$, + over $\mathbb R$ and over $\mathbb C$: + \[ p(x) = x^6+x^5-2x^4-3x^3-x^2+2x+2 \] + \item Find the determinant, the trace and the characteristic polynomial + of the following matrix: + \[A= + \left(\begin{array}{rrrrr} + 2 & 3 & 0 & 1 & 2 \\ + 1 & 0 & \frac{1}{2} & 1 & -1 \\ + 0 & 0 & -1 & 0 & 0 \\ + 0 & -1 & -1 & 0 & -1 \\ + -1 & -1 & -1 & -1 & 0 + \end{array}\right) + \] + \item Find the solutions of the linear system $A\mathbf x=\mathbf 0$, where + $A$ is the matrix above and $\mathbf 0$ is the zero vector. + \item Find the points of intersection of the circle of equation $x^2+y^2=4$ + ad the ellipse of equation $\left(\frac{x}{2}\right)^2+(2y)^2=4$ + \item Plot the graph of the function $f(x)=\sqrt{1-x^6}$. + \item Find the area between the $x$-axis and the grap of the function of + the previous point. + \item Find the derivative, a primitive (integral) and the Taylor series + expansion up to order 4 of the function $h(x)=\log(1+x+x^2)$. + \item Use Sage to get the Latex code for the objects you computed in + the previous point. + \item Find a solution for the differential equation with initial conditions + \[ + \begin{cases}g'(x)&=\frac{1}{3}g(x) - 7\\g(1)&=30\end{cases} + \] + \item Draw a bar chart of the data set $[0,1,3,7,5,7,2,8,9,3]$, like the foll + following: + \begin{center} + \includegraphics[scale=0.5]{bc.png} + \end{center} +\end{enumerate} + +\newpage +\section*{Grading} + +\vspace{0.3cm} +\textbf{Exercise 1 (5 points).} +\begin{itemize} + \item A correct use of the \texttt{\textbackslash newtheorem} command is + worth 2 out of 5 points. + \item A correct use of the labelling and reference system is worth 2 points. + \item Reproducing correctly the mathematical formulas is worth 1 point. +\end{itemize} + +\vspace{0.3cm} +\textbf{Exercise 2 (5 points).} +\begin{itemize} + \item Part (a) is worth 2 points: 1 point for having the letters + $M$, $N$ and $A$ in the correct position and the arrows + pointing between them and 1 point for the style of the arrows + and the labels $f$, $i$ and $\tilde f$ in the correct position. + \item Part (b) is worth 3 points: 1 point for drawing a triangle, 1 + point for the other decorative elements (colored dots, grid lines + and labels) and 1 point for having the point $C$ correctly set + as a macro so that it can be changed easily. +\end{itemize} + +\vspace{0.3cm} +\textbf{Exercise 3 (10 points).} +\begin{itemize} + \item Each of the 10 parts is worth 1 point. +\end{itemize} + + + +\end{document} -- cgit v1.3