From 5fec40e88d4e560f325a95f9e2403a45e2b294e6 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Wed, 2 Feb 2022 10:28:03 +0100 Subject: Removed retake source (for now) --- src/Retake/Retake1.tex | 208 ------------------------------------------------- 1 file changed, 208 deletions(-) delete mode 100644 src/Retake/Retake1.tex (limited to 'src/Retake/Retake1.tex') diff --git a/src/Retake/Retake1.tex b/src/Retake/Retake1.tex deleted file mode 100644 index bbff371..0000000 --- a/src/Retake/Retake1.tex +++ /dev/null @@ -1,208 +0,0 @@ -\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