Retake1.tex (6911B)
1 \documentclass[12pt,a4paper]{article} 2 \usepackage[utf8]{inputenc} 3 \usepackage{amsmath} 4 \usepackage{amsfonts} 5 \usepackage{amssymb} 6 \usepackage{amsthm} 7 \usepackage{tikz,tikz-cd} 8 \usepackage{enumitem} 9 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry} 10 11 \title{Mathematical software - homework 1} 12 \author{Sebastiano Tronto} 13 14 \newtheorem*{thm}{Theorem} 15 \newtheorem{prop}{Proposition} 16 17 \theoremstyle{definition} 18 \newtheorem{ex}{Exercise} 19 20 \theoremstyle{definition} 21 \newtheorem{remark}{Remark} 22 23 \begin{document} 24 25 \noindent\hrulefill 26 27 \begin{center} 28 \Huge{\textbf{Mathematical Software - Retake}} 29 \end{center} 30 31 \noindent\hrulefill 32 \begin{center} 33 \begin{tabular}{lcr} 34 \texttt{sebastiano.tronto@uni.lu} & \qquad \qquad \qquad \qquad & 35 \textbf{Deadline: Wednesday, February 9th} 36 \end{tabular} 37 \end{center} 38 39 \vspace{0.3cm} 40 41 \begin{center} 42 \emph{\large For exercises 1 and 2 submit a .tex and a .pdf file. 43 For exercises 3 and 4 submit your Sage code either in text format (.txt or 44 .sage) or as a Jupyter Notebook file (.ipynb). 45 } 46 \end{center} 47 48 49 \section*{Exercise 1} 50 Write a short Latex document that contains the following theorem-like 51 environments using the \texttt{\textbackslash newtheorem} command of the 52 \texttt{amsthm} package (the box around the text is not needed): 53 \begin{center} 54 \fbox{\parbox{0.95\textwidth}{ 55 \begin{prop}[Fundamental Theorem of Algebra] 56 \label{prop:fta} 57 Let \(p(x)\) be a non-constant polynomial with coefficients in 58 $\mathbb C$. Then there is \(z\in\mathbb C\) such that $p(z)=0$. 59 \end{prop} 60 61 \begin{remark} 62 Proposition \ref{prop:fta} is not true for polynomials with 63 coefficients in $\mathbb R$. For example 64 \begin{align} 65 p(x) = x^2+1 66 \end{align} 67 does not have real roots. 68 \end{remark} 69 70 \begin{thm} 71 If $X$ and $Y$ are $\sigma$-finite measure spaces and $f:X\times Y\to 72 \mathbb R$ is measurable and such that 73 \begin{align*} 74 \int_{X\times Y}|f(x,y)|\mathrm d(x,y) < \infty 75 \end{align*} 76 then 77 \begin{align} 78 \label{eq:fubini} 79 \int_X\left(\int_Yf(x,y)\mathrm d y\right)\mathrm d x = 80 \int_Y\left(\int_Xf(x,y)\mathrm d x\right)\mathrm d y = 81 \int_{X\times Y} f(x,y)\mathrm d(x,y)\,. 82 \end{align} 83 \end{thm} 84 85 \begin{remark} 86 In practice, equation \eqref{eq:fubini} means that we can switch the 87 order of integration in a double integral. 88 \end{remark} 89 }} 90 \end{center} 91 Notice that Propositions, Remarks and some of the equations are numbered, 92 and some of them are referred to in the Remarks. This numbering should change 93 accordingly if more numbered Theorems and equations are added before this 94 part of the text. 95 96 \newpage 97 98 \section*{Exercise 2} 99 Create a Latex document containing the following pictures: 100 \begin{enumerate} 101 \item[(a)] The following commutative diagram: 102 \begin{center} 103 \begin{tikzcd} 104 M \ar[r, "f"] \ar[d,swap,"i",hook] & A \\ 105 N \ar[ur, swap, "\tilde f", dashed] 106 \end{tikzcd} 107 \end{center} 108 \item[(b)] A triangle with verteces on a grid, as below. 109 Moreover, the position of the vertex $C$ below must be easy to 110 change at will: you should use the 111 \texttt{\textbackslash pgfmathsetmacro} command to set a value 112 for its coordinates at the beginning, so that changing only 113 those numbers makes the whole picture change accordingly (sides, 114 dots, labels). 115 \begin{center} 116 \begin{tikzpicture}[scale=1] 117 \pgfmathsetmacro{\ax}{2} 118 \pgfmathsetmacro{\ay}{1} 119 \pgfmathsetmacro{\bx}{8} 120 \pgfmathsetmacro{\by}{1} 121 \pgfmathsetmacro{\cx}{9} 122 \pgfmathsetmacro{\cy}{7} 123 124 \draw[lightgray!30,thin] (0,0) grid (10,10); 125 \draw[-] (\ax, \ay) -- (\bx, \by) -- (\cx, \cy) -- cycle; 126 \filldraw[blue] (\ax, \ay) circle[radius=0.1] node[below left] {$A$}; 127 \filldraw[blue] (\bx, \by) circle[radius=0.1] node[below right] {$B$}; 128 \filldraw[red] (\cx, \cy) circle[radius=0.1] node[above] {$C$}; 129 130 \end{tikzpicture} 131 \end{center} 132 133 \end{enumerate} 134 135 \newpage 136 \section*{Exercise 3} 137 Use SageMath to solve the following problems: 138 \begin{enumerate}[label=(\arabic*)] 139 \item Find the roots of the following polynomial over $\mathbb Q$, 140 over $\mathbb R$ and over $\mathbb C$: 141 \[ p(x) = x^6+x^5-2x^4-3x^3-x^2+2x+2 \] 142 \item Find the determinant, the trace and the characteristic polynomial 143 of the following matrix: 144 \[A= 145 \left(\begin{array}{rrrrr} 146 2 & 3 & 0 & 1 & 2 \\ 147 1 & 0 & \frac{1}{2} & 1 & -1 \\ 148 0 & 0 & -1 & 0 & 0 \\ 149 0 & -1 & -1 & 0 & -1 \\ 150 -1 & -1 & -1 & -1 & 0 151 \end{array}\right) 152 \] 153 \item Find the solutions of the linear system $A\mathbf x=\mathbf 0$, where 154 $A$ is the matrix above and $\mathbf 0$ is the zero vector. 155 \item Find the points of intersection of the circle of equation $x^2+y^2=4$ 156 ad the ellipse of equation $\left(\frac{x}{2}\right)^2+(2y)^2=4$ 157 \item Plot the graph of the function $f(x)=\sqrt{1-x^6}$. 158 \item Find the area between the $x$-axis and the grap of the function of 159 the previous point. 160 \item Find the derivative, a primitive (integral) and the Taylor series 161 expansion up to order 4 of the function $h(x)=\log(1+x+x^2)$. 162 \item Use Sage to get the Latex code for the objects you computed in 163 the previous point. 164 \item Find a solution for the differential equation with initial conditions 165 \[ 166 \begin{cases}g'(x)&=\frac{1}{3}g(x) - 7\\g(1)&=30\end{cases} 167 \] 168 \item Draw a bar chart of the data set $[0,1,3,7,5,7,2,8,9,3]$, like the foll 169 following: 170 \begin{center} 171 \includegraphics[scale=0.5]{bc.png} 172 \end{center} 173 \end{enumerate} 174 175 \newpage 176 \section*{Grading} 177 178 \vspace{0.3cm} 179 \textbf{Exercise 1 (5 points).} 180 \begin{itemize} 181 \item A correct use of the \texttt{\textbackslash newtheorem} command is 182 worth 2 out of 5 points. 183 \item A correct use of the labelling and reference system is worth 2 points. 184 \item Reproducing correctly the mathematical formulas is worth 1 point. 185 \end{itemize} 186 187 \vspace{0.3cm} 188 \textbf{Exercise 2 (5 points).} 189 \begin{itemize} 190 \item Part (a) is worth 2 points: 1 point for having the letters 191 $M$, $N$ and $A$ in the correct position and the arrows 192 pointing between them and 1 point for the style of the arrows 193 and the labels $f$, $i$ and $\tilde f$ in the correct position. 194 \item Part (b) is worth 3 points: 1 point for drawing a triangle, 1 195 point for the other decorative elements (colored dots, grid lines 196 and labels) and 1 point for having the point $C$ correctly set 197 as a macro so that it can be changed easily. 198 \end{itemize} 199 200 \vspace{0.3cm} 201 \textbf{Exercise 3 (10 points).} 202 \begin{itemize} 203 \item Each of the 10 parts is worth 1 point. 204 \end{itemize} 205 206 207 208 \end{document}