commit f24ed39222d0a822aa9dc86aafd052eda637c586
parent 5fec40e88d4e560f325a95f9e2403a45e2b294e6
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Thu, 10 Feb 2022 10:39:21 +0100
Added retake
Diffstat:
6 files changed, 795 insertions(+), 1 deletion(-)
diff --git a/.gitignore b/.gitignore
@@ -1,3 +1,2 @@
Administration
Homework_submissions
-src/Retake
diff --git a/src/Retake/Retake1.aux b/src/Retake/Retake1.aux
@@ -0,0 +1,4 @@
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diff --git a/src/Retake/Retake1.log b/src/Retake/Retake1.log
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diff --git a/src/Retake/Retake1.pdf b/src/Retake/Retake1.pdf
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diff --git a/src/Retake/Retake1.tex 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}
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