commit 5fec40e88d4e560f325a95f9e2403a45e2b294e6
parent 462dca8585cb64bcb032ad8b1b8f3b2a6b02e172
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Wed, 2 Feb 2022 10:28:03 +0100
Removed retake source (for now)
Diffstat:
6 files changed, 1 insertion(+), 795 deletions(-)
diff --git a/.gitignore b/.gitignore
@@ -1,2 +1,3 @@
Administration
Homework_submissions
+src/Retake
diff --git a/src/Retake/Retake1.aux b/src/Retake/Retake1.aux
@@ -1,4 +0,0 @@
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diff --git a/src/Retake/Retake1.log b/src/Retake/Retake1.log
@@ -1,583 +0,0 @@
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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
@@ -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}
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