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% This top part of the document is called the 'preamble'.  Modify it with caution!
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% The real document starts below where it says 'The main document starts here'.

\documentclass[12pt]{article}

\usepackage{amssymb,amsmath,amsthm}
\usepackage[top=1in, bottom=1in, left=1.25in, right=1.25in]{geometry}
\usepackage{fancyhdr}
\usepackage{enumerate}

%% Import a package useful for displaying graphics.
\usepackage{graphicx}

% Comment the following line to use TeX's default font of Computer Modern.
\usepackage{times,txfonts}

\newtheoremstyle{homework}% name of the style to be used
  {18pt}% measure of space to leave above the theorem. E.g.: 3pt
  {12pt}% measure of space to leave below the theorem. E.g.: 3pt
  {}% name of font to use in the body of the theorem
  {}% measure of space to indent
  {\bfseries}% name of head font
  {.}% punctuation between head and body
  {2ex}% space after theorem head; " " = normal interword space
  {}% Manually specify head
\theoremstyle{homework} 

% Set up an Exercise environment and a Solution label.
\newtheorem*{exercisecore}{\@currentlabel}
\newenvironment{exercise}[1]
{\def\@currentlabel{#1}\exercisecore}
{\endexercisecore}

\newcommand{\localhead}[1]{\par\smallskip\noindent\textbf{#1}\nobreak\\}%
\newcommand\solution{\localhead{Solution:}}

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% Stuff for getting the name/document date/title across the header
\makeatletter
\RequirePackage{fancyhdr}
\pagestyle{fancy}
\fancyfoot[C]{\ifnum \value{page} > 1\relax\thepage\fi}
\fancyhead[L]{\ifx\@doclabel\@empty\else\@doclabel\fi}
\fancyhead[C]{\ifx\@docdate\@empty\else\@docdate\fi}
\fancyhead[R]{\ifx\@docauthor\@empty\else\@docauthor\fi}
\headheight 15pt

\def\doclabel#1{\gdef\@doclabel{#1}}
\doclabel{Use {\tt\textbackslash doclabel\{MY LABEL\}}.}
\def\docdate#1{\gdef\@docdate{#1}}
\docdate{Use {\tt\textbackslash docdate\{MY DATE\}}.}
\def\docauthor#1{\gdef\@docauthor{#1}}
\docauthor{Use {\tt\textbackslash docauthor\{MY NAME\}}.}
\makeatother

% Shortcuts for blackboard bold number sets (reals, integers, etc.)
\newcommand{\Reals}{\ensuremath{\mathbb R}}
\newcommand{\Nats}{\ensuremath{\mathbb N}}
\newcommand{\Ints}{\ensuremath{\mathbb Z}}
\newcommand{\Rats}{\ensuremath{\mathbb Q}}
\newcommand{\Cplx}{\ensuremath{\mathbb C}}
%% Some equivalents that some people may prefer.
\let\RR\Reals
\let\NN\Nats
\let\II\Ints
\let\CC\Cplx



% The following commands set up the material that appears
% in the header.
\doclabel{Math 426: Demo Homework}
\docauthor{Your name here.}
\docdate{August 21, 2020}

\begin{document}

\begin{exercise}{Chapter 37:  3}
If $a$ and $b$ are even integers, then so is $a+b$.
\end{exercise}
\begin{proof}
Let $a$ and $b$ be even integers.  Then there exist integers
$j$ and $k$ such that $a=2j$ and $b=2k$.  But then
\begin{equation}
a+b = 2j+ 2k = 2(j+k).
\end{equation}
Since $j+k\in\Ints$, $a+b$ is even.
\end{proof}

\begin{exercise}{Worksheet: 9}
Plot $\sin(x)$ and $\cos(x)$ for $-\pi \le x \le \pi$ on the same graph.
Make sure the graph is labeled nicely.
\end{exercise}
\solution
\begin{verbatim}
octave:1> x=[-pi:0.01:pi];
octave:2> plot(x,sin(x),x,cos(x));
octave:3> set(gca, "fontsize", 14 )
octave:4> xlabel("x");ylabel("y");title("sin and cos");
octave:5> legend("sin","cos");
\end{verbatim}
\begin{center}
\includegraphics[width=5in]{HWDemo-prob2.pdf}
\end{center}
\end{document}