\documentclass[11pt]{amsart} \usepackage{graphicx} \usepackage{amssymb} \textwidth = 6.5 in \textheight = 8.3 in \oddsidemargin = 0.0 in \evensidemargin = 0.0 in \topmargin = 0.0 in \headheight = 0.0 in %\headsep = 0.1 in \parskip = 0.2in \parindent = 0.0in \newcommand{\pchoose}[2]{\begin{pmatrix}#1\\ #2\end{pmatrix}} \newcommand{\bchoose}[2]{\begin{bmatrix}#1\\ #2\end{bmatrix}} \title{Assignment \#3} \author{Date: January 29, 2020 \hskip .5in Due: February 12, 2020 } \begin{document} \maketitle Your assignment should include complete sentences and explanations and not just a few equations or numbers. A solution will not receive full credit unless you explain what your answer represents and where it came from. You may discuss the homework with other students in the class, but please write your own solutions. \begin{enumerate} \item Prove or disprove the statements: \begin{enumerate} \item If $x$ is a real number such that $\left|x+2\right|+\left|x\right| \leq 1$, then $\left|x^{2}+2x-1\right| \leq 2$. \item If $x$ is a real number such that $\left|x+2\right|+\left|x\right| \leq 2$, then $\left|x^{2}+2x-1\right| \leq 2$. \item If $x$ is a real number such that $\left|x+2\right|+\left|x\right| \leq 3$, then $\left|x^{2}+2x-1\right| \leq 2$. \item If $x$ is a real number such that $\left|x+2\right|+\left|x\right| \leq 5$, then $\left|x^{2}+2x-1\right| \leq 2$. \end{enumerate} \item Prove or disprove the statements: \begin{enumerate} \item If $z$ is a complex number such that $|z + 1| + |z - 1| \leq 3$, then $| z^2 - 1 | \leq 2$. \item If $z$ is a complex number such that if $| z^2 - 1 | \leq 2$, then $|z + 1| + |z - 1| \leq 3$. \end{enumerate} \item A clock with a face that has the numbers 1 through 12 has three hands that indicate the second, minute and hour of the day. Assume that the center of the clock is at position $(0,0)$, and at noon the end points of the hands are (respectively) at $(0,1)$, $(0,3/4)$, $(0,1/2)$. \begin{enumerate} \item Give the position of the end points of each of the hands at time $t$ where $t$ represents the number of seconds after noon in both polar and rectangular coordinates (make sure that you label which you are using clearly). \item At what times do your equations say that the hands of the clock will all align? \end{enumerate} \end{enumerate} \end{document}