\documentclass[12pt]{report} 

\usepackage{palatino}
\usepackage{epsfig}
\usepackage[margin=0.8in]{geometry}

\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{patterns,arrows,snakes}

\pagestyle{empty} %%% this results in no pagenumbers (footer is empty}

\setlength{\parindent}{0in}
\setlength{\parskip}{0.05in}

\newcommand{\textbox}[1]{
\raisebox{-.06 in}{\begin{tikzpicture}
\node [draw] at (0,0) {#1};
\end{tikzpicture}}\!
}

\baselineskip=20pt

\tikzstyle{printersafe}=[snake=snake,segment amplitude=0 pt]

\newcommand{\dsps}{\displaystyle}
\newcommand{\pp}{\par \noindent}
\newcommand{\newp}{\vfil \eject}

\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 A - Spring 2022 \hfil \pp
\hfil Midterm Exam Number One\hfil  \pp
\hfil April 21st, 2022 \hfil \pp
}}

\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.5in} Student ID no. : \hrulefill

\vfil

\hfil Signature: \hrulefill \hrulefill \hrulefill \hspace{0.5in} Section: \hrulefill 
 
\begin{center}
\qquad \qquad {\Huge
\begin{tabular}{||c|c|r||} \hline 
 1 & 15 &\hspace{10mm} \hfil\\ \hline 
 2 & 15 &       \\ \hline
 3 & 15 &       \\ \hline
 4 & 15 &       \\ \hline
Total & 60 & \\ \hline 
\end{tabular}
}\begin{tikzpicture}
\node [scale=.7] at (0,0) {\em This grid is purely decorative.};
\node [scale=.7] at (0,-.5) {\em The exam is graded online.};
\end{tikzpicture}
\end{center}

\begin{itemize}
\item This exam consists of {\bf FOUR} problems on {\bf FOUR} double-sided pages. The fourth page is left blank for scratch work.
\item Show all work for full credit.
\item You may use a TI-30X IIS (or equivalent) calculator during this
exam. Other calculators and electronic devices are not permitted.
\item You do not need to simplify your answers.
\item If you use a trial-and-error or guess-and-check method
when a more rigorous method is available, you will not receive full credit.
\item \textbox{Draw a box} around your final answer to each problem.
\item {\bf Do not write within 1 centimeter of the edge!} Your exam will be scanned for grading.
\item If you run out of room, write on one of the scratch work pages {\bf and indicate that you have done so}. If you still need more room, raise your hand and ask for an extra page.
\item You may use one hand-written double-sided 8.5" by 11" page of notes.
\item You have 50 minutes to complete the exam.
\end{itemize}


\newpage

You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see first page'' below a problem.

\newpage


\begin{enumerate}

% Circle and line
\item {\bf [15 points]} Jim is standing at the center of a circular ring of radius $10$ meters.\\[.1 in]
At time $t=0$, Naomi is $16$ meters west and $2$ meters south of Jim.\\[.1 in]
Naomi runs in a straight line towards the northernmost point of the ring.\\[.1 in]
Naomi runs at a constant speed of $5$ meters per second while she's outside the ring, but slows down to $3$ meters per second once she enters the ring.
\begin{enumerate}
\item When does Naomi enter the ring?

\vskip 3.5 in

\item When is Naomi closest to Jim?
\end{enumerate}

\newpage

% Parametric stuff
\item {\bf [5 points per part]} Gregg and Bea are walking around the coordinate plane.
\begin{enumerate}
\item Gregg starts at the point $(5,-2)$, and walks towards the point $(-13,7)$ in a straight line at a constant speed, reaching it after $12$ seconds.\\[.1 in]
Write parametric equations for Gregg's location after $t$ seconds.
\vskip 2.5 in
\item Bea starts at the point $(-3,4)$, and walks towards the point $(3,-4)$ at a constant speed of $4$ units per second.\\[.1 in]
Write parametric equations for Bea's location after $t$ seconds.

\vskip 2.5 in
\item When is Gregg due north of Bea?
\end{enumerate}

\newpage

% Multipart function w/ graph
\item For this problem, consider the following multipart function:
\[
f(x) = \begin{cases}
x+6 & \text{if } -6 \leq x < -3\\
-1+\sqrt{25-(x-2)^2} & \text{if } -3 \leq x < 2\\
-1 & \text{if } 2 \leq x < 5
\end{cases}
\]

\begin{enumerate}
\item {\bf [6 points]} Sketch a graph of $f$ here:
\vspace{-.3 in}
\begin{center}
\begin{tikzpicture}
\draw [line width=2pt,<->] (-7.5,0) -- (7.5,0) node[right] {$x$};
\draw [line width=2pt,<->] (0,-2.5) -- (0,4.5) node[above] {$f(x)$};
\foreach \x in {-7,...,7} \draw [dash pattern=on .15 cm off .1 cm,dash phase=.075 cm,printersafe] (\x,-2.25) -- (\x,4.25);
\foreach \y in {-2,...,4} \draw [dash pattern=on .15 cm off .1 cm,dash phase=.075 cm,printersafe] (-7.25,\y) -- (7.25,\y);
\foreach \x in {-7,...,7} \node [scale=.8,anchor=south west] at (\x,0) {\x};
\foreach \y in {-2,-1,1,2,3,4} \node [scale=.8,anchor=south west] at (0,\y) {\y};
\end{tikzpicture}
\end{center}

\item {\bf [3 points]} What is the range of $f$?
\vskip 1.5 in

\item {\bf [6 points]} Find all values of $x$ such that $f(x) = 2$.

\end{enumerate}

\newpage

% Parabola
\item Merlin is selling orbs. His profit is a quadratic function of how much he charges.\\[.1 in]
If Merlin gives away the orbs for free, he'll {\bf lose} \$$200$.\\[.1 in]
If Merlin charges \$$10$ per orb, he'll {\bf earn} a profit of \$$280$.\\[.1 in]
If Merlin charges \$$20$ per orb, he'll {\bf earn} a profit of \$$700$.
\begin{enumerate}
\item {\bf [12 points]} Write a function $f(x)$ for Merlin's profit if he charges \$$x$ per orb.
\vskip 5 in
\item {\bf [3 points]} How much should Merlin charge to maximize his profit?
\end{enumerate}

\newpage

You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see back page'' below a problem.

\newpage

You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see back page'' below a problem.

\end{enumerate}

\end{document}
