\begin{homework}\
\label{homework:1}

%%% To update
\medskip

\exercise %1
For each pair of points $A(x_1,y_1)$ and $B(x_2,y_2)$ find (i) $\Delta x$
and $\Delta y$ in going from $A$ to $B$, (ii) the slope of the line joining
$A$ and $B$, (iii) the equation of the line joining $A$ and $B$ in the form
$y=mx+b$, and (iv) the distance from $A$ to $B$:

\smallskip

\begin{centering}
\begin{tabular}{ll}
(a) $A(2,0)$, $B(4,3)$; &(b) $A(1,-1)$, $B(0,2)$;\\
 (c) $A(0,0)$, $B(-2,-2)$; &(d) $A(-2,3)$, $B(4,3)$;\\
 (e) $A(-3,-2)$, $B(2,3)$; & (f) $A(0.01,-0.01)$, $B(-0.01,0.05)$. \\
\end{tabular}\\
\end{centering}

\smallskip

\exercise %2
Graph each of the following lines, after changing to the form $y=mx+b$,
and also find the $y$-intercept and $x$-intercept:

\smallskip

\begin{centering}
(a) $y-2x=2$; (b) $x+y=6$; (c) $x=2y-1$; (d) $3=2y$; (e) $2x+3y+6=0$.\\
\end{centering}

\smallskip

\exercise %3
Find the equation of the circle of radius 3 centered at: (a) $(0,0)$;
(b) $(5,6)$; (c) $(-5,-6)$; (d) $(0,3)$; (e) $(0,-3)$; (f) $(3,0)$.

\smallskip

\exercise %4
Graph the circles

\smallskip

\begin{centering}
 (a)  $x^2+y^2+10y=0$;  (b) $x^2-10x+y^2=24$; (c) $x^2-6x+y^2-8y=0$.\\
\end{centering}

\smallskip

\exercise %5
Let $x$ stand for temperature in degrees Celsius (centigrade), and let
$y$ stand for temperature in degrees \index{ Fahrenheit} Fahrenheit.  A temperature of $0^\circ$C
(freezing of water) corresponds to $32^\circ$F, and a temperature of
$100^\circ$C (boiling of water) corresponds to $212^\circ$F.  Find the
equation of the line that relates temperature Fahrenheit $y$ to
temperature \index{ Celsius} Celsius $x$.  Graph the line, and find the $y$- and $x$-intercepts.
What is the practical meaning of the intercepts?

\smallskip

\exercise %6
A \index{ car rental} car rental firm has the following charges for a certain type of car:
\$25 per day with 100 free miles included, \$0.15 per mile for more than
100 miles.  Suppose you want to rent a car for one day, and you know you'll
use it for more than 100 miles.  What is the equation relating the cost
$y$ to the number of miles $x$ that you drive the car?

\smallskip

\exercise %7
An instructor gives a 100-point final exam, and decides that a score
90 or above will be a grade of 4.0, a score of 40 or below will be a grade
of 0.0, and between 40 and 90 the grading will be linear.  Let $x$ be
the exam score, and let $y$ be the corresponding grade.  Find a formula
of the form $y=mx+b$ which applies to scores $x$ between 40 and 90.

\smallskip

\exercise %8
A\index{ photocopy} photocopy store advertises the following prices: 5\cnt~per copy for
the first 20 copies, 4\cnt~per copy for the 21st through 100th copy, and
3\cnt~per copy after the 100th copy.  Let $x$ be the number of copies, and let
$y$ be the total cost of photocopying.  (a) Graph the cost as $x$ goes from 0
to 200 copies.  (b) Find the equation in the form $y=mx+b$ that tells you the
cost of making $x$ copies when $x$ is more than 100.

\smallskip

\exercise 
In the Kingdom of Zog the tax system works as follows.  Someone who
earns less than 100 gold coins per month pays no tax.  Someone who earns
between 100 and 1000 \index{ golds coins} golds coins pays tax equal to 10\% of the amount over
100 gold coins that he or she earns.  Someone who earns over 1000 gold coins
must hand over to the King all of the money earned over 1000 in addition to
the tax on the first 1000.  (a) Draw a
graph of the tax paid $y$ versus the money earned $x$, and give
formulas for $y$ in terms of $x$ in each of the regions $0\le x\le 100$,
$100\le x\le 1000$, and $x\ge 1000$.  (b) Suppose that the King of Zog
decides to use the second of these line segments (for $100\le x\le 1000$)
for $x\le 100$ as well.  Explain in practical terms what the King is doing,
and what the meaning is of the $y$-intercept. 

%%\smallskip
%\newpage

\exercise 
Market research tells you that if you set the price of an item at
\$1.50, you will be able to sell 5000 items; and for every 10 cents you
lower the price below \$1.50 you will be able to sell another 1000 items.
Let $x$ be the number of items you can sell, and let $P$ be the price of
an item.  (a)~Express $P$ linearly in terms of $x$, in other words,
express $P$ in the form $P=mx+b$.  (b)~Express $x$
linearly in terms of $P$.

\smallskip

\exercise 
The \index{ tax} tax for a single taxpayer is described in the figure below. Use this
information to graph tax versus taxable income (i.e., $x$ is the
amount on Form 1040, line 37, and $y$ is the amount on Form 1040, line 38).
Find the slope and $y$-intercept of each line that makes up the \index{ polygonal graph} polygonal graph.

\bigskip

\begin{centering}
\begin{tiny}
\begin{tabular}{rrrr|rrrr}
\multicolumn{8}{l}{\bf \large 1990 Tax Rate Schedules} \\
\multicolumn{8}{l}{ \begin{minipage}{6.2in}{\bf Caution:} 
{\em use ONLY if your taxable income (form 1040, line 37) is \$50,000 or
more. If less, use the {\bf Tax Table}. (Even though you cannot use the
tax rate schedules below if your taxable income is less than \$50,000, we
show all levels of taxable income so that taxpayers can see the tax rate
that applies to each level.)}\end{minipage}}\\ \hline
\multicolumn{4}{l}{{\bf \large Schedule X---}%
Use if your filing status  is {\bf   Single}}
&
\multicolumn{4}{|l}{{\bf \large Schedule Z---}%
Use if your filing status is {\bf  Head of}}\\
&&&& \multicolumn{4}{|l}{\ \hspace{0.95in}{\bf  household}} \\
\begin{minipage}{0.75in} 
If the amount on Form 1040, line 37, is:

\ 
\ 

Over---
\end{minipage}
&
\begin{minipage}{0.5in} 
\ 

\

\

\

But not

over---
\end{minipage}
&
\begin{minipage}{0.75in} 
Enter on

Form 1040,

line 38
\ 

\ 
\end{minipage}
&
\begin{minipage}{0.5in} 
\

\

\

of the

amount

over---
\end{minipage}
&
\begin{minipage}{0.75in} 
If the amount on Form 1040, line 37, is:\\
\

\

Over---
\end{minipage}
&
\begin{minipage}{0.5in} 
\

\ 

\

\

But not

over---
\end{minipage}
&
\begin{minipage}{0.75in} 
Enter on

Form 1040,

line 38

\ 

\ 
\end{minipage}
&
\begin{minipage}{0.5in} 
\

\

\

of the 

amount

over---
\end{minipage}\\
\multicolumn{4}{c}{\ \hrulefill \ }
&
\multicolumn{4}{|c}{\ \hrulefill \ }\\
\$0   &\$19,450 &\dotfill{\bf 15\%} & {\bf \$0}
&
\$0   &\$26,050 &\dotfill{\bf 15\%} &{\bf \$0}\\
19,450& 47,050&{\bf\$2,917.50+28\%} &{\bf19,450}&
26,050& 67,200&{\bf\$3,907.50+28\%} &{\bf26,050}\\
47,050& 97,620&{\bf\$10,645.50+33\%} &{\bf47,050}&
67,200&134,930&{\bf\$15,429.50+33\%} &{\bf67,200}\\
&& \multicolumn{2}{c|}{\ \hrulefill\ } 
&& \multicolumn{2}{c}{\ \hrulefill\ }\\ 
97,620& \ \dotfill \ &\begin{minipage}{1.0in}{\scriptsize%
Use {\bf Worksheet}\\ below to figure\\ your tax.}
\end{minipage} 
&
&134,930& \ \dotfill \ &\begin{minipage}{1.0in}{\scriptsize%
Use {\bf Worksheet}\\ below to figure\\ your tax.}
\end{minipage} & \\ \hline
\end{tabular}\\
\end{tiny}
\end{centering}

%------------------------------

%\end{homework}
%------------------------------

%
%HOMEWORK PROBLEMS ADDED FOR MA113 CALC1
%
%
\smallskip

\exercise Do the points shown in the following table below lie on a line?  If so, 
find the equation for this line.  \\ \\ \begin{tabular}{|c|c|c|c|c|c|c|} \hline
$x$ & 5.2 & 5.3 & 5.4 & 5.6 & 5.9 & 6.5 \\ \hline
$y$ & 27.8 & 29.2 & 30.6 & 33.4 & 37.6 & 46.0 \\ \hline
\end{tabular}

\smallskip
\smallskip

\exercise If the point $(3,k)$ lies on the line with slope $m = -2$ passing through
the point $(2,5)$, find $k$.

\smallskip


\exercise Does the point $(3, -2)$ lie on the line through the points $(8,0)$ and
 $(-7,-6)$?
\smallskip

\exercise Use the slopes to determine whether the points $(7,-1)$, $(10,1)$, and 
$(6,7)$ are the vertices of a right triangle. 
\smallskip
 
\exercise Is there  a value of k so that the graphs of  $x^3-x$ and $kx^2 -1$ meet in three distinct points? In four points? In 2 points? In one point? Do not meet? If so give specific values of k; if not, explain why.

\smallskip

\end{homework}

