\documentclass[12pt]{report}
\everymath{\displaystyle}
\hbadness = 10000
\tolerance = 10000

\def\smallans{ \hbox{\hskip2.5in Ans. \vrule height .4pt depth 0 pt width
1.5in}}
\def\ans{ \hbox{\hskip2.5in Ans. \vrule height .4pt depth 0 pt width 2.5in}}
\def\bigans{ \hbox{\hskip1in Ans. \vrule height .4pt depth 0 pt width 4in}}
\def\name{\hbox{\hskip2.25in  {\bf Name:} \vrule height .4pt depth 0 pt
width 3in}
               \vskip.25in}
\def\med{\medskip}
\def\small{\smallskip}
\def\superskip{\bigskip \  \medskip}
\def\c{\centerline}
\def\eq{\ = \ }
\def\and{\hbox{\quad and \quad}}
\def\andd{\hbox{\qquad and \qquad}}
\def\page{\vfill \eject \noindent}
\def\next{\smallskip \noindent}


\begin{document}
\noindent
{\bf Math 105}  \hfil {\bf Review questions --- test 1}

\bigskip  \noindent
1. Find the slope of each of the following lines:

\smallskip
(a) through $(1,-1)$ and~$(3,2)$

(b) given by the equation $2x + 3y = 5$

(c) through $(1, -2)$, parallel to the line with equation $x - 2y = 1$.

 (d) horizontal, through $(-4,7)$

\next
2.  Find the equation of each of the following lines:

\smallskip
(a) Through $(1/2, 1)$ with slope $-2/3$.

(b) Through $(0,-1)$ and $(-2, 5)$

(c) Through $(1,1)$ and $(-1, 3)$

(d) Through $(-3,-1)$ and $(5,-1)$ \hfil (e) Through $(-3,-1)$ and $(-3, 2)$.

(f)  Through the point $(4, -1)$ and
parallel to the line $\ 2x + 5y = 6\,.$

\next
3.  Determine whether each of the following functions is continuous.  If
not, find the
discontinuities.

\bigskip \noindent
(a) $f(x) = \cases{2x-1 &if $x < 0$ \cr 1-x & if $0 \le x \le 1$ \cr  x-1 &
if $x > 1$ \cr}$
\hfil
(b) $f(x) = \cases{1-x^2 & if $x < 1$ \cr x-1 &if $x \ge 1$ \cr}$

\bigskip \noindent
(c) $f(x) = \cases{x^2 + 1 &if $x \le 0$ \cr 1 &if $0 < x < 1$ \cr x^2 &if
$x \ge 1$ \cr}$
\hfil
(d) $f(x) = \cases{x^2  &if $x \le 0$ \cr 1 &if $0 < x < 1$ \cr x^2 + 1 &if
$x \ge 1$ \cr}$

\next
4.  Find the natural domain of each of the following functions.

\medskip
(a) $f(x) = \sqrt{x-2}$  \hfil (b) $\displaystyle f(x) = {1\over{\sqrt{x-2}}}$

\bigskip
(c) $\displaystyle f(x) = {{\sqrt{x}}\over{x-2}}$  \hfil
(d) $\displaystyle {x\over{x^2 + 4}}$

\bigskip
(e) $\displaystyle {x\over{x^2 - 4}}$ \hfil
(f) $\displaystyle {x\over{\sqrt{x^2 - 4}}}$

\next
5. Find the vertical and horizontal asymptotes, if any, for the graphs of the
following functions.  If there are none, write ``none.''

\medskip
(a)  $f(x) = {{x^2 - 1}\over{x^2 + 1}}$
\hfil
(b) $f(x) = {x\over{x^2 - 4}}$

\bigskip
(c) $f(x) = {{x^4}\over{x^2 - 100}}$
\hfil
(d) $f(x) = {{x+2}\over{3x-4}}$

\bigskip \hfil
(e) $f(x) = {{3x^4 + 15x^3 + 10x^2 + 5x + 20}\over{x^4 + 1}}$ \hfil

\next
6.  A ball is thrown upward from a height of 15 feet with an initial speed
of 8 feet
per second.  The height of the ball is given by the formula
$$h = -16t^2 + 8t + 15$$
where $h$ is the height (in feet) and $t$ is the time (in seconds).

\smallskip
(a) When does the ball reach its highest point?

(b) How high does it go?

(c) When does it hit the ground?

\next
7.  Complete the squares of each of the following quadratics.  Then state
whether the graph has
a maximum or a minimum point and find the coordinates of the point.

\smallskip
(a) $2x^2 + 12x + 13$  \hfil (b) $-3x^2  + 10x - 8$

\next
8.  A company that makes Adirondack chairs has fixed costs of \$5,000 per month
plus a cost of \$30 per chair. The company plans to sell the chairs for
\$50 each.  Suppose they
produce $q$ chairs a month, and assume that they sell every chair they produce.

\smallskip
(a) Find the monthly  profit function.

(b) Find the break-even point.

\next
9.  A small company manufacturing tennis rackets has fixed costs of \$700 per
week.  For the first 50 rackets they make, each one costs \$32  to produce.
Every racket over
50 they make costs \$38.  If $x$ is the number of rackets they produce
in a given week, find the cost function $C(x)$.

\next
10.  A company offers dinner cruises on the Chicago River.  The company has
found
that the average number of passengers per night is 80 if the price is \$45
per person. At a price
of \$30 per person, the average number of passengers per night is 125. Let
$p$ denote the price
and $q$ denote the demand (the average number of passengers).

\smallskip
(a) Assuming that the demand is a linear function of the price, write the
demand as a function
of the price.

(b)  Write the nightly revenue $R$ as a function of the price.

(c) What price should the company charge to maximize the revenue?  (Round
your answer off to
the nearest dollar.)

\newpage
\noindent 
11.  Compute each of the following without a calculator.  Your answer
should be an integer or
simple fraction.

\smallskip
(a) $\displaystyle\left({1\over9}\right)^{1/2}$ \hfil
(b) $\displaystyle\left({1\over9}\right)^{-2}$ \hfil
\hfil
(c) $(0.1)^3$  \hfil (d) $(0.1)^{-3}$

\next 
12. Let $\displaystyle f(x) = {1 \over (x+2)^{2}}$.  Compute the
following.

\smallskip
(a) $f(1)$ \hfil
(b) $f(0)$ \hfil
(c) $f(a-1)$

\vfil \eject \noindent
\underline{\bf Answers:} 

\medskip \noindent
{\bf 1.} (a) 3/2 \ (b) $-2/3$ \ (c) 1/2 \ (d) 0

\medskip \noindent
{\bf 2.} (a) $y = -(2/3)x + (4/3)$

\medskip \noindent
(b) $y = -3x-1$ \ \ (c) $y = -x+2$ \ \ (d) $y=-1$ \ \ (e) $x=-3$ \ \
(f) $2x + 5y = 3$

\medskip \noindent
{\bf 3.} (a) disc. at $x=0$ \  (b) cont. \  (c) cont. \ (d) disc. at $x=0, 1$

\medskip \noindent
{\bf 4.} (a) $x \ge 2$ \ (b) $x > 2$
\
(c) $x \ge 0, \ x \ne 2$ \
 (d) all $x$ \ (e) $x \ne \pm 2$ \ (f) $|x| > 2$

\medskip \noindent
{\bf 5.} (a) h.a. $y=1$, no v.a. \ \ (b) h.a. $y=0$, v.a. $x = 2, \ x = -2$

\medskip \noindent
 (c) no h.a., v.a. $x=10, \ x=-10$ \ \ (d) h.a. $y=1/3$, v.a. $x=4/3$

\medskip \noindent
(e) h.a. $y=3$, no v.a. 

\medskip \noindent
{\bf 6.} (a) 1/4 sec \ (b) 16 ft \ (c) 5/4 sec

\medskip \noindent
{\bf 7.} (a) $2(x+3)^2 - 5$, min at $(-3,-5)$

\medskip \noindent
(b) $-3\left(x - (5/3)\right)^2 + (1/3)$, max at $(5/3, 1/3)$

\medskip \noindent
{\bf 8.} (a) $P(x) = 20x - 5000$ \ \ (b) 250

\medskip \noindent
{\bf 9.} $C(x) = \cases{700 + 32x &if $0 \le x \le 50$ \cr 2300 + 38(x-50)
                 &if $50 < x$ \cr}$

\medskip \noindent
{\bf 10.} (a) $q = -3p + 215$ \ (b) $R = -3p^2 + 215p$ \ (c) 36 


\medskip \noindent
{\bf 11.} (a) 1/3 \ \  (b) 81 \ \ (c) 0.001 \ \  (d) 1000

\medskip \noindent
{\bf 12.} (a) 1/9 \ \  (b) 1/4  \ \   (c) $\displaystyle {1 \over (a+1)^{2}}$


\end{document}

