\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\title{Brief Article}
\author{The Author}
\begin{document}
\maketitle




\def\picture #1 by #2 (#3){
  \vbox to #2{
    \hrule width #1 height 0pt depth 0pt
    \vfill
    \special{picture #3} % this is the low-level interface
    }
  }

\def\scaledpicture #1 by #2 (#3 scaled #4){{
  \dimen0=#1 \dimen1=#2
  \divide\dimen0 by 1000 \multiply\dimen0 by #4
  \divide\dimen1 by 1000 \multiply\dimen1 by #4
  \picture \dimen0 by \dimen1 (#3 scaled #4)}
  }

\def\trough{\scaledpicture .5in by 0in (trough scaled 700)}
\def\circprob{\scaledpicture .5in by 0in (circle scaled 700)}


\title
\noindent
{\bf Math 119: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl December 19, 1994}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed.  Hand in this answer page only.  Record your answers 
to the multiple choice problems by  placing an $\times$ through one letter for
each problem on this answer sheet. There are  30 multiple choice questions, worth 5
points each.  \vskip .25cm \noindent {\bf You are taking this exam under the
honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}


\note %1
\problem
Which of the following is the equation of the line through the point $(1,2)$ and
parallel to the line $2x-3y=4$?
\smallskip

\correct
$2x-3y=-4$
\wrong
$2x-3y=6$
\wrong
$3x-2y=-1$
\wrong
$3x-2y=0$
\wrong
$2x+3y=8$

\note %2
\problem
Find $\displaystyle {d \over {dx}} (x^3 - 2)^{2/3}$
\correct
$\displaystyle {{2x^2} \over {(x^3 - 2)^{1/3}}}$
\wrong
$\displaystyle {{2} \over {3(x^3 - 2)^{1/3}}}$
\wrong
$\displaystyle {{3x^2} \over {(x^3 - 2)^{1/3}}}$
\wrong
$\displaystyle 2x^2 (x^3 - 2)^{1/3}$
\wrong
$\displaystyle (3x^2 )^{-1/3}$

\note %3
\problem Where is the graph of the function
$$f(x) \ = \ x^3 - 6x^2 + 9x - 2$$
concave up, and where is it concave down?
\correct concave up on $(2,\infty)$, concave down on $(-\infty,2)$
\wrong concave up on $(-\infty,1)$ and $(3,\infty)$, concave down on $(1,3)$
\wrong concave up on $(1,3)$, concave down on $(-\infty,1)$ and $(3,\infty)$
\wrong concave up on $(0,2)$, concave down on $(-\infty,0)$ and $(2,\infty)$
\wrong concave up on $(-\infty,2)$, concave down on $(2,\infty)$

\note %4
\problem 
Suppose
$$\lim_{x\to a}\,f(x) = 0 \hbox{\qquad and \qquad} \lim_{x\to a}\,g(x) = 0\,.$$
Which, if any, of the following statements has to be true  about $\displaystyle
\lim_{x\to a}\, {f(x) \over g(x)} \>?$
\correct You can't tell from the given information what $\displaystyle \lim_{x\to
a}\, {f(x) \over g(x)}$ is.
\wrong $\displaystyle \lim_{x\to a}\,{{f(x)} \over {g(x)}}$ does not exist.
\wrong $\displaystyle \lim_{x\to a}\,{{f(x)} \over {g(x)}} = 0$
\wrong $\displaystyle \lim_{x\to a}\,{{f(x)} \over {g(x)}} = 1$
\wrong Either \ \ \ $\displaystyle \lim_{x\to a}\,{{f(x)} \over {g(x)}} = \infty$ \
\
\ or
\ \ \
$\displaystyle \lim_{x\to a}\,{{f(x)} \over {g(x)}} = -\infty$

\note %5
\problem Find $\displaystyle \lim_{h \to 0} {{{(2h-3)}^2 - 9} \over h}$
\correct $-12$
\wrong $2$
\wrong $-6$
\wrong $0$
\wrong $-3$

\note %6
\problem
Find $\displaystyle {d \over {dx}} \left ( x^2 \cos x \right )$
\correct
$\displaystyle -x^2 \sin x + 2x\cos x$
\wrong
$\displaystyle x^2 \sin x + 2x\cos x$
\wrong
$\displaystyle -2x \sin x$
\wrong
$\displaystyle x^2 \sin x - 2x\cos x$
\wrong
$\displaystyle 2x \sin x$

\note %7
\problem Find the slope of the tangent line to the graph of the equation
$$x^2 - y^2 + xy = -1$$
at the point $(1,2)$
\correct $\displaystyle {4 \over 3}$
\wrong $\displaystyle {5 \over 3}$
\wrong $\displaystyle {2 \over 3}$
\wrong $1$
\wrong $3$

\note %8
\problem Find $\displaystyle \int_{0}^1 {(x^2 - 2x)} dx$
\correct $\displaystyle {-2 \over 3}$
\wrong $\displaystyle {-1 \over 3}$
\wrong $-1$
\wrong $\displaystyle {1 \over 3}$
\wrong $\displaystyle {4 \over 3}$

\note %9
\problem  If $\displaystyle {\int {f(x) dx}} = {x^3 - 3x^2 + 2x + C}$, what is
$f'(x)$?
\correct $6x - 6$
\wrong $3x^2 - 6x + 2$
\wrong $\displaystyle {{x^4} \over 4} - x^3 + x^2$
\wrong $x^3 - 3x^2 + 2x $
\wrong None of the above

\note %10
\problem Find the area under the graph of the function
$$f(x) = \sqrt x$$
from $x =0$ to $x=4$.
\correct ${{16}\over3}$
\wrong 1
\wrong ${{14}\over3}$
\wrong ${1\over4}$
\wrong ${{11}\over2}$

\note %11
\problem 
Find $\displaystyle \int_{0}^{2\pi} \sin x \ dx$
\correct 0
\wrong 2
\wrong $-2$
\wrong $2\pi$
\wrong 1

\note %12
\problem
Find $\displaystyle \lim_{x \to \infty} \sqrt{x^2 - x + 1} - x$
\correct $-1 / 2$
\wrong $\infty$
\wrong 0
\wrong 1
\wrong $-1$

\note %13 
\problem
Let $f(x)$ be a function which is continuous on the interval $[a,b]$, but not
necessarily positive on the whole interval.  Which of the following statements {\bf
must} be true:

\medskip

\hskip .5in I. $f(x)$ has a derivative which is defined at every point of $(a,b)$

\hskip .5in II. $f(x)$ has an antiderivative which is defined at every point of
$(a,b)$

\hskip .5in III. $\displaystyle \int_a^b f(x) dx$ is equal to the area between the
graph of $f(x)$ and the $x$-axis.

\correct Only II.
\wrong Only I.
\wrong Only III.
\wrong II.\ and III.
\wrong All three statements must be true

\note %14
\problem Find all the asymptotes of the function $\displaystyle {{x-1} \over
{(x-2)(x-3)}}$.
\correct vertical asymptotes $x=2$, $x=3$; horizontal asymptote $y=0$
\wrong vertical asymptotes $x=2$, $x=3$; no horizontal asymptote
\wrong vertical asymptotes $x=2$, $x=3$; horizontal asymptote $y=1/2$
\wrong vertical asymptote $x=1$; no horizontal asymptote
\wrong vertical asymptotes $x=2$, $x=3$;  horizontal asymptote $y = 1/6$

\note %15
\problem Suppose $f$ and $g$ are functions satisfying the following conditions:
\bigskip
\hskip 1in $f(0) = 1$  \hfil  $f(1) = 0$  \hfil  $g(0) = 3$  \hfil  $g(1) = -2$ 
             \hskip 1in
\bigskip
\hskip 1in $f'(0) = 2/3$  \hfil  $f'(1) = - 1/2$  \hfil  $g'(0) = -5$ 
                    \hfil  $g'(1) = 1/3$ \hfil
\bigskip
If $h(x) = g\big(f(x)\big)$, what is $h'(1)$?
\correct $5/2$
\wrong $-{1/6}$
\wrong $2/9$
\wrong $-{10/3}$
\wrong 0

\note %16
\problem
A particle is moving in the positive direction along the $x$-axis so that
its position on the axis at $t$ seconds is given by the formula
$$s(t) \ = \ {t^2 \over{t+1}}$$  What is the instantaneous velocity of the
particle after one second?
\correct $3/4$ 
\wrong $1/2$ 
\wrong 2 
\wrong 1
\wrong $0$

\note %17
\problem
What is the natural domain of the function $\displaystyle f(x) = \cot 2x$?
\correct All $x$ except $\displaystyle 0, \pm {\pi \over 2}, \pm \pi, \pm {{3\pi}
\over 2}, \pm 2\pi \dots$
\wrong All $x$ except $\displaystyle 0, \pm \pi, \pm 2\pi, \pm 3\pi \dots$
\wrong All $x$ except $\displaystyle \pm {\pi \over 2}, \pm {{3\pi} \over 2}, \pm
{{5\pi} \over 2}, \dots$
\wrong All $x$ except $\displaystyle 0, \pm 2\pi, \pm 4\pi, \pm 6\pi \dots$
\wrong All $x$ except $\displaystyle 0,  \pi,  2\pi, 3\pi \dots$

\note %18
\problem
Find the radius of the circle given by the equation $$ x^2 + 2x +
y^2 + 4y = 4$$
\stet
\wrong 1
\wrong 2
\correct 3
\wrong 4
\wrong 5

\note %19
\problem
A water trough is 5 ft long and its ends have the shape of isosceles triangles
that are 3 ft high and 6 feet across the top (see diagram).  Water is being
poured in at a rate of 20 $\hbox{ft}^3$/min.  At a certain instant, the water is
rising at a rate of $3$ ft/min.  How high is the water in the trough at that
instant?

\vskip 1.3in
\hskip 1.1in\trough
\bigskip

\correct 2/3 ft
\wrong 2 ft
\wrong 1 ft
\wrong 1/2 ft
\wrong 3 ft

\note %20 
\problem Which of the following limits
represents the slope of the graph of $$y = \sin x$$ at
$\displaystyle x= {\pi \over 6} \ $?
\correct $\displaystyle \lim_{h \to 0} {1 \over h} \left [ \sin \left ( {\pi \over
6} + h \right ) - {1 \over 2} \right ]$
\wrong  $\displaystyle \lim_{h \to 0} {1 \over h} \left [ \sin \left ( {\pi \over
6} + h \right ) - {{\sqrt{3}} \over 2} \right ]$
\wrong
 $\displaystyle \lim_{h \to 0} {1 \over h} \left [ \sin \left ( h \right ) - {1
\over 2} \right ]$
\wrong
 $\displaystyle \lim_{h \to 0} {1 \over h} \left [ \sin \left ( {\pi \over
6} + h \right ) - \sin h \right ]$
\wrong
$\displaystyle \lim_{h \to 0} {{\sin h} \over h}$

\note %21
\problem
Where is the following function continuous and where is it differentiable
(i.e.\ where does its derivative exist)?
$$f(x) = \cases{{0}&if $\ x<0$\cr
                 x^2 &if $\ 0 \leq x \leq 1$\cr
                  x&if $\ x>1$\cr}$$
\correct continuous everywhere; differentiable everywhere except at $x=1$
\wrong continuous everywhere; differentiable everywhere
\wrong continuous everywhere except at $x=0$; differentiable everywhere except at
$x=0$ and $x=1$
\wrong continuous everywhere except at $x=0$ and $x=1$; differentiable everywhere
except at $x=0$ and $x=1$
\wrong continuous everywhere; differentiable everywhere except at $x=0$ and $x=1$

\note %22
\problem
Which of the following is the upper sum for the function $g(x) = 1 - x^2$ over
the interval $[0,2]$ (i.e.\ $0 \leq x \leq 2$) if the interval is divided into 4
equal segments?
\correct
$\displaystyle {1 \over 2} \left [ 1 + {3 \over 4} + 0 - {5 \over 4} \right ]$
\wrong
$\displaystyle {1 \over 2} \left [ {3 \over 4} + 0 - {5 \over 4} - 3 \right ]$
\wrong
$\displaystyle {1 \over 2} \left [ 1 + {3 \over 4} + 0 + {5 \over 4} \right ]$
\wrong
$\displaystyle {1 \over 2} \left [ 1 + {3 \over 4} - {5 \over 4} - 3 \right ]$
\wrong
$\displaystyle {1 \over 2} \left [ {1 \over 2} + 1 + {3 \over 2} + 2 \right ]$

\note %23
\problem
Let $f(x) = \sin x$ for $0 \leq x \leq \pi$.  Find $A_0^x (f)$ (the area function)
for $0 \leq x \leq \pi$.
\correct $-\cos x + 1$
\wrong $- \cos x$
\wrong $\cos x$
\wrong 2
\wrong $\pi$

\note %24
\problem
Let $$f(x) = \cases{\sqrt{x} + 1 &if $\ 0 \leq x \leq 1$\cr
                \ \ 2 &if $\ 1 \leq x \leq 2$\cr}$$
Find the area function $\displaystyle A_0^x (f)$ for $0 \leq x \leq 2$.
\correct
$\displaystyle A_0^x (f) = \cases{{2 \over 3} x^{3/2} + x &if $\ 0 \leq x
                         \leq 1$\cr
                \ \ 2x - {1 \over 3} &if $\ 1 \leq x \leq 2$\cr}$
\wrong
$\displaystyle A_0^x (f) = \cases{{2 \over 3} x^{3/2} + x &if $\ 0 \leq x
                         \leq 1$\cr
                \ \ 2x + {5 \over 3} &if $\ 1 \leq x \leq 2$\cr}$
\wrong
$\displaystyle A_0^x (f) = \cases{{2 \over 3} x^{3/2} + x + 1 &if $\ 0 \leq x
                         \leq 1$\cr
                \ \ 2x + {5 \over 3} &if $\ 1 \leq x \leq 2$\cr}$
\wrong
$\displaystyle A_0^x (f) = \cases{{2 \over 3} x^{3/2} + x &if $\ 0 \leq x
                         \leq 1$\cr
                \ \ 2x  &if $\ 1 \leq x \leq 2$\cr}$
\wrong
$\displaystyle A_0^x (f) = \cases{{1 \over 2} x^{-1/2}  &if $\ 0 \leq x
                         \leq 1$\cr
                \ \ 0 &if $\ 1 \leq x \leq 2$\cr}$

\note %25
\problem
A circle is expanding in such a way that its radius is growing at a constant rate
of 3 ft/sec.  At what rate is its area increasing when the radius is 5 ft?
\correct $30 \pi \ \hbox{ft}^2$/sec
\wrong $6 \pi \ \hbox{ft}^2$/sec
\wrong $10 \pi \ \hbox{ft}^2$/sec
\wrong $15 \pi \ \hbox{ft}^2$/sec
\wrong $25 \pi \ \hbox{ft}^2$/sec

\note %26
\problem
Let $f(x) = 6 x^{1/3} - {3 \over 4} x^{4/3}$.  Notice that $$f'(x) = 2 x^{-2/3} -
x^{1/3} = \left ( {1 \over {x^{2/3}}} \right ) (2-x)$$ Find all the local maxima
and local minima of $f(x)$.

\correct local maximum at $x=2$; no local minimum
\wrong local maximum at $x=2$; local minimum at $x=0$
\wrong local maximum at $x=2$ and at $x=0$; no local minimum
\wrong no local maximum; local minimum at $x=2$
\wrong no local maximum; local minimum at $x=0$ and at $x=2$

\note %27
\problem
If $F(x)$ is an antiderivative for $f(x) = x^2 + 2x + 1$ satisfying $F(3) = 25$,
what is $F(0)$?
\correct 4
\wrong 2
\wrong 1
\wrong $\displaystyle {7 \over 3}$
\wrong 0

\note %28
\problem
Find the point on the graph of the function $y = \sqrt{x}$ which is closest to
the point $(3,0)$.
\correct $\displaystyle \left ( {5 \over 2}, \sqrt{{5 \over 2}} \right )$
\wrong $(4,2)$
\wrong $(1,1)$
\wrong $(2, \sqrt{2})$
\wrong $(3, \sqrt{3})$

\note %29
\problem
A rectangle is to be inscribed in a circle of diameter 10 (see
diagram).  Your mission is to  \break determine the dimensions of the
rectangle which maximize its area.  What function would you differentiate to solve
that problem?  (You don't have to figure out the final answer.)

\vskip 1.3in
\hskip 1.7in \circprob

\bigskip

\correct $x\sqrt{100 - x^2}$
\wrong $\sqrt{100-x^2}$
\wrong $x^2 + y^2$
\wrong $x^2 \sqrt{100-x^2}$
\wrong $\displaystyle {{x^2} \over {\sqrt{100-x^2}}}$

\note %30 
\problem
Find $\displaystyle \lim_{h \to 0} {{\sin(\pi + h)} \over {2h}}$.
\correct $-1/2$
\wrong $1/2$
\wrong $1$
\wrong $-1$
\wrong $\displaystyle {\pi \over 2}$

\appendix
%That's all folks!


 \end{document}
