\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\rect{\scaledpicture .5in by 0in (rect scaled 700)}


\title
\noindent
{\bf Math 119: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl December 16, 1996}\hfill
Time of MWF class:\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  
Let 
$$
f(x) = \cases{{{x^2 - 1} \over {x-1}}, &if $x \neq 1$;\cr
              c, &if $x=1$.}
$$
For what value of $c$ is $f(x)$ continuous for all $x$?
\correct 2
\wrong 0
\wrong 1 
\wrong $-1$
\wrong $-2$

\note %2
\problem
Suppose you know that $\displaystyle 0 \leq x \leq {{\pi} \over 2}$ and
$\displaystyle 
\sin^2 x = {1 \over 2}$.  Find $\tan x$.

\correct 1
\wrong $-1$
\wrong $\displaystyle {1 \over {\sqrt{3}}}$
\wrong $\displaystyle \sqrt{3}$
\wrong $\displaystyle {{\sqrt{2} \over 2}}$

\note %3
\problem
Find $\displaystyle \lim_{x \rightarrow 0} {{\sin^2 ( 5x)} \over {\sin^2 (4x)}}$

\correct $\displaystyle {{25} \over {16}}$
\wrong  $\displaystyle {{16} \over {25}}$
\wrong  $\displaystyle {{5} \over {4}}$
\wrong  $\displaystyle {{4} \over {5}}$
\wrong  The limit does not exist.

\note %4
\problem
If $f(x) = \tan x$, find the slope of the tangent line to the graph of $y = f(x)$
at $\displaystyle x = {{5 \pi} \over 4}$.

\correct 2
\wrong $-2$
\wrong 1
\wrong $-1$
\wrong 0

\note %5
\problem
Let $f(x) = \sin (\cos x)$.  Find $\displaystyle f' ( {{\pi} \over 2} 
)$.

\correct $-1$
\wrong 0
\wrong 1
\wrong $\displaystyle {{\sqrt{2}} \over 2}$
\wrong $\displaystyle {1 \over 2}$

\note %6
\problem
If $xy + y^2 + x^2 = y^3$, find $\displaystyle {{dy} \over {dx}}$.

\correct $\displaystyle {{-y -2x} \over {x + 2y - 3y^2}}$
\wrong $\displaystyle {{x+y} \over {y^2}}$
\wrong $\displaystyle {{-x-y} \over {x+y-3y^2}}$
\wrong $\displaystyle {{-x-2y} \over {x+3y^2}}$
\wrong $\displaystyle {{-2x} \over {x-3y^2}}$

\note %7
\problem 
Let $\displaystyle g(x) = x + {1 \over x}$.  Find $g^{(4)} (x)$ (the fourth
derivative).

\correct $\displaystyle {{24} \over {x^5}}$
\wrong $\displaystyle 1- {{24} \over {x^5}}$
\wrong $\displaystyle x^4 + {{24} \over {x^4}}$
\wrong $\displaystyle x^5 - {{24} \over {x^5}}$
\wrong $\displaystyle {{24} \over {x^4}}$

\note %8
\problem
An object dropped from a 100-foot tower on a certain planet has height $h(t) =
-4t^2 + 100$ at time $t$.  Find the acceleration due to gravity on that planet.

\correct $-8$
\wrong $-4$
\wrong $-2$
\wrong $-16$
\wrong $-32$

\note %9
\problem
A particle is moving with constant speed around the circle $x^2 + y^2 = 5$.  At
the instant when it reaches the point $(1,2)$, the rate of change of the
$x$-coordinate is 
$-6$ units/second.  What is the rate of change of the $y$-coordinate at that
instant?

\correct 3 units/second
\wrong $-3$ units/second
\wrong $\displaystyle {5 \over 2}$ units/second
\wrong $\displaystyle -{5 \over 2}$ units/second
\wrong 2 units/second

\note %10
\problem 
A cart is being rolled up a $30^\circ$ ($= \pi / 6$ radians) ramp.  Its velocity
along the ramp is 6 ft/sec.  How quickly is its height from the ground
increasing when it is 12 feet up the ramp?

\correct 3 ft/sec
\wrong 6 ft/sec
\wrong 12 ft/sec
\wrong $6 \sqrt{3}$ ft/sec
\wrong $12 \sqrt{3}$ ft/sec

\note Questions 11 and 12 refer to the function $\displaystyle f(x) = {{x+1}
\over {\sqrt{x^2 - 1}}}$.  Note that the function has domain
$(-\infty, -1 ) \cup (1, \infty)$.

\bigskip
%11
\problem
Exactly two of the following four statements are correct.  Which two?

$$
\matrix{ \displaystyle
{\bf I.} \ \lim_{x \rightarrow 1^+} f(x) = \infty & \hbox{\hskip 2cm}
& \displaystyle {\bf II.} \ \lim_{x \rightarrow 1^+} f(x) = 0 \cr
\displaystyle {\bf III.} \ \lim_{x \rightarrow -1^-} f(x) = -\infty & \hbox{\hskip
2cm} & \displaystyle {\bf IV.} \ \lim_{x \rightarrow -1^-} f(x) = 0 
}$$

\correct {\bf I} \& {\bf IV}
\wrong {\bf I} \& {\bf III}
\wrong {\bf II} \& {\bf III}
\wrong {\bf II} \& {\bf IV}
\wrong {\bf I} \& {\bf II}

\note %12
\problem 
Exactly two of the following four statements are correct.  Which two?

$$
\matrix{ \displaystyle
{\bf I.} \ \lim_{x \rightarrow - \infty} f(x) = 1 & \hbox{\hskip 2cm}
& \displaystyle {\bf II.} \ \lim_{x \rightarrow -\infty} f(x) = -1 \cr
\displaystyle {\bf III.} \ \lim_{x \rightarrow \infty} f(x) = 1 &
\hbox{\hskip 2cm} & \displaystyle {\bf IV.} \ \lim_{x \rightarrow \infty} f(x) =
-1  }$$

\correct {\bf II} \& {\bf III}
\wrong {\bf I} \& {\bf III}
\wrong {\bf I} \& {\bf IV}
\wrong {\bf II} \& {\bf IV}
\wrong {\bf III} \& {\bf IV}

\note %13
\problem 
A stone which is dropped from the top of a certain building takes 3 seconds to hit
the ground.  How tall is the building?  (Recall that the constant acceleration
due to gravity is $-32$ ft/sec.)

\correct 144 ft
\wrong 72 ft
\wrong 216 ft
\wrong 432 ft
\wrong 600 ft

\note %14
\problem
The following is known about the function $f(x)$:
$$
f'(x) = 8x^3 + 9x^2 + 8x \ ; \ \ \ \ \ f(1) = 10
$$
Find $f(0)$.

\correct 1
\wrong $-1$
\wrong 0
\wrong 2
\wrong $-2$

\note %15
\problem
Both ends of a 12-inch long piece of wire are bent to form $90^\circ$ angles,
resulting in a ``topless'' rectangle as shown in the figure.  What is the largest
area which can be obtained for the \break rectangle?

\vskip .8in

\hskip 1.2in \rect

\bigskip

\correct 18 sq.\ inches
\wrong 20 sq.\ inches
\wrong 16 sq.\ inches
\wrong 22 sq.\ inches
\wrong 24 sq.\ inches

\note %16
\problem
The absolute minimum and maximum values of the function 
$$
f(x) = x^2 - 6x + 8, \ \ \ \ 0 \leq x \leq 8
$$
are:

\correct $\displaystyle 
\matrix{
\hbox{minimum value} & -1 \cr
\hbox{maximum value} & 24
}$
\wrong $\displaystyle 
\matrix{
\hbox{minimum value} & 0 \cr
\hbox{maximum value} & 24
}$
\wrong $\displaystyle 
\matrix{
\hbox{minimum value} & -8 \cr
\hbox{maximum value} & 8
}$
\wrong $\displaystyle 
\matrix{
\hbox{minimum value} & 1 \cr
\hbox{maximum value} & 8
}$
\wrong $\displaystyle 
\matrix{
\hbox{minimum value} & 1 \cr
\hbox{maximum value} & 24
}$

\note %17
\problem
The set of critical points of the function $\displaystyle f(x) = (x^2 - 1)^{4/3}$
consists exactly of the points:

\correct $\{ -1,0,1 \}$
\wrong $\{ 0 \}$
\wrong $\{ -1,1 \}$
\wrong $\{ 0,1 \}$
\wrong $\{ 0, -1 \}$

\note %18
\problem
The set of points where the graph of the function
$$
f(x) = x^4 - 6x^3 - 24 x^2 + 7x + 8
$$
is concave upward is:

\correct $(-\infty, -1) \cup (4, \infty)$
\wrong $(-\infty, 1) \cup (-4, \infty)$
\wrong $(-\infty, -1) \cup (-1,4) \cup (4, \infty)$
\wrong $ (4, \infty)$
\wrong none of the above

\note %19
\problem
Which of the following expressions in sigma notation is equivalent to the sum 
$$
9
+ 16 + 23 + 30 + 37 + \cdots + 100?
$$

\correct $\displaystyle \sum_{n=1}^{14} (2 + 7n)$
\wrong $\displaystyle \sum_{n=0}^{14} (9 + 7n)$
\wrong $\displaystyle \sum_{n=3}^{10} n^2$
\wrong $\displaystyle \sum_{n=9}^{100} n$
\wrong $\displaystyle \sum_{n=3}^{100} n$

\note %20
\problem
If $f(x) = {{x} \over {x^2+1}}$ then what is $f'(x)$?

\correct $\displaystyle {{1-x^2} \over {(x^2+1)^2}}$
\wrong $\displaystyle {{1} \over {2x}}$
\wrong $\displaystyle {{{-1} \over {x^2}} + 1}$
\wrong $\displaystyle {{x^2-1} \over {x^2+1}}$
\wrong $\displaystyle {{3x^2+1} \over {(x^2+1)^2}}$

\note %21
\problem
What is the radius and center of the circle given by the equation
$x^2-6x+y^2+4y-3=0$?

\correct $\displaystyle
\matrix{
\hbox{radius} & 4 \cr
\hbox{center} & (3,-2)
}$
\wrong $\displaystyle
\matrix{
\hbox{radius} & 16 \cr
\hbox{center} & (-6,4)
}$
\wrong $\displaystyle
\matrix{
\hbox{radius} & \sqrt{3} \cr
\hbox{center} & (3,-2)
}$
\wrong $\displaystyle
\matrix{
\hbox{radius} & 3 \cr
\hbox{center} & (-3,2)
}$
\wrong $\displaystyle
\matrix{
\hbox{radius} & 6 \cr
\hbox{center} & (-3,2)
}$

\note %22
\problem
Evaluate $\displaystyle \lim_{h \rightarrow 0} {{(1+h)^{1/5}-1}
\over {h}}$. Hint: Interpret the limit as the derivative of some
function at a particular point.

\correct $\displaystyle {{1} \over {5}}$
\wrong $\displaystyle {0}$
\wrong $\displaystyle {1}$
\wrong $\displaystyle {{-4} \over {5}}$
\wrong The limit does not exist

\note %23
\problem
Find the equation of the line passing through the point $(2,3)$
and parallel to the line $y=5x+7$.

\correct $\displaystyle {y-3=5(x-2)}$
\wrong $\displaystyle {y+3=5(x+2)}$
\wrong $\displaystyle {y-3={{-1} \over {5}}(x-2)}$
\wrong $\displaystyle {y-2=5(x-3)}$
\wrong $\displaystyle {y-2=7(x-3)}$

\note %24
\problem
Find the equation of the tangent line to the graph of the
equation $y=\sin (4x) + x + 2$ when $x=0$.

\correct $\displaystyle {y-2=5x}$
\wrong $\displaystyle {y-5=5x}$
\wrong $\displaystyle {y-2=x}$
\wrong $\displaystyle {y-5=2x}$
\wrong $\displaystyle {y+2=-5x}$

\note %25
\problem
Find the intervals on which the function $f(x)=x^3-6x^2+2$ is
increasing.

\correct $(-\infty, 0) \cup (4,\infty)$
\wrong $(-\infty, \infty)$
\wrong $(-\infty, 4)$
\wrong $(0, \infty)$
\wrong $(0,4)$

\note %26
\problem
Let 
$$
f(x)=\cases{x^2+1 & if $x<0$ \cr
 2x+1 & if $0\leq x \leq 1$ \cr
 x^2+2 & if $x\geq 1$.}
$$
Which of the following statements is true?

\correct $f(x)$ is differentiable at $x=1$ but not at $x=0$
\wrong $f(x)$ is differentiable at both $x=0$ and $x=1$
\wrong $f(x)$ is differentiable at $x=0$ but not at $x=1$
\wrong $f(x)$ is NOT differentiable at both $x=0$ and $x=1$
\wrong $1+1=37$

\note %27
\problem
Which one of the following is an antiderivative of
$f(x)=x^{-2}+\sin (3x)$?

\correct $-x^{-1}-{{\cos (3x)} \over {3}}$
\wrong $-x^{-1}+\cos(3x)$
\wrong ${{-1} \over {3}} x^{-3}+cos ({{3} \over {2}}x^2)$
\wrong $-x^{-1}-\cos ({{3} \over {2}}x^2)$
\wrong ${{1} \over {3}} x^{-3}+3\cos (3x)$

\note %28
\problem
Which of the following formulas is equivalent to $\displaystyle
\sum_{i=1}^{n} (5+2i)$?

\correct $6n+n^2$
\wrong ${{(5+2n)(6+2n)} \over {2}}$
\wrong $5n+2n^2$
\wrong $5+2n$
\wrong $5+10n+4n^2$

\note %29
\problem
If $F(x)=\int_{-3}^x \sqrt {4+\sin t}\  dt$, what is
$F'(0)$?  (Hint: first find $F'(x)$.)

\correct 2
\wrong 1
\wrong 3
\wrong 0
\wrong 4

\note %30
\problem
Evaluate: $\int_0^4 \sqrt {16-x^2}\  dx$. Hint: think
geometrically!

\correct $4\pi$
\wrong ${{2} \over {3}}$
\wrong ${{128} \over {3}}$
\wrong $16\pi$
\wrong $4$
 \appendix
%That's all folks!

 \end{document}
