\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




\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 15, 1995}\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  29 multiple choice questions, worth 5
points each.  An additional 5 points will be given for your correct tutorial
section number.  \vskip .25cm \noindent {\bf You are taking this exam under the
honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}

\note %1  
\problem  Find the equation of the line through the points $(3,2)$ and $(2,5)$.
\correct $3x+y = 11$
\wrong $3x+y = 9$
\wrong $x+3y = 9$
\wrong $x+3y = 11$
\wrong  $-3x +y = -7$

\note %2
\problem Find the domain of the function
$\displaystyle f(x) = \sqrt{x^2 -1} 
$ .
(Recall that $A \cup B$ means all $x$ which are either in $A$ or $B$ or both.)

\correct $(- \infty ,-1] \cup [1,\infty)$
\wrong $[-1,1]$
\wrong all $x$
\wrong $[1,\infty)$
\wrong $[0,\infty)$

\note %3
\problem Is the following the equation of a circle?  If so, what is its center?
$$
x^2 - 2x + y^2 + 2y - 3 = 0
$$
\correct yes, center $(1,-1)$
\wrong no, it's not a circle
\wrong yes, center $(-1,1)$
\wrong yes, center $(1,1)$
\wrong yes, center $(-1,-1)$

\note %4
\hskip 3.5in \xsqr
\vskip -.6in
\problem \parshape=1 0in 3.2in  Consider the graph of the function $f(x) =
x^2$.  (See the diagram to the right.)  What is the slope of the secant line
joining the points on the graph corresponding to $x=1$ and
$x=1+h$?
\correct $2+h$
\wrong 2
\wrong $\displaystyle {{(1+h)^2} \over h}$
\wrong $(1+h)^2$
\wrong $\displaystyle {{1} \over {(1+h)^2}}$

\note %5
\problem Find $\displaystyle \lim_{t \rightarrow 1} {{t^3 - t} \over {t-1}}$.
\correct 2
\wrong this limit does not exist
\wrong 1
\wrong 0
\wrong $\displaystyle {1 \over 2}$

\note %6
\problem Suppose $f$ and $g$ are functions satisfying the following conditions:
$$
f(0) = 2 \ \ \ \ \ f'(0) = 3 
$$
$$
g(0) = 4 \ \ \ \ \ g'(0) = 5
$$
Let $\displaystyle h(x) = {{f(x)} \over {g(x)}}$.  Find $h'(0)$.
\correct $\displaystyle {1 \over 8}$
\wrong can't be determined from the given information
\wrong $\displaystyle {{11} \over 8}$
\wrong $\displaystyle {3 \over 5}$
\wrong $\displaystyle -{{1} \over 8}$

\note %7
\problem For what value of the constant $c$ is the following function $f$
continuous on
$(-\infty,\infty)$:
$$
f(x) = \cases{x^2 +1 &if $x \leq 1$; \cr
              cx+2 &if $x>1$.}
$$
\correct 0
\wrong 1
\wrong 2
\wrong $-1$
\wrong $\displaystyle {1 \over 2}$

\note %8
\problem Let $\theta$ be an angle between $\displaystyle {{\pi} \over 2}$ and
$\pi$, such that $\displaystyle \sin \theta = {1 \over 4}$.  Find $\cos \theta$. 
(Hint: don't forget I said $\displaystyle {{\pi} \over 2} \leq \theta \leq \pi$.)
\correct $\displaystyle -{{\sqrt{15}} \over 4}$
\wrong $\displaystyle {{\sqrt{15}} \over 4}$
\wrong $\displaystyle {{\sqrt{3}} \over 4}$
\wrong $\displaystyle -{{\sqrt{3}} \over 4}$
\wrong $\displaystyle -{1 \over {\sqrt{15}}}$

\note %9
\problem Convert  $210^\circ$ (degrees) to radians.
\correct $\displaystyle {{7\pi} \over 6}$
\wrong $\displaystyle {{5\pi} \over 6}$
\wrong $\displaystyle {{5\pi} \over 2}$
\wrong $\displaystyle {{3\pi} \over 2}$
\wrong $\displaystyle {{11\pi} \over 6}$

\note %10
\problem Find $\displaystyle \lim_{t \rightarrow 0} {{\sin^2 4t} \over {t^2}}$.
\correct 16
\wrong 4
\wrong 2
\wrong 1
\wrong 8

\note %11
\problem Find $\displaystyle {d \over {dx}} \csc (x^2 )$.
\correct $- 2x\csc (x^2 ) \cot (x^2)$
\wrong $-\csc (2x) \cot (2x)$
\wrong $2x \csc x - x^2 \csc x \cot x$
\wrong $-\csc (x^2 ) \cot (x^2)$
\wrong $2x \csc (x^2 )$

\note %12
\problem Find $\displaystyle {d \over {dx}} \sqrt{1 + x^2}$.
\correct $\displaystyle {x \over {\sqrt{1 + x^x}}}$
\wrong $\displaystyle {1 \over 2 \sqrt{1 + x^2}}$
\wrong $\displaystyle {{2x} \over {\sqrt{1 + x^x}}}$
\wrong $\displaystyle {{x^2} \over {\sqrt{1 + x^x}}}$
\wrong $\displaystyle {{x} \over {2\sqrt{1 + x^x}}}$

\note %13
\problem Find $\displaystyle {{dy} \over {dx}}$ by implicit differentiation:
\hskip 1cm
$\displaystyle 
x^2 -xy + y^3 = 8.
$
\correct $\displaystyle {{-2x+y} \over {-x+3y^2}}$
\wrong $\displaystyle {{-2x-y} \over {-x+3y^2}}$
\wrong $\displaystyle {{8-2x+y} \over {-x+3y^2}}$
\wrong $\displaystyle {{8-2x-y} \over {-x+3y^2}}$
\wrong $\displaystyle {{-2x-y} \over {-x+3y}}$

\note %14
\problem Let $f(x)$ be some function whose first derivative is $\displaystyle f'(x)
= {x \over {x^2 +1}}$.  Find $f''(1)$. (Note that's $f''(1)$, not $f'(1)$.)
\correct 0
\wrong $\displaystyle {1 \over 2}$
\wrong 1
\wrong $\displaystyle {1 \over 4}$
\wrong 4

\note %15
\problem Let $\displaystyle f(x) = x^{100} + x^{50} + x^5 +1$.  Find $f^{(50)}
(0)$.
\correct $50!$
\wrong $\displaystyle {{100!} \over {50!}} + 50!$
\wrong $100! + 50!$
\wrong $100! + 50! + 5!$
\wrong 0

\note %16
\problem A particle is moving along the $x$-axis so that its position on the
$x$-axis is given by the formula $\displaystyle s(t) = {{t-1} \over {t+1}}$,
where we are only interested in $t \geq 0$.  Notice that 
$$
s'(t) = {2 \over {(t+1)^2}} \ \ \ \ \ \ \ \ \ \ s''(t) = {{-4} \over {(t+1)^3}}
$$
(you don't have to verify these facts).  Which of the following is/are true:
\bigskip

\itemitem{I.} The particle is always moving in the positive direction.
\itemitem{II.} The particle is always decelerating (i.e.\ slowing down).
\itemitem{III.} The particle is getting closer and closer to the position $x=1$,
but never quite reaches it.

\correct all three are true
\wrong only I.\ and II.\ are true
\wrong only I.\ is true
\wrong only I.\ and III.\ are true
\wrong only II.\ and III. are true

\note %17
\problem A large spherical balloon is being inflated in such a way that its volume
is increasing at a rate of 100 $\hbox{ft}^3$/sec.  At what rate is its radius
increasing when the radius is 5 ft?  You may use the fact that the volume of a
sphere is given by the formula $\displaystyle V = {4 \over 3} \pi r^3$.

\correct ${1 \over {\pi}}$ ft/sec
\wrong ${5 \over {\pi}}$ ft/sec
\wrong $\pi$ ft/sec
\wrong ${{500 \pi} \over {3}}$ ft/sec
\wrong ${1 \over {5\pi}}$ ft/sec

\note %18
\problem A particle is moving with the following data:
$$
a(t) = 12t^2 \hskip 1in v(0) = 3 \hskip 1in s(0) = 1
$$
Find the position of the particle at time $t=1$.
\correct 5
\wrong 1
\wrong  24
\wrong 2
\wrong 4

\note %19
\problem Let $f(x) = 1 + (x+1)^2$.  Find the absolute maximum and minimum values
of $f(x)$ on the interval $[-2,2]$.
\correct $\displaystyle \matrix{
\hbox{absolute maximum} = 10 \hfill\cr
\hbox{absolute minimum} = 1 \hfill }$
\wrong $\displaystyle \matrix{
\hbox{absolute maximum} = 10 \hfill\cr
\hbox{absolute minimum} = 2 \hfill }$
\wrong $\displaystyle \matrix{
\hbox{absolute maximum} = 9 \hfill\cr
\hbox{absolute minimum} = 0 \hfill }$
\wrong $\displaystyle \matrix{
\hbox{no absolute maximum} \hfill\cr
\hbox{absolute minimum} = 1 \hfill }$
\wrong $\displaystyle \matrix{
\hbox{absolute maximum} = 10 \hfill\cr
\hbox{no absolute minimum} \hfill }$

\note %20
\problem Let $f(x)$ be some function whose derivative is 
$$
f'(x) = {{3x(x-4)} \over {(x+1)^2}}
$$
Find the intervals on which $f(x)$ is increasing or decreasing.  (You don't have
to figure out what $f(x)$ is. Recall that $A \cup B$ means all $x$ which are
either in $A$ or $B$ or both.)
\correct $\displaystyle \matrix{
\hbox{increasing on } (-\infty, -1) \cup (-1,0] \cup [4,+\infty) \hfill\cr
\hbox{decreasing on } [0,4] \hfill }$
\wrong $\displaystyle \matrix{
\hbox{increasing on }  (-1,0] \cup [4,+\infty) \hfill\cr
\hbox{decreasing on } (-\infty, -1) \cup [0,4] \hfill }$
\wrong $\displaystyle \matrix{
\hbox{increasing on } (-\infty, -1) \cup [0,4]  \hfill\cr
\hbox{decreasing on } (-1,0] \cup [4,+\infty) \hfill }$
\wrong $\displaystyle \matrix{
\hbox{increasing on } [0,4] \hfill \cr
\hbox{decreasing on }(-\infty, -1) \cup (-1,0] \cup [4,+\infty) \hfill}$
\wrong $\displaystyle \matrix{
\hbox{increasing on } (-\infty, -1) \cup (-1,0] \hfill \cr
\hbox{decreasing on }[0,+\infty) \hfill}$

\note %21
\problem Let $\displaystyle f(x) = (x^2 -1)^2$.  Notice that 
$$
f'(x) = 4x(x+1)(x-1) \hskip .25in \hbox{and} \hskip .25in  f''(x) = 4(\sqrt{3}
x-1)(\sqrt{3} x+1).
$$
(You don't have to verify these facts.  Note that when I write $\sqrt{3} x$,
the $x$ is not inside the square root sign.)  On what interval(s) is $f$ concave
up?  (Recall that $A \cup B$ means all $x$ which are either in $A$ or $B$ or both.)
\correct $\displaystyle (-\infty, {{-1} \over {\sqrt{3}}}) \cup ({{1} \over
{\sqrt{3}}}, +\infty)$
\wrong $\displaystyle ({{-1} \over {\sqrt{3}}},{{1} \over {\sqrt{3}}})$
\wrong it's concave up everywhere
\wrong it's never concave up
\wrong $\displaystyle (-1,0) \cup (1,+\infty)$

\note %22
\problem Find $\displaystyle \lim_{x \rightarrow -\infty} {{\sqrt{x^2 +1}} \over
{2x-5}}$ (Note that's a $-\infty$, not a $+\infty$.)
\correct $\displaystyle -{1 \over 2}$
\wrong $\displaystyle {1 \over 2}$
\wrong $\displaystyle {5 \over 2}$
\wrong $\displaystyle -{1 \over 5}$
\wrong $\displaystyle {1 \over 5}$

\note %23 
\problem If 4800 $\hbox{cm}^3$ of material is available to make a box with a
square base and an open top, find the length of one side of the base of the box
with largest possible volume.  (I.e.\ find the value of $x$ in the diagram below
which maximizes the volume.)

\vskip 1.5in
\hskip 1.7in \openbox

\correct 40 cm
\wrong 20 cm
\wrong 100 cm
\wrong 50 cm
\wrong 25 cm

\note %24
\problem Write the following sum in sigma notation:
$$
2x + 3x^2 + 4x^3 + \cdots + nx^{n-1}
$$
\correct $\displaystyle \sum_{i=1}^{n-1} (i+1)x^i$
\wrong $\displaystyle \sum_{i=1}^{n} (i+1)x^i$
\wrong $\displaystyle \sum_{i=1}^{n} (i)x^i$
\wrong $\displaystyle \sum_{i=1}^{n} (i)x^{i-1}$
\wrong $\displaystyle \sum_{i=2}^{n} (i)x^i$

\note %25
\problem Let 
$$
f(x) = \cases{x +1 &if $0 \leq x \leq 1$; \cr
              2x &if $1 \leq x \leq 2$.}
$$
Find $\displaystyle A^x_0 (f)$ (the area function).
\correct $\displaystyle 
A^x_0 (f) = \cases{{1 \over 2}x^2 +x &if $0 \leq x \leq 1$; \cr
              x^2 + {1 \over 2} &if $1 \leq x \leq 2$.}$
\wrong $\displaystyle 
A^x_0 (f) = \cases{{1 \over 2}x^2 +x -{1 \over 2} &if $0 \leq x \leq 1$; \cr
              x^2  &if $1 \leq x \leq 2$.}$
\wrong $\displaystyle 
A^x_0 (f) = \cases{{1 \over 2}x^2 +x &if $0 \leq x \leq 1$; \cr
              x^2  &if $1 \leq x \leq 2$.}$
\wrong $\displaystyle 
A^x_0 (f) = \cases{{1 \over 2}x^2 +x &if $0 \leq x \leq 1$; \cr
              x^2 - {1 \over 2} &if $1 \leq x \leq 2$.}$
\wrong $\displaystyle 
A^x_0 (f) = \cases{{1 \over 2}x^2 +x + {1 \over 2} &if $0 \leq x \leq 1$; \cr
              x^2 + 1 &if $1 \leq x \leq 2$.}$

\note %26
\problem Let $f(x) = 9-x^2$ on the interval$[-2,3]$.  If the interval is divided
into 5 equal subintervals (i.e.\ $n=5$) and $x_i^* =$ right endpoint, find the
sum of the areas of the corresponding approximating rectangles.
\correct 30
\wrong 35
\wrong 44
\wrong $\displaystyle {{100} \over 3}$
\wrong 25

\note %27
\problem Evaluate the following integral by interpreting it in terms of areas:
$\displaystyle 
\int_0^2 \sqrt{4-x^2}
$
\correct $\pi$
\wrong $2\pi$
\wrong $\displaystyle {{\pi} \over 2}$
\wrong $4\pi$
\wrong $\displaystyle {{\pi} \over 4}$

\note %28
\problem A streetlight is at the top of a 15 ft.\ pole.  A man 6 ft.\ tall walks
away from the pole at a rate of 4 ft/sec.  At what rate is his shadow
lengthening when he is 20 ft.\ from the base of the pole?
\correct $\displaystyle {8 \over 3}$ ft/sec
\wrong 6 ft/sec
\wrong $\displaystyle {4 \over 3} $ ft/sec
\wrong 8 ft/sec
\wrong $\displaystyle {8 \over 5} ft/sec$

\note %29
\problem Find the distance between the points $(-2,3)$ and $(6,-3)$.
\correct 10
\wrong 100
\wrong 4
\wrong $\sqrt{52}$
\wrong $\sqrt{14}$

\appendix
\noindent I enjoyed teaching this class.  Have a great Christmas vacation, and
enjoy the Orange Bowl.  See you next semester.


 \end{document}
