%\magnification 1200
          \def\marknumbers#1{}
\marknumbers{1,20}
\def\frac#1#2{{{#1}\over{#2}}}
\def\textfrac#1#2{{{\textstyle#1}\over{\textstyle#2}}}
\def\scriptfrac#1#2{{{\scriptstyle#1}\over{\scriptstyle#2}}}
\def\s{\hskip 3em}
\def\m{\hskip 1em}
\def\d{\displaystyle}   %%%equivalent to \dsize in ams.tex
\def\pt(#1,#2){\hbox{\mathsurround=0pt$\left(#1,#2\right)$}}
\def\nms#1{{\mathsurround=0pt{#1}}}
\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=1.7cm
\multiplechoiceskip=10pt

\pageannouncements={none}

%\marknumbers{1,20}
%\twocolumn{16}
%\twocolumn{p.5}
%\def\section{{\bf Class Time:}}

\def\HonorCodeStatement{ 
\hbox{I have not violated the Honor Code in any way with regard to this
work.}\hskip -4in 
\lower .8cm \hbox{Signature: \vrule height .2pt depth 0pt width 3.74in} 
}
\answersheetfootline{\HonorCodeStatement}

\input exam.perm1
\setglobalperms{\perm2:edcba}
%%%\def\st{\perm1:abcde \perm2:edcba}
\title{Math 105{--}Test II}
\date{March 3, 1994}
\lastversion=2

%%%%%%Date: Fri, 18 Feb 94 19:06:04 CST
%%%%%%From: andrew@math.uic.edu
%%%%%%To: Bei.Hu.1@nd.edu
%%%%%%Subject: test2.tex

%%%%%\input psfig

\global\let\app\longrightarrow

%%%%%%%\global\vertmultchoice=2.5in
\global\pagenumberstyle=2
\global\let\ansborder=\boxborder

%\lastversion=1
\lineskip=15pt
%%%%%%%\normalafterproblemskip=15mm
%%%%%%%\Professor{Andrew Mathas}
%%%%%%%\setglobalperms{\perm1:abcde%
%%%%%%%                \perm2:cdeab%
%%%%%%%                \perm3:adebc%
%%%%%%%                \perm4:edbac}

%%%%%%\comment{\baselineskip12pt\hbox to
%%%%%%\answersheetlinewidth{\vbox{%
%%%%%%\centerline{\bf Read these instructions carefully!}\medskip
%%%%%%1. Calculators are not allowed.\smallskip
%%%%%%2. This exam consists of 20 multiple choice questions each
%%%%%%worth 5 points.  There is no partial credit.\smallskip
%%%%%%3. For each question, mark an X through the one correct
%%%%%%answer on this page. Do not circle the answer.\smallskip
%%%%%%4. This test will be exactly 60 minutes in length. When you
%%%%%%are told to begin, but not before, glance through the entire
%%%%%%test and make sure that your copy of the test contains all
%%%%%%20 questions.\smallskip
%%%%%%5. This test is conducted under the Honor Code.}}}


\problem What is the derivative of the function
$f(x)=(x^2+1)^3(2x+4)$?
\correct $6x(x^2+1)^2(2x+4)+2(x^2+1)^3$
\wrong $3(x^2+1)^2(2x+4)+2(x^2+1)^3$
\wrong $6x(x^2+1)^2(2x+4)+2(x^2+1)^3(2x+4)$
\wrong $2x(x^2+1)^2$
\wrong $6x(x^2+1)^2(2x+4)$

\problem Let $f(x)=x^4-2x^3-12x+9$. Then $f(x)$ has a point(s)
of inflection at:
\wrong $(0,9)$
\wrong $(1, -4)$
\wrong $(2,-39)$ 
\correct $(0,9)$ and $(1,-4)$
\wrong No points of inflection.

\problem What is the value of
$\left.\frac{d\,}{ds}\bigg(3s^3+2s^2-6s+1\bigg)\right|_{s=-1}$?
\wrong $-2$
\correct $-1$
\wrong $0$
\wrong $1$
\wrong $2$

\problem What is the largest interval on which the function 
$f(x)=24+10x-x^2$ is decreasing?
\wrong $(10,\infty)$
\correct $(5,\infty)$
\wrong $(-\infty,5)$
\wrong $(-5,\infty)$
\wrong $(-\infty, -3)$

\problem  What is the {\it second\/} derivative of the function
$f(x)=(x^2+1)\sqrt{x^2+1}$?
\wrong $\frac{3}{4\sqrt{x^2+1}}$
\wrong $\frac{3}{2\sqrt{x^2+1}}$
\correct $3\sqrt{x^2+1}+\frac{3x^2}{\sqrt{x^2+1}}$
\wrong $\frac 3 4\sqrt{x^2+1} +\frac{3x}{4\sqrt{x^2+1}}$
\wrong $\sqrt{(x^2+1}+\frac{3x^2}{4\sqrt{x^2+1}}$

\problem If $f(t)=4t^8-2t^5+4t^3+4$ then what is 
$f'(-1)$?
\correct $-30$
\wrong $-20$
\wrong $15$
\wrong $0$
\wrong $11$

\problem Suppose that for a function $f(x)$ it is known that
$f(4)=26$ and that $f'(4)=-0.5$. Use the first
derivative to estimate $f(4.1)$.
\wrong $26$
\wrong $26.5$
\wrong $25.5$
\wrong $26.05$
\correct $25.95$

\problem On what interval(s) is the function 
$f(x)=2x^3-9x^2+12x-11$ increasing?
\wrong $(-\infty,1)$
\correct $(-\infty,1)$ and $(2,\infty)$
\wrong $(1,2)$
\wrong $(3,\infty)$
\wrong $(1,\infty)$

\problem Which of the following graphs could be the graph of
the function 

$f(x)=2x^3+3x^2-12x+12$?

\vskip 1.5in \m
\special{postscriptfile Exam2.5.eps}

\wrong (I) only
\correct (IV) only
\wrong (I) and (IV)
\wrong (I), (II), and (IV)
\wrong (II) and (IV)


\problem Suppose that a manufacturer's profit on $t$--items is
given by the function 

$P(t)=400t-4t^2$. How many items should
she produce in order to maximise her profit?
\wrong $400$ items
\wrong $100$ items
\correct $50$ items
\wrong $200$ items
\wrong $800$ items
 
\problem If $f(x)=\sqrt{x}(x^2+1)^3$ then $f'(x)$ is:
\correct $6x\sqrt{x}(x^2+1)^2+\frac{(x^2+1)^3}{2\sqrt{x}}$
\wrong $\frac{6x(x^2+1)^2}{2\sqrt{x}}$
\wrong $6x\sqrt{x}(x^2+1)^2$
\wrong $\frac{(x^2+1)^3}{2\sqrt{x}}+\sqrt{x}(x^2+1)^2$
\wrong $3\sqrt{x}(x^2+1)^2 + \frac{(x^2+1)^3}{2\sqrt{x}}$

\problem A farmer wants to build two holding pens
next to the side of a barn as shown in the diagram below. If
$120$ yards of fencing material is available, what is the
maximum area that the two holding pens can have?

\vskip 1.5in \hskip 2in
\special{postscriptfile Exam2.barn.eps}

\wrong $120$ square yards
\correct $1200$ square yards
\wrong $1000$ square yards
\wrong $1400$ square yards
\wrong $800$ square yards

\problem If $f(x)=x^2+2x$ and $g(x)=\frac{1}{x-4}$ then
what is $\frac{d\,}{dx}(f\circ g)(x)$?
\wrong $\frac{2x+2}{(x^2+2x-4)^2}$
\wrong $\frac{-(2x+2)}{(x^2+2x-4)^2}$
\correct $\frac{-2}{(x-4)^3}+\frac{-2}{(x-4)^2}$
\wrong $\frac{1}{(x-4)^3}+\frac{-2}{(x-4)^2}$
\wrong $\frac{-3}{(x-4)^3}+\frac{-2}{(x-4)^2}$

\problem If $f(x)=x^3-15x^2+250$, then which of the
following statements about the behaviour of $f(x)$ at the
point $(5,0)$ is {\it true}.
\correct $(5,0)$ is a point of inflection for $f(x)$.
\wrong $f(x)$ has a relative minimum at $(5,0)$
\wrong $f(x)$ is increasing at $(5,0)$ and is concave up.
\wrong $f(x)$ is decreasing at $(5,0)$ and is concave down.
\wrong $f(x)$ has a relative maximum at $(5,0)$

\problem If $f'(x)>0$ on the intervals $(-\infty,-1)$ and 
$(2,\infty)$ and $f'(x)<0$ on the interval $(-1,2)$, which
of the following graphs could be the graph of $y=f(x)$?

\vskip 1.5in \m
\special{postscriptfile Exam2.11.eps}

\wrong (II) only
\wrong  (III) only
\wrong  (I) and (IV) only
\correct (II) and (IV) only
\wrong (II), (III), and (IV) only

\problem If $f(x)=x^3-6x^2-15x+12$ which of the following
statements is {\it true}:
\wrong $f(x)$ does not have a point of inflection.
\wrong $f(x)$ is increasing on the interval $(-1,5)$
\correct $f(x)$ is concave up on the interval $(2,\infty)$
\wrong $f(x)$ has relative extrema at $x=1$ and $x=-5$
\wrong $f(x)$ is concave down on the interval $(-\infty,4)$

\problem A tennis ball is thrown straight up in the air and its
height after $t$ seconds is given by the equation 
\hbox{$s(t)=-10t^2+80t+7$}. How many seconds will it take before 
the balls reaches its maximum height?
\wrong $1$ second
\wrong $2$ seconds
\correct $4$ seconds
\wrong $8$ seconds
\wrong $16$ seconds

\problem Recall that ${}^3\sqrt{125}=5$. Use the first derivative to
give an estimate of ${}^3\sqrt{124}$.
\correct $4\frac{74}{75}$
\wrong $4\frac{24}{25}$
\wrong $4\frac{23}{25}$
\wrong $5\frac{1}{25}$
\wrong $5\frac{1}{75}$

\problem Find the $x$--coordinate(s) of the relative extrema of 
the function  

$f(x) = \frac{1}{(2x^3-15x^2+36x)^3}$
\wrong $x=1$
\wrong $x=2$
\wrong $x=3$
\correct $x=2$ and $x=3$
\wrong $f(x)$ has no relative extrema.

\problem A flu epidemic is spreading through a school in such
a way that after $t$ days the number of children who have the
flu is given by the formula $P(t)=-t^3+10t^2+10t$. At what 
rate is the flu spreading in the school after $5$ days?
\wrong $45$ students per day
\correct $35$ students per day
\wrong $30$ students per day
\wrong $25$ students per day
\wrong $20$ students per day


\end
