\input LaTest.tex
\documentstyle{latest}
\textwidth 6.3in\textheight 8.5in\parindent 0cm
\topmargin -1cm\oddsidemargin 0.7cm

\begin{document}
\baselineskip 0.5cm\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=2.5cm\pageannouncements={usual}
                                         \input exam.perm 
                                         \lastversion=1
                                         \date{{\large November 20, 1997}}
                                         \title{{\large\bf Math 105, Test 3}}
\comment{\large 
\begin{minipage}[c]{6in}
Calculators are not allowed.  Please do not remove this answer sheet-you will return the
whole exam.  There are 15 multiple choice questions each worth 5 points. Record your
multiple choice problems by placing an X through one letter for each problem on this answer
sheet.  In addition there are 2 partial credit problems each worth 10 points.  Please show
all of your work for these two problems in the space provided.  In addition 5 points will
be given for spelling your name correctly.  Good Luck! {\bf You are taking this exam under
the honor code.} 
\end{minipage}}
 \twocolumn{11}  \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm\pagenumberstyle=1
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\problem  
If  $y(x)=(x+5)\sqrt{2x+2},$ calculate $y'(1)$.

  
\correct 5
\wrong  2  
\wrong   1/2
\wrong   7/2
\wrong 10

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\problem  

Suppose that $f,g$ are differentiable functions and that $f(4)=7, f'(2)=3,
g(1)=2, g'(1)=6$.  Compute $\frac{d}{dx}f(g(x))$ at $x=1$.



\correct 18
\wrong  21
\wrong  42
\wrong  6
\wrong  Can not be determined

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\problem  
Find $\frac{d}{dx} (x+2)^2(x-3)^{3}.$ 

\correct $5x(x+2)(x-3)^2 $
\wrong $ 20(x+2)(x-3)^2 $
\wrong $  2(x+2)(x-3)^2(3x-4) $
\wrong $ (x+2)(x-3)^{2}(6x-5) $
\wrong  $2(x+2)^2(x-3)^{3}(x-1)$


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\problem  
Find $\frac{dy}{dx}$ if $y$ is defined as a function of $x$ by the equation $x^2y^2=2$.

\correct $\frac{-y}{x} $
\wrong $ \frac{x}{y} $
\wrong $  2xy $
\wrong $ \frac{1-x^2y}{xy^2} $
\wrong $ \frac{2+xy}{xy} $

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\problem  

Say that the price (in dollars) of Notre Dame/West Virginia tickets is $p(t)$, 
while the daily sales (in tickets), $x(t)$ satisfy the demand equation:
$$
\frac{1}{3}p^3 +500 x^2 = 3000.
$$
Here t is measured in days.  Determine the rate (in dollars per day) at which the price is
changing when
$x=5,p=50,$ and the  daily sales are increasing at the rate of 1 ticket per day.

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


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\problem  

Solve the following equation for $x$:
$$
 2^{x+9}=8.
$$

\correct $x=-6$
\wrong $x=0$
\wrong $x=-3$
\wrong $x=e$
\wrong $x=4$


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\problem  

If $f(x)=(xe^x-e^x)^2,$ find all $x$ such that $f'(x)=0$.

\correct  $x=0,1$
\wrong $ x=-1,1 $
\wrong $ x=e,-e $
\wrong $ x=1+e, 1-e $
\wrong $ x=0,-1 $



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\problem  

Differentiate $\frac{\ln x }{x }$.



\correct $\frac{1-\ln x}{x^2}$
\wrong  $\frac{\ln x -1}{x^2}$
\wrong  $\ln x +1$
\wrong  $-e^{-x}$
\wrong  $\ln x$

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\problem  

Simplify $\ln (e^{\ln 2}).$

\correct $\ln 2 $
\wrong $  e^2 $
\wrong $  \ln (\ln 2 ) $
\wrong $  1 $
\wrong  $ 0$


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\problem  
Solve the following equation for $x$:
$$
4-3e^{x+6}=0.
$$

\correct $\ln(4/3)-6 $
\wrong $  \frac{1}{3}\ln4 -6 $
\wrong $  \ln(12)-6 $
\wrong $ e^{4/3} -6 $
\wrong  $\ln 7+6$

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\problem
Differentiate $e^x(e^{-x}+\ln e).$
\correct $e^x$
\wrong $-e^{-x}$
\wrong $e^x+1$
\wrong $e^{2x}$
\wrong $\ln x$


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\problem  

What is the derivative of  $\ln (5x)+e^{7x^2}?$

\correct $ \frac{1}{x}+14xe^{7x^2} $
\wrong $  \frac{1}{5x}+14xe^{7x^2} $
\wrong $  \frac{1}{5x}+14e^{7x^2} $
\wrong $  \frac{1}{x}+xe^{7x^2} $
\wrong $  \frac{5}{x}+e^{7x^2} $


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\problem  

What is the slope of the curve $y=e^x\ln x$ at $x=1$?


\correct $ e $
\wrong $ 2e $
\wrong $ e^2/2 $
\wrong $ e+1 $
\wrong $ e+\ln 2 $

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\problem  

Simplify $4 \ln 2 -\frac{3}{2}\ln 4  .$

\correct $ \ln 2 $
\wrong $ \ln (1/2) $
\wrong $ 2\ln 4 $
\wrong $ 0 $
\wrong $ e^2 $


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\problem  

Differentiate $f(x)=x^{\ln x}$.


\correct $ \frac{2}{x} x^{\ln x} \ln x $
\wrong $  x^{\ln x -1} \ln x $
\wrong $ \ln x $
\wrong $ (x-1) x \ln x $
\wrong $  2 x^{\ln x -1} $

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\problem  


A 5 foot ladder is leaning against a vertical wall.  If the foot of the ladder
is being pushed toward the wall at 6 ft/sec, how fast is the top end of the 
ladder sliding along the wall at the moment when the foot of the ladder is 4 feet
away from the wall?

\vspace*{17cm}


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\problem  

If $y(t)$ satisfies the equation $y'(t)=3y$ and $y(0)=10$, find an equation for $y(t).$

\vspace*{8cm}

Suppose $y(t)$ from above represents the number of people with the flu after $t$ days.  Then
how many people will have the flu after 2 days? (Hint: the answer may involve the constant
$e$.) 

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\endtest
