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\def\header{{\hfill Exam 2 \hfill A. Miller \hfill Fall 2011
                     \hfill Math 211 \hfill}}
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\begin{document}



\bigskip

No books, no notes, no
electronic devices of any kind.
Show all your work.  Simplify
your answer and circle it.

\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\bigskip

\begin{minipage}[b]{2.8in}

Circle your TA's name and the time

your section starts.

\bigskip
\begin{tabular}{llrr}
Clement, Nathan  &  M &11:00  &   1:20 \\
Jefferis, Leland &  M & 1:20  &   2:25 \\
Makuluni, Edson  &  T & 8:50  &   9:55 \\
Nan, Ting-Ting   &  T &11:00  &  12:05 \\
Wang, Kejia      &  M & 8:50  &   9:55 \\
Yao, Chengjian   &  T & 1:20  &   2:25 \\
You, Qian        &  T &11:00  &  12:05 \\
Zhao, Yongqiang  &  M & 8:50  &  11:00 \\
\end{tabular}

\bigskip
Hand in to your TA.

\end{minipage}
\begin{minipage}[c]{2in}
\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Page & Points & Score \\  \hline\hline
  \input{exam.tbl}

  \hline \hline
 \end{tabular}
\end{center}
\end{minipage}
\immediate\openout\examTBLFile=exam.tbl

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%   begin problems  %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\setcounter{page}{0}


\prob{7} % 5.6- 13
The region under the curve
$$y=\frac1{\sqrt{x}}$$
for $x$ such that $1\leq x\leq 2$ is rotated about
the $x$ axis.
What is the volume of the
solid of revolution?

Circle your answer.

\answer $\pi\ln(2)$

\prob{7} % 4.2- 31
Solve the following equation for $x$:
$$ 2\ln(x)=\ln(12)-\ln(3)$$
Simplify your answer. Circle your answer.

\answer $x=2$

\prob{7} % 3.1- 31
Determine the critical numbers of the
function:
$$g(t)=\frac{t^2}{t^2+t+2}$$
Using the first derivative only, determine
whether each critical point is a relative minimum,
relative maximum, or neither.
Circle your answer.

\answer $t=0$ is rel min, $t=-4$ is rel max

\prob{7} % 3.2- 31
Use the second derivative test to find the relative
maxima and minima of $f$:
$$f(x)=x+1+\frac{4}x$$
Circle your answer.

\answer $(2,5)$ rel min, $(-2,-3)$ rel max

\def\del{\triangle}
\def\spacer{\phantom{A}}

\prob{7} % 3.4- 25
Suppose $q$ is the demand and $p$ is the price of
a certain commodity.  The elasticity of demand, $E$,
is the percentage change of $q$ divided by the
percentage change of $p$ taken to the limit:

$$E=\lim_{\del p\to 0} \;\frac{\spacer\frac{\del q}q\spacer}{\frac{\del p}p}$$

If the demand is related to the price by
$q=100-p^2 \mbox{ for } 0\leq p\leq 10$,
find the price $p$ for which $E=-2$, i.e., a  small percentage raise in price
leads to a drop in demand by twice that percentage.
Circle your answer.

\answer $p=\sqrt{50}$

\prob{7} % 3.5- 29
It is estimated that the cost of constructing an office
building that is $n$ floors high is
$$C(n)=2n^2+500n+200$$
thousand dollars. How many floors should the building
have in order to minimize the average cost per floor?
Circle your answer.

\answer $n=10$

\prob{7} % 4.1- 39
How much money should be invested now at 5\% nominal annual
interest to obtain \$10000 in 10 years if interest is
compounded continuously?
Circle your answer.

\answer $10000\;e^{-.5}$

\prob{7} % 4.3- 35
Find $f^\prime(x)$ given that
$$f(x)=\frac{x}{2^x}$$
Circle your answer.

\answer $f^\prime(x)=2^{-x}-x2^{-x}\ln(2)$

\prob{7} % 4.4- 11
Graph the function $$f(x)=xe^x$$

Determine where it
is increasing or decreasing, concave up or down.  Show
as many key features as possible: high and low points,
points of inflection, vertical and horizontal asymptotes,
intercepts, cusps, vertical and horizontal tangents.

\answer $f(0)=0$
  \par    increasing for $x>-1$
  \par    decreasing for $x<-1$
  \par    global min at $x=-1$ and horizontal tangent
  \par    concave up for  $x>-2$
  \par    concave down for $x<-2$
  \par    inflection point at $x=-2$
  \par    limit at $\infty$ is $\infty$
  \par    limit at $-\infty$ is $0^{-}$ $\;\;\;\;x$-axis is an asymptote

\prob{8} % 5.1- 33
Solve
$$\frac{dy}{dx}=\frac1x-\frac2{x^2}
\;\;\;\;\;\;\mbox{ where $y=5$ when $x=1$.}$$
Circle your answer.

\answer $y=\ln(x)+\frac2x+3$


\prob{7} % 5.2- 31
Find the integral:
$$\int \frac{4x}{2x+1}\; dx$$
Circle your answer.

\answer $2x+1-\ln|2x+1|+C$

\prob{7} % 5.3- 31
Find $$\int^2_1 2f(x)+5 g(x)\;\; dx$$
given that

\def\spc{\;\;\;\;\;}
\bigskip
$$\int^2_{0} f(x)\;dx\;=5 \spc
\int^2_{0} g(x)\;dx\;=2  \spc
\int^1_{0} f(x)\;dx\;=1   \spc
\int^1_{0} g(x)\;dx\;=4 $$
 Circle your answer.

\answer $-2$

\prob{7} % 5.4- 13
Find the area of the region between the curves
$$y=x^3-x^2 \;\;\; \mbox{ and } \;\;\; y=x^2$$
Circle your answer.

\answer $\frac43$

\prob{8} % 5.5- 15
$$D(q)=450-q^2\;\;\;\;\; S(q)=150+ 2 q^2$$
are the demand and supply functions for a particular
commodity.  Specifically $q$ units will be
demanded (sold) at a price $p=D(q)$ dollars per unit, while
$q$ units will be supplied by producers when
the price is $p=S(q)$ dollars per unit.

(a) Find the equilibrium point $(q_e,p_e)$ where supply equals demand.

(b) Find the consumer's surplus at equilibrium.  This is the
total amount consumers are willing to spend minus what consumers
have to spend.

\answer $(10,350)$ and $CS=666 \frac23$

\printanswers

\finishtbl

\end{document}



