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\def\header{{\hfill Final \hfill A. Miller \hfill Fall 2011
                     \hfill Math 211 \hfill}}
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\probno=0
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% \def\answer{\par\bigskip\bigskip\bigskip\noindent Answer: } % comment out

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{\begin{list}{(\alph{enumi})}
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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.

\bigskip\bigskip

Detach pages 14-16 from your exam and take them home with you.

\bigskip

{\bf Hand in only pages 0-13.}

\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{10}{ 2.1 - 29 exam 1 problem 5(from exam 1)}
$$f(x)=x^2$$ 
Find the derivative of $f$ using the definition
of derivative.  Begin by stating the definition of $f^\prime(x)$.
Do {\bf not} use differentiation rules. This problem is 
from Section 2.1.  In all other problems you may use
the differentiation rules to compute derivatives.

\answer Define $$f^\prime(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
then
$$f(x+h)-f(x)=(x+h)^2-x^2=x^2+2hx+h^2-x^2=2hx+h^2$$
so
$$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=
\lim_{h\to 0}\frac{2xh+h^2}h =\lim_{h\to 0}2x+h =2x$$


\prob{10}{ 2.5,6 (From Exam 1)}
Suppose $y$ is implicitely defined as a function of $x$ by
the equation:
$$e^{xy}+x+y=2$$

Find $\frac{dy}{dx}$ when $x=0$ and $y=1$.
Circle you answer.

\answer $-2$



\prob{10}{ 3.3- 39}
Note that this is the derivative of $f$:
$$f^\prime(x)=x^2(x-1)^3(e^x+1)$$

Find the intervals where $f$ is increasing or decreasing.
Circle your answer.

\answer decreasing for $x<1$ and increasing for $x>1$.

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

\prob{10}{  Exam 2  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
$q=300-p^2$, 
% \mbox{ for } 0\leq p\leq \sqrt{300}$
find the price $p$ for which $E=-1$, i.e.,
of unit elasticity.
Circle your answer.

\answer $p=10$

\prob{10} { 5.5 - 15 exam 2}
$$D(q)=450-q^2\;\;\;\;\;\;\; S(q)=150+ 2 q^2$$
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 producer's surplus at equilibrium.

\noindent Circle your answers.

\answer $(10,350)$ and $PS=\frac{4000}3$

\prob{10}{  6.1-17}
Find the value of the definite integral:
$$\int_0^1 \frac{x}{e^{2x}}\; dx $$

\answer $\frac14-\frac34 e^{-2}$

\prob{10}{ 6.2-23}
Find the particular solution of the differential equation
$$ \frac{dy}{dx}= \frac{2x}{\; y^2}$$
satisfying $y=3$ when $x=2$. Circle your answer.

\answer $y=(3x^2+15)^{\frac13}$


\prob{10}{ 7.2 - 31}
Find the second partials (including the mixed partials) of
$$z=e^{xy^2}$$
Circle your answers.

\answer $$\frac{\partial^2 z}{\partial x\partial y}=2y(y^2x+1)e^{xy^2}$$

\prob{10}{ 7.3 - 15}
Find the critical points of
$$f(x.y)=x^3-6xy+y^3$$
and classify each as a relative minimum,
relative maximum, or saddle point.
Circle your answer.

\answer rel min at $(2,2)$, saddle at the origin.

\prob{10}{ 7.4 - 5}
Given the points 
$$(1,2)\;\; \;\;(2,4)\;\; \;\;(4,4)\;\; \;\;$$ find
the least-squares line.
Circle your answer.

\answer $y={\frac47} x+ 2$

\prob{10}{ 7.5 - 15}
Find the minimum and maximum values of
$$f(x,y,z)=x+2y+3z$$
subject to the constraint 
$x^2+y^2+z^2=14$.
Circle your answers.

\answer max 14, min -14

\prob{10}{ 7.6 -  61}
Find the volume of the solid under the surface:
$$f(x,y)= 2x+2y$$
and over the region $R$ where $R$ is bounded
by $y=x$, $y=2-x$, and $y=0$. Circle your answer.

\answer $\frac83$

\newpage
Detach pages 14-16 from your exam and take them home with you.  

{\bf Hand in only pages 0-13.}

Record your answers to problems 13-17 here.

\bigskip

Circle your answers below.

\bigskip
\def\sp{$\;\;$}
\def\choice{\sp\sp\sp 0 \sp  1 \sp 2 \sp $\infty$ \sp $-\infty$ 
\sp\sp doesn't-exist \sp\sp none-of-these}

{\bf 13.}
\begin{enum}
\item\choice
\item\choice
\item\choice
\item\choice
\item\choice
\item\choice
\item\choice
\item\choice
\item\choice
\item\choice
\end{enum}



\def\letters{\sp\sp a \sp\sp b \sp\sp c \sp\sp d \sp\sp e \sp\sp
\par\bigskip}

\bigskip
{\bf 14.} \letters

{\bf 15.} \letters

{\bf 16.} \letters

{\bf 17.} \letters

\newpage

Put your answers on page 13.

\bigskip

\hprob{10}{ 1.5-6 exam 1 problem 10 }
\begin{enum}
\item $\lim_{x\to\infty}  \;\;\frac1x$
\item $\lim_{x\to -\infty}  \;\;\frac1x$
\item $\lim_{x\to 0^+}  \;\;\frac1x$
\item $\lim_{x\to 0^-} \;\;\frac1x$
\item $\lim_{x\to 0} \;\;\frac1x$
\item $\lim_{x\to 0} \;\;\frac1{x^2}$
\item $\lim_{x\to \infty} \;\; e^x$
\item $\lim_{x\to -\infty} \;\; e^{x}$
\item $\lim_{x\to 0^+} \;\; \ln(x)$
\item $\lim_{x\to \infty} \;\; \ln(x)$
\end{enum}

\answer 0 0 $\infty$ $-\infty$ doesn't-exist \sp\sp $\infty$ $\infty$ 0 
$-\infty$ $\infty$
\sp\sp\sp 14e 15e 16a 17c


\hfill

\hprob{5}{6.3}
The improper integral $\int_2^\infty\frac1{x^2}$ 

a. is $\infty$.

b. is $1$.

c. diverges but not to $\infty$.

d. is $-\frac12$.

e. none of above

\hfill


\newpage

Put your answers on page 13.

\bigskip

\hprob{5}{6.4}
The value of the trapazoid rule to estimate the
integral $\int_1^2 \frac1x \; dx$ with $n=4$ is

a. undefined

b. $\infty$

c. $\frac{320987}{22}$

d. $\ln(2)$

e. none of above

\hfill

\hprob{5}{1.6 - 53 From Exam 1}
For what value of $A$ (if any) is the following function
$f$ continuous for every $x$?
$$f(x)=\left\{
\begin{array}{ll}
\ln(x) & \mbox { if } x > 1 \\
x^2+Ax-2 & \mbox { if } x \leq 1 \\
\end{array}\right.$$

a. $A=1$ 

b. no $A$ 

c. all $A\leq 1$

d.  $A=e$

e.  none of above.

\hfill

\hprob{5}{7.1 - 19}
Describe the domain of the function:
$$f(x,y)=\sqrt{y-x^2}$$

a. All ordered pairs $(x,y)$ of real numbers.

b. All real numbers $y$ which are nonnegative.

c. All ordered pairs $(x,y)$ of real numbers
such that $y\geq x^2$.

d. All ordered pairs $(x,y)$ of real numbers
such that $y = x^2$.

e. None of above.

\hfill

\newpage
Use this sheet for scratch paper.

\printanswers

\finishtbl

\end{document}



