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\pagestyle{myheadings}
\def\header{{\hfill Exam 1 \hfill A. Miller \hfill Fall 2011
                     \hfill Math 211 \hfill}}
\markboth\header\header
\newcount\probno
\newcount\total
\newwrite\examTBLFile

\probno=0
\def\prob#1{\advance\probno by 1
%      \ifodd\probno
%        \par\vfill\newpage
%      \else \vskip 3.8in \fi
    \newpage
    \noindent\the\probno . (#1 pts)\advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}
\def\finishtbl{
    \immediate\write\examTBLFile{Total \string& \the\total
    \string& \qquad \string\\ \string\hline}
     }
\newwrite\ans
\immediate\openout\ans=answers
\outer\def\answer{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\answerx{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
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  \immediate\write\ans{}
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\def\copytoblankline{\begingroup\setupcopy\copyans}
\def\setupcopy{\def\do##1{\catcode`##1=12}\dospecials
     \catcode`\|=12\obeylines}
{\obeylines \gdef\copyans#1
  {\def\next{#1}%
  \ifx\next\empty\let\next=\endgroup%
  \else\immediate\write\ans{\next}\let\next=\copyans\fi\next}}
\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

% \def\answer{\par\bigskip\bigskip\bigskip\noindent Answer: }

% \outer\def\answer{ {\par\bigskip Circle your answer.} \par\medbreak
%  \def\myskip{\vskip 2in}
%  \def\np{\vfill\newpage}
%  \def\vf{\vfill}
%  \def\vs#1{\par\vskip #1in}



\begin{document}



\bigskip

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

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

\bigskip
Circle your TA's name and the time of your section. 

\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

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Page & Points & Score \\  \hline\hline
  \input{exam.tbl}

  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

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

\setcounter{page}{0}

\prob{10} % 1.1 -23
For each of the following  
circle your answer.

\bigskip
(a) Determine the domain of the function $g(x)=\frac x{x^2-1}$.

\vfill

(b) Determine the domain of the function $h(x)=(x^2-1)(x-3)$.

\vfill

\answer (a) all real numbers $x$ except $x=1$ and $x=-1$.
(b) all real numbers


\prob{10} % 1.2 -31
Find the points of intersection (if any) of the curves:
$$y=x^2 \mbox{ and } y=x+2$$
Draw the graphs of these curves and
label the points of intersection.

\answer $(-1,1)$ and $(2,4)$

\prob{10} % 1.3 -29
Find the equation for the line thru the points
$(1,-2)$ and $(3,4)$.  Draw the graph of this line and
label these two points and label the $x$ and $y$ intercepts.

\answer $y=3x-5$

\prob{10} % 1.6 - 53 
For what value of $A$ (if any) is the following function
$f$ continuous for every $x$?   Circle your answer.

$$f(x)=\left\{
\begin{array}{ll}
\frac3x & \mbox { if } x\geq 1 \\
x^2+Ax-2 & \mbox { if } x < 1 \\
\end{array}\right.$$

\answer $A=4$

\prob{10} % 2.1 - 29
Find the derivative of $f$ using the definition
of derivative.  Begin by stating the definition of $f^\prime(x)$.
$$f(x)=x^2-3x$$ 
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
$$f(x+h)-f(x)=((x+h)^2-3(x+h))-(x^2-3x) $$
$$=\;\;x^2+2hx+h^2-3x-3h-x^2+3x $$
$$=\;\;2hx+h^2-3h$$
so
$$\frac{((x+h)^2-3(x+h))-(x^2-3x)}h=\frac{2hx+h^2-3h}h=2x+h-3$$
so
$$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}(2x+h-3) =2x-3$$

\prob{10} % 2.2 -37 2.4-47
Find the derivative $\frac{dz}{dt}$,
simplify it, and circle your answer.

\bigskip\bigskip
$$z=\frac3{t^2}\;\;-\;\;\sqrt[5]{3t^2-1} $$

\answer $$\frac{dz}{dt}=-6t^{-3}-\frac65t(3t^2-1)^{-\frac45}$$

\prob{10} % 2.3 - 47
Find the second derivative of $y$ with respect to $x$.
Use the correct notation for the second derivative.
Simplify your answer and circle it.
$$y=x^5+3x^3-x^2-2x+12$$
\answer $$\frac{d^2y}{dx^2}= 20x^3+18x-2$$

\prob{10} % 2.5,6
Suppose $y$ is implicitely defined as a function of $x$ by
the equation:
$$y^2+2xy+x^3=13$$

(a) Find $\frac{dy}{dx}$ when $x=2$ and $y=1$.

(b) Use it to estimate the change in $y$ when $x$
is changed from $2$ to $2\frac17$.

\answer (a) $-\frac73\;\;\;\;$ (b) $-\frac13$

\prob{10} % 2.6- 47
A tumor is modeled as being roughly spherical with radius $r$.
If the radius of tumor is currently $r=5$ milometers and growing at
the rate of $\frac12$ milometer per month what is the corresponding rate
of change of its volume $V=\frac43 \pi r^3$?

\answer $50\pi$

\prob{10} % 1.5-6
For each of these circle the best answer.

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

\bigskip\item  $\lim_{x\to 2^+} \;\; \frac1x$
\choice
\bigskip\item  $\lim_{x\to\infty}  \;\;\frac1x$
\choice
\bigskip\item  $\lim_{x\to -\infty}  \;\;\frac1x$
\choice
\bigskip\item  $\lim_{x\to -\infty}  \;\;x^2+x+1$
\choice
\bigskip\item  $\lim_{x\to 0^+}  \;\;\frac1x$
\choice
\bigskip\item  $\lim_{x\to 0^-} \;\;\frac1x$
\choice
\bigskip\item  $\lim_{x\to \infty}  \;\;\frac1{1-x^2}$
\choice
\bigskip\item $\lim_{x\to 0} \;\;\frac1x$
\choice
\bigskip\item $\lim_{x\to 0} \;\;\frac1{x^2}$
\choice
\bigskip\item $\lim_{x\to 1^-} \;\;{x^2+1}$
\choice

\end{enumerate}

\answer none-of-these 0 0 $\infty$ $\infty$ $-\infty$ $-\infty$ doesn't-exist
$\infty$ 2


\printanswers

\finishtbl

\end{document}



