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\def\header{{\hfill Final \hfill A. Miller \hfill Fall 2006
                     \hfill Math 211 \hfill}}
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\begin{document}


\bigskip

No books, no notes, no calculators, no cell phones, no pagers, no
electronic devices of any kind.
However you can bring to the exam

    one  $8.5$ by $11$

 cheat sheet with anything you want written or printed on both sides.

Put your answer in the box provided if there is a box. Show
every step of your solution.  Erase or mark out anything false in your work.
Simplify your answers as much as you can.  It may not be sufficient to have a
right answer to get full credit.  Some graders may be stricter than others.

Exam and answers will be posted shortly after the exam:
 www.math.wisc.edu/$\sim$miller

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

\bigskip\bigskip\bigskip

\begin{minipage}{4in}
Circle the number (321-333) 
of your discussion section and 
hand in to your TA:
\par\bigskip
\par\noindent
\begin{tabular}{lll}
 321 &   7:45  & Kumar, Rohini \\
 322 &   8:50  & Kumar, Rohini \\
 323 &   9:55  & Mantilla Soler, Guillermo Arturo \\
 324 &  11:00  & Wang, Bing \\
 325 &  12:05p & Wang, Bing \\
 326 &  12:05p & Potluri, Vijaya Kranthi \\
 327 &   1:20p & Mantilla Soler, Guillermo Arturo \\
 328 &   1:20p & Umarji, Pallavi Anand \\
 329 &   2:25p & Yin, Weidong \\
 330 &   2:25p & Umarji, Pallavi Anand \\
 331 &   3:30p & Yin, Weidong \\
 332 &   2:25p & Christodoulopoulou, Konstantina \\
 333 &   3:30p & Christodoulopoulou, Konstantina \\
\end{tabular}
\par\bigskip\bigskip Bing Wang's students hand in to
Myoungjean Bae.
\end{minipage}
\begin{minipage}{2in}
\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
   \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}
\end{minipage}

\immediate\openout\examTBLFile=exam.tbl

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

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\put(10,25){\text{Put your answer}}
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\put(10,15){\text{Show work below.}}
\end{picture}
}

\prob{10} % 1.3

Find the number $c$ that makes $f(x)$ continuous for every $x$:
$$f(x)=\left\{
\begin{array}{cl}
{(1-x)^2\over {1-x^2}} & \mbox{ if } x\neq 1 \\
c & \mbox{ if } x=1\\
\end{array}\right .$$

\answerbox

\answer $c=0$

\prob{10} % 2.2
Suppose you open an account paying 12\% nominal annual interest compounded
continuously.  How much should you deposit to ensure that there will be \$
5,000 in the account after 6 months?

\answerbox

\answer $5000 e^{-.06}$

\prob{10} % 3.8
Find the equation for the tangent line to the curve
$$xy^2+2y+3x = 6$$
at the point $(1,1)$.

\answerbox

\answer $y=-x+2$

\prob{10} % 4.2-3
Find the inflection points of the function $f(x)=e^{-x^2}$.

\answerbox

\answer $x=\pm \sqrt{1\over 2}$


\prob{10} % 5.5
Suppose
$$G(t)=t^4e^t +\int_0^t x^4 e^x\;dx$$
What is $G^\prime(t)$?

\answerbox

\answer $4t^3 e^t + 2t^4e^t$

\prob{10} % 5.7- 1   from exam 2  material

Use the midpoint rule with $n=2$ to approximate the integral
$$\int_0^1 (1+x^2)\;dx.$$
Find the computed value $C$, the true value $T$ of the integral, and
the error, $E$.

\answerbox

\answer $T={64\over 48}$, $C={63\over 48}$, $E=\pm {1\over 48}$ 

\prob{10}  % 6.3

Find the unique solution for the initial value problem:
$${dy\over dx}={y+1\over x-1}\;\;\mbox{ and } \;\; y(0)=1$$

\answerbox

\answer $y=-2x+1$

%%%%%%%%%%%%%%%  post exam 2 

\prob{10} % 6.4-23
Park rangers have been studying the population dynamics of a
the rabbits in a small park.  They calculate that 
the logistic equation
$${dy\over dt}=ry-ky^2$$
is a good model for the number of rabbits $y(t)$ at time $t$
if they take the damping factor $k=.003$ and the annual growth rate is
$r=.51$.
Suppose $y(0)$ is not zero and
find the longterm number of rabbits in the park, i.e., 
$$\lim_{t\to\infty}y(t)$$

\answerbox

\answer $170$

\prob{10} % 6.5-9
Compute the following improper integral or show that it diverges:
$$\int_1^\infty x e^{-x^2}\;dx$$

\answerbox

\answer ${1\over 2} e^{-1}$

\prob{10} % 7.1-35
Sketch the level curves $z_0=1,2,3$ and the cross sections
$x_0=2$ and $y_0=0$ for the surface 
$$z=x^2+y^2$$

\answer circles $x^2+y^2=1$, $x^2+y^2=2$, $x^2+y^2=3$
\par parbola $z=y^2+4$ 
\par parbola $z=x^2$.



\prob{10} % 7.2-55
Find all first-order partial derivatives for the function 
$$f(x,y,z)=xy^2e^z$$


\answer $f_x=y^2e^z$, $f_y=2xye^z$, $f_z=xy^2e^z$

\prob{10} % 7.3- 3
Find all critical points for the function
$$f(x,y)=x^2+3y^2+3y-2x-5$$
and determine whether each is a local minimum, local maximum,
or saddle point.

\answerbox

\answer $(1,-{1\over 2})$ is a local minimum.


\prob{10} % 7.4- 3
Find the least squares line for the data:
$$(0,1)\;\;\;\;(1,2)\;\;\;\;(2,5)$$

\answerbox

\answer $L(x)=2x+{2\over 3}$


\prob{10} % 7.5-15
The function $f(x,y,z)=x^2+2y^2+z^2$ has a minimum subject to
the constraint $x-y+2z=2$.  Find it by the method of Lagrange
multipliers.

\answerbox

\answer $({4\over 11},-{2\over 11},{8\over 11})$ where $f$ has
the value ${88\over 121}$

\prob{10} % 

\def\myskip{\bigskip\bigskip\bigskip\bigskip}

\def\blankline#1{\underline{\phantom{{#1} x {#1}}}}
\def\blankline#1{\underline{{#1}}}

Fill-in the blanks with the best possible answer:

\bigskip
1. A limit
$$\lim_{x\to 0}{f(x)\over g(x)}$$
is an  indeterminate form, if
$${{\lim_{x\to 0}{f(x)}}\over
{\lim_{x\to 0}{g(x)}}}
=\blankline{{{0\over 0}}}
\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{ or }\;\;\;\;\;\;\;\;\;\;\;\;\;
{{\lim_{x\to 0}{f(x)}}\over
{\lim_{x\to 0}{g(x)}}}
=\blankline{{{\infty\over \infty}}}$$

\myskip

If $f(x)$ is a continuous function
on the interval $[a,b]$, $f(a)<0$, and $f(b)>0$, then for some
$c$ in $(a,b)$ we must have that \blankline{$f(c)=0$}.

\myskip
2. The following limit equals
$$\lim_{n\to\infty}(1+{1\over n})^n=\blankline{e}.$$


\myskip
3. The derivative of a function $f(x)$ is a new function $f^\prime(x)$ that
gives the \blankline{slope} of the tangent line to the graph at the point
$x$.
It is defined by the following limit:
$$f^\prime(x)=
\blankline{\lim_{h\to 0}{{f(x+h)-f(x)}\over h}}$$

\myskip

If a function $f$ is differentiable, then it must be \blankline{continuous}.

\myskip

%The marginal profit $MP(x)$ is the \blankline{derivative} of the profit 
%function $P(x)$.

\myskip
4. If $f^\prime(x)>0$ for all $x$ in 
an interval $(a,b)$, then the function $f(x)$ is \blankline{increasing}
on the interval.

\myskip

The second derivative test says that if $f^\prime(c)=0$ and
\blankline{$f^{\prime\prime}(c)<0$}, then $f$ has a local maximum at $c$.

\myskip

Suppose $x_0$ is in the interval $(a,b)$ and for every $x$ in the interval
$(a,b)$ we have that $f(x)\leq f(x_0)$, then we say that $f(x_0)$ is the
\blankline{global maximum} of the function $f$ in the interval.

\myskip
5. An antiderivative of a given function $f(x)$ is a function $F(x)$ such
that 
$$\blankline{F^\prime(x)=f(x)}$$  

\myskip

The Fundamental Theorem of Calculus states that if $f(x)$ is continuous
on the interval $[a,b]$ and $F(x)$ is an antiderivative of $f(x)$, then
$$\blankline{\int^b_a\;f(x)\;dx ={F(b)-F(a)}}$$

\myskip

\noindent If a function $f$ is \blankline{continuous} on the interval
$[a,b]$, then the integral $\int^b_a\;f(x)\;dx$ exists, i.e., the
Riemann sums approach a finite limit.

\myskip

6.  Suppose $f(x)$ and $g(x)$ are continuous functions on the
interval $[a,\infty)$,  $0\leq f(x)\leq g(x)$ for all $x$ in
$[a,\infty)$, and $\int_a^\infty f(x)\;dx$ diverges.  Then $\int_a^\infty g(x)$
\blankline{diverges}

\myskip
7.  To find the maximum or minimum of $w=f(x,y,z)$,
we look for all points $(x,y,z)$ such that
$$\blankline{{{\partial w} \over {\partial x}} =0 \mbox{ and }
{{\partial w} \over {\partial y}} =0
\mbox{ and }
{{\partial w} \over {\partial z}} =0}
$$


\answer See problem.

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