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\def\header{{\hfill Exam 1 \hfill A. Miller \hfill Fall 2006
                     \hfill Math 211 \hfill}}
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% \def\answer{\par\bigskip\bigskip\bigskip\noindent Answer: }
% \outer\def\answer{ {\par\bigskip Circle your answer.} \par\medbreak
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\begin{document}

\bigskip

Show all work.
Simplify your answers as much as you can.
% Circle your answer.
No books, no notes, no calculators, no cell phones, no pagers, no
electronic devices of any kind.
Use the last two pages of the exam for scratch paper or ask a TA for
extra scratch paper.
% However you can bring to the exam
%    one  $8.5$ by $11$
% cheat sheet with anything you want written or printed on both sides.
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
Circle the number (321-333) of your discussion section and hand in
to your TA:

\begin{verbatim}
 321    7:45    TR    B329 VAN VLECK  Kumar, Rohini
 322    8:50    TR    6322 SOC SCI    Kumar, Rohini
 323    9:55    TR    348 BIRGE       Mantilla Soler, Guillermo Arturo
 324   11:00    TR    1333 Sterling   Wang, Bing
 325   12:05p   TR    348 BIRGE       Wang, Bing
 326   12:05p   TR    B337 VAN VLECK  Potluri, Vijaya Kranthi
 327    1:20p   TR    B135 VAN VLECK  Mantilla Soler, Guillermo Arturo
 328    1:20p   TR    B123 VAN VLECK  Umarji, Pallavi Anand
 329    2:25p   TR    B211 VAN VLECK  Yin, Weidong
 330    2:25p   TR    B115 VAN VLECK  Umarji, Pallavi Anand
 331    3:30p   TR    B135 VAN VLECK  Yin, Weidong
 332    2:25p   TR    223 INGRAHAM    Christodoulopoulou, Konstantina
 333    3:30p   TR    B115 VAN VLECK  Christodoulopoulou, Konstantina
\end{verbatim}

\bigskip

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Page & Points & Score \\  \hline\hline
%   \input{exam.tbl}
1& 14 & \qquad\\ \hline
2& 14 & \qquad\\ \hline
3& 14 & \qquad\\ \hline
4& 14 & \qquad\\ \hline
5& 14 & \qquad\\ \hline
6& 14 & \qquad\\ \hline
7& 14 & \qquad\\ \hline
8&  2 & \qquad\\ \hline \hline
Total & 100& \hskip 2em\relax \\ \hline
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

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

\setcounter{page}{0}


\prob{7} % 1.1-43
Assume that $\lim_{x\to 0}f(x)=5$ and $\lim_{x\to 0}g(x)=-1$
find the limit:
$$\lim_{x\to 0}{{xf(x)-g(x)}\over {f(x)-g(x)}}$$

\answer $1/6$

\prob{7} % 1.2-23
Find the limit:
$$\lim_{x\to -\infty}{{x^3}\over {1-x^2}}$$

\answer $\infty$

\prob{7} % 1.3-17
Determine where the function $f$ is continuous and where it is
discontinuous.
$$f(x)=\left\{
\begin{array}{ll}
{{x^2-9}\over x-3} & \mbox{ if } x < 3 \\
& \\
  6                        & \mbox{ if } x \geq 3 \\
\end{array}\right.
$$

\answer The function $f$ is continuous at every point on the real line.

\prob{7} % 2.1-15
Suppose $b$ is a positive number and $u$ is
a fixed number for which it is known
that $$b^u=5$$  What is $b^{2u}$?

\answer $25$

\prob{7} % 2.2- 7
Suppose you put \$2000 dollars in an account paying an annual nominal
interest rate of 3.6\% compounded continuously.
How much will you have in the account at the end of 6 months?

\answer The amount $A$ in the account after $t$ years satisfies
$$A=2000e^{.036t}$$   Since 6 months is a half of a year you will have
  $2000 e^{(.036)(1/2)}=2000 e^{.018}$


\prob{7} % 2.3-39
Suppose that $B$ and $a$ are numbers such that
$\log_2(B)=a$.  Write the logarithm below as a function of $a$.
$$\log_2(\root 3 \of {4B})$$

\answer $\log_2(\root 3 \of {4B})= $
$\log_2({(4B)}^{1\over 3})= $
${1\over 3} \log_2({4B}) = $
${1\over 3} (\log_2({4}) +\log_2({B})= $
${1\over 3}(2+a)$

\prob{7} % 2.4- 3
Rewrite $x^y$ as an expression using the natural logarithm and
the natural exponentiation function.  (Suppose your calculator
has no button for computing $x^y$ but does have buttons
for $\ln(x)$ and $e^x$.  How could you use these to compute $x^y$?)

\answer $e^{y\ln(x)}$

\prob{7} % 3.1-11
Find the equation of line tangent to the curve $y=x^2$ at
the point $(2,4)$.

\answer $y=4x-4$.

\prob{7} % 3.2-27
Suppose that $f(x)$ is a differentiable function whose graph goes through the
point $(1,2)$ and whose tangent line at that point has the equation
$x+y=3$.  Without computing, find each of the following limits:

\bigskip
(a) $\lim_{h\to 1} f(h)$

\bigskip
(b) $\lim_{\triangle x\to 0} {{f(1+\triangle x)-2}\over \triangle x}$

\bigskip
(c) $\lim_{x\to 1} {{f(x)-2}\over {x-1}}$

\answer (a) Since $f$ is continuous this is $f(1)=2$.
(b) and (c) are equivalent definitions of $f^{\prime}(1)$.  Since
the slope of the tangent line is $-1$, these are both $-1$.


\prob{7} % 3.3- 1
A miniature water rocket is launched straight upward from the surface
of the earth. Its height $s$ in feet at time $t$ in seconds
is given by the formula:
$$s= -16t^2+40t $$
Find the height and velocity of the rocket at the end
of 1 second.


\answer The velocity is ${ds\over dt}=-32t+40$ at $t=1$ we have
that $s=24$ and ${ds\over dt}=8$.

\prob{7} % 3.4-45
A real estate broker charges a commission of 7\% percent on any
house sold for \$100,000 or less.  On sales that exceed \$100,000
the broker charges a flat fee of \$5,000 plus 3\% of the sale price.
Let $x$ be the amount of a given sale and $f(x)$ the brokerage fee.

(a) Write a (multi-line) formula for $f(x)$ and sketch the graph.

(b) At what points, if any, is $f(x)$ discontinuous?
At what points, if any, does $f(x)$ fail to be differentiable?

\answer $f$ is discontinuous at $x_0=100,000$ because
$$\lim_{x\to x_0^-} f(x)=\lim_{x\to x_0^-} (.07)x  = 7000$$
$$\lim_{x\to x_0^+} f(x)= \lim_{x\to x_0^+} (.05)x + 5000 =8000$$
Since differentiable
implies continuous is not differentiable at $x_0$.  Everywhere
else it is differentiable and continuous. (Ignore $x\leq 0$.)
$$f(x)=\left\{
\begin{array}{ll}
(.07)x   &      \mbox{ if } x\leq 100000 \\
(.05)x + 5000 & \mbox{ if } x > 100000
\end{array}\right.$$


\prob{7} % 3.5-27
Find the derivative $ dy\over dx $ of
$$y={{\ln(x)+2} \over {3}}$$

\answer $1\over 3x$

\prob{7} % 3.6-15
Find the derivative $ dy\over dx $ of
$$y={{ x^2+1}\over{ 1+e^x}}$$

\answer ${(2x)(1+e^x)-(x^2+1)e^x}\over {( 1+e^x)^2}$

\prob{7} % 3.7-21
Find the derivative $ dy\over dx $ of
$$y=(2x^4-x^3+1)^7$$

\answer $7(2x^4-x^3+1)^6(8x^3-3x^2)$

\prob{2} Circle either True or False:

\bigskip (a) For any real valued function $f$ on the real line, if
$f$ is continuous at a point $a$, then $f$ must be differentiable at the
point $a$. \hfill True \hskip .5in False


\bigskip (b) There is a real valued function $f$ defined for every real number
which is continuous at every point but has no derivative at any point.
\hfill True \hskip .5in False

\answer (a) False (b) True

\bigskip




%  \prob{2} Circle the letter (a) (b) (c) of the correct answer:
%
%  % p. 131
%
%  \bigskip Leonhard Euler (1707-1783) despite many health problems and failing
%  eyesight, which eventually led to blindness, was one of the most inventive
%  and prolific mathematicians of all time. He made significant contributions to
%  virtually every area of mathematics, including the calculus, number theory,
%  differential equations, and combinatorics.  It was in his honor that the
%  irrational number below was named:
%
%  (a) $\sqrt{2}$
%
%  (b) $\pi$
%
%  (c) $e$
%
%
%  % p.141
%
%  \bigskip John Napier (1550-1617) was the son of one of the most important
%  families in  Scotland and held extensive estates, which he managed with great
%  skill. He was also known for being a great genius at inventing as well as for
%  his religious fervor, which involved him in a number of heated controversies.
%  He is credited with inventing the concept of the
%
%  (a) logarithm.
%
%  (b) hydrogen bomb.
%
%  (c) 2 for 1 drink special at Billy-Bobs Bar and Barbecue.
%
%
%  % p.151
%
%  \bigskip Marie Skiodowska Curie (Madame Curie) 1867-1934 discovered
%  the radioactive substance, radium, and was the
%
%  (a) first woman who walked on the surface of the moon.
%
%  (b) first woman who won the Nobel Prize and the first person to win the Nobel
%  twice.
%
%  (c) first woman to be elected President of the United States of America.
%
%  \bigskip Sir Isaac Newton (1643-1727) who discovered the law of gravity and
%  Gottfried Wilhelm von Leibnitz (1646-1716) who made many contributions to
%  philosophy and mathematics, each simultaneously discovered what important
%  mathematical theory causing much bitterness and rivalry between them:
%
%  (a) the differential and integral calculus.
%
%  (b) the theory that 17 is the smallest random integer.
%
%  (c) catastrophe theory or the mathematical theory that explains why when you
%  accidentally drop a piece of toast it always lands with the jelly side down.
%
%  \answer caba
%
%

\printanswers
\finishtbl
\end{document}



