%$Modified: Fri Jun  8 16:26:08 2001 by wgd $

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\def\whichexam{Math 120, Exam III}
\def\examdate{Tuesday, April 24, 2001}
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\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own.
\itemitem{$\bullet$} The exam lasts for {\bf one hour and fifteen minutes}.
\itemitem{$\bullet$} Be sure that your name is on every page in case
pages become detached. 
\itemitem{$\bullet$} Be sure that you have all 9 pages of the
test. Good luck!
%\vskip5pt
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PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
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%
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%
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%
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%
\newchoice{Multiple Choice}
\five
%
%
\ifsampletest\par\noindent{\bf This test consists of $12$ 
multiple-choice questions and four partial credit questions.
The correct answer to each multiple choice question is (a).
}\par\smallskip\fi
%
%
\ifproofreading\par\noindent{\bf 
\centerline{PROOFREADING COPY. TOP SECRET!} 
The correct answer to each multiple-choice question is (a). }
\par\smallskip\fi

\three
\newp 1.
If $A$ dollars are invested at an annual
rate $r$ compounded monthly, how much money will
be in the investment account after 2 years?
\ANp
(a) $A(1+\frac{r}{12})^{24}$
(b) $A(1+\frac{r}{6})^{12}$
(c) $A(1+r)^{2}$
(d) $(A+\frac{r}{12})^{12}$
(e) $Ae^{24r}$
\Pra42315


\three
\newp 2. 
For what values of $r$ is the function $y=e^{rx}$  a solution of the differential 
equation $y''-3y'+2y=0\,\,$?
\ANp
(a) $r=1,2$
(b) $r=0,2,-2$
(c) $r=2,3$
(d) $r=0$
(e) $r=-1$
\Pra54312

\three
\newp 3.
What amount should be invested at an annual interest rate of $4\%$, 
compounded continuously, in order for the investment to reach \$100,000 in 25 
years?
\ANp
(a) $100,000\cdot e^{-1}$
(b) $100,000\cdot e$
(c) $100,000\cdot 1.04^{-25}$
(d) $100,000\cdot 1.04^{25}$
(e) $100,000\cdot 25^{-1.04}$
\Pra21354

\three
\newp 4.
A bacteria culture, growing exponentially, starts with 500 bacteria. After 
2 hours there are 1200 bacteria. Find the number of bacteria after 5
hours. In the answers, the notation $\exp(x)$ stands for the exponential $e^x$.
\ANp
(a) $500\cdot  \exp\left({\frac{5}{2} \ln\frac{12}{5}}\right)$
(b) $500\cdot  \exp\left({\frac{2}{5} \ln\frac{5}{12}}\right)$
(c) $500 \cdot \left(\frac{12}{5}\right)^3$
(d) $500 \cdot \exp\left({\frac{5}{12}}\right)$
(e) $2250$
\Pra41325

\three
\newp 5.
A radioactive substance has half life of 100 years. If a sample has a mass 
of 500g, when will it be reduced to 20g?
\ANp
(a) $\frac{100\ln 25}{\ln 2}$ years.
(b) $100\ln 50$ years.
(c) $1,250$ years.
(d) $\frac{100\ln 2}{\ln 5}$ years.
(e) $2,500$ years.
\Pra31542


\newp 6.  
Find the coefficient of $a^2b^7$ in the expansion of $(a+b)^9$.
\ANp
(a) 36
(b) 21
(c) 7
(d) 84
(e) 14
\Pra54321

\ifcompressedspacing\newpage\fi
\one
\def\LP{\left(}
\def\RP{\right)}
\def\CDOT{}
\newp 7.
A certain die has a shape of a triangle, with
sides labelled 1,2,3. When it is rolled, each side
appears on the bottom with probability of
$\frac{1}{3}$. If die is rolled 4 times, what is
the probability that the side labelled ``2" will
appear on the bottom at least 2 times?
\ANp
(a) $6\CDOT\LP\frac{1}{3}\RP^2\CDOT\LP\frac{2}{3}\RP^2 + 
4\CDOT\LP\frac{1}{3}\RP^3\CDOT\LP\frac{2}{3}\RP + \LP\frac{1}{3}\RP^4$
(b) $\LP\frac{1}{3}\RP^4$
(c) $2\CDOT\LP \frac{1}{3}\RP ^2\CDOT\LP \frac{2}{3}\RP ^2+3\CDOT
\LP \frac{1}{3}\RP ^3\CDOT\LP \frac{2}{3}\RP  + 4\CDOT\LP \frac{1}{3}\RP ^4$
(d) $6\CDOT\LP \frac{1}{3}\RP ^2+4\CDOT\LP \frac{1}{3}\RP ^3+\LP \frac{1}{3}\RP ^4$
(e) $\LP \frac{1}{3}\RP ^2\CDOT\LP \frac{2}{3}\RP ^2$
\Pra32154


\newp 8.
What approximation of $\cos(0.6)$ does the 2nd degree Taylor polynomial of 
the function $f(x)=\cos(x)$ give?
\ANp
(a) $0.82$
(b) $1.36$
(c) $0.83$
(d) $0.825$
(e) $1$
\Pra15243

\one
\newp 9.
Find the third degree Taylor polynomial of the function $f(x)=2\sin(x) 
+\cos(x)$.
\ANp
(a) $1 + 2x - \frac{1}{2!}x^2 - \frac{2}{3!}x^3$
(b) $1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3$
(c) $x -  \frac{2}{3!}x^3$
(d) $2  - \frac{1}{2!}x^2 + \frac{1}{3!}x^3$
(e) $1 + 2x + \frac{1}{2!}x^2$
\Pra23154

\ifcompressedspacing\newpage\fi
\newp 10.
 Which of the following represents the $k$th Taylor coefficient of the 
function $f(x)=e^{3x}+e^x$?
\ANp
(a) $\frac{3^k+1}{k!}$
(b) $\frac{4^k}{k!}$
(c) $\frac{3k+1}{(3k)!}$
(d) $\frac{4}{k!}$
(e) $3k+1$
\Pra43512

\one
\newp 11.
Consider the differential equation $y' = y - 2\sin x$. Of the
following two functions
\par
\centerline{(I) $y = \cos x + \sin x$}
\smallskip
\centerline{(II) $y = \cos x + e^{-x}$}
\smallskip
\noindent which, if any, are solutions of the differential equation?
\ANp
(a) (I) only
(b) (II) only
(c) (I) and (II)
(d) Neither is a solution
(e) {\hsize1.8in\vtop{\noindent It is impossible to determine from the given information.}}
\Pra31254

\newp 12.
Find the centroid (center of mass) of the region bounded by $y = 1-x^4$ and $y=0$.
\ANp
(a) $(0,\frac{4}{9})$
(b) $(0,\frac{1}{2})$
(c) $(\frac{1}{3},0)$
(d) $(\frac{1}{3},\frac{8}{9})$
(e) $(\frac{1}{2},0)$
\Pra12534


\newchoice{Partial Credit}\multichoicefalse
\newp 13.
Represent $0.152323232323232\ldots$ as a fraction or sum of fractions.

\newp 14.
Use the linear approximation to the function 
$f(x)=\sqrt{9+x}$ to approximate $\sqrt{10}$, and estimate the error
involved in making the approximation.


\newp 15. 
In many card games each person
receives a 13-card hand dealt at random from a
52-card deck. You should give the answers to the following questions
in terms of binomial coefficients. {\it Don't attempt to express the
binomial coefficients as single integers!} The numbers involved may
turn out to be very large. 

\smallskip\noindent
(a) How many different 13-card hands are
possible?

\smallskip\noindent
(b) A regular deck contains 16 face cards (4
in each suit).  How many 13 card hands are
there which contain $no$ face cards?

\smallskip\noindent
(c) If you are dealt a 13-card hand from a shuffled
deck, what is the probability of not getting any
face cards?


\newp 16.
Suppose that $|5y| < 1$. Calculate  the sum 
$$\sum_{k=1}^{\infty} (5y)^k\,.$$
\vfill\eject}
\exam{}
%\exam{Byers}
%\exam{Dwyer}
%\exam{Wong}
%\exam{Xavier}
%\exam{Vassiliev}
%\exam{Neusel}
%\marked{}
\end
