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\begin{center}
{\bf APPM 1360 --- FALL 1997}   \\
\medskip

\underline{\bf Final Exam} \\
\end{center}

\noindent
ON THE EXAM BOOKLET PLEASE:~~  (1) Write your name,  (2) Student ID,
(3) Lecture Number
(L010 --- Herod, L020 --- Halburd),
and (4) Recitation Instructor.

\noindent
\underline {\bf SHOW ALL WORK}.  A correct answer with no relevant work may
receive no credit, while an incorrect answer accompanied by some correct
work may receive partial credit.  All graphs should be neatly drawn.
\begin{center}
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\end{center}

\vskip 1.5cm
\begin{enumerate}
\item {\it (20 points)}
Find ${\displaystyle dy\over\displaystyle dx}$ for the following
$$
\mbox{(a)}\ \ y={\ln (2^x)\over x}+\ln 3,\qquad
\mbox{(b)}\ \ y=(\sin x)^{({\rm e}^x)}.
$$
\item {\it (20 points)}
Calculate the eccentricity of the hyperbola
$$ 4x^2-y^2=4.$$
{NEATLY} graph this curve.  Find the foci, directrices, and asymptotes,
and CLEARLY LABEL them on your graph. 
\item {\it (20 points)}
\begin{enumerate}
\item Find the Cartesian equation ({\it i.e.\/}  an equation in
rectangular or $xy$-coordinates)
for the curve traced by the parametric representation
$$
x(t)=-\sqrt{t}; \quad y(t)=t,\qquad t\ge 0.
$$
Hence graph the path traced by this parameterization and clearly label any
initial or final points.
\item Find the arclength ({\it i.e.\/} length of the curve) of
$$
x(t)=\cos^3t;\quad y(t)=\sin^3t,\qquad 0\le t\le\pi/2.
$$
\end{enumerate}
\item {\it (20 points)}
\begin{enumerate}
\item Graph the following curves 
and find all points of intersection:
$$
r=1-\cos\theta,\qquad
r=\cos\theta.
$$
\item
Convert the polar equation
$$ r={5\over \sin\theta - 2\cos\theta} $$
to a 
Cartesian
equation.  Hence identify the curve.
\end{enumerate}
\item {\it (20 points)}  Find the following integrals
$$
 \begin{array}{lll}
& \mbox{(a)  }\ds \int_0^{1/2} \sin^{-1}x\, dx,\raisebox{-2em}{\ }\qquad
& \mbox{(b)  }\ds \int {2x-2\over (x+1)(x^2+1)}\, dx.\qquad
\end{array}
$$
\item  {\it (20 points)}
Which of the following integrals converge?  Name any test you use.
$$
\mbox{(a)  }\int_1^\infty {2+\sin x\over x}\, dx,\qquad
\mbox{(b)  }\int_{-\infty}^\infty {\rm e}^{-|x-1|}\,dx.\qquad
$$
\item {\it (20 points)}
\begin{enumerate}
\item 
State whether the following two series converge
    absolutely, converge conditionally, or diverge.  
Name any test you use.
    \[ (i) ~~ \sum_{n=1}^\infty \frac{(-1)^n 3 n^2}{n^3 - 4},
    ~~~~~~~~~~~~~~~~~~ (ii) ~~ \sum_{n=2}^\infty \frac{1}{n (\ln n)^2}.\]
\item  Find the sum of the series $\ds \sum_{n=0}^\infty
    \frac{2^{n-1} - 3^n}{5^n}$.
\end{enumerate}
\item {\it (20 points)}
\begin{enumerate}
\item Find the interval of convergence of 
$$\ds
    \sum_{n=0}^\infty \frac{x^{n+1}}{(n+1) 2^n}.
$$  State where
    the convergence is absolute  and where it is conditional.  Be sure to name
    which convergence test you are using.
\item The first few terms in the Maclaurin series for $f(x)$ and $g(x)$ are
$$
f(x)=2+x^2+3x^3+\cdots,\mbox{  and  }
g(x)=4-x-2x^3+\cdots.
$$
Use series multiplication to find the first four terms in the Maclaurin series
generated by $f(x)g(x)$.
\end{enumerate}
\item {\it (20 points)}
Estimate the error that results from approximating $\ds
  f(x) = e^x$ by $\ds P(x) = 1 + x + \frac{1}{2} x^2$ on the
  interval $\ds |x| \leq 0.1$.  You do not have to give 
a decimal expression for
this error.
\item {\it (20 points)}
A radioactive element decays at a rate proportional to the amount of
material present.
A GAO report suggests that there is about 60 lbs of
  plutonium dust in the ductwork at the Rocky Flats facility.  Suppose
  that it takes 10 lbs of plutonium to build a bomb and that the
  half-life of plutonium is about 24,000
  years ({\it i.e.\/} half the remaining plutonium decays every 24,000 years).
  How long will it be before the plutonium in the ductwork
  has decayed to the point where there is only 10 lbs left?  (You may
  leave your answer in terms of $\ln$'s of numbers).
\end{enumerate}
\vskip 6mm
{\bf Extra Credit:} {\it (10 points)}
To what function does the series
$$ 
\sum_{n=0}^\infty(-1)^n(2n)x^{2n-1}
$$
converge? 
\vskip 3mm
{\bf Hint:} Can you think of an operation that will generate a geometric
series from the one given above?
\end{document}