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\begin{document}

\title{Mathematics 2C, Section A, Spring 1997}
\author{Marcel Oliver}
\date{Final Exam}

\maketitle

\begin{enumerate}

\item Compute the following limits:
\begin{enumerate}

\item $\displaystyle \lim_{n \to \infty}
   n \cdot \sin \left( \frac2n \right)$

\item $\displaystyle \lim_{x \to 0}
   \frac{\sqrt{1+x} - \dfrac1{\sqrt{1+x}}}{\ln (1+x)}$

\item $\displaystyle \sum_{n=0}^\infty
   \frac{(-1)^n}{n!}$

\end{enumerate}
\hfill (5+5+5)

\item Determine the set of values for $x$ for which the given series
converges.
\begin{enumerate}

\item $\displaystyle \sum_{n=2}^\infty
   \frac{x^n}{\ln n}$

\item $\displaystyle \sum_{n=0}^\infty
   \frac{x^{3n}}{n!}$

\item $\displaystyle \sum_{n=0}^\infty (\cos x)^n$

\end{enumerate}
\hfill (5+5+5)

\item Let $|x|<1$.  Verify the identity
\begin{equation*}
  \int_0^x \frac1{1-u^2} \, du
  = \dfrac12 \, \ln \left( \frac{1+x}{1-x} \right)
\end{equation*}
by writing both sides of the equation as a power series.
\hfill (10)

\item
\begin{enumerate}

\item Find the $n$th-degree Taylor polynomial $P_n(x)$ at the point
$a=0$ for the function
\begin{equation*}
  f(x) = \sin(x) .
\end{equation*}

\item Show that the $n$th-degree remainder term $R_n(x)$ converges to
zero as $n \to \infty$ for any fixed value of $x$. What does this tell
you about the Taylor series of $f$?

\end{enumerate}
\hfill (5+5)

\item Find the area of the surface generated by revolving around the
$y$-axis the parametric curve
\begin{align*}
  x &= t^3 \\
  y &= 2 \, t + 3
\end{align*}
for $t \in [-1,1]$.
\hfill (10)

\item Find a unit tangent vector $\boldsymbol{T}$ and a unit normal
vector $\boldsymbol{N}$ to the parametric curve
\begin{align*}
  x &= t^2 \\
  y &= t^3
\end{align*}
at the point $x=1$, $y=-1$.
\hfill (5)

\item A point moves on the unit circle $x^2 + y^2 = 1$.  Use polar
coordinates to show that the position vector $\boldsymbol{r}(t)$ and
the velocity vector $\boldsymbol{v}(t)$ are always perpendicular.

Note: Do not assume that the point is moving with constant speed!
\hfill (10)

\item Find the angle between the plane
\begin{gather*}
  x - y + 2\,z = 1
\intertext{and the line}
  \frac{x-1}1 = \frac{y-2}{-1} = \frac{z+1}{2}.
\end{gather*}
\hfill (10)

\item The motion of a point on a curve in space is given in
cylindrical coordintes by $\rho = \rho(t)$, $\theta = \theta
(t)$ and $z = z(t)$.
\begin{enumerate}

\item Show that the speed of the moving point is given by
\begin{equation*}
  |\boldsymbol{v}(t)|^2 =
    \left( \frac{d\rho}{dt} \right)^2 + \rho^2
    \left( \frac{d\theta}{dt} \right)^2 +
    \left( \frac{dz}{dt} \right)^2.
\end{equation*}

\item Let $\rho(t) = 1$, $\theta(t)=\sin t$ and $z(t)= \cos t$, where
$t\in [0, 2\pi]$.  Use the result from part (a) to find the arclength
of this curve.

\item Describe the curve in part (b).  Is the curve closed?  Is it
simple (non-self\/intersecting)?  What else can you say?
\end{enumerate}
\hfill(5+5+5)


\end{enumerate}

\end{document}
