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\begin{center}
{\LARGE {Mathematics 2C, Spring 1997, Section A }}
\end{center}
\begin{center}
  \large  Midterm Exam II
\end{center}
\medskip

\begin{enumerate}

\item
\begin{enumerate}
\item Compute the first three non-zero terms of the power series of
\begin{equation*}
  f(x) = \sin x \, \cos x .
\end{equation*}

\item Compute, by using the power series from part (a) or otherwise,
\begin{equation*}
  \lim_{x \to 0} \frac{\sin(3x) \, \cos(3x)}{x+x^2-x^{12}}.
\end{equation*}

\item Compute
\begin{equation*}
  \lim_{n \to \infty} n^2 \cdot
    \ln \left( 1 + \frac2{n^2 + 2\,n} \right).
\end{equation*}
\end{enumerate}
\hfill (10+10+10)

\item
\begin{enumerate}
\item Write down an integral formula for computing the volume of the
solid of revolution generated by rotating around the $x$-axis the
graph of a function $y=y(x)$.  Then use substitution to show that if
$x$ and $y$ are given by the parametric equation
\begin{equation*}
  x= f(t), \qquad y=g(t), \qquad t \in [\alpha,\beta],
\end{equation*}
the formula for the volume of the solid of revolution becomes
\begin{equation*}
  V = \pi \int_\alpha^\beta g^2(t) \, f'(t) \, dt.
\end{equation*}
Note: Be explicit about the limits of integration in every step.

\item Find the volume of the solid of revolution generated by
rotating around the $x$-axis the parametric curve
\begin{equation*}
  x= \cos(t), \qquad y=e^t, \qquad t \in [0,\pi].  \tag{*}
\end{equation*}

\item Eliminate the parameter in (*).
\end{enumerate}
\hfill (10+10+5)

\item Let $\boldsymbol{u} = 6\boldsymbol{i} - 2\boldsymbol{j} + 3
\boldsymbol{k}$ and $\boldsymbol{v} = 3\boldsymbol{j} + 4
\boldsymbol{k}$. Compute:
\begin{enumerate}
\item $\boldsymbol{u}+2\boldsymbol{v}$
\item $|\boldsymbol{v}|$
\item $\boldsymbol{u} \times \boldsymbol{v}$
\item $\boldsymbol{v} \times \boldsymbol{u}$
\item $\cos \theta$, where $\theta$ is the angle between
$\boldsymbol{u}$ and $\boldsymbol{v}$.
\end{enumerate}
\hfill(5+5+5+5+5)

\item We have seen in class that the position of a point on the unit
circle $C$, when $C$ rolls along the $x$-axis with constant speed, is
given by
\begin{equation*}
  \boldsymbol{r}(t) = (t - \sin t, 1 - \cos t)
\end{equation*}
(this curve is called a ``cycloid'').
\begin{enumerate}
\item Compute the velocity and acceleration vectors for this motion.

\item What are the times $t$ at which the rate of change of the speed
is zero?

Hint:  The speed remains constant whenever velocity and acceleration
are perpendicular.
\end{enumerate}
\hfill(10+10)

\end{enumerate}

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