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\textbf{Math 126: Calculus II}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Final}\quad \textsl{December 17, 2002}
\hfill\makebox[2.5in][r]{Instructor:\hrulefill}

\bigskip

There are 12 problems on 10 pages (including the cover page)
 worth a total of 140 points.  You start
with 10 points. Unless otherwise specified,
each part of a problem is worth the same number of points.

\medskip
You may use a calculator if you wish.

\medskip
To receive full credit for a problem, you must \emph{give an exact
  answer}
unless otherwise specified.
To receive partial credit on a problem, you must \emph{show your work
and all important steps}. No credit will be given for an answer if no
work is shown.

\bigskip

\ 
 
\bigskip

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 \problemspace

\begin{enumerate}

\item (15 points) Evaluate the integral:
\[ \int_{1}^{4} e^{\sqrt{x}}\,dx.\]


\answerspace

\item (10 points) Find $ \displaystyle \lim_{x \to \infty} \left( \frac{x+1}{x}
\right) ^{4x}. $


\answerspace

\problemspace

\item (10 points)
Find the centroid of the region enclosed by the graphs of $y=4$ and
        $y=x^{2}$.

\answerspace

\problemspace

\item (15 points) Solve the initial value problem:
\[ (1+x^{2}) y' + x y^{2} = x, \hspace{20pt} y(0) = 0. \]
Hint: This is a separable equation.

\problemspace


\item (10 points)  A bacteria culture starts with 600 bacteria and grows at a
rate proportional to its size. If after 3 hours there are
12,000 bacteria, how long will it take until the population
reaches 45,000.

\answerspace


\item (10 points)
For which $ p > 0$ is the series 
$$ \sum _{n=1}^\infty
\frac{2+e^{-n}}{n^p}$$ 
convergent?  For which is it divergent?
Justify your answer.


\answerspace

\problemspace

\item (10 points)
 Use the integral test to examine the series
$$\sum _{n=2}^\infty \frac{1}{n(\ln n)^3}$$
for convergence or divergence.

\answerspace

\item (10 points) 
Find the radius of convergence and the interval of convergence
  of the power series 
  \begin{math}
    \sum _{n=1}^\infty \frac {x^n}{n^5}.
  \end{math}

\answerspace

\problemspace


\item (15 points)

\begin{enumerate}

\item   Find the Maclaurin series of 
  \begin{math}
    f(x) = x^2 \cos x.
  \end{math}

\answerspace

\item   Find
  \begin{math}
    f^{(2002)} (0).
  \end{math}

\answerspace

\end{enumerate}

\problemspace

\item (15 points) 
\begin{enumerate}

\item Find a power series for 
  \begin{math}
    \int _0 ^x e^{-t^2}\,dt.
  \end{math}

\answerspace

\item Evaluate 
  \begin{math}
    \int _0 ^1 e^{-t^2}\,dt
  \end{math}
with an error less than 
\begin{math}
  10^{-2}.
\end{math}

\answerspace


\end{enumerate}

\problemspace

\item (10 points) Consider the parametric curve 
  \begin{math}
    x= 2 \cos t,\ y=3 \sin t, \ 0\le t \le \pi.
  \end{math}

\medskip

  \begin{enumerate}
  \item Sketch this curve.

\answerspace

  \item Set up an integral to compute the length of this curve.
Do \textbf{NOT} evaluate the integral.


\answerspace

  \end{enumerate}

\problemspace

\item (10 points) Find the area of the region that lies inside the curve 
  \begin{math}
    r=2 - 2\cos \theta
  \end{math}
but outside the curve 
\begin{math}
  r=2.
\end{math}

\answerspace

\end{enumerate}

\end{document}


