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\markright{Math 132, Midterm 3}
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\textbf{College of the Holy Cross, Spring Semester, 2005} \\
\textbf{Math 132, Makeup Midterm~3}\\
\textbf{Monday, May 2}
\end{center}

\vfil

\noindent\textbf{Name}: \underline{\phantom{Name Name Name Name Name}}
\vskip 0pt plus .3fil

\paragraph{Instructions} Please write your answers in the spaces
provided, and show work on the test itself. \textbf{For possible
  partial credit, you must show work}.  Use the back of the preceding
page if you need more space for scratch work.

\vfil

\noindent\textbf{Please do not write in the space below}

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\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
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1  & \hfil/\  15\hfil \Strut \\
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2  & \hfil/\  10\hfil \Strut \\
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3  & \hfil/\  10\hfil \Strut \\
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4  & \hfil/\  15\hfil \Strut \\
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5  & \hfil/\  10\hfil \Strut \\
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6  & \hfil/\  15\hfil \Strut \\
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7  & \hfil/\  15\hfil \Strut \\
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8  & \hfil/\  10\hfil \Strut \\
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Total   & \hfil/100\hfil \Strut \\
\hline
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\be
  
\item{} [5~points each]\quad Suppose $f(0)=0.5$, $f'(0)=-1$ and
$f''(0)=-2$.
  
  (a) Write down the Taylor polynomial of degree~$2$ for~$f$ near
  $a=0$.

  \vfil
  \MyAnswer{Answer:}{$\phantom{A pretty long answer goes here}$}
  \sep
  
  (b) Use your answer to part~(a) to estimate~$f(0.2)$.

  \vfil
  \Answer{Answer:}
  \sep

  (c) Which (if any) of the following could be the graph of~$f$?
  (Recall that $f(0)=0.5$, $f'(0)=-1$ and $f''(0)=-2$. More than one
  answer may be correct.)

  $\bigcirc$~(i)\qquad
  $\bigcirc$~(ii)\qquad
  $\bigcirc$~(iii)\qquad
  $\bigcirc$~(iv)\qquad
  $\bigcirc$~None
  \bigskip

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    \input{midterm3_2i.eepic}\hfil
    \input{midterm3_2ii.eepic}\hfil
    \input{midterm3_2iii.eepic}\hfil
    \input{midterm3_2iv.eepic}
  \end{center}

  \eject
  
\item{} [10~points]\quad Evaluate (find the sum of) the series
  $\ds 4 - \frac{8}{3} + \frac{16}{9} - \frac{32}{27} + 
  \frac{64}{81} - \cdots$

  \vfil
  \Answer{Answer:}
  \sep

\item{} [10~points]\quad Use the power series for~$e^x$ about $a=0$ to
  find the power series for~$z^2 e^{-z}$ about $a=0$. Express your
  answer both in summation form, and by writing out the first four
  nonzero terms.

  \vfil\vfil
  \MyAnswer{Summation form:}{$\phantom{\Bigg|A pretty much very long
  answer goes here}$}%

  \MyAnswer{First four terms:}{$\phantom{\Bigg|A pretty much very long
  answer goes here}$}% 

  \eject

\item{} The distribution of length in a certain population of
  inchworms is described by the pdf $p(x)=c(3x^2-x^3)$ for $0\leq
  x\leq 3$.

  (a) [8~points]\quad Find the value of the constant~$c$.

  \vfil
  \Answer{Answer:}
  \sep

  (b) [7~points]\quad Find the proportion of inchworms with length at
  betweem 1~cm and 2~cm.

  \vfil
  \Answer{Answer:}
  \eject
  
\item{} [5~points each]\quad Use the indicated test to determine
  whether the given series converges or diverges. \textbf{Justify your
  answers.}
  
  (a) $\ds\sum_{n=1}^\infty\frac{n}{n^2+4}$; integral test

  \vfil
  $\bigcirc$\quad Converges\hspace*{1in}
  $\bigcirc$\quad Diverges
  \sep
  
  (b) $\ds \sum_{n=1}^\infty\frac{n}{3n^3+n^2+1}$; comparison test

  \vfil
  $\bigcirc$\quad Converges\hspace*{1in}
  $\bigcirc$\quad Diverges
  \eject
  
\item{} [15~points]\quad Let $f(x)=\sqrt{x}$. Use \textbf{the
    definition} to calculate its Taylor polynomial of degree~$3$
  at $a=1$.

  \vfil
  \MyAnswer{Answer:}{$\phantom{\Bigg|A pretty much very long
  answer goes here}$}% 
  \eject
  
\item{} Both parts refer to the power series $\ds\sum_{n=1}^\infty
  \frac{(x-2)^n}{n\cdot7^n}$.
  
  (a) [10~points]\quad Use the ratio test to find the radius of
  convergence.

  \vfil
  \Answer{Answer:}
  \sep
  
  (b) [5~points]\quad Investigate the endpoint behavior, and determine
  the interval of convergence.

  \vfil
  \MyAnswer{Answer:}{$\phantom{\Bigg| My very long
  answer here}$}
  \eject
  
\item{} [5~points each]\quad Suppose 
  \[
  p(x)=\frac{4}{\pi}\cdot\frac{1}{1+x^2}
  \]
  is the probability density function for the quantity~$x$, for
  $0\leq x\leq 1$.

  (a) Find the mean of~$x$.

  \vfil
  \Answer{Answer:}
  \sep

  (b) Find the median of~$x$.

  \vfil
  \Answer{Answer:}
  \eject
  
  \ee

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