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\textbf{College of the Holy Cross, Spring Semester, 2005} \\

\textbf{Math 132, Midterm~3} (All Sections) \\

\textbf{Wednesday, April~27, 6~PM}

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\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.



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



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\begin{tabular}{|l||r|}

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Problem & Points/Poss    \Strut \\

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1  & \hfil/\  14\hfil \Strut \\

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2  & \hfil/\  12\hfil \Strut \\

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3  & \hfil/\  12\hfil \Strut \\

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4  & \hfil/\  10\hfil \Strut \\

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5  & \hfil/\  12\hfil \Strut \\

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6  & \hfil/\  14\hfil \Strut \\

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7  & \hfil/\  14\hfil \Strut \\

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8  & \hfil/\  12\hfil \Strut \\

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Total   & \hfil/100\hfil \Strut \\

\hline

\end{tabular}

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\be

  

\item{} [7~points each]\quad Suppose that the length~$x$ (in cm) of a

  certain type of inchworm has density $p(x)=c(3x^2-x^3)$ for $0\leq
x\leq 3$.

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  (a) Find the value of the constant~$c$.



  \vfil

  \Answer{$c=$}

  \sep



  (b) Find the proportion of inchworms with length between

  1~and 2~cm.



  \vfil

  \Answer{Fraction of population between 1~and 2~cm in length}

  \eject

  

\item{} [6~points each]\quad Suppose 

  \[
  P(x)=\frac{4}{\pi}\arctan(x)
  \]

  is the cumulative distribution function for the quantity~$x$, for $0\leq x\leq 1$.



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



  \vfil

  \Answer{Median of~$x$}

  \sep



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



  \vfil

  \Answer{Mean of~$x$}

  \eject

  

\item{} [6~points each]\quad Use the indicated test to determine whether 
the given series converges or diverges. \textbf{Justify your answers, but
do not attempt to compute the sum of either series.}

  

  (a) $\ds\sum_{n=1}^\infty\frac{n^2}{n^3+1000000}$; integral test



  \vfil

  $\bigcirc$\quad Converges\hspace*{1in}

  $\bigcirc$\quad Diverges

  \sep

  

  (b) $\ds \sum_{n=1}^\infty\frac{n}{4n^3+3n^2+5}$; comparison test



  \vfil

  $\bigcirc$\quad Converges\hspace*{1in}

  $\bigcirc$\quad Diverges

  \eject

  

\item{} [5~points each]\quad Use the geometric series formula to

  evaluate (find the sum of) the following:



  (a) $\ds 5 - \frac{10}{3} + \frac{20}{9} - \frac{40}{27} + \frac{80}{81} - \cdots$

% $5-5+5-5+5-5+\cdots$



  \vfil

  \Answer{Sum}

  \sep



 (b)  $\ds\sum_{n=1}^\infty \frac{3^n+7}{4^n}$



 \vfil

 \Answer{Sum}

 \eject



\item{} Suppose $f(0)=0.5$, $f'(0)=-1$ and $f''(0)=-2$. 

  

  (a) [9~points]\quad Write down the Taylor polynomial of degree~$2$ for~$f$ near $a=0$.



  \vfil

  \MyAnswer{Taylor polynomial:}{$\phantom{A pretty long answer goes here}$}

  \sep

  

  (b) [3~points]\quad Which (if any) of the following could be the

  graph of~$f$? (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{} [14~points]\quad Let $f(x)=\sqrt{x}$. Use \textbf{the definition} 
to calculate the Taylor polynomial of degree~$3$ for~$f$ centered at $a=4$.



  \vfil\eject

  

\item{} [7~points each]\quad Both parts refer to the power series

  $\ds\sum_{n=1}^\infty \frac{(x-1)^n}{n\cdot5^n}$.

  

  (a) Use the ratio test to find the radius of convergence.



  \vfil

  \Answer{Radius:}

  \sep

  

  (b) Investigate the endpoint behavior, and determine the interval of convergence.



  \vfil

  \Answer{Interval of Convergence:}

  \eject

  

\item{} (a) [6~points]\quad Use the power series for~$\sin x$ about $a=0$ 
to find the power series for~$\sin(x^2)$ about $a=0$. Express your answer 
both in summation form, and by writing out the first four nonzero terms.



  \vfil

  \MyAnswer{Summation form:}{$\phantom{A pretty long answer goes here}$}%



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

  \sep

  

  (b) [6~points]\quad Use the first four nonzero terms in the result
from part~(a) to estimate the value of the integral

\[ \int_0^1\sin(x^2)\,dx.\]



  \vfil

  \Answer{Estimate:}

  \eject

  \ee



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