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\markright{Math 132, Midterm 2}

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\textbf{College of the Holy Cross, Fall Semester, 2004} \\
\textbf{Math 132, Midterm 3} (All Sections) \\
\textbf{Wednesday, April 27, 6 PM}
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\noindent\textbf{Name}: \underline{\phantom{Name Name Name Name Name}}
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\paragraph{Instructions} Please write your answers in the spaces
provided, and show work on the test itself. 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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Problem & Points/Poss    \Strut \\
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1  & \hfil/\  16\hfil \Strut \\
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2  & \hfil/\  12\hfil \Strut \\
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3  & \hfil/\  18\hfil \Strut \\
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4  & \hfil/\  14\hfil \Strut \\
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5  & \hfil/\  14\hfil \Strut \\
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6  & \hfil/\  14\hfil \Strut \\
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7  & \hfil/\  12\hfil \Strut \\
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Total   & \hfil/100\hfil \Strut \\
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\begin{enumerate}
  \item Suppose that the length $x$ (in cm) of a certain type of inchworm has density 
  $p(x)=c(3x^2-x^3)$ for $0\leq x\leq 3$. 
  \begin{enumerate}
    \item Find the value of the constant $c$.
    \item Find the proportion of inchworms with length greater than $1$cm.
  \end{enumerate}
  
  \item Suppose 
    \[
    P(t)=\frac{4}{\pi}\arctan(x)
    \]
    is the cumulative distribution function for the quantity $x$, for $0\leq x\leq 1$. 
    \begin{enumerate}
      \item Find the mean of $x$.
      \item Find the median of $x$.
    \end{enumerate}
  
  \item Use the indicated test to determine whether the given series converges or diverges. 
  You do not need to compute the sum of the series.
  \begin{enumerate}
    \item $\displaystyle \sum_{n=1}^\infty\frac{n^2}{n^3+1000000}$; integral test
    \item $\displaystyle \sum_{n=1}^\infty\frac{n}{4n^3+3n^2+5}$; comparison test
  \end{enumerate}
  
  \item Evaluate the following geometric series.
  \begin{enumerate}
    \item $\displaystyle\sum_{n=1}^\infty \frac{3^n+7}{4^n}$
    \item $5-5+5-5+5-5+\cdots$
  \end{enumerate}
  
  \item Let $f(x)=\sqrt{x}$. Find the Taylor polynomial of degree $3$ for $f$ near 
  $x=4$.
  
  \item Suppose $f(3)=4$, $f'(3)=-2$ and $f''(3)=6$. 
  \begin{enumerate}
    \item Write down the Taylor polynomial of degree $2$ for $f$ near $x=3$.
    \item Which (if any) of the following graphs could be the graph of $f$?
  \end{enumerate}
  
  \item Consider the power series
    \[
    \sum_{n=1}^\infty \frac{(x-1)^n}{n\,5^n}.
    \]
    \begin{enumerate}
      \item Use the ratio test to find the radius of convergence of this series.
      \item What is the interval of convergence of the power series? Be sure to 
      consider the behavior at the endpoints.
    \end{enumerate}
    
    \item 
    \begin{enumerate}
      \item Use the power series for $\sin x$ near $x=0$ to find the power series 
      for $\sin(x^2)$ near $x=0$. Express the series in summation form, and write out 
      the first four nonzero terms.
      \item Use the first four nonzero terms in the result from part (a) to estimate 
      the value of the integral
        \[
        \int_0^1\sin(x^2)\,dx.
        \] 
    \end{enumerate}
    
    
    
\end{enumerate}
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