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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2007} \\
\textbf{Math 132, Midterm Exam 1} (All Sections) \\
\textbf{Wednesday, February 20, 6 PM}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 0pt plus .3fil
\noindent\textbf{Your Section}: 

\vskip 15pt
\leftline{Little (8:00am) $\underline{\qquad}$ Ballantine (9:00am) $\underline{\qquad}$} \leftline{DeStefano (10:00am) $\underline{\qquad}$ DeStefano (noon) $\underline{\qquad}$}

\vskip 30pt
\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all work} on the test pages and place your final answer
in the box provided for the problem.  Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 10\  \hfil \\
\hline
II  & \hfil/ 10\  \hfil \\
\hline
III  & \hfil/ 15\  \hfil \\
\hline
IV  & \hfil/ 35\  \hfil \\
\hline
V  & \hfil/ 30\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I. (10) State the definition of the definite integral $\displaystyle \int_a^b f(x)\ dx$.  

\vskip 3.5truein
\noindent
II.  Compute the derivatives of each of the following functions defined by integrals.
\begin{enumerate}
\item[A.]  (5) ${\displaystyle f(x) = \int^x_1 e^{t^2}\ dt}$
\vskip 1.0truein
\MyAnswer{$f'(x) = $ }{$\phantom{A very long answer here}$}%
\item[B.] (5) ${\displaystyle g(x) = \int_{x^2}^4 \frac{\cos(t)}{t^2}\ dt}$
\vskip 1.0truein
\MyAnswer{$g'(x) = $ }{$\phantom{A very long answer here}$}%
\vfil\eject
\end{enumerate}

\noindent
III.  The following graph (made up of straight line segments)
shows $y = f(t)$ for $0 \le t \le 4$.

\begin{center}
\includegraphics[width=2.3truein,height=2.3truein]{132P1F1.ps} 
\end{center}

\noindent
Given: $f(1) = 2$, $f(2) = -2$, $f(3) = -1$, and $f(4) = 0$,  
The function $F$ is defined by $F(x) = \int_0^x f(t)\ dt$.  
\begin{enumerate}
\item[A.] (5) Determine the values $F(x)$ for $x = 0,1,2,3,4$ and enter them in the following table.

\vskip 1.0truein

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
   $x$ & 0 & 1 & 2 & 3 & 4\\ \hline
       &   &   &   &   &     \\
$F(x)$ & $\qquad\qquad$  & $\qquad\qquad$  & $\qquad\qquad$  & $\qquad\qquad$  & $\qquad\qquad$  \\ 
       &           &           &           &           &           \\  \hline 
\end{tabular}
\end{center}

\item[B.]  (5)  Does $F(x)$ have any critical points?  If so, say where.  If not say why not.
\vskip 10pt
\MyAnswer{Critical point(s) of $F(x)$}{$\phantom{A very very very long answer here}$}%

\vskip 15 pt
\item[C.]  (5)  Over which interval(s) is $F(x)$ concave down?
\vskip 10pt
\MyAnswer{Concave down on:}{$\phantom{A very very very long answer here}$}%
\end{enumerate}
\vfil\eject

\noindent
IV.  
\begin{enumerate}
\item[A.] (5)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int_0^1 (4x^3 + 1)^{3/5} x^2\ dx}$.

\vskip 2.5truein
\MyAnswer{Answer:}{$\phantom{A very very very very very long answer here}$}%

\vskip 10pt
\item[B.]  (5)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int \frac{x\sin(3x^2)}{\cos(3x^2)+1}\ dx}$.

\vskip 3.5truein
\MyAnswer{Answer:}{$\phantom{A very very very very very long answer here}$}%
\vfill\eject
\item[C.] (7.5)  Integrate by parts:
${\displaystyle \int x^2 \sin(5x)\ dx}$
\vskip 8.0truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\vfill\eject
\item[D.] (7.5) Integrate with the partial fraction method:  
${\displaystyle \int \frac{3x^2 + 1}{x^2 + 5x + 6}\ dx}$

\vskip 8.0truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\vfill\eject

\item[E.] (10)  Integrate via trigonometric substitution:
${\displaystyle \int \sqrt{36 - x^2}\ dx}$

\vskip 8.0truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%

\end{enumerate}
\vfill\eject

\noindent
V.  Integrate using any applicable method or the table.  If you 
do use the table, give the number of the entry you are using.

\begin{enumerate}
\item[A.] (10) ${\displaystyle \int \tan x \sec^4 x\ dx}$.

\vskip 3.5truein 
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%

\vskip 10pt
\item[B.] (10) ${\displaystyle \int \frac{\sqrt{9 + e^{2x}}}{e^x}\ dx}$

\vskip 2.5truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\vfill\eject
\item[C.] (10) ${\displaystyle \int\frac{dx}{x^2 + 8x + 17}}$.
\vskip 8.0truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\end{enumerate}
\end{document}

