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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2014} \\
\textbf{Math 136, section 1, Midterm Exam 1} \\
\textbf{Friday, February 21}
\end{center}

\vfil

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


\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/ 25\  \hfil \\
\hline
II  & \hfil/ 10\  \hfil \\
\hline
III  & \hfil/ 15\  \hfil \\
\hline
IV  & \hfil/ 50\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I. Let $f(x) = x^3 + 1$ on the interval $[a,b] = [1,3]$. 
\begin{enumerate}
\item[A.] (10)  Evaluate the Riemann sum for $f$ on this interval using $n = 4$ and 
$x_i^* = $ right endpoints.
\vskip 2.0truein
\MyAnswer{Right-hand sum}{$\phantom{A very long answer here}$}%
\item[B.] (10) Now repeat part A, but using the left endpoints.
\vskip 2.0truein
\MyAnswer{Left-hand sum}{$\phantom{A very long answer here}$}%
\item[C.] (5) One of your answers in parts A and B is definitely an 
overestimate for the value of $\int_1^3 x^3 + 1\ dx$.  Which is it?
\vskip 1.0truein
\MyAnswer{Overestimate}{$\phantom{A very long answer here}$}%
\vfil\eject
\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.]  (10)
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.] (15)  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.] (10) Integrate with any applicable method we have discussed: 
${\displaystyle \int \frac{x}{\sqrt{x^2 + 1}} dx}$

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

\item[E.] (10)  Integrate with any applicable method we have discussed:
${\displaystyle \int e^x \cos(4x)\ dx}$

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

\end{enumerate}

\end{enumerate}
\end{document}

