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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2020} \\
\textbf{Math 136, section 1, Midterm Exam 1} \\
\textbf{Friday, February 21}
\end{center}

\vskip 30pt

\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.

The trigonmetric reduction formulas are provided on a separate
sheet


\vskip 200pt


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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 20\  \hfil \\
\hline
II  & \hfil/ 30\  \hfil \\
\hline
III  & \hfil/ 30\  \hfil \\
\hline
IV  & \hfil/ 20\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject
\noindent
I. 
\begin{enumerate}
\item[A.] (10)  Evaluate the $L_4$ Riemann sum for $f(x) = e^{x^2}$ on the interval $[a,b] = [0,2]$. 
\vskip 3.5truein
\MyAnswer{$L_4 = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.] (10)  The following limit of a sum would equal the definite integral 
$\int_a^b f(x)\ dx$ for
some function $f(x)$ on some interval $[a,b]$.  What function and what interval?
\[
\lim_{N\to\infty} \sum_{j=1}^N \left(\sqrt{\left( \frac{5(j-1)}{N} \right)^3 + 4}\right) \cdot \frac{5}{N}.
\]
\vskip 60pt
\MyAnswer{$f(x) = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\vskip 10pt
\MyAnswer{$[a,b] = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
II.  All parts of this problem refer to:
$$A(x) = \int_0^x t(5-t)\ dt$$  
\begin{enumerate}
\item[A.]  (10)  Is $A(4)$ a positive or negative number?  How can you tell?
(It's not necessary to compute the value to tell -- why not?)
\vskip 75pt
\MyAnswer{Answer:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (10) Where does $A(x)$ have critical points?  Is each of them a local maximum, 
an local minimum, or neither?

\vskip 75pt
\MyAnswer{Critical points at $x = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\MyAnswer{local max/min/neither}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\item[C.]  (10)  If $B(x) = A(\tan(x)) = \int_0^{\tan(x)} t(5-t)\ dt$, find $B'(x)$.

\vskip 120pt

\MyAnswer{$B'(x) = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
III.  
\begin{enumerate}
\item[A.] (10)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int (6x^5 + 1)^{2/5} x^4\ dx}$.
\vskip 3truein
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (10) Integrate by parts:  ${\displaystyle \int x^2 \cos(4x)\ dx}$
\vskip 3truein 
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\vfill\eject
\item[C.] (10) Integrate:
${\displaystyle \int \sec^5(x) dx}$
\vskip 7truein
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
IV.  (20) Integrate ${\displaystyle \int_{-6}^6 x^2\sqrt{36 - x^2}\ dx}$
\vskip 7truein
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{document}

