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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{Math 136, section 2, Midterm Exam 1} \\
\textbf{Friday, September 23}
\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 following power sum formulae may be useful:

\begin{eqnarray}
\sum_{j=1}^N j &=& \frac{N^2}{2} + \frac{N}{2}\\
\sum_{j=1}^N j^2 &=& \frac{N^3}{3} + \frac{N^2}{2} + \frac{N}{6}\\
\sum_{j=1}^N j^3 &=& \frac{N^4}{4} + \frac{N^3}{2} + \frac{N^2}{4}.
\end{eqnarray}

\vskip 30pt


\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/ 20\  \hfil \\
\hline
IV  & \hfil/ 30\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.  
\begin{enumerate}
\item[A.] (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?
\[
\lim_{N\to\infty} \sum_{j=1}^N \left(37 - 3\left(1 + \frac{2j}{N}\right)^2\right)\cdot \frac{2}{N}.
\]

\vskip 20pt
\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}%

\item[B.] (5) For any given $N$ (i.e. without taking the limit), is the sum in part 
A greater than or less than the integral? Explain how you can tell.

\vskip 150pt 

\item[C.] (5) Would you expect the Midpoint Riemann sum or the sum in part A (i.e. before taking
the limit as $N\to\infty$) to be closer to the value of your integral?  Explain.

\vskip 150pt
\end{enumerate} 
\vfill\eject
\noindent
II. All parts of this problem refer to $f(x) = x^2 + 1$ on the interval $[a,b] = [0,2]$. 
\begin{enumerate}
\item[A.] (10)  Evaluate the $R_4$ Riemann sum for $f$ on this interval.
\vskip 150pt
\MyAnswer{$R_4 = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\item[B.] (15) Find a formula for $R_N$ not involving a summation; use it to compute
$\int_0^2 x^2 + 1\ dx$.
\vskip 200pt  
\MyAnswer{$R_N = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\item[C.] (5) Use Part II of the Fundamental Theorem of Calculus to check your answer from part B.   
\vskip 50pt

\end{enumerate}
\vfill\eject
\noindent
III.  All parts of this problem refer to $f(x) = (x - 2)^2 (x - 4)$ and 
$$A(x) = \int_0^x f(t)\ dt$$ 
for this $f$.  
\begin{enumerate}
\item[A.]  (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[B.]  (5)  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[C.]  (5)  If $B(x) = \int_1^x f(t)\ dt$, how are $A(x)$ and $B(x)$ related to each other?

\vskip 75pt

\MyAnswer{Answer:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
IV.  
\begin{enumerate}
\item[A.] (10)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int_0^1 (4x^3 + 1)^{3/5} x^2\ dx}$.

\vskip 150pt

\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (10)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int \frac{x\sin(3x^2)}{\cos(3x^2)+1}\ dx}$.
\vskip 150pt 
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[C.] (10) Integrate with any applicable method we have discussed: 
${\displaystyle \int_0^1 \frac{x}{\sqrt{x^2 + 1}} dx}$
\vskip 120pt
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\end{document}

