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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2018} \\
\textbf{Math 134 Makeup Midterm Exam 1} \\
\textbf{Monday, February 19}
\end{center}

\vskip 100pt

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 200pt


\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:

\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/ 30\  \hfil \\
\hline
II  & \hfil/ 10\  \hfil \\
\hline
III  & \hfil/ 20\  \hfil \\
\hline
IV  & \hfil/ 40\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject
\noindent
I. Both parts of this problem refer to $f(x) = 3x^2 + 1$ on the interval $[a,b] = [1,3]$. 
\begin{enumerate}
\item[A.] (15)  Evaluate the $L_4$ Riemann sum for $f$ on this interval.
\vskip 200pt
\MyAnswer{$L_4 = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\item[B.] (10) Use Part I of the Fundamental Theorem of Calculus to 
evaluate $\int_1^3 3x^2 + 1\ dx$.   
\vskip 200pt
\MyAnswer{$\int_1^3 3x^2 + 1\ dx = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\item[C.] (5) Your answer to part B should be larger than your answer to part A.  Explain
how you could know it would turn out that way even without computing the numerical values.  
\end{enumerate}
\vfill\eject
\noindent
II.  (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?
\[
\lim_{N\to\infty} \sum_{j=1}^N \left(2 + \cos\left(\frac{2\pi j}{N}\right)\right)\cdot \frac{2\pi}{N}.
\]


\vskip 450pt
\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}%

\vfill\eject
\noindent
III.  Both parts of this problem refer to $f(x) = \displaystyle{\frac{x}{x^4 + 2}}$.  
\begin{enumerate}
\item[A.]  (10) Let $\displaystyle A(x) = \int_0^x f(t)\ dt$. What is $A'(x)$?

\vskip 150pt
\MyAnswer{$A'(x) = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\item[B.]  (10) Now consider $\displaystyle B(x) = \int_1^{x^2} f(t)\ dt$.  What is $B'(x)$?

\vskip 150pt
\MyAnswer{$B'(x) = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
 
\end{enumerate}
\vfill\eject
\noindent
IV.  Compute the following integrals.  
\begin{enumerate}
\item[A.] (10) $\displaystyle \int 3\sqrt{x} + \frac{4}{x} + e^x\ dx$
\vskip 200pt
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.] (10) $\displaystyle \int \sin(x) + \frac{1}{1 + x^2}\ dx$
\vskip 200pt
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\begin{enumerate}
\item[C.] (10)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int_0^1 (x^3 + 3)^{3/8} x^2\ dx}$.
\vskip 300pt

\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[D.]  (10)
Integrate with a suitable $u$-substitution:  ${\displaystyle \int (\sec(x))^2 \sec(x)\tan(x) \ dx}$.
\vskip 200pt 
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\end{document}

