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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2019} \\
\textbf{Math 135, Section 1, Midterm 1}\\
\textbf{Friday, September 27}
\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}: You must show all work to justify your
answers.  Enter answers in the boxes when they are provided. 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.

\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/\  25\hfil \\
\hline
II  & \hfil/\  25\hfil \\
\hline
III  & \hfil/\  20\hfil \\
\hline
IV  & \hfil/\  30\hfil \\
\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfill\eject

\vskip 10pt
\noindent
I.   The following table contains values for
three different functions: $f(x),g(x),h(x)$.  
$$\begin{array}{ccccccc}
x  &\vline&2.5   &3.4  &4.3  &5.2   &6.1\\ \hline
       f(x)  &\vline& 3 & 9 & 27  & 81 & 243\\
       g(x)  &\vline& 12.1 & 13.2 & 14.3  & 15.4 & 16.5\\\
       h(x)  &\vline& 5.4   & 3.6  &1.3   &5.3   & 7.9
\end{array}$$
\begin{enumerate}
\item[A)] (15) One of these is a linear function. 
Explain how you can tell which one it is, and give a formula for it.
\vskip 4truein
\item[B)] (10) One of these functions is {\it neither linear nor exponential}.  
Explain which one that is and why.
\end{enumerate}
\vfill\eject

\noindent
II.
\begin{enumerate}
\item[A)] (10)  Simplify using properties of logarithms and exponents.  (No credit
will be given for only an approximate calculator value.)  
$$\log_3\left(\frac{\sqrt{3}}{\sqrt[4]{27}}\right)$$
\vskip 2.5truein
\MyAnswer{Simplified form:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B)] (15)  The population of a city (in millions) at time $t$ (years) 
is $P(t) = 3.7 e^{0.04 t}$.  When will the population reach $6.3$ million?  
\vskip 2.5truein
\MyAnswer{Time:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}


%%%%%%%%%% Combining functions and finding domains %%%%%%%%%%%%

\vfill\eject
\noindent
III. Given $\displaystyle{f(x)=\frac{1}{x^2 - 6x + 8}}$ 
and $g(x)=\tan(x)$, but defined \emph{only for $x$ in the interval} 
$-3 \le x \le 3$.  Answer the following questions.

\begin{enumerate}
\item[A)] (10) Which $x$ between $-3$ and $3$ must be removed to obtain 
\vskip 3.5truein
\MyAnswer{the domain of $f$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\vskip 10pt
\MyAnswer{the domain of $g$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B)] (10) Using the Limit Laws, determine $\displaystyle{\lim_{x\to 1} f(x)g(x)}$.
\vskip 2truein
\MyAnswer{limit:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}

\vfill\eject
\noindent
IV.  (Make sure your calculator is set in radian mode for this problem.) 
\begin{enumerate}
\item[A)] (20)  Let $\displaystyle f(x) = \frac{\sin(5x)}{8x}$.   
Compute the  values 
at $x = \pm \pi/10$, $x = \pm \pi/100$, $x = \pm \pi/1000$ 
accurate to three decimal places and fill in the table 
below:  
\vskip 3truein
$$\begin{array}{clclclclclclcl}
x &\vline &-\pi/10 &\vline &-\pi/100 &\vline &-\pi/1000 &\vline & \pi/1000 &\vline & \pi/100&\vline & \pi/10&\vline\\
\hline
f(x) &\vline &\qquad\qquad&\vline &\qquad\qquad&\vline &\qquad\qquad&\vline &\qquad\qquad&\vline &\qquad\qquad&\vline &\qquad\qquad&\vline\\
\hline
\end{array}$$
\item[B)] (10) What's your estimate of the value of the limit
$\displaystyle \lim_{x\to 0} \frac{\sin(5x)}{8x}$ based on this numerical information?
\vskip 20pt
\MyAnswer{limit:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\end{document}
