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\begin{center}

\textbf{College of the Holy Cross, Fall 2019} \\
\textbf{Math 135, Section 1, Solutions for Midterm 1}\\
\textbf{Friday, September 27}
\end{center}

\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.
\item[] {\it Solution:}  The function $g(x)$ is linear since equal increments
in $x$ produce equal increments in $g(x)$.  Alternatively, the slopes between
all pairs of points $(x,g(x))$ from the table are equal.  The slope is 
$$m = \frac{13.2 - 12.1}{3.4 - 2.5} = \frac{1.1}{.9} \doteq 1.22$$
and the equation of the line can be found by the point-slope form:
$$y - 12.1 \doteq (1.22)(x - 2.5)$$
or $y \doteq 1.22 x + 9.05$ (the intercept is closer to $9.04$ if more decimal
places in the slope are used).  
\item[B)] (10) One of these functions is {\it neither linear nor exponential}.  
Explain which one that is and why.
\item[] {\it Solution:}  Exponential and linear functions are either always increasing
or always decreasing.  The third function $h(x)$ is neither, so it is the one that
is neither linear nor exponential.
\end{enumerate}

\vskip 10pt
\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.)  
$$\log_3\left(\frac{\sqrt{3}}{\sqrt[4]{27}}\right)$$
\item[] {\it Solution:}  By rules for exponents, this the same as
$$\log_3\left(\frac{3^{1/2}}{27^{1/4}}\right) = \log_3\left(\frac{3^{1/2}}{3^{3/4}}\right)
= \log_3(3^{-1/4}) = \frac{-1}{4}$$
\MyAnswer{Simplified form:}{$-1/4$}%
\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?  
\item[]{\it Solution:}  We need to solve the equation 
$$3.7 e^{0.04 t} = 6.3$$
for $t$.  We divide by $3.7$, then take natural logs to solve for the $t$ in
the exponent:
$$t = \frac{\ln(6.3/3.7)}{.04} \doteq 13.3$$ 
\MyAnswer{Time:}{\text{(a bit more than) } 13.3 \text{ years}}%
\end{enumerate}


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


\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 
\item[] {\it Solution}:  The $x$-values where $f(x)$ are undefined can 
be found by either factoring:
$$x^2 - 6x + 8 = (x - 2)(x - 4)$$
or using the quadratic formula.  So $f(x)$ is undefined at $x = 2,4$, but 
only $x = 2$ is contained in $-3 \le x \le 3$. The function 
$g(x) = \tan(x) = \sin(x)/\cos(x)$ is undefined at 
all the odd integer multiples of $\pi/2$ (the zeroes of $\cos(x)$).  Of those,
only $\pm \pi/2$ are contained in the interval $-3 \le x \le 3$.  
\vskip 10pt
\MyAnswer{the domain of $f$:}{x = 2 \text{ must be removed}}%
%\end{enumerate}
%\end{document}
\vskip 10pt
\MyAnswer{the domain of $g$:}{x = \pm \pi/2 \text{ must be removed}}%
\item[B)] (10) Using the Limit Laws, determine $\displaystyle{\lim_{x\to 1} f(x)g(x)}$.
\item[] {\it Solution:}  By the Limit Product Law, and then the Limit Quotient Law and
Limit Sum Law for $f(x)$,
\begin{eqnarray*}
\lim_{x\to 1} f(x) g(x) &=& \lim_{x \to 1} f(x) \cdot \lim_{x\to 1}g(x)\\
& = & (\lim_{x\to 1} \frac{1}{x^2 - 6x + 8})\cdot \lim_{x\to 1} \tan(x)\\
& = & \frac{1}{3}\cdot \tan(1)\\
& \doteq & .519
\end{eqnarray*}
\vskip 10pt
\MyAnswer{limit:}{\text{The exact value is } \frac{1}{3}\tan(1) \doteq .519}%
\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:  
$$\begin{array}{clclclclclclcl}
x &\vline &-\pi/10 &\vline &-\pi/100 &\vline &-\pi/1000 &\vline & \pi/1000 &\vline & \pi/100&\vline & \pi/10&\vline\\
\hline
f(x) &\vline & .398 &\vline &.622 &\vline & .625 &\vline &.625 &\vline &.622 &\vline &.398 &\vline\\
\hline
\end{array}$$
\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:}{\text{Probably around } .625}%
\end{enumerate}
\end{document}
