\section*{Review Lecture 1: Functions; Linear,
Polynomial, and Rational Functions}\label{sup1}

{\bf \Large Supplemental Exercises 1}

Suppose you are given the following data:

\vspace{0.5in}
\begin{center}
\begin{tabular}{|c|c|}
\hline \bfseries $x$                &   $f(x)$                      \\
\hline 1                            &   -1                          \\
\hline 2                            &   41                          \\
\hline 3                            &   219                         \\
\hline 4                            &   701                         \\
\hline 5                            &   1727                        \\
\hline
\end{tabular}
\end{center}

These data points are points on the graph of a function $f(x)$ of
the form
\[
    f(x) = Ax^4 + Bx^3 + Cx^2 + Dx + F
\]
where $A$, $B$, $C$, $D$, and $F$ are constants.  Determine values
of these constants to fit the data points.

Perform all of your computations {\bf by hand}.  You may use a
calculator for {\bf simple arithmetic only}.

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\section*{Review Lecture 3: Trigonometric Functions}

{\bf \Large Supplemental Exercises 2}\label{sup2}

\begin{enumerate}

\item   Determine the amplitude, period, phase shift, and vertical
        shift of

        {\bf a.} $f(x) = \cos(2x + 1) +5$

        {\bf b.} $f(x) = 7\sin(\pi - 3\pi x) - 3$

        {\bf c.} $f(x) = \cos(x) + 1$

\item   Determine a formula for a trigonometric function $f(x)$

        {\bf a.} with period 7, amplitude 12, and vertical shift 9 that passes through the point $(3,
        9)$.

        {\bf b.} that repeats itself every $5\pi$ $x$-units,
        oscillates between a minimum of $-1$ and a maximum of $8$, and reaches its maximum
        value at $x = 1$.

\end{enumerate}

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\newpage
\section*{Data Lecture 1: Projectile Motion}

A small, heavy ball bearing is thrown off of a bridge at a height of
50 meters with an initial velocity of 18 m/s at a $45^\circ$ angle
from the horizontal. The height of the object is recorded each
second until the ball hits the ground.  The height measurements are
accurate to one tenth of a meter.

\vspace{0.5in}
\begin{center}
\begin{tabular}{|c|c|}
\hline \bfseries Time (seconds)     &   \bfseries Height (meters)   \\
\hline 0                            &   50                          \\
\hline 1                            &   54.1                        \\
\hline 2                            &   48.4                        \\
\hline 3                            &   32.9                        \\
\hline 4                            &   7.6                         \\
\hline
\end{tabular}
\end{center}

{\bf \noindent Questions}

\begin{itemize}
\item   At what time does the object hit the ground?
\item   What is the vertical velocity of the object at $t = 0$
        seconds?  At $t = 1$ seconds?  $t = 2$ seconds?
\item   What is the maximum height of the object?  At what time is
        this height obtained?
\end{itemize}


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\newpage
\section*{Data Lecture 2: Sulfuric Acid pH}

Several solutions of sulfuric acid ($\mbox{H}_2\mbox{SO}_4$) are
mixed with different concentrations.  The pH of each solution is
measured with a pH meter and probe, and the results are given in the
following table.  In each case, the concentration is measured in
{\bf molarity, M}: moles of solute (sulfuric acid) per liters of
solution (water).  A {\bf mole} is a unit of measurement that gives
a specific number of molecules of a compound: a mole of sulfuric
acid corresponds to 50 grams.  The pH values are accurate to the
nearest hundredth.

\vspace{0.5in}
\begin{center}
\begin{tabular}{|c|c|}
\hline \bfseries Concentration (M)  &   \bfseries pH                  \\
\hline 0.00001                      &   4.70                          \\
\hline 0.00005                      &   4                             \\
\hline 0.0001                       &   3.70                          \\
\hline 0.0006                       &   2.92                          \\
\hline 0.0008                       &   2.80                          \\
\hline 0.001                        &   2.70                          \\
\hline
\end{tabular}
\end{center}

{\bf \noindent Questions}

\begin{itemize}

\item   How does the pH appear to change as the concentration
        changes?

\item   What are the units of the rate of change?  What does the
        rate of change tell you?

\item   How does the rate of change appear to behave?  Does a slight
        increase in concentration make more or less of a difference
        at large concentrations compared to small concentrations?

\end{itemize}


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\newpage
\section*{Data Lecture 3: Coffee Sales}

The owner of a coffee shop is well aware of the fact that the sales
of her hot beverages vary with the seasons.  In an attempt the model
this variation, she has collected the following data of daily sales
figures.  She chose every 40th day as a sufficient representative,
but also included day 357, the day on which the sales are maximal,
and a few other useful days. In the following data, the days are
measured from the beginning of the calender year (so that January
1st, 2004 corresponds to $t = 0$).

\vspace{0.5in}
\begin{center}
\begin{tabular}{|c|c|}
\hline \bfseries Day (days)    &   \bfseries Sales (\$)          \\
\hline 0                       &   355.29                        \\
\hline 40                      &   329.13                        \\
\hline 80                      &   276.48                        \\
\hline 84                      &   271.00                        \\
\hline 120                     &   221.33                        \\
\hline 160                     &   188.82                        \\
\hline 200                     &   193.75                        \\
\hline 240                     &   233.89                        \\
\hline 280                     &   290.94                        \\
\hline 320                     &   338.90                        \\
\hline 357                     &   356.00                        \\
\hline 360                     &   355.92                        \\
\hline 400                     &   334.24                        \\
\hline 440                     &   283.75                        \\
\hline
\end{tabular}
\end{center}

{\bf \noindent Questions}

\begin{itemize}

\item   Describe the behavior of the sales as a function of time.
        What trends do you expect?

\item   On what day do you expect the minimum sales?

\item   Describe the behavior of the rate of change of sales.

\item   On what day are the sales figures changing the fastest?

\end{itemize}


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\newpage
\section*{Supplemental Data Lecture 1: Compound Interest}

A student has two bank accounts. The first is compounded annually,
and the second is compounded continuously. The student deposited
$\$100$ into each account.  The tables below shows the balance of
each account every year for five years. Each value is rounded to the
nearest cent

\vspace{0.5in}
\begin{center}
Compounded Annually

\begin{tabular}{|c|c|}
\hline \bfseries Time (years)      &   \bfseries Balance (dollars)     \\
\hline 0                            &   100                           \\
\hline 1                            &   105.40                        \\
\hline 2                            &   111.09                        \\
\hline 3                            &   117.09                        \\
\hline 4                            &   123.41                        \\
\hline 5                            &   130.08                        \\
\hline
\end{tabular}
\end{center}

\vspace{0.5in}
\begin{center}
Compounded Continuously

\begin{tabular}{|c|c|}
\hline \bfseries Time (years)      &   \bfseries Balance (dollars)     \\
\hline 0                            &   100                           \\
\hline 1                            &   105.55                        \\
\hline 2                            &   111.40                        \\
\hline 3                            &   117.59                        \\
\hline 4                            &   124.11                        \\
\hline 5                            &   131.00                        \\
\hline
\end{tabular}
\end{center}

{\it \noindent   Refer to Mathematica notebook {\texttt M1\_ L4\_
bankaccounts.nb}.}


{\bf \noindent Questions}

\begin{itemize}

\item   Compare the rates at which the balances of the two accounts
        are increasing.

\item   For each of the two accounts, how is the rate of increase
        changing?

\item   Is there a significant difference between the growth of the
        two accounts?  Does the difference become more or less
        significant with time?

\item   What is the value of each account after 20 years?  30 years?

\end{itemize}



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\newpage
\section*{Supplemental Data Lecture 2: Ferris Wheel Height}

Students are trying to model the height of a car in a Ferris wheel.
The students take several measurements of the height of a car in the
Ferris wheel that starts at a (minimum) height of 5.4 m at time 0.
The students can only measure the height of the car when it less
than 15 meters above the ground. The measurements they are able to
take are in the table below.  They are accurate to the nearest
hundredth of a meter.

\vspace{0.5in}
\begin{center}
\begin{tabular}{|c|c|}
\hline \bfseries Time (min:sec)
                             &   \bfseries Height (m)                 \\
\hline 0:00                  &   5.40                           \\
\hline 0:12                  &   5.71                           \\
\hline 0:16                  &   5.97                           \\
\hline 0:25                  &   6.80                           \\
\hline 0:40                  &   8.99                           \\
\hline 11:40                 &   8.74                           \\
\hline 11:48                 &   7.50                           \\
\hline 12:00                 &   6.17                           \\
\hline 12:18                 &   5.40                           \\
\hline 12:30                 &   5.71                           \\
\hline 12:47                 &   7.28                           \\
\hline
\end{tabular}
\end{center}

{\noindent \it Refer to Mathematica notebook {\texttt M1\_ L7\_
ferriswheel.nb}.}

{\bf \noindent Questions}

\begin{itemize}

\item   What is the period of the function?

\item   What is the phase shift?

\item   How much information is required in order to determine the
        amplitude?

\end{itemize}



