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\markright{Chapter 1: Worksheet 2}
\begin{document}

\Bf{\Star\ \Star\ Chapter 1: Worksheet 2  \hfill Jack K. Cohen \hfill Colorado School of Mines}

\vspace{1.5ex}
\Cb{Functions, Symbol Problems}
\vspace{1.5ex}

\noindent \Bf{Suggested Problems}
Section 1.1: 27, 45, 55

\begin{enumerate}
\item It is important to understand the difference between a \Em{formula} and a \Em {function}.  Some functions are specified by formulas---but there are also other ways to specify a function.  And some formulas do \Em{not} specify functions.

\noindent Some formulas follow.  Which specify a function? For those that do specify a function, what is the ``natural'' domain and range (see page 6 just below the red line)?  On some problems, a graph might help.  You can get one painlessly with \Mma\ as follows:

\noindent \Tt{Plot[x + 5, \{x, -5, 5\}]}

\noindent \Tt{Plot[Sin[x] + Cos[x], \{x, -2Pi, 2Pi\}]}

\noindent and so forth.

        \begin{enumerate}
        \item   $y = x + 5$
        \item   $y = \sin x + \cos x$
        \item   $y = 1 / x$
        \item   $y - x - 5 = 0$
        \item   $x^2 + y^2 - 4 = 0$
        \item   $x^2 + y^2 + 4 = 0$
        \end{enumerate}

\item Functions can sometimes be specified by tables.
        \begin{enumerate}
        \item The ``Denver Diary'' for July 15, 1991 was:
        
                \begin{tabular}{ll|ll}  \hline
                {\bf Time} & {\bf Temp} & {\bf Time} & {\bf Temp} \\
                Mdnt & 72 & Noon & 87 \\
                1 am & 69 & 1 pm & 92 \\
                2 am & 67 & 2 pm & 93 \\
                3 am & 63 & 3 pm & 95 \\
                4 am & 61 & 4 pm & 96 \\
                5 am & 58 & 5 pm & 97 \\
                6 am & 57 & 6 pm & 96 \\
                7 am & 64 & 7 pm & 86 \\
                8 am & 70 & 8 pm & 81 \\
                9 am & 76 & 9 pm & \\
                10am & 80 & 10pm & \\
                11am & 84 & 11pm & \\   \hline
                \end{tabular}
                
\vspace{1 ex}
\noindent First, if you are new to Colorado, note the temperature range in a single day!  Next: Is Temp a function of Time?  Is Time a function of Temp?  If you answered ``yes'' to either or both questions, what are the domains and ranges?

        \item The ``Never on Sunday'' data set is:

                \begin{tabular}{l|l}    \hline
                {\bf Day} & {\bf Answer} \\
                Sunday & No \\
                Monday & Yes \\
                Tuesday & Yes \\
                Wednesday & Yes \\
                Thursday & Yes \\
                Friday & Yes \\
                Saturday & Yes \\       \hline
                \end{tabular}

\vspace{1 ex}
\noindent Is Answer a function of Day?  Is Day a function of Answer?  If you answered ``yes'' to either or both questions, what are the domains and ranges?
        \end{enumerate}

\item Functions can sometimes be specified by giving (domain, range) pairs. 
        \begin{enumerate}
        \item Is this  set of pairs a function: \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\}  If you said ``yes'', domain? range?
        \item Is this set of pairs a function: \{(1, 1), (2, 2), (2, 3), (4, 4), (1, 5)\}  If you said ``yes'', domain? range?
        \end{enumerate}

\item Functions can sometimes be specified by a graph.  Which of the following represent a function $y = f(x)$?  Is it feasible to get the domain and range from the graph?
                \begin{figure}[htb]
                \epsfysize 100pt
                \centerline{\epsffile{ws2p4a.eps}}
                \caption{Graph  a} 
                \end{figure}

                \begin{figure}[hbt]
                \epsfysize 100pt
                \centerline{\epsffile{ws2p4b.eps}}
                \caption{Graph b} 
                \end{figure}

\item  Functions can sometimes be specified in words.
        \begin{enumerate}
        \item Does ``the straight line passing through the points (1,3), (6, 12)'' describe a function?
        \item What about ``the circle with center (3,4) and radius 5''?
        \item How about ``the monthly population of Vail, Colorado between 1950 and the present''?
        \end{enumerate}

\item Well, even with all these alternatives, it can't be denied that a formula is a useful thing.  So now, I'll ask you for some!

        \begin{enumerate}
        \item A straight line is determined by two points, right?  So if
                the points are $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, quick
                now, write a formula for the straight line.  Any special
                cases to note?
        
        \item A straight line is also determined by a point and a slope.  So if the point is $P_1(x_1, y_1)$ and the slope is $m$, write a formula for the straight line.
        
        \item A straight line is also determined by its slope and ``intercept''.  But isn't this just a special case of the previous case?  Anyway, if the slope is $m$ and the $y$-intercept is $b$, write down a formula for the straight line.
        \end{enumerate}

\end{enumerate}
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