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%TCIDATA{<META NAME="Title" CONTENT="Exercises">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:02:20 1997}
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\section{Exercises\label{Exercises}}

\begin{enumerate}
\item[1.]  \textsf{Define} $a=5$. \textsf{Define} $b=a^{2}$. \textsf{Evaluate%
} $b$. Now \textsf{Define} $a=\sqrt{2}$. Guess the value of $b$ and check
your answer by evaluation.

\item  \textsf{Define} $f(x)=x^{2}+3x+2$. \textsf{Evaluate} 
\[
\frac{f(x+h)-f(x)}{h}
\]
and \textsf{Simplify} the result. Do computations in place to show
intermediate steps in the simplification.

\item  Rewrite the function $f(x)=\max \left( x^{2}-1,7-x^{2}\right) $ as a
piecewise-defined function.

\item  Experiment with the Euler phi function $\varphi (n)$, which counts
the number of positive integers $k\leq n$ such that $\gcd (k,n)=1$. Use%
\textsf{\ Define + Define Maple Name} to open a dialog box. Type \texttt{%
phi(n)} as the Maple name, $\varphi (n)$ as the \textsl{Scientific Notebook}
name, and check the \textsf{Maple Library Numtheory} box. Test the statement
``If $\gcd (n,m)=1$ then $\varphi (nm)=\varphi (n)\varphi (m)$'' for several
specific choices of $n$ and $m$.

\item  Define $d(n)$ by typing \texttt{divisors(n)} as the Maple name, $d(n)$
as the\textsl{\ Scientific Notebook} name, and check the \textsf{Maple
Library Numtheory} box. Explain what the function $d(n)$ produces. This is
an example of a \emph{set-valued function}, since the function values are
sets instead of numbers.
\end{enumerate}

\section{\label{Solutions}Solutions}

\begin{enumerate}
\item  If $a=5$ then defining $b=a^{2}$ produces $b=25.$ Now define $a=\sqrt{%
2}$. The value of $b$ is now $b=\allowbreak 2$.

\item  Evaluate followed by Simplify yields 
\begin{eqnarray*}
\frac{f(x+h)-f(x)}{h} &=&\allowbreak \frac{\left( x+h\right) ^{2}+3h-x^{2}}{h%
} \\
&=&\allowbreak 2x+h+3
\end{eqnarray*}
Select the expression $=\allowbreak \frac{\left( x+h\right) ^{2}+3h-x^{2}}{h}
$ and with the \textsc{ctrl } key down drag the expression to create a copy.
Select the expression $\left( x+h\right) ^{2}$ and with the \textsc{ctrl }
key down choose \textsf{Expand}. Add similar steps (use \textsf{Factor} to
rewrite $2xh+h^{2}+3h$) until you have the following: 
\begin{eqnarray*}
\frac{f(x+h)-f(x)}{h} &=&\allowbreak \frac{\left( x+h\right) ^{2}+3h-x^{2}}{h%
} \\
&=&\allowbreak \frac{x^{2}+2xh+h^{2}+3h-x^{2}}{h} \\
&=&\allowbreak \frac{2xh+h^{2}+3h}{h} \\
&=&\allowbreak \frac{h\left( 2x+h+3\right) }{h} \\
&=&\allowbreak 2x+h+3
\end{eqnarray*}

\item  To rewrite $f(x)=\max \left( x^{2}-1,7-x^{2}\right) $ as a
piecewise-defined function, first note that the equation $x^{2}-1=7-x^{2}$
has the solutions $x=-2$ and $x=2$. The function $f$ is given by 
\[
g(x)=\left\{ 
\begin{array}{lll}
x^{2}-1 & \text{if} & x<-2 \\ 
7-x^{2} & \text{if} & -2\leq x\leq 2 \\ 
x^{2}-1 & \text{if} & x>2
\end{array}
\right. 
\]
As a check, note that $f(-5)=\allowbreak 24$, $g(-5)=\allowbreak 24$, $%
f(1)=\allowbreak 6$, $g(1)=\allowbreak 6$, $f(3)=\allowbreak 8$, and $%
g(3)=\allowbreak 8$.

\item  Construct the following table: 
\[
\begin{tabular}{rrrrrrrr}
$n$ & $\varphi (n)$ &  & $n$ & $\varphi (n)$ &  & $n$ & $\varphi (n)$ \\ 
\hline
$1$ & $1$ &  & $11$ & $10$ &  & $21$ & $12$ \\ 
$2$ & $1$ &  & $12$ & $4$ &  & $22$ & $10$ \\ 
$3$ & $2$ &  & $13$ & $12$ &  & $23$ & $22$ \\ 
$4$ & $2$ &  & $14$ & $6$ &  & $24$ & $8$ \\ 
$5$ & $4$ &  & $15$ & $8$ &  & $25$ & $20$ \\ 
$6$ & $2$ &  & $16$ & $8$ &  & $26$ & $12$ \\ 
$7$ & $6$ &  & $17$ & $16$ &  & $27$ & $18$ \\ 
$8$ & $4$ &  & $18$ & $6$ &  & $28$ & $12$ \\ 
$9$ & $6$ &  & $19$ & $18$ &  & $29$ & $28$ \\ 
$10$ & $4$ &  & $20$ & $8$ &  & $30$ & $8$%
\end{tabular}
\]
Notice, for example, that 
\begin{eqnarray*}
\varphi (4\cdot 5) &=&\allowbreak 8=\varphi (4)\varphi (5) \\
\varphi (4\cdot 7) &=&\allowbreak 12=\varphi (4)\varphi (7) \\
\varphi (3\cdot 8) &=&\allowbreak 8=\varphi (3)\varphi (8)
\end{eqnarray*}

\item  We have the following table: 
\[
\begin{tabular}{llllll}
$n$ & $d(n)$ & $n$ & $d(n)$ & $n$ & $d(n)$ \\ \hline
\multicolumn{1}{r}{$1$} & \multicolumn{1}{c}{$\left\{ 1\right\} $} & 
\multicolumn{1}{r}{$11$} & \multicolumn{1}{c}{$\left\{ 1,11\right\} $} & 
\multicolumn{1}{r}{$21$} & \multicolumn{1}{c}{$\left\{ 1,3,7,21\right\} $}
\\ 
\multicolumn{1}{r}{$2$} & \multicolumn{1}{c}{$\left\{ 1,2\right\} $} & 
\multicolumn{1}{r}{$12$} & \multicolumn{1}{c}{$\left\{ 1,2,3,4,6,12\right\} $%
} & \multicolumn{1}{r}{$22$} & \multicolumn{1}{c}{$\left\{ 1,2,11,22\right\} 
$} \\ 
\multicolumn{1}{r}{$3$} & \multicolumn{1}{c}{$\left\{ 1,3\right\} $} & 
\multicolumn{1}{r}{$13$} & \multicolumn{1}{c}{$\left\{ 1,13\right\} $} & 
\multicolumn{1}{r}{$23$} & \multicolumn{1}{c}{$\left\{ 1,23\right\} $} \\ 
\multicolumn{1}{r}{$4$} & \multicolumn{1}{c}{$\left\{ 1,2,4\right\} $} & 
\multicolumn{1}{r}{$14$} & \multicolumn{1}{c}{$\left\{ 1,2,7,14\right\} $} & 
\multicolumn{1}{r}{$24$} & \multicolumn{1}{c}{$\left\{
1,2,3,4,6,8,12,24\right\} $} \\ 
\multicolumn{1}{r}{$5$} & \multicolumn{1}{c}{$\left\{ 1,5\right\} $} & 
\multicolumn{1}{r}{$15$} & \multicolumn{1}{c}{$\left\{ 1,3,5,15\right\} $} & 
\multicolumn{1}{r}{$25$} & \multicolumn{1}{c}{$\left\{ 1,5,25\right\} $} \\ 
\multicolumn{1}{r}{$6$} & \multicolumn{1}{c}{$\left\{ 1,2,3,6\right\} $} & 
\multicolumn{1}{r}{$16$} & \multicolumn{1}{c}{$\left\{ 1,2,4,8,16\right\} $}
& \multicolumn{1}{r}{$26$} & \multicolumn{1}{c}{$\left\{ 1,2,13,26\right\} $}
\\ 
\multicolumn{1}{r}{$7$} & \multicolumn{1}{c}{$\left\{ 1,7\right\} $} & 
\multicolumn{1}{r}{$17$} & \multicolumn{1}{c}{$\left\{ 1,17\right\} $} & 
\multicolumn{1}{r}{$27$} & \multicolumn{1}{c}{$\left\{ 1,3,9,27\right\} $}
\\ 
\multicolumn{1}{r}{$8$} & \multicolumn{1}{c}{$\left\{ 1,2,4,8\right\} $} & 
\multicolumn{1}{r}{$18$} & \multicolumn{1}{c}{$\left\{ 1,2,3,6,9,18\right\} $%
} & \multicolumn{1}{r}{$28$} & \multicolumn{1}{c}{$\left\{
1,2,4,7,14,28\right\} $} \\ 
\multicolumn{1}{r}{$9$} & \multicolumn{1}{c}{$\left\{ 1,3,9\right\} $} & 
\multicolumn{1}{r}{$19$} & \multicolumn{1}{c}{$\left\{ 1,19\right\} $} & 
\multicolumn{1}{r}{$29$} & \multicolumn{1}{c}{$\left\{ 1,29\right\} $} \\ 
\multicolumn{1}{r}{$10$} & \multicolumn{1}{c}{$\left\{ 1,2,5,10\right\} $} & 
\multicolumn{1}{r}{$20$} & \multicolumn{1}{c}{$\left\{ 1,2,4,5,10,20\right\} 
$} & \multicolumn{1}{r}{$30$} & \multicolumn{1}{c}{$\left\{
1,2,3,5,6,10,15,30\right\} $}
\end{tabular}
\]
Notice that $d(n)$ consists of all the divisors of $n$.
\end{enumerate}

\subsection{Related topics}

\begin{itemize}
\item  \hyperref{Function and Expression Names}{}{}{DM5-1.tex#Function name}

\item  \hyperref{Defining Variables and Functions}{}{}{DM5-2.tex#Define}

\item  \hyperref{Handling Definitions}{}{}{DM5-3.tex#Definitions}

\item  \hyperref{Maple Functions}{}{}{DM5-5.tex#Maple functions}

\item  \hyperref{Tables of \textsl{Scientific Notebook} Equivalents}{}{}{%
DM5-6.tex#Constants}
\end{itemize}

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\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special{language
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{\small Index entries: \hyperref{Accessing maple functions}{}{}{%
DMIndex.tex#Accessing maple functions}, \hyperref{Define}{}{}{%
DMIndex.tex#Define}, \hyperref{Functions}{}{}{DMIndex.tex#Functions}, 
\hyperref{Maple constants}{}{}{DMIndex.tex#Maple constants}, \hyperref{Maple
functions}{}{}{DMIndex.tex#Maple functions}, \hyperref{Maple menu}{}{}{%
DMIndex.tex#Maple menu commands}}

\end{document}
