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%TCIDATA{<META NAME="Title" CONTENT="Searching for Solutions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Mar 26 00:11:52 1997}
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\begin{document}


\section{Searching for Solutions}

\subsubsection{Solving Equations Exactly}

\begin{quote}
$\blacktriangleright $ To solve an equation
\end{quote}

\begin{itemize}
\item  Place the insertion point in an equation, such as $2x-3=7x$ (or
select the equation with the mouse).

\item  Choose \textsf{Maple + Solve + Exact}. (You will get the response
``Solution is : $\left\{ x=-\frac{3}{5}\right\} $.'')
\end{itemize}

\begin{quotation}
\emph{Hint}\quad {\small To make definitions, it is often most convenient to
select the equation with the mouse (``highlight'' it) before making the
definition. To define }$x=-\frac{3}{5}${\small \ from the solution above,
for example, you can either remove the braces first or select the equation
with the mouse. Simply placing the insertion point in the equation would
select the braces as well as the equation, and this may not allow you to
make the definition.}
\end{quotation}

\begin{quote}
$\blacktriangleright $ To solve the system of equations $2x+3y=41$\ and $%
x+y=146$
\end{quote}

\begin{itemize}
\item  Choose \textsf{Insert + Display} (or click the \textsf{Display}
button \FRAME{itbpF}{0.3304in}{0.3157in}{0.0701in}{}{}{display.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.3304in;height 0.3157in;depth
0.0701in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'Display.wmf';file-properties "XNPEU";}} on the \textsf{Math} toolbar) to
open a display.

\item  Enter the equation $2x+3y=41$ in the input box.

\item  With the insertion point on the right of the equation, press enter to
create a new line in the display.

\item  Enter the equation $x+y=146$ in the input box in the second line of
the display.

\item  With the insertion point in the display, 
\begin{eqnarray*}
2x+3y &=&41 \\
x+y &=&146
\end{eqnarray*}
choose \textsf{Solve + Exact}. (\textsl{Scientific Notebook} will return the
result, Solution is : $\left\{ y=-251,x=397\right\} $.)
\end{itemize}

\begin{quotation}
\emph{Note}{\small \quad You can enter a system of equations in any
one-column ``matrix'' with the number of rows equal to the number of
equations. The display feature is a convenient way to do this, but you could
also begin with Insert + Matrix and specify one column and two rows. This
matrix would appear ``in-line'' rather than displayed.}
\end{quotation}

\subsubsection{Solving Equations Numerically}

In some of the activities you will want to search for solutions to equations
on specified intervals. For this purpose, use the command \textsf{Solve +
Numeric} and the notation 
\[
x\in \left( a,b\right) 
\]
to indicate that $x$ is in the interval $\left( a,b\right) $, that is, that $%
a<x<b$. (You will find the symbol `$\in $' in the \textsf{Binary Relations}
panel that drops down when you click the button \FRAME{itbpF}{0.3009in}{%
0.3009in}{0.0701in}{}{}{binrels.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file
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17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Binrels.wmf';file-properties "XNPEU";}} on the
math toolbar.) The general procedure for specifying an interval for a
solution is as follows.\bigskip 

\begin{quote}
$\blacktriangleright $ To locate a solution to an equation $\mathbf{c=f(x)}$%
\ by specifying a search interval $\mathbf{(a,b)}$
\end{quote}

\begin{itemize}
\item  Choose \textsf{Insert + Display}

\item  Enter $f(x)=c$ into the input box and press \QTR{KeyName}{enter}.

\item  Enter $x\in \left( a,b\right) $ into the second input box.

\item  Choose \textsf{Solve + Numeric}.
\end{itemize}

\section{The Search for the Answer}

The graph of $y=\sin (x+x\cos x)$ has many $x$-intercepts. Find the $x$%
-intercept(s) given by \textsf{Solve + Exact} and \textsf{Solve + Numeric}
and compare the results. Specify a search interval and use \textsf{Solve +
Numeric} to find the $x$-intercept between $6$ and $6.4$. Find a solution to 
$\sin (x+x\cos x)=0.5$ with $x$ between $2$ and $3$. Explain how you would
find all the $x$-intercepts in an interval $a\leq x\leq b$. Include a graph
for the interval $a=-2$, $b=8$ in your discussion.

\subsection{Sally's Solution}

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Applying \textsf{Solve + Exact} to the equation $\sin \left( x+x\cos
x\right) =0$ gives the $x$-intercepts $\left\{ x=0\right\} ,\allowbreak
\left\{ x=\pi \right\} $. Applying \textsf{Solve + Numeric} to the equation $%
\sin \left( x+x\cos x\right) =0$ gives the $x$-intercept $\left\{
x=3.\,1416\right\} $. In this example, \textsf{Solve + Exact} gave two exact
solutions, and \textsf{Solve + Numeric} found an approximation to one of the
same solutions. As you will see in the graph below, there are actually many $%
x$-intercepts.

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{%
With the insertion point in the equation $\sin \left( x+x\cos x\right) =0$,
choose \textsf{Solve + Exact}.
\par
With the insertion point in the equation $\sin \left( x+x\cos x\right) =0$,
choose \textsf{Solve + Numeric}.}

To find an $x$-intercept between $6$ and $6.4$, we apply \textsf{Solve +
Numeric} to the display 
\begin{eqnarray*}
\sin \left( x+x\cos x\right) &=&0 \\
x &\in &(6,6.4)
\end{eqnarray*}
This gives the solution $\left\{ x=6.\,2832\right\} $.

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{%
Choose \textsf{Insert + Display.}
\par
Type $\sin \left( x+x\cos x\right) =0$ in the input box and press \textsf{%
Enter}. Type $x\in \left( 6,6.4\right) $ in the input box.
\par
With the insertion point in the display 
\begin{eqnarray*}
\sin \left( x+x\cos x\right) =0 \\
x\in \left( 6,6.4\right)
\end{eqnarray*}
choose \textsf{Solve + Numeric}.}

To find a solution to the equation $\sin \left( x+x\cos x\right) =0.5$
between $2$ and $3$, we apply \textsf{Solve + Numeric} to the display 
\begin{eqnarray*}
\sin \left( x+x\cos x\right) &=&0.5 \\
x &\in &(2,3)
\end{eqnarray*}
This gives the solution $\left\{ x=2.\,4797\right\} $.

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{%
Choose \textsf{Insert + Display.}
\par
Type $\sin \left( x+x\cos x\right) =0.5$ in the input box and press \textsf{%
Enter}.
\par
Type $x\in \left( 2,3\right) $ in the input box and choose \textsf{Solve +
Numeric}.}

If there are multiple solutions, \textsf{Solve + Exact} may not find them
all. In this case it found only two solutions when there are obviously
infinitely many. For the equations we are looking at, \textsf{Solve + Numeric%
} finds only one solution at a time but it allows you to find different
solutions by specifying an interval the solution lies in. Plotting the
function makes it obvious that there are many $x$-intercepts for the
function $y=\sin (x+x\cos x)$ and shows approximately where they are. $%
y=\sin \left( x+x\cos x\right) $\FRAME{dtbpFU}{3in}{2.0003in}{0pt}{\Qcb{$%
y=\sin \left( x+x\cos x\right) $}}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\sin \left( x+x\cos x\right) $};linecolor
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"8";xviewmin "-2.2";xviewmax "8.204";yviewmin "-1.04";yviewmax
"1.041";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}%
\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
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TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"XNPEU";}}]{Margin Hint}{%
With the insertion point in the expression $\sin (x+x\cos x)$, choose 
\textsf{Maple + Plot2D + Rectangular.}
\par
Click the plot and choose \textsf{Edit + Properties. }
\par
Click the \textsf{Labeling} tab and type $y=\sin (x+x\cos x)$ in the \textsf{%
Caption Text} box.
\par
Click the \textsf{Plot Components} tab and change the \textsf{View Interval}
to $-2<x<8$%
\par
Choose \textsf{OK}.}

To find all the solutions in an interval $a\leq x\leq b$, I would first plot
the function on that interval. Then from the graph I could locate $x$%
-intercepts approximately, and by specifying an interval about each $x$%
-intercept that contains only one crossing, I would find a numerical
approxim ation to each $x$-intercept in the interval. In the graph above, I
used the interval $-2<x<8$. For this interval, I would then specify eight
different subintervals to find all of the $x$-intercepts between $-2$ and $8$%
.\bigskip

\begin{quotation}
\emph{Note}{\small \quad You can enter an equation and specify an interval
for its solution in a one-column, two-row matrix instead of in a two-line
display, if you prefer. Remember, you must use Solve + Numeric for this
technique (not Solve + Exact).\medskip }
\end{quotation}

\subsection{Activity}

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Find all of the solutions to $\sin \left( x+x\cos x\right) =0$ between $22$
and $23$. First use the zoom-in tool to estimate the solutions to as many
decimal places as you want, then specify an interval for the solution and
use \textsf{Solve + Numeric}. Which method do you prefer and why?\bigskip 

\begin{quotation}
\emph{Reminder}{\small \quad To create the graph, place the insertion point
in the expression }$\sin (x+x\cos x)${\small \ and choose Plot2D +
Rectangular. To find the zoom-in tool, double click the graph. Three
plotting tools (Pan, Zoom in, and Zoom out) appear in the upper right corner
of the frame. Select the large mountain (zoom in tool), place the mouse
cursor over a point in the graph, press and hold the mouse button, drag to
the right, and release the mouse button. This will create a box and the
contents of the box will expand to fill the frame.}
\end{quotation}

\end{document}
