%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
\usepackage{amssymb}

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\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="31">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Formulas">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:01:24 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="author" HREF="DM0-1.tex">}
%TCIDATA{<LINK REL="first" HREF="DM5-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="copyright" HREF="DM0-2.tex">}
%TCIDATA{<LINK REL="end" HREF="DM5-7.tex">}
%TCIDATA{<LINK REL="index" HREF="DMIndex.tex">}
%TCIDATA{<LINK REL="next" HREF="DM5-5.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM5.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM5-3.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}
%TCIDATA{MapleDefs=
%$a=\ln x $
%$b=e^{x} $
%$x=20.56 $
%$y=18.92 $
%$\limfunc{a1}=2 $
%$\limfunc{b1}=5 $
%}


\input{tcilatex}
\begin{document}


\section{Formulas\label{Formulas}}

The \textsf{Formula} dialog provides a way to enter an expression and a
Maple operation. What appears on the screen is the result of the operation
and depends upon active definitions of variables that appear in the formula.
Formulas remain active in your document--that is, changing definitions of
relevant variables will change the data on the screen.

\strut

When \textsf{Helper Lines} are turned on, a \textsf{Formula} is identified
by a yellow background.

\begin{quote}
$\blacktriangleright $ To insert a formula
\end{quote}

\begin{enumerate}
\item  Click\FRAME{itbpF}{0.3208in}{0.3208in}{0.0701in}{}{}{formula.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3208in;height 0.3208in;depth 0.0701in;original-width
19.5625pt;original-height 18.8125pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'formula.wmf';file-properties "XNPEU";}} on the 
\textsf{Field} toolbar

- or -\medskip 

From the \textsf{Insert} menu, choose \textsf{Field} and then choose \textsf{%
Formula}.

\item  In the \textsf{Formula} area, enter a mathematics expression.

\item  In the \textsf{Operation} area, enter the operation you want to
perform on the expression. (Click the arrow to the right for a list of
available operations.)

\item  Choose \textsf{OK}.\medskip 
\end{enumerate}

\textsl{Scientific\ Notebook} displays the results of the operation on the
screen.

\strut

\begin{example}
Choose \textsf{Insert + Field + Formula}. In the \textsf{Formula} box, type
\ $a$, and under \textsf{Operations} choose \textsf{evaluate}. Choose 
\textsf{OK}.\FRAME{dtbpF}{330.5pt}{179.875pt}{0pt}{}{}{Figure }{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 330.5pt;height 179.875pt;depth 0pt;original-width
326.75pt;original-height 176.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Formula2.wmf';file-properties "XNPEU";}}The $a$
will appear on your screen at the position of the insertion point. Now, at
any point in your document, define $a=\sin x$. The formula $a$ will be
replaced by the expression $\sin x$. Make another definition for $a$. The
formula will be replaced by the new definition.
\end{example}

\strut

\begin{example}
Insert a $2\times 2$ matrix. With the insertion point in the first input
box, click \FRAME{itbpF}{0.3001in}{0.2888in}{0.0701in}{}{}{formula.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3001in;height 0.2888in;depth 0.0701in;original-width
19.5625pt;original-height 18.8125pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'formula.wmf';file-properties "XNPEU";}}. In the 
\textsf{Formula} box, type $a$. Under \textsf{Operations}, choose \textsf{%
evaluate}. Choose \textsf{OK}. Repeat for each matrix entry, typing $b$, $%
a+2b$, and $\left( a-b\right) ^{2}$ respectively in the formula box to get
the following matrix: 
\[
\left[ 
\begin{array}{cc}
a & b \\ 
a+2b & \left( a-b\right) ^{2}
\end{array}
\right] 
\]
Now define $a=\sin x$ and $b=\cos x$. The matrix will be replaced by the
following matrix. 
\[
\left[ 
\begin{array}{cc}
%TCIMACRO{\FORMULA{a}{\sin x}{evaluate} }
%BeginExpansion
\sin x%
%EndExpansion
& 
%TCIMACRO{\FORMULA{b}{\cos x}{evaluate} }
%BeginExpansion
\cos x%
%EndExpansion
\\ 
%TCIMACRO{\FORMULA{a+2b}{\sin x+2\cos x}{evaluate} }
%BeginExpansion
\sin x+2\cos x%
%EndExpansion
& 
%TCIMACRO{
%\FORMULA{\left( a-b\right) ^{2}}{\left( \sin x-\cos x\right) ^{2}}{evaluate}}
%BeginExpansion
\left( \sin x-\cos x\right) ^{2}%
%EndExpansion
\end{array}
\right] 
\]
Define $a=\ln x$ and $b=e^{x}$. The matrix will be replaced by the following
matrix. 
\[
\left[ 
\begin{array}{cc}
\ln x & e^{x} \\ 
\ln x+2e^{x} & (\ln x-e^{x})^{2}
\end{array}
\right] 
\]
\end{example}

\strut

\begin{example}
Insert a table with 2 columns and 5 rows. Insert formulas $x$, $y$, and $%
x+y+z$ in the columns on the right. 
\[
\begin{tabular}{|r|r|}
\hline
Date & Income \\ \hline\hline
1/31/96 & $%
%TCIMACRO{\FORMULA{x}{x}{evaluate} }
%BeginExpansion
x%
%EndExpansion
$ \\ \hline
2/28/96 & $%
%TCIMACRO{\FORMULA{y}{y}{evaluate} }
%BeginExpansion
y%
%EndExpansion
$ \\ \hline
3/31/96 & $%
%TCIMACRO{\FORMULA{z}{z}{evaluate} }
%BeginExpansion
z%
%EndExpansion
$ \\ \hline
Total & $%
%TCIMACRO{\FORMULA{x+y+z}{x+y+z}{evaluate} }
%BeginExpansion
x+y+z%
%EndExpansion
$ \\ \hline
\end{tabular}
\]
Define $x=20.56$, $y=18.92$, $z=23.45$ to get the table 
\[
\begin{tabular}{|r|r|}
\hline
Date & Income \\ \hline\hline
1/31/96 & $20.56$ \\ \hline
2/28/96 & $18.92$ \\ \hline
3/31/96 & $23.45$ \\ \hline
Total & $62.\,\allowbreak 93$ \\ \hline
\end{tabular}
\]
\end{example}

\begin{example}
Multiple choice examinations with variations can be constructed using
formulas. This example outlines a way for constructing them manually. For
more information on an automatic way for creating such examinations, look
for references to on line quizzes in \hyperref{Welcome}{}{}{%
..\Docs\Welcome.tex}.

The questions depend on definitions that are made globally for each
document---they're not local to each question or variant. This means that
you should use \hyperref{math names}{}{}{DM5-1.tex#Function name} instead of
single character names for variables. A sample question is shown below. The
variables $\limfunc{a1}$ and $\limfunc{b1}$ shown in this question are math
names. They should be entered as formulas--use \textsf{Insert + Field +
Formula} followed by \textsf{Insert + Math Name}, or click \FRAME{itbpF}{%
0.3001in}{0.2888in}{0.0701in}{}{}{formula.wmf}{\special{language "Scientific
Word";type "GRAPHIC";display "PICT";valid_file "F";width 0.3001in;height
0.2888in;depth 0.0701in;original-width 19.5625pt;original-height
18.8125pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'formula.wmf';file-properties "XNPEU";}} and then click \FRAME{itbpF}{%
0.3009in}{0.3009in}{0.0701in}{}{}{function.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'FUNCTION.WMF';file-properties "XNPEU";}}.

You can create an examination with variations by making different
definitions for the variables such as the $\limfunc{a1}$ and $\limfunc{b1}$
shown in the following question. Turn on \textsf{Helper Lines} to check that
all appropriate entries are formulas.\medskip
\end{example}

\begin{enumerate}
\item  For which values of the variable $x$ is $\limfunc{a1}x-\limfunc{b1}<0$%
?

\begin{enumerate}
\item  $~x<\limfunc{b1}/\limfunc{a1}$

\item  $~x>\limfunc{b1}/\limfunc{a1}$

\item  $~x>\limfunc{b1}$

\item  $~x<\limfunc{b1}$

\item  $~$None of these
\end{enumerate}
\end{enumerate}

\strut

\emph{First variation}: Define $\limfunc{a1}=2$ and $\limfunc{b1}=5$ by
placing the insertion point in each equation and choosing Define + New
Definition.\medskip

\begin{enumerate}
\item  For which values of the variable $x$ is $%
%TCIMACRO{\FORMULA{\limfunc{a1}}{2}{evaluate}}
%BeginExpansion
2%
%EndExpansion
x-%
%TCIMACRO{\FORMULA{\limfunc{b1}}{5}{evaluate}}
%BeginExpansion
5%
%EndExpansion
<0$?

\begin{enumerate}
\item  $~x<%
%TCIMACRO{\FORMULA{\limfunc{b1}}{5}{evaluate}}
%BeginExpansion
5%
%EndExpansion
/%
%TCIMACRO{\FORMULA{\limfunc{a1}}{2}{evaluate}}
%BeginExpansion
2%
%EndExpansion
$

\item  $~x>%
%TCIMACRO{\FORMULA{\limfunc{b1}}{5}{evaluate}}
%BeginExpansion
5%
%EndExpansion
/%
%TCIMACRO{\FORMULA{\limfunc{a1}}{2}{evaluate}}
%BeginExpansion
2%
%EndExpansion
$

\item  $~x>%
%TCIMACRO{\FORMULA{\limfunc{b1}}{5}{evaluate}}
%BeginExpansion
5%
%EndExpansion
$

\item  $~x<%
%TCIMACRO{\FORMULA{\limfunc{b1}}{5}{evaluate}}
%BeginExpansion
5%
%EndExpansion
$

\item  $~$None of these
\end{enumerate}
\end{enumerate}

\strut

{\small After printing a quiz, make different definitions for} $\limfunc{a1}$
and $\limfunc{b1}$ {\small to obtain additional variations of the quiz.}

\strut

\subsection{\ Related topics}

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

\item  \hyperref{Exercises and Solutions}{}{}{DM5-7.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 38.875pt;height 24.5pt;depth
6pt;original-width 36.875pt;original-height 22.5625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'index.wmf';file-properties
"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Define}{}{}{DMIndex.tex#Define}, \hyperref{Field}{}{}{%
Index.tex#Field}, \hyperref{Formula}{}{}{DMIndex.tex#Formula}, \hyperref{%
Formula}{}{}{Index.tex#Formula}}

\end{document}
