Autonomous solution sketch Flash Applet Sample Problem: Difference between revisions

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</p>
</p>
<p style="background-color:#93BED2;border:black solid 1px;padding:3px;">This applet and WeBWorK problem are based upon work supported by the National Science Foundation under Grant Number DUE-0941388.</p>
<p style="background-color:#93BED2;border:black solid 1px;padding:3px;">This applet and WeBWorK problem are based upon work supported by the National Science Foundation under Grant Number DUE-0941388.</p>
<p style="background-color:#ffccff;border:black solid 1px;padding:3px;">Click here to see a problem like this in action:  [https://testcourses.webwork.maa.org/webwork2/FlashAppletDemos/differentialEquations/26/?effectiveUser=practice2&displayMode=MathJax&key=nRUZJzWgAGUNEqrRIcW112raR05OYPt3&user=practice2]</p>
<p style="background-color:#ffccff;border:black solid 1px;padding:3px;">Click here to see a problem like this in action:  [https://testcourses.webwork.maa.org/webwork2/FlashAppletDemos/differentialEquations/26/?user=practice1&effectiveUser=practice1&key=zMDBULgyqcEAyZAZoJSZsDDCG6Fmf33g&displayMode=MathJax&showOldAnswers=0]</p>
<p>
<p>
A standard WeBWorK PG file with an embedded applet has six sections:
A standard WeBWorK PG file with an embedded applet has six sections:
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[[Hint Applet (Trigonometric Substitution) Sample Problem]]<br>
[[Hint Applet (Trigonometric Substitution) Sample Problem]]<br>
[[phasePortrait Flash Applet Sample Problem 1]]<br>
[[phasePortrait Flash Applet Sample Problem 1]]<br>
[[autonomous solution sketch Flash Applet Sample Problem]]<br>
Other useful links:<br>
Other useful links:<br>
[[Flash Applets Tutorial]]<br>
[[Flash Applets Tutorial]]<br>
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| <pre>
| <pre>
##DESCRIPTION
##DESCRIPTION
##  Graphical limits
##  Sketch autonomous solutions to polynomial differential equation
##    Sample problem to illustrate
##    the use of the GraphLimit.swf
##    Flash applet
##ENDDESCRIPTION
##ENDDESCRIPTION


## KEYWORDS('limits')
##KEYWORDS('logistic', 'population')


## DBsubject('Calculus')
## DBsubject('Differential Equations')
## DBchapter('Limits')
## DBchapter('Introduction')
## DBsection('Graphical limits')
## DBsection('Autonomous Differential Equations')
## Date('7/5/2011')
## Date('8/9/2013')
## Author('L. Felipe Martins')
## Author('Barbara Margolius')
## Author('Barbara Margolius')
## Institution('Cleveland State University')
## Institution('Cleveland State University')
## TitleText1('')
## TitleText1('Differential Equations')
## EditionText1('2011')
## EditionText1('2')
## AuthorText1('')
## AuthorText1('Ricardo')
## Section1('')
## Chapter('1')
## Problem1('')
## Problem1('1_1')
###########################################
###########################################
# This work is supported in part by  
# This work is supported in part by  
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|- style=" background-color:#ddffdd;"
|- style=" background-color:#ddffdd;"
| <pre>
| <pre>
DOCUMENT();      
DOCUMENT();    


loadMacros(
loadMacros(
  "PGstandard.pl",
  "PGstandard.pl",     # Standard macros for PG language
  "AppletObjects.pl",
  "MathObjects.pl",
  "MathObjects.pl",
  "parserPopUp.pl",
  "AppletObjects.pl",
  "AnswerFormatHelp.pl",
  "PGasu.pl",
);
);
sub BPINK {
  MODES(TeX => '{\\color{pink} ', HTML => '<span style="color:pink">');
};
sub EPINK {
  MODES( TeX => '}', HTML => '</span>');
};
sub BBLUE {
  MODES(TeX => '{\\color{blue} ', HTML => '<span style="color:blue">');
};
sub EBLUE {
  MODES( TeX => '}', HTML => '</span>');
};
sub BYELLOW {
  MODES(TeX => '{\\color{yellow} ', HTML => '<span style="color:yellow">');
};
sub EYELLOW {
  MODES( TeX => '}', HTML => '</span>');
};
# Print problem number and point value (weight) for the problem
TEXT(beginproblem());
# Show which answers are correct and which ones are incorrect
$showPartialCorrectAnswers = 1;
</pre>
</pre>
| <p>
| <p>
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|- style=" background-color:#ffffdd;"
|- style=" background-color:#ffffdd;"
| <pre>
| <pre>
# Set up problem
##############################################################
$qtype='limits';
#
#  Setup
#
#
Context("Numeric");
Context()->variables->add(y=>"Real");
$a = -random(1,4,1);
$expr = Formula("y*(y+$a)")->reduce();
$ymax = Compute("-$a+4");
 
$I1=Compute("(-infinity,0)");
$I2=Compute("(0,-$a)");
$I3=Compute("(-$a,infinity)");
$cup = Compute("(0,-$a/2)U(-$a,infinity)");
$cdown = Compute("(-infinity,0)U(-$a/2,-$a)");
$lim1 = Compute("0");
$popup1 = PopUp(["?", "extinction", "equilibrium",
  "explosive growth"], "extinction");
$popup2 = PopUp(["?", "extinction", "equilibrium",
  "explosive growth"], "explosive growth");
$popup3 = PopUp(["?", "extinction", "equilibrium",
  "explosive growth"], "equilibrium");
 
$inc = Compute("$I1 U $I3");
 
$lim2 = Compute("infinity");
 
$boardMessage = "Sketch three solutions to this
  differential equation using the information
  given in the problem.";
# applet adds a point for each error it detects. 
# It records 100 if the graphs are not drawn
$ans = Compute("0");


$showHint = 0;
#+++++++++++++++++++++++++++++++++++++++++++++
if(time>$dueDate){
# Designate characteristics of pink curve
  $showHint=1;
# min is IC, max is ymax; concave up on ($IC,ymax);
}
# increasing on ($IC,ymax)
# note intervals are in y not t
$pinkICy = Compute("-$a+1");
$pinkIntervalsIncLow = $pinkICy;
$pinkIntervalsIncHigh = $ymax;
$pinkIntervalsCupLow = $pinkICy;
$pinkIntervalsCupHigh = $ymax;
$pinkMin = $pinkICy;
#+++++++++++++++++++++++++++++++++++++++++++++
# Designate characteristics of blue curve
# min is IC, max is ymax; concave down on (0,$IC);  
# decreasing on (0,$IC)  
# note intervals are in y not t
$blueICy = Compute("-$a/2");
$blueIntervalsDecLow = 0;
$blueIntervalsIncHigh = $blueICy;
$blueIntervalsCupLow = 0;
$blueIntervalsCupHigh = $blueICy;
$blueMax = $blueICy;
$blueMin = 0;
#+++++++++++++++++++++++++++++++++++++++++++++
# Designate characteristics of blue curve
# min is IC, max is IC; horizontal line
# note intervals are in y not t
$yellowICy = Compute("-$a");
$yellowMax = $yellowICy;
$yellowMin = $yellowICy;


$x1=random(-8,-2,1);
$x2=$x1+random(2,4,1);
$x3=$x2+random(2,3,1);
$x4=random($x3+2,7,1);
</pre>
</pre>
| <p>
| <p>
The GraphLimits.swf applet will accept four different question types, specified with the <code>$qtype</code> variable.  These are: limits, continuity, first_derivative and second_derivative.  This sample problem is set to 'limits'.
The sketch_3.swf applet requires the student to sketch three solution curves.  The problem author specifies initial conditions, intervals of increase, intervals of decrease, intervals of concavity, and the maximum and minimum possible values of the curves.
</p>
<p>
The applet has solution/hint information embedded in it.  When <code>$hintState=0</code>, this information is not shown.  When <code>$hintState=1</code>, this information is revealed.  The <code>time</code> parameter tracks the current date and time.  The conditional compares that to the due date for the problem set (in the <code>$dueDate</code> scalar variable) and sets <code>$hintState</code> to 1 if the due date has passed and leaves <code>$hintState</code> set to 0 if the assignment is not yet due.
</p>
<p>
The four variables <code>$x1</code>, <code>$x2</code>, <code>$x3</code> and <code>$x4</code> are the x-coordinates of four points on the graph that the applet will set to be a removable discontinuity, a jump discontinuity or a cusp.  The order of these phenomena is random as are the y-values chosen.  The x-coordinates must be between -10 and 10.
</p>
</p>
|- style=" background-color:#ccffff;"
|- style=" background-color:#ccffff;"
| <pre>
| <pre>
#######################################
###################################
#  How to use the Graph_Test applet.
# Create link to applet  
#    Purpose:  The purpose of this
###################################
#      applet is to ask graphical
     $appletName = "sketch_3";
#      limit questions
    $applet =  FlashApplet(
#    Use of applet:  The applet
       codebase              => findAppletCodebase("$appletName.swf"),
#      state consists of the
       appletName            => $appletName,
#      following fields:
       appletId              => $appletName,
#    qType - question type: limits,
       setStateAlias        => 'setXML',
#      continuity, first_derivative,
       getStateAlias        => 'getXML',
#      second_derivative
       setConfigAlias        => 'setConfig',
#    hintState - context sensitive
       getConfigAlias       => 'getConfig',
#      help is either on or off. 
       maxInitializationAttempts => 5,  # number of attempts to initialize applet
#      Generally turned on after
       answerBoxAlias       => 'answerBox',
#      dueDate
      height                => '500',
#    problemSeed - the seed sets
      width                => '650',
#      the random parameters that
       bgcolor              => '#ededed',
#      control which graph is
       debugMode            => 0,
#      chosen.  If the seed is
       submitActionScript  => 
#      changed, the graph is
  qq{getQE("answerBox").value=getApplet("$appletName").getAnswer() },
#      changed.
    );
#######################################
#     qType = limits
#      right_limits - returns a
#       list of points (a,b)
#       such that
#       lim_{x\to a^-}f(x)=b,  
#       but
#       lim_{x\to a^+}f(x)\= b
#      left_limits - returns a
#       list of points (a,b)
#       such that
#       lim_{x\to a^+}f(x)=b,  
#        but
#       lim_{x\to a^-}f(x)\= b
#      neither_limits - returns
#       a list of points (a,b)
#       such that
#       lim_{x\to a^-}f(x)\=
#          lim_{x\to a^+}f(x)\=  
#          f(a)=b
#      get_intervals returns a
#       list of intervals on
#        which f(x) is continuous.
#     get_f_of_x - given x value,  
#       returns f(x).  
#        returns NaN for x notin
#        [-10,10].
#      getf_list - given x value
#       and string returns
#        "function" - returns f(x)
#        "leftlimit" - returns
#          lim_{x->a^-}f(x)
#        "rightlimit" - returns
#          lim_{x->a^+}f(x)
#        "limit" - returns
#          lim_{x->a}f(x) or "DNE"
</pre>
</pre>
| <p>This is the <strong>Applet link section</strong> of the problem.   
| <p>This is the <strong>Applet link section</strong> of the problem.   
</p><br>
</p><br>
<p>
<p>
Those portions of the code that begin
You must include the section that  
the line with <code>#</code>
are comments and can be omitted or
replaced with comments appropriate
to your particular problem.
</p>
|- style=" background-color:#ccffff;"
| <pre>
#
#  What does the applet do?
#    The applet draws a graph
#    with jumps, a cusp and
#    discontinuities
#    When turned on, there is
#    context sensitive help.
################################
    ############################
    # Create  link to applet
    ############################
    $appletName = "Graph_Limit";
    $applet =  FlashApplet(
      codebase             
        => findAppletCodebase
        ("$appletName.swf"),
      appletName           
        => $appletName,
      appletId             
        => $appletName,
      setStateAlias       
        => 'setXML',
      getStateAlias       
        => 'getXML',
      setConfigAlias       
        => 'setConfig',
      maxInitializationAttempts
        => 10, 
      height               
        => '475',
      width               
        => '425',
      bgcolor             
        => '#e8e8e8',
      debugMode           
        =>  0,
      submitActionScript 
        =>    qq{
getQE("func").value=getApplet
  ("$appletName").getf_list($x1,"function");
getQE("rlimit").value=getApplet
  ("$appletName").getf_list($x2,"rightlimit");
getQE("llimit").value=getApplet
  ("$appletName").getf_list($x3,"leftlimit");
getQE("limit").value=getApplet
  ("$appletName").getf_list($x4,"limit");
  },
    );
</pre>
| <p>You must include the section that  
follows <code># Create  link to  
follows <code># Create  link to  
applet</code>.  If you are embedding  
applet</code>.  If you are embedding  
a different applet, from the Graph_Limit
a different applet, from the sketch_3
applet, put your applet name in place of  
applet, put your applet name in place of  
'Graph_Limit' in the line <code>$appletName =</code>
'sketch_3' in the line <code>$appletName =</code>
<code> "Graph_Limit";</code>.   
<code> "sketch_3";</code>.   
Enter the height of the applet  
Enter the height of the applet  
in the line <code>height => '475',</code>  
in the line <code>height => '500',</code>  
in place of 475 and the width in the line   
in place of 500 and the width in the line   
<code>width => '425',</code> in place of 425.
<code>width => '650',</code> in place of 650.
</p><br>
</p>
 
<p>The code <code>qq{ </code><code>
getQE("func").value=getApplet</code>
<code>("$appletName").getf_list($x1,"function");</code><code>
getQE("rlimit").value=getApplet</code>
<code>("$appletName").getf_list($x2,"rightlimit");</code><code>
getQE("llimit").value=getApplet</code>
<code>("$appletName").getf_list($x3,"leftlimit");</code><code>
getQE("limit").value=getApplet</code>
<code>("$appletName").getf_list($x4,"limit");</code><code>
  }</code>
is called when the 'Submit Answers'
button in the problem is pressed. 
There is an external interface function
designed inside the applet.  The function
name is 'getf_list'.  These lines of code
call the function with javascript. 
<code>getf_list</code>, takes two arguments: 
the x-coordinate of a point, and a string
value.  The string may be any of the
following four alternatives:  "function",
"rightlimit", "leftlimit", "limit". 
<code>getf_list</code> returns either
the value of the function at the x-coordinate,
or the specified limit.  The line <code>
getQE("func").value=getApplet</code>
<code>("$appletName").getf_list($x1,"function");</code>
gets the value of the function at <code>$x1</code>
and stores this value in the hidden javascript
form field named "func".</p>
|- style=" background-color:#ccffff;"
|- style=" background-color:#ccffff;"
| <pre>
| <pre>
###################################
$config_string = <<"ENDCONFIG";
# Configure applet
<XML>
###################################
    <boardMessage>$boardMessage</boardMessage>
# configuration consists of
    <xmin>0</xmin><xmax>13</xmax><ymin>-2</ymin><ymax>$ymax</ymax>
# hintState, question type, and
    <depVar>y</depVar><indVar>t</indVar>
# random seed, and x-coordinates of
    <showSolution>false</showSolution>
# four points where jumps,
    <blueIntervalsCup>
# discontinuities or cusps
      <interval left='0' right='$blueICy'></interval></blueIntervalsCup>
# occur.
    <blueIntervalsDec>
$applet->configuration(qq{<xml>
      <interval left='0' right='$blueICy'></interval>
<hintState>$hintState</hintState>
    </blueIntervalsDec>
<qtype>limits</qtype>
    <pinkIntervalsInc>
<seed>$problemSeed</seed>
      <interval left='$pinkICy' right='$ymax'></interval>
<xlist x1='$x1' x2='$x2'
    </pinkIntervalsInc>
x3='$x3' x4='$x4' /></xml>});
    <pinkIntervalsCup>
$applet->initialState(qq{<xml>
      <interval left='$pinkICy' right='$ymax'></interval>
<hintState>$hintState</hintState>
    </pinkIntervalsCup>
<qtype>limits</qtype>
    <blueMax>$blueMax</blueMax>
<seed>$problemSeed</seed>
    <blueMin>$blueMin</blueMin>
<xlist x1='$x1' x2='$x2'  
    <pinkMin>$pinkMin</pinkMin>
x3='$x3' x4='$x4' /></xml>});
    <yellowMin>$yellowMin</yellowMin>
 
    <yellowMax>$yellowMax</yellowMax>
TEXT( MODES(TeX=>'object code',
    <pinkICy>$pinkICy</pinkICy>
  HTML=>$applet->insertAll(
    <blueICy>$blueICy</blueICy>
  debug=>0,
    <yellowICy>$yellowICy</yellowICy>
  includeAnswerBox=>0,
    <pinkMaxX>0.2</pinkMaxX>
  )));
</XML>
 
ENDCONFIG
TEXT(MODES(TeX=>"", HTML=><<'END_TEXT'));
<input type="hidden"
  name="func" id="func" />
<input type="hidden"
  name="llimit" id="llimit" />
<input type="hidden"
  name="rlimit" id="rlimit" />
<input type="hidden"
  name="limit" id="limit" />
END_TEXT
 
$answerString1 =
$inputs_ref->{func};
my $correctAnswer1 =
Compute("$answerString1");
 
$answerString2 =
$inputs_ref->{rlimit};
my $correctAnswer2 =
Compute("$answerString2");
 
$answerString3 =
$inputs_ref->{llimit};
my $correctAnswer3 =
Compute("$answerString3");
 
$answerString4 =
$inputs_ref->{limit};
my $correctAnswer4 =
Compute("$answerString4");
 


$applet->configuration($config_string);
$applet->initialState($config_string);
</pre>
</pre>
| <p>  The lines <code>$applet->configuration</code>
| <p>
<code>(qq{<xml><hintState>$hintState</hintState></code>
The XML here conveys to the applet what each curve should look like.  The code <code><pinkMaxX>0.2</pinkMaxX></code> specifies that the pink curve should cover at least 20% of the domain.  The default is 100% and so is not specified for the blue or yellow curves which should go all the way across the graph.
<code><qtype>$qtype</qtype></code>
<code><seed>$problemSeed</seed></code>
<code><xlist x1='$x1' x2='$x2'</code>
<code> x3='$x3' x4='$x4' /></code>
<code></xml>});</code> and <code>$applet</code>
<code>->initialState</code><code>(qq{<xml></code>
<code><hintState>$hintState</hintState></code>
<code><qtype>$qtype</qtype></code><code><seed>$problemSeed</seed></code>
<code><xlist x1='$x1' x2='$x2'</code><code> x3='$x3' x4='$x4' /></code>
<code></xml>});</code> configure the applet. 
The configuration of the applet is done in xml.   
The hintState is set to the variable
<code>$hintState</code>, the question type is set to
<code>$qtype</code> and the problem seed is the
WeBWorK environmental variable <code>$problemSeed</code>. 
The variables <code>$x1</code>, <code>$x2</code>,
<code>$x3</code> and <code>$x4</code> are also
passed to the applet.
</p><br>
<p>
The hidden form fields are created in the code block:
<code>
TEXT(MODES(TeX=>"", HTML=><<'END_TEXT'));
<input type="hidden"</code>
<code> name="func" id="func" /></code>
<code>
<input type="hidden"</code>
<code> name="llimit" id="llimit" /></code>
<code>
<input type="hidden"</code>
<code> name="rlimit" id="rlimit" /></code>
<code>
<input type="hidden"</code>
<code> name="limit" id="limit" /></code>
<code>
END_TEXT
</code>
The line <code>TEXT(MODES(TeX=>"", HTML=><<'END_TEXT'));</code>
prevents the hidden fields from becoming part of the hard copy.
</p><br>
 
<p>
<code>TEXT( MODES(TeX=>'object code',</code>
<code> HTML=>$applet->insertAll(</code>
<code>
  debug=>0,</code>
<code>
  includeAnswerBox=>0,</code>
<code>
  reinitialize_button=>$permissionLevel>=10,</code>
<code>
  )));</code> actually embeds the applet
in the WeBWorK problem.
</p><br>
</p><br>
<p>When the submit button is pressed, the
hidden form fields defined in this block are
filled with information from the applet.
</p>
<p>
The data from the hidden form fields is used
in these simple perl subroutines to define the
correct answers to the four questions that are
part of this WeBWorK problem.</p>
<p>The WeBWorK variable $answerString1 is the
content of the hidden form field "func". 
$correctAnswer1 is the solution to the first
question.  The solutions for the next two
questions are defined in a similar way. 
The final question also has 'DNE' as a possible
correct answer for the student to enter.  The
way that the applet is designed, the left and
right limits always exist.</p>
|- style=" background-color:#ccffff;"
|- style=" background-color:#ccffff;"
| <pre>
| <pre>
Line 450: Line 301:
|- style=" background-color:#ffdddd;"
|- style=" background-color:#ffdddd;"
| <pre>
| <pre>
##############################################################
#
#  Text
#
#
Context()->texStrings;
BEGIN_TEXT
BEGIN_TEXT
Even before you learn techniques for solving differential
equations, you may be able to analyze equations $BITALIC
qualitatively$EITALIC.  As an example, look at the
nonlinear equation \(\frac{dy}{dt}=$expr.\)  You are
going to analyze the solutions, \(y\), of this equation
without actually finding them.  You will be asked to
sketch three solutions of the differential equation on
the graph below based on qualitative information from
the differential equation.
$BR
$BR
In what follows, picture the \(t-\)axis running
horizontally and the \(y-\)axis running vertically.
$BR$BR
a) For what values of \(y\) is the graph of \(y\) as
a function of \(t\) increasing? Use interval notation
for your answer.  \{ AnswerFormatHelp("intervals") \}
\{ ans_rule(20) \}$BR
$BR
$BR


b) For what values of \(y\) is the graph of \(y\)
concave up? \{ ans_rule(20) \}
\{AnswerFormatHelp("intervals") \}  $BR
For what values of \(y\) is it concave down?
\{ ans_rule(20) \} \{AnswerFormatHelp("intervals") \}
$BR
What information do you need to answer a question about
concavity?  Remember that \(y\) is an implicit function
of \(t\).  $BR
$BR
$BR
The graph shown is for the
function \(f(x)\). 
$BR Compute the following
quantities:
$BR
$BR
a)
 
\(f($x1)=\)
c) Say you are given the initial condition 
\{ans_rule(35) \}
\(y(0)=$blueICy\).  Use the information found in parts
(a) and (b) to sketch the graph of \(y\) in the applet
provided.  Draw the curve in
$BBOLD\{ BBLUE() \} blue\{ EBLUE() \}$EBOLD. 
What is the $BITALIC long-term$EITALIC behavior of
\(y(t)\)?  That is, what is \(\lim_{t\to\infty}y(t)\)? 
\{ ans_rule(10) \}$BR
$BR
$BR
b)
END_TEXT
\(\lim_{x\to {$x2}^+}f(x)=\)
TEXT( MODES(TeX=>'object code', HTML=>$applet->insertAll(
\{ans_rule(35) \}
  debug=>0,
  includeAnswerBox=>1,
#  reinitialize_button=>$permissionLevel>=10,
  )));
 
BEGIN_TEXT
$BR$BR


d) Say you are given the initial condition 
\(y(0)=$pinkICy\).  Use the information found
in parts (a) and (b) to sketch the graph of \(y\).
Draw the curve on the applet used in part (c). 
Draw the curve in $BBOLD\{ BPINK() \}pink
\{ EPINK() \}$EBOLD(pink).  What is the
$BITALIC long-term$EITALIC behavior of
\(y(t)\)?  That is, what is
\(\lim_{t\to\infty}y(t)\)?  \{ ans_rule(10) \}$BR
$BR
$BR
$BR
c)
\(\lim_{x\to {$x3}^-}f(x)=\)
\{ans_rule(35) \}


e) Sketch the graph of \(y\) if \(y(0)=$yellowICy\). 
(Look at the original equation.)  Sketch this curve in
$BBOLD\{ BYELLOW() \}yellow\{ EYELLOW() \}
$EBOLD(yellow).  $BR
$BR
$BR
$BR
d)
\(\lim_{x\to {$x4}}f(x)=\)
\{ans_rule(35) \}


f) If \(y(t)\) represents the population of
some animal species, and if units on the \(y-\)axis
are in thousands, interpret the results of (c),
(d) and (e). 
$BR
The solution to part (c) (sketched in
$BBOLD\{ BBLUE() \} blue\{ EBLUE() \}
$EBOLD) represents: \{ $popup1->menu() \}$BR
$BR
The solution to part (d) (sketched in
$BBOLD\{ BPINK() \}pink\{ EPINK() \}
$EBOLD)(pink) represents: \{ $popup2->menu() \}$BR
$BR
$BR
The solution to part (e) (sketched in
$BBOLD\{ BYELLOW() \}yellow\{ EYELLOW() \}
$EBOLD(yellow)) represents: \{ $popup3->menu() \}$BR
$BR
Adapted from $BITALIC A Modern Introduction to
Differential Equations, 2nd Ed., $EITALIC
Henry J. Ricardo, 2009.
END_TEXT
END_TEXT
Context()->normalStrings;
 
</pre>
</pre>
| <p>
| <p>
This is the <strong>text section</strong>  
This is the <strong>text section</strong>  
of the problem.  The  
of the problem.  The <code>Context()-&gt;texStrings</code>  
<code>TEXT(beginproblem());</code> line
displays a header for the problem, and
the <code>Context()-&gt;texStrings</code>  
line sets how formulas are displayed in the  
line sets how formulas are displayed in the  
text, and we reset this after the text section.   
text, and we reset this after the text section.   
Line 513: Line 437:
|- style=" background-color:#eeccff;"
|- style=" background-color:#eeccff;"
| <pre>
| <pre>
#############################
Context()->normalStrings;
 
##############################################################
#
#
#  Answers
#  Answers
#
#
## answer evaluators
#
 
ANS($inc->cmp);
ANS($cup->cmp);
ANS($cdown->cmp);
 
ANS($lim1->cmp);
NAMED_ANS('answerBox'=>$ans->cmp());
 
ANS($lim2->cmp);
ANS( $popup1->cmp() );
ANS( $popup2->cmp() );
ANS( $popup3->cmp() );


ANS( $correctAnswer1->cmp() ); 
#checks AnSwEr00001
ANS( $correctAnswer2->cmp() ); 
#checks AnSwEr00002
ANS( $correctAnswer3->cmp() ); 
#checks AnSwEr00003
ANS(num_cmp($correctAnswer4,
strings=>['DNE'])); 
#checks AnSwEr00004


COMMENT('This problem requires that Flash be enabled
on your device.  Click try it to see what students will see.');


ENDDOCUMENT();    
ENDDOCUMENT();        
</pre>
</pre>
| <p>
| <p>
This is the <strong>answer</strong>  
This is the <strong>answer</strong>  
section of the problem.  The problem answer  
section of the problem.  The answer to
is set by the <code>ANS( $correctAnswer1->cmp() );</code>,
the applet portion of the problem is
<code>ANS( $correctAnswer2->cmp() );</code>,
evaluated by the line
<code>ANS( $correctAnswer3->cmp() );</code>,
<code>NAMED_ANS('answerBox'=>$ans->cmp());</code>.
and <code>ANS(num_cmp</code>
Other <code>ANS</code> lines pertain to the
<code>($correctAnswer4,</code>
answer blanks appearing in the problem.   
<code>strings=>['DNE']));</code> lines. 
The answer evaluators are in the order
These compare the student's answer with the  
of the corresponding blanks in the problem.
answers returned from the applet.  Answers 1-3
follow the same basic structure.  The fourth
answer allows for either a numeric answer or
the string 'DNE' for limits that do not exist.  
</p>
</p>
<p>
<p>The <code>COMMENT</code> line warns the instructor
The solution is embedded in the applet and
using the OPL that this problem requires Flash.
becomes available when the due date has passed.
Flash is available for virtually any desktop
or laptop, but is not available in a practical
way for most mobile devices.
</p>
</p>
<p>
<p>
Line 556: Line 485:
</p>
</p>
|}
|}
==The complete error message for the sketch shown==
<p>
Your sketch of the pink curve is not consistent with the differential equation.  For at least one y-value, you are showing a solution with more than one slope for the tangent line.</p>
<p>
The pink curve contains some points that are either larger or smaller than is possible with the given differential equation and initial condition.</p>
<p>
The pink is not increasing for all values of y for which the derivative is positive.  Erase your pink sketch and try again being sure to sketch an increasing graph for those values of y for which the derivative is positive.</p>
<p>
The pink curve is not concave up for all values of y for which the second derivative is positive.  For minor errors in concavity, you may be able to correct the graph by pressing the smooth button several times.  Otherwise, erase your pink sketch and try again being sure to sketch a concave up graph for those values of y for which the second derivative is positive.</p>
<p>
The yellow curve contains some points that are either larger or smaller than is possible with the given differential equation and initial condition.
</p>
<p>
The blue and yellow curves intersect.  Think about why this is impossible for the given differential equation, and redraw your graphs.  (Hint:  what is the derivative equal to at the y-value where the graphs intersect.  Do both curves appear consistent with this derivative value?)
</p>
== License ==
The Flash applets developed under DUE-0941388 are protected under the following license: 
[http://creativecommons.org/licenses/by-nc/3.0/ Creative Commons Attribution-NonCommercial 3.0 Unported License].
[[Category:Sample Problems]]
[[Category:Applets]]
[[Category:Problem_Techniques]]
[[Category:Flash Applets]]

Latest revision as of 15:03, 10 August 2013

Flash Applets embedded in WeBWorK questions Autonomous solution sketch Example

Sample Problem with sketch_3.swf embedded

This sample problem shows how to use this versatile applet.

This applet and WeBWorK problem are based upon work supported by the National Science Foundation under Grant Number DUE-0941388.

Click here to see a problem like this in action: [1]

A standard WeBWorK PG file with an embedded applet has six sections:

  1. A tagging and description section, that describes the problem for future users and authors,
  2. An initialization section, that loads required macros for the problem,
  3. A problem set-up section that sets variables specific to the problem,
  4. An Applet link section that inserts the applet and configures it, (this section is not present in WeBWorK problems without an embedded applet)
  5. A text section, that gives the text that is shown to the student, and
  6. An answer and solution section, that specifies how the answer(s) to the problem is(are) marked for correctness, and gives a solution that may be shown to the student after the problem set is complete.

The sample file attached to this page shows this; below the file is shown to the left, with a second column on its right that explains the different parts of the problem that are indicated above. A screenshot of the applet embedded in this WeBWorK problem is shown below:

There are other problems using applets:
Derivative Graph Matching Flash Applet Sample Problem
GraphSketch Flash Applet Sample Problem 1
USub Applet Sample Problem
trigwidget Applet Sample Problem
solidsWW Flash Applet Sample Problem 1
solidsWW Flash Applet Sample Problem 2
solidsWW Flash Applet Sample Problem 3
Hint Applet (Trigonometric Substitution) Sample Problem
phasePortrait Flash Applet Sample Problem 1
Other useful links:
Flash Applets Tutorial
Things to consider in developing WeBWorK problems with embedded Flash applets

PG problem file Explanation 
##DESCRIPTION
##  Sketch autonomous solutions to polynomial differential equation
##ENDDESCRIPTION

##KEYWORDS('logistic', 'population')

## DBsubject('Differential Equations')
## DBchapter('Introduction')
## DBsection('Autonomous Differential Equations')
## Date('8/9/2013')
## Author('L. Felipe Martins')
## Author('Barbara Margolius')
## Institution('Cleveland State University')
## TitleText1('Differential Equations')
## EditionText1('2')
## AuthorText1('Ricardo')
## Chapter('1')
## Problem1('1_1')
###########################################
# This work is supported in part by 
# the National Science Foundation 
# under the grant DUE-0941388.
###########################################

This is the tagging and description section of the problem. Note that any line that begins with a "#" character is a comment for other authors who read the problem, and is not interpreted by WeBWorK.

The description is provided to give a quick summary of the problem so that someone reading it later knows what it does without having to read through all of the problem code.

All of the tagging information exists to allow the problem to be easily indexed. Because this is a sample problem there isn't a textbook per se, and we've used some default tagging values. There is an on-line list of current chapter and section names and a similar list of keywords. The list of keywords should be comma separated and quoted (e.g., KEYWORDS('calculus','derivatives')). 

DOCUMENT();      

loadMacros(
   "PGstandard.pl",     # Standard macros for PG language
   "MathObjects.pl",
   "parserPopUp.pl",
   "AppletObjects.pl",
   "AnswerFormatHelp.pl",
   "PGasu.pl",
);

sub BPINK { 
  MODES(TeX => '{\\color{pink} ', HTML => '<span style="color:pink">'); 
};
sub EPINK { 
  MODES( TeX => '}', HTML => '</span>'); 
};
sub BBLUE { 
  MODES(TeX => '{\\color{blue} ', HTML => '<span style="color:blue">'); 
};
sub EBLUE { 
  MODES( TeX => '}', HTML => '</span>'); 
};
sub BYELLOW { 
  MODES(TeX => '{\\color{yellow} ', HTML => '<span style="color:yellow">'); 
};
sub EYELLOW { 
  MODES( TeX => '}', HTML => '</span>'); 
};

# Print problem number and point value (weight) for the problem
TEXT(beginproblem());

# Show which answers are correct and which ones are incorrect
$showPartialCorrectAnswers = 1;

This is the initialization section of the problem. The first executed line of the problem must be the DOCUMENT(); command. Note that every command must end with a semicolon.

The loadMacros command loads information that works behind the scenes. For our purposes we can usually just load the macros shown here and not worry about things further. 

##############################################################
#
#  Setup
#
#
Context("Numeric");
Context()->variables->add(y=>"Real");
$a = -random(1,4,1);
$expr = Formula("y*(y+$a)")->reduce();
$ymax = Compute("-$a+4");

$I1=Compute("(-infinity,0)");
$I2=Compute("(0,-$a)");
$I3=Compute("(-$a,infinity)");
$cup = Compute("(0,-$a/2)U(-$a,infinity)");
$cdown = Compute("(-infinity,0)U(-$a/2,-$a)");
$lim1 = Compute("0");
$popup1 = PopUp(["?", "extinction", "equilibrium", 
   "explosive growth"], "extinction");
$popup2 = PopUp(["?", "extinction", "equilibrium", 
   "explosive growth"], "explosive growth");
$popup3 = PopUp(["?", "extinction", "equilibrium", 
   "explosive growth"], "equilibrium");

$inc = Compute("$I1 U $I3");

$lim2 = Compute("infinity");

$boardMessage = "Sketch three solutions to this 
   differential equation using the information 
   given in the problem.";
# applet adds a point for each error it detects.  
# It records 100 if the graphs are not drawn
$ans = Compute("0");

#+++++++++++++++++++++++++++++++++++++++++++++
# Designate characteristics of pink curve
# min is IC, max is ymax; concave up on ($IC,ymax); 
# increasing on ($IC,ymax)
# note intervals are in y not t
$pinkICy = Compute("-$a+1");
$pinkIntervalsIncLow = $pinkICy;
$pinkIntervalsIncHigh = $ymax;
$pinkIntervalsCupLow = $pinkICy;
$pinkIntervalsCupHigh = $ymax;
$pinkMin = $pinkICy;
#+++++++++++++++++++++++++++++++++++++++++++++
# Designate characteristics of blue curve
# min is IC, max is ymax; concave down on (0,$IC); 
# decreasing on (0,$IC) 
# note intervals are in y not t
$blueICy = Compute("-$a/2");
$blueIntervalsDecLow = 0;
$blueIntervalsIncHigh = $blueICy;
$blueIntervalsCupLow = 0;
$blueIntervalsCupHigh = $blueICy;
$blueMax = $blueICy;
$blueMin = 0;
#+++++++++++++++++++++++++++++++++++++++++++++
# Designate characteristics of blue curve
# min is IC, max is IC; horizontal line
# note intervals are in y not t
$yellowICy = Compute("-$a");
$yellowMax = $yellowICy;
$yellowMin = $yellowICy;

The sketch_3.swf applet requires the student to sketch three solution curves. The problem author specifies initial conditions, intervals of increase, intervals of decrease, intervals of concavity, and the maximum and minimum possible values of the curves. 

###################################
# Create  link to applet 
###################################
    $appletName = "sketch_3";
    $applet =  FlashApplet(
       codebase              => findAppletCodebase("$appletName.swf"),
       appletName            => $appletName,
       appletId              => $appletName,
       setStateAlias         => 'setXML',
       getStateAlias         => 'getXML',
       setConfigAlias        => 'setConfig',
       getConfigAlias        => 'getConfig',
       maxInitializationAttempts => 5,   # number of attempts to initialize applet
       answerBoxAlias        => 'answerBox',
       height                => '500',
       width                 => '650',
       bgcolor               => '#ededed',
       debugMode             =>  0,
       submitActionScript  =>  
  qq{getQE("answerBox").value=getApplet("$appletName").getAnswer() },
     );

This is the Applet link section of the problem.


You must include the section that follows # Create link to applet. If you are embedding a different applet, from the sketch_3 applet, put your applet name in place of 'sketch_3' in the line $appletName = "sketch_3";. Enter the height of the applet in the line height => '500', in place of 500 and the width in the line width => '650', in place of 650. 

$config_string = <<"ENDCONFIG";
<XML>
    <boardMessage>$boardMessage</boardMessage>
    <xmin>0</xmin><xmax>13</xmax><ymin>-2</ymin><ymax>$ymax</ymax>
    <depVar>y</depVar><indVar>t</indVar>
    <showSolution>false</showSolution>
    <blueIntervalsCup>
      <interval left='0' right='$blueICy'></interval></blueIntervalsCup>
    <blueIntervalsDec>
      <interval left='0' right='$blueICy'></interval>
    </blueIntervalsDec>
    <pinkIntervalsInc>
      <interval left='$pinkICy' right='$ymax'></interval>
    </pinkIntervalsInc>
    <pinkIntervalsCup>
      <interval left='$pinkICy' right='$ymax'></interval>
    </pinkIntervalsCup>
    <blueMax>$blueMax</blueMax>
    <blueMin>$blueMin</blueMin>
    <pinkMin>$pinkMin</pinkMin>
    <yellowMin>$yellowMin</yellowMin>
    <yellowMax>$yellowMax</yellowMax>
    <pinkICy>$pinkICy</pinkICy>
    <blueICy>$blueICy</blueICy>
    <yellowICy>$yellowICy</yellowICy>
    <pinkMaxX>0.2</pinkMaxX>
</XML>
ENDCONFIG

$applet->configuration($config_string);
$applet->initialState($config_string);

The XML here conveys to the applet what each curve should look like. The code <pinkMaxX>0.2</pinkMaxX> specifies that the pink curve should cover at least 20% of the domain. The default is 100% and so is not specified for the blue or yellow curves which should go all the way across the graph. 


TEXT(MODES(TeX=>"", HTML=><<'END_TEXT'));
<script>
if (navigator.appVersion.indexOf("MSIE") > 0) {
    document.write("<div width='3in' 
    align='center' style='background:yellow'>
    You seem to be using Internet Explorer.
    <br/>It is recommended that another 
    browser be used to view this page.</div>");
}
</script>
END_TEXT

The text between the <script> tags detects whether the student is using Internet Explorer. If the student is using this browser, a warning is issued and the student is advised to use another browser. IE mis-sizes the applets. Some will work correctly when displayed at the wrong size, but others will fail. We do not recommend using IE with WeBWorK problems with Flash embedded.

##############################################################
#
#  Text
#
#

Context()->texStrings;
BEGIN_TEXT
Even before you learn techniques for solving differential 
equations, you may be able to analyze equations $BITALIC 
qualitatively$EITALIC.  As an example, look at the 
nonlinear equation \(\frac{dy}{dt}=$expr.\)  You are 
going to analyze the solutions, \(y\), of this equation 
without actually finding them.  You will be asked to 
sketch three solutions of the differential equation on 
the graph below based on qualitative information from 
the differential equation.

$BR
$BR
In what follows, picture the \(t-\)axis running 
horizontally and the \(y-\)axis running vertically.
$BR$BR

a) For what values of \(y\) is the graph of \(y\) as 
a function of \(t\) increasing? Use interval notation 
for your answer.  \{ AnswerFormatHelp("intervals") \} 
\{ ans_rule(20) \}$BR
$BR
$BR

b) For what values of \(y\) is the graph of \(y\) 
concave up? \{ ans_rule(20) \} 
\{AnswerFormatHelp("intervals") \}   $BR 
For what values of \(y\) is it concave down? 
\{ ans_rule(20) \} \{AnswerFormatHelp("intervals") \} 
$BR
What information do you need to answer a question about 
concavity?  Remember that \(y\) is an implicit function 
of \(t\).  $BR
$BR
$BR

c) Say you are given the initial condition  
\(y(0)=$blueICy\).  Use the information found in parts 
(a) and (b) to sketch the graph of \(y\) in the applet 
provided.  Draw the curve in 
$BBOLD\{ BBLUE() \} blue\{ EBLUE() \}$EBOLD.  
What is the $BITALIC long-term$EITALIC behavior of 
\(y(t)\)?  That is, what is \(\lim_{t\to\infty}y(t)\)?  
\{ ans_rule(10) \}$BR
$BR
END_TEXT
TEXT( MODES(TeX=>'object code', HTML=>$applet->insertAll(
  debug=>0,
  includeAnswerBox=>1,
#   reinitialize_button=>$permissionLevel>=10,
   )));

BEGIN_TEXT
$BR$BR

d) Say you are given the initial condition  
\(y(0)=$pinkICy\).  Use the information found 
in parts (a) and (b) to sketch the graph of \(y\). 
Draw the curve on the applet used in part (c).  
Draw the curve in $BBOLD\{ BPINK() \}pink
\{ EPINK() \}$EBOLD(pink).  What is the 
$BITALIC long-term$EITALIC behavior of 
\(y(t)\)?  That is, what is 
\(\lim_{t\to\infty}y(t)\)?  \{ ans_rule(10) \}$BR
$BR
$BR

e) Sketch the graph of \(y\) if \(y(0)=$yellowICy\).  
(Look at the original equation.)  Sketch this curve in 
$BBOLD\{ BYELLOW() \}yellow\{ EYELLOW() \}
$EBOLD(yellow).  $BR
$BR
$BR

f) If \(y(t)\) represents the population of 
some animal species, and if units on the \(y-\)axis 
are in thousands, interpret the results of (c), 
(d) and (e).  
$BR
The solution to part (c) (sketched in 
$BBOLD\{ BBLUE() \} blue\{ EBLUE() \}
$EBOLD) represents: \{ $popup1->menu() \}$BR
$BR
The solution to part (d) (sketched in 
$BBOLD\{ BPINK() \}pink\{ EPINK() \}
$EBOLD)(pink) represents: \{ $popup2->menu() \}$BR
$BR
The solution to part (e) (sketched in 
$BBOLD\{ BYELLOW() \}yellow\{ EYELLOW() \}
$EBOLD(yellow)) represents: \{ $popup3->menu() \}$BR
$BR

Adapted from $BITALIC A Modern Introduction to 
Differential Equations, 2nd Ed., $EITALIC 
Henry J. Ricardo, 2009.

END_TEXT

This is the text section of the problem. The Context()->texStrings line sets how formulas are displayed in the text, and we reset this after the text section. Everything between the BEGIN_TEXT and END_TEXT lines (each of which must appear alone on a line) is shown to the student.

Mathematical equations are delimited by \( \) (for inline equations) or \[ \] (for displayed equations); in these contexts inserted text is assumed to be TeX code.

There are a number of variables that set formatting: $PAR is a paragraph break (like \par in TeX). This page gives a list of variables like this. Finally, \{ \} sets off code that will be executed in the problem text. Here, ans_rule(35) is a function that inserts an answer blank 35 characters wide. 

Context()->normalStrings;

##############################################################
#
#  Answers
#
#

ANS($inc->cmp);
ANS($cup->cmp);
ANS($cdown->cmp);

ANS($lim1->cmp);
NAMED_ANS('answerBox'=>$ans->cmp());

ANS($lim2->cmp);
ANS( $popup1->cmp() );
ANS( $popup2->cmp() );
ANS( $popup3->cmp() );


COMMENT('This problem requires that Flash be enabled 
on your device.  Click try it to see what students will see.');

ENDDOCUMENT();

This is the answer section of the problem. The answer to the applet portion of the problem is evaluated by the line NAMED_ANS('answerBox'=>$ans->cmp());. Other ANS lines pertain to the answer blanks appearing in the problem. The answer evaluators are in the order of the corresponding blanks in the problem.

The COMMENT line warns the instructor using the OPL that this problem requires Flash. Flash is available for virtually any desktop or laptop, but is not available in a practical way for most mobile devices.

The ENDDOCUMENT(); command is the last command in the file.

The complete error message for the sketch shown

Your sketch of the pink curve is not consistent with the differential equation. For at least one y-value, you are showing a solution with more than one slope for the tangent line.

The pink curve contains some points that are either larger or smaller than is possible with the given differential equation and initial condition.

The pink is not increasing for all values of y for which the derivative is positive. Erase your pink sketch and try again being sure to sketch an increasing graph for those values of y for which the derivative is positive.

The pink curve is not concave up for all values of y for which the second derivative is positive. For minor errors in concavity, you may be able to correct the graph by pressing the smooth button several times. Otherwise, erase your pink sketch and try again being sure to sketch a concave up graph for those values of y for which the second derivative is positive.

The yellow curve contains some points that are either larger or smaller than is possible with the given differential equation and initial condition.

The blue and yellow curves intersect. Think about why this is impossible for the given differential equation, and redraw your graphs. (Hint: what is the derivative equal to at the y-value where the graphs intersect. Do both curves appear consistent with this derivative value?)

License

The Flash applets developed under DUE-0941388 are protected under the following license: Creative Commons Attribution-NonCommercial 3.0 Unported License.

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