Autonomous solution sketch Flash Applet Sample Problem

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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:
Sketch WWprob.png
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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