HeavisideStep1

Using the Heaviside Step Function

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This PG code shows how to add a the Heaviside step function step(x), which takes the value 1 if x > 0, and the value 0 if x ≤ 0, to the context. It also shows how to add a named function u(x) to the context that has a reliable answer checker and can stand in for the Heaviside step function when the student answer is a function.

• Download file: File:HeavisideStep1.txt (change the file extension from txt to pg when you save it)
• File location in NPL: FortLewis/Authoring/Templates/DiffEq/HeavisideStep1.pg

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PG problem file	Explanation
Problem tagging data	Problem tagging:
DOCUMENT(); loadMacros( "PGstandard.pl", "MathObjects.pl", "AnswerFormatHelp.pl", "parserFunction.pl", ); TEXT(beginproblem());	Initialization: We load parserFunction.pl to make adding a named function to the context easier.
$ftex = "5 u(t-3)"; Context("Numeric"); Context()->variables->are(t=>"Real"); Context()->functions->add( step => { class => 'Parser::Legacy::Numeric', perl => 'Parser::Legacy::Numeric::do_step' }, ); $f = Formula("5*step(t-3)"); $answer1 = List($f->eval(t=>2),$f->eval(t=>3),$f->eval(t=>4));	Setup 1: We add the step function to the context with the name step. The function step(t) is the Heaviside function and takes the value 1 when t > 0, and the value 0 when t ≤ 0. We will use the function step when evaluating the Heaviside function to obtain an answer that is a number. We will not use the function step when the student's answer is a function. Since answers are checked numerically by comparing the student answer to the correct answer at several randomly points in the domain (the default is 5 points) in an interval (the default is [-1,1]), the function step is not very robust when checking answers. For example, if a student types in the answer step(t-0.1) and the correct answer is step(t), there is a good chance that the student's answer will be marked correct, since the probability that a test point was chosen in the interval (0,0.1) is much less than 100%. Also, if the correct answer were step(t-5), then a student could enter the answer 0 and be marked correct because the correct answer is identically zero on the interval [-1,1]. For more details, see the forum discussion on the Heaviside step function
Context()->texStrings; BEGIN_TEXT Let \( u(t) \) be the Heaviside step function defined by \( \displaystyle u(t) = \left\lbrace \begin{array}{lcl} 0 && \mbox{ if } x > 0, \\ 1 && \mbox{ if } x \leq 0. \end{array} \right. \) $BR $BR (a) Evaluate the function \( $ftex \) when \( t \) is 2, 3, and 4 and enter your answer as a comma separated list. $BR \{ ans_rule(20) \} \{ AnswerFormatHelp("numbers") \} END_TEXT Context()->normalStrings;	Main Text 1:
$showPartialCorrectAnswers=1; ANS( $answer1->cmp(ordered=>1) );	Answer Evaluation 1:
Context("Numeric"); Context()->variables->are(t=>"Real"); Context()->flags->set( limits=>[-5,5], num_points=>10, ); parserFunction("u(t)" => "1.5*sin(e*t)+5*pi/e+arctan(t)"); $answer2 = Formula("5 u(t-3)");	Setup 2: Using parserFunction.pl, we add another function named u(t) to the context. The reason for adding this function to the context is that step, which is the Heaviside function, will be used when the answer is a number, and u will be used when the answer is a function. Notice that the function u(t) is never zero, is not constant, is differentiable, and takes moderately sized values, which makes its answer checking very robust. Further, because of the arctangent, it is not periodic and so u(t)-u(t-a) should never be identically zero. The formula for u(t) is not something students are likely to input as an answer out of nowhere. The function u(t) is great as a named function that stands in for the Heaviside function when the answer is a function. However, if the answer is a number obtained by evaluating the Heaviside function, then step(t) should be used instead of u(t) for obvious reasons. Note: Currently, functions that are added to the context do not work with test_points, but they do work with changes to the domain limits=>[a,b] and the number of points num_points=>c set using Context flags.
Context()->texStrings; BEGIN_TEXT $BR $BR (b) Enter the function \( $ftex \). $BR \{ ans_rule(20) \} \{ AnswerFormatHelp("formulas") \} END_TEXT Context()->normalStrings;	Main Text 2:
ANS( $answer2->cmp() );	Answer Evaluation 2:
Context()->texStrings; BEGIN_SOLUTION ${PAR}SOLUTION:${PAR} Solution explanation goes here. END_SOLUTION Context()->normalStrings; COMMENT('MathObject version.'); ENDDOCUMENT();	Solution:

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