Heaviside or Step Function

Setup

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. The step function will be used when evaluating the Heaviside function to obtain an answer that is a number.

For more details on adding the Heaviside function to the context, see the forum discussion on the Heaviside step function.

For the second question, since answers are checked numerically by comparing the student answer to the correct answer at several random points in the domain (the default is 5 points) in an interval (the default is [-1,1 ]), the function step(t) = u(t) is not very robust when checking answers using these defaults. For example, if a student types in the answer u(t - 0.1) and the correct answer is u(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 u(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].

To make the answer checking robust, in $answer2 a larger domain centered at $a is specified using limits, five test points are specified, and 10 test points are required to be used (the five that were specified and five others generated at random). Notice that the construction $f->with(...) is used to do this (using $f->{test_at} = [ [1], [2] ] would generate an error because the functions added to the context are not “blessed” with enough permissions to modify $f in that way).

In part (b), since the students never actually see the values of the function u(t), the function could be defined as

parserFunction("u(t)" => "1.5 * sin(e * t) + 5 * pi / 3 + arctan(t)");

If u(t) had been defined in this way, the function step(t) would not need to be added to the context and the defaults for the answer checker could be used. 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 using the defaults for the answer checker. Further, because of the arctangent, it is not periodic and so u(t) - u(t - a) should never be identically zero. Also, 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 or the function u(t) should be properly defined as the Heaviside function for obvious reasons.