nitpicky arcsecant issue

[Alex Jordan]

I'm programming WW questions, where nitpicky details can pop up in
order to code the actual correct answer as correct. An example of this
is section 6.1 #47:

Antidifferentiate
5/sqrt(x^4 - 16x^2)

If we factor out x^2 in the radical, and pull it out as x, we have:
5/[x sqrt(x^2 - 16)]
and we could use the formula from the section, getting
5/4 asec(|x|/4)

But since sqrt(x^2) = |x|, if we pull out that factor, we should have:
5/[|x| sqrt(x^2 - 16)]
and I think the antiderivative would be:
5/4 asec(x/4)

None of this matters except if you are considering negative x in the domain.

Question 1: Greg, what do you want the answer to this to be reported
as? Like, in the list of answers in the back of the book? It is
currently 5/4 asec(|x|/4).

Question 2: How do we want the WW answer checker to behave for an
automatically graded answer? It could easily accept both, just by
restricting the domain upon which it checks to positive x. Or it could
distinguish between these.

Question 3: Is it worth modifying the question since this subtlety is
not addressed in the section? It could be presented with the x
factored out already. Or it could specify "x > 4".

[Sean Fitzpatrick]

I think we've defined arcsecant using the "trig/calculator-friendly"
definition,
which puts the range of arcsecant as [0,pi/2)U(pi/2,pi].

That lets you define arcsec(x)=arccos(1/x), which I understand is useful
for people who want to evaluate arcsecant using a graphing calculator.
(Having never used a graphing calculator in my life, I was unaware such
people existed ;-)  )

But it gives you the derivative with the absolute value. In that case,
5/sqrt(x^2-16x^2)=5/[|x|sqrt(x^2-16)], and the antiderivative is
(5/4)arcsec(x/4). (If I've remembered correctly where the 4 is supposed
to go.)

There is also the "calculus-friendly" definition of arcsecant, with
range [0,pi/2)U[pi,3pi/2). Since tangent is positive on this range,
there's no sign ambiguity and the derivative does not have an absolute
value in it.
But this isn't the convention used in APEX, so moot.

Greg can correct me, but my understanding here is that the x^2 should be
factored out of the radical as |x|.

[gregory...@gmail.com]

Ugh. What an ugly problem. (Who wrote that?) I think it should be rewritten.

The point of the problem, clearly, is to have student do a small amount of manipulation before applying the cookbook answer of "this is arcsecant and a=4". And yet the result is terrible - the answer given in the back of the book is *wrong*, though intuitive to all people (like me) who simplify sqrt(x^2) to just x. 

Adding something like "x > 4" will just confuse students - we haven't restricted domains of integrals before, right? They'll wonder "What am I to do differently here?"

Accepting the wrong answer in WW is just ... wrong. I understand the sentiment, but long term, we won't be happy with ourselves. But only accepting the right answer will be more of a frustration to faculty/students than a help.

I think we should just factor the x out. To keep the "manipulate first" flavor of the problem, what do you think of:
5 / (x Sqrt((x-4)(x+4))   ? 

[Alex Jordan]

What if it were:

5x/sqrt(x^6 - 16x^4)

Now you factor out x^4 inside the radical, and it's correct to bring
that out as x^2. Some x's cancel and you are left with:

5/[x sqrt(x^2 - 16)]

as desired. Still need to do that bit of algebra manipulation. It's a
tiny bit "more" than before though.

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[Sean Fitzpatrick]

Hmm. The sqrt(x^2)=x mistake is one of those common errors that I always
try to address at the beginning of term. So in my case, I would
absolutely expect students to know that sqrt(x^4-4x^2) = |x|sqrt(x^2-4).

And in a WeBWorK context (where they have unlimited attempts) getting a
wrong answer because they forgot the absolute value is a useful learning
opportunity to reinforce the correction of this common error.

But that might just be me. ;-)

[Alex Jordan]

I think the issue is that in the exposition, integrating
1 / [|x| sqrt(x^2 - 4)]
is not demonstrated. So you can expect them to do the algebra
correctly and have |x|, but they might be confused about what to do
next. Especially when it so closely resembles the one without the abs
value.

[Yes, you could ask them to recall the derivative of arcsecant, but my
guess is one would expect this "type" to have been demonstrated in
6.1.]

I've coded the WW problems to distinguish between ln(x) and ln(|x|)
where that matters, so fwiw there will be ample automated reminders to
students that they are forgetting abs value bars.

Another question for Greg. There are a few exercises where the
antiderivative might have like:
ln(|x^2 - 2x + 10|)

That quadratic comes from:
(x - 1)^2 + 9
which is always positive. So in these cases, the above is the same as
ln(x^2 - 2x + 10)

WW won't care; both are correct. But which one would you like to have
/displayed/ as "the" correct answer in that back of the book?

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[gregory...@gmail.com]

We should expect students to know how to properly simplify Sqrt[x^2], but we haven't really covered how to integrate functions with abs values. I could give lots of prompts to students to guide them through finding an antiderivative of sin(|x|), but they'd certainly stumble at first. So practicing a new integration type along with abs values seems like a lot.

I'm fine with Alex's suggested replacement problem. 

When f(x) > 0 for all x, then I'd prefer we write " ln( f(x) )" vs "ln( |f(x)| )". Some students will definitely wonder, and that'd make a good teaching moment. And I agree with Alex that WW should accept both.