the nature of concave up

[Alex Jordan]

Here's a thing that initiated with a report from a local instructor, who was accessing Sean's latest version that has interactive WW exercises, where there is answer checking.

For example:
3x^2 is increasing on [0,inf)
according to:
https://opentext.uleth.ca/apex-video/sec_incr_decr.html#def_incr_decr
which is a standard definition, except maybe in some books "increasing on an interval" is only defined for open intervals.

So for f(x)=x^3, f' is increasing on [0,inf).

So according to:

https://opentext.uleth.ca/apex-video/sec_concavity.html#def_concavity

x^3 is concave up on [0,inf)

I draw your attention to the closed endpoint. This is how I coded WW problems: what are the maximal intervals where the function is concave up? And the official answer would be [0,inf) here.

I'm happy to make the WW problems accept either (0,inf) or [0,inf). So my first question is, does anyone object to that, and would want only one of them accepted?

My next question is, does anyone object to "the" answer to the question being [0,inf)? This is what would be reported in, say, an answer manual.

Lastly, if [0,inf) is the actual correct answer (which for many reasons, I believe it is) then the examples are not consistent with this view. This one does not use interval notation, but it implies f is concave up on (0,inf) rather than [0,inf). If it's possible to edit the examples to "correctly" conclude with closed endpoints without making things super pedantic and wordy, are there objections to editing the examples that way?

-----

Related observations:

The definition for increasing has nothing to do with derivatives. It is fundamental in a pre-calculus sense, and applies to nondifferentiable functions just as much as to differentiable ones.

The APEX definition for concave up relies on f'. I don't object to that. Personally I think of concave up in terms of chords: that the midpoint of a chord within the interval is higher than the function's value at the midpoint of the x-values:  [f(a) + f(b)] / 2  >  f([a + b] / 2) for all a!=b in the interval. Is there any interest in moving to this sort of pre-calculus definition for concave up?

Is abs(x^3+x) concave up on all of (-inf,inf)? I would say yes. But there is no tangent line at 0.

People conflate having these features on an interval with having these features at each and every point within the interval. This makes it feel wrong to some people for x^3 to be concave up on [0, inf), because it is also concave down on (-inf,0], so "therefore" it is both concave up and concave down at x=0. But really, concave up at a point should mean there exists a surrounding open interval that satisfies the concave up definition for intervals. And then there is no contradiction. Is there any value in adding a paragraph to point out that increasing/decreasing/concave-up/concave-down on an interval does /not/ mean that the function is increasing/decreasing/concave-up/concave-down at each point within that interval?

[Sean Fitzpatrick]

This is ultimately a question for Greg, but here are my thoughts:

1. I always tell students that there are differing conventions on this, and I don't really care if they include the endpoint or not, because the goal at this level is being able to sketch a graph.

(One person's "increasing" is another's "strictly increasing", and so on. Some get bothered by the thought that a constant function could be both increasing and decreasing at the same time.)

2. I always define concavity in terms of whether the tangent line lies above or below the graph, as a measure of "deviation from linearity". But that doesn't capture your example, I suppose.

3. In the section on continuity, Greg has already gone to the trouble of pointing out the difficulty with using union is intervals: a step function is continuous on [n, n+1) for all n, but not on the union.

With that precedent established, it seems like it would not be a big leap to include endpoints for intervals when discussing increasing, decreasing, and concavity.

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[gregory...@gmail.com]

Behavior at boundaries is one of the wonderful, and frustrating, things about math. The endpoints of intervals are a difficult issue that has cropped up in these discussions quite a few times. Not a universal truth, but most books I looked at avoided these difficulties by simply defining everything to occur on open intervals. 

APEX defines increasing/decreasing in a non-calculus way. It then makes the statement "If f' > 0, then f is increasing." True; it leaves open the possibility that f is increasing AND f' = 0, though this possibility is never explored. (Not even the case of y = x^3.) Rather, we answer problems like "Determine the intervals on which f(x) = x^2 is incr/decr" by looking at where f' > 0 & f' < 0, arriving at answers of (0, infty) and (-infty, 0), respectively. But, as Alex points out in the case of the concavity, the answer that follows from the def. is [0, infty) and (-infty, 0].

One way of handling this discrepancy is to ask a different question: "Give the *open* intervals on which f is incr/decr." But that seems lazy. 

Another is to fix both 3.3 and 3.4. 

When optimizing / looking for abs. max & min, we already have established the principle of "use calculus to do the bulk of the work, and also check the endpoints." We can establish that principle earlier in 3.3 & 3.4.

3.3: Keep our non-calc def. of incr/decr, and keep the theorem that relates to the first derivative. Then we note that the full interval of incr/decr will take into account endpoints. This will make intuitive sense; f(x) = sqrt(x) has a min at x=0 because it is increasing on [0,infty). Currently, I effectively make a leap by saying "f is incr. on (0,infty), with a min at 0."

3.4:  As incr/decr is defined without calculus; we could define concavity in a similar way (which would also mean not using Sean's tangent-line definition). The chord-midpoint def. is fine and visual. Then we can introduce a thm. that is similar to what is used in 3.3: if f'' > 0, then f' is increasing and f is concave up. (Allowing for the converse to sometimes be true.) We still need to check endpoints. And we can toss in the tangent line definition as an exercise for those interested.

I'm suggesting changes with broad strokes, and the devil's in the details. The current text is "fine" when viewed from afar, but the details show inconsistencies; I'm sure that if I'm not careful the above will just introduce a different set of inconsistencies. 

[Alex Jordan]

> It then makes the statement "If f' > 0, then f is increasing." True; it leaves open the possibility that f is increasing AND f' = 0, though this possibility is never explored.

Also the possibility that f is increasing *and* f' is undefined here
and there, like cases{x, x ≤ 0; 2x, otherwise}. Or even that f is
increasing and f is not even continuous, like cases{x, x ≤ 0; x+1,
otherwise}. I wouldn't want to present such functions to intro calc
students expressed this way. But maybe I would graphically.

> "Give the *open* intervals on which f is incr/decr."

I acknowledge it's a bit pedantic, but for example the function x^2 is
increasing on (1,2), and on (1,7), and on (3,infinity), etc. So I
phrased the exercises to ask for the "maximal" intervals. They could
alternatively ask for "maximal open" intervals. Maybe "maximal" is too
math-majory for this, and it should say "largest" or something more
friendly. "largest" by containment though, not by measure...

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[gregory...@gmail.com]

I think "maximal interval" is a fine term, especially if it is used in the text; "largest interval" is probably a better choice. (?) Yes, we could get pedantic and force a definition of "maximal" and "largest", or let common sense be enough.

[Sean Fitzpatrick]

One thing I would add: for the exercises, my preference would be to accept both open and closed intervals as solutions for the WeBWorK versions.
That way the problems work, even if an instructor doesn't follow the book to the letter. (Since no instructor ever does.)

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[Sean Fitzpatrick]

This issue is next on my list of edits. But I think it has tentacles throughout the book.

How thorough do we want to be on addressing this? (And does someone else want to take a crack at it?)

We can just update 3.3 and 3.4, or we can carefully check for consequences in later chapters. (I think there are not too many.)

There are two things to update:

1. The definition of concave up/down. We have Alex's definition in terms of chords, which is good.

But I might suggest expanding a little: the definition in terms of chords tells us what it means to be concave up on an *interval*.

Here is a discussion, which could go into an aside?

What does it mean to be concave up/down at a *point* x=a? Well, this amounts to letting b --> a in the chord definition.

But if we let b --> a, a chord becomes a tangent line! [Related note: rather than introducing "chord", should we stick with "secant line"?]

This leads to a definition you'll sometimes encounter in other calculus books:

The graph of f is concave up at c if the tangent line at (c,f(c)) lies below the graph y=f(x).

And concave down if the tangent line is above the graph.

However, with this tangent line definition, we are back to a graph being neither concave up nor down at an inflection point (which brings back open intervals).

2. All the open intervals in examples would need to be changed to closed intervals. (Or half-open, when there are points of discontinuity or asymptotes.)

And I suppose we would do this for both increasing/decreasing and concave up/down.

Another option to consider: we improve the definition, and add a discussion pointing out that while we could extend the definition to use closed intervals,

open intervals are common practice, and for practical purposes (classifying critical points, curve sketching, etc.) the distinction doesn't really matter.

That allows us to address the issue without tracking down all the places where intervals need to be adjusted.

[Alex Jordan]

My take: being ____ at a point means that there is an open interval
containing that point on which the function is ____.

So it's still not "calculus" based. Not relying on derivatives or limits.

Then it's a theorem that f'(a) > 0 implies f is increasing at a,
f''(a) > 0 implies f is concave up at a, etc.

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[Sean Fitzpatrick]

We can certainly use that convention!
But that brings open intervals back into the picture.

Could we give a "non-calculus" definition that still uses open intervals,
then add the appropriate theorem about the sign of some derivative,
and also add an aside pointing out that it would not be inappropriate to
use a closed interval instead?

That is easy to implement and doesn't require rewriting several examples
and exercises.

[Alex Jordan]

An observation: the tangent line idea doesn't say anything for
functions like |x| + 2x at x=0, where you might want to say that it is
increasing at that point.

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[Sean Fitzpatrick]

Oh, yeah! But the tangent line definition refers to concavity, not increase/decrease.

But the same issue could arise there.

Now, for the larger structural issue: do we want to improve the definition (and the discussion surrounding it) and leave the rest, or do we also want to comb through the text to identify all the places where an open interval could be replaced by a closed interval?

I think I prefer open intervals, but I also tell students they're not wrong is they use closed.

[gregory...@gmail.com]

I wanted to attend the PreTeXt workshop to discuss this today. Then I blew it and totally forgot - my sincere apologies.

Here's an attempt to unify some of the various thoughts throughout this thread, if only for my own good.

1. a. We define `increasing/decreasing' without calculus. The definition allows for saying a function can be incr/decr on closed intervals. 

    b. We then state that "If f is cont. on [a,b], and diff. on (a,b), and f' > 0 on (a,b), then f is increasing on [a,b]." 

    c. We don't explore it, but this leaves open the possibility that a function is increasing without being differentiable.

    d. For the rest of the text, I am pretty sure that we only use f' > 0 to determine increasing; it is sufficient.

2. a. I think we should define 'concave up/down'  without calc. I think the definition "A function f(x) is concave up on [a,b] if [f(x1) + f(x2)] / 2  >  f([x1 + x2] / 2) for all x1 != x2 in [a,b]" is a good start. This allows for f being concave up on a closed interval. (Right? - am I missing something? So y = x^3 is concave up on [0,\infty). )

    b. We then state that "If f is continuous on [a,b] and both f & f' are diff. on (a,b), and f'' > 0 on (a,b), then f is concave up on [a,b]." (I haven't thought too hard about extending to the closed interval here, but it feels that it should work.)

    c. We need not explore it, but we can have concave up & not differentiable (or not doubly differentiable).

    d. For purposes of the rest of the text, we only use f'' > 0 to determine concave up. (Our theorem allows that y = sqrt(x) is concave down on [0,infty).

Am I right on #1? And does #2 sufficiently summarize where we've been heading in this thread? Does it resolve endpoint issues?

3. Exercises: how do we word exercises asking for intervals of concave up/down, incr/decr? 

   a. There is a desire to have both open/closed versions of answers be accepted in webwork. 

   b. There is a suggestion that we phrase problems with "Give the maximal (or, largest) *open* interval on which f is incr/decr., concave up/down." If we use that language, then it doesn't make sense to accept both open/closed interval answers. 

   c. If we say "Give the maximal/largest interval on which f is incr/decr, etc.", we can write the largest (i.e., often closed) interval in the back of the book to align with the definition & thms. We could also code webwork to accept both, BUT this may upset some instructors who want to make sure students are using the closed definition.

How does this sit with people? 

[Alex Jordan]

Greg's summary sounds right on all points as far as where I think this
discussion has led.

> c. If we say "Give the maximal/largest interval on which f is incr/decr, etc.", we can write the largest (i.e., often closed) interval in the back of the book to align with the definition & thms. We could also code webwork to accept both, BUT this may upset some instructors who want to make sure students are using the closed definition.

Using WW with APEX is a "new" feature. It shouldn't upset an
instructor in the sense of something changing. But maybe the behavior
is still not what some instructors would like.

Option (a): these exercises never care whether a student enters an
interval using open or closed delimiters (except they still disallow
closed at infinity). The exercises print "the" correct answer as
whatever we settle on wrt the definitions, theorems, phrasing of the
exercise statement, and specifics of the function in the exercise.

Option (b): "the" correct answer is whatever we settle on wrt the
definitions, theorems, phrasing of the exercise statement, and
specifics of the function in the exercise. If that ends up being
closed, it can give credit for the maximally open subset of that, and
also produce a small slap on the wrist feedback message that says
really, the answer is this slightly larger set but fine we are giving
you credit here in WeBWorK for now.

Option (a) is easier to code, but option (b) is workable too. It's not
all that many exercises in the end.

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[Sean Fitzpatrick]

My opinion:

The ambiguity around endpoints is a feature, not a bug. Instructors should embrace the inherent messiness as an opportunity to point out that in math, it is sometimes ok to say that two different answers can both be considered correct, depending on your point of view.

The WeBWorK code supports this take.

What I want my students to understand:

1. We can't say that f is increasing on [1, 3] if there's a vertical asymptote at x=2. In this case, using [1, 2)U(2, 3] is necessary.

(And maybe we ditch using union for the same reason that we do for continuity?)

2. If f'(x)>0 on [1, 2)U(2, 3], and f'(2)=0, it *is* correct to say f is increasing on [1, 3].

3. If f'(x)>0 on (1, 2), you can say f is increasing on (1, 2), or on [1, 2] (assuming continuity at the endpoints) and either answer is acceptable.

What I think I would do:

- Add a new definition of concave up/down.

- Add a corresponding theorem involving the second derivative

- Add remarks (or margin notes) in both 3.3 and 3.4 that address the open vs closed question

- Leave examples and exercises written as they are, and let WeBWorK allow both answers.

[gregory...@gmail.com]

I'm 95% ok with Sean's suggestion. The minor tweaks:

The notes following the thms of 3.3 and 3.4 should mention that we'll default to using open intervals. (Maybe already Sean's intent.)

The current wording of the examples and exercises is "find the intervals on which f is XYZ", with the expectation that the answer will be an open interval, as per the notes. We could also make a note at the end (in the text, or as marginal notes) of the examples that certain closed intervals would be acceptable. In Example 3.3.1, we could have written (-infty,-1], [-1,1/3], [1/3,infty). In Example 3.3.2, though, we acknowledge the asymptote at x=1 and must accept either (1,3) or (1,3]. 

[Sean Fitzpatrick]

That would work for me. Let's see what Alex thinks.

(He's busy wrestling with some of the technical guts of PreTeXt though)

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[Sean Fitzpatrick]

By the way, the "Test for increasing/decreasing" theorem already states that for f(x) continuous on [a,b] and differentiable on (a,b), if f'(x)>0 on (a,b), then f is increasing on [a,b].

(So the theorem has a closed interval.) We can do the same for concave up/down.

But then we still add a note for both that although we can often state things in terms of closed intervals, we cave to convention and use open intervals as the default when stating results.

[Sean Fitzpatrick]

Change is now made at https://github.com/APEXCalculus/APEXCalculusPTX/pull/136/commits/3d44f57013fb459d42372f76a74ef2e42fea2c6b

Let me know what you think. I'm going to run an HTML build shortly so we can get a better look at the changes.