That's one of my test questions where I lost most of my points. I had a long debate with my teacher about it. His point of view is that the graph is decreasing at a decreasing rate because the slope of the graph from left to right is numerically decreasing. My point of view is that the graph is decreasing at an increasing rate; this simply makes logical sense. Rate of change only includes a sign to show direction; as such, if a rate of change is increasing, then it could be getting more positive (numerically increasing) OR more negative (numerically decreasing), and if a rate of change is decreasing, then it's getting close to 0, regardless of its sign. It is clear that the slope of the graph is numerically decreasing from left to right, but that is not what the question is asking. What do you guys think? (Sorry for posting here, but I don't really have anyone else to ask when I'm disagreeing with my math teacher)

The derivative is negative, so the rate of change is negative, but the rate of decrease at any given point, is positive, because decrease already has an implied negative. And since it decreases more quickly over time, the rate of decrease increases over time.

It looks like your teacher is just wrong.* Is this, by any chance, the same math teacher you complained about before with the clock question? It sounds like you have a bad teacher.

*Well, it's possible to make a case for it being the other choice, too, but that choice is certainly not more correct than the one you chose.

Quote: Original post by cowsarenotevil
The derivative is negative, so the rate of change is negative, but the rate of decrease at any given point, is positive, because decrease already has an implied negative. And since it decreases more quickly over time, the rate of decrease increases over time.

Okay, thanks for the clarification. That's definitely a better way of explaining it than mine.

Quote:
It looks like your teacher is just wrong.

He claims it's a fact in calculus.

Quote: Is this, by any chance, the same math teacher you complained about before with the clock question? It sounds like you have a bad teacher.

Funny that you ask... small story: I was in Math B until January, which is when I took the Math B regents and skipped half-way into PreCalc H. Now I have a different teacher than the clock guy. Ironically enough, I talked to my previous teacher (he's my APCS teacher so we're on pretty good terms other than the clock thing), and he agrees with me on this question.

That's a terrible question because it's ambiguous. I would have asked for clarification if I had encountered that on a test.

[Edit: I toned down the... tone of this post.]

[Edited by - nilkn on May 29, 2010 8:30:53 PM]

I'm surprised that he didn't give you any points back for providing a decent argument as to why your answer was correct, especially since there weren't any hard calculations and the answers were all qualitative. Seriously, if a student were able to argue their way out of what I thought was an incorrect answer (and their arguments were all valid), they'd get points back.

The question was very badly worded, or simply meant as a trick question. Given that your argument is sound, your teacher should've upped you 5 pts.

The graph is clearly decreasing at a decreasing rate though.
Look at the rate of change in consumption at certain points as you move right on the graph.

At t=0, the rate is negative close to zero.
At t=1, the rate is a negative number lower than the year before, say -1.
At t=41, the rate seems to be ~ -100, a much lower number than at t=1.

Thus the number decreases, and the rate with which it deceases, also decreases. A decreasing number at an increasing rate would "flatten out" instead.

But I'm no math teacher at all, and the question is ambiguous at best.

More interestingly, the graph predicts that in 2013, no americans will be smoking cigarettes. Hmm, could this be another subtle sign of the 2012 armageddon/ragnarok?

Yeah I would have answered the same as you - the magnitude of the rate of change is clearly getting bigger (thus, increasing). The rate of change is also clearly becoming more negative (thus, decreasing?).

I'd prefer "increasing" to mean "further away from zero" (regardless of sign), not "more positive" (like your teacher says), but I guess this use of the word is ambiguous. Do any of your texts provide a formal definition of "increasing"?

I think both answers are correct. For example, cowsarenotevil explained clearly how your interpretation is correct.

I believe, however, that your professor's interpretation of the sentence is different and that what he meant was to write "it is decreasing AND the rate of change is decreasing" (And if he wrote the question that way, then his answer would be right without any ambiguity).

I think the second interpretation is what will naturally come up in the mind of a trained mathematician, because it is the way in which we normally interpret these kind of sentences in more abstract settings. Poorly worded question indeed.

Quote: Original post by nullsquared
Rate of change only includes a sign to show direction; as such, if a rate of change is increasing, then it could be getting more positive (numerically increasing) OR more negative (numerically decreasing), and if a rate of change is decreasing, then it's getting close to 0, regardless of its sign.

That's not mathematically sound, however. If a value is increasing, then it is becoming more positive. If a value is decreasing, then it is getting more negative, regardless of what that value "represents". Terms like "increasing" and "decreasing" are defined very precisely in mathematics and their meaning is unambiguous, contrary to their counterpart in everyday english usage.

What you are saying would make perfect sense if you replaced "rate of change" with "the absolute value of the rate of change".

[Edited by - Hedos on May 25, 2010 11:12:00 PM]

Interestingly enough, it is a known problem.

And as per vox populi, (iii) is correct.

You even have a case:

Quote: Such students must never be marked down for failure to know another "language" that is so precisely opposite of their own, unless great pains are taken by the teacher to reeducate the students on the new math lingo, a process that will certainly aggravate students determined to uphold the integrity of their language.

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After some consideration, this is a well known problem in anything computer related, but there the majority vote has won long ago. It's what "regular user" speaks, that goes.

There are commonly two groups of people: the in crowd, and everyone else. People involved in a domain speak different language. "My thingy with button and blue broke that gizmo with thingamajig - ... oh, your email is broken again" Obviously, a better bug report would be: "The IMAP server is giving me 0x80000A2E error".

But that isn't happening.

Context should always be emphasized, and situations where ambiguity arises should be treated as special case and pointed out. I know of several courses where accurate terminology was required to pass, but the assignments were given out appropriate to that. There was nothing worse than getting an ambiguous problem description.


In this regard the newfangled mingling of hard science with populist presentation is a disaster. Areas should be treated as pure or populistic, not arbitrary mix of two.

Climate change is prime example of what happens if this is violated. In scientific terms, the results of studies were either inconclusive or "the qi parameter of n-dimensional temporal model was shown to have a negative second derivative, thereby confirming the hypothesis - up to domain experts to interpret what that means, or if it's significant".

[Edited by - Antheus on May 26, 2010 6:01:58 AM]

Quote: Original post by Antheus
Interestingly enough, it is a known problem.

That's such a good find that I just rated you up even though this thread has absolutely nothing to do with me.