MATH debate: does X^3 increase? Options

[Blade]

Paying attention to x=0, where d/dx(x^3) = 0, along with the graph does X^3 increase in all points?

And, as related, should the definition of increasing functions f(x) be that f'(x) > 0 or f'(x) >= 0?

State why for each answer.

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[Crosis]

I'd say it's constantly increasing because if you go a tiny bit to the right of zero it goes up, and if you go a tiny bit to the left it goes down (in other words, the secant lines always have positive slope). For me, as long as it's a single point and not a region where f' = 0, it's still increasing. It has to be on some interval where f' = 0 for me to consider it non-increasing there.

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I'd be happy to stop contradicting you, as soon as you start being right.

[big hungry joe]

I'd say this is a matter of the definition.

Monotonically increasing: x>=y implies f(x)>=f(y)
Strictly increasing: x>y implies f(x)>f(y)

Another definition might be:

Bladian increasing: df(x)/dx > 0, for all x


Pick your definition...

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

Edited by - big hungry joe on Nov 21 2001 6:26:00 PM

[Blade]

quote:

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I'd say it's constantly increasing because if you go a tiny bit to the right of zero it goes up, and if you go a tiny bit to the left it goes down (in other words, the secant lines always have positive slope). For me, as long as it's a single point and not a region where f' = 0, it's still increasing. It has to be on some interval where f' = 0 for me to consider it non-increasing there.

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Do you htink the definition of increasing should hence involve >=?

quote:

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Monotonically increasing: x>=y implies f(x)>=f(y)
Strictly increasing: x>y implies f(x)>f(y)

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Clarify what you mean by those terms.

quote:

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Bladian increasing: df(x)/dx > 0, for all x

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. So I'm an advocate of the ">" instead of the ">="? Why?

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[big hungry joe]

Monotonically increasing: x>=y implies f(x)>=f(y)
So for all x and all y, if x>=y, then it MUST be true that f(x)>=f(y).

f(x)=x^3
Let x=y+e (e being epsilon, a small positive number), so x>y
f(y)=y^3
f(x)=x^3=y^3+3ey^2+3ye^2+e^3
So f(x)>f(y)
Now let e=0
f(y)=y^3
f(x)=x^3=y^3
f(x)=f(y)

So the function is monotonically increasing. But moreover, you can see that from t he first part of the proof, where e was positive, f(x) is strictly increasing as well, a subset of all monotonically increasing functions.

Bladian increasing is when df(x)/dx > 0 for all x. (and as to why I picked that I just did more or less randomly)

f(x) is *not* Bladian increasing, for reasons you showed in the first post.

Increasing is all in how you define it. In the math world, dictionary definitions are often worthless - you have to strictly define each definition. The word 'increasing' can mean many things. So mathematicians have many different types of 'increasing' - each one meaning a single thing

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

[Blade]

Many mathematical terms and concepts are defined by everyday-used words because of an intuitive connection between the mathematical and real world regarding it. I'm sure we all agree as to an idea of increase, and the issue is to have we should define it in math.

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[big hungry joe]

quote from Blade:

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Many mathematical terms and concepts are defined by everyday-used words because of an intuitive connection between the mathematical and real world regarding it. I'm sure we all agree as to an idea of increase, and the issue is to have we should define it in math.

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My point was, the word 'increase' in English is very vague, mathematically speaking. Has my father's income increased over the last 30 years? Yes. Has there ever been a period in which it has gone down? Yes.

Math definitions need to be totally precise. I see no reason we must need the exact word 'increase' to map to a specific definition, and I see less need to argue about it. For purposes of a certain topic, sure, let's agree that when you say 'increasing' you mean monotonically increasing. In general though, I don't see a need for it.

I'm sure there are lots who disagree with me; the above was just my opinion on the matter.

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

[Blade]

quote from bhj:

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My point was, the word 'increase' in English is very vague, mathematically speaking. Has my father's income increased over the last 30 years? Yes. Has there ever been a period in which it has gone down? Yes.

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Of course. This does make the question, or at least the heart of the question, difficult to determine.

quote:

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Math definitions need to be totally precise. I see no reason we must need the exact word 'increase' to map to a specific definition, and I see less need to argue about it. For purposes of a certain topic, sure, let's agree that when you say 'increasing' you mean monotonically increasing. In general though, I don't see a need for it.

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Is there any special definition for increasing to be used generally that willbe more convenient and effective to use?

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[big hungry joe]

To your latter point, I don't know. I'd lean towards increasing meaning strictly increasing, though.

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

[Blade]

As in Monotonical or Bladistic?

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[big hungry joe]

quote from Blade:

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As in Monotonical or Bladistic?

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No, as in strictly increasing, where x>y implies f(x)>f(y).

It is possible to be strictly increasing yet not Bladian (like f(x)=x^3 demonstrates) but IMHO the Bladian definition doesn't add anything interesting to its 'increasingness'. I think the interesting property is that for any x and y, x>y implies f(x)>f(y) and it can do that while having df(x)/dx=0 at point(s).

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

[Blade]

How?
(btw what specifically do you refer to by f(y), are we not merely discussing whether f(x) is increasing?)

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[big hungry joe]

Let f(x)=x^3.

Then:
f(z)=z^3
f(2)=2^3 = 8
f(5)=5^3 = 125
f( (2x+3) ) = (2x+3)^3 = ...whatever that expands out to

It's basically variable replacement. The 'x' is unimportant as anything other than a variable placeholder. The 'f' is the name of the function, and the <variable>^3 is what it evaluates to. 'x' is a 'generic' variable. I could as easily say let f(Q)=Q^3

You ask why a function can be strictly increasing and yet still have df(x)/dx=0 at a point - I already showed in a previous reply, I'll do it again:

f(x)=x^3

Let y=z+e (where e is a small positive number)
Note that y>z

f(z) = z^3
f(y) = y^3 = (y+e)^3 = y^3 + 3*y^2*e + 3*y*e^2 + e^3

So it is clear, since e>0, that f(y)>f(z)

So f is strictly increasing. And yet it still has df/dx=0 at a point.

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

Edited by - big hungry joe on Nov 22 2001 01:17:23 AM

[Crosis]

I take kind of a middle road with Blade's definitions. According to BHJ, I'd say "strictly increasing." The slope of the secant lines is always positive, even with one end at 0, so I'd say it's always increasing. It's just that it doesn't increase much around 0.

As for Blade's definitions, if it is only a point at which f'(x)=0 and it goes positive on both sides, then it's increasing. If it's a larger region (if the function has a multi-part definition and one part's a constant over an interval, for example), then it's not increasing there.

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I'd be happy to stop contradicting you, as soon as you start being right.

[Blade]

quote from bhj:

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No, as in strictly increasing, where x>y implies f(x)>f(y).

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Are there any benefits for one as opposed to another in applications?
It seems that the "Bladian" definition is probably more useful for purposes like finding inflection points, etc.


Btw, my friend(who posted post 11, I let him post a few posts while I was at the bathroom a week ago) wishes to know which method, strictly or d/dx is accepted generally by mathematicians. AFAIK, there is wide debate but the latter is generally accepted for "increasing." Is this correct?

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All human actions are equivalent … and … all are on principle doomed to failure. - Jean-Paul Sartre
Man is a useless passion - Jean-Paul Sartre
All the cruelty and torment of which the world is full is in fact merely the necessary result of the totality of the forms under which the will to live is objectified. - Arthur Schopenhauer
Let us beware of saying that death is the opposite of life. The living being is only a species of the dead, and a very rare species.
- Friedrich Nietzsche

[big hungry joe]

I have no idea which is accepted as the definition for the word 'increasing'. To tell the truth I really doubt any single definition is, and that is why there are so many. In my math classes, a prof has outlined what he will mean when he says 'increasing', and he only uses the word 'increasing' because it is a pain to say montonically increasing or strictly increasing every time. I've never heard of a single definition for 'increasing' - perhaps go ask on the sci.math newsgroup.

As far as which is the most important, I have no idea. I believe that Bladian increasing is a subset of strictly increasing, and strictly increasing is a subset of montonically increasing

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...but that they would not understand, that the whole thing would be regarded as a momentary aberration whereas in truth it was my fate. -- Demian

Edited by - big hungry joe on Nov 25 2001 9:49:29 PM

[weezer]

Looks like I stepped into the wrong forum again...sorry.

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