Need some understanding.

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xepher
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Need some understanding.

Postby xepher » Mon May 03, 2010 2:10 pm UTC

I've been reading through a calc book for a while, and I'm having trouble understanding two things: the law of the mean and L'Hopital's rule. I don't get either of them.

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OverBored
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Re: Need some understanding.

Postby OverBored » Mon May 03, 2010 2:32 pm UTC

What particularly are you having issue with. The result or the method to get there?
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Yakk
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Re: Need some understanding.

Postby Yakk » Mon May 03, 2010 8:35 pm UTC

How did you approach these things -- epsilon-delta approach? Hand waving?

The law of the mean states that, given a continuous function, the area under the curve in some interval is the width of the interval times some height that the continuous function actually attains in that interval.

In short, if I drove 1 thousand km in 8 hours, then the area under my speed/time curve is 1 thousand km. That means at some point I must have been going at 1000/8 = 125 km/h. It is possible that I was moving at exactly 125 km/h the entire time, or I could have been going slower at some point (and faster at others).

l'Hopital's rule is a rule about what happens when you take the limit of f(x)/g(x) at some point a where f(x)->0 and g(x)->0.

It states that the limit is the limit of f'(x) / g'(x) (the derivative of f and g, both with respect to x) as x goes to the same spot a, if that limit exists. Together with the result that "if your function is continuous and g is non-zero at a, then limit as x goes to x of f(x)/g(x) = f(a)/g(x)", it makes it easy to evaluate entire categories of limit.

This is a quite powerful result. You can see how it sort of makes sense: with f' and g' being the slopes of f and g, as you approach the point a the ratio of the functions gets close to the ratio of their slopes! As you are really close to a, f(x) approximates 0 + f`(a)*x, and g(x) approximates 0 + g'(a)*x. It goes further than this, which justifies recursively using l'Hopitals rule in the case where the derivatives also end up evaluating to 0.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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dean.menezes
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Re: Need some understanding.

Postby dean.menezes » Mon May 03, 2010 10:29 pm UTC

Yakk wrote:l'Hopital's rule is a rule about what happens when you take the limit of f(x)/g(x) at some point a where f(x)->0 and g(x)->0.


Actually it also works if g(x) tends to either +∞ or to -∞.

To see why L'hopital's rule works, multiply both the numerator and denominator by 1/(x-a):
[math]{ f(x) \over g(x) } = {f(x)/(x-a) \over g(x)/(x-a)}[/math]
Then if f(a) -> 0 and g(a) -> 0 we can rewrite this as:
[math]{f(x) - f(a) \over (x - a)} \over {g(x) - g(a) \over (x - a)}[/math]
which is the same as f'(x) / g'(x). This doesn't cover the case where g(x) tends to +∞ or to -∞ and it also doesn't show why you can use the rule recursively.


Also "L'Hôspital's rule" has an circumflex on the o.

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Eebster the Great
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Re: Need some understanding.

Postby Eebster the Great » Tue May 04, 2010 6:32 am UTC

dean.menezes wrote:Also "L'Hôspital's rule" has an circumflex on the o.

Don't you mean "L'Hôpital's rule?"

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Yakk
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Re: Need some understanding.

Postby Yakk » Tue May 04, 2010 5:37 pm UTC

Hopital bought the theorem. Unlike Möbius, who earned the use of the Umlaut, I don't think paying a mathematician to name a theorem after you is worthy of bothering with character codes. :)

http://en.wikipedia.org/wiki/L%27Analys ... es_Courbes
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: Need some understanding.

Postby lu6cifer » Tue May 04, 2010 6:07 pm UTC

Eebster the Great wrote:
dean.menezes wrote:Also "L'Hôspital's rule" has an circumflex on the o.

Don't you mean "L'Hôpital's rule?"


Actually, I've seen it spelled either way.
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Re: Need some understanding.

Postby Syrin » Tue May 04, 2010 6:35 pm UTC

Yakk wrote:Hopital bought the theorem. Unlike Möbius, who earned the use of the Umlaut, I don't think paying a mathematician to name a theorem after you is worthy of bothering with character codes. :)

http://en.wikipedia.org/wiki/L%27Analys ... es_Courbes


I always thought he published anonymously?

lu6cifer wrote:Actually, I've seen it spelled either way.

I've seen "piqued" spelt "peaked" - doesn't make it correct.

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Re: Need some understanding.

Postby Token » Tue May 04, 2010 6:37 pm UTC

Yakk wrote:Hopital bought the theorem. Unlike Möbius, who earned the use of the Umlaut, I don't think paying a mathematician to name a theorem after you is worthy of bothering with character codes. :)

I'm confused as to why the story you give is nothing like the one on the wiki page you linked...
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Yakk
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Re: Need some understanding.

Postby Yakk » Tue May 04, 2010 7:31 pm UTC

Yes, I was being facetious. Exaggerating for the sake of effect. Being silly.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Syrin
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Re: Need some understanding.

Postby Syrin » Tue May 04, 2010 10:34 pm UTC

Humour isn't allowed on a board created for mathematical discussion. Outright silliness is deplorable.

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Yakk
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Re: Need some understanding.

Postby Yakk » Tue May 04, 2010 11:58 pm UTC

Doing a bit of the old knock knock. On the paddle-wilky. Twiddling my toes in the shallow end.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Eebster the Great
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Re: Need some understanding.

Postby Eebster the Great » Wed May 05, 2010 3:08 am UTC

Yakk wrote:Hopital bought the theorem. Unlike Möbius, who earned the use of the Umlaut, I don't think paying a mathematician to name a theorem after you is worthy of bothering with character codes. :)

http://en.wikipedia.org/wiki/L%27Analys ... es_Courbes


Ah, but umlauts are at a premium, while circumflex diacritics are a dime a dozen.

AllSaintsDay
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Re: Need some understanding.

Postby AllSaintsDay » Thu May 06, 2010 6:25 pm UTC

Eebster the Great wrote:
dean.menezes wrote:Also "L'Hôspital's rule" has an circumflex on the o.

Don't you mean "L'Hôpital's rule?"

L'Hôpital and L'Hospital are both correct. L'Hôspital is not. The circumflex, among other uses, came into French as shorthand for "there's an s after this vowel that we don't pronounce anymore."
For example, the Latin word noster became French nôtre and the English word isle comes from Old French that became the modern île. Timingwise, the contemporary spelling of his name was L'Hospital.


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