Real Analysis | The Generalized Mean Value Theorem and One part of L'Hospital's rule.

[@goodplacetostop2973]

Oh, empty set for that video... So you love the empty set, uh? Name three of its elements then.

[@myrus5722]

, ,and

[@ujjwal2473]

What you know about your great great grandpa, what you don't know about your great great grandpa, what you know about Illuminati

[@ganeshbhawal6118]

You are best teacher. Your teaching inspires me. Thank you so much. Keep up the good work 👍🏻

[@heliocentric1756]

I feel betrayed 😣 Where is That's good place to stop??

[@tgx3529]

LHospital If all understand, it is assumed here that the functions f, g are the derivative of the neighbourhood of ​​point a. If this property is also expected to point a, so we can write lim((f(x)-f(a))/(x-a))/lim((g(x)-g(a)/(x-a))=f'(a)/g'(a).

[@huubmooren2553]

Your videos are highly appreciated Michael! L'Hopital seems to contradict the direct trig approach in the limit ((sin(a)-sin(b))/((tan(a)-tan(b)) when a goes to b. I must do something wrong here..

[@buttkicksHORIZON13]

thanks for carrying me through my exams (havent done em but I feel more confident now :D)

[@okhan5087]

Great video. Very informative.

[@aashleshbhaskar7380]

why don't use LMVT rule, F ' (C)={F(b)-F(a)}/b-a and G ' (C)= {G(b)-G(a)}/b-a and then divide them to get the required result.

[@marceloborjasbernaola7167]

the c's could be different

[@aashleshbhaskar7380]

@@marceloborjasbernaola7167 gotcha !

[@MountPanda]

I think the proof of L'Hospital's rule is not entirely valid - in the last equality step where you go from lim f'(a_n) / g'(a_n) to lim f'(x) / g'(x) you are assuming the two functions f' and g' to be continuous - i.e. f,g are not only differentiable, but continously differentiable (at a). Wikipedia says that the more general case holds, but the proof is (likely) more involved.

[@dbmalesani]

Indeed, the proof seems to assume continuous derivatives. The Wikipedia page even singles out this special case as simpler: en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule#Special_case

[@carl3260]

I'm not sure about this. It is assumed that lim h(x) = L, where h(x)= f'(x)/g'(x), i.e. the limit of h(x) at a exists, so (by sequential def of limit) for any sequence {a_n}, a_n=/=a, that converges to a, {h(a_n)} converges to L. The proof creates such a sequence. Key thing is to consider h as a whole, not f', g' separately. It seems a more rigorous way of using GMVT to say lim_{x->a} (f(x)-f(a))/(g(x)-g(a)) = lim_{c->a}f'(c)/g'(c), which is L by def (where c's exists by GMVT and converge, to a, by "squeeze" rule).

[@MountPanda]

@@carl3260 Yeah I think you're right, I think it's more natural to think of the equations going from right to left than left to right.

[@anakinkylo.thepomenerianan9084]

Cauchy's mean value theorem

[@unonovezero]

So that was not a good place to stop

[@adad-nerari4117]

Thanks for these explanations,but I wonder what is the difference between a rule and a theorem.

[@ziruitao1281]

1:57: "if g'(c) NOT equal to 0"

[@CM63_France]

Hi, 0:56 : you say that the fact that f(a)=f(b) "makes the average change value = 0", I don't agree. For fun: 3 "let's may be go ahead", including 1 "let's may be go ahead and do that", 1 "let's go ahead and", 1 "so I'll go ahead and", 1 "so now I'll go ahead and", 1 "let's go ahead", 1 "the next thing that I want to do", "that finishes this proof" instead of "that's a good place to stop".

[@jonaskoelker]

> you say that the fact that f(a)=f(b) "makes the average change value = 0", I don't agree. f is the antiderivative of its derivative. But then if you integrate f' from a to b you get f(b) - f(a) = 0 (by assumption). So the summed/integrated total change is zero (f' is the change in f). By any reasonable definition of average, the average is also zero. If you (still) disagree, could you articulate why?

[@mrflibble5717]

Brilliant! A very useful result, explained as usual, with clarity and careful detail. Thank you Michael.

[@datsmydab-minecraft-and-mo5666]

can we generalize this to other h(x)?

[@wjun0131]

But where's a good place to stop?

[@elgourmetdotcom]

Where’s the good place to stop?? 😱

[@JalebJay]

He probably had a 3rd theorem he wanted to prove, but instead decided to cut it off early.

[@elgourmetdotcom]

Jaleb yeah perhaps the other L’Hôpital’s case

[@stephenbeck7222]

Nicolás Ángel Damonte or the more general form of the 0/0 case, which doesn’t require continuity or defined function values at x=a but only in the deleted neighborhood.

[@Rob-oj9bj]

Maybe it's not a good place to stop? We'll have to watch more of his videos until it is.

[@abderrahmanyousfi5565]

👍🏻👍🏻

[@momed_nikon]

didn't understand anything