
Portuguese: 
Temos muitos vídeos sobre
o teorema do valor médio,
mas eu vou revisar
isto um pouco,
para que possamos ver como
isto se conecta com o teorema
valor médio que aprendemos
em Cálculo Diferencial.
Como se conecta com
o que aprendemos
sobre valor médio de uma função
usando integrais definidas.
Assim, o teorema do valor médio nos diz
que se eu tenho alguma função f, que é
contínua, contínua no intervalo
fechado, então está
incluindo os pontos finais de a para b.
E é diferenciável,
é diferenciável,
de modo que a derivada é definida no
intervalo aberto de a para b,
então não necessariamente
precisa estar definida a
diferenciável nos limites desde que
seja diferenciável entre o limites.
Então nós sabemos.
Então sabemos que existe
algum valor, ou algum número ,

Bulgarian: 
Имаме много видео уроци за
теоремата за средните стойности,
но ще направя кратък преговор,
така че да можем
да видим връзката между
теоремата за средните стойности,
която учихме в 
диференциалното смятане,
как тя се свързва с това,
което учим за
средната стойност на функция
при определените интеграли.
Теоремата за средните
стойности твърди, че ако имаме
някаква функция, която е
непрекъсната в затворен интервал,
значи интервал, който включва
крайните точки, от а до b,
и е диференцируема, така че
производната ѝ
е определена в отворения
интервал от а до b,
не е задължително да е
диференцируема
в крайните точки, стига да е
диференцируема
между крайните точки,
тогава ние знаем, че

Korean: 
평균값의 정리에 관한 많은 강의가 있지만,
이번에 다시 복습해서
이 정리가 저희가 미분학에서 배운것과는
어떻게 연관되는지 보려고 합니다
그 정리가 정적분을 이용해 함수의
평균을 구하는 것과 연관시켜봅시다
평균값 정리는, f가 연속이고,
닫힌 구간에서 연속인,
a에서 b까지 끝점을 포함하는 구간을 말합니다
그리고 미분가능한, 미분 가능한
그러니까 도함수가 열린구간 a에서 b까지
정의되어 있을때, 그러니까
양 끝 구간에서는 정의되지 않아도 됩니다
양 끝 점에서의 미분가능성은
사이 구간에서 미분 가능하다면 상관 없습니다
그렇다면
어떤 값, 어떤 숫자가 존재하여,
구간에 존재하는 숫자,

Thai: 
เรามีวิดีโอหลายอันเรื่องทฤษฎีบทค่าเฉลี่ย
แต่ผมจะทบทวนสักหน่อย เราจะได้
รู้ว่ามันเชื่อมโยงกับทฤษฎีบทค่าเฉลี่ย
ที่เราเรียนในแคลคูลัสเชิงอนุพันธ์อย่างไร
มันเชื่อมโยงกับสิ่งที่เราเรียน
เรื่องค่าเฉลี่ยของฟังก์ชัน
โดยใช้อินทิกรัลจำกัดเขตอย่างไร
ทฤษฎีบทค่าเฉลี่ยบอกเราว่า ถ้าผมมี
ฟังก์ชัน f ที่ต่อเนื่องบนช่วงปิด
มันจะรวมจุดปลายด้วย จาก a ถึง b
และมันหาอนุพันธ์ได้ อนุพันธ์
นิยามบนช่วงเปิด จาก a ถึง b
มันไม่จำเป็นต้องหาอนุพันธ์ได้
ตรงขอบ ตราบใดที่มันหาอนุพันธ์ได้
ระหว่างขอบ แล้วเรารู้ว่า
มันมีค่า หรือจำนวน c ตัวหนึ่ง

English: 
- [Voiceover] We have many
videos on the mean value theorem,
but I'm going to review it
a little bit, so that we can
see how this connects
the mean value theorem
that we learned in differential calculus,
how that connects to what we learned about
the average value of a function
using definite integrals.
So the mean value theorem
tells us that if I have
some function f that is
continuous on the closed interval,
so it's including the
endpoints, from a to b,
and it is differentiable,
so the derivative
is defined on the open
interval, from a to b,
so it doesn't necessarily
have to be differentiable
at the boundaries, as long
as it's differentiable
between the boundaries, then we know that
there exists some value, or some number c,

English: 
such that c is between the
two endpoints of our interval,
so such that a < c < b, so
c is in this interval, AND,
and this is kind of the meat of it,
is that the derivative of
our function at that point,
you could use as the
slope of the tangent line
at that point, is equal
to essentially the average
rate of change over the
interval, or you could even
think about it as the slope
between the two endpoints.
So the slope between the
two endpoints is gonna be
your change in y, which is going to be
your change in your function value,
so f of b, minus f of a, over
b minus a, and once again,
we go into much more depth in this

Portuguese: 
algum número c tal que
c está entre os dois pontos
finais de nosso intervalo.
Então, a é menor que c que é menor
que b, c está, portanto, neste intervalo.
E esta é a parte mais importante.
A parte importante é que a derivada,
a derivada da nossa função naquele ponto,
a derivado da função naquele ponto,
vocẽ pode usar como a
inclinação da linha tangente
naquele ponto, é igual, essencialmente, a
taxa média da variação sobre o intervalo.
Ou você pode pensar nisto como a
inclinação entre os dois pontos finais.
A inclinação entre os dois pontos finais
será a variação em y que será
a variação do valor de sua função, então
será f de b menos f de a, sobre b menos a.
E mais uma vez fazemos
isto, nós entramos em

Bulgarian: 
съществува някаква стойност,
някакво число с,
такова, че е между крайните
точки на интервала,
значи а < c < b,
с принадлежи на този интервал, И
– това е същината –
производната на тази функция
в тази точка –
може да я разглеждаш като
наклона на допирателната в тази точка,
е равна на средната скорост
на изменение
в този интервал, или
можеш даже да я разглеждаш
като наклона между двете
крайни точки.
Значи наклонът между двете
крайни точки ще бъде равен
на изменението на у, което
ще бъде
изменението на стойността
на твоята функция.
Значи (f(b) – f(а))/(а – b).
Разгледахме това 
по-задълбочено,

Korean: 
이 구간에 존재하는c를
저희는 잡을 수 있습니다
a와 b 사이의 c를
그리고, 이게 핵심입니다
이 정리의 핵심은, 도함수가,
점 c에서의 도함수가,
여기서 c에서의 도함수를
c에서의 접선의 기울기라고 할 수 있습니다
접선의 기울기가 구간에서의
평균 기울기와 같도록 c를 잡을 수 있다
양 끝 점의 기울기라고도 볼 수 있습니다
그러므로, 양 끝 점의 기울기는
y의 값의 변화, 함수값의 변화인
f(b) 빼기 f(a)를 b 빼기 a로 나눈 것입니다

Thai: 
โดยที่ c อยู่ระหว่างจุดปลายสองจุดของช่วง
โดยที่ a น้อยกว่า c น้อยกว่า b
c อยู่ในช่วงนี้ และ
นี่คือเนื้อของทฤษฏีนี้
อนุพันธ์ของฟังก์ชันที่จุดนั้น
คุณมองมันเป็นความชันของเส้นสัมผัส
ที่จุดนั้นได้ เท่ากับค่าเฉลี่ย
อัตราการเปลี่ยนแปลงเฉลี่ย
ตลอดช่วงนั้น หรือคุณ
คิดถึงมันเป็นความชัน
ระหว่างจุดปลายสองจุดได้
ความชันระหว่างจุดปลายสองจุดจะเป็นการ
เปลี่ยนแปลงของ y ซึ่งก็คือ
การเปลี่ยนแปลงค่าฟังก์ชัน
f ของ b ลบ f ของ a, ส่วน b ลบ a
เราลงรายละเอียดเรื่องนี้

Thai: 
ตอนเราพูดถึงครั้งแรกในแคลคูลัสเชิงอนุพันธ์
แต่เพื่อให้คุณเห็นภาพ
เพราะผมว่ามันมีประโยชน์
ทฤษฎีบทค่าเฉลี่ยที่เราเรียนไป
ในแคลคูลัสเชิงอนุพันธ์
บอกเราว่า เฮ้ ดูสิ ถ้านี่คือ a นี่คือ b
ผมมีฟังก์ชันทำสิ่งที่น่าสนใจ
นี่คือ f ของ a นี่คือ f ของ b ปริมาณนี้
ตรงนี้ เราหาการเปลี่ยนแปลง
ค่าของฟังก์ชัน แล้วอันนี้ตรงนี้คือ
f ของ b ลบ f ของ a คือการเปลี่ยนแปลง
ของค่าฟังก์ชัน หารด้วยการเปลี่ยนแปลง
ของแกน x มันก็คือการเปลี่ยนแปลงของ y
ส่วนการเปลี่ยนแปลงของ x
มันจะให้ความชัน อันนี้ตรงนี้ให้
ความชันของเส้นตรงนี้ ความชันของเส้นตรง
ที่เชื่อมสองจุดนี้ นั่นคือปริมาณนี้
และทฤษฎีบทค่าเฉลี่ยบอกเราว่า มันมี c
ระหว่าง a กับ b ที่คุณมีความชันเท่ากัน
มันมีอย่างน้อยหนึ่งแห่ง มันอาจ

Bulgarian: 
когато за пръв път го срещнахме
в диференциалното смятане,
но само за илюстрация,
защото мисля, че винаги
е полезно:
теоремата за средните стойности,
която учихме в диференциалното смятане
ни казва, че...
това е а, това е b,
функцията прави нещо
интересно,
значи това е f(а), а това е f(b),
и тази стойност ето тук,
където имаме изменението на
стойността на функцията,
това точно тук е
f(b) – f(а), тази 
промяна на
стойността на функцията,
делена на изменението
по оста х, значи изменението
на у върху изменението на х,
това ни дава наклона,
ето тук ни дава
наклона на тази права,
която свързва
тези две точки,
тази стойност.
Теоремата за средната стойност
ни казва, че има някаква точка с
между a и b, където 
ще имаме същия наклон,
така че има ПОНЕ
едно място, може да е

Portuguese: 
muito mais detalhes
quando cobrimos isto
em cálculo diferencial,
mas só para lhe dar
uma visualização disto, porque
acho que é sempre útil.
O teorema do valor médio que aprendemos
em Cálculo Diferencial
apenas nos diz, olhe,
você sabe, se isto é a, e isto é b,
eu tenho minha função
fazendo algo interessante.
Então, isto é f de a, isto é f de b.
Portanto, esta quantidade bem aqui.
Onde está tomando a variação
do valor da função dividido.
Portanto, isto bem aqui é f de b.
f de b menos f de a é esta variação
no valor da nossa função,
dividida pela variação em nosso eixo x, é
nossa variação em y sobre variação em x.
Isso nos dá a inclinação, isto aqui
nos dá a inclinação desta linha.
A inclinação da linha que
conecta estes dois pontos.
Isto é essa quantidade, e este
teorema do valor médio
nos diz que há algum c entre
a e b onde você terá a mesma inclinação,
então poderá ser pelo menos um lugar.

English: 
when we covered it the first
time in differential calculus,
but just to give you
a visualization of it,
'cause I think it's always handy, the
mean value theorem that we
learned in differential calculus
just tells us, hey look,
if this is a, this is b,
I've got my function doing
something interesting,
so this is f of a, this is
f of b, so this quantity
right over here, where
you're taking the change
in the value of our function,
so this right over here is
f of b, minus f of a,
is this change in the
value of our function,
divided by the change
in our x-axis, so it's a
change in y over change in x,
that gives us the slope,
this right over here gives us
the slope of this line,
the slope of the line
that connects these two
points, that's this quantity,
and the mean value theorem
tell us that there's some c
in between a and b where you're
gonna have the same slope,
so it might be at LEAST
one place, so it might be

Korean: 
그리고 다시 깊이있게 본다면
이것을 미분학의 첫시간에 했지만
쉽게 보기 위해서 다시
그림을 그려보겠습니다
미분학에서 배운
평균값 정리는 다음을 알려줍니다
이게 a고, 이게 b일때,
흥미로운 점이 있다
이게 f(a), 이게 f(b)일때,
이곳의 값
함수값의 변화를 나누려면
여기 이곳이 f(b)입니다
f(b) 빼기 f(a)는 함수의 변화량, x축의 변화로 나누면
y의 변화량 나누기 x의 변화량입니다
이게 기울기, 여기 이것이 바로 기울기입니다
이 두점을 잇는 선분의 기울기는
평균값정리에 의하면
a와 b사이의 구간에 있는 점 c에서의
기울기와 똑같다고 말할 수 있습니다

English: 
right over there, where you
have the exact same slope,
there exists a c where the
slope of the tangent line
at that point is going to be
the same, so this would be
a c right over there, and
we actually might have
a couple of c's, that's
another candidate c.
There's at least one c where the slope
of the tangent line is the
same as the average slope
across the interval, and
once again, we have to assume
that f is continuous,
and f is differentiable.
Now when you see this, it
might evoke some similarities
with what we saw when
we saw how we defined,
I guess you could say, or the formula
for the average value of a function.
Remember, what we saw for the
average value of a function,
we said the average value
of a function is going to be
equal to 1 over b minus a,
notice, 1 over b minus a,
you have a b minus a in
the denominator here,
times the definite integral
from a to b, of f of x dx.
Now this is interesting, 'cause
here we have a derivative,

Portuguese: 
Poderia ser bem ali, onde você
tem a mesma inclinação.
Existe um c, em que a inclinação
da linha tangente nesse ponto será
a mesma, isto aqui seria ac,
nós, na verdade, 
teríamos um par de c's
É outro candidato a c.
Há pelo menos um c onde
a inclinação da linha
tangente é a mesma que a inclinação
média através do intervalo.
E, mais uma vez, temos que assumir que
f é contínua e f é diferenciável.
Agora, quando você ver
isto, isto pode
despertar algumas semelhanças
com o que vimos quando vimos
ou quando nós definimos,
acho que poderíamos dizer,
a fórmula para o valor
médio de uma função.
Lembre-se, o que vimos para o valor
médio da função, nós dissemos que,
o valor médio de uma função será
igual a um sobre de b menos a.
Note, um sobre b menos a,
você tem um b menos a no
denominador aqui, vezes a integral
definida de a até b de f de x dx.
Isto é interessante, porque aqui temos
uma derivada, aqui temos uma integral,

Korean: 
적어도 한 점이 있으므로, 같은 기울기인 여기겠네요
값 c가 존재하여, 접선의 기울기가 같을 것입니다
c는 여기 있겠네요
그리고 여러 개의 c가 존재할 수 있습니다
여기 또다른 c 후보가 있습니다
적어도 한 c가 존재하여, 접선의 기울기가
구간 전체에서의 평균 기울기와 같음을 알 수 있습니다
f가 연속이고, 미분가능할 때만 확신할 수 있습니다
그리고, 평균값 정리와 비슷한 것을 봤다는
생각이 들 것입니다
바로 함수의 평균을 정의할 때일 것입니다
기억할 것은, 함수의 평균을 정의할 때
함수의 평균은 1/(b-a)
1/(b-a)가 분모에 있고,
곱하기 a에서 b까지 f(x)의 정적분입니다
흥미롭게도, 여긴 도함수, 여긴 적분이 있습니다

Thai: 
อยู่ตรงนี้ ที่คุณมีความชันเท่ากันพอดี
มันมี c ที่ความชันของเส้นสัมผัส
ที่จุดนั้น จะเท่ากับ อันนี้จะ
เป็น c ตรงนี้ และเราอาจมี
c ได้หลายตัว นั่นอาจเป็น c อีกตัว
มันมี c อย่างน้อยหนึ่งตัวที่ความชัน
ของเส้นสัมผัสเท่ากับความชันเฉลี่ย
ตลอดช่วง ย้ำอีกครั้ง เราต้องสมมุติ
ว่า f ต่อเนื่อง และ f หาอนุพันธ์ได้
ทีนี้ เมื่อคุณเห็นอันนี้ 
คุณอาจเห็นความคล้ายคลึง
ที่เราเห็น เวลาเราเห็นว่าเรานิยามค่า
หรือสูตร
สำหรับค่าเฉลี่ยของฟังก์ชัน
นึกดู สิ่งที่เราเห็นสำหรับค่าเฉลี่ยของฟังก์ชัน
เราบอกว่า ค่าเฉลี่ยของฟังก์ชันจะ
เท่ากับ 1 ส่วน b ลบ a สังเกตว่า 1 ส่วน b ลบ a
คุณมี b ลบ a ในตัวส่วนตรงนี้
คูณอินทิกรัลจำกัดเขตจาก a ถึง b
ของ f ของ x dx
ทีนี้ อันนี้น่าสนใจ เพราะตรงนี้เรามีอนุพันธ์

Bulgarian: 
ето тук, където
имаме точно същия наклон,
съществува точка с, в която
наклонът на допирателната
в тази точка е равен...
това тук е с, и
всъщност може да има
няколко точки с, това е
друг кандидат за точка с.
Но има поне една точка с,
където наклонът
на допирателната е равен
на средния наклон
в целия интервал, и повтарям,
ние приемаме, че
функцията f е непрекъсната,
т.е. е диференцируема.
Когато видиш това, може би
това ти напомня
за случая, когато видяхме
как да дефинираме...
или бих казал на формулата
за средната стойност
на една функция.
Спомни си, че средната
стойност на една функция...
казахме, че средната
стойност на една функция е равна
на 1/(b – а), обърни внимание,
1/(b – а)
тук в знаменателя имаме b – а,
по определен интеграл от
а до b на f(х) dх.
Това е интересно, защото
тук имаме производна,

Portuguese: 
mas talvez se pudéssemos ligá-los, talvez
poderíamos conectar estas duas coisas.
Mas coisa que poderíamos dizer para você,
é que talvez pudéssemos reescrever, talvez
pudéssemos reescrever este numerador
aqui nesta forma, de alguma maneira.
E eu encorajo você a pausar o vídeo e ver
se você pode, e eu vou
lhe dar grande dica.
Ao invés de ser um f de x aqui, o que
acontece se houver um f linha de x?
Então, eu encorajo você a tentar de novo.
Mais uma vez, isto é, deixe-me reescrever.
Isto será igual a, isto aqui é
exatamente a mesma coisa que a
integral definida de a até b
de f linha de x dx.
Pense nisso.
Você tomará a anti-derivada de
f linha de x, que será f de x.
E você vai calculá-lo em b, f de b.
E depois, disto você irá subtrair
pelo calculado em a, menos f de a.
Estas duas coisas são idênticas.
E então você pode, é claro, dividi-lo.
Dividir por b menos a.

Thai: 
ตรงนี้เรามีอินทิกรัล แต่เราเชื่อมโยงพวกมันได้
เราอาจจะเชื่อมสองอย่างนี้ได้
สิ่งหนึ่งที่คุณอาจคิดได้ทันที
คือว่าเราอาจเขียน
ตัวเศษนี่ตรงนี้ใหม่ในรูปนี้ได้สักวิธี
และผมแนะนำให้คุณหยุดวิดีโอนี้ 
แล้วลองดูว่าคุณทำได้ไหม
และผมจะให้คำใบ้สำคัญ
แทนที่จะเป็น f ของ x ตรงนี้ จะเกิดอะไรขึ้นถ้ามี
f ไพรม์ของ x แทน ผมแนะนำให้คุณลองดู
เหมือนเดิม ขอผมเขียนทั้งหมดนี้ใหม่
อันนี้จะเท่ากับ --
อันนี้ตรงนี้เท่ากับ
อินทิกรัลจำกัดเขตจาก a ถึง b 
ของ f ไพรม์ของ x dx
คิดดู
คุณจะหาปฏิยานุพันธ์ของ f ไพรม์ของ x
ซึ่งก็คือ f ของ x และคุณจะ
หาค่ามันที่ b, f ของ b, แล้วจากนั้น
คุณจะลบพจน์นั้นหาค่าที่ a, ลบ f ของ a
สองตัวนี้เหมือนกันเป๊ะ
แล้ว คุณก็หารด้วย b ลบ a ได้

Bulgarian: 
тук имаме интеграл, но може би 
можем да намерим връзка между тях.
Може би можем да
намерим връзка между тях.
Може би тук ти хрумва, че
може би можем да преработим
този числител ето тук
някак до този вид.
Насърчавам те да спреш видеото
и да опиташ самостоятелно,
а аз ще дам една
голяма подсказка:
ако вместо f(х) тук е f'(х),
какво ще стане –
опитай да го направиш.
Сега отново ще напиша 
всичко това,
това ще е равно на...
Това ето тук е
съвсем същото като
определен интеграл от а до b
от f'(х)dх.
Помисли.
Ще намерим примитивната
функция на f'(х),
която е f(х), и после
трябва
да я изчислим за b,
f(b), от което
вадим стойността ѝ за а,
минус f(а).
Тези двете са идентични.
После можем да разделим 
на (b – а).

English: 
here we have an integral, but
maybe we could connect these.
Maybe we could connect these two things.
Well one thing that might jump out at you
is maybe we could rewrite
this numerator right over
here in this form somehow.
And I encourage you to pause
the video and see if you can,
and I'll give you actually
quite a huge hint,
instead of it being an f of x
here, what happens if there's
an f prime of x there, so I
encourage you to try to do that.
So once again, let me rewrite all of this,
this is going to be equal to...
This over here is the exact same thing as
the definite integral from
a to b, of f prime of x dx.
Think about it.
You're gonna take the
anti-derivative of f prime of x,
which is going to be f
of x, and you're going to
evaluate it at b, f of
b, and then from that
you're going to subtract it
evaluated at a, minus f of a.
These two things are identical.
And then, you can of
course divide by b minus a.

Korean: 
어쩌면 이 두 식을 연관지을 수 있을 것입니다
한 가지 들 수 있는 생각은, 다시 쓰면
분자를 이 형태로 다시 쓸 수 있습니다
영상을 잠깐 멈추시고, 찾을 수 있다면,
엄청난 힌트를 드리겠습니다
f(x)대신, f'(x)가 여기 왔다면 어떻게 될까요?
그러니, 다시 한 번 시도해보시길 바랍니다
이 모든 것을 다시 써보겠습니다
여기 이 식은 a부터 b까지의
f'(x)의 정적분과 같을 것입니다
생각해봅시다
f'(x)의 원함수를 잡으면, f(x)가 되겠죠
b에서의 값은, f(b)
여기서 f(a)를 뺍니다
이 두개가 동일합니다
그리고, 이것을

Bulgarian: 
Сега започва да става
интересно.
Единият начин да го разглеждаме
е, че трябва да има с,
което има средна стойност,
трябва да има такова с, че когато
намираш производната за с,
тя представлява средната
стойност на производната.
Друг начин да го разглеждаме е,
ако напишем g(х) = f'(х),
тогава сме много близко
до това, което получихме тук,
защото това ето тук 
ще бъде g(с),
запомни, че f'(с)
е равно на g(с),
е равно на 1/(b – а),
така че съществува с,
когато g(с) = 1/(b – а),

English: 
Now this is starting to get interesting.
One way to think about
it is, there must be a c
that takes on the average value of,
there must be a c, that when you
evaluate the derivative at c,
it takes on the average
value of the derivative.
Or another way to think about it,
if we were to just write g of
x is equal to f prime of x,
then we get very close to
what we have over here,
because this right over
here is going to be g of c,
remember, f prime of c is
the same thing as g of c,
is equal to 1 over b minus
a, so there exists a c
where g of c is equal to 1 over b minus a,
times the definite integral
from a to b, of g of x dx,

Korean: 
b-a로 나눕니다
이제 조금 흥미로워졌습니다
한가지로는, c가 무조건 있어야 합니다
평균값을 가지는 c가 있어야 합니다
c가 존재하여, c에서의 도함수를
계산하면, 도함수의 평균 꼴이 나와야 합니다
아니면, 직접 써보면
g(x)를 f'(x)로 잡아보면
여기 이 식과 매우 비슷한게 나옵니다
이 값이 g(c)가 되고, f'(c)가 g(c)입니다,
이는 1/(b-a)
1/(b-a), 그러니까 c가 존재하여, g(c)가 1/(b-a) 곱하기

Thai: 
ทีนี้ มันเริ่มน่าสนใจแล้ว
วิธีคิดอย่างหนึ่งคือว่า มันต้องมี c
ที่เท่ากับค่าเฉลี่ย
มันต้องมี c ที่หากคุณ
หาค่าอนุพันธ์ที่ c
มันจะเท่ากับค่าเฉลี่ยของอนุพันธ์
หรือวิธีคิดอีกอย่าง
ถ้าเราเขียน g ของ x เท่ากับ f ไพรม์ของ x
แล้วเราจะใกล้กับสิ่งที่เรามีตรงนี้มาก
เพราะอันนี้ตรงนี้จะเท่ากับ g ของ c
นึกดู f ไพรม์ของ c เท่ากับ g ของ c
เท่ากับ 1 ส่วน b ลบ a จึงมี c
ที่ g ของ c เท่ากับ 1 ส่วน b ลบ a
คูณอินทิกรัลจำกัดเขตจาก a ถึง b, 
ของ g ของ x dx

Portuguese: 
Agora, isto está começando
a ficar interessante.
Uma maneira de pensar sobre
isto é, deve haver um c,
deve haver um c que
assuma o valor médio.
Deve haver um c que, quando você calcula
a derivada em c, ela admite
o valor médio da derivada.
Uma outra maneira de
pensar sobre isso,
se fôssemos apenas escrever
g de x é igual a f linha de x,
então, chegaríamos muito
perto do que temos aqui.
Porque isto aqui será g de c,
lembre-se, f linha de c é a
mesma coisa que g de c,
é igual a um b menos a.
Então existe um c, onde g de c
é igual a um sobre b menos a,

Korean: 
a에서 b까지 g(x)의 정적분을 만족합니다
f'(x)이 g(x)입니다
다른 방법으로 생각해보면, 이건 사실
평균값 정리의 다른 형태인데, 적분을 이용한
평균값 정리입니다
적분을 이용한 평균값 정리
이게 평균이고, 앞글자만 쓰겠습니다
적분을 이용한 평균값 정리입니다
그리고, 더 정형화된 꼴을 쓰자면,
함수 g가 있을때
만약 g가, 더 깊은 내용으로 들어가면
g(x)가 연속, 닫힌 구간에서 연속이라면
a에서 b까지의 구간, 그러면 c가 존재하여
g(c)가, 뭐랑 같나요?

Bulgarian: 
по определен интеграл
от а до b от g(х)dх,
f'(х) е равно на g(х).
Друг начин да го разглеждаме е,
че това всъщност е
друг вид на теоремата
за средната стойност,
наречена теорема за средната
стойност за интеграли.
Ще запиша съкратено –
теорема за средната стойност за 
интеграли или за интегриране,
което, за да бъде
малко по-формално,
ако имаш функция g,
ако g е...
ще слеза малко надолу –
което ни казва, че
ако g(х) е непрекъсната
в този затворен интервал
от а до b, тогава съществува
точка с в този интервал,
в която g(с) е равна на...
какво е това?

Thai: 
f ไพรม์ของ x เท่ากับ g ของ x
วิธีคิดอีกอย่างคือว่า นี่ก็คือ
ทฤษฎีบทค่าเฉลี่ยอีกรูปหนึ่ง
มันเรียกว่าทฤษฎีบทค่าเฉลี่ยสำหรับอินทิกรัล
ผมจะเขียนตัวย่อนะ
ทฤษฎีบทค่าเฉลี่ยสำหรับอินทิกรัล 
หรือการอินทิเกรต
ซึ่งก็คือ เขียนอย่างเป็นทางการคือว่า
ถ้าคุณมีฟังก์ชัน g ถ้า g คือ
ขอผมลงมาข้างล่างหน่อย มันบอกเราว่า
ถ้า g ของ x ต่อเนื่องบนช่วงปิดนี้
จาก a ถึง b จะมี c ในช่วงนี้
ที่ g ของ c เท่ากับ อันนี้คืออะไร?

Portuguese: 
vezes a integral definida de a
até b, de g de x dx.
f linha de x é a mesma coisa que g de x.
Uma outra maneira de pensar nisto,
esta é, realmente, uma outra forma
do teorema do valor médio, é o chamado
teorema do valor médio para integrais.
Teorema do valor médio para integrais.
Então, este é o-- eu vou 
apenas escrever a sigla--
teorema do valor médio para
integrais, ou integração,
que essencialmente,
para lhe dar um sentido
um pouco mais formal, se você
tem alguma função g, então se g é,
na verdade, deixe-me descer um pouco.
Que nos diz que se g de x é contínua,
contínua neste intervalo fechado.
Indo de a até b, então existe
um c neste intervalo, onde
g de c é igual a, o que é isso?

English: 
f prime of x is the same thing as g of x.
So another way of thinking
about it, this is actually
another form of the mean value theorem,
it's called the mean value
theorem for integrals.
I'll just write the acronym,
mean value theorem for
integrals, or integration,
which essentially, to
give it in a slightly more
formal sense, is if you have
some function g, so if g is,
let me actually go down a
little bit, which tells us that
if g of x is continuous
on this closed interval,
going from a to b, then there
exists a c in this interval
where g of c is equal to, what is this?

Thai: 
นี่คือค่าเฉลี่ยของฟังก์ชันเรา
มันมี c โดยที่ g ของ c เท่ากับ
ค่าเฉลี่ยของฟังก์ชันเราบนช่วงนั้น
นี่คือนิยามค่าเฉลี่ยของฟังก์ชัน
เอาล่ะ นี่คือวิธีกล่าวทฤษฎีบท
ค่าเฉลี่ยของอินทิกรัลอีกวิธี
เพื่อแสดงว่ามันเกี่ยวข้องกันอย่างยิ่ง
มันใช้สัญลักษณ์คนละอย่างก็จริง แต่มัน
มันก็คือแนวคิดเดียวกัน
กับทฤษฎีบทค่าเฉลี่ยที่คุณเรียน
ในแคลคูลัสเชิงอนุพันธ์
แต่ด้วยสัญลักษณ์ใหม่
คุณก็ตีความได้ต่างออกไปเล็กน้อย
เรากำลังคิดถึงมันในแคลคูลัสเชิงอนุพันธ์
เราคิดถึงการมีจุด
ที่ความชันของเส้นสัมผัสฟังก์ชันที่จุดนั้น
เท่ากับอัตราเฉลี่ย นั่นคือ
ตอนเราคิดในแง่อนุพันธ์ และเรา
คิดในแง่ของความชัน 
และความชันของเส้นสัมผัส
ตอนนี้ ตอนเราคิดในแง่ปริพันธ์ เรากำลังคิด
ในแง่ของค่าเฉลี่ย
ค่าเฉลี่ยของฟังก์ชัน มันมี c
โดย g ของ c มันมี c

English: 
This is the average value of our function.
There exists a c where g of c is equal to
the average value of your
function over the interval.
This was our definition of the
average value of a function.
So anyway, this is just
another way of saying you might
see some of the mean value
theorem of integrals,
and just to show you that
it's really closely tied,
it's using different
notation, but it's usually,
it's essentially the
same exact idea as the
mean value theorem you learned
in differential calculus,
but now your different
notation, and I guess
you could have a slightly
different interpretation.
We were thinking about it
in differential calculus,
we're thinking about having a point where
the slope of the tangent line
of the function of that point
is the same as the average
rate, so that's when
we had our kind of differential
mode, and we were kind of
thinking in terms of slopes,
and slopes of tangent lines,
and now, when we're in
integral mode, we're thinking
much more in terms of average value,
average value of the
function, so there's some c,
where g of c, there's some c,

Bulgarian: 
Това е средната стойност
на нашата функция.
Съществува с, за което
g(с) е равно на
средната стойност на функцията
в този интервал.
Това беше нашето определение
за средна стойност на функция.
Това е просто друг начин
на формулиране, по който може
да видиш теоремата за
средната стойност на интегралите.
И само да ти покажа, че
те са наистина много близко свързани,
използват различен начин
на записване, но всъщност
това е една и съща идея като в
теоремата за средната стойност,
която учим в диференциалното смятане,
но сега се използва различен
начин за записване, което дава
малко по-различно
тълкуване.
Разглеждахме го в 
диференциалното смятане,
там разглеждахме една точка, където
наклонът на допирателната
на функцията в тази точка
е същият като средното изменение,
така че когато
сме в режим на диференциране,
и мислим за наклони,
за наклоните на допирателните,
а сега в режим на интегриране
ние разглеждаме
много повече средната стойност,
средната стойност
на функцията, така че тук има точка с,
в която g(с)...
има някаква точка с,

Korean: 
함수의 평균과 같음을 만족합니다
c가 존재하여 g(c)가 적분의
평균과 같아집니다
이게 함수의 평균값의 정의였습니다
그러니까, 이 방법은
적분을 이용한 평균값 정리를 알려주고
다른 방법으로 표기하는 것과
밀접한 관련이 있다는 것을 알려줍니다
하지만 보통은, 결국엔 미분학에서 배운
평균값 정리와 똑같은 개념입니다
표기 방법이 다를 뿐입니다
그리고 살짝 다르게 쓸 수도 있습니다
미분학에서 생각하고 있었습니다
어떤 점에서, 접선의 기울기가
평균 증가량과 같을 때를 생각했었습니다
여기서 미분적인 방법이었고, 저희는
기울기와, 접선의 기울기에 집중했습니다
이제 적분 방법은, 더 평균값과,
함수의 평균값에 집중합니다
어떤 c가 존재하여, c가 존재하여

Portuguese: 
Isto é o valor médio da função.
Existe um c em que
g de c é igual ao
valor médio de sua
função sobre a integral.
Esta foi a nossa definição de
valor médio de uma função.
Isto é apenas outra maneira de dizer
que você pode ver
alguns teoremas de
valor médio, e apenas
para lhe mostrar
que estão intimamente ligados,
estão usando notações diferentes.
Mas são, essencialmente,
as mesmas idéias do
teorema do valor médio
que você aprendeu no
Cálculo Diferencial, mas
agora com notação diferente.
E acho que poderia ter uma
interpretação diferente.
Estávamos pensando sobre
isso em Cálculo Diferencial.
Estávamos pensando em ter um ponto
onde a inclinação da linha tangente
da função naquele ponto
é a mesma que a taxa média.
Então, quando estávamos
no "modo diferencial",
pensávamos em termos de inclinações,
e inclinações das linhas tangentes.
E agora, quando estamos no
"modo integral", pensamos
mais em termos de valor médio,
valor médio da função.
Portanto, há algum c, onde g de c,

Portuguese: 
onde a função calculada neste
ponto é igual ao valor médio.
Então, uma outra maneira
de pensar sobre isto,
se eu fosse desenhar,
se eu fosse desenhar um g de x,
então, isto é x, este é o meu eixo y.
Este é o gráfico de y é igual a g de x,
que era a mesma coisa f linha de x,
nós apenas a reescrevemos agora para ser
mais consistente com nossa
fórmula de valor médio.
E estamos falando sobre
o intervalo de a até b.
Nós já vimos como calcular o valor médio.
Já vimos como calcular o valor médio,
portanto o valor médio talvez seja
isto aqui, que é g médio.
Então, o nosso valor médio é isto.
O teorema do valor médio
para integrais nos diz
que há algum c onde a nossa
função deve assumir, onde nossa

Bulgarian: 
където функцията, изчислена
в тази точка
е равна на средната стойност,
така че друг начин
да го разглеждаме е, че
ако начертая g(х),
това е оста х, това е
графиката на у = g(х),
което е същото нещо като
f'(х), но ние току-що
го преработихме,
за да бъде по-сходно
с формулата за средната стойност,
и разглеждаме интервала
от а до b.
Вече знаем как да изчислим
средната стойност,
така че може би средната
стойност е ето тук,
значи това е g средно, значи
средната стойност е това.
Теоремата за средната стойност 
на интегралите ни казва, че
има някакво с, където
нашата функция трябва да приеме

Korean: 
그 점에서의 함수값이 평균값과 같다
다른 말로,
만약 그려야 한다면,
g(x)를 그려야 한다면,
이게 x고, 이게 y축이 됩니다
이것은 y는 g(x)의 그래프입니다
이는 f'(x)와 같은 것이었죠
방금 저희는
평균값에 맞게 다시 썼습니다
그리고 구간 a에서 b에서 말하고 있습니다
평균값을 계산하는 방법은 이미 압니다
이미 평균값을 계산하는 방법을 아므로, 아마도
평균값은, 여기가 g의 평균일 것입니다
따라서 평균값은 이것입니다
적분을 이용한 평균값정리는 구간안에
c가 존재하여 평균값과 똑같은

Thai: 
โดยที่ฟังก์ชันหาค่าที่จุดนั้น
เท่ากับค่าเฉลี่ย วิธีคิด
อีกอย่างคือว่า ถ้าผมวาด g ของ x
นั่นคือ x นั่นคือแกน y นี่คือกราฟของ
y เท่ากับ g ของ x ซึ่งแน่นอนว่าเหมือนกับ
f ไพรม์ของ x แต่เราเขียนมันใหม่
ให้สอดคล้องกับสูตรค่าเฉลี่ย
และเราจะพูดถึงช่วงจาก a ถึง b
เราเห็นวิธีคำนวณค่าเฉลี่ยไปแล้ว
บางทีค่าเฉลี่ยคือค่านั่นตรงนั้น
แล้ว g เฉลี่ย ค่าเฉลี่ยของเราคือค่านี้
ทฤษฎีบทค่าเฉลี่ยสำหรับอินทิกรัลบอกเรา
ว่ามี c ฟังก์ชันของเรามีค่าตามนั้น

English: 
where the function
evaluated at that point,
is equal to the average
value, so another way
of thinking about it, if
I were to draw g of x,
that's x, that is my
y-axis, this is the graph of
y is equal to g of x, which
of course is the same thing
as f prime of x, but we've
just rewritten it now
to be more consistent with
our average value formula,
and we're talking about
the interval from a to b,
we've already seen how to
calculate the average value,
so maybe the average value
is that right over there,
so that is g average, so
our average value is this,
the mean value theorem for
integrals just tells us
there's some c where our
function must take on

English: 
that value at c, whereas that c is inside,
where the c is in that interval.

Portuguese: 
função deve assumir este
valor em c, considerando que
c está dentro, onde c
está neste intervalo.
Legendado por [Raul Guimaraes].

Korean: 
함수값을 가져야 함을 말합니다

Thai: 
ที่ c โดย c นั้นอยู่ข้างใน
โดย c อยู่ในช่วงนั้น

Bulgarian: 
такава стойност за с, като
с принадлежи на този интервал.
