
Thai: 
 
ลองดูถ้าเราจะเข้าใจเรื่อง
ทฤษฎีบทค่าเฉลี่ยโดยสัญชาตญาณได้ไหม
อย่างที่เราจะเห็น เมื่อคุณเข้าใจ
ภาษาและสัญลักษณ์ทางคณิตศาสตร์แล้ว
มันจะเป็นทฤษฎีที่ค่อนข้างตรงตามสัญชาตญาณ
ลองคิดถึงฟังก์ชัน f ตัวหนึ่ง
สมมุติว่าผมมีฟังก์ชัน f
และเรารู้ข้อมูลบางอย่างเกี่ยวกับฟังก์ชันนี้
เรารู้ว่ามันต่อเนื่องตลอดช่วงปิด
ระหว่าง x เท่ากับ a กับ x เท่ากับ b
แล้วเมื่อเราใส่วงเล็บเหลี่ยมตรงนี้
มันหมายความว่าเป็นช่วงปิด
เมื่อเราใส่วงเล็บเหลี่ยมตรงนี้
มันหมายความว่าเรารวมจุด a
และถ้าผมใส่วงเล็บเหลี่ยมทางขวามือ
แทนที่จะเป็นวงเล็บโค้ง นั่นหมายความว่าเรา
รวมจุด b ด้วย
และต่อเนื่อง หมายความว่าเราไม่

Czech: 
Podívejme se, zda rozumíme
Větě o střední hodnotě.
A jak uvidíme, až si rozebereme
některé matematické pojmy a symboly,
jedná se o docela
intuitivní větu.
Pojďme si představit
nějakou funkci f.
A o této funkci víme
několik věcí.
Víme, že je spojitá na
uzavřeném intervalu,
kde x je mezi 'a' a 'b'.
Když použijeme tyto závorky,
mluvíme o uzavřeném intervalu.
Tedy zahrnujeme bod 'a' z této
strany a bod 'b' z pravé strany.
Pozn. překladatele: v češtině
označujeme jako .
A spojitá znamená
jen to,

English: 
Let's see if we
can give ourselves
an intuitive understanding
of the mean value theorem.
And as we'll see, once you parse
some of the mathematical lingo
and notation, it's actually
a quite intuitive theorem.
And so let's just think
about some function, f.
So let's say I have
some function f.
And we know a few things
about this function.
We know that it is
continuous over the closed
interval between x equals
a and x is equal to b.
And so when we put
these brackets here,
that just means closed interval.
So when I put a
bracket here, that
means we're including
the point a.
And if I put the bracket on
the right hand side instead
of a parentheses,
that means that we
are including the point b.
And continuous
just means we don't

Korean: 
평균값 정리에 대해 알아보겠습니다
수학 용어와 기호에 대해 잘 이해한다면
직관적으로 알 수 있는 정리입니다
함수 f에 대해 이야기해봅시다
함수 f가 있다고 하고
이 함수에 대해 몇 가지 사실들이 주어집니다
a와 b 사이 닫힌 구간에서 연속이고
대괄호는
닫힌 구간이라는 의미입니다
여기에 대괄호를 써서
점 a를 포함시키고
오른쪽에도 대괄호를 쓰면
점 b도 포함시킨다는 의미입니다

Bulgarian: 
Да видим дали можем
да разберем добре
теоремата за средните стойности.
И както ще видим, щом веднъж разгледаме добре някои от математическите понятия
и символиката, това ще е една доста разбираема за нас теорема.
Та нека разгледаме дадена функция f.
Да кажем, че имам някаква функция f.
Знаем няколко неща за тази функция.
Знаем, че тя е непрекъсната в рамките на затворения
интервал между х, равно на а, и х, равно на b.
И когато поставим тези скоби тук,
това означава затворен интервал.
Когато напиша скобата така, това
означава, че включваме точката а.
А ако поставя скобата от дясната страна, вместо
кръгли скоби, това означава, че
включваме точката b.
И "непрекъсната" функция означава, че

Portuguese: 
Veremos se podemos entender intuitivamente
o teorema do valor médio.
E,como veremos, uma vez que você processar
parte dos termos matemáticos e da notação,
verá que é um teorema bastante intuitivo.
Vamos pensar sobre uma função f.
Digamos que tenho uma função f
e sabemos algumas coisas 
sobre esta função.
Sabemos que é contínua 
sobre o intervalo fechado
entre x igual a a e x igual a b.
Quando colocamos estes colchetes 
quer dizer que o intervalo é fechado.
Quando coloco um colchete aqui,
significa que estou incluindo o ponto a.
E se coloco o colchete à direita, 
em vez de parênteses,
significa que estamos incluindo o ponto b.

English: 
have any gaps or jumps in
the function over this closed
interval.
Now, let's also assume that
it's differentiable over
the open interval
between a and b.
So now we're saying,
well, it's OK
if it's not
differentiable right at a,
or if it's not
differentiable right at b.
And differentiable
just means that there's
a defined derivative,
that you can actually
take the derivative
at those points.
So it's differentiable over the
open interval between a and b.
So those are the
constraints we're
going to put on ourselves
for the mean value theorem.
And so let's just try
to visualize this thing.
So this is my function,
that's the y-axis.
And then this right
over here is the x-axis.
And I'm going to--
let's see, x-axis,
and let me draw my interval.
So that's a, and then
this is b right over here.

Korean: 
그리고 연속은 a와 b 사이에서 
함수가 끊기지 않고이어진다는 뜻입니다
또한 열린 구간 (a,b)에서 미분 가능하다고 합시다
이 말은,
점 a에서는 미분가능하지 않아도 되며
b에서도 가능하지 않아도 된다는 뜻입니다
미분 가능하다는 말은 a와 b 사이에서 
도함수가 존재한다는 뜻입니다
a와 b 사이 열린 구간에서 미분 가능하고
이것이 평균값 정리를 쓸 수 있는 조건입니다
그림으로 나타내볼게요
이것이 함수의 y 축이고
여기는 x 축 입니다
그리고 구간을 그려볼게요
여기는 a로 잡고,
여기를 b로 잡을게요

Portuguese: 
Continuidade significa que não temos
nenhuma descontinuidade ou saltos
neste intervalo fechado.
Também vamos assumir que a função
é diferenciável no intervalo aberto
entre a e b.
Estamos dizendo que não há problema
se a função não é diferenciável em a
ou em b.
E diferenciável significa que há 
uma derivada definida; que você pode
calcular a derivada nesses pontos.
Então, a função é diferenciável
no intervalo aberto entre a e b.
Essas são as condições que vamos impor
para o teorema do valor médio.
Vamos tentar visualizar isto.
Esta é minha função, este é o eixo y
e este é o eixo x. Deixe-me desenhar
meu intervalo. Isso é a e isto aqui é o b.

Thai: 
มีช่องว่างหรือการกระโดดในฟังก์ชันตลอดช่วงปิดนี้
 
ทีนี้ ลองสมมุติด้วยว่ามันหาอนุพันธ์ได้ตลอด
ช่วงเปิดระหว่าง a กับ b
ตอนนี้ เราจะบอกว่า มันไม่เป็นไร
ถ้ามันหาอนุพันธ์ไม่ได้ที่ a
หรือถ้ามันหาอนุพันธ์ไม่ได้ที่ b
และการหาอนุพันธ์ได้ หมายความว่า
มันอนุพันธ์ที่นิยามได้ ที่คุณ
หาอนุพันธ์ที่จุดเหล่านั้นได้จริงๆ
มันหาอนุพันธ์ได้ตลอดช่วงเปิดระหว่าง a กับ b
พวกนั้นคือเงื่อนไขที่เรา
จะกำหนดในทฤษฎีบทค่าเฉลี่ย
ลองมามองภาพอันนี้กัน
นี่คือฟังก์ชันของผม นั่นคือแกน y
แล้วค่านี่ตรงนี้คือแกน x
และผมจะ -- ลองดู แกน x
ขอผมวาดช่วงของผมนะ
นั่นคือ a แล้วนี่ก็คือ b ตรงนี้

Czech: 
že funkce nemá žádné mezery nebo
skoky na tomto uzavřeném intervalu.
Pojďme dále předpokládat,
že je funkce diferencovatelná
na otevřeném intervalu
mezi body 'a' a 'b'.
Teď tedy říkáme,
že je v pořádku,
pokud funkce není diferencovatelná
přímo v bodech 'a' a 'b'.
Diferencovatelná znamená,
že existuje definovaná derivace.
Že funkci mezi těmito
body lze derivovat.
Je tedy diferencovatelná na
otevřeném intervalu (a;b).
Toto jsou tedy omezení,
se kterými budeme pracovat
ve Větě o střední hodnotě.
Pojďme si to zkusit
představit.
Toto je moje funkce.
Toto je moje osa y
a zde je osa x.
A nakreslím náš interval.
Toto je a, toto je b.

Bulgarian: 
нямаме никакви празноти или прескачания във функцията в рамките на този затворен
интервал.
Сега, нека също приемем, че функцията е диференцируема в рамките
на отворения интервал между а и b.
Сега казваме, ами, хубаво е ако
не е диференцируема точно в а,
или ако не е диференцируема точно в b.
A диференцируема означава, че е налице
определена проиводна, така че можем всъщност
да намерим производната в тези точки.
Така че функцията е диференцируема в рамките на отворения интервал между а и b.
A това са ограниченията, които
ще си наложим при теоремата за средните стойности.
Нека се опитаме да си представим нагледно това.
Това е моята функция, това е ординатната ос.
И тогава това тук е абсцисната ос.
И аз ще - да видим, абсцисната ос,
нека начертая моя интервал.
Така, това е а, а това тук е b.

English: 
And so let's say our function
looks something like this.
Draw an arbitrary
function right over here,
let's say my function
looks something like that.
So at this point right over
here, the x value is a,
and the y value is f(a).
At this point right
over here, the x value
is b, and the y value,
of course, is f(b).
So all the mean
value theorem tells
us is if we take the
average rate of change
over the interval,
that at some point
the instantaneous rate
of change, at least
at some point in
this open interval,
the instantaneous
change is going
to be the same as
the average change.
Now what does that
mean, visually?
So let's calculate
the average change.
The average change between
point a and point b,
well, that's going to be the
slope of the secant line.

Bulgarian: 
И нека нашата функция е нещо такова.
Тук чертая произволна функция,
да кажем, че моята функция изглежда така.
Така че в тази точка тук, стойността на х е а,
а стойността на у е f (а).
f(a).
В тази точка тук, стойността на х
е b, а стойността на у, естествено, е f(b).
f(b).
Така че всичко, което ни казва теоремата за средните стойности,
е това, че ако вземем средната степен на изменение
в рамките на интервала така, че в дадена точка
моменталната степен на изменение, поне
в някаква точка от този отворен интервал,
моменталното изменение
ще е същото като средното изменение.
Как можем да представим това нагледно?
Нека пресметнем средното изменение.
Средното изменение между точка а и точка b,
ами, това ще е наклонът на пресечната права.

Czech: 
A řekněme, že funkce
vypadá nějak takto.
Můžeme nakreslit
libovolnou funkci.
Řekněme, že moje funkce
vypadá nějak takto.
V tomto bodě je hodnota
na ose x rovna a.
A hodnota na
ose y je f(a).
V tomto bodě je hodnota
na ose x rovna b
a hodnota na
ose y je rovna f(b).
To, co nám Věta o
střední hodnotě říká je,
že když vezmeme průměrnou
změnu na intervalu,
tak v nějakém bodě, alespoň v
nějakém z tohoto otevřeného intervalu,
je okamžitá rychlost změny stejná 
jako průměrná rychlost změny.
Teď si znázorníme,
co to znamená.
Pojďme spočítat
průměrnou změnu.
Průměrná změna mezi
bodem 'a' a bodem 'b'

Thai: 
และสมมุติว่าฟังก์ชันของเราเป็นแบบนี้
วาดฟังก์ชันใดๆ ตรงนี้
สมมุติว่าฟังก์ชันของผมเป็นแบบนั้น
ที่จุดนี่ตรงนี้ ค่า x คือ a
และค่า y คือ f(a)
 
ที่จุดนี่ตรงนี้ ค่า x
คือ b และค่า y แน่นอนคือ f(b)
 
ทฤษฎีบทค่าเฉลี่ยบอกเรา
ว่าถ้าเรานำอัตราการเปลี่ยนแปลงเฉลี่ย
ตลอดช่วง ที่จุดหนึ่ง
อัตราการเปลี่ยนแปลงชั่วขณะ อย่างน้อย
หนึ่งจุดในช่วงเปิดนี้
อัตราการเปลี่ยนแปลงชั่วขณะ จะ
เท่ากับอัตราการเปลี่ยนแปลงเฉลี่ย
แล้วมันหมายความว่ายังไงในแง่ของภาพ?
ลองคำนวณอัตราการเปลี่ยนแปลงเฉลี่ยกัน
การเปลี่ยนแปลงเฉลี่ยระหว่างจุด a กับจุด b
มันจะเท่ากับความชันของเส้นตัด

Korean: 
그리고 함수가 이렇게 생겼다고 할게요
이 점의 x 값은 a이고
y 값은 f(a)입니다
이 점의 x 값은 b이고
y 값은 f(b)입니다
평균값 정리는 a와 b 사이 구간의 평균변화율과
어떤 점에서의 순간 변화율이
열린 구간 사이 적어도 한 점에서는
순간 변화율이 평균 변화율과 같을 것입니다
시각적으로 나타내볼게요
평균 변화율을 구해봅시다
점 a와 점 b 사이 평균 변화율은

Portuguese: 
Digamos que minha função
parece algo assim.
Neste ponto aqui, o valor de x é a 
e o valor de y é f de a.
Neste ponto, o valor de x é b
e o valor de y é, obviamente, f de b.
Tudo que o teorema do valor médio diz
é que se calcularmos a taxa de variação
média do intervalo, em algum ponto
a taxa de variação instantânea
será igual a taxa de variação média.
O que isso significa visualmente?
Vamos calcular a taxa de variação média.
A variação média entre o ponto a
e o ponto b será o coeficiente angular

Portuguese: 
da reta secante. Esta é a reta secante.
Pense sobre o coeficiente angular.
O que o teorema do valor médio nos diz
é que em algum ponto deste intervalo
o coeficiente angular instantâneo
da reta tangente será igual
ao coeficiente angular da reta secante.
Podemos observar que aqui
parece que o coeficiente angular
da reta tangente é igual
ao coeficiente angular da reta secante.
E parece que aqui também 
o coeficiente angular
da reta tangente é igual ao coeficiente
angular da reta secante.
Isso faz sentido intuitivamente.
Em algum ponto, o seu
coeficiente angular instantâneo
será igual ao seu coeficiente
angular médio. Como escreveríamos isso
matematicamente? Vamos calcular
o coeficiente angular 
médio deste intervalo.
O coeficiente angular médio 
deste intervalo, ou a taxa
de variação média - o coeficiente angular
da reta secante - será nossa diferença

Thai: 
 
มัน -- นี่คือเส้นตัด
คิดถึงความชันกัน
ทฤษฎีบทค่าเฉลี่ยบอกเรา
ว่า ที่จุดหนึ่งในช่วงนี้
ความชันของเส้นสัมผัส
จะเท่ากับความชันของเส้นตัด
และเราเห็นได้ จากภาพ ดูเหมือนว่าตรงนี้
ความชันของเส้นสัมผัส มันดูจะเท่ากับ
ความชันของเส้นตัด
มันดูเหมือนว่าจะเป็นตรงนี้ด้วย
ความชันของเส้นสัมผัสเท่ากับความชัน
ของเส้นตัด
และมันตรงตามสัญชาตญาณ
ที่จุดหนึ่ง ความชันชั่วขณะของคุณ
จะเท่ากับความชันเฉลี่ย
แล้วเราเขียนในทางคณิตศาสตร์ได้ว่าอะไร?
ลองคำนวณอัตราการเปลี่ยนแปลงเฉลี่ย
ตลอดช่วงนี้กัน
ความชันเฉลี่ยตลอดช่วงนี้
หรือการเปลี่ยนแปลงเฉลี่ย ความชันของเส้นตัด
จะเท่ากับการเปลี่ยนแปลงของ y 
-- การเปลี่ยนแปลงของ y
ตรงนี้ -- ส่วนการเปลี่ยนแปลงของ x

Bulgarian: 
Наклонът на пресечната права.
Така че това е - това е пресечната права.
Нека помислим за нейния наклон.
Всичко, което ни казва теоремата за средните стойности,
е това, че в някаква точка от този интервал,
моментният наклон на допирателната
ще е същият като наклона на пресечната права.
И можем да видим нагледно, че това изглежда така,
наклонът на допирателната изглежда е същият
като този на пресечната права.
Прилича на този случай тук.
Наклонът на допирателната е равен на наклона
на пресечната права.
И това е нещо разбираемо.
В дадена точка, моменталният наклон
ще е същият като средния наклон.
Как ще запишем това математически?
Ами нека пресметнем средния наклон
на този интервал.
Средният наклон на този интервал,
или средното изменение, наклона на пресечната права,
ще е изменението в у - нашето изменение в у
тук на това място - върху изменението в х.

Korean: 
할선의 기울기겠죠
이것이 할선이고
기울기를 생각해봅시다
평균값 정리는 이 구간 사이 어떤 점에서의
접선의 기울기가 할선의 기울기와 같아진다는 것입니다
여기쯤이 되겠네요
할선과 접선의 기울기가 같아집니다
여기도 같아지겠네요
접선의 기울기는 할선의 기울기와 같아집니다
직관적인 감으로도 알 수 있습니다
어떤 한 점에서의 순간변화율은 평균변화율과 같아질 것입니다
수학적으로는 어떻게 정리할까요?
일단 구간의 평균변화율을 구해봅시다
평균변화율은 할선의 기울기와 같습니다

Czech: 
bude sklon sečny.
Toto je tedy sečna.
Vše, co nám Věta o
střední hodnotě říká, je,
že v nějakém bodě
tohoto intervalu
bude okamžitý sklon tečny
roven sklonu této sečny.
A pohledově můžeme
vidět, že zde to vypadá,
že skon tečny bude
stejný jako sklon sečny.
Také to vypadá, na
stejný případ zde.
Sklon tečny bude zde
stejný jako sklon sečny.
A to intuitivně
dává smysl.
V nějakém bodě bude okamžitý
sklon stejný jako průměrný sklon.
Jak bychom toto
zapsali matematicky?
Pojďme spočítat průměrný
sklon na tomto intervalu.
Průměrná změna,
sklon sečny,

English: 
So that's-- so this
is the secant line.
So think about its slope.
All the mean value
theorem tells us
is that at some point
in this interval,
the instant slope
of the tangent line
is going to be the same as
the slope of the secant line.
And we can see, just visually,
it looks like right over here,
the slope of the tangent line
is it looks like the same
as the slope of the secant line.
It also looks like the
case right over here.
The slope of the tangent
line is equal to the slope
of the secant line.
And it makes intuitive sense.
At some point, your
instantaneous slope
is going to be the same
as the average slope.
Now how would we write
that mathematically?
Well, let's calculate
the average slope
over this interval.
Well, the average slope
over this interval,
or the average change, the
slope of the secant line,
is going to be our change
in y-- our change in y
right over here--
over our change in x.

Portuguese: 
em y sobre nossa diferença em x.
Qual é nossa diferença em y?
A nossa diferença em y
é f de b menos f de a
- sobre nossa diferença em x-
sobre b menos a.
Farei isso em vermelho.
Vamos recordar o que está acontecendo.
Isto aqui é o gráfico de y,
que é igual a f de x.
Estamos dizendo que o coeficiente angular
da reta secante, ou nossa taxa de variação
média do intervalo entre a e b,
é nossa variação em y
--a letra grega delta significa variação--
sobre nossa variação em x.
Que, obviamente, é igual a isso daqui.
O teorema do valor médio nos diz

Czech: 
bude změna y zde
lomeno změnou x.
Jaká je naše změna y?
Změna y je
f(b) minus f(a).
Toto celé lomeno
změna x.
Tedy lomeno
b minus a...
Napíši to
správnou barvou.
...Připomeňme si,
o co tady jde.
Toto zde je graf
funkce y rovno f(x).
Říkáme, že sklon sečny,
neboli průměrná změna
na intervalu (a;b)
je změna y...
toto je řecké písmeno delta,
označení pro změnu,
...lomeno změna
na ose x.
Což je samozřejmě
rovno tomuto.
A Věta o střední
hodnotě nám říká,

Bulgarian: 
Върху изменението в х.
А какво е изменението в у?
Изменението в у е f(b) минус f(a),
и това ще е върху изменението в х.
Върху b минус b минус а.
Ще оцветя това в червено.
Така, нека си припомним какво се случва тук.
Това, което виждаме, е графиката на у,
равно на f(x).
Казваме, че наклонът на пресечната права,
или средната стойност на изменението в интервала от а до b,
представлява изменението в у - гръцката буква делта служи
за кратък запис на изменение в у -върху изменението в х.
...
Което, естествено, е равно на това.
И теоремата за средните стойности ни казва,

English: 
Well, what is our change in y?
Our change in y is
f(b) minus f(a),
and that's going to be
over our change in x.
Over b minus b minus a.
I'll do that in that red color.
So let's just remind ourselves
what's going on here.
So this right over here,
this is the graph of y
is equal to f(x).
We're saying that the
slope of the secant line,
or our average rate of change
over the interval from a to b,
is our change in y-- that the
Greek letter delta is just
shorthand for change in
y-- over our change in x.
Which, of course,
is equal to this.
And the mean value
theorem tells us

Korean: 
x 변화율 분의 y 변화율이 되겠죠
y 변화율은 어떻게 구할까요?
y 변화율은 f(b) - f(a)입니다
이것을 x 변화율로 나누면 됩니다
b - a로 나누면 됩니다
지금까지 한 것을 정리해볼게요
이것은 y = f(x)의 그래프입니다
할선의 기울기
즉, a와 b 사이의 평균변화율은
y의 변화율
이 그리스 알파벳 델타는 변화율을 나타냅니다
나누기 x의 변화율입니다
그리고 이 값과 같아요
평균값 정리는

Thai: 
 
ทีนี้ การเปลี่ยนแปลงของ y คืออะไร?
การเปลี่ยนแปลงของ y คือ f(b) ลบ f(a)
และมันจะเป็น ส่วนการเปลี่ยนแปลงของ x
ส่วน b ลบ, b ลบ a
ผมจะเขียนด้วยสีแดงนั่นนะ
ลองทบทวนสิ่งที่เกิดขึ้นตรงนี้
รูปนี่ตรงนี้ นี่คือกราฟของ y
เท่ากับ f(x)
เรากำลังบอกว่า ความชันของเส้นตัด
หรืออัตราการเปลี่ยนแปลงเฉลี่ย
ตลอดช่วงจาก a ถึง b
คือการเปลี่ยนแปลงของ y -- ตัวอักษรกรีกเดลต้า
ก็แค่ตัวย่อการเปลี่ยนแปลงของ y --
ส่วนการเปลี่ยนแปลงของ x
 
ซึ่ง แน่นอน เท่ากับค่านี้
และทฤษฎีบทค่าเฉลี่ยบอกเรา

Thai: 
ว่ามันมี -- ถ้าเรารู้ว่าฟังก์ชันเป็นไปตาม
สองข้อนี้ แล้วจะมี
ค่า x บางค่าระหว่าง a กับ b
ในช่วงเปิดระหว่าง a กับ b จะมี c ค่าหนึ่ง
จะมี c ค่าหนึ่ง และเราบอกได้
ว่ามันเป็นสมาชิกของช่วงเปิดระหว่าง a กับ b
 
หรือเราบอกได้ว่า มี c ที่ a น้อยกว่า c
ซึ่งน้อยกว่า b
มี c ในช่วงนี้
c อยู่ระหว่างพวกมัน โดยที่อัตราการเปลี่ยนแปลง
ชั่วขณะที่ค่า x นั้น เท่ากับ
อัตราการเปลี่ยนแปลงเฉลี่ย
มี c บางค่าในช่วงเปิดนี้
ที่อัตราการเปลี่ยนแปลงเฉลี่ย

English: 
that there exists-- so
if we know these two
things about the
function, then there
exists some x value
in between a and b.
So in the open interval between
a and b, there exists some c.
There exists some
c, and we could
say it's a member of the open
interval between a and b.
Or we could say some c
such that a is less than c,
which is less than b.
So some c in this interval.
So some c in between it
where the instantaneous rate
of change at that
x value is the same
as the average rate of change.
So there exists some c
in this open interval
where the average
rate of change is

Portuguese: 
que existe -- então se sabemos
essas duas coisas sobre a função--
um valor de x entre a e b. Então,
no intervalo aberto entre a e b
existe um valor c. E podemos dizer
que pertence ao intervalo aberto
entre a e b. Ou podemos dizer que algum
valor c tal que a é menor que c e
c é menor que b. Algum valor c onde
a taxa de variação instantânea
nesse valor x é igual a taxa
de variação média.
Então existe um c neste intervalo aberto
no qual a taxa de variação média é igual

Korean: 
만약 함수에 대해 이 두 가지를 알고 있다면
a와 b 사이의 어떤 x 값에서
즉, 열린구간 a와 b 사이 어떤 점c가
또는 c는 a보다 크고 b보다 작다고 할 수 있습니다
이 구간 (a,b)에 속하는 어떤 점c에서의 접선의 기울기는 평균변화율과 같습니다

Czech: 
že když víme o těchto
dvou vlastnostech funkce,
tak zde existuje nějaká
hodnota x, mezi body 'a' a 'b'.
Tedy v otevřeném intervalu (a;b)
existuje nějaká hodnota c.
Můžeme říci, že hodnota c
leží na otevřeném intervalu (a;b).
Nebo můžeme říci, že existuje hodnota c,
kde a je menší než c a to je menší než b.
Tedy nějaké c
v tomto intervalu,
kde okamžitá změna
v této hodnotě x
je stejná jako průměrná
rychlost změny.
Existuje tedy nějaké c
v tomto otevřeném intervalu,

Bulgarian: 
че съществува- ако знаем тези два
елемента на функцията, тогава
съществува някаква стойност на х между a и b.
И в отворения интервал между a и b съществува дадено с.
Съществува дадено с, и можем
да кажем, че то е член на отворения интервал между а и b.
Между а и b.
Или можем да кажем, че с, такова, че а е по-малко от с,
което пък е по-малко от b.
Та дадено c е в този интервал.
Дадено c е тук вътре, където моментната степен
на изменението за тази стойност на х е равна
на средната степен на изменението.
Един вид в този отворен интервал съществува дадено c,
за което средната степен на изменението

Czech: 
kde průměrná rychlost změny je rovna
okamžité rychlosti změny v tomto bodě.
To je celé, co
věta říká.
A jak jsme viděli na
tomto grafu,
toto by mohlo být naše c nebo
toto by také mohlo být naše c.
Podívejme se, máme
f spojitou na uzavřeném intervalu ,
f diferencovatelnou na
otevřeném intervalu (a;b)
a pak platí tento zápis.
Co to vlastně
znamená?
Vše, co nám
to říká je,
že v nějakém bodě tohoto intervalu
je okamžitá rychlost změny
stejná jako průměrná rychlost
změny na celém intervalu.
V dalším videu vám zkusím ukázat
reálný příklad, kde to využít.

Portuguese: 
a taxa de variação instantânea
nesse ponto. É tudo o que está dizendo.
Assim como vimos neste diagrama,
este poderia ser o nosso c. Ou este.
Você diria que f é continua
em a e b, no intervalo fechado,
e diferenciável no intervalo aberto.
e você vê toda esta notação.
O que isso nos diz?
Diz que em algum ponto no intervalo,
a taxa de variação instantânea será igual
a taxa de variação média
sobre todo o intervalo
No próximo vídeo, tentaremos
dar um exemplo da vida real
sobre quando isso faz sentido.
[legendado por: Pilar]

English: 
equal to the instantaneous
rate of change at that point.
That's all it's saying.
And as we saw this diagram right
over here, this could be our c.
Or this could be our c as well.
So nothing really--
it looks, you
would say f is continuous over
a, b, differentiable over-- f
is continuous over the closed
interval, differentiable
over the open interval, and
you see all this notation.
You're like, what
is that telling us?
All it's saying is at some
point in the interval,
the instantaneous
rate of change is
going to be the same as
the average rate of change
over the whole interval.
In the next video,
we'll try to give you
a kind of a real life example
about when that make sense.

Korean: 
이 말은, 구간 사이의 평균변화율은 
어떤 점 c에서의 순간변화율과 같을 것입니다
이 그래프에서 여기가 c가 될 수도 있고
또는 여기가 c가 될 수도 있습니다
함수 f는 a와 b사이 닫힌 구간에서 연속이고
열린 구간에서 미분가능하고
수학적으로 표기한 것을 보면
여러분들은 이것이 무엇을 의미하나요?라고 물을 것입니다
정리하자면,
어떤 한 점에서의 순간변화율은 
이 구간 어딘가에서 평균변화율과 같아진다는 것입니다
다음 강의를 통해 평균값 정리에 대해 좀더 알아봅시다

Thai: 
เท่ากับอัตราการเปลี่ยนแปลงชั่วขณะ ณ จุดนั้น
นั่นคือสิ่งที่มันบอก
และเราเห็นจากแผนภาพตรงนี้ นี่เป็น c ของเราได้
หรือนี่คือ c ของเราได้
ไม่มีอะไร -- มันดู
คุณบอกได้ว่า f ต่อเนื่องบนช่วง a, b
หาอนุพันธ์ได้ตลอดช่วง -- f
ต่อเนื่องบนช่วงปิด หาอนุพันธ์ได้
บนช่วงเปิด และคุณเห็นสัญลักษณ์ทั้งหมดนี้
คุณอาจจะถามว่า มันบอกอะไรเรา?
มันก็แค่บอกว่า ที่จุดหนึ่งบนช่วง
อัตราการเปลี่ยนแปลงชั่วขณะ
จะเท่ากับอัตราการเปลี่ยนแปลงเฉลี่ย
ตลอดช่วงทั้งหมด
ในวิดีโอหน้า เราจะพยายาม
ยกตัวอย่างในชีวิตจริง ว่ามันใช้ได้เมื่อใด

Bulgarian: 
е равна на моментната степен на изменението в тази точка.
Това е всичко, което се казва.
И както видяхме тази диаграма тук, това може да е споменатото c.
Или и това може да е c.
И реално нищо - изглежда
можем да кажем, че f е непрекъсната между а и b, диференцируема в - f
е непрекъсната в рамките на затворения интервал, диференцируема
в отворения интервал, вижда се всичката символика тук.
И в крайна сметка какво се има предвид това?
Това означава, че в дадена точка от интервала,
моментната степен на изменението
ще е равна на средната степен на изменението
в целия интервал.
Следващия път ще се опитаме да дадем
един реален пример от живота, когато това има смисъл.
