
Korean: 
 
지난 영상에서 함수의
도함수를 그리는 법을 배웠습니다
이번 영상의 목적은
함수의 원시함수를 그리는 것입니다
 
원시함수는
미분하기 전의 함수를
일컫는 단어입니다
보통 f(x)의 원시함수는
대문자 F를 사용하여
F(x)로 표기합니다
앞으로도 이 방법으로
원시함수를 표시할 것입니다
F(x)의 특징은
미분하였을 때 원래의 함수
f(x)가 되는 것입니다
그래프와 같이
f(x)가 주어졌을 때

Portuguese: 
No último vídeo, nós vimos uma função
e tentamos desenhar sua derivada.
Agora, neste vídeo, vamos
olhar uma função
e tentar desenhar sua antiderivada.
Antiderivada soa como
uma palavra muito chique,
mas é o mesmo que dizer que
a antiderivada de uma função
é uma função cuja a
derivada é aquela função.
Assim, por exemplo,
se temos f de x,
e dizemos que a 
antiderivada de f de x
é F maiúsculo de x.
E isto tende a ser a 
notação, quando você
fala sobre uma antiderivada.
Isto apenas significa que a derivada
de F de x, que é igual a, 
você poderia dizer
F linha de x,
é igual a f de x.
Vamos tentar fazer isto aqui,
temos nosso f de x.
E nós vamos tentar
pensar sobre qual

Bulgarian: 
В последното видео 
разгледахме една функция
и опитахме да начертаем
нейната производна.
В това видео ще разгледаме
една функция
и ще опитаме да начертаем
нейната примитивна функция.
Това звучи много сложно,
но означава просто, че
примитивната функция
е такава функция, на която
нашата функция е производна.
Например, ако имаме f(х),
и нека да кажем, че примитивната
функция на f(х)
е F(х) (главно F).
Това е начинът, по който се записва 
примитивна функция.
Това просто означава, че 
производната
на F(х), която е F'(х)
е равна на f(х).
Сега тук ще опитаме...
имаме графиката на f(х)
и ще опитаме да определим
коя

Czech: 
V předchozím videu jsme měli funkci
a snažili jsme se nakreslit její derivaci.
V tomto videu máme zadanou funkci
a pokusíme se k ní nakreslit
primitivní funkci.
Zní to honosně,
ale ve skutečnosti to znamená:
Pro primitivní funkci k dané funkci platí,
že její derivace je ta původní funkce.
Například, mějme funkci f(x) a řekněme,
že její primitivní funkcí je F(x).
To je konvenční značení primitivní funkce.
To znamená,
že derivace F(x) je rovna f(x).
O co se budeme snažit zde je,
že pro funkci f(x) budeme hledat funkci,

Spanish: 
en el pasado video observamos la funcion e intentamos dibujar las lineas de la derivada
ahora en este video obersevaremos la funcion e intentaremos dibujar su antiderivada
que suena como una palabra elegante
pero solo digamos que la antiderivada es una funcion cuya derivada es esa funcion
asi que por ejemplo si tenemos f(x)
y digamos que su antiderivada de f(x) es (F) mayuscula de (x)
esto significa, literalmente significa
que la derivada de (F) mayuscula, lo cual es igual a (F) prima mayuscula
es igual a f(x), entonces lo que tratamos de hacer aqui
tenemos nuestra f(x) y vamos a tratar de pensar

Thai: 
 
ในวิดีโอที่แล้ว เราได้ดูฟังก์ชัน
แล้วพยายามหาอนุพันธ์ของมัน
ทีนี้ ในวิดีโอนี้ เราจะดูฟังก์ชัน
แล้วพยายามวาดปฏิยานุพันธ์ของมันบ้าง
 
ซึ่งฟังดูเป็นคำหรูหรามาก
แต่มันก็แค่บอกว่า ปฏิยานุพันธ์ของฟังก์ชัน
คือฟังก์ชันที่มีอนุพันธ์เท่ากับฟังก์ชันนั้น
ตัวอย่างเช่น ถ้าเรามี f ของ x
และสมมุติว่าปฏิยานุพันธ์ของ f ของ x
เท่ากับ F ใหญ่ของ x
อันนี้มักเป็นสัญลักษณ์ที่ใช้ เวลาคุณ
พูดถึงปฏิยานุพันธ์
นั่นหมายความว่าอนุพันธ์
ของ F ใหญ่ของ x ซึ่งเท่ากับ คุณจะบอกว่า
F ใหญ่ไพรม์ของ x ก็ได้ เท่ากับ f ของ x
สิ่งที่เราพยายามทำตรงนี้คือ เรามี f ของ x
และเราพยายามหาว่า

English: 
In the last video we
looked at a function
and tried to draw
its derivative.
Now in this video, we're
going to look at a function
and try to draw
its antiderivative.
Which sounds like
a very fancy word,
but it's just saying the
antiderivative of a function
is a function whose
derivative is that function.
So for example,
if we have f of x,
and let's say that the
antiderivative of f of x
is capital F of x.
And this tends to be the
notation, when you're
talking about an antiderivative.
This just means
that the derivative
of capital F of x, which is
equal to, you could say capital
f prime of x, is
equal to f of x.
So we're going to try to do
here is, we have our f of x.
And we're going to try
to think about what's

Bulgarian: 
евентуално е функцията, на
която това може да е производна.
Ще изучаваш това много
по-задълбочено,
когато стигнеш до
интегралното смятане.
Има много възможни
функции,
за които това може
да е тяхна производна.
Целта ни в това видео е просто
да начертаем една
убедителна възможност.
Да поразсъждаваме.
Нека тук отгоре да начертая
у = F(х).
Ще се опитаме да начертаем
функцията, за която е възможно
производната ѝ да изглежда 
по този начин.
Това, което правим, когато
отиваме от тази графика
до тази графика тук,
е намирането на производната.
Да помислим как може
да изглежда тази функция.
Когато гледаме тази производна
в този първи интервал ето тук,
ще го направя в цикламено,
това е в този интервал
от х = 0 до тази стойност
на х ето тук,
виждаме, че наклонът 
е постоянен, равен е на +1.

Thai: 
ฟังก์ชันอะไรมีอนุพันธ์เท่านี้ได้บ้าง?
และคุณจะได้เรียนอย่างละเอียด
เวลาคุณเริ่มดูแคลคูลัสเชิงปริพันธ์
แต่มีฟังก์ชันที่เป็นไปได้มากมาย
ที่ฟังก์ชันนี้เป็นอนุพันธ์ได้
และเป้าหมายในวิดีโอนี้คือแค่
วาดตัวหนึ่งที่เป็นไปได้
ลองคิดกันหน่อย
ตรงนี้ ด้านบน วาด
y เท่ากับ F ใหญ่ของ x
สิ่งที่เราพยายามวาด
คือฟังก์ชันที่มีอนุพันธ์
เป็นแบบนี้
สิ่งที่เราจะทำ คือว่า
เมื่อเราไปจากสิ่งที่เราวาดตรงนี้
เรากำลังหาอนุพันธ์
ลองคิดดูว่าฟังก์ชันนี้เป็นอย่างไรได้
เวลาเราดูอนุพันธ์นี้
มันบอกว่า ตลอดช่วงนี้ ตลอดช่วงแรกนี่ตรงนี้
ขอผมใช้สีม่วงนะ
มันบอกว่า ตลอดช่วงนี้จาก x
เท่ากับ 0 ไปจนถึงค่า x นี้
ตรงนี้ มันบอกว่าความชันเท่ากับ
บวก 1 คงที่

Czech: 
která by po zderivování dala f(x).
Toto prozkoumáte mnohem hlouběji
v integrálním počtu matematické analýzy.
Je zde ve skutečnosti více možných funkcí,
které po zderivování dají tu samou f(x).
Naším cílem je nakreslit
nějakou hezkou možnost.
Zamysleme se nad tím.
Zde nahoru budeme kreslit
y je rovno F(x).
Snažíme se nakreslit takovou funkci,
jež po derivování vypadá takto.
V podstatě děláme to,
že vezmeme, co je nahoře,
a derivujeme to.
Zamysleme se nad tím,
jak taková funkce může vypadat.
Podíváme-li se na derivaci…
Na tomto intervalu…
Udělám to fialově.
Na tomto intervalu od 0 až po tento bod
je sklon roven kladné konstantě 1.

Portuguese: 
função possível poderia ser a derivada?
E você vai estudar isto mais profundamente
quando começar a ver cálculo integral.
Mas existem realmente
muitas funções possíveis
que poderiam ser a derivada desta função.
E nosso objetivo neste
vídeo é apenas
desenhar uma possibilidade razoável.
Vamos pensar um pouco sobre isto.
Vamos desenhar aqui no topo
y igual a F maiúsculo de x.
Assim o que nós vamos
tentar desenhar
é uma função cuja derivada
pareça com isto.
O que essencialmente estamos fazendo é,
quando vamos do que desenhamos
aqui em cima para isto,
estamos lidando com a derivada.
Vamos pensar sobre o que
esta função poderia parecer.
Quando olhamos esta derivada,
ela diz sobre esse intervalo
sobre este primeiro intervalo
aqui, deixe-me fazer
isso em roxo,
Ela diz sobre esse
intervalo de x
igual a zero até qualquer
que seja o valor de x,
isso aqui, isso diz que
a inclinação é
uma constante positiva 1.

English: 
a possible function that this
could be the derivative of?
And you're going to study
this in much more depth
when you start looking
at integral calculus.
But there's actually
many possible functions
that this could be
the derivative of.
And our goal in
this video is just
to draw a reasonable
possibility.
So let's think about
it a little bit.
So let's, over here
on the top, draw
y is equal to capital F of x.
So what we're going
to try to draw
is a function where
its derivative
could look like this.
So what we're
essentially doing is,
when we go from what we're
draw up here to this,
we're taking the derivative.
So let's think about what
this function could look like.
So when we look at
this derivative,
it says over this interval
over this first interval right
over here, let me
do this in purple,
it says over this
interval from x
is equal to 0 all the way
to whatever value of x,
this right over here, it
says that the slope is
a constant positive 1.

Korean: 
원시함수를 추측해 봅시다
더 자세한 내용은
미적분학에서 배웁니다
실제로 여러 함수의 도함수가
f(x)가 될 수 있습니다
이번에는 가능한 경우를
그려보도록 합시다
 
위쪽에 F(x)를 그리겠습니다
위쪽에 F(x)를 그리겠습니다
즉 어떤 함수가
f(x)를 도함수로 가질지
추측하는 것입니다
반대로 말하면
위 함수를 미분했을 때
아래의 함수가 될 것입니다
위 함수의 개형을 판단합시다
첫 번째 구간에서
도함수를 살펴봅시다
 
x가 0인 점부터
첫 구간이 끝날 때까지
기울기가 1로
일정합니다

Spanish: 
cual es la posible funcion de la cual esta podria ser la derivada
y vas a estudiar mas esta definicion cuando empieces
con calculo integral
pero hay muchas funciones de las cuales esta podria ser su derivada
y lo que vamos a hacer en este video
es encontrar una posibilidad razonable
asi que vamos a pensarlo un poco
aqui arriba tenemos que y es igual (F) mayuscula de (x)
lo que intenteramos dibujar es una funcion que su derivada
pueda verse asi

Bulgarian: 
Ще начертая права с постоянен
наклон +1.
Мога да поставя тази права
по-нагоре или по-надолу.
Има много възможни
примитивни функции.
Но аз ще избера една.
Може да имаме права
като тази.
Искам да направя наклона
+1 максимално добре.
Да кажем, че изглежда
ето так.
Мога да направя функцията
дефинирана или недифинирана.
Производната е недефинирана
в тази точка.
Мога да направя функцията
дефинирана или недефинирана,
както предпочитам.
Това вероятно е точка на прекъсване 
на оригиналната функция.
Не е задължително, но аз
просто се опитвам
да начертая една 
възможна функция.
Да кажем, че всъщност
е дефинирана в тази точка ето тук.
Но след като е прекъсната,
производната ще бъде
недефинирана в тази точка.
Това е първият интервал.
Сега да разгледаме втория
интервал.
Вторият интервал, от там, където
първият интервал свършва,
чак до ето тук.

English: 
So let me draw a line with a
slope of a constant positive 1.
And I could shift
that line up and down.
Once again, there's many
possible antiderivatives.
I will just pick
a reasonable one.
So I could have a line that
looks something like this.
I want to draw a slope of
positive 1 as best as I can.
So let's say it looks
something like this.
And I could make the function
defined here or undefined here.
The derivative is
undefined at this point.
I could make the function
defined or undefined
as I see fit.
This will probably be a
point of discontinuity
on the original function.
It doesn't have to
be, but I'm just
trying to draw a
possible function.
So let's actually
just say it actually
is defined at that
point right over there.
But since this is going
to be discontinuous,
the derivative is going to
be undefined at that point.
So that's that first interval.
Now let's look at
the second interval.
The second interval, from where
that first interval ended,
all the way to right over here.

Thai: 
ขอผมวาดเส้นตรงที่มีความชันเป็นบวก 1 คงที่นะ
และผมเลื่อนมันขึ้นลงได้
ย้ำอีกครั้ง มันมีปฏิยานุพันธ์ได้หลายตัว
ผมจะเลือกสักค่าหนึ่ง
ผมก็มีเส้นตรงที่เป็นแบบนี้
 
ผมอยากวาดความชันเป็นบวก 1
ให้ดีที่สุดเท่าที่ทำได้
สมมุติว่ามันเป็นแบบนี้
และผมทำให้ฟังก์ชันนิยามตรงนี้ หรือไม่นิยามก็ได้
อนุพันธ์ไม่นิยามที่จุดนี้
ผมทำให้ฟังก์ชันนิยามหรือไม่นิยามก็ได้
ตามที่ผมเห็นสมควร
อันนี้น่าจะเป็นจุดไม่ต่อเนื่อง
บนฟังก์ชันเดิม
มันไม่ต้องเป็นอย่างนั้นก็ได้ ผมแค่
พยายามวาดฟังก์ชันที่เป็นไปได้ตัวหนึ่ง
ลองสมมุติว่ามัน
นิยามที่จุดนั่นตรงนั้น
แต่เนื่องจากอันนี้ไม่ต่อเนื่อง
อนุพันธ์จะไม่นิยามตรงจุดนั้น
นั่นคือช่วงแรกนั่น
ทีนี้ ลองดูช่วงที่สองกัน
ช่วงที่สอง จากจุดที่ช่วงแรกจบ
ไปจนถึงตรงนี้

Portuguese: 
Deixe-me desenha uma linha com
inclinação constante e positiva de 1.
E poderia muda a linha
para cima ou para baixo.
De novo, existem muitas
antiderivadas possíveis.
Vou apenas escolher uma razoável.
Eu poderia ter uma linha que
se parece com algo assim.
Quero desenhar uma inclinação de 1
positiva o melhor que posso.
Digamos que se pareça
com algo assim.
E poderia fazer a função
definida ou indefinida aqui.
A derivada é indefinida
nesse ponto.
Poderia fazer a função
definida ou indefinida.
Conforme vejo encaixar.
Esse provavelmente é um ponto
de descontinuidade
na função original.
Não precisa ser, mas vou
tentar desenhar uma 
função possível.
Vamos dizer que
ela é, na verdade,
definida nesse ponto
aqui.
Mas como ela será
descontínua,
a derivada será 
indefinida nesse ponto.
Então esse é o primeiro intervalo
Agora vamos ver o 
segundo intervalo.
O segundo intervalo, vai de onde
o primeiro acabou,
o caminho todo até aqui.

Korean: 
기울기가 1인 그래프지만
y절편은 모릅니다
따라서 y축 방향으로
평행이동하더라도
기울기만 같다면 가능합니다
한 가지 가능성을
골라서 그리겠습니다
구간의 끝점에서
도함수가 정의되지 않았으니
원시함수도 정의되지 않을 수 있습니다
따라서 적절하게
함수를 정의해야 합니다
아마 이 점에서
원시함수가 불연속일 것입니다
단정지을 수는 없지만
가능성을 고려하여
이 점에서 F(x)가
정의되었다고 합시다
그렇지만 불연속적이므로
도함수는 정의되지 않습니다
첫 번째 구간은 끝났습니다
두 번째 구간에서도
주어진 범위 안에서
함숫값이 일정합니다

Czech: 
Nakreslím tedy přímku
s kladným konstantním sklonem 1.
Mohl bych tu přímku
posouvat nahoru či dolů,
je zde mnoho možností.
Vyberu si nějakou rozumnou možnost.
Mohu mít například takovou přímku.
Snažím se to nakreslit nejlépe,
jak to dovedu.
Řekněme, že to může vypadat nějak takto.
Funkci bych tu mohl nechat
definovanou či nikoliv.
Derivace není v tomto bodě definována.
Mohu se rozhodnout, jak chci.
Toto by mohl být
bod nespojitosti původní funkce.
Nemusel by být, kreslím jednu z možností.
Řekněme, že to bude
v tomto bodě definované.
Jelikož to bude nespojité,
derivace nebude v tomto bodě definována.
To je první interval.
Pojďme na další.
Druhý interval, od bodu,
kde končil první, až sem.

Korean: 
-2를 일정하게 가집니다
역시 기울기가 -2로 일정한
직선 그래프가 될 것입니다
 
첫 구간에서 기울기의
두 배로 그려야 합니다
가능한 경우를 살펴보는 중이므로
이 점에서 연속적이라 가정하면
기울기는 두 배, 길이는 절반이므로
같은 값에서 끝날 것입니다
같은 값에서 끝날 것입니다
출발했던 값으로 돌아오되
기울기는 두 배가 되게
그래프를 그립시다
 
첫번째 구간에서 기울기는
도함수의 함숫값인 1이고
두 번째 구간에서 기울기는
마찬가지로 -2입니다
함수가 연속일 필요는 없으니

Thai: 
อนุพันธ์เป็นลบ 2 คงที่
นั่นหมายความว่าตรงนี้ ผมจะ
มีเส้นตรงที่มีความชันเป็นลบคงที่
หรือความชันเป็นลบ 2 คงที่
มันจะชันเป็น 2 เท่าของค่านี่
ตรงนี้
ผมก็แค่วาดมัน
ผมทำให้มันเป็นฟังก์ชันต่อเนื่องได้
ผมแค่ทำให้ความชันเป็นลบ 2 แบบนี้
มันดูเหมือนว่าช่วงนี้จะยาวเป็นครึ่งหนึ่ง
ของช่วงนี้
มันจึงได้จุดเดิมพอดี
มันอาจเป็น -- ขอผม
วาดให้สวยหน่อย -- แบบนี้
 
ความชันตรงนี้เท่ากับ 1
เราเห็นมันตรงนี้ในอนุพันธ์
แล้วความชันตรงนี้เท่ากับลบ 2
เราเห็นว่าในอนุพันธ์
ทีนี้ สิ่งต่างๆ เริ่มน่าสนใจ

Czech: 
Derivace je konstantně -2.
To znamená, že zde budu mít
klesající přímku se sklonem -2.
Bude dvakrát strmější, než byla ta první.
Nakreslím ji.
Mohu tu funkci udělat spojitou.
Mohu udělat sklon -2 takto.
Zdá se, že je tento interval
dvakrát kratší, než byl ten první.
Dostane se tedy do stejného bodu.
Mohlo by to vypadat nějak takto.
Sklon zde je roven 1,
to vidíme z průběhu derivace.
Sklon zde je roven -2,
to rovněž vidíme z průběhu derivace.
Teď se začnou dít věci!

Bulgarian: 
Производната е константа, –2.
Това означава, че тук
ще имаме права с постоянен
отрицателен наклон или
наклонът е равен на константата –2.
Значи това ще е два пъти
по-стръмно от това.
Мога просто да я начертая.
Мога да направя това
непрекъсната функция
с наклон –2, ето така.
Изглежда, че интервалът е
наполовина на този интервал.
Може би стига до същата точка.
Може би изглежда като...
ще я начертая малко
по-старателно, ето така.
Наклонът точно тук
е равен на 1,
виждаме това тук при
производната.
После наклонът ето тук е
равен на –2.
Виждаме отново това
от производната.
Сега нещата стават 
интересни.

Portuguese: 
Essa derivada é uma constante
negativa de 2.
Isso significa que aqui,
vou
ter uma linha de inclinação
constante negativa,
ou uma inclinação
negativa constante de 2.
E será duas vezes mais inclinada
que essa
aqui.
Então eu posso apenas
desenhar ela.
Eu poderia fazer uma
função contínua,
Poderia fazer uma inclinação
negativa de 2, bem assim.
E parece que esse intervalo
tem metade do tamanho
desse intervalo.
Talvez ela chegue ao
mesmo ponto.
Então poderia se parecer
alguma coisa-- deixe-me
desenhar isso mais limpo--
assim.
A inclinação aqui é
igual a 1,
vemos isso aqui na
derivada.
E a inclinação aqui é
igual a 2 negativo.
Vemos isso na derivada.
Agora fica interessante.

English: 
The derivative is a
constant negative 2.
So that means over
here, I'm going
to have a line of
constant negative slope,
or constant negative 2 slope.
So it's going to be twice
as steep as this one right
over here.
So I actually
could just draw it.
I could make it a
continuous function,
I could just make a negative
2 slope, just like this.
And it looks like this
interval is about half as long
as this interval.
So it maybe gets to
the exact same point.
So it could look
something-- let me
draw it a little bit
neater-- like this.
The slope right over
here is equal to 1,
we see that right over
there in the derivative.
And then the slope right over
here is equal to negative 2.
We see that in the derivative.
Now things get interesting.

Czech: 
Opět mohu tuto modrou přímku
libovolně posunout svisle.
Nemusím sestrojit spojitou funkci,
ale dělám to tak pro zábavu.
Existuje více primitivních funkcí.
Co se tedy bude dít na třetím intervalu?
Udělám to oranžově.
Sklon začíná na vysoké hodnotě,
začíná na +2.
Pak klesá a zde se dostane na 0.
Zde je sklon roven 0.
A pak je více a více záporný.
Zkusím pro zábavu
sestrojit spojitou funkci.
Opět připomínám, nemusí být.
Zde je sklon velké kladné číslo.
Je to +2.
Bude to tedy…
Toto je -2,
takže toto bude +2.
Pak je to zápornější až sem…
Měl bych raději říct,
že je to méně kladné.
I zde je sklon stále kladný.
Zde se dostane do 0.

Bulgarian: 
Пак мога да преместя тази
синя права нагоре или надолу.
Не съм длъжен да правя функцията
непрекъсната по този начин.
Но го правя за удоволствие.
Има много възможни
примитивни функции
за тази функция.
Какво се случва
в следващия интервал?
Ще използвам оранжево.
Наклонът започва 
от много висока стойност,
започва от +2.
После намалява, намалява,
тук става нула.
Наклонът става 0 ето тук.
И после започва да
става все по-отрицателен
и по-отрицателен.
Ще се опитам да направя
това непрекъсната функция,
просто ей така, за забавление.
Но повтарям, че
не е задължително.
Тук наклонът е много 
положителен, +2.
После наклонът става,
да видим, това е –2,
значи ще бъде +2.
И после става все 
по-отрицателен до тук,
или по-добре да кажа
все по-малко положителен.
Дори тук наклонът е
положителен.
Ето тук става нула.

Portuguese: 
De novo, eu poderia mover
essa linha para cima ou para baixo.
Eu não preciso construir uma função
contínua assim.
Mas vou fazer por diversão.
Existem muitas possíveis
antiderivadas
dessa função.
O que está acontecendo
no próximo intervalo?
E vou fazer isso de laranja.
A inclinação começa em um
valor muito alto.
Começa em 2 positivo.
E continua decrescendo até
chegar em zero, bem aqui.
A inclinação chega até
zero aqui.
E começa a ficar mais
e mais e mais
e mais negativa.
Vou tentar fazer uma
função contínua, só
por diversão.
De novo, ela 
não precisa ser.
Aqui a inclinação
é muito positiva.
É um 2 positivo.
Nossa inclinação vai ser como,
isso é um 2 negativo,
Então será um 
2 positivo.
E ela fica mais e mais
e mais negativa até esse,
ou, ela fica menos e menos
e menos positiva eu diria.
Até aqui a inclinação é positiva.
E ela chega ao zero 
bem aqui.

English: 
Once again, I could have shifted
this blue line up and down.
I did not have to construct a
continuous function like this.
But I'm doing it just for fun.
There's many possible
antiderivatives
of this function.
Now what's going on
over the next interval?
And I'll do it do it in orange.
The slope starts off
at a very high value.
It starts off at positive 2.
Then it keeps decreasing, and
it gets to 0, right over here.
The slope gets to
0 right over there.
And then it starts becoming
more and more and more
and more negative.
So I'll just try to make this
a continuous function, just
for fun.
Once again, it does
not have to be.
So over here the slope
is very positive.
It's a positive 2.
So our slope is going to be
like, this is negative 2,
so it's going to
be a positive 2.
And then it gets more and more
and more negative up to this,
or, it becomes less and less
and less positive I should say.
Even here the slope is positive.
And it gets to 0
right over there.

Thai: 
เหมือนเดิม ผมเลื่อนเส้นสีฟ้าขึ้นลงได้
ผมไม่ต้องสร้างฟังก์ชันต่อเนื่องแบบนี้
แต่ผมทำเพื่อความสนุก
มันมีปฏิยานุพันธ์ที่เป็นไปได้หลายอย่าง
สำหรับฟังก์ชันนี้
ทีนี้ ช่วงต่อไปจะเป็นอย่างไร?
 
และผมจะทำด้วยสีส้มนะ
ความชันเริ่มต้นด้วยค่าสูงมาก
มันเริ่มต้นที่บวก 2
แล้วมันลดลงไปเรื่อยๆ มันไปยัง 0 ตรงนี้
ความชันเป็น 0 ตรงนี้
แล้วมันเริ่มเป็นลบมากขึ้น มากขึ้น มากขึ้น
และมากขึ้น
ผมจะวาดมันเป็นฟังก์ชันต่อเนื่องนะ เพื่อความสนุก
 
ย้ำอีกครั้ง เราไม่จำเป็นต้องทำอย่างนั้น
ตรงนี้ ความชันเป็นบวกมาก
มันเป็นบวก 2
ความชันจะเป็นแบบนี้ นี่คือลบ 2
มันจึงเท่ากับบวก 2
แล้วมันเป็นลบมากขึ้น มากขึ้น เรื่อยๆ ถึงตรงนี้
หรือมันเป็นบวกน้อยลงเรื่อยๆ จะว่างั้นก็ได้
ถึงแม้ตรงนี้ความชันจะเป็นบวก
และมันเป็น 0 ตรงนี้

Korean: 
그래프를 평행이동해도
문제가 되지 않습니다
그러나 원시함수는 애초에
여러 가능성이 있으므로
편리한 대로 그립시다
 
 
 
그 다음 구간은
기울기 2에서 시작하여
0을 지나 계속 감소합니다
기울기가 0인 점을 표시하겠습니다
이후로도 계속 감소하여
음수의 기울기를 갖습니다
이 점 역시 편의를 위해
연속성을 가정합시다
 
처음에는 기울기가 2로
상대적으로 큽니다
두 번째 구간에서의 -2와 비교하면
이런 식으로 그려질 것입니다
그러다가 점점 음수로 변합니다
기울기가 작아진다는 것입니다
0이 되는 점을 기준으로
이전에는 양수, 이후는 음수입니다

Korean: 
이 점에서 기울기가 0일 것이고
전체적인 개형은 포물선입니다
아래로 오목하게 그립니다
처음 기울기는
2에서 시작하고
점점 감소하여
0까지 도달합니다
그 후 음수가 됩니다
 
이러한 개형을 가질 것이라
예상할 수 있습니다
 
대칭성을 살려 그리면
시작했던 기울기와
기울기의 크기가 같습니다
0인 점을 기준으로
대칭적으로 그릴 수 있습니다
 
마지막 구간에서는
기울기가 0입니다
역시 연속함수일 필요는 없으나

English: 
So maybe it gets to 0
maybe right over here.
And so what we have, we could
have some type of a parabola.
So the downward facing parabola.
So notice, the slope is
a very positive value.
It's a positive 2
right over here,
then it becomes less and less
and less and less positive,
all the way to 0
right over there.
And then the slope
starts turning negative.
And so our function
could look something
like that over the interval.
Let me draw it a
little bit neater.
This is symmetric.
How ever positive
our slope was here,
it's equally negative here.
So our curve should probably
be symmetric as well.
So let me draw it like
this over that interval.
It could look like that.
And then finally, over
this last interval,
or I guess we could say, it
keeps going, our slope is 0.
And once again I don't have
to draw a continuous function.

Czech: 
Asi někde zde.
Co jsme dostali?
Je to parabola.
Parabola otevřená směrem dolů.
Všimněte si, sklon je kladné číslo.
Je to zde +2,
pak je to méně kladné číslo,
až se dostane do 0.
Pak je sklon záporný.
Naše funkce by tedy na tomto intervalu
mohla vypadat takto.
Nakreslím to lépe.
Toto je symetrické.
Tak jak to zde bylo kladné,
tady to musí být stejně záporné.
Naše křivka by tedy
měla být také symetrická.
Nakreslím to tedy takto.
Mohlo by to vypadat takto.
Nakonec tento poslední interval.
Mohl bych říct, že funkce má nulový sklon.
Opět říkám, nemusím
sestrojit spojitou funkci.

Bulgarian: 
Може би става нула ето тук.
Значи ще имаме
някаква парабола.
Парабола, обърната надолу.
Обърни внимание, тук
наклонът е много положителен.
Тук е плюс 2,
после става по-малко и
по-малко положителен,
тук става нула.
После наклонът започва
да става отрицателен.
Така че нашата функция
ще изглежда като
нещо подобно в този интервал.
Ще го начертая малко по-спретнато.
Това е симетрично.
Не е възможно наклонът
да стигне дотук,
това е отрицателно, но
със същата стойност.
Кривата ни вероятно
също е симетрична.
Затова ще я начертая
по този начин в интервала.
Ще изглежда ето така.
И накрая остана
последният интервал,
или можем да кажем,
че наклонът тук е нула.
Пак повтарям, че не е задължително 
функцията да е непрекъсната.

Thai: 
บางที มันอาจเป็น 0 ตรงนี้
แล้วสิ่งที่เรามี เรามีพาราโบลาอยู่
พาราโบลาคว่ำ
สังเกตว่า ความชันมีค่าบวกมาก
มันเป็นบวก 2 ตรงนี้
แล้วมันเป็นบวกน้อยลงเรื่อยๆๆๆ
จนถึง 0 ตรงนี้
แล้วความชันก็เริ่มเป็นลบ
 
และฟังก์ชันของเราจะเป็น
แบบนี้ตลอดช่วง
ขอผมวาดให้สวยขึ้นหน่อยนะ
อันนี้มันสมมาตร
ความชันของเราเป็นบวกตรงนี้เท่าไหร่
มันจะเป็นลบเท่านั้นตรงนี้
เส้นโค้งของเราจึงน่าจะสมมาตรด้วยเช่นกัน
ขอผมวาดมันแบบนี้ตลอดช่วงนั้น
มันอาจเป็นแบบนั้น
แล้วสุดท้าย ตลอดช่วงสุดท้ายนี้
จะเรียกว่างั้นก็ได้ มันไปต่อ ความชันเป็น 0
ย้ำอีกครั้ง ผมไม่ต้องวาดฟังก์ชันต่อเนื่องก็ได้

Portuguese: 
Talvez ela chegue ao zero
talvez aqui.
Então o que temos, podemos
ter algum tipo de parábola.
Assim, a parábola voltada para baixo.
Note que a inclinação é
um valor muito positivo.
É um 2 positivo 
bem aqui,
Então ela fica menos e menos
e menos positiva,
o caminho todo até o zero
bem aqui.
Então a inclinação começa
a ficar negativa.
Então nossa função poderia
parecer com algo
assim nesse intervalo.
Deixe-me desenhar de maneira mais clara.
Isso é simétrico.
O quanto a inclinação
fosse positiva,
seria igualmente negativa.
Nossa curva provavelmente
também seria simétrica.
Deixe-me desenhar isso assim
sobre aquele intervalo.
Poderia se parecer com isso.
E finalmente, sobre esse
último intervalo,
ou poderíamos dizer, ela 
continua indo, nossa inclinação é zero.
E de novo eu não preciso desenhar
uma função contínua.

Bulgarian: 
Но когато наклонът е 0, това
означава просто, че
имаме права с нулев наклон.
Това е хоризонтална права.
Мога да начертая тази
хоризонтална права тук,
и тогава получаваме
прекъсване.
Но мога да начертая
хоризонталната права ето тук,
и така примитивната
функция ще стане
непрекъсната функция.
Повтарям отново, че мога
да преместя всяка от тези отсечки
нагоре или надолу, и да получа 
съвсем същата производна.
Но тогава няма да имаме
непрекъсната функция като тази.
Но ние успяхме да
построим
възможна примитивна
функция на тази f(х).
И само да напомня, 
това f(х) е –
че примитивната функция е функция,
на която
f(х) може да е производна.

English: 
But when the slope
is 0, that just
means that I have a
line with slope 0.
I have a horizontal line.
And I could draw that horizontal
line up here, in which case
I'd have to say
its discontinuous.
Or I could draw that horizontal
line right over here,
and try to make
my antiderivative
a continuous function.
So once again, I could have
shifted any of these segments
up or down, and gotten
the exact same derivative.
But then I would not have
gotten a continuous function
like this.
But we have been
able to construct
a possible antiderivative
for f of x.
And just as a reminder, that's
just saying this f of x is--
the antiderivative is
a function that f of x
could be the derivative of.

Portuguese: 
Mas quando a inclinação
é zero, isso só
significa que eu tenho uma
linha com inclinação zero.
É uma linha horizontal.
E poderia desenhar essa linha 
horizontal aqui em cima, que é o caso
eu teria que dizer que ela
é descontínua.
Ou eu poderia desenhar a linha
horizontal bem aqui,
E tentar fazer da minha
antiderivada
uma função contínua.
De novo, eu poderia mover
qualquer desses segmentos
para cima ou para baixo, e teria
a mesma antiderivada.
Mas eu não teria uma função
contínua
como essa.
Mas conseguimos construir
uma antiderivada possível
para f de x.
E um lembrete, apenas dizendo que
essa f de x é--
a antiderivada é
uma função que f de x
poderia ser sua derivada.
Traduzido por [Eduardo Rebelo]

Thai: 
แต่เมื่อความชันเป็น 0 มัน
หมายความว่าผมมีเส้นตรงที่มีความชันเป็น 0
ผมมีเส้นแนวนอน
และผมวาดเส้นแนวนอนบนนี้ได้ ซึ่ง
ผมบอกว่า มันไม่ต่อเนื่องก็ได้
หรือผมจะวาดเส้นแนวนอนตรงนี้
และพยายามให้ปฏิยานุพันธ์ของผม
เป็นฟังก์ชันต่อเนื่อง
ย้ำอีกครั้ง ผมเลื่อนส่วนใดๆ ในนี้
ขึ้นลงได้ และยังได้อนุพันธ์เท่าเดิม
แต่ผมจะไม่ได้ฟังก์ชันต่อเนื่อง
อย่างนี้
แต่เราสามารถสร้าง
ปฏิยานุพันธ์สำหรับ f ของ x ที่เป็นไปได้อันหนึ่ง
และเพื่อเป็นการทบทวน มันก็แค่บอกว่า 
F ของ x นี้ --
ปฏิยานุพันธ์คือฟังก์ชันที่ f ของ x
ไปเป็นอนุพันธ์ของมันได้
 

Czech: 
Je-li však sklon nulový,
dostanu vodorovnou přímku.
Mohl bych ji nakreslit zde,
ale měl bych nespojitou funkci.
Nebo bych ji mohl nakreslit zde
a pokusit se o spojitou primitivní funkci.
Každý z těchto úseků bych mohl libovolně
posunout a dostal bych tu samou derivaci.
Neměl bych však takto
krásnou spojitou funkci.
Sestrojili jsme tedy jednu
z možných primitivních funkcí.
Pro připomenutí, primitivní funkce
k dané funkci je taková funkce,
jejíž derivací je původní daná funkce.

Korean: 
기울기가 0이면
단순히 수평선이 됩니다
 
말했듯이 연속일 필요가 없으므로
수평선의 위치는 알 수 없습니다
다만 연속적인 함수를 가정하여
수평선의 높이를
이 위치로 정합시다
이 부분들 중 어느 것이라도
수직하게 이동할 수 있습니다
그 경우 보이는 것처럼
연속하지 않을 것입니다
우리가 구한 것은
f(x)의 원시함수 중 하나입니다
정확히 말하자면 F(x)는
f(x)를 도함수로 가지는
수많은 함수 중 하나입니다
 
