
Bulgarian: 
Дадена е функцията f(х), равна
на натурален логаритъм от х^2.
"Нека L(х) да е линейното приближение
на функцията f за а = е.
Каква е формулата на L(х)?"
Само да се уверим, че разбираме
какво ни питат, като аз копирах
и поставих това упражнение тук, 
така че да можем да работим по него.
Нека да начертаем какво
представлява f(х).
И какво представлява L(х).
Каква е линейната апроксимация, ако
я центрираме около х = е.
Ако кажем, че има множество,
когато ни казват "а",
това е просто приетото означение
за центъра на апроксимацията.
Каква е областта, каква е стойността
на х, около която апроксимираме.
Просто ще избера няколко точки тук,
само за да визуализирам f(х).
Да кажем, че това е 1, това е 2,
всъщност ще ги разпръсна 
малко повече.

Portuguese: 
Considere a função f de x igual ao
logaritmo natural de x ao quadrado
Suponha que L de x seja a linearização 
local de f, para a igual a e.
Qual é a sentença para L de x?
Primeiro vamos nos certificar da 
compreensão sobre o que estamos querendo.
Copiei e colei o exercício em um rascunho 
para trabalharmos.
Vamos inicialmente traçar um esboço 
da f de x,
A reta L de x representa a
linearização local,
e estamos centrando em x igual a e.
Se estamos centrando em um determinado
valor, chamado por convenção de a,
ele será o valor ao redor do qual 
faremos aproximações.
Vamos determinar alguns pontos apenas para
visualizar f de x.
Na escala temos o um, o dois
vamos espalhar um pouco

English: 
Consider the function F of X is equal to
the natural log of X squared.
Let L of X be the local linearization of F
at A is equal to E.
What is a rule for L of X?
Let's just make sure we understand what
they're asking for, and I've copied and
pasted this exact exercise on my little
scratchpad so we can work through it.
So, let's just visualize what f of x looks
like.
And what l of x is.
What the local linearization, if we are
centering it at x equals e.
If we're saying that there's the set, when
they say
a that's just a convention for what are we
approximating around.
What the locality, what the x value that
we are approximating around.
So lets just, let me grab some points here
just to visualize f of x.
So let's say this is 1, this is 2, this
is, actually let me spread them out a
little bit more.

Korean: 
ln(x²)과 같은 함수 f(x)를 생각해봅시다
a가 e일 때 L(x)가 함수 f의
 부분 선형화가 되도록 하세요
함수 L(x)는 무엇인가요?
질문하는 것이 무엇인지를 이해하도록 합시다
그리고 이 문제를 해결할 수 있도록
제 작은 스크래치 패드에 복사해서 붙여 넣었어요
일단 함수 f(x)가 어떤지 그려 봅시다
그리고 함수 L(x)가 무엇인지도요
부분적 선형화란, 우리가 
x와 e가 같다는 것에 초점을 맞출 때
세트가 있다고 말하면
그것은 단지 우리가 근접한 것에 대한 관습일 뿐입니다.
즉 우리가 구하려는 x의 값 근처인거죠
x를 나타내기 위해 몇 개의 점들을 잡아 보도록 할게요
이것을 1 이것을 2라고 한다면
조금만 더 넓혀 볼게요

Thai: 
พิจารณาฟังก์ชัน f ของ x เท่ากับ
ล็อกธรรมชาติของ x กำลังสอง
ให้ L ของ x เป็นการประมาณเชิงเส้นท้องถิ่น
ของ f ที่ a เท่ากับ e
กฎสำหรับ L ของ x จะเป็นอะไร?
เพื่อให้แน่ใจว่าเราเข้าใจสิ่งที่เข้าถาม
ผมได้ลอกและวาง
แบบฝึกหัดเดียวกันนี้บนกระดานทด
เราจะได้เขียนลงไปได้
ลองมองภาพว่า f ของ x เป็นอย่างไร
และ L ของ x คืออะไร
การประมาณเชิงเส้นท้องถิ่น ถ้าเรา
ให้ศูนย์กลางอยู่ที่ x เท่ากับ e
ถ้าเราบอกว่า มีเซต เวลาเขาบอกว่า
a มันก็แค่ธรรมเนียมว่าเราะประมาณแถวไหน
และท้องถิ่น คือค่า x ที่เราจะประมาณ
ลองดู ขอผมเลือกจุดตรงนี้
เพื่อสร้างภาพ f ของ x
สมมุติว่านี่คือ 1 นี่คือ 2 นี่คือ
ขอผมกระจายมันกว้างกว่านี้หน่อย

Thai: 
นี่คือ 1 นี่คือ 2 นี่คือ 3
f ของ 1 จะเป็น 0
ล็อกธรรมชาติของ 1 กำลังสองเป็น 0
แล้วลองดู เราคิดถึง f
ของ e นั่นคือจุดที่เป็น
ศูนย์กลางการประมาณเชิงเส้น
เราจะคิดถึงในท้องถิ่นนั้น
e อยู่ประมาณตรงนี้
e อยู่ตรง ประมาณตรงนี้
และ f ของ e เท่ากับล็อกธรรมชาติของ
e กำลังสอง มันจะเท่ากับ 2
และอันนี้จะเท่ากับ เราจได้ 1 กับ 2 แล้ว อันนี้
ก็คือจุดบนกราฟ และมันจะเป็นแบบนี้
เป็นแบบ เป็นแบบนั้น
นั่นคือสิ่ง นั่นคือกราฟของ y เท่ากับ f ของ x
y เท่ากับ f ของ x
ทีนี้ เขาหมายถึงอะไรเวลาเขาบอกว่าท้องถิ่น
การประมาณเชิงเส้นท้องถิ่นของ f 
ที่ a เท่ากับ e
ธรรมเนียมตรงนี้ คือเขาใช้ตัวอักษร a

Portuguese: 
Então temos o um, o dois, o três.
f de um vale zero.
O logaritmo natural de um ao 
quadrado vale zero.
E podemos também pensar em f de e,
onde centralizaremos nossa linearização.
Vamos pensar em valores nessa região.
o número "e" está mais ou menos aqui.
E f de e é igual ao logaritmo natural de 
e ao quadrado, que vale dois.
Temos um e dois, e assim
outro ponto do gráfico, que ficará 
parecido com isto.
Este é o esboço do gráfico de y igual a
f de x.
E qual o significado da linearização
local de f em a, para a igual a e?

English: 
So this is 1, this is 2, this is 3.
F of 1 is gonna be 0.
The natural log of 1 squared is just 0.
And let's see, we could think of F of
E, and that's actually where we're
centering our linearization.
We're gonna think about values in that
locality.
So E is roughly right over here.
E is right, roughly right over here.
And F of E is equal to natural log of
E squared is, well, that's just going to
be 2.
So this is going to be, you're gonna have
1 and 2 and so, this is also
going to be a point on the graph and it's
going to look something like this.
Gonna look something, something like that.
That's what, that's the graph of Y is
equal to F of X.
y is equal to f of x.
Now, what do they mean when they say the
local,
the local linearizaton of f at a, is equal
to e.
Well the convention here, is they use the
letter a

Bulgarian: 
Значи това е 1, това е 2
а това е 3.
f(1) е равно на 0.
Натурален логаритъм от 1
на квадрат е просто 0.
Да видим, ако разглеждаме f(е),
това е центърът на нашата
линейна апроксимация.
Ще разгледаме стойностите
в тази област.
Значи "е" е приблизително
ето тук.
f(е) е равно на натурален
логаритъм от e^2,
което е равно просто на 2.
Значи това е точка (1;0), тя е
точка от графиката на функцията.
Някъде тук е точка (е; 2), която 
също е точка от графиката.
Значи графиката на функцията 
ще изглежда нещо такова.
Ето това е графиката на 
функцията у = f(х).
Какво имат предвид, когато
казват линейната апроксимация
на f в точката а = е?
Прието е, че когато използват 
буквата "а",

Korean: 
그러니깐 이것은 1 이것은 2  이것은 3입니다
f(1)은=0입니다
또한 ln(1²)= 0입니다
우리는 f(e)에 대해 생각할 수 있습니다
이것이 우리가 부분적 선형화를 찾을 부분이죠
그 근처에서의 값들을 생각해 볼겁니다
e는 대충 이쯤이구요
e는 오른쪽 대충 이쯤이구요
f(e)=ln(e²)입니다
그건 2가 될거구요
그러니깐 1과 2 등등의 값들이 있어요 이것은 또
그래프위의 점이 될 거고 이렇게 생겼을 거에요
딱 이렇게 생겼겠네요
이것은 y=f(x)의 그래프입니다
y는 f(x)와 같습니다
이제 부분이라고 함은
a에서의 f의 부분 선형화는 e와 같아요
글쎄요, 여기서 관례는 문자를 사용하나요?

Korean: 
부분 선형화에 대해 이야기 할 때
이것은 우리가 찾는 부분 근방인거죠
이게 우리가 구한 값입니다
그러니깐 이 지점에서의 접선을 생각하는 거죠
아니면 이 지점이죠.(e, f(e))
L(x)는
바로 이 선이겠네요
이 함수는 y=L(x)입니다
이 함수를 부분 선형화를 할 형태로 바꿀 것입니다
그리고 e 주위의 x에 대한 함수값을 알아볼 것입니다
이게 무슨 의미일까요?
함수 f(x)의 값을 구하기 위해
f(x)과 L(x)가 유사하다는 것을 이용하는 것이죠
a 근처의 x에 대해서 생각하고 있으니
f(a)를 가질 것입니다

Thai: 
เวลาเขาพูดถึง เวลาเขาพูดถึง
การประมาณเชิงเส้นท้องถิ่น
บอกว่ โอเค นี่คือสิ่งที่เราจะประมาณรอบๆ
นี่คือค่า a ตรงนี้
เราจะคิดถึงความชันของเส้นสัมผัส
ที่จุดนั้น หรือจุด (e, f(e))
แล้ว L ของ x จะบรรยาย L
ของ x จะบรรยายเส้นตรงนี่ตรงนี้
นี่คือ อันนี้ตรงนี้ คือ y เท่ากับ L ของ x
แต่เราจะเขียนมัน
ตามแบบที่เหมาะกับการประมาณเชิงเส้น
ท้องถิ่นเพื่อประมาณ
ค่าฟังก์ชันแถวนั้น สำหรับค่า x แถวๆ e
เราหมายความว่าอะไร?
วิธีบรรรยาย L ของ x โดยทั่วไป
คือ L ของ x เท่ากับฟังก์ชันหาค่าแล้ว
ที่จุดนั้น คุณตั้งศูนย์กลางที่ a ที่ a
มันจะเป็นอันนี้ มันจะเป็นค่านี่ตรงนี้

Portuguese: 
A convenção é usar a letra a quando 
tratamos de linearização local.
Tudo bem, uma vez que esse será o
valor onde faremos
as aproximações.
E aqui temos o nosso valor de a.
Essencialmente vamos procurar a inclinação
da reta tangente no ponto (e, f de e).
L de x descreverá a reta a seguir.
Isto aqui será y igual a L de x, mas 
escreveremos de uma forma
mais apropriada para a linearização local
aproximando os valores da função para 
valores de x ao redor de e.
O que queremos dizer com isso?
Bem, a forma genérica para se descrever 
L de x diz que
L de x é igual à função
calculada no ponto
onde estamos centralizando,
no caso, o ponto a
então será igual a este valor
que aparece no gráfico

Bulgarian: 
когато става въпрос за
линейна апроксимация,
това е стойността, около която
правим апроксимацията.
Значи това е тази стойност
ето тук.
Така че принципно ни трябва
наклона на допирателната
в тази точка, или точката
(е; f(е)).
Значи L(х) описва тази
права ето тук.
Така че това е, ето тук, това
е у = L(х), но ние ще го запишем
по един начин, който е по-подходящ 
за линейна апроксимация,
стойността на функцията за стойности
на х около "е".
Какво означава това?
Обичайният начин да опишем L(х)
е, че L(х) е равно на
изчислената стойност на функцията.
В точката, когато се центрираме
около а, в точка а,
значи това ще е ето тази стойност.

English: 
when they're talking about, when they're
talking about local linearization.
Say okay, this is what we're approximating
around.
So this our a value right over here.
So we're essentially just gonna think
about the slope of the tangent
line at that point, or this point E comma
F of E.
So the L of X is going to describe, L
of X is going to describe this line right
over here.
So this is, this right over here, is Y is
equal to L of x, but we write it in a
way that is well suited for local
linearization, for approximating
the value of the function around, well,
for x values around e.
So what do we mean by that?
Well, the general way of describing L of x
say
that L of x is equal to the function
evaluated.
At that point that you're centering around
at a, at a, so
it's going this, it's going to be this
value right over here.

Korean: 
거기에 a에서의 접선의 기울기와
거기에 a에서의 접선의 기울기와
근사치인 x와 a의 차를 곱한 값을 더해줍니다
그리고 다시 한 번 더 강조하자면, 이게 왜 가능한거죠?
여기에서 이 값의 반 만큼을
측정하려고 한다고 가정합시다
측정된 함수는 여기일거구요 하지만 우리는
부분선형화를 사용하고 싶으니까 그러니깐 L(x)의 값은 무엇이죠?
그러니깐 정확히 이 값이 무엇인지 알고 싶은 거죠
f(a)에서 시작해 봅시다
f(a) +f'(a)(x-a)를 하게 될 것입니다
f(a) +f'(a)(x-a)를 하게 될 것입니다
x- a는 이 거리 만큼이구요
이 값에 기울기를 곱하면 y 값의 변화를 줄 것입니다
y의 변화값을 f(a)에 더해주면 원하는 답을 줍니다
f(x)의 근사값을요
여기에요
이 방식은 근사를 할 때 유용하죠
ln(e+0.2)²을

Thai: 
บวกความชันของเส้นสัมผัสที่ a
บวกความชันของเส้นสัมผัส
ที่ a คูณระยะ คูณผลต่างระหว่าง
ค่า x ที่คุณกำลังประมาณกับ a
ย้ำอีกครั้ง ทำไมอันนี้ถึงถูกต้อง?
สมมุติว่าคุณกำลังพยายาม
ประมาณครึ่งหนึ่งของค่านี่ตรงนี้
ฟังก์ชันหาค่าตรงนี้จะอยู่ตรงนี้ แต่เราอยากใช้
การประมาณเชิงเส้นท้องถิ่น เราอยากบอกว่า
L ของค่า x นี้เป็นเท่าใด?
เราอยากหาว่าค่านี่ตรงนี้คืออะไร
คุณเริ่มที่ f ของ a ซึ่งก็คือตรงนี้
แล้วคุณคูณ หรือจากค่านั้น คุณบวก
ความชันคูณผลต่างระหว่างสองค่านี้
x ลบ a คือผลต่างนี้
คุณคูณมันด้วยความชัน 
มันจะได้การเปลี่ยนแปลงของ y
การเปลี่ยนแปลงของ y บวกค่า y นี่ตรงนี้
จะให้ค่า y ของจุดนั้น
ตรงนั้น
อันนี้มีประโยชน์เวลาประมาณค่า
สมมุติคุณอยากประมาณว่า ว่า

Bulgarian: 
Плюс наклона на допирателната тук,
по разстоянието между
стойността х, която 
апроксимираме, и "а".
Какво означава това?
Да кажем, че тук опитваме
да апроксимираме тази
стойност ето тук.
Функцията, изчислена тук, ще бъде 
ето тук, но ние искаме да използваме
линейна апроксимация, така че
принципно търсим L за тази стойност на х.
Търсим колко е това ето тук.
Значи започваме от f(а), което
е ето това.
После умножаваме по това,
и по това, прибавяме
наклона по разликата
на тези две стойности.
Това разстояние е (х – а).
Умножаваме го по наклона,
което представлява промяната на у.
Значи промяната на у,
плюс това у ето тук,
това е у-стойността
на ето това.
Това може да е полезно
за апроксимиране,
да кажем, че искаме да
апроксимираме това,

Portuguese: 
mais a inclinação da reta tangente em a,
vezes a diferença entre
o valor de x que estamos aproximando e a.
E porquê tudo isso faz sentido?
Digamos que você esteja tentando descobrir
o valor aproximado da função neste ponto.
A função avaliada estaria aqui, mas 
queremos usar a linearização local
nos perguntando: qual é 
o valor de L deste x?
Então precisamos descobrir isto aqui.
Começamos em f de a, aqui
e então somamos à inclinação 
da curva multiplicada
à diferença entre os dois valores.
x menos a equivale a esta distância.
Multiplicando pela inclinação 
temos a variação em y.
A variação em y, mais o valor de y
nos dará o valor que procuramos.
Bem aqui.
Isto pode ser útil em aproximações.
Digamos que que queira 
saber o valor aproximado

English: 
Plus the slope of the tangent line today,
plus the slope of the tangent line
today times how far, times the difference
between
the x value that you're approximating and
a.
And once again why does this make sense?
Well let's say that you're trying to
approximate half of this value right over
here.
So the function evaluated there would be
right over there, but we want to use
the local linearization, so we essentially
want to say, what is l of this x value?
So we want to figure out what is this
right over here.
Well you start at f of a, which is this.
And then you multiply, or to that, you add
the slope times the difference between
these two values.
X minus a is this distance.
You multiply that times the slope, that's
gonna give you your change in y.
So your change in y, plus this y right
over
here, is going to give you the y value of
that.
Right over there.
So this could be useful for approximating,
let's
say you want to approximate what, what
the, what

Portuguese: 
do logaritmo natural de e
mais 0.2 ao quadrado,
e podemos determiná-lo.
Vamos analisar este caso particular.
Sabemos que a é igual a e.
Substituímos nos três locais.
E quanto a f de e?
f de e será o logaritmo natural 
de e ao quadrado.
E quanto a f linha de e?
Para determinar o valor de
f linha de x, usaremos a regra da cadeia.
Será a derivada do logaritmo natural de x
ao quadrado em relação a x ao quadrado, 
então teremos
um sobre x ao quadrado, vezes a derivada 
de x ao quadrado em relação a x,
que vale dois x. 
Simplificando teremos dois sobre x.

Korean: 
구하고 싶다고 한다면
이 과정은 유의미 합니다
하지만 일단 이 특정한 경우를 해결해 봅시다
우리는 a와 e가 같다는 것을 알아요
그러니 a에 e를 대입해 봅시다
이것도 e가 될거구요 이것도 e 이것도 e일 거에요
그럼 f(e)는 무엇이죠?
f(e)=ln(e²)일 것이고
f(e)=ln(e²)일 것이고
그러면 f'(e)는 무엇일까요?
연쇄법칙을 사용하면 f'(x)를 구할 수 있죠
이것은 ln(x²)의 도함수이고
그러니 (1/x² )(x² 미분값)이 됩니다
그러니 (1/x² )(x² 미분값)이 됩니다
x² 의 미분값은 2x이므로 결과는 2/x가 됩니다
f'(e)=2/e가 될 것이고

Bulgarian: 
колко е натурален логаритъм
от (е + 0,2)^2,
тогава това може
да е полезно.
Но нека да разгледаме това
за конкретния случай.
Знаем, че а = е.
Да заместим, това трябва да е "е",
това трябва да е "е" и това също е "е".
ЕИ колко е сега f(е)?
Това тук е 
натурален логаритъм от
е^2 и сега колко е f'(е)?
Да го изчислим тук долу.
f'(х) можем да намерим като
производна на сложна функция.
Това е производната на 
натурален логаритъм от х^2,
спрямо х^2, така че това
ще бъде 1/х^2,
по производната на x^2
спрямо х.
Значи става 2х, и това
ще е равно на 2/х.
f'(е) ще е равно на 2/е,

Thai: 
ล็อกธรรมชาติของ 
e บวก 0.2 กำลังสองคืออะไร
มันจึงมี มีประโยชน์ทีเดียว
แต่ลองคิดอันนี้สำหรับกรณีเฉพาะนี้กัน
เรารู้ว่า เราบอกว่า a เท่ากับ e
ลองแทนค่า อันนี้ควรเป็น e อันนี้ควรเป็น e
และอันนี้ควรเป็น e
แล้ว f ของ e คืออะไรตอนนี้?
อันนี้คืออันนี้ตรงนี้ คือล็อก
ธรรมชาติของ e กำลังสอง แล้วตอนนี้
f ไพรม์ของ e คืออะไร?
ลองคำนวณมันข้างล่างตรงนี้
f ไพรม์ของ x เราใช้กฎลูกโซ่ได้
มันคืออนุพันธ์ของล็อกธรรมชาติ
ของ x กำลังสองเทียบกับ x กำลังสอง นั่นก็คือ
1 ส่วน x กำลังสอง 
คูณอนุพันธ์ของ x กำลังสองเทียบกับ x
คูณ 2x และอันนี้จะเท่ากับ 2 ส่วน x
f ไพรม์ของ e จึงเท่ากับ 2 ส่วน e แล้ว

English: 
the natural log of e plus 0.2 squared is,
and so this could, could be, could be
valuable.
But let's just work through this for this
particular case.
We know that, so we're saying that a is
equal to e.
So let's replace, this should be e, this
should be e, and this should be e.
And now what is f of e?
Well this is this right over here is the
natural
log of e squared and now whats f-prime of
e?
Well let's just calculate that down here.
So f-prime of x we can just use the chains
rules.
It's the derivative of the natural log of
x squared with respect to x squared, so
that's
going to be 1 over x squared, times the
derivative of x squared with respect to x.
So times 2x and so this is going to be
equal to 2 over x.
So f prime of e is going to be two over e,
and

Bulgarian: 
после по (х – е), това е от
правилото, от определението за L(х).
И сега да видим кое от тези...
Разбира се можем да опростим 
натурален логаритъм от e^2.
Степента, на която трябва
да повдигнем, за да получим е^2,
това е разбира се 2.
Значи 2 + 2 върху е по (х – е).
Това е този вариант ето тук.
Само да поясня, как това
може да е полезно, да кажем, че
опитваме да апроксимираме, опитваме 
да апроксимираме f(е + 0,1).
Опитваме да апроксимираме
ето това.
Ако просто заместя (е + 0,1)
и го повдигна на квадрат,
и трябва да намеря натурален
логаритъм от това
без калкулатор, това 
изглежда доста трудно.
Но сега можем да използваме
линейна апроксимация.

Portuguese: 
Assim, f linha de e será dois sobre e,
vezes x menos e.
Vamos simplificar os termos da 
sentença para L de x,
Assim, o logaritmo natural de e 
ao quadrado
equivale à potência que devemos usar em e
para obter e ao quadrado, 
o que obviamente vale dois.
Portanto dois mais dois sobre 
e, vezes x menos e.
É esta opção aqui.
Apenas para deixar claro como pode
ser útil, suponha que
estejamos tentando encontrar o valor
aproximado de f de e mais 0.1
Substituindo teremos e mais 0.1
ao quadrado e precisamos determinar o
valor do seu logaritmo natural
sem o uso de uma calculadora, o que me 
parece ser muito difícil.
Mas podemos usar a linearização local.
Ela será aproximadamente igual a dois mais

Thai: 
คูณ x ลบ e นั่นก็คือ นั่นคือกฎของเรา
นิยามสำหรับ L
ของ x แล้วดู ลองดูว่าข้อใด
โอ้ แน่นอน เราลดรูปล็อก e กำลังสองได้
กำลังที่ผมต้องยก e ให้ได้
e กำลังสอง แน่นอนว่ามันคือ 2
2 บวก 2 ส่วน e คูณ x ลบ e
2 บวก 2 ส่วน e คูณ x ลบ e
มันคือตัวเลือกนั่นตรงนั้น
แต่เพื่อให้เห็นชัดว่าอันนี้มีประโยชน์อย่างไร
ลองสมมุติว่าเราอยาก
ประมาณ เราพยายามประมาณ 
f ของ e บวก 0.1
เราพยายามประมาณค่านั้น
ทีนี้ ถ้าผมใส่ e บวก 0.1 แล้วผมยกกำลังสอง
และผมต้องหาล็อกธรรมชาติของค่านั้น
โดยไม่ใช้เครื่องคิดเลข 
มันจะเป็นเรื่องยากสำหรับผม
แต่ตอนนี้ เราใช้การประมาณเชิงเส้นท้องถิ่นได้
อันนี้จะประมาณเท่ากับ 2 บวก

English: 
then times x minus e so that's, that's our
rule, our definition for l
of x and lets see which one of these lets
see which one of
these oh and of course we can simplify the
natural log of e squared.
The power that have to raise e to get to
e squared, well this is of course equal to
2.
So 2 plus 2 over e times x minus e.
So 2 plus 2 over e times x minus e.
It's that choice right over there.
But just to make it clear, how this could
be useful, let's say that we're trying to
approximate, we're trying to approximate f
of e plus 0.1.
So, we're trying to approximate that.
Now, like if I just put e plus 0.1 and I
squared and I have
to figure out the natural log of that
without a calculator, that seems hard to
me.
But now we can use a local linearization.
This is going to be approximately equal to
2 plus,

Korean: 
여기에 (x-e)를 곱해주면 함수 L(x)가 됩니다
이 식이 L(x)의 방정식이 됩니다
ln(e²)은 2로 단순화 될 수 있습니다
ln(e²)= 2로 나타낼 수 있습니다
정리하자면 2+2(x-e)/e 입니다
2+2(x-e)/e 입니다
2+2(x-e)/e 입니다
따라서 답은 4번이 됩니다
이것이 어떻게 쓰이는 지 알아봅시다
우리가 f(e+0.1)을 근사한다고 가정해 봅시다
우리가 f(e+0.1)을 근사한다고 가정해 봅시다
만약에 단순히 e+0.1 대입했다고 하면
계산기 없이는 어려울 것입니다
계산기 없이는 어려워 보이네요
하지만 이제는 부분 선형화를 사용할 수 있잖아요
f(e+0.1)은 2+2(e+0.1-e)/e와 비슷할 것입니다

Thai: 
2 ส่วน e คูณ e บวก 0.1 ลบ e คืออะไร
มันก็แค่ 0, 0.1
ผมว่า คุณอาจจะยังต้องใช้เครื่องคิดเลขหาค่านี้
หรือคุณจะปล่อยไว้อย่างนั้นก็ได้
ไม่ว่ายังไง อันนี้เป็นการประมาณที่ดี
ว่าล็อกธรรมชาติของ e บวก 0.1 กำลังสอง
จะเท่ากับอะไร
แล้วลอง ลองตรวจคำตอบกัน
มันคือข้อนั้นตรงนั้น
และเราตอบถูกด้วย

Portuguese: 
dois sobre e, vezes e mais 0.1, menos e.
Teremos então apenas 0.1
Você precisará de uma calculadora 
para determinar esse valor,
ou pode apenas deixar como está, 
pois é uma aproximação bem razoável
para o valor do logaritmo natural 
de e mais 0.1 ao quadrado.
Voltando ao nosso problema, 
vamos verificar a resposta
que será esta aqui
E está correta.
Legendado por: Tatiana Ferrari D'Addio.
Revisado por: Raiza Carlos de Souza

Korean: 
f(e+0.1)은 2+2(e+0.1-e)/e와 비슷할 것입니다
e+0.1-e=0.1입니다
계산을 위해서 계산기가 필요하겠지만
단순 대입보다는 훨씬 효율적입니다
ln(e+0.1)²=2+2x0.1/e이 됩니다
그럼 이제 답을 확인해 볼까요
답은 4번이죠
네 정답입니다

Bulgarian: 
Това ще е приблизително равно
на 2 плюс (2/е),
по, колко е "е" + 0,1 минус "е", 
умножено по...
’e’ минус ’e’ е нула, значи остава просто 0,1. Като резултат получаваш, че f(e + 0,1) = 2 + (2/e).(0,1)
Но предполагам, че все пак
ти трябва калкулатор, за да го сметнеш,
или можеш да го оставиш така,
тъй като това е добра апроксимация
за стойността на натуралния логаритъм от (е + 0,1) на втора степен.
Само да проверим отговора.
Това е ето това тук.
И отговорът ни е верен.

English: 
2 over e, times, well what's e plus 0.1,
minus e.
Well it's just gonna be 0, 0.1.
And I guess you probably still need a
calculator to figure this out,
or you could just leave it here, anyway
this is a pretty good approximation.
For what the natural log of e plus 0.1
squared is going to be equal to.
And let's just, let's just check our
answer.
So, it is that one right there.
And we got it right.
