
Bulgarian: 
Нека да кажем, че ни интересува
 да определим приблизително
на какво е равно квадратен
 корен от 4,36.
Искаме да намерим
приближение на това число 
и нямаме калкулатор под ръка.
Един начин да мислиш за това, е, че
знаем на колко е равно 
квадратен корен от 4.
Знаем, че е равно на плюс 2.
Квадратен корен от 4 
е равно на плюс 2.
Тогава корен от даденото число 
ще бъде равно на малко повече от 2.
Но да кажем, че искаме 
да сме малко по-точни
и това, което ще ти покажа 
в настоящия урок,
е метод именно за това. Тоест за 
приблизително определяне стойността на
функция, която е близка 
до известна ни вече стойност.
Е, за какво става дума?
Нека просто да си представим, 
че имаме дадена функция.
Имаме функция f от х, която е равна 
на квадратен корен от х,
което представлява 
същото като х на степен 1/2.
Знаем на какво е равно f от 2.

Korean: 
우리가 근사하는 것에 관심있다고 해봅시다
√4.36이 무엇인지 근사하는것에
우리는 이것의 근사값을 계산하고 싶은데
계산기가 없습니다
근사값을 생각하는 한 가지 방법은
우리가 알고있는 √4를 이용하는 것입니다
우리는 √4가 2인 것을 알고 있습니다
√4는 2입니다
때문에 저것은 2보다 약간 클 것입니다
우리가 더 정확한 계산을 하고 싶다고 해봅시다
제가 지금 이 영상에서 보여줄것은 그것의 방법입니다
우리가 이미 알고있는 값 근처의
함숫값을 근사하는
제가 지금 무엇을 말하고 있나요?
함수를 하나 생각해봅시다
f(x)=√x 라는 함수가 있습니다
f(x)=x^(1/2)와 동일한
우리는 f(2)가 무엇인지 압니다

Czech: 
Řekněme, že bychom chtěli odhadnout,
čemu je rovna druhá odmocnina z 4,36.
Zajímá nás tedy odhad tohoto,
aniž bychom použili kalkulačku.
Jeden pohled je, že víme,
kolik je druhá odmocnina ze 4.
Víme, že toto je plus 2.
Odmocnina ze
4 je rovna 2.
Takže toto bude
o trochu více než 2.
Řekněme, že chceme být
trochu přesnější.
V tomto videu vám tedy ukážu
metodu, jak to udělat.
Metodu, jak odhadnout hodnotu funkce
poblíž bodu, jehož hodnotu už známe.
Tedy co tím myslím.
Představme si, že máme funkci
f(x) je rovna druhé odmocnině z x.
To je to samé jako
x na jednu polovinu.

English: 
Let's say that we're interested in
approximating what
the square root of 4.36 is equal to.
So we want to figure apro, we want to
figure out
an approximation of this, and we don't
have a calculator at hand.
Well, one way to think about it is we know
what, we know what the square root of 4
is.
We know that this is positive 2.
The principle root of 4 is positive 2.
So, okay, this is going to be a little bit
more than two.
Well, let's say that we wanna get a little
bit more
accurate, and so what I'm going to show
you in this video
is a method for doing that, for
approximating the value of
a function near, near a value where we
already know the value.
So what am I talking about?
So let's just imagine that we had the
function.
We have the function f of x is equal to
the square root of
x, which is, of course, the same thing as
x to the one-half power.
So we know what f of 2 is.

Thai: 
สมมุติว่าเราต้องการประมาณว่า
รากที่สองของ 4.36 มีค่าเท่าใด
คือเราต้องการหาค่าประมาณ
ค่าประมาณของจำนวนนี้ โดยไม่มีเครื่องคิดเลข
วิธีคิดอย่างหนึ่งคือ
เรารู้ค่าของกรณฑ์ที่สองของ 4
เรารู้ว่ามันคือ 2
กรณฑ์ที่สองของ 4 คือ 2
แสดงว่า มันจะต้องมากกว่า 2 นิดหน่อย
สมมุติว่าเราอยากให้ใกล้เคียงกว่านี้หน่อย
นั่นคือสิ่งที่ผมจะทำในวีดิโอนี้
คือวิธีการสำหรับประมาณค่า
ฟังก์ชันที่อยู่ใกล้ค่าที่เรารู้ค่าฟังก์ชันแล้ว
ผมกำลังพูดถึงอะไร
ลองนึกภาพว่าเรามีฟังก์ชัน
เรามีฟังก์ชัน f ของ x เท่ากับ
กรณฑ์ที่สองของ x
ซึ่งก็เหมือนกับ x ยกกำลังเศษหนึ่งส่วนสอง
เรารู้ว่า f ของ 2 เป็นเท่าไร

Portuguese: 
Vamos supor que queremos calcular o valor
aproximado da raiz quadrada de 4,36.
Queremos uma aproximação sem o auxílio
de uma calculadora.
Uma maneira de pensar é:
Nós sabemos qual a raiz quadrada de quatro
Sabemos que é dois positivo.
E então isso será um pouco maior que dois.
Mas queremos ser mais precisos.
Neste vídeo vou te mostrar um método para
aproximar o valor de uma função
a um valor já conhecido.
Sobre o que estou falando?
Vamos imaginar que temos a função f de x
igual a raiz quadrada de x.
Que é o mesmo que x elevado a um meio.

Korean: 
아 죄송합니다
우리는 f(4)의 값을 알고 있습니다
f(4)는 2와 같은 √4입니다
혹은 4의 제곱근입니다 +2와 같은
우리가 근사하고 싶은 것은
f(4.36)의 값입니다
이것은 그냥
이 영상을 시작하면서 했던 질문과 정확히 같습니다
함수를 하나 생각해봅시다
잠깐만 생각해봅시다
제가 축을 좀 그리겠습니다
이것이 y축입니다
이것은 x축이고 y=f(x)의 그래프를 그려봅시다
이렇게 생겼다고 해봅시다
y=f(x)는 저렇게 생겼습니다
꽤 잘 그린 것 같습니다
저기 있는것이 y=f(x)입니다

Portuguese: 
Sabemos o que é f de quatro.
Será a raiz quadrada de quatro.
A raiz positiva de quatro que é dois.
Queremos descobrir a raiz de 4.36
É apenas uma outra forma de fazer
a pergunta feita no início do vídeo.
Vamos imaginar nossa função
por um momento.
Deixe-me desenhar os eixos.
Este será o eixo y.
Este será o eixo x.
Vamos fazer o gráfico de f de x.
Será mais ou menos assim.
Assim está bom.
Isso aqui é y igual a f de x.

English: 
We know, we know that f of 2, I'm sorry,
we know that f of 4 is.
We know that f of 4 is the square root of
4, which is
going to be equal to 2 or the principle
root of 4, which is
equal to positive 2 and what we want to
approximate, we wanna figure out
what f of, we wanna figure out what f of
4.36 is equal to.
This is just another way of framing the
exact same question that we started off
this video.
So let's just imagine our function.
Let's just imagine it for a second.
So, let me draw some axes.
This is my y-axis.
This is my, this is my x-axis, and let's
graph y is equal to f of x.
So let's say it looks something like this.
Y equals f of x looks something like that.
That's pretty decent.
All right, so that's, that right there is
y is equal to

Bulgarian: 
Знаем...Извинявай! Знаем, на какво е равно f от 4.
Знаем, че f от 4 е равно 
на квадратен корен от 4,
което ще бъде равно на 2, 
или квадратен корен от 4
е равно на плюс 2, а това,
 което искаме да определим,
е на какво ще бъде равно f от 4,36.
Това е просто друг начин 
да зададем
точно същия въпрос, с който
 започнахме този урок.
Нека да си представим
 дадената функция.
Нека просто за секунда 
да си я представим.
Ще начертая някакви оси.
Това е моята ос у.
Това е моята ос х и ще начертая
 y равно на f от х.
Нека да предположим, че
 изглежда като нещо такова.
y равно на f от х изглежда
като нещо такова.
Това е сравнително добре начертано.
Добре, това, което начертах тук 
е y равно

Thai: 
เรารู้ค่าของ f ของ 2
โทษที ผมหมายถึงค่าของ f ของ 4
เรารู้ว่า f ของ 4 คือรากที่สองของ 4 ซึ่ง
จะเท่ากับ 2
ที่จริงคือกรณฑ์ที่สองของ 4 ซึ่ง
เท่ากับ บวก 2 และเราต้องการประมาณ
ค่าของ f ของ 4.36
นี่เป็นแค่วิธีบรรยาย
ปัญหาเดียวกันกับตอนที่เราเริ่มวีดิโอ
ลองนึกภาพฟังก์ชันของเรา
ลองนึกภาพสักแป๊บ
ผมจะวาดแกนตรงนี้
นี่คือแกน y
นี่คือแกน x
ทีนี้ มาวาดกราฟสำหรับ y เท่ากับ f ของ x กัน
มันดูหน้าตาแบบนี้
y เท่ากับ f ของ x ดูหน้าตาแบบนั้น
ใกล้เคียงใช้ได้อยู่
เอาล่ะ นั่นคือกราฟของ y เท่ากับ

Czech: 
My víme,
čemu je rovno f(4).
Víme, že f(4) je druhá odmocnina
ze 4, tedy to bude rovno 2.
A to, co chceme odhadnout,
to, co chceme zjistit je,
čemu je rovna druhá
odmocnina z 4,36.
Toto je jen jiný způsob formulování úplně
stejné otázky, jakou jsme začali video.
Představme si na 
chvíli naši funkci.
Takže, nakreslím si
osy souřadnic.
Toto je osa y,
toto je osa x.
A pojďme nakreslit
y je rovna f(x).
Řekněme, že vypadá
nějak takto.
y je rovna f(x), vypadá
nějak takto.
Je to celkem poctivé.
Toto zde, je
y rovno f(x).

Czech: 
Víme, že
f(4) je rovno 2.
Zde je x rovno 4, nenakreslil 
jsem to přesně v měřítku,
ale věřím, že je to
dostatečně jasné.
Toto je tedy 2.
To je f(4).
A to, co se snažíme
odhadnout, je f(4,36).
Takže 4,36 bude
přibližně zde.
Chceme tedy odhadnout
tuto hodnotu y.
Toto odhadujeme, 
to je f(4,36).
A předpokládáme, že nemáme
po ruce kalkulačku.
Jak to tedy můžeme udělat s
využitím toho, co víme o derivacích?
Co kdybychom zjistili
rovnici tečny tohoto bodu.
Tedy rovnici tečny,
kde x je rovno 4.

Thai: 
f ของ x และเรารู้ว่า
f ของ 4 เท่ากับ 2
f ของ 4 เท่ากับ 2
นี่คือเมื่อ x เท่ากับ 4
ผมวาดไม่ค่อยตรงสัดส่วนเท่าไร
แต่หวังว่าจะชัดเจนพอ
ตรงนี้จะเท่ากับ 2
นั่นคือ f ของ 4
และเราต้องการประมาณค่าของ f ของ 4.36
ซึ่ง 4.36 จะอยู่ประมาณแถว ๆ นี้
และเราต้องการประมาณค่า
เราต้องการประมาณค่า y ตรงนี้
เราต้องการประมาณค่านั้น
ตรงนี้คือ f ของ 4.36 และ
อย่าลืมว่าเราไม่มีเครื่องคิดเลขตอนนี้
เราจะทำอย่างไรโดยใช้ความรู้เรื่องอนุพันธ์
อย่างนี้เป็นไง เราหาสมการสำหรับเส้นตรง
ที่สัมผัสกราฟที่จุดตรงนี้ก่อน
สมการของเส้นสัมผัสที่ x=4 แล้วเราก็ใช้

Korean: 
그리고 우리는 f(4)가 2인 것을 알고 있습니다
f(4)는 2이고 이것은 x가 4일때의 값입니다
실제 크기 비율로 그리지는 않았지만 꽤 분명합니다
저기 저 점이 2입니다
저것이 f(4)입니다
우리는 f(4.36)의 값을 근사하고 싶은 것이고 4.36은 저기 쯤 될 것입니다
우리는 근사하고 싶습니다
우리는 저기 y값을 근사하고 싶습니다
우리는 저것을 근사하고 싶습니다
여기가 f(4.36)이고
우리는 계산기가 없다고 가정합시다
우리가 미분에 대해 알고 있는 것을 어떻게 이용할 수 있을까요?
우리가 이 점의 접선의 방정식을 구한다면
바로 이 점에서 접하는
x=4에서의 접선의 방정식은

Bulgarian: 
на f от х и знаем, че 
f от 4 е равно на 2.
f от 4 е равно на 2, така че това 
се получава, когато х е равно на 4.
Не съм начертал графиката мащабно, 
но надявам се, че е достатъчно ясно.
Това ето тук 
ще бъде равно на 2.
А това е f от 4.
А това, което искаме да определим 
приблизително, е f от 4,36.
Числото 4,36 може 
да се намира около...
Точно около това място 
и искаме да определим
приблизително съответната
 у стойност ето тук.
Искаме да определим 
приблизително тази стойност.
Точно ето тук се намира
 f от 4,36. Още веднъж,
ние правим предположение, защото 
нямаме калкулатор под ръка.
А как може да решим задачата като използваме 
това, което знаем за производните?
Какво ще стане ако трябваше 
да намерим уравнението
на допирателната към точката,
 която се намира точно ето тук?
Уравнението на допирателната, когато
 х е равно на 4, като тук ще използваме

Portuguese: 
E sabemos que f de quatro é igual a dois.
Aqui é quando x vale quatro.
O desenho está um pouco fora de escala,
espero que não prejudique a compreensão.
Aqui será dois.
Essa é f de quatro.
O que queremos aproximar é f de 4.36.
4.36 estará mais ou menos aqui.
Queremos aproximar esse valor de y aqui.
Isso é f de 4.36.
Novamente, estamos fazendo tudo
isso sem recorrer a uma calculadora.
Como fazemos isso usando o
que sabemos sobre derivadas?
E se descobríssemos a equação da
reta tangente a este ponto aqui?
A equação da tangente
onde x é igual a quatro.

English: 
f of x, and we know f of 4 is equal to 2.
F of 4 is equal to 2, so this is when x is
equal to 4.
I haven't drawn it really to scale, but
hopefully, this is clear enough.
So that right over here is going to be 2.
That's f of 4.
And what we wanna approximate is f of
4.36, so 4.36 might be right around, right
around there, and so we want to
approximate,
we wanna approximate this y value right
over here.
We want to approximate that.
Right over here is f of 4.36, and, once
again, we're assuming we don't have a
calculator at hand.
So, how can we do that using what we know
about derivatives?
Well, what if we were to figure out an
equation for the line
that is tangent to the point, to tangent
to this point right over here.
So the equation of the tangent line at x
is equal to 4, and then we use

Thai: 
การประมาณการเชิงเส้นเพื่อหาค่าใกล้ ๆ จุดนี้
เทคนิคนี้เรียกว่า การประมาณการเชิงเส้นท้องถิ่น
(local linearization)
ที่ผมจะบอกก็คือ เรามาหาสมการของเส้นตรงนี้
เราจะเรียกว่า L ของ x
และใช้สิ่งนี้เพื่อประมาณค่า
เราหาค่ามันที่ 4.36 ได้ และหวังว่า
มันจะทำได้ง่ายกว่าการหาค่าตรงนี้ตรง ๆ
มันจะทำได้ง่ายกว่าการหาค่าตรงนี้ตรง ๆ
แล้วเราทำยังไง
วิธีคิดวิธีหนึ่งคือ
ที่จริงมันมีหลายวิธีนะ แต่วิธีหนึ่งก็คือ
เอาล่ะ มันจะเท่ากับ f ของ 4 ซึ่งเท่ากับ 2
มันจะเท่ากับ f ของ 4 บวกความชัน
ความชันที่ x เท่ากับ 4
ซึ่งแน่นอนว่าเท่ากับอนุพันธ์ของ f ที่ 4
มันคือความชันของเส้นตรงนี้สำหรับ L ของ x
คือ f ไพรม์ ของ 4
ให้ผมทำให้ชัดนะ

Korean: 
우리는 선형화를 사용하고 지역 값을 근사하기 위해 근사되었기 때문에
이 기술은 지역 선형화라고 부릅니다
제가 지금 말하는 것은 이 선의 방정식을 구해보자는 것입니다
우리가 그것을 L(x)라고 합시다
우리는 저 식을 사용할 수 있습니다
우리는 4.36에서 계산할 수 있고
약간 쉬울꺼라 바랍니다
저기서 값을 계산하는것 보다는
그래서 어떻게 할까요?
생각할 수 있는 한가지 방법은
선을 표현하는 방법은 많지만 그 중 한가지 방법은
L(x)는 2인 f(4)
L(x)는 f(4)더하기 x=4에서의 기울기
물론 f'(4)입니다
L(x)의 기울기가 f'(4)입니다
분명히 해봅시다

English: 
that linearization, that linearization
defined to approximate values local
to it, and this technique is called local
linearization.
So what I'm saying is, let's figure out
what this, the equation of this line is.
Let's call that l of x.
And then we can use that to appro, and
then we can evaluate that at 4.36, and
hopefully
that will be a little bit easier to do
than to try and figure out this right over
here.
So how would we do that?
Well, one way to think about it, and
obviously, there are
many ways to express a line, but one way
to think about it is,
okay, it's going to l of x is going to be
f of 4, which is 2.
It's going to be f of 4 plus the slope,
the slope at, at x equals
4, which is, of course, the derivative f
prime of 4, so
that's going to be the slope of this line
of l of x is f prime of 4.
Let me make that clear.

Bulgarian: 
линейно приближение, дефинирано за
 намиране на съседни стойности.
Тази техника се нарича линейно 
приближение (локална линеаризация).
Аз предлагам да намерим 
уравнението на тази допирателна.
Нека да я наречем L от х.
И сега може да използваме 
тази права като приближение
и да изчислим стойността на функцията 
в точката 4,36. Надявам се, че това
ще бъде малко по-лесно 
да го направим,
отколкото да се опитваме 
да изчислим стойността ето тук.
А как ще го направим?
Един начин да мислиш 
за това е... Очевидно има
много начини да се представи права, но 
един начин да мислиш за това е следният.
L от х ще бъде равно на f от 4, 
което е равно на 2...
L от х ще бъде равно на f от 4, 
плюс наклона в точката
х равно на 4, което разбира се, 
е производната f' от 4,
така че на това ще бъде равен наклонът
 на тази права L от х, или f' от 4.
Нека да го изясня.

Portuguese: 
E então usamos essa linearização para
aproximar os valores locais.
Essa técnica se chama linearização local.
O que digo é, vamos descobrir
a equação desta reta.
Podemos chamá-la de L de x.
Podemos calcular isso em 4.36.
Fazendo assim será mais fácil
do que tentar descobrir isso aqui.
Como fazemos isso?
Há muitas formas de expressar
uma linha mas uma delas é:
L de x será f de quatro, que é dois,
mais a inclinação em x igual a quatro,
que é a derivada.
f linha de quatro será a inclinação
de L de x.
Deixe-me ser mais claro.

Czech: 
A potom použili tuto linearizaci
k odhadu hodnoty v okolí.
Tato technika se nazývá
aproximace pomocí tečny.
Naším úkolem tedy je zjistit,
jaká je rovnice této přímky.
Nazvěme ji L(x).
Potom ji můžeme použít
k určení hodnoty 4,36.
A doufejme, že to bude snadnější,
než výpočet tohoto zde.
Jak to tedy
můžeme udělat?
Jeden způsob, jakým se na
to můžeme podívat, je tento.
L(x) bude f(4), tedy 2,
plus sklon v bodě f(4).
Což je samozřejmě
derivace v f(4).
To bude tedy sklon
této přímky f'(4).

English: 
So this right over here is the slope.
The slope when x is at, at x equals 4, so
this is a slope of this entire line, and
so any
other point on this is gonna be f' of 4
plus
the slope times how far you are away from
x equals 4.
So it's going to be times x minus 4.
Let's just, let's just validate that this
makes sense.
When we put 4.36 here, when we put 4.36
here, actually let me
zoom in on this graph just to make things
a little bit clearer.
So, if this is, so I'm gonna do a zoom in.
I'm gonna do a zoom in.
I'm gonna try to zoom in into this region
right over here.
So, this is the point.
This is the point (4, f of 4), and we are
going to graph l of x.
So let me do that.
So this right over here is l of x.
That's l of x.

Czech: 
Aby to bylo jasné,
toto je sklon pro x rovno 4.
Je to tedy sklon
této přímky.
A jakýkoli jiný bod
na této přímce bude
f(4) plus sklon křivky krát
vzdálenost od bodu x rovno 4.
Bude to tedy
krát (x minus 4).
Pojďme ověřit, že
to dává smysl.
Když sem
dosadíme 4,36...
Jen si přiblížím tento graf,
aby to bylo jasnější.
Přiblížím tuto oblast.
Toto je tedy
bod [4; f(4)].
Nakreslíme L(x).

Portuguese: 
Isso aqui é a inclinação
quando x é igual a quatro.
A inclinação desta linha.
Qualquer outro ponto aqui
será f de 4 mais a inclinação
vezes o quão distante de x
igual a quatro você está.
Então será vezes x menos quatro.
Vamos verificar a validade disso.
Quando colocamos 4.36 aqui--
deixe-me dar um zoom neste
gráfico pra ficar mais claro--
Darei um zoom nessa região aqui.
Aqui é o ponto quatro vírgula f de quatro.
Vou fazer o gráfico de L de x.
Deixe-me fazer isso.
Isso aqui é L de x.

Thai: 
พจน์ตรงนี้คือความชัน
ความชันเมื่อ x เท่ากับ 4
มันคือความชันของเส้นตรงนี้ทั้งเส้น
และจุดอื่น ๆ บนเส้นนี้จะเป็น f ของ 4 บวก
ความชัน คูณด้วยความห่างจากตำแหน่ง x=4
ก็คือคูณด้วย x ลบ 4
เรามาตรวจสอบว่ามันสมเหตุสมผลจริง
เวลาเราแทนค่า 4.36 ตรงนี้ ที่จริงให้ผม
ซูมกราฟเข้าไปอีกนิด จะได้ทำให้เห็นชัดขึ้น
ผมซูมเข้าไปนะ
ผมซูมเข้าไปนะ
ผมซูมเข้าไปตรงพื้นที่นี้
เอาล่ะ นี่คือจุด
มันคือจุด (4, f ของ 4) 
และเราจะวาดกราฟ L ของ x
ให้ผมทำนะ
เส้นนี้ก็คือ L ของ x
เส้นนี้ก็คือ L ของ x

Bulgarian: 
Това ето тук е наклонът.
Наклонът, когато х е равно на 4,
т.е. това е наклонът на цялата линия,
и следователно всяка точка от нея 
ще бъде равна на f от 4 плюс наклона,
умножен по разстоянието
от теб до точката х = 4.
Следователно ще се получи 
по х минус 4.
Нека само да потвърдим, че 
в това има логика.
Когато поставим 4,36 тук, 
то всъщност...
Нека да увелича изображението, така че 
нещата да се виждат малко по-ясно.
Ако това е така...
Нека да увелича изображението.
Нека да начертая по-голям 
избрания участък.
Ще се опитам да увелича 
ето този участък точно тук.
И така, това е точката...
Това е точката (4; f от 4) 
и ще начертаем правата L от х.
Нека го направя.
Това ето тук е L от х.
Това е L от х.

Korean: 
여기 있는 것이 기울기입니다
x=4에서의 기울기는
이것이 이 선 전체의 기울기입니다
이 선의 모든 점에서의 기울기가 f'(4)입니다
그 기울기 곱하기 x=4에서 얼마나 멀리 떨어져 있는지 입니다
따라서 곱하기 (x-4)가 될 것입니다
이것이 말이 되도록 해봅시다
여기에 4.36을 넣으면
분명히 하기 위해 그래프를 확대해봅시다
만약 이것이
확대하겠습니다
확대하겠습니다
이 부분을 확대 해보겠습니다
이것이 점입니다
이것이 (4,f(4))이고 L(x)를 그릴것입니다
해봅시다
이것이 L(x)입니다

Czech: 
A tento bod zde
je bod [4,36; f(4,36)].
A k tomu, abychom
odhadli tuto hodnotu,
potřebujeme zjistit
tuto hodnotu zde.
Tato hodnota bude
[4,36; L(4,36)].
Tato přímka v bodě
x je rovno 4,36.
Čemu se toto
bude rovnat?
Pojďme sem dosadit.
L(4,36) bude to f(4), tedy 2 plus 
derivace, sklon této přímky.
Tedy plus derivace v f(4)
krát x minus 4.
Tedy 4,36 minus 4,
to bude krát 0,36.
To dává smysl.
Začínali jsme
v bodě 2.

Korean: 
저것이 L(x)입니다
이 점이 (4.36,f(4.36))이고
이 값을 근사하고 싶습니다
이 값이 무엇이 될지 알고 싶습니다
이것이 어떻게 될까요?
이 점은 (4.36,L(4.36))이 될것입니다
x=4.36일때 저것이 어떻게 될까요?
저 값이 무엇일까요?
한번 봅시다
그냥 계산해봅시다
L(4.36)은 f(4)
그니깐 2 더하기 기울기인 f'(4) 곱하기
(x-4)입니다
(4.36-4)는 0.36이 될 것이고 말이 됩니다

Thai: 
และนี่ จุดตรงนี้
คือจุด (4.36, f ของ 4.36) และ
วิธีที่เราจะประมาณค่า ก็คือ
หาค่าตรงนี้คือเท่าไร
และค่าตรงนี้จะเป็นเท่าไร
จุดตรงนี้ก็คึอ (4.36, L ของ 4.36)
เส้นตรงนี้ แทนค่าที่ x เท่ากับ 4.36 จะได้เท่าไร
แล้วนั่นจะเท่ากับเท่าไร
มาดูกัน
ลองคำนวณออกมา
L ของ 4.36 จะเท่ากับ f ของ 4
มันจึงเท่ากับ 2 บวก อนุพันธ์ 
ก็คือความชันของเส้นนี้
บวก f ไพรม์ของ 4 คูณ (x ลบ 4)
ซึ่ง 4.36 ลบ 4 จะเท่ากับ 0.36 ซึ่งก็มีเหตุผล

English: 
And let's say this, right over here, this
right over
here, is the point (4.36, f of 4.36), and
the way
that we're, we're gonna approximate this
value is to figure
out what, to figure out, what this value
is right over here.
And what is this one going to be?
This right over here is going to be, this
is going to be (4.36, L of 4.36).
This line evaluated when x is equal to
4.36, and what is that going to be?
What is that going to be equal to?
Well, let's see.
Let's just evaluate it.
L of 4.36 is going to be f of 4.
So, it's going to be 2 plus the
derivative, so the
slope of this line plus f prime of 4 times
x minus 4.
So, 4.36 minus 4 is going to be times
0.36, and that makes sense.

Bulgarian: 
И нека да кажем следното...
Точно ето тук
се намира точката 
(4,36; f(4,36))
и начинът, по който 
ще определим тази стойност,
е да намерим на какво е равна 
тази стойност ето тук.
А на какво ще бъде равна тя?
Тази точка тук ще бъде (4,36; L от 4,36).
Тоест точка от правата,
 изчислена, когато х = 4,36.
И на какво ще бъде равно това?
На какво ще бъде равно това?
Е, нека да видим.
Нека просто да го изчислим.
L от 4,36 ще бъде равно на f от 4...
Тоест ще бъде равно на 2
 плюс производната,
т.е. наклонът на тази линия, 
плюс f' от 4 по х минус 4.
И така, 4,36 минус 4, т.е. ще умножим 
по 0,36 и това има смисъл.

Portuguese: 
E digamos que aqui seja o ponto
4.36 vírgula f de 4.36.
E vamos aproximar esse valor
descobrindo este aqui.
O que será esse?
Será 4.36 vírgula L de 4.36.
O quanto vale essa linha em 4.36?
Vejamos.
Basta calcular.
L de 4.36 será L de quatro que vale dois
mais a derivada, f linha de quatro,
vezes x menos quatro.
4.36 menos quatro, será então vezes 0.36.

Czech: 
Změna hodnoty
x je rovna 4,36.
Změna hodnoty y tedy bude
sklon přímky krát změna x,
abychom dostali
tuto hodnotu.
Pojďme zjistit, čemu je
roven tento výraz.
Abychom to zjistili, potřebujeme
vědět, čemu je rovna derivace f(4)...
Pojďme zpět nahoru,
abychom si to lépe představili.
Derivace f v bodě x je
jedna polovina x na minus jednu polovinu.
Používám zde vzorec
pro derivování mocniny.
Tedy derivace f(4) je rovno jedné polovině
krát 4 na minus jednu polovinu.
Což je samozřejmě rovno
jedna polovina krát jedna polovina.
4 na jednu
polovinu jsou 2.

Korean: 
여러분은 2에서 시작합니다
x를 4.36으로 변화시킵니다
y변화량은 기울기 곱하기 x의 변화량입니다
우리가 저 값을 알기 위한 것입니다
이것이 무엇인지 알아봅시다
이것이 무엇인지
우리는 f'(4)를 먼저 구해야합니다
이것을 여기에 시각화해봅시다
f'(x)는 급수의 법칙을 사용해보면
(1/2)*x^(-1/2)입니다
때문에 f'(4)는 (1/2)*4^(-1/2)입니다
물론
1/2 * 1/2 입니다
4^(1/2)는 2입니다

Portuguese: 
Faz sentido. Começamos em dois
e a variação em x é 4.36.
Então a variação em y será a inclinação
vezes a variação em x.
Para obtermos aquele valor ali.
Vamos descobrir o que isso realmente é.
Para fazer isso precisamos de f linha
de quatro...vamos voltar aqui em cima.
Deixarei isso aqui visível.
Então f linha de x será um meio
x elevado a menos um meio.
Usei a regra do expoente aqui.
Logo, f linha de quatro será meio
vezes quatro elevado a menos meio.
Que é igual a meio vezes meio.

Thai: 
คุณเริ่มที่ 2 แล้วบอกว่า เอาล่ะ
ค่า x ที่เปลี่ยนไปคือ 4.36
[ที่ถูกต้องเป็น 0.36]
ดังนั้นค่า y ที่เปลี่ยนไปจึงจะเท่ากับ
ความชันคูณค่า x ที่เปลี่ยนไป
เพื่อให้ได้ค่านั้น เพื่อให้ได้ค่าตรงนี้
เอาล่ะ มาหากันว่ามันได้เท่าไร
เอาล่ะ มาหากันว่ามันได้เท่าไร
ในการนี้ เราต้องหาค่า f ไพรม์ของ 4 
ถ้างั้นก็กลับมาตรงนี้
ผมพยายามปล่อย ผมจะปล่อยภาพไว้ตรงนี้
มาดูกัน f ไพรม์ของ x จะเท่ากับ
ครึ่งหนึ่งของ x ยกกำลังลบ 1/2
ตามกฎฟังก์ชันยกกำลังตรงนี้
f ไพรม์ของ 4 ก็จะเท่ากับ
1/2 คูณ 4 ยกกำลัง ลบ 1/2 ซึ่งก็คือ
1/2 คูณ 1/2
4 ยกกำลัง 1/2 ได้ 2

Bulgarian: 
Започваш в точката 2 и си казваш:
моето изменение за х
 е равно на 4,36.
Тогава моето изменение за у ще бъде равно на 
наклона, умножен по това изменение за х,
за да получа тази стойност, т.е. 
за достигна ето тази стойност тук.
Добре, нека да намерим...
Нека да намерим
на какво всъщност е равен 
ето този израз.
За да направим това, трябва да намерим 
f' от 4, така че нека да се върнем тук.
Ще се опитам да оставя 
този увеличен чертеж тук.
Нека да видим...
f' от х ще бъде равно
на 1/2 по х на степен минус 1/2,
просто прилагаме правилото за 
намиране производна на степен.
Следователно f' от 4 е равно на 1/2
умножено по 4 на степен –1/2, което
разбира се, е равно на 1/2 по 1/2.
4 на степен 1/2 е равно на 2.

English: 
You're starting at 2 and you're saying,
okay, my
change in x, my change in x is 4.36.
So, my change in y is going to be my slope
times that change
in x to get me that value, to get me that
value right over there.
So, let's figure out, let's figure out
what this, let's
figure out what this thing, what this
thing actually is.
So to do that, we need to figure out f
prime of 4, so let's go back up here.
I'll try to leave actually, I'll leave
this little visualization here.
So let's see, f prime, f prime of x is
going to be
one-half x to the negative one-half just
using the power rule over here.
So f prime of 4, f prime of 4 is equal to
one-half
times 4 to the negative one-half, which
is, of
course, equal to one-half times one-half.
4 to the one-half would be 2.

Portuguese: 
Quatro elevado a menos meio é meio.
Então isso será um quarto.
Merecemos um rufar de 
tambores agora... L de 4.36
é igual a f de quatro
mais f linha de quatro.
Usarei amarelo aqui.
Mais f linha de quatro vezes 4.36--
deixe-me usar outra cor--
vezes 4.36 menos quatro.
Deixe-me fazer os quatros da mesma cor.

Bulgarian: 
4 на степен –1/2 
ще бъде равно на 1/2.
Тоест f' от 4 e равно на 1/4.
Е, сега заслужаваме поздравление, 
защото L от 4,36 е равно
на f от 4...Нека просто 
да го запиша по следния начин.
f от 4 плюс f' от 4 плюс...
О, защо избрах този цвят!
Нека да го запиша в жълто.
Плюс f' от 4, умножено...
Умножено по 4,36.
4,36
Нека да запиша последния израз 
с друг цвят, за да се отличава.
L от 4,36...Получава се 
по 4,36 минус 4...Минус 4.
Всъщност, нека да направя
 всички числа 4 в един цвят,
за да се вижда, че 
са едно и също нещо.

Czech: 
4 na minus jednu polovinu 
je rovno jedné polovině.
Toto je tedy rovno
jedné čtvrtině.
Dostaneme tedy
L v bodě 4,36 je rovno
f(4) plus f s čárkou (4)
krát (4,36 minus 4).
Toto je vždy stejné.

Korean: 
4^(-1/2)는 1/2가 될것입니다
그래서 이 값은 1/4가 됩니다
L(4.36)은 f(4)
이것을 다시 써봅시다
f(4)+f'(4)
이런 제가 왜 색깔을 바꾼거죠?
노란색으로 해봅시다
+f'(4)곱하기
4.36
4.36
새로운 색으로 해봅시다
(4.36-4)를 곱하면 됩니다
모든 4를 같은색으로 만들어봅시다
같은 것으로 보입니다

Thai: 
4 ยกกำลัง ลบ 1/2 จึงเป็น 1/2
มันก็จะเท่ากับ 1/4
เราเตรียมตีกลองฉลองกันหน่อยตอนนี้
เราได้ L ของ 4.36 เท่ากับ
f ของ 4 ซึ่งเท่ากับ -- ขอผมเขียนใหม่นะ
มันคือ f ของ 4 บวก f ไพรม์ของ 4 บวก
อึ๋ย ทำไมผมเปลี่ยนไปสีนั้นล่ะเนี่ย
ขอผมใช้สีเหลืองนะ
บวก f ไพรม์ของ 4 คูณ
คูณ 4.36
คูณ 4.36
ผมใช้สีใหม่ตรงนี้นะ จะได้เห็นชัด ๆ
เอาล่ะ 4.36 ลบ 4
ที่จริงผมเขียนเลข 4 ทุกตัว
ให้เป็นสีเดียวกันดีกว่า
คุณจะได้เห็นว่ามันเหมือนกันหมดแบบนี้

English: 
4 to the negative one-half is going to be
one-half.
So, this is equal to 1 4th.
So L of, we deserve a little bit of a drum
roll now, L of 4.36 is equal
to f of 4, is equal to f of 4, which is
eq, let me just rewrite it.
It's f of 4 plus f prime of 4 plus, gee,
why am I switching to that color.
Let me do the yellow.
Plus f prime of 4, times, times,
times, 4.36.
4.36.
Let me make this actually a new color,
just so we see it.
So, 4.36, so times 4.36 minus 4, minus
4, and actually let me make all the 4s one
color,
too, so you see it's the same, so just
like that.

Portuguese: 
Quanto será isso?
Isso já sabemos, é dois.
Isso também--usarei amarelo--
vale um quarto.
E esta parte é 0.36.
Isso será igual a dois mais,
um quarto vezes 0.36 é 0.09.
Isso será igual a 2.09.
Essa é nossa aproximação.
Ela é um pouco maior que o valor
real da raiz quadrada de 4.36.
Vamos escrever aqui em cima.
Isso será aproximadamente...
escreverei aqui embaixo.

Korean: 
무엇이 될까요?
이것은 2라고 미리 계산했습니다
저것은 계산한 바에 의하면
1/4이고 저것은 0.36입니다
그래서 2+(1/4)*0.36입니다
즉 0.08입니다
그래서 2.09가 될 것입니다
이것이 근사값입니다
그래프를 기반으로 구한 값이라
실제 √4.36 보다는 약간 큽니다
이것을 이 위에 쓰겠습니다
이것은 근사적으로
이렇게 써봅시다
루트
여기 밑에 쓰겠습니다

English: 
So what is this going to be?
Well, this we already established is
positive 2.
This we already established, and we do
this in a yellow color.
This, we already established is 1 4th, and
this part right over here is 0.36.
So, this is going to be equal to 2 plus 1
4th times
0.36, is 0.9 or is 0.09, so this is going
to be equal to, this is going to be equal
to 2.09.
So that is our approximation and should be
at
least the way, based on how I graphed it,
a
little higher than the actual value of the
square
root of 4.36, but we could write that up
here.
This is going to be approximately, let me
just write it this way.
The square root, I'll just write it down
here.

Thai: 
แล้วมันจะได้เท่าไร
พจน์นี้เราได้ดูไปก่อนแล้วว่าเท่ากับ 2
พจน์นี้เราดูไปก่อนแล้วด้วยสีเหลือง
มันคือ เศษ 1 ส่วน 4
และส่วนตรงนี้คือ 0.36
ที่ก็จะเท่ากับ 2 บวก เศษ 1 ส่วน 4 คูณ
0.36 ซึ่งคือ 0.09 ดังนั้น
นี่จะเท่ากับ 2.09
นี่คือค่าประมาณของเรา และอย่างน้อย
จากวิธีที่ผมวาดกราฟ
ควรมากกว่าค่าจริงของรากที่สอง
ของ 4.36 แต่เราเขียนมันบนนี้
อันนี้จะประมาณ ขอผมเขียนมันแบบนี้นะ
รากที่สอง ผมจะเขียนมันลงไปตรงนี้

Czech: 
Kolik to tedy bude?
Toto už víme, že jsou plus 2,
toto už víme, že je jedna čtvrtina
a tato část zde je 0,36.
Toto celé tedy bude rovno
2 plus 1/4 krát 0,36.
Tedy 0,09.
Dostáváme tedy
2,09.
Toto je proto náš odhad.
Mělo by to být lehce nad skutečnou
hodnotou druhé odmocniny z 4,36.
Napišme to sem nahoru.
Toto bude přibližně...
Napíšu to raději sem.

Bulgarian: 
И на какво ще бъде 
равен този израз?
Определихме, че f от 4 
е равно на плюс 2.
f' от 4 вече е определено 
и ще го запиша в жълт цвят.
Определихме, че е равно на 1/4, 
а този израз ето тук е равен на 0,36.
Следователно всичко 
ще бъде равно на 2,
плюс 1/4 по 0,36, 
което е равно на 0,09.
И това ще бъде равно на 2,09.
Това се получава за нашето 
приближение и така трябва да бъде,
поне по начина, по който
 съм го изобразил.
Малко по-високо 
от действителната стойност
на квадратен корен от 4,36, 
но може да го запишем ето тук горе.
Това ще бъде приблизително равно...
Просто ще го запиша по следния начин.
Квадратен корен...
Ще го запиша ето тук долу.

Portuguese: 
A raiz quadrada de 4.36 que 
é o mesmo que f de 4.36
é aproximadamente igual a 2.09.
Vamos agora usar uma calculadora
pra ver se nossa aproximação é boa.
Aqui está ela.
A raiz quadrada de 4.36 dá 2.088.
Se aproximarmos à segunda casa
teremos uma excelente aproximação.
Como vimos nesse gráfico indicativo aqui,
nossa aproximação foi de fato maior.
Legendado por: [Vitor Tocci]
Revisado por: [Rodrigo Melges]

Czech: 
...druhá odmocnina z 4,36,
což je to samé jako f(4,36).
Toto je přibližně
2,09.
Řekněme, že jsme
zrovna našli kalkulačku.
Pojďme se, jen pro zajímavost,
podívat, jak dobrý tento odhad je.
Chceme spočítat druhou
odmocninu z 4,36.
A dostáváme 2,088.
Když to tedy zaokrouhlíme na setiny,
získali jsme velmi dobrý odhad.
Přesně tak, jak jste viděli
v tomto názorném grafu,
náš odhad byl trošku vyšší
než skutečná hodnota.

English: 
So we could say the square root of 4.36,
which is the same thing as f of 4.36.
This is approximately equal to 2.09.
Now let's just say we happen to find a
calculator, and just
out of curiosity let's see how good of an
approximation that is.
Let's get a calculator out.
And so, we wanna do the square root of
4.36, and we get 2.088.
So, we actually, if we round to the
nearest hundredths,
we got a pretty good approximation just
like we saw.
In this, in this indicative graph right
over here, it is,
our approximation was indeed a little bit
higher than the actual value.

Korean: 
√4.36 즉 f(4.36)은
근사적으로 2.09입니다
계산기가 생겼다고 해봅시다
근사를 얼마나 잘했는지 확인해 봅시다
계산기를 꺼내봅시다
√4.36을 계산해보면 2.088이 나옵니다
소수점 2자리 정도 수준으로 비슷합니다
볼 수 있듯이 근사를 꽤 잘한것 같습니다
이 그래프에서
실제 값보다 약간 높은 근사값을 구했습니다

Thai: 
เราบอกได้ว่า รากที่สองของของ 4.36
ซึ่งเท่ากับ f ของ 4.36
อันนี้ประมาณเท่ากับ 2.09
ทีนี้ ลองสมมุติว่าเราเกิดหาเครื่องคิดเลขได้
ลองหาว่าเราประมาณได้ดีแค่ไหนกัน
จะได้หายสงสัย
ลองเอาเครื่องคิดเลขออกมา
แล้ว เราหารากที่สองของ 4.36
ได้ 2.088
ที่จริง ถ้าเราปัดเป็นทศนิยม
สองตำแหน่งที่ใกล้ที่สุด
เราจะได้ค่าประมาณดีทีเดียวอย่างที่เห็น
ในนี้ ตามที่กราฟบอกตรงนี้
ค่าประมาณของเรามากกว่าค่าจริงนิดหน่อยจริง

Bulgarian: 
Може да заявим, че квадратен корен от 4,36,
което е същото нещо като f от 4,36,
ще бъде приблизително равно на 2,09.
А сега нека да приемем, че 
сме намерили калкулатор и
сме любопитни колко точно 
е нашето приближение.
Нека да вземем един калкулатор.
Искаме да намерим квадратен корен 
от 4,36 и получаваме 2,088.
Следователно, ако закръглим
 до най-близките стотни,
то ние сме получили много добро
 приближение. И както може да видим,
на тази показателна графика ето тук,
нашето приближение действително беше 
малко по-високо от истинската стойност.
