
English: 
- So, let's say we've been given all
this information about the function g
and it's derivative
evaluated at x equals two.
We know g of two is equal to three.
G prime of two is equal to one.
The second derivative of g
evaluated two is negative one.
The third derivative of g
evaluated at two is two.
Given that, what we're
being tasked with is
we want to use the second
degree Taylor polynomial
centered at x equals two to
approximate g prime of one.
Not g of one, g prime of
one and so I encourage you
to pause this video and try
to think about it on your own.
I'm assuming you've had a go at it.
Let's just remind ourselves what a second
degree Taylor polynomial
centered at x equals two
would look like for a
general function f of x.

Korean: 
다음은 x=2에 대하여
함수 g와
그 도함수들에 대한 정보를
나타낸 것입니다
g(2) = 3
g'(2) = 1
g''(2) = -1
g'''(2) = 2
이러한 정보가
주어진 상황에서
g'(1)의  근삿값을
구하기 위하여
x=2를 중심으로
2차 테일러 다항식을 이용합니다
g(1)이 아닌 g'(1) 입니다
강의를 멈추고
스스로 생각해 보세요
여러분이 생각해 봤다고
가정하겠습니다
x=2가 중심인
이차 테일러 다항식이
일반적인 함수
f(x)에 대하여
어떤 모습인지
상기시켜 봅시다

Bulgarian: 
Да кажем, че ни е дадена цялата 
тази информация за функцията g
и нейната производна, 
изчислена за х = 2.
Знаем, че g(2) е равна на 3.
g'(2) е равно на 1.
Втората производна g''(2)
е равна на –1.
Третата производна на g,
изчислена за 2, е равна на 2.
Като е дадено това, от нас не иска
да използваме ред  
на Тейлър от втора степен,
центриран около х = 2, за да
апроксимираме g'(1).
Не g(1), а g' (прим) от едно. 
Препоръчвам ти
да спреш видеото и да опиташ
да го решиш самостоятелно.
Предполагам, че опита.
Само да си припомним какво е
ред на Тейлър от втора степен,
центриран около х = 2
за някаква произволна 
функция f(х).

Thai: 
สมมุติว่าเราได้ข้อมูลทั้งหมด
นี้เกี่ยวกับฟังก์ชัน g
และอนุพันธ์ของมันหาค่าที่ x เท่ากับ 2
เรารู้ว่า g ของ 2 เท่ากับ 3
g ไพรม์ของ 2 เท่ากับ 1
อนุพันธ์อันดับสองของ g 
หาค่าที่ 2 เท่ากับลบ 1
อนุพันธ์อันดับสามของ g หาค่าที่ 2 เท่ากับ 2
จากข้อมูลนั้น ภารกิจของเราคือ
เราอยากใช้พหุนามเทย์เลอร์ดีกรีสอง
ที่มีศูนย์กลางที่ x เท่ากับ 2 
เพื่อประมาณ g ไพรม์ของ 1
ไม่ใช่ g ของ 1, g ไพรม์ของ 1 
แล้วผมแนะนำให้คุณ
หยุดวิดีโอนี้ ลองคิดเองก่อน
ผมถือว่าคุณได้ลองแล้วนะ
ลองทบทวนกันว่าพหุนามเทย์เลอร์
ดีกรีสองที่มีศูนย์กลางที่ x เท่ากับ 2
จะเป็นอย่างไรสำหรับ
ฟังก์ชัน f ของ x โดยทั่วไป

Portuguese: 
Digamos que recebemos
algumas informações sobre a função g
e o valor da sua derivada
em x igual à dois.
Sabemos que g de dois é igual a três.
g linha de dois é igual à um.
A segunda derivada de g
de dois é menos um.
A terceira derivada de g em dois é dois.
Com essas informações,
queremos usar o polinômio de Taylor
de segundo grau
em x igual à dois para aproximar
o valor de g linha de um.
Não g de um, mas g linha de um.
Recomendo que você pause o vídeo
e tente fazer isso sozinho.
Estou assumindo que você já tentou.
Vamos recordar o que é
um polinômio de Taylor
de segundo grau em x igual à dois
para um função qualquer f de x.

Thai: 
f ของ x จะประมาณเท่ากับ
มันจะเป็น f ของ 2 บวก f ไพรม์ของ 2
คูณ x ลบ 2 บวก f ไพรม์ไพรม์ของ 2
คูณ x ลบ 2 กำลังสอง 
ทั้งหมดนั้นส่วน 2 แฟคทอเรียล
มันจะให้เทอมดีกรีสอง
เพราะมันคือ x ลบ 2 กำลังสอง
นี่จะให้พหุนามดีกรีสองแก่เรา
นี่คือกรณีทั่วไป
ถ้าเราอยากหาค่าประมาณ
สำหรับ f ที่ศูนย์กลาง
อยู่ที่ x แต่เราอยากหาสำหรับ g ไพรม์
ขอผมเขียนตัวนี้ลงไปนะ
เอาล่ะ ผมจะใช้สีฟ้า
เรามี g ไพรม์ของ x คือสิ่งที่เรา
จะลอง และประมาณ แล้วเรา
จะหาค่ามันที่ x เท่ากับ 1

English: 
F of x would approximately be equal to,
it would be f of two plus f prime of two
times x minus two plus
f prime prime of two
times x minus two squared, all
of that over two factorial.
That would get us to a second degree place
because it's x minus two squared.
This is gonna give us a
second degree polynomial.
This is the general case.
If we want to find the
approximation for f centered
at x equals but we're
gonna do it for g prime.
Let me write this down.
Alright, so I'll do it in blue.
So, we have g prime of
x is what we're going
to try and approximate and then we're
going to evaluate it at x equals 1.

Portuguese: 
f de x seria aproximadamente igual -
seria f de dois mais f linha de dois
vezes x menos dois
mais f linha linha de dois
vezes x menos dois ao quadrado
sobre dois fatorial.
Obteríamos um polinômio de segundo grau,
pois temos x menos dois ao quadrado.
Obteremos um polinômio de segundo grau.
É o caso geral.
Se queremos fazer uma aproximação
para f em x igual a dois-
mas faremos isso para g linha.
Deixe-me escrever isso. Farei em azul.
g linha de x é o que tentaremos aproximar
e calcular seu valor em x igual à um.

Korean: 
f(x)는
f(2) + f'(2)(x - 2) + f''(2)(x - 2)²/2!
f(2) + f'(2)(x - 2) + f''(2)(x - 2)²/2!
f(2) + f'(2)(x - 2) + f''(2)(x - 2)²/2!
이차식이 나올 것입니다
(x - 2)²이 있기 때문이죠
이를 이용하여
이차 다항식을 구합니다
이는 일반적인 경우입니다
x의 값을 중심으로
근사하는 경우입니다
하지만 우리는
g'에 대하여 해결해야 합니다
한번 적어볼게요
파란색으로 하죠
g'(x)가 있는데
이를 근사하고
x=1에 대하여
계산할 것입니다

Bulgarian: 
f(х) ще е приблизително 
равно на
f(2) + f'(2) по (х – 2)
плюс f''(2) по (х – 2)^2
цялото върху 2 факториел.
Това ни дава втора степен,
защото имаме (х – 2)^2.
Така получаваме полином 
от втора степен.
Това е общият случай,
когато искаме да намерим
апроксимацията на f, центрирана
около х = 2, но ще трябва
да го направим за g'.
Ще го запиша.
Ще го направя със синьо.
Значи g'(х) е това, което
се опитваме да апроксимираме,
и после ще го изчислим 
за х = 1.

Portuguese: 
g linha de x será aproximadamente igual à-
será a mesma função
que estou tentando aproximar
em x igual à dois.
Estou estimando f de x, que é
o valor dessa função em x igual à dois.
Se estou estimando g linha de x,
seria o valor dessa função
em x igual à dois,
mais a primeira derivada disto,
que é a segunda derivada de g.
g linha de dois vezes x menos dois
mais a segunda derivada da nossa função
que estou tentado aproximar,
mas a segunda derivada
de g linha será a terceira derivada de g.
Será g três linhas de duas vezes x
menos dois ao quadrado
sobre dois fatorial.
Essas expressões são dadas.
-Usarei outras cores-
Eles nos dizem que g de dois
é igual à três.

English: 
G prime of x is going to
be approximately equal
to, well same thing, it's
going to be the function
that I'm going to try to approximate
evaluated at two so g prime of two.
Notice, so I'm approximate f of x,
it's that function evaluated at two.
If I'm approximating g prime of x,
it's that function evaluated at two.
Then, plus the first
derivative of this thing
which is the second derivative of g.
G prime prime of two times x minus two
and then plus the second
derivative of the function
that I'm trying to approximate
but the second derivative
of g prime is going to be
the third derivative of g.
It's going to be g prime prime
prime of two times x minus
two squared, all of
that over two factorial.
Now, they tell us what these things are.
Let me use some new colors here.
They tell us g of two is equal to three.
This right here, oh, actually no,

Thai: 
g ไพรม์ของ x จะประมาณเท่ากับ
เหมือนเดิม มันจะเป็นฟังก์ชัน
ที่ผมพยายามประมาณ
หาค่าที่ 2 แล้ว g ไพรม์ของ 2
สังเกตดู ผมประมาณ f ของ x
มันคือฟังก์ชันนั้นหาค่าที่ 2
ถ้าผมประมาณ g ไพรม์ของ x
มันคือฟังก์ชันนั้นหาค่าที่ 2
แล้ว บวกอนุพันธ์อันดับหนึ่งของตัวนี้
ซึ่งก็คืออนุพันธ์อันดับสองของ g
g ไพรม์ไพรม์ของ 2 คูณ x ลบ 2
แล้วบวกอนุพันธ์อันดับสองของฟังก์ชัน
ที่ผมพยายามประมาณอยู่ แต่อนุพันธ์
อันดับสองจะเป็นอนุพันธ์อันดับสามของ g
มันจะเท่ากับ g ไพรม์ไพรม์ไพรม์ของ 2 
คูณ x ลบ
2 กำลังสอง ทั้งหมดนั้นส่วน 2 แฟคทอเรียล
ทีนี้ เขาบอกเราว่าพวกนี้คืออะไร
ขอผมใช้สีใหม่ตรงนี้นะ
เขาบอกเราว่า g ของ 2 เท่ากับ 3
ค่านี่ตรงนี้ โอ้ ไม่

Bulgarian: 
g'(х) е приблизително равно на...
същото нещо, ще бъде
функцията,
която се опитваме 
да апроксимираме
изчислена за 2, така че g'(2).
Обърни внимание, че 
апроксимирам f(х),
това е тази функция, 
изчислена за х = 2.
Ако апроксимирам g'(х),
това е тази функция,
изчислена за 2.
После плюс първата 
производна на това,
което е втората производна на g.
g''(2) по (х – 2)
и после плюс втората
производна на функцията,
която апроксимирам,
но втората производна
на g' e всъщност
третата производна на g.
Това е равно на g'''(прим, прим, прим)
от 2 по (х – 2)^2,
цялото върху 2 факториел.
Тук са ни дали на какво 
са равни тези неща.
Ще използвам нов цвят.
Казват ни, че g(2) е равно на 3.
Ето това тук, всъщност не,

Korean: 
g'(x)는 다음 식에 근사합니다
2에 대하여
근삿값을 구합니다
2에 대하여
근삿값을 구합니다
g'(2)
주의하세요
f(x)를 근사하고
x=2에 대하여 계산합니다
만약 g'(x)를 근사하고 있다면
x=2에 대하여
계산하면 됩니다
g'(x)의 일계도함수는
g''(x)입니다
g''(2)(x - 2)
함수의 이계도함수를 더하는데
g'의 이계도함수는
g의 삼계도함수입니다
g'''(2)(x - 2)³ / 2!
g'''(2)(x - 2)³ / 2!
이들이 무엇인지
알고 있습니다
새로운 색깔을 써보죠
g(2) = 3
앗! 여기가 아니네요

Portuguese: 
Na verdade, não é o que queremos usar.
Estamos usando g de dois.
Eles nos dizem que g linha de dois
é igual à um.
g linha linha de dois é igual à menos um.
É menos um.
A terceira derivada de g
em x igual à dois, é dois.
Dois sobre dois fatorial
é duas vezes um.
Esses dois termos se cancelam.
Com o que ficamos na nossa aproximação
do polinômio de segundo
grau de g linha de x
em x igual à dois?
Ficamos com g linha de x é aproximadamente
um menos x menos dois.
Um menos x menos dois. Poderia escrever

Bulgarian: 
няма да използваме това.
Казват ни да използваме g(2).
Казват ни, че g'(2) = 1.
g'(2), това тук е равно на 1.
g''(2) е равно на –1.
Това ето тук е –1,
и накрая третата производна
от g, изчислена за 2, е 2.
Две върху 2!; две факториел
е равно на 2 по 1, значи е просто 2.
Това и това се съкращават.
Какво ни остава 
като апроксимация,
апроксимацията от втора степен на
g'(х), центрирана около х = 2?
Остава ни g'(х) е приблизително 
равно на 1 – (х – 2).
1 – (х – 2), предполагам, че
можем да го запишем като

Thai: 
นั่นไม่ใช่สิ่งที่เราอยากใช้
เขาบอกเราว่าเราจะใช้ g ของ 2
เขาบอกเราว่า g ไพรม์ของ 2 เท่ากับ 1
g ไพรม์ของ 2 ค่านี่ตรงนี้คือ 1
g ไพรม์ไพรม์ของ 2 เท่ากับลบ 1
นี่คือลบ 1 ตรงนี้
แล้วสุดท้าย อนุพันธ์อันดับสาม
ของ g หาค่าที่ 2 เป็น 2
2 ส่วน 2 แฟคทอเรียล, 2 แฟคทอเรียล
คือ 2 คูณ 1 หรือก็คือ 2
ตัวนั้นกับตัวนั้นตัดกัน
สิ่งที่เหลือจากการประมาณ
การประมาณดีกรีสองของ g
ไพรม์ของ x ศูนย์กลางที่ x เท่ากับ 2 คืออะไร?
เราเหลือ g ไพรม์ของ x ประมาณ
เท่ากับ 1 ลบ x ลบ 2
1 ลบ x ลบ 2 ผมว่าผมเขียน

Korean: 
이 값을 사용하면 안됩니다
g(2)가 아닌
g'(2) = 1 을 이용합니다
따라서 이 값은 1입니다
g''(2) = -1 이므로
이 값은 -1이고
마지막으로
g'''(2) = 2 이므로
마지막으로
g'''(2) = 2 이므로
2/2!
2! = 2 × 1 = 2 이므로
둘은 상쇄됩니다
이차다항식인 g의 도함수를
x=2를 중심으로 근사하는데
남은 식을 정리하면
어떻게 될까요?
g'(x)는
다음 식에 근사합니다
1 - (x - 2)
1 - (x - 2)

English: 
that's not what we're gonna wanna use.
They tell us we're using g of two.
They're telling us g prime
of two is equal to one.
G prime of two, so this right
over here is equal to one.
G prime prime of two is
equal to negative one.
This is negative one right over here
and then finally the third derivative
of g evaluated at two is two.
Two over two factorial, two factorials is
two times one or it's just two.
So that and that cancel out.
What are we left with
for our approximation
our second degree approximation of g
prime of x centered at x equals two?
We are left with g prime
of x is approximately
equal to one minus x minus two.
One minus x minus two,
I guess I could write

Korean: 
이렇게 나타낼 수 있습니다
1 + 2 - x
-(x - 2)는 2 - x 입니다
+ (x - 2)²
더 간단히
할 수 있을 것 같네요
3 - x + (x - 2)²
이제 계산할 수 있습니다
g'(1)의 근삿값은
g'(1)의 근삿값은
식의 x에
1을 대입합니다
식의 x에
1을 대입합니다
3 - 1 + (1 - 2)²
3 - 1 = 2 이고
1 - 2 = -1 이지만
제곱하면 +1이 됩니다
따라서 1이 되고
2 + 1 = 3이 됩니다
다시 한번
이 값은 g'(1)의 근삿값입니다

English: 
that as plus two minus
x so plus two minus x.
The negative of x minus
two is two minus x.
Plus x minus two squared and obviously I
could simplify this even more.
This is three minus x plus x minus
two squared and now I can evaluate it.
If I wanna approximate g prime of one,
I could say g prime of one
is going to be approximately
equal to, wherever I see the x is here
I put in a one there, so it's going
to be three minus one plus
one minus two squared.
Well, this is going to be two and then one
minus two is negative one but then
if I square it I get a positive one.
So this whole thing is one.
Two plus one is equal to three.
Once again, this is an
approximation for g prime of one.

Portuguese: 
como dois menos x,
Menos um vezes x menos dois
é dois menos x.
Mais x menos dois ao quadrado.
Posso simplificar
essa expressão ainda mais.
Isto é três menos x
mais x menos dois ao quadrado.
Posso calcular isso.
Se quero aproximar
o valor de f linha de um -
posso dizer que g linha de um
será aproximadamente igual à-
sempre que tiver um x, coloco um,
então será três menos um
mais um menos dois ao quadrado.
Ficará dois - um menos dois é negativo,
mas ao quadrado obtenho um positivo.
Tudo isso é igual a um.
Dois mais um é igual a três.
Mais uma vez, isto é uma aproximação
de g linha de um.

Thai: 
เป็นบวก 2 ลบ x ได้ บวก 2 ลบ x
ลบของ x ลบ 2 คือ 2 ลบ x
บวก x ลบ 2 กำลังสอง และแน่นอนผม
จัดรูปตัวนี้ต่อได้อีก
นี่คือ 3 ลบ x บวก x ลบ
2 กำลังสอง แล้วตอนนี้ผมหาค่ามันได้
ถ้าผมอยากประมาณ g ไพรม์ของ 1
ผมบอกได้ว่า g ไพรม์ของ 1 จะประมาณ
เท่ากับ เมื่อใดก็ตามที่ผมเห็น x
ผมจะใส่ 1 ลงไป มันจะ
เท่ากับ 3 ลบ 1 บวก 1 ลบ 2 กำลังสอง
ค่านี้จะเป็น 2 แล้ว 1
ลบ 2 คือลบ 1 แต่
ถ้าผมกำลังสองมัน ผมจะได้บวก 1
ทั้งหมดนี้ก็คือ 1
2 บวก 1 เท่ากับ 3
ย้ำอีกครั้ง นี่คือการประมาณค่า g ไพรม์ของ 1

Bulgarian: 
плюс 2 минус х.
–(х – 2) е равно на (2 – х).
Плюс (х –2)^2 и очевидно
можем да опростим даже още.
Това е 3 – х + (х – 2)^2.
Сега можем да го изчислим.
Ако искаме 
да апроксимираме g'(1),
можем да кажем, че g'(1)
ще е приблизително равно на,
навсякъде, където има х тук,
замествам с 1, така че става
3 – 1 + (1 – 2)^2.
Това е 2, а тук (1– 2) е равно на –1,
но това е на квадрат,
така че става +1.
Това цялото е 1.
2 плюс 1 е равно на 3.
Това е апроксимацията на g'(1).

Korean: 
여기서 무엇을 했죠?
테일러 급수를 구했습니다
x = 2를 중심으로
g'(x)의 이차 테일러 급수를
근사하였습니다
그 다음 x = 1에 대하여
근삿값을 계산하여
g'(1)을 구했습니다
어찌됐든
즐거움을 느꼈기를 바랍니다

Thai: 
เราทำอะไรตรงนี้ไป?
เราหาอนุกรมเทย์เลอร์
การประมาณอนุกรมเทย์เลอร์อันดับสอง
สำหรับ g ไพรม์ของ x ที่ศูนย์กลางอยู่รอบ
x เท่ากับ 2
แล้วเราหาค่าประมาณนั้น
ที่ x เท่ากับ 1 เพื่อประมาณค่า g ไพรม์ของ 1
เอาล่ะ หวังว่าคุณคงสนุกนะ

Bulgarian: 
Какво направихме?
Намерихме реда на Тейлър –
апроксимация с ред 
на Тейлър от втора степен
за g'(х), центрирана
около х = 2,
и после изчислихме тази
апроксимация
за х = 1, за да получим
апроксимацията на g'(1).
Надявам се, че ти беше забавно.

Portuguese: 
O que fizemos aqui?
Achamos a série de Taylor.
A aproximação da série de Taylor
de segundo grau
para g linha de x em x igual a dois,
e calculamos o valor em x igual a um
para aproximar o valor de g linha de um.
Espero que tenha gostado.
[Legendado por: Pilar Dib]

English: 
What did we do here?
We found the Taylor series.
The second degree Taylor
series approximation
for g prime of x centered
around x equals two
and then we evaluated that approximation
at x equals one to
approximate g prime of one.
Anyway, hopefully, you found that fun.
