Taylor & Maclaurin polynomials intro (part 1) | Series | AP Calculus BC | Khan Academy

Taylor & Maclaurin polynomials are a very clever way of approximating any function with a polynomial. Learn how these polynomials work. Created by Sal Khan.

AP Calculus BC on Khan Academy: Learn AP Calculus BC - everything from AP Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP Test

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[michaelmarsh]

I hope I'm not the only one who burst out laughing at "well Sal that's a hoarrible approximation".

[dybiosol]

"...IN A NEW COLOUR!"
*Fails to click on a colour*
"That is not 'a new colour'!"
😂😂😂

PS. This was the best explanation of MacLaurin's Series I've seen till date. When we did this in college, our prof seemed to use Aladin's magic lamp to bring this series into existence. Thank you for this.

[Friemelkubus]

Sometimes you need these tiny details, these tiny bits of intuition to get the picture. You really help with this. Thanks.

[sabiansmasher]

honestly, this guy may not show difficult examples that I may be tested on, however his theory lessons are what makes hime so great!
Thank you very much Sal :)

[tokkia]

“TRY TO DO ANY BETTER USING A HORIZONTAL LINE THEN” 😂 loved that

[xkaili]

Khan Academy just has this magic that touches parts of everyone's brains and gets it working like no one else could and would ...  

[Dalton-ccnf]

Hey Sal, I think I am in love with you. This is me shooting my shot. I have a winning personality, fun hobbies, and I will spend every moment with you like it is my last. Much love xoxo <3

[Zswxde]

I dont know what I would do without you :')

[Anonymoose]

I love the "aha!" moment you get in the middle of watching the video. Nothing quite like it!

[Jking]

Amazing! When my college professor explained this he didn't even bother to tell us what we were actually doing/finding. He just basically gave us the formula. Thanks so much!

[chas-onjt]

I really, really like this.

Derivation of the Maclaurin Series from "CORE MATHS for A-level" by L. Bostock and S. Chandler, published by Stanley Thornes (Publishers) Ltd:

A power function of f(x) = (a + x)^n

Where, when considered with the general binomial theorem, gives:

a^n + a^(n-1) * x + a^(n-2) * x^2 + ...

Where a^n, a^(n-1), a^(n-2)... are all constants, to be reconsidered as:

a0 + a1 * x + a2 * x^2 + ...

f(x) = (a + x)^n = a0 + a1 * x + a2 * x^2 +...

f'(x) = a1 + (2)a2 * x +...

f''(x) = (2)a2 + ... (3)(2)a3 * x +...

f''(x) = (3)(2)a3 + (4)(3)(2)a4 * x +...


When x = 0:

f(0) = a0

f'(0) = a1

f''(0) = (2)a2 -> a2 = f''(x) / 2 = f''(x) / 2!

f'''(0) = (3)(2)a3 -> a3 = f'''(x) / (3)(2) = f'''(x) / 3!

...

f^n(0) = f^n(x) / n(n-1)(n-2)... = f^n(x) / n!


Therefore:
f(x) = (a + x)^n = a0 + a1 * x + a2 * x^2 +... = f(0) + f'(0) * x + [f''(0) / 2!] * x^2 + [f'''(0) / 3!] * x^3 + ... [f^n(0) / n!] * x^n

[UberCuba]

@Yakushii not really, because you wouldn't be able to find the second derivative etc, what it does allow you to do is turn a function into a polynomial ie one just involving x to certain powers, so you can make sin(x) into a function of lots of x's of different powers added together.

[sarahxboxbeara]

You can evaluate at any number, but a Maclaurin series is evaluated at zero. That's the only distinction. Zero can be replaced with 'a' or any variable holding the place for any number.

[anorman]

I was really bad at Maclauren and Taylor series when I was in Calc 1, and now I'm learning about Laurent series in Complex Analysis. This video helped me tremendously to understand it because I'm too lazy/exhausted to read about it. Thanks again, Mr. Khan.

[afiatabassum]

Sal! You are just amazing. You know what? You are on the list of my most most most favourite teachers in my life.
You are that amazing.

[ayushshah]

i cant's take it in but watching again again & finally get it

[beegdigit]

Khan breaks everything down in understandable chunks yet without losing the generalization rigor of Maths - wonderful

[parthsinghal]

Funny how Indian students do this in class 11 while in the US college students struggle in AP calc 😅

[wtfyman]

@dickie4thepeople i think of it as an approximation. First we write an equation, then we add an equation for the gradient, then we add an equation for the rate of how the gradient changes (using differentiation) etc... so our approximation for the rest of the line is becoming better.... sorry i dont completely grasp this either. but i hope this kinda helps!

[idreeskhan-zpey]

After reading the comments I envy all of you that you all have mastered the Taylor theorem but I am still struggling!!😂😂