So where we left off in the

last video, we kept trying to approximate this purple f

of x with a polynomial. And we at first said we’ll just

make the polynomial a constant and set it — it’s just going

to intersect f of 0 at x is equal to 0. So that’s a first — you can

kind of all think of it as a 0 of order approximation

of the function. Then we said, oh, what if not

only do they intersect at x is equal to 0, but let’s say that

their slope is the same as x is equal to 0, and that’s

that approximation? And that’s about as good as

you’re going to do with a line, especially as

you get close to 0. And we said, OK, that’s good,

but what if their second derivative is the same? And that’s where we ended

up with — we added this term here. And I hinted that we’ll just

keep doing this process. And so you could imagine, if I

want the third derivative to be the same, I could add another

term right here, plus, where I know what the value of f of

x’s third derivative is at 0. So I’ll write that as f to

the third derivative at 0 times x to the third. Now what do you think is

going to be down here? What’s going to be

he denominator? You might be tempted to say

that we’ll put a 3 down here. But it turns out you’re going

to put a 3 times a 2, which is a 6, or 3 factorial. Now why is that? Let me just take a little

departure here and I think you’ll start to understand

why you put a 6 down here. Why this isn’t a 3

and you put a 6. Here you put a 2, but 2 is

also 2 factorial, right? 2 factorial is 2

times 1, right? Hopefully you remember what

factorial — actually, let me tell you what factorial

is just in case. 10 factorial is equal to 10

times 9 times 8 times 7, dah, dah, dah, dah, times 2 times 1. So you’re multiplying all of

the numbers up to that number. 4 factorial — and the numbers

get big very, very fast — is 4 times 3 times 2 times 1. 2 factorial is equal

to 2 times 1. 1 factorial is equal to 1. Now this is kind of

a weird definition. It comes out of combinatorics. Actually it works for what

we’re doing is, well, 0 factorial is also equal to 1. I know that might be a

little un-intuitive. This is just a definition. It’s like saying that i squared

is equal to negative 1. It is a definition that it

makes formulas be more general, I guess is a

simple way to put it. But let me erase all of this

because that was just a divergence just because I

realized I was going to use factorial, so you should

know what a factorial is. But I think that’s a fairly

straightforward concept. So going back to

what we were doing. I was asking you why do I put

a 6 down here instead of a 3, like we put a 2 here? Well, let’s just take

this term alone and take its third derivative. So if I have the term and it’s

f, the third derivative at 0, x to the third over — and

let me just write 6 as 3 times 2 or 2 times 3. That’ll make it a

little more clear. What’s the first

derivative here? What happens when I take

the derivative once? Well, I’m going to multiply the

whole thing by the 6 exponent and decrement the

exponent, right? So I’m going to multiply the

whole thing times 3 times f, the third derivative, x

squared over 2 times 3. So that first time I did

it, this 3 and this 3 cancel out, right? That red’s looking a

little bit too demonic. Let me pick another color. And then when I take the

second derivative what am I going to get? Well, the 3’s gone, now I just

have a 2 in the denominator, so I multiply the whole thing by 2

times f prime prime prime of 0 times, and I decrement the

exponent, x to the 1 over 2. Well, now the 2’s

cancel out, right? So the reason why you’re

putting a factorial there is every time you take a

derivative you’re decrementing the exponent 1, and multiplying

the whole expression by the exponent. So if you’re going to take n

derivative, you’re essentially going to be multiplying this

expression times n factorial. So you don’t want an n

factorial out here. You put an n factorial

at the bottom. Hopefully that makes sense. Play around with it yourself

and it should start to make a little bit more sense. So in general, if we just

kept doing this process forever, what would the

function look like? The reason why I’m covering

this is because going this way we’re going to be able to prove

what I think is the most mind-bending concept

in mathematics. And it will make you love

mathematics, hopefully. Some people actually —

well, I won’t go into the spiritual aspects of it. So what would be this, if I

just kept saying that I’m just going to keep taking

derivatives and adding them to this term, this polynomial? Well, the polynomial would

become p of x is equal to f of 0 plus f prime of 0 x. And let’s just divide it by 1

factorial, just to make it clear that that’s a

1 factorial, right? And that’s an x

to the 1, right? That’s just this term,

but I just wrote it a little differently. This term right here, this

is f of 0 times x to the 0. I know that’s really

messy, but hopefully you see what I’m saying. And that’s over 0

factorial, right? 0 factorial is 1, x to the 0

is 1, so it’s just f of 0. And then plus the second

derivative at 0 times x squared over 2 factorial plus —

and we just keep adding. The third derivative at x is

equal to 0 of x to the third over 3 factorial, and

we just keep going on. So we could do

this to infinity. And actually we will do

it, and this is called the Maclaurin Series. So if we just wanted to

approximate this as hard as we can, essentially take the

infinite derivatives of it, we get the Maclaurin Series. So we are going to define

this polynomial p of x. It’s going to be the infinite

series, the infinite sum. Let’s start with n is equal

to 0, and we’re going to go to infinity. What is each term? It’s going to be f of — well

it’s going to be f, the nth derivative of f evaluated at

0 times x to the n over n factorial. This is the Maclaurin Series. We’re later learn that the

Maclaurin Series is a specific example of the Taylor Series,

which is a specific example of a power series. But anyway, this might seem

very complicated to you. I have all the sigma notation. Just remember, this is

essentially just that and I just keep going to infinity. And if you play around with

it it should make sense. But I think this will become a

lot more concrete when I do this with a specific f of x. This is where it gets cool. In case you don’t think

it’s already cool. So let’s pick f of x to be,

to me, the most amazing function of them all. If I ever built a shrine or

a church or something or skyscraper, I would somehow

make this function show up all over the place, and then years

from now people would be awed by the mysticism of it all. But anyway, let’s try to

approximate e to the x with a Maclaurin Series. You know that sigma thing,

that’s hard to memorize. Just remember you want all the

derivatives to be the same. So let’s make the

approximation of this. Actually, I won’t prove it. It’s out of the scope of

what we’re doing right now. But the approximation, even

when it’s centered at 0, actually equals the function

when you take the infinite sum. But let’s just see

what it looks like. Because this is pretty cool. Before we start building the

polynomial, let’s just figure out a couple of things. So what is f prime of x? That’s also e to the x, right? What’s f prime prime of x? Well that also

equals e to the x. We have learned and actually

recently did a proof that the derivative of e to

the x is e to the x. But that also needs a second

derivative and the third and the fourth and the nth

derivative of e to the x is equal to e to the x. I could take an arbitrary

number of derivatives of e to the x and it equals e to

the x, which is amazing. The rate of change of the

function at any point is equal to the function. The rate of change of the rate

of change of the function at any point is equal

to the function. I mean that’s — I want to just

go some place and ponder it, but I’m too busy making videos. But anyway, back to

what we were doing. So what is f of 0? f of 0 is equal to e to the 0,

which is equal to 1, right? Well that’s also going

to be f prime of 0. That’s also e to the 0,

which is equal to 1. So all of the derivatives, the

nth derivative at 0 is going to equal 1 for this specific case

of f of x, for e to the x. And this is why

this is so cool. But actually, it actually

gets even more amazing. So, you hopefully realize

that f of 0 and all of its derivatives at

0 are equal to 1. So now we can do the powers

of the Maclaurin Series in the next video. See you soon.

yeh thnx for the video sal

good vid….

Lol, gotta love his spills about e^x. =)

Starting at 8:55

It blows my mind that I've been taking math for who knows how many years and it's first now that I've realised what's so special with e. Thanks for pointing out what no one else, including my math teachers, has bothered to say.

Remember, he's not taking the derivative of f(x), he's taking the derivative of f(0), which is just a constant number, not a variable.

Like say you have f(x) = 1+x

In that case, f(0) = 1, so f(0) can just be treated as 1

a shrine of e^x. hahaha.. nice.

haha "that red is looking a little bit too demonic"

Sal, you never fail to put a smile on my face. OH and I can't thank you enough for the enlightening lessons you provide.

Thanks, Jon

Please remove yourself from his penis.

you will love mathematics. some people even – well I won't go into the spiritual aspects of it.

hhahahhahah i was like "woah! dalai lama solving differential equations?"

I just realized that x=1! = Integral of x

(x^2)/2! = twice the Integral of x

(x^3)/3!= three times the Integral of x

So you can change the Summation a little bit if you prefer it that way.

There's a mistake in the first line, x/1! = Integral of 1 dx.

Oh and the mistake went for the ride, in all the lines, there should be Integral of 1 dx.

Tiny typo in title: it's Maclaurin, not Maclauren series, according to the wikipedia.

Thanks for the all the videos. Watched 228 ones already and keeping watching 🙂 I've never believed I can learn so much.

@Genomatekk – Yeah made me laugh too XD that exact comment!

omg Sal! ahhahaha "lets pick x to be the most amazing function of them all. if i ever built a shrine or church or something or skyscraper i would somehow make this function show up all over the place and years from now people would be awed by the mysticism of it all" gold.

Love the passion 😀 Thnx for the lecture ^^

is there a video on factorials of non-whole numbers?

Approximating so hard right now.

I've struggled with math my entire life, and yet you have made me love it.

what are the spiritual aspects of math?

Please good sir, do discuss spiritual aspects of this.

We appreciate you postponing your desires to ponder the wonders of e^x so that you can single handedly teach the world maths…. Maybe one day, we will have learned enough to take over the teaching responsibilities… So you can go somewhere and ponder away

Ah 240p we meet again

Excellent work

Hats off

Anyone ever tried to refine some graphics or in-depth as well as an intuitive explanation towards this method? Like the inventor of the Laplace transform once said sth like intuition is more fun than theoretical proof.

good explanation sir. but i think on degree two polynomial here is must open upward

We would love to hear you give advices about life other than math❤ so don't stop yourself please