Sal's old Maclaurin and Taylor series tutorial
Polynomial approximation of functions (part 2) Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series)
Polynomial approximation of functions (part 2)
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- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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