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Let's say I have this bizarre-looking function.

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It's just some arbitrary function.

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And we'll call that f of x.

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So this function right there is f of x.

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And what we're going to do in this video is, it's not an

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experiment, but we're going to play around a little bit, and

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we're going to try to approximate this function

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using a polynomial with some coefficients.

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And this polynomial we're going to do, we're going to keep

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adding terms to the polynomial, so that we can better and

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better approximate this function.

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And that's actually called a power series.

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And we'll do more about that later, but we're going to

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specifically try, in this case, to approximate the function

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around x is equal to 0.

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So around this point.

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So the easiest way to approximate it is to say, well,

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the simplest polynomial is just a constant, right?

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Let's say this is my polynomial, let me call

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my polynomial p of x.

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The simplest polynomial is just a constant, and it would just

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be a horizontal line someplace.

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So if I just wanted this one term polynomial, what would be

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my best approximation for this function, at least

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at this point?

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Well, I could just set p of x is equal to f of 0.

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And in that case, p of x would just look like a horizontal

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line going through f of 0.

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It would just look like that.

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I'm going to erase that now, just so I don't dirty up

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this picture too much.

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But that's, you could say, a very rough approximation

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of f of x, right?

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So that's a start.

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Well, what would be one way to approximate it even more?

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Well, what if not only does p of x equal f of 0 at x is equal

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to 0, so that horizontal line we did, we got p of 0 is equal

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to f of 0, so we knew that at x equals 0, at least that

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horizontal line is the same value of f of x, that's a

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very rough approximation.

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But what if we set it up so that the derivative of p of 0

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is equal to the derivative of the function at 0?

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All of a sudden, this could be a little bit more interesting.

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So how could we set it up like that?

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Well, what if we set p of x, and I'm doing it very general,

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and we're going to do specific examples, and actually, the

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first example we're going to do is probably the coolest one.

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So what if p of x is equal to, well, the constant term is f of

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0, and then it's plus the derivative of this function, so

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the slope of this function at that point, f prime

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of 0 times x.

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Let's say I'm defining, so this is a polynomial.

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I just added a first degree term here.

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I had a constant, and now I'm adding a first degree term.

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And let me confirm that this will have the same derivative.

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So let's see.

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First of all, let's confirm that p of 0 is equal to f of 0.

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Well, p of 0 is equal to f of 0 plus f prime of 0 times 0.

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Well, this last term just goes to, is nothing, right?

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Times 0.

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So that checks out.

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At x is equal to 0, the two functions are

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equal to each other.

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Now let me confirm that their derivative, their first

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derivatives are the same.

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So what's the first derivative of p?

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p prime of x is equal to, well, the derivative of a

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constant term is 0, right?

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Plus, and what's the derivative of a next term, of a

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first degree term?

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Well, it's just f prime of 0.

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So this checks out.

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My new polynomial that I've defined right here is equal to

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the function f at x is equal to 0, and its derivative is equal

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to the function f at x is equal to 0.

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So what would it look like?

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Well, it would intersect, at x is equal to 0, the two

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functions would overlap.

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And also, they would have the same slope at that point.

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So whatever f of x is doing, that function's

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going to be doing.

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So it's going to look something like, I'm going to try my best

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to, it's going to look something like that.

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And so that is a better approximation, if we had to use

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a line, that's as good as any, especially around 0,

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of what f of x is.

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f of x might have been some really crazy weirdo function,

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but we were able to approximate it reasonably well with this

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very simple linear equation.

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Well, that's all good, but let's approximate it with a

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quadratic equation, with adding another x squared term.

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And we're going to do that way, but we're going to say, well,

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we said, when at x is equal to 0, the functions

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equal each other.

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That's what we did here.

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Then we said, the derivatives equal each other, and

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so we added this term.

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And now I'm going to say, what happens when their second

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derivatives equal each other?

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So what has to happen for their second derivatives

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to equal each other?

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Well, let's try out this, and I think you'll start

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to see the intuition here.

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Let me define my new p of x, so let me add another term.

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p of x, the first terms are going to be the same.

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They're going to be f of 0.

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Remember, this is just a constant term.

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Plus f prime of 0, the first derivative at 0,

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the slope at 0 times x.

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Plus f prime prime, the second derivative of the function at

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0, times x squared over 2.

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Now you're probably saying, why are you multiplying

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it by 1/2 here?

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And you'll see, and maybe you'll even realize it, when

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you take a second derivative, what happens, right?

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You multiply the expression by the exponents so you can have a

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2 come down, it's going to cancel out with the 1/2.

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And that's why I put the 1/2 there.

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So that when I take the derivative, that 2 and the 1/2

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cancel out, and I'm just left with the second derivative

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of the original function.

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So let me confirm that.

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So p of 0 is equal to f of 0 plus, well when x is equal to

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0, that's 0, this term is 0, and when x is equal to 0,

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that term is 0, right?

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So that checks out.

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What's the first derivative of p?

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The first derivative of p is going to be, up here, this

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was the first derivative of p at 0, right?

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So what's the first derivative of p?

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Well, the constant term becomes 0, plus-- oh, actually, no,

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this was actually of x, sorry.

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I shouldn't go back on my work, I know it best when I'm doing

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it the first time around.

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Anyway.

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The first derivative of p of x, this is my current p of x, this

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constant term, derivative of 0.

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This x term, the derivative is f prime of 0.

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And then, what's the derivative of this term?

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Well, we take the exponent, multiply it by the expression.

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We have 2 times 1/2, that cancels out.

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So we're just left with f prime prime of 0 x.

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Right?

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You take the exponent, multiply by the whole thing, and then

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decrement the exponent by 1.

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So what is p prime of 0?

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What is the derivative at 0?

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Well, it equals, this is nothing.

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It equals f prime of 0 plus, and, well, this

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term's going to be 0.

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So that checks out.

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And so what's the third derivative?

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Let me clean up some of the stuff on the top.

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Since this is our current f of x anyway, I can clean

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up all of this stuff.

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Let me clean up all of this.

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So what is the third derivative of this p that I defined here?

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This is our current p.

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Well, the third derivative is going to be, so p prime prime

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prime of x, we could have also written p3 of x, is equal

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to the derivative of this.

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Oh, sorry, we're not on the third, we're only on

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the second derivative.

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p, and I'll write prime prime, I was going to write a 2 there.

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p prime prime of x.

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That equals what?

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That's the derivative of this.

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So there was a 0 here, that goes to nothing.

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This is now a constant term, that becomes nothing.

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And then we take the derivative of this term.

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Well, we just, it's a constant times x.

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Remember, this might look like a function or some variable.

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It's just a constant.

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Because we evaluated this curvy function, it's

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second derivative 0, so we just got a number here.

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So this derivative is just this number.

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So it equals f prime prime of 0.

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And so our current p of x has the same value when x is equal

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to 0 as f of x, it has the same first derivative at xis equal

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to zero as f of x, it has the same second derivative.

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And I don't, this is getting beyond my visualization

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ability, especially for an arbitrary function like this,

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but I could guess that maybe it looks something like this.

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I don't know.

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Maybe it looks, maybe our new function will curve, and it'll

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approximate it a little bit better, and then maybe

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it'll come down like that.

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I don't know.

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I'm just guessing.

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But around x is equal to 0, it's going to be a better

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approximation of f of x.

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Well, we could just keep doing this, and actually, we will

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keep doing this, and you know, just saying, well, the zeroth

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derivative, or at the value, is the same the first derivative

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is the same at 0, the second derivative is the same at 0,

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we'll say the third derivative, the fourth derivative, and

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we'll keep doing that.

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And I only have 20 seconds left in this video, so we will

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continue that in the next video.

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