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So where we left off in the last video, we kept trying to

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approximate this purple f of x with a polynomial.

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And we at first said we'll just make the polynomial a constant

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and set it -- it's just going to intersect f of 0

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at x is equal to 0.

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So that's a first -- you can kind of all think of it as a

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0 of order approximation of the function.

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Then we said, oh, what if not only do they intersect at x is

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equal to 0, but let's say that their slope is the same as x is

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equal to 0, and that's that approximation?

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And that's about as good as you're going to do with a

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line, especially as you get close to 0.

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And we said, OK, that's good, but what if their second

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derivative is the same?

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And that's where we ended up with -- we added

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this term here.

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And I hinted that we'll just keep doing this process.

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And so you could imagine, if I want the third derivative to be

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the same, I could add another term right here, plus, where I

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know what the value of f of x's third derivative is at 0.

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So I'll write that as f to the third derivative at

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0 times x to the third.

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Now what do you think is going to be down here?

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What's going to be he denominator?

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You might be tempted to say that we'll put a 3 down here.

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But it turns out you're going to put a 3 times a 2, which

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is a 6, or 3 factorial.

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Now why is that?

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Let me just take a little departure here and I think

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you'll start to understand why you put a 6 down here.

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Why this isn't a 3 and you put a 6.

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Here you put a 2, but 2 is also 2 factorial, right?

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2 factorial is 2 times 1, right?

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Hopefully you remember what factorial -- actually, let

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me tell you what factorial is just in case.

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10 factorial is equal to 10 times 9 times 8 times 7, dah,

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dah, dah, dah, times 2 times 1.

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So you're multiplying all of the numbers up to that number.

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4 factorial -- and the numbers get big very, very fast -- is

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4 times 3 times 2 times 1.

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2 factorial is equal to 2 times 1.

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1 factorial is equal to 1.

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Now this is kind of a weird definition.

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It comes out of combinatorics.

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Actually it works for what we're doing is, well, 0

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factorial is also equal to 1.

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I know that might be a little un-intuitive.

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This is just a definition.

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It's like saying that i squared is equal to negative 1.

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It is a definition that it makes formulas be more

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general, I guess is a simple way to put it.

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But let me erase all of this because that was just a

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divergence just because I realized I was going to use

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factorial, so you should know what a factorial is.

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But I think that's a fairly straightforward concept.

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So going back to what we were doing.

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I was asking you why do I put a 6 down here instead of a

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3, like we put a 2 here?

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Well, let's just take this term alone and take

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its third derivative.

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So if I have the term and it's f, the third derivative at 0, x

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to the third over -- and let me just write 6 as

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3 times 2 or 2 times 3.

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That'll make it a little more clear.

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What's the first derivative here?

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What happens when I take the derivative once?

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Well, I'm going to multiply the whole thing by the 6 exponent

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and decrement the exponent, right?

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So I'm going to multiply the whole thing times 3 times f,

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the third derivative, x squared over 2 times 3.

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So that first time I did it, this 3 and this

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3 cancel out, right?

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That red's looking a little bit too demonic.

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Let me pick another color.

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And then when I take the second derivative what

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am I going to get?

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Well, the 3's gone, now I just have a 2 in the denominator, so

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I multiply the whole thing by 2 times f prime prime prime of 0

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times, and I decrement the exponent, x to the 1 over 2.

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Well, now the 2's cancel out, right?

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So the reason why you're putting a factorial there is

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every time you take a derivative you're decrementing

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the exponent 1, and multiplying the whole expression

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by the exponent.

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So if you're going to take n derivative, you're essentially

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going to be multiplying this expression times n factorial.

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So you don't want an n factorial out here.

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You put an n factorial at the bottom.

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Hopefully that makes sense.

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Play around with it yourself and it should start to make

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a little bit more sense.

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So in general, if we just kept doing this process

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forever, what would the function look like?

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The reason why I'm covering this is because going this way

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we're going to be able to prove what I think is the most

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mind-bending concept in mathematics.

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And it will make you love mathematics, hopefully.

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Some people actually -- well, I won't go into the

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spiritual aspects of it.

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So what would be this, if I just kept saying that I'm

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just going to keep taking derivatives and adding them to

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this term, this polynomial?

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Well, the polynomial would become p of x is equal to f

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of 0 plus f prime of 0 x.

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And let's just divide it by 1 factorial, just to make it

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clear that that's a 1 factorial, right?

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And that's an x to the 1, right?

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That's just this term, but I just wrote it a

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little differently.

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This term right here, this is f of 0 times x to the 0.

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I know that's really messy, but hopefully you

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see what I'm saying.

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And that's over 0 factorial, right?

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0 factorial is 1, x to the 0 is 1, so it's just f of 0.

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And then plus the second derivative at 0 times x squared

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over 2 factorial plus -- and we just keep adding.

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The third derivative at x is equal to 0 of x to the third

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over 3 factorial, and we just keep going on.

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So we could do this to infinity.

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And actually we will do it, and this is called

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the Maclaurin Series.

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So if we just wanted to approximate this as hard as we

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can, essentially take the infinite derivatives of it, we

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get the Maclaurin Series.

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So we are going to define this polynomial p of x.

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It's going to be the infinite series, the infinite sum.

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Let's start with n is equal to 0, and we're going

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to go to infinity.

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What is each term?

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It's going to be f of -- well it's going to be f, the nth

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derivative of f evaluated at 0 times x to the n

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over n factorial.

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This is the Maclaurin Series.

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We're later learn that the Maclaurin Series is a specific

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example of the Taylor Series, which is a specific example

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of a power series.

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But anyway, this might seem very complicated to you.

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I have all the sigma notation.

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Just remember, this is essentially just that and I

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just keep going to infinity.

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And if you play around with it it should make sense.

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But I think this will become a lot more concrete when I do

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this with a specific f of x.

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This is where it gets cool.

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In case you don't think it's already cool.

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So let's pick f of x to be, to me, the most amazing

152
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function of them all.

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If I ever built a shrine or a church or something or

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skyscraper, I would somehow make this function show up all

155
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over the place, and then years from now people would be awed

156
00:07:55,220 --> 00:07:56,960
by the mysticism of it all.

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But anyway, let's try to approximate e to the x

158
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with a Maclaurin Series.

159
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You know that sigma thing, that's hard to memorize.

160
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Just remember you want all the derivatives to be the same.

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So let's make the approximation of this.

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163
00:08:13,959 --> 00:08:14,794
Actually, I won't prove it.

164
00:08:14,795 --> 00:08:16,819
It's out of the scope of what we're doing right now.

165
00:08:16,819 --> 00:08:19,029
But the approximation, even when it's centered at 0,

166
00:08:19,029 --> 00:08:22,659
actually equals the function when you take the infinite sum.

167
00:08:22,660 --> 00:08:24,470
But let's just see what it looks like.

168
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Because this is pretty cool.

169
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Before we start building the polynomial, let's just figure

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00:08:28,939 --> 00:08:29,550
out a couple of things.

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So what is f prime of x?

172
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That's also e to the x, right?

173
00:08:37,470 --> 00:08:40,540
What's f prime prime of x?

174
00:08:40,539 --> 00:08:42,169
Well that also equals e to the x.

175
00:08:42,169 --> 00:08:45,139
We have learned and actually recently did a proof that

176
00:08:45,139 --> 00:08:47,049
the derivative of e to the x is e to the x.

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00:08:47,049 --> 00:08:48,719
But that also needs a second derivative and the third and

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00:08:48,720 --> 00:08:50,946
the fourth and the nth derivative of e to the x

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00:08:50,946 --> 00:08:52,600
is equal to e to the x.

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00:08:52,600 --> 00:08:56,040
I could take an arbitrary number of derivatives of e

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00:08:56,039 --> 00:08:59,699
to the x and it equals e to the x, which is amazing.

182
00:08:59,700 --> 00:09:04,300
The rate of change of the function at any point is

183
00:09:04,299 --> 00:09:05,359
equal to the function.

184
00:09:05,360 --> 00:09:07,360
The rate of change of the rate of change of the function at

185
00:09:07,360 --> 00:09:08,659
any point is equal to the function.

186
00:09:08,659 --> 00:09:12,569
I mean that's -- I want to just go some place and ponder it,

187
00:09:12,570 --> 00:09:14,400
but I'm too busy making videos.

188
00:09:14,399 --> 00:09:16,319
But anyway, back to what we were doing.

189
00:09:16,320 --> 00:09:20,040
So what is f of 0?

190
00:09:20,039 --> 00:09:25,230
f of 0 is equal to e to the 0, which is equal to 1, right?

191
00:09:25,230 --> 00:09:28,389
Well that's also going to be f prime of 0.

192
00:09:28,389 --> 00:09:31,559
That's also e to the 0, which is equal to 1.

193
00:09:31,559 --> 00:09:37,349
So all of the derivatives, the nth derivative at 0 is going to

194
00:09:37,350 --> 00:09:41,522
equal 1 for this specific case of f of x, for e to the x.

195
00:09:41,522 --> 00:09:43,990
And this is why this is so cool.

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00:09:43,990 --> 00:09:47,129
But actually, it actually gets even more amazing.

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00:09:47,129 --> 00:09:50,450


198
00:09:50,450 --> 00:09:54,240
So, you hopefully realize that f of 0 and all of

199
00:09:54,240 --> 00:09:57,320
its derivatives at 0 are equal to 1.

200
00:09:57,320 --> 00:10:00,530
So now we can do the powers of the Maclaurin Series

201
00:10:00,529 --> 00:10:01,529
in the next video.

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00:10:01,529 --> 00:10:03,031
See you soon.