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Welcome to the video on the Taylor Theorem and

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Taylor polynomials.

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And we've actually already touched on this.

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When we did the videos on approximating functions with

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polynomials, we used Maclaurin series, which is actually a

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special case of the Taylor polynomial or the

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Taylor Theorem.

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And we just pick, we approximate the function

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around x equals 0 when we did the Maclaurin series.

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But in general, you could approximate a function

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around any value.

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And if you do it around something other than 0, it's

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kind of the more general case, and we're dealing

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with the Taylor polynomial.

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So what is that?

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Let me just write the definition down, and then we'll

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do a couple of examples, and then we'll graph it to

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get the intuition.

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So a Taylor polynomial says that if I have a differentiable

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function f of x, and I want to approximate it with a

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polynomial at c, so at some value of x equals c, I want to

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approximate this function.

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So let me just draw a quick and dirty one, and we'll actually

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draw an accurate one later.

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So let's say that that's my axes, this is my

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function f of x.

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So I could pick some value c, some value x is equal to

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c, maybe it's right there.

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So that's c.

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And I would want to approximate it, I would want to create a

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polynomial that can approximate the function around this point.

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And the Taylor Theorem tells us that the Taylor polynomial to

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approximate this is, and then I'll give you the intuition

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for it in a second.

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p of x.

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And this looks really complicated, but when you

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do some examples, you'll see it's not so bad.

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p of x is equal to f of c plus f prime of c times x minus c

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plus f prime prime of c over, they say 2 factorial, which is

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just 2, but OK, I'll write 2 factorial.

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They do that so that you see the pattern that emerges.

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This is over 1 factorial, really, and this is over

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0 factorial, really.

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Times x minus c squared plus, I'm already running out of

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space, f the third derivative, I think at this point people

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just write a 3 in parentheses, of the function evaluated at c

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over 3 factorial times x minus c to the third, and you could

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just keep adding terms.

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You could go on like this for infinity.

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But let me give you the intuition of what this is.

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Let me just show you, just to hit the point home.

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Then you could plus the fourth derivative of the function

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evaluated at c times over 4 factorial times x minus

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c to the fourth.

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Now what's the intuition?

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So first of all, what happens to this polynomial at c?

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So what's p of c?

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p of c is equal to-- if p of c, everywhere where you see an x

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here, you have to put a c, right?

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So this term would be c minus c, so that would go to

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zero, or it would be 0.

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This term would be c minus c, it would be 0.

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This term would be c minus c, so it would be 0.

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This term would be c minus c, 0.

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And all you'd be left with is this f of c.

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So great.

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We already know that, at least at the value of c, the

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polynomial is equal to the function.

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So it's going to intersect this line.

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Right?

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And actually, if we just had a Taylor polynomial with just

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that first term, what would it look like?

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Well, it would just be a horizontal line right there.

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So it would be a pretty bad approximation.

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But, what does this second term do us?

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Because we know that if we just evaluated c.

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All these other terms just drop out.

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So what do they do for us?

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Well, the second term actually ensures that the derivative of

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this polynomial, evaluated at c, is equal to the derivative

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of this function, evaluated at c.

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What do I mean there?

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Well, what's p prime of x?

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p prime of x is equal to, well, this is just a constant term.

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It might look like a function, but it's a function evaluated

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at c, so it's just a constant term.

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And so, that's 0.

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And then, what is this?

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What's the derivative of this?

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Well, we could use, this is a constant term, and the

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derivative of this is just 1.

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So you could almost just view this as f prime of c times x,

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minus f prime of c times c which is a constant, whatever.

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So the derivative of this expression is f of c, and then

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plus the derivative of this expression, and that's

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equal to what?

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2 times 2 divided by 2 factorial, which is just 1.

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So it's f prime prime of c, times x minus c, and

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then plus, let's see.

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3 over 3 factorial, so that's 3 over 6, we'll just have

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a 2 in the denominator.

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f prime prime prime of c over, what was it, 2

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times x minus c squared--

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And you don't have to worry about all of this.

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And then, we could just keep going.

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But I wanted just to show you one thing.

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What is t prime at c?

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t prime at c.

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What is the derivative of this polynomial when

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you evaluate it at c?

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Well, when you put c into this derivative function, all these

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other terms are going to drop off, and you're just

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left with this one.

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Right?

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Because this x minus c-- sorry, just had some walnuts.

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I should have some water with it.

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If you put the c here, they drop out.

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So the derivative of this function evaluated at c,

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is equal to f prime of c.

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So as you can see, what's neat about this Taylor polynomial

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is, it's equal to the function at c, its derivative is equal

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to the function at c, the second derivative is equal

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to the function at c.

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And every term you add to the Taylor polynomial actually

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makes it so that that term, derivative, of the polynomial

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evaluated at c, is equal to the function.

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Hope I didn't confuse you.

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The big picture is, the whole thinking behind, I guess, what

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Taylor thought of, was, wow.

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If this function is infinitely differentiable, meaning that I

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can take the first, second, third, fourth, you know, all

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the way to infinity derivative of this function, I could

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construct a polynomial like this, and i can just keep going

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by adding more and more terms, so that this polynomial's, you

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know, zeroth derivative, which is means the function, the 0,

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first, second, third, fourth, all of this polynomial's

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derivatives are going to be equal to the function.

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At least around that point.

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And actually, we'll see that there's actually a whole class

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of functions, that the Taylor polynomial, if you were to take

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the infinite series, is actually equal to that

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function at all points.

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But anyway.

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And actually, I talk a little bit about that when I prove

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that e to the i pi is equal to negative 1, which to me was the

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most amazing result in mathematics.

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Whatever, whatever.

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This might have been a little confusing for you, so let's

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do a particular example.

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The particulars are always the more fun.

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I think when you see me do an example, you'll see

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that it's not so bad.

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I'm even going to erase this fourth term.

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So let's approximate, I don't know, cosine of x.

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So let's say that f of x is equal to, let me do

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it in a different color.

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We want to approximate f of x is equal to cosine of access.

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And let's pick some arbitrary number.

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Let's not pick some number that works well with

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trigonometric functions.

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Let's pick around, let's say c is equal to 2.

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No, 1.

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So we're going to approximate cosine of x around 1.

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So what is the Taylor approximation, or the

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Taylor polynomial?

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Well, we could just chug through this one.

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p of x, I'll do it in yellow.

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p of x is equal to f of c.

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So the function evaluated at c is just cosine of 1, right?

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plus s prime of c.

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Well, what is the derivative of cosine of x?

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It's minus sine of x, right?

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Minus sine of x, and we want to evaluate it at c.

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So it's minus sine of 1, right? c is one, that's

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we're approximating around.

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Times x minus c.

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And then, plus the second derivative of x.

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So what's the second derivative?

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What's going to be the derivative of minus sine,

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which is minus cosine of x?

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So it's minus cosine.

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But we're evaluating it at c, so there's actually going

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to be a number, right?

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So c is 1.

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Cosine of 1.

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Over 2, right?

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2 factorial is just 2.

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Times x minus 1 squared, oh, sorry, this should

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be a one, right?

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I said, c is equal to 1.

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Times x minus 1 squared.

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Let's keep going.

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Plus the third derivative, plus, what's the third

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derivative of cosine?

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Well, it's the derivative of minus cosine, so

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that's plus sign.

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So plus sine evaluated at 1.

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Sine of 1.

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Divided by 3 factorial, so that's 6 over 6 times

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x minus 3 to third.

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Sorry, my brain is really, I ate too many walnuts.

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Undo, edit, undo. x minus 1 to the third, right?

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And then, let's do one more term, just for fun.

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So then we're going to take the fourth derivative, which is the

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derivative of the third derivative, so the third

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derivative was positive sine, so now we're going

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to be plus cosine.

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Plus cosine evaluated at 1 over 4 factorial--

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what's 4 factorial?

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It's 3 factorial times 4.

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So over 24.

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Times x minus 1 to the fourth.

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And we could just keep going.

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The fifth derivative over 5 evaluated at 1, over 5

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factorial times x minus 1 to the fifth, and just keep

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adding, but then it will take us forever, et

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cetera, et cetera.

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So, what does this thing look like?

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And what I'm going to do, is I'm going to show you how

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this polynomial develops as we add terms.

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So let's see.

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I have this graphing calculator that I-- so this thing I

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got from, just to give them credit, it's my.hrw.com.

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And this is the graph of cosine of x.

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So just the first term here.

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Cosine of 1.

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If we were to just to graph the first term of this polynomial,

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what does it look like?

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So I'll just type in cosine of 1, and graph it.

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So there you go.

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Just the first term of the polynomial.

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If all of these terms weren't here, the polynomial would

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just be a constant, right?

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Cosine of one.

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And it's a pretty bad approximation, but at least

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it equals the function at this point.

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So it gives a something.

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But let's add some terms.

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Let's add the second term to it.

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So what was the second term?

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It was sine of 1 minus sine of 1 times x minus 1.

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Let me add that.

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Graph it.

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There you go.

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So this is neat.

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So when you just added 2 terms, what did we say?

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The polynomial will be equal to the function at x equals 1.

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And now the slope is also equal to the function.

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The slope of the polynomial is also equal to the slope of the

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function at x is equal to 1.

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So this is a better approximation.

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At least if we stay pretty close to our chosen c, it's

259
00:12:51,549 --> 00:12:55,490
a decent approximation for the function.

260
00:12:55,490 --> 00:12:58,620
Obviously, if we get far away, out here, this is a horrible

261
00:12:58,620 --> 00:12:59,840
approximation for the function.

262
00:12:59,840 --> 00:13:01,200
But let's keep adding terms.

263
00:13:01,200 --> 00:13:04,000
As you can see, I just want to show you that I'm just

264
00:13:04,000 --> 00:13:07,950
typing in the actual terms.

265
00:13:07,950 --> 00:13:09,850
So let me type in the next term.

266
00:13:09,850 --> 00:13:12,830
just so you believe that I'm doing it.

267
00:13:12,830 --> 00:13:15,050
So the next term, we'll have to see it.

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00:13:15,049 --> 00:13:16,049
Let me type it in.

269
00:13:16,049 --> 00:13:27,439
So the next term is minus cosine of 1 divided by 2

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00:13:27,440 --> 00:13:35,420
times x minus 1 squared.

271
00:13:35,419 --> 00:13:38,659
And let me graph it.

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00:13:38,659 --> 00:13:39,600
OK.

273
00:13:39,600 --> 00:13:42,220
So now, just to show you, I just typed in the second

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00:13:42,220 --> 00:13:45,980
term, and now let's look at the graph.

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00:13:45,980 --> 00:13:47,970
Now this is neat, right?

276
00:13:47,970 --> 00:13:50,720
So the first term got us a horizontal line that just

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00:13:50,720 --> 00:13:53,220
intersected the point at cosine of 1, and it was a really

278
00:13:53,220 --> 00:13:54,190
bad approximation.

279
00:13:54,190 --> 00:13:56,590
Then the second term made sure that at least the first

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00:13:56,590 --> 00:13:57,910
derivative was the same.

281
00:13:57,909 --> 00:14:00,759
And so then we, the line was just essentially the tangent

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00:14:00,759 --> 00:14:02,210
line, we only had 2 terms.

283
00:14:02,210 --> 00:14:06,320
Now the third term makes sure that the second derivative of

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00:14:06,320 --> 00:14:11,980
our polynomial at x equals 1 is equal to the second derivative

285
00:14:11,980 --> 00:14:14,810
of the polynomial of the function.

286
00:14:14,809 --> 00:14:19,099
And notice that this green graph is concave

287
00:14:19,100 --> 00:14:20,389
downwards, right?

288
00:14:20,389 --> 00:14:25,110
Which means that, and so is the function at 1.

289
00:14:25,110 --> 00:14:26,269
So this is this is pretty neat.

290
00:14:26,269 --> 00:14:28,350
We're getting a little bit, so it's kind of approximating

291
00:14:28,350 --> 00:14:30,060
the curve here.

292
00:14:30,059 --> 00:14:31,059
It's getting a little bit better.

293
00:14:31,059 --> 00:14:32,949
Remember when we went out far to the left?

294
00:14:32,950 --> 00:14:34,910
Starting to approximate the function better around here.

295
00:14:34,909 --> 00:14:36,309
It's closer, at least.

296
00:14:36,309 --> 00:14:36,519
Right?

297
00:14:36,519 --> 00:14:38,559
The last time, the line just went up, and here it was a

298
00:14:38,559 --> 00:14:40,349
really bad approximation.

299
00:14:40,350 --> 00:14:42,080
But let's add another term.

300
00:14:42,080 --> 00:14:43,120
Let's add our third term.

301
00:14:43,120 --> 00:14:46,860
Our third term, I can see it, it's right there.

302
00:14:46,860 --> 00:14:56,870
So plus sine of 1 divided by 6 times x minus 1

303
00:14:56,870 --> 00:14:59,950
to the third power.

304
00:14:59,950 --> 00:15:01,860
Just to show you, I just typed it in, right there.

305
00:15:01,860 --> 00:15:06,090


306
00:15:06,090 --> 00:15:09,060
Let me graph it.

307
00:15:09,059 --> 00:15:10,589
That is neat.

308
00:15:10,590 --> 00:15:16,240
Just with three terms on our polynomial, well, actually,

309
00:15:16,240 --> 00:15:18,860
that's the fourth term, officially.

310
00:15:18,860 --> 00:15:22,039
But the first term was essentially-- well,

311
00:15:22,039 --> 00:15:23,279
you get the point.

312
00:15:23,279 --> 00:15:25,269
But we're already starting to approximate this

313
00:15:25,269 --> 00:15:26,960
pretty well, right?

314
00:15:26,960 --> 00:15:33,320
Now the third derivative of the polynomial is equal to the

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00:15:33,320 --> 00:15:36,330
third derivative of the function at the point x is

316
00:15:36,330 --> 00:15:38,530
equal to 1, and we haven't even studied third derivatives.

317
00:15:38,529 --> 00:15:41,769
That's kind of like the concativity of the

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00:15:41,769 --> 00:15:43,490
derivative, or whatever.

319
00:15:43,490 --> 00:15:44,990
But as we can see, it approximates the

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00:15:44,990 --> 00:15:45,970
function even better.

321
00:15:45,970 --> 00:15:48,290
Obviously, though, when we go further away, it starts

322
00:15:48,289 --> 00:15:49,000
to break down again.

323
00:15:49,000 --> 00:15:50,600
But pretty close.

324
00:15:50,600 --> 00:15:52,149
If all you saw is from here to here, it would be

325
00:15:52,149 --> 00:15:53,879
hard to tell them apart.

326
00:15:53,879 --> 00:15:57,179
Let's add that last term we calculated.

327
00:15:57,179 --> 00:15:58,779
And this should be pretty neat.

328
00:15:58,779 --> 00:15:59,519
Let's see.

329
00:15:59,519 --> 00:16:01,649
The last term.

330
00:16:01,649 --> 00:16:07,730
Plus cosine sign of 1 divided by 24.

331
00:16:07,730 --> 00:16:11,870
And notice, every term, the scaling factor, right?

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00:16:11,870 --> 00:16:15,830
Here's 1, then 1/2, then 1/6, 1/24-- it becomes

333
00:16:15,830 --> 00:16:17,160
a smaller impact on it.

334
00:16:17,159 --> 00:16:20,219
And it only starts to matter as you move really, really far

335
00:16:20,220 --> 00:16:22,090
away from your chosen c.

336
00:16:22,090 --> 00:16:24,450
In this case 1, right.

337
00:16:24,450 --> 00:16:27,129
The further out you go, when you're close to your

338
00:16:27,129 --> 00:16:28,720
point that you've picked.

339
00:16:28,720 --> 00:16:31,040
these other terms don't matter much, right?

340
00:16:31,039 --> 00:16:33,459
Because you're doing 1/24, and then 1 over 5 factorial,

341
00:16:33,460 --> 00:16:34,090
et cetera et cetera.

342
00:16:34,090 --> 00:16:37,399
But as you get further and further away, these terms

343
00:16:37,399 --> 00:16:39,399
become more significant, right?

344
00:16:39,399 --> 00:16:42,789
As x gets further and further away from 1, and then that's

345
00:16:42,789 --> 00:16:44,399
where these start to play in, and you see that in

346
00:16:44,399 --> 00:16:45,709
the approximation.

347
00:16:45,710 --> 00:16:48,330
Anyway, let me graph it.

348
00:16:48,330 --> 00:17:00,450
So cosine of 1 divided by 24 times x minus 1 to the fourth.

349
00:17:00,450 --> 00:17:01,400
Let me graph it.

350
00:17:01,399 --> 00:17:05,240


351
00:17:05,240 --> 00:17:06,220
Even neater!

352
00:17:06,220 --> 00:17:08,470
And if you have some spare time, you might just want to

353
00:17:08,470 --> 00:17:10,049
keep adding terms to this.

354
00:17:10,049 --> 00:17:12,109
So that's all the Taylor polynomial is.

355
00:17:12,109 --> 00:17:13,439
And I realize, this is probably one of longest

356
00:17:13,440 --> 00:17:14,370
videos I've done.

357
00:17:14,369 --> 00:17:15,789
I'm pushing 17 minutes.

358
00:17:15,789 --> 00:17:17,869
It's a little confusing at first, because it gives you

359
00:17:17,869 --> 00:17:20,729
this huge formula, and they give you the c, and you're

360
00:17:20,730 --> 00:17:22,710
like, what is that c, and how do I take the derivative?

361
00:17:22,710 --> 00:17:24,600
But when you actually try to chug through it, you just

362
00:17:24,599 --> 00:17:25,769
have to realize, oh.

363
00:17:25,769 --> 00:17:30,759
All this is, is saying, we are constructing a polynomial that,

364
00:17:30,759 --> 00:17:35,450
at some point c that we've picked, this polynomial's

365
00:17:35,450 --> 00:17:38,740
zeroth, first, second, third, fourth, fifth, and so on-th

366
00:17:38,740 --> 00:17:42,039
derivative is going to be equal to our function.

367
00:17:42,039 --> 00:17:45,586
And actually, if we did 10 terms, or if we did all of the

368
00:17:45,586 --> 00:17:48,170
derivatives, these would start to actually equal each other.

369
00:17:48,170 --> 00:17:49,730
So hopefully that didn't confuse you.

370
00:17:49,730 --> 00:17:53,980
I know when you see the formula at first, it can be kind of

371
00:17:53,980 --> 00:17:57,769
daunting, and especially, sometimes it's even more

372
00:17:57,769 --> 00:17:59,599
daunting when someone even explains it to you.

373
00:17:59,599 --> 00:18:01,419
But hopefully that gave you some intuition.

374
00:18:01,420 --> 00:18:03,640
If it didn't, ignore this video.

375
00:18:03,640 --> 00:18:05,142
See you soon.