Two on. All right, 9.2, Taylor polynomials. We have been talking about the Taylor form of a line all year long, right?
So y'all are used to that. We call the Taylor form of a line, right? It could be Taylor form of a line.
There you go, squeeze an R in there. Is this, right? We called it L equals, it's Y sub one plus M times X minus X sub one.
And I actually wrote it like that. OK, this is a very specific type of what belongs to the family of Taylor polynomials. Right. This happens to be a first degree polynomial.
I notice when we write it like this, we're writing this in increasing order of degree. Right. When you when you looked at MX plus B back in the day and when you wrote all your quadratics in standard form.
You wrote them in descending order, right? The largest term was in the front, and then you worked down. Well, when we do Taylor form, we like to write them in increasing order.
So this would be your linear Taylor polynomial, okay? And notice what it involves. It involves the input x sub 1, the output y sub 1, and the slope at that point, yeah? So how does that pertain to the derivatives, right? For us, it's going to be if we have like...
f of a equals b, then this would end up being b equals f prime of a plus x minus a, right? That makes it specific now. Because really all we needed was a function and an input, right?
We can find the output by plugging it in, and then we can find the derivative output by finding the derivative and plugging it into there. Well... It turns out that for some functions, we can actually develop a Taylor polynomial that is bigger than first degree, bigger than linear.
We can develop a quadratic Taylor polynomial, a cubic polynomial, so on and so forth. So to explore that idea, we're going to look at sine of x, all right? Sine of x is what we call a transcendental function, transcendental. All the trig functions are e to the x is, natural log of x is. They're not polynomials, okay?
So I want to approximate, starting with the end in mind, I want to approximate what sine of 0.2 is. OK, the way we do this is going to be similar to also how your calculator does it, because your calculator doesn't actually use sine when you type in sine of a number. It actually uses the Taylor polynomial. OK, so let's just review.
This is review. I think it's from section 5.7. We need to let Landon in, by the way. Thank you all for reminding me. Landon, land, we're done.
Landon, landon, we're done. I almost forgot you, but not really. Not really. They were saying, leave them out there.
Leave them out there. It'll raise the average of the rest of us. We just started 9.2 without you. I'm sorry.
All right. So here's what it says on example one. Find the equation of the tangent line.
and use it to approximate sine of point two. So what do we need for the equation of a tangent line? Slope and a point, right? Slope and a point.
So let's go ahead and find the point. Here's the x value. Here's the function value.
So what is f of zero? It's sine of zero, which is what? Zero. So our point is the origin, zero, zero.
Now we need the slope with that point. So before I find f prime of zero, I gotta find f prime of x. And what's the derivative of sine?
Cosine. Good. And so f prime of zero now is cosine of zero, which is one.
So remember that sine of x and tangent of x both launch out of the origin at a 45 degree angle. OK, so we say L of x, we give it a name because we're going to use it, is the y value of zero plus the slope one times x minus the x value of zero. Right. That's Taylor form.
And of course, that simplifies down to just plan all X, doesn't it? And so now to approximate sine of point two, which is only two tenths from our point of tangency, we can march out along that line. So here's what we say using the notation. They gave sine of X the name F. So we'll say F of zero point two.
is approximated by squiggles, right, L of 0.2, which equals. Now, if you leave it in Taylor form, when it doesn't simplify this nicely. Um, it's easy because you're going to have your center minus 0.2, which is close.
So for us, it's just 0.2. Okay. Now we can also determine if this is an over or an under approximation, right? What is it in this case?
It's an over. Why? Okay.
I like the second part, but what's it? What's concave down? No, you had it right. Because sine of X is concave down.
Okay, so let's go ahead and say this is less than F of 0.2 since sine of X or F of X is CC down on, bless you, the neighborhood from 0 to 0.2. Remember, it's always from your point of tangency, which we started calling the center of the approximation, up to the point we want. And we know it's concave down because we know what Sahala looks like.
All right. Yeah. So just to just to get a quick visual here.
Sahala looks like this and like that. And we found the tangent line to be this. Right.
There it is. And we'll just call this point two. So we found this value and we really wanted this value.
Okay. So we also learned another method. If we wanted, if we wanted, well, first of all, let's do this.
I'm curious. Aren't you curious? You should always be curious.
Let's see what the calculator says sine of 0.2 is. Yeah, I know it's boring, right, Bentley? No, that just means you're oxygenating your brain.
Right, getting ready for the next big idea. What's the big idea yawning in my class? Getting ready for it, sir. All right, so I'm in radian mode, sine of 0.2. Look, it's 0.198.
Is 0.198 less than 0.2? Yeah, it's not bad. It's smaller. So that's a decent approximation.
Now, we also had a way. to get a better approximation but still using lines what was that Euler or we call it Euler's method right he pronounces it Euler but he's not here to defend himself so it looks like Euler Euler's method basically breaks that step up into more than one step and we tiptoe out along tangent lines still okay and we can hug the curve by breaking it up but there's another way to hug the curve so that you get a better approximation with the single calculation And that is to use a higher order polynomial. So what I'm doing here now is I'm switching to this graphic. Did y'all download the version that has that on there?
No. So you can look at this graphic. You could download this graphic from the internet. Just look at, and I probably should have updated it, but I did it here recently. Normally we do this with a calculator.
But just look up Taylor series graph. for sine of x. Okay, and then you can screenshot it and bring it in.
Taylor series graph for sine of x. No, no, but the piece of paper will be in your head, filed safely in your head. So the black graph is sine of x.
What we just did was the red graph, okay, but there is a cubic polynomial that looks like the orange graph that exists. That is the exact same value assigned at the center at zero, but it hugs the curve a little bit more closely. So if you notice the orange curve, it actually almost is not differentiable or you can't differentiate it from the actual curve until it starts leaving, which is near the high point at pi halves, right?
So that cubic polynomial evaluated at 0.2 would give you a better approximation. Now, there is a tradeoff, right? Because evaluating the cubic polynomial, you now have to cube numbers. And remember back in the day without a calculator, that was kind of a challenge. So there was always a tradeoff between how accurate you want to be and how many terms you want to use.
OK, but computers nowadays, they can plug and chug. They have no problem doing something cubed. But if you tell them to take sine of a number, they're like, what do you mean, boss? Well, guess what?
If a cubic polynomial in orange approximates it. Maybe there's a fifth degree polynomial in green that approximates it even better. And you can see it kind of wraps around the curve.
Looks like it wraps around the curve well past by halves, right? Well, maybe if it's not a fifth degree polynomial, maybe there's a sixth degree polynomial in, I guess, the seventh degree, sorry, would be in blue, right? Look at the blue graph.
It starts hugging the curve even sooner, okay? And then there would be a ninth degree polynomial that would be this purple one that comes in the top, hugs the curve, and goes out the bottom. And then there's an 11th degree polynomial in this light purple that comes in the bottom. And it almost looks like it grabs a whole cycle of sine, doesn't it? It almost looks like it wraps around an entire cycle of sine both ways before it departs.
Now, you can imagine if 11th works, then maybe there's a 13th degree polynomial, so on and so forth. But notice that because sine has rotational symmetry, it's an odd function. All of the polynomials that are approximating it are odd degree, linear, cubic. fifth degree, so on and so forth, because they come in and go out opposite side. That's one of the characteristics of odd degree polynomials.
They have opposite end behaviors. If it's an even degree polynomial, they come in and go out the same way, like x squared, okay? So what we're learning in this section is, how do we come up with the equations for those higher order polynomials so that we can get better approximations than just the straight old line, okay?
Well, That comes, that's what we did on example two. We just did it on example two. All right, so here it is. This is very important.
Like for the rest of the year, if you don't learn anything else, this little piece right here is the golden ticket. It's the kernel of wisdom that you need right there. That's the nth term.
So here's what it says. Definition of an nth degree polynomial, Taylor polynomial. If f has n derivatives at x equals c. We're going to call C the point of tangency.
We're going to call it C for the center of approximation. Then this right here, P for polynomial sub n, okay? So like we did the partial sum, we did S sub n. This is going to be the polynomial P sub n is the degree of the polynomial is this. It's the y value plus the derivative there times x minus c.
What do you notice about those first two terms alone? You recognize those first two terms? That's the Taylor form of a line, isn't it?
It's the y value at c plus the derivative times x minus the c. But look, I can keep adding terms in this format. The second derivative at c over 2 factorial times x minus c squared.
The third term is going to be the third derivative at c over 3 factorial times x minus c cubed. The fourth term is going to be the fourth derivative at the center over 4 factorial times x minus c to the fourth. The fifth derivative will be what?
Or the fifth term will be the fifth derivative at the center over 5 factorial times x minus c to the fifth. OK, in general, the nth term in the polynomial is going to be the nth derivative evaluated at the center times x minus c to the n over n factorial. Or you could think of these as being the coefficients, right?
These are the coefficients, the nth derivative of the center over n factorial, and then times x minus c to the n. These are your variables. Okay. Now, Brooke Taylor was an English mathematician.
He's the guy that kind of came up with this. That's why they're called Taylor polynomials, or later on they're called Taylor series. There is another mathematician that kind of.
He was born after, but he did the work before. His name is McLaurin. And McLaurin, you may hear that term McLaurin series. A McLaurin polynomial or a McLaurin series is nothing more than a Taylor series whose center is, guess what, zero.
Like the one we just did for sign, that would be a Taylor polynomial centered at zero, also known as a McLaurin series. Now, the nice thing about Taylor's version is he expanded it to include. any center we want, right?
If I was trying to evaluate sine of 5,000.2, I would not center it at zero, right? I would probably center it further down the line at like 5,000, okay? Even though I can come up with a polynomial that might wrap itself around all the way to 5,000.
So that's important. You can read a little bit about both those guys here if you want, but this is super important. This will be in every section the rest of the way.
To get... the nth term to the nth power in a polynomial, it's the nth derivative at the point of tangency at the center over n factorial. times x minus the center to the nth power.
Now, when we write the equation of the tangent line, right, this is Taylor form of a line, it fits the pattern, right? Watch this f of c is going to be the zero derivative at c bless you over zero factorial times x minus the center to the 0 power. Now, we don't really call the 0th derivative the 0th derivative. Guess what we call the 0th derivative?
Just the function itself, right? What is 0 factorial? We talked about it earlier in the year.
It's 1. And what's x minus c to the 0? 1. So this pattern reduces down to just the y value. This here would be the first derivative at the center over 1 factorial times x minus c to the first. And of course, what is one factorial?
One. And anything to the first power, you don't need the exponent. So the pattern doesn't really pick up noticeably until the second power, right? Now you have the second derivative at the center over two factorial over x minus c squared.
So notice the pattern is still there. So when I teach you to write these Taylor polynomials, I'm going to teach you this way. Always write out the Taylor form of the line first. and then kick in the pattern at the second derivative, okay?
Because that's when it's visible. And to write it out like the blue pieces here every time, which you could, that reinforces the rule of the pattern, but it's a lot of simplifying at the end, okay? So the y value plus the derivative times x minus the x value, and then pick up the pattern, the Taylor pattern. And one last thing, if the center is zero, if c is zero, what's another name for this Taylor polynomial? Maclaurin.
mclorin that's a term that might be thrown out on my or the ap exam it'll say write a mclorin polynomial and you know oh the center is zero now what we're going to do is um we're going to actually look at example four starting next time because example three uh well we might do example three as well but you got the idea right and that's important and there's the bell so have a great sea lunch i'm going to open up the ap classroom so