2015 AP Calculus BC 6c

- [Voiceover] Write the first nonzero terms of the Maclaurin series for e to the x. Use the Maclaurin series for e to the x to write the third-degree Taylor polynomial for g to the x is equal to e to the x times f of x about x equals zero. So Maclaurin series, if that looks familiar to you, watch the videos on Khan Academy on Maclaurin series. I'll give a little bit of primary here. For taking the Maclaurin series of f of x, that's going to be equal to f of zero. We could view this f of zero times x to the zero over zero factorial, if you assume zero factorial is equal to one, and then plus f prime of zero times x to the first over one factorial plus f prime prime, the second derivative evaluated at zero, times x squared over two factorial. I think you see the pattern here. Plus f prime prime prime, the third derivative evaluated at zero of x to the third power over three factorial. I think you see where this is going. And of course, this first one, x to the zero over zero factorial, that's just one. So oftentimes it will just be written f of zero. And this term one factorial is just one. So oftentimes it's just a zero, and this f prime of zero times x, so on and so forth. So, actually, let me just write approximately right over there. And so let's do it for e to the x. So e to the x is approximately equal to, well it's going to be e to the zero which is one. Plus, you might already know that f of x. If f of x is equal to e to the x, then f prime of x is also equal to e to the x. That's one of the magical things about e to the x. The slope of the tangent line at any point is, well, equal to the value is equal to the x value there. And if you take the third derivative or second derivative, it's also, you're gonna get as many derivatives as you want. You'll still get e to the x. That's one of the special things about e. So the first derivative evaluated zero, well that's still e to the zero power times x to the first. So plus x. And then we have the second derivative evaluated at zero. Well, that's still one. So it's going to be times, or so it's one times x squared over two factorial. So we just say plus x squared over two, plus x squared over two. Two factorials is two times one and then plus. Once again, the third derivative evaluated at zero, that's just e to the x evaluated at zero which is e to the zero which is one. So x to the third over three factorial. So plus x to the third over three factorial. We could write it as three factorial or that's three times two times one. That's equal to six. And then we keep going. So we just wrote the first four nonzero terms of the Maclaurin series for e to the x. That's one, two, three, four nonzero terms. Now we wanna use, so let me just underline that. That's part of what they're asking us to do. Then they say use the Maclaurin series for e to the x to write the third-degree Taylor polynomial for g of x which is equal to the product of e to the x and f of x. So what I'm gonna do is I'm gonna write our original f of x. Let's write down the first few terms of it. And then what we could do this thing about, well, how can we multiply those two polynomials? And we just have to know enough about the multiplication of those two polynomials to just get us our terms that are no higher than third degree. So f of x is approximately equal to, let's see, it's x, I have a bad memory, it is x minus 3/2 x squared. X minus 3/2 x squared and then, plus three x to the third power, plus three x to the third power. Plus three x to the third power. Plus, and actually you could say minus if you like because that is gonna be, you are gonna have, it's plus minus plus minus, however you want us to do it. And that's enough. And, why do I feel confident that that's enough? Well, we only want we only want to write the third degree Taylor polynomial. So if we multiply this and we involve terms that are higher than third degree, well, we're gonna, that's going to give us the terms that are polynomial that are higher than third degree. So let's just think about it. What's gonna be the product? So e to the x times f of x. That's going to be approximately equal to, well let's see. We are going to multiply this infinite polynomial times this infinite polynomial and that might seem intimidating to you at first. What you could do is you could go for each of these terms. Must start multiplying it times each of these terms at here. Essentially, you know, when you're multiplying polynomials, you're just repeatedly doing the distributive property so you take this and distribute it on that. But we should only worry about the terms up to third degree because anything beyond that, well, that's going to add up to a higher than third degree term. So x times one is x. X times x squared, or sorry, x times x is x squared so plus x squared. X times x squared over two is x to the third over two. So I'll just write plus 1/2 x to the third power. And I'm gonna stop there because if I do x times that, that's gonna be fourth degree term and I don't wanna worry about that. We are writing the third-degree Taylor polynomial. So that's x, x squared, x to the third over two. So now let's distribute the negative 3/2 x squared. I'll use another color here just to help explain it a little bit. So if you multiply that times one, that's going to be minus 3/2 x squared. I'm just going to a second line here so I can line things up and add them nicely. And then this times this is negative 3/2 x to the third. Negative 3/2 x to the third. And once again, I'm gonna start there. If I multiply these two, I'm gonna get a fourth degree term. I don't care about the fourth degree terms. And then, and then, let's do, let's worry about this guy. And so let's start distributing. So if I multiply it times this one, I'm gonna get three x to the third power. Three x to the third power. And I'm gonna stop there. Because then if I start multiplying it times that guy, it's getting me a fourth degree term and then a fifth degree term and then a sixth degree term which I don't need to worry about. So these are all the pieces that are going to make up that third degree polynomial. And so, what is that going to be equal to? What is that going to be equal to? This is a little bit of, you know, getting your math intuition for multiplying infinite polynomials. Let's see, you're gonna have x. And then you're going to have, this is one x squared minus 1 1/2 x squared. So that's going to be negative 1/2 x squared. Now let's see. Here, you have plus 1/2 minus 1 1/2 which would give you negative two and then plus three, plus, no, sorry, 1/2 minus 1 1/2 which would be negative one plus three is positive two plus two x to the third power. So we could say e to the x times f of x, the third degree Taylor polynomial for this is x minus 1/2 x squared plus two x to the third power. And we are done. So this was a little bit tricky. You have to appreciate how to, and besides calculus, it's a little bit of just algebra. I just appreciate, okay, I only need the third degree. I don't have to distribute this times the infinite number of terms. Because, at first you might say, "That's super hard. "How do I multiply two infinite polynomials?" The key is we only worry about the third degree, up to the third degree.