PROFESSOR: Welcome
back to recitation. In this video what
I'd like us to do is practice Taylor series. So I want us to write
the Taylor series for the following
function, f of x equals 3 x cubed plus 4 x
squared minus 2x plus 1. So why don't you
pause the video, take some time to work on
that, and then I'll come back and show you what I get. All right, welcome back. Well, we want to find the Taylor
series for this polynomial f of x equals 3 x cubed plus
4 x squared minus 2x plus 1. So what I'm going to do is
I'm just going to write down Taylor's-- or the expression
we have for the sum, for the Taylor series in general
and then I'm going to start computing what I need and
I'm going to see what I get. So what do I need to remember? Well let's remind ourselves
what the formula is. We should get f of x is
equal to the sum from n equals 0 to infinity of the
nth derivative of f at 0 over n factorial times x to the n. So that's what we want. So what I obviously
need to start doing is figuring out
derivatives of f at 0. And so what I'm going
to do is I'm just going to make myself a little table. So let's see, we're going
to say f 0 at 0, f 1 at 0, f 2 at 0, f 3 at 0, f 4 at 0. And I'm getting
tired of writing, so I'm going to stop there. OK, so let's take the function--
the zeroth derivative of f is just the function itself,
so let's come back here. What is the function if I
evaluate it at x equals 0? 0, 0, 0, 1. I get 1. All right. What is the first derivative? So I'm going to write
out the first derivative and then I'm going to say I'm
evaluating it at x equals 0. So the first derivative looks
like it's 9 x squared plus 8 x minus 2. So I'm gonna write this down. 9 x squared plus 8 x minus 2. Evaluate it at x equals 0. 0, 0. I get negative 2. All right, well, let me
take the second derivative. OK let's see what I get here. I get 18x plus 8. And I'm going to evaluate
that at x equals 0. This is just a way
to write, I'm going to evaluate what's here
at x equal this number, so if you haven't
seen that before. So I get 8. OK and then the third
derivative is 18, oh just 18. Evaluate it at x equals 0. I get 18. And then the fourth derivative. What's the derivative
of a constant function? It's 0. What do you think the fifth
derivative is evaluated at 0? Looks like it'll be 0. You take the sixth derivative. Looks like everything
bigger than 3-- so the nth derivative at 0 is
equal to 0 for n bigger than 3. So it looks like we should
only have 4 terms in this. So that maybe seems a little
weird, but let's keep going and see what happens. Let's start plugging things in. So again, let's
remember the formula. I'm going to walk
over here to the right and I'm going to start
using that formula and using these numbers that
I have and writing it out. So the first term is going to
be the function evaluated at 0 divided by 0 factorial times 1. 0 factorial is 1,
so it's just going to be the function
evaluated at 0 times 1. The function evaluated
at 0, we said was 1, so that's the first term
in the Taylor series. OK what's the next term? The next term, remember,
is the first derivative evaluated at 0 divided
by 1 factorial, which is still 1, times x. So the first derivative,
if I come back over here, evaluated
at 0, I get negative 2. So I'm going to get minus 2x. The next term, so I had zeroth
derivative, first derivative, now I'm at the
second derivative. Now it's getting confusing. I'm going to start
writing the things above. The second derivative
evaluated at 0 divided by 2 factorial
times x squared. That's what I should have here. Let's come over here. Second derivative
evaluated at 0 was 8. So it's going to be 8 over
2, 'cause 2 factorial is 2, x squared. So it's going to be
plus 4 x squared. And then I have to have the
third derivative evaluated at 0 divided by 3
factorial times x cubed. What's 3 factorial? 3 factorial is 6. What was the third
derivative evaluated at 0? It was 18. 18 divided by 6 is 3. So I get plus 3 x cubed. And all the other terms were 0,
so I'll just stop writing them. OK now if you watched the
video all the way through here, at some point maybe you said
"Christine, this is madness." Well why is it madness? Because what is this? Well this is the
function again, right? It's exactly what
we started with. The order is opposite of
what it was before 'cause now the powers go up
instead of down, but it's the same polynomial. OK we talked about
this briefly, I think, when we were doing some
quadratic approximations. And I mentioned way back
that quadratic approximations of polynomials at x equals 0
are just the polynomials again. This is the exact same
kind of thing happening. Because what is
the Taylor series? It's just better and
better approximations as n gets larger and larger. So if I wanted to
have a fourth order approximation of this
function f of x at x equals 0, I would get the
same function back. That's really the idea
of what's happening here. So maybe you saw the sort
of trick in this question, and when you saw this
problem you laughed at me and you said, "Well
I'm just going to write down the function
again and I'll be done." Maybe you didn't
see that right away, and if you didn't see
that right away that's OK. I bet you're in good company. And it's totally fine
because now you've seen this. You've seen how it works out. And you know, hey, now any
time I see a polynomial and I want to do the Taylor
series for this polynomial, I just have to write down
the polynomial again. So that was the main
goal of this video. It took us a long time to get
there, but I think we got it. So the answer to the ultimate
answer to the question of write the Taylor series
of this function, it's just this function again. All right, that's
where I'll stop.
