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PROFESSOR: Welcome back
to recitation.

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In this video what I'd
like us to do is

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practice Taylor series.

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So I want us to write the
Taylor series for the

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following function, f of x
equals 3x cubed plus 4x

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squared minus 2x plus 1.

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So why don't you pause the
video, take some time to work

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on that, and then I'll come back
and show you what I get.

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All right, welcome back.

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Well we want to find the
Taylor series for this

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polynomial f of x equals 3x
cubed plus 4x squared

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minus 2x plus 1.

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So what I'm going to do is I'm
just going to write down

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Taylor's, or the expression we
have for the sum for the

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Taylor series in general and
then I'm going to start

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computing what I need and I'm
going to see what I get.

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So what do I need to remember?

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Well let's remind ourselves
what the formula is.

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We should get f of x is equal to
the sum from n equals 0 to

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infinity of the nth derivative
of f at 0 over n factorial

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times x to the n.

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So that's what we want.

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So what I obviously need to
start doing is figuring out

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derivatives of f at 0.

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And so what I'm going to do is
I'm just going to make myself

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a little table.

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So let's see, we're going to say
f0 at 0, f1 at 0, f2 at 0,

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f3 at 0, f4 at 0.

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And I'm getting tired
of writing, so I'm

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going to stop there.

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OK, so let's take the function,
the 0's derivative

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of f is just the function
itself, so

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let's come back here.

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What is the function if
I evaluate it at x

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equals 0, 0, 0, 1.

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I get 1.

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All right.

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What is the first derivative?

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So I'm going to write out the
first derivative and then I'm

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going to say I'm evaluating
it at x equals 0.

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So the first derivative looks
like it's 9x squared

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plus 8x minus 2.

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So I'm gonna write this down.

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9x squared plus 8x minus 2.

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Evaluate it at x
equals 0, 0, 0.

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I get negative 2.

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All right, well, let me take
the second derivative.

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OK let's see what I get here.

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I get 18x plus 8.

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And I'm going to evaluate
that at x equals 0.

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This is just a way to write, I'm
going to evaluate what's

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here at x equal this
number, so if you

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haven't seen that before.

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So I get 8.

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OK and then the third derivative
is 18, oh just 18.

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Evaluate it at x equals 0.

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I get 18.

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And then the fourth
derivative.

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What's the derivative of
a constant function?

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It's 0.

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What do you think the fifth
derivative is evaluated at 0?

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Looks like it'll be 0.

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You take the sixth derivative.

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Looks like everything bigger
than 3, so the nth derivative

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at 0 is equal to 0 for
n bigger than 3.

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So it looks like we should only
have 4 terms in this.

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So that maybe seems a little
weird, but let's keep going

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and see what happens.

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Let's start plugging
things in.

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So again let's remember
the formula.

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I'm going to walk over here to
the right and I'm going to

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start using that formula and
using these numbers that I

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have and writing it out.

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So the first term is going to be
the function evaluated at 0

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divided by 0 factorial
times 1.

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0 factorial is 1, so it's just
going to be the function

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evaluated at 0 times 1.

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The function evaluated at 0, we
said was 1, so that's the

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first term in the
Taylor series.

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OK what's the next term?

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The next term remember is the
first derivative evaluated at

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0 divided by 1 factorial, which
is still 1, times x.

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So the first derivative, if I
come back over here, evaluated

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at 0, I get negative 2.

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So I'm going to get minus 2x.

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The next term, so I had zeroth
derivative, first derivative,

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now I'm at the second
derivative.

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Now it's getting confusing.

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I'm going to start writing
the things above.

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The second derivative evaluated
at 0 divided by 2

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factorial times x squared.

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That's what I should
have here.

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Let's come over here.

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Second derivative evaluated
at 0 was 8.

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So it's going to be
8 over 2, 'cause 2

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factorial is 2x squared.

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So it's going to be
plus 4x squared.

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And then I have to have the
third derivative evaluated at

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0 divided by 3 factorial
times x cubed.

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What's 3 factorial?

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3 factorial is 6.

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What was the third derivative
evaluated at 0?

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It was 18.

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18 divided by 6 is 3.

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So I get plus 3x cubed.

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And all the other terms
were 0, so I'll just

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stop writing them.

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OK now if you watched the video
all the way through

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here, at some point maybe you
said "Christine this is

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madness." Well why
is it madness?

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Because what is this?

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Well this is the function
again, right?

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It's exactly what
we started with.

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The order is opposite of what
it was before 'cause now the

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powers go up instead of down,
but it's the same polynomial.

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OK we talked about this briefly
I think when we were

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doing some quadratic
approximations.

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And I mentioned way back that
quadratic approximations of

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polynomials at x equals 0 are
just the polynomials again.

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This is the exact same kind
of thing happening.

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Because what is the
Taylor series?

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It's just better and better
approximations as n gets

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larger and larger.

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So if I wanted to have a fourth
order approximation of

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this function f of x at x equals
0, I would get the same

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function back.

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That's really the idea of
what's happening here.

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So maybe you saw the sort of
trick in this question, and

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when you saw this problem you
laughed at me and you said,

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"Well I'm just going to write
down the function again and

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I'll be done." Maybe you didn't
see that right away,

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and if you didn't see that
right away that's OK.

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I bet you're in good company.

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And it's totally fine because
now you've seen this.

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You've seen how it works out.

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And you know, hey, now any time
I see a polynomial and I

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want to do the Taylor series
for this polynomial, I just

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have to write down the
polynomial again.

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So that was the main
goal of this video.

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It took us a long time to get
there, but I think we got it.

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So the answer to the ultimate
answer to the question of

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write the Taylor series of this
function, it's just this

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function again.

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All right, that's
where I'll stop.

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