Visualizing Taylor Series Approximations

I've talked a lot
about using polynomials

to approximate functions, but
what I want to do in this video

is actually show you that
the approximation is actually

happening.

So right over here-- and I'm
using WolframAlpha for this.

It's a very cool website.

You can do all sorts of crazy
mathematical things on it.

So WolframAlpha.com-- I got this
copied and pasted from them.

I met Steven Wolfram at a
conference not too long ago.

He said yes, you
should definitely

use WolframAlpha in your videos.

And I said, great.

I will.

And so that's what
I'm doing right here.

And it's super useful,
because what it does

is-- and we could
have calculated

a lot of this on
our own, or even

done it on a graphic calculator.

But you usually can do it just
with one step on WolframAlpha--

is see how well we can
approximate sine of x using--

you could call it a
Maclaurin series expansion,

or you could call it a
Taylor series expansion--

at x is equal to 0 using
more and more terms.

And having a good
feel for the fact

that the more terms we add, the
better it hugs the sine curve.

So this over here in
orange is sine of x.

That should hopefully look
fairly familiar to you.

And in previous
videos, we figured out

what that Maclaurin
expansion for sine of x is.

And WolframAlpha does
it for us as well.

They actually calculate
the factorials.

3 factorial is 6, 5 factorial
is 120, so on and so forth.

But what's interesting
here is you

can pick how many of
the approximations

you want to graph.

And so what they did
is if you want just one

term of the
approximation-- so if we

didn't have this whole thing.

If we just said that our
polynomial is equal to x,

what does that look like?

Well, that's going to be
this graph right over here.

They tell us which
term-- how many terms we

used by how many dots there are
right over here, which I think

is pretty clever.

So this right here, that is the
function p of x is equal to x.

And so it's a very
rough approximation,

although for sine of x,
it doesn't do a bad job.

It hugs the sine curve
right over there.

And then it starts to veer
away from the sine curve again.

You add another term.

So if you have the x minus
x to the third over 6.

So now you have two
terms in the expansion.

Or I guess we should say we
were up to the third-order term,

because that's how their
numbering the dots.

Because they're not talking
about the number of terms.

They're talking about
the order of the terms.

So they have one
dot here, because we

have only one first-degree term.

When we have two
terms here, since we--

when you do the
expansion for sine of x,

it doesn't have a
second-degree term.

We now have a third-degree
polynomial approximation.

And so let's look
at the third-degree.

We should look for three dots.

That's this curve
right over here.

So if you just have
that first term,

you just get that straight line.

You add the negative x to
the third over 6 to that x.

You now get a curve
that looks like this.

And notice it starts hugging
sine a little bit earlier.

And it keeps hugging
it a little bit later.

So once again, just
adding that second term

does a pretty good job.

It hugs the sine
curve pretty well,

especially around
smaller numbers.

You add another term.

And now we're at an order five
polynomial, right over here.

So x minus x to the third over
6 plus x to the fifth over 120.

So let's look for the five dots.

So that's this one right over
here-- one, two, three, four,

five.

So that's this curve
right over here.

And notice it begins hugging
the line a little bit earlier

than the magenta
version, and it keeps

hugging it a little bit longer.

Then it flips back up like this.

So it hugged it a
little bit longer.

And you can see I'll keep going.

If you have all these
first four terms,

it gives us a seventh
degree polynomial.

So let's look for the
seven dots over here.

So they come in just like this.

And then once again, it hugs the
curve sooner than when we just

had the first three terms.

And it keeps hugging the
curve all the way until here.

And then the last one.

If you have all of these
terms up to x to the ninth,

it does it even more.

You start here.

It hugs the curve
longer than the others.

And goes out.

And if you think about
it, it makes sense,

because what's happening here is
each successive term that we're

adding to the expansion,
they have a higher degree

of x over a much, much,
much, much larger number.

So for small x
value-- so when you're

close to the origin
for small x values,

this denominator is
going to overpower

the numerator, especially
when you're below 1.

Because when you
take something that

has absolute value
less than 1 to a power,

you're actually
shrinking it down.

So when you're
close to the origin,

these latter terms
don't matter much.

So you're kind of
not losing some

of the precision of some
of the earlier terms.

When these tweaking
terms come in,

these come in when
the numerator can

start to overpower
the denominator.

So this last term, it starts
to become relevant out here,

where all of a sudden x to the
ninth can overpower 362,880.

And the same thing
on the negative side.

So hopefully this
gives you a sense.

We only have one, two,
three, four, five terms here.

Imagine what would
happen if we had

an infinite number of terms.

I think you get a
pretty good sense

that it would kind of hug the
sine curve out to infinity.

So hopefully that makes you feel
a little bit better about this.

And for fun, you
might want to go type

in-- you can type in
Taylor expansion at 0

and sine of x, or Maclaurin
expansion or Maclaurin

series for sine of x,
cosine of x, e to the x,

at WolframAlpha.com.

And try it out for a bunch
of different functions.

And you can keep adding
or taking away terms

to see how well
it hugs the curve.