Let’s say we’ve got the function

f of x is equal to e to the x. And just to get a sense

of what that looks like, let me do a rough drawing of

f of x is equal to e to the x. It would look

something like this. So that is e to the x. And what I want to do is

I want to approximate f of x is equal to e to the

x using a Taylor series approximation, or a

Taylor series expansion. And I want to do it not

around x is equal to 0. I want to do it around

x is equal to 3, just to pick another

arbitrary value. So we’re going to do it

around x is equal to 3. This is x is equal to 3. This right there. That is f of 3. f of 3 is e to the third power. So this is e to the third

power right over there. So when we take the

Taylor series expansion, if we have a 0 degree

polynomial approximating it, the best we could probably do

is have a constant function going straight through

e to the third. If we do a first

order approximation, so we have a first

degree term, then it will be the tangent line. And as we add more and

more degrees to it, we should hopefully be

able to kind of contour or converge with the curve

better and better and better. And in the future, we’ll

talk a little bit more about how we can

test for convergences and how well are we converging

and all that type of thing. But with that said, let’s

just apply the formula that hopefully we

got the intuition for in the last video. So the Taylor series

expansion for f of x is equal to e to the

x will be the polynomial. So what’s f of c? Well, if x is equal

to 3, we’re saying that c is 3 in this situation. So if c is 3, f of 3 is

e to the third power. So it’s e to the third power

plus– what’s f prime of c? Well f prime of x is also

going to be e to the x. You take the derivative of e

to the x, you get e to the x. That’s one of the super cool

things about e to the x. So this is also f prime of x. Frankly, this is the same thing

as f the nth derivative of x. You could just keep

taking the derivative of this and you’ll

get e to the x. So f prime of x is e to the x. You evaluate that at 3, you

get e to the third power again times x minus 3, c is

3, plus the second derivative our function is

still e to the x, evaluate that at 3, you get

e to the third power over 2 factorial times x minus

3 to the second power. And then we could keep going. The third derivative

is still e to the x. Evaluate that at 3. c

is 3 in this situation. So you get e to the third

power over 3 factorial times x minus 3 to the third power. And we can keep going

with this, but I think you get the general idea. But what’s even more interesting

than just kind of going through the mechanics of

finding the expansion, is seeing how as we add

more and more terms, it starts to approximate e

to the x better and better and better. And our approximation gets

good further and further away from x is equal to 3. And to do that, I

used WolframAlpha, available at wolframalpha.com. And I think I typed in

Taylor series expansion e to the x and x equals 3. And it just knew what

I wanted and gave me all of this business

right over here. And it actually

calculated the expansion. And you can see it’s

the exact same thing that we have over here, e to the

third plus e to the third times x minus 3. We have e to the third plus e

to the third times x minus 3 plus 1/2. They actually expanded

out the factorial. So instead of 3 factorial,

they wrote a 6 over here. And they did a bunch

of terms up here. But what’s even more interesting

is that they actually graph each of these polynomials

with more and more terms. So in orange, we

have e to the x. We have f of x is

equal to e to the x. And then they tell us,

“order n approximation shown with n dots.” So the order one

approximation, so that should be the situation where we

have a first degree polynomial, so that’s literally– a

first degree polynomial would be these two

terms right over here. Because this is a 0-th degree,

this is a first degree. We just have x to the

first power involved here. If we just were to plot this–

if this was our polynomial, that is plotted with 1 dot. And that is this one right

over here, with one dot, and they plot it

right over here. And we can see that

it’s just a tangent line at x is equal to 3. That is x is equal to

3 right over there. And so this is the tangent line. If we add a term, now we’re

getting to a second degree polynomial, because we’re

adding an x squared. If you expand this out,

you’ll have an x squared term, and then you’ll

have another x term, but the degree of the polynomial

will now be a second degree. So let’s look for two dots. So that’s this one

right over here. So let’s see, two dots. Two dots coming in. See, you’ll notice

one, two dots. So you have two dots,

and it comes in. And this is a parabola. It’s a second degree

polynomial, and then it comes back like this. But notice it does a better job,

especially around x equals 3, of approximating e to the x. It stays with the curve

a little bit longer. You add another term– let me

do this in a new color, a color that I have not used. You add another term. Now you have a third

degree polynomial. If you have all

of these combined, if this is your polynomial, and

you were to graph that– and so let’s look for the three

dots right over here. So one, two, three. So it’s this curve. Third degree polynomial is

this curve right over here. And notice, it

starts contouring e to the x a little bit sooner

than the second degree version. And it stays with it

a little bit longer. And so you have

it just like that. You add another term to it,

you add the fourth degree term to it. So now we have all of

this plus all of this. If this is your

polynomial, now you have this curve right over here. Notice every time

you add a term, it’s getting better and

better at approximating e to the x further and further

away from x is equal to 3. And then if you add another

term, you get this one up here. But hopefully that

satisfies you, that we are getting closer and

closer, the more terms we add. So you can imagine it’s a

pretty darn good approximation as we approach adding an

infinite number of terms.

@khanacademy could you make a video of a antiderivative because i'm having some trouble whit it and it would help me a lot (and sry about my English it's not my native language) thank you in advance

@MrTopa92 go to calculus playlist and ya will find them there "The Indefinite Integral or Anti-derivative "

Excellent video sir

Can you do a video on the lagrange remainder and taylor's inequality?

Well, this is very clear, and it's fascinating to watch the close convergences develop. Humbly, I must say, I find Taylor Series to be awe-inspiring. Modeling functions with infinite polynomials blows my mind out the door and into the next county. I suspect my mind could use some extra elastic proteins added that Taylor must have had in abundance in his rather incredible, plastic fantastic, and drastically insightful brain!

Do you have anything on parametrizations?

With the testing for convergece stuff around 1:25, it sounds like you might be getting into a bit of Numerical Analysis. Awesome.

@LV07TSK Thanks! Glad you enjoyed the comments. I try to keep my sense of humor engaged in dealing with math. I wish there was an infinite polynomial breakfast cereal that would make the topic more accessible to the general public. Taylor polynomials definitely require mental physical fitness. I'd best get over to the neurotransmitter gym before I look them over again.

awesome explanation….. please uploading visualization videos like this…

what's confusing: the graph of successive taylor terms looks great around the point of interest, but what about points further away? The approximation looks horrible out there! What gives?

what's wrong with the Eng sub, it does not show up

If we look at each term except the first it looks like each one will be zero. If x = 3, and each term other than the first has a factor of (x – 3) then why wouldn't they be zero?

This guy makes me miss Barrack

Hello ! I was wondering what the last term on the graph's expression was for 0((x-3)^6) : I know that it is a "negligeable" sign (or at least that is what it is called in french) but I do not see it's use in the taylor series. Could anyone help me out ?

thank you sal

Hey sal!!!!

can't u make a video on Gaussian integral…