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Let's say we've got the function f(x) is equal to e to the x,

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just to get a sense of what that looks like

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we do a rough drawing of f(x) is equal to e to the x.

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It would look something like this. It would look something

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like that. So that is e to the x, and what I want to do is approximate f(x) is equal to e to the x

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using a taylor series approximation or taylor series expansion, and I want to do it

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not around is equal to 0, I want to do it around x is equal to

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3, just to pick another arbitrary value, so we're going to do it around x is

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equal to 3, this is x is equal to 3, this right there, that is f of 3

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, f of 3 is e to the third power. So this is e to the third power

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over there. So when we take the taylor series expansion, if we have a

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zero degree polynomial approximating it, the best we can do is have a constant

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function going straight through e to the third. If we do a first order approximation

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so we have a first degree term, then it will be the tangent line

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. As we add more and more degrees to it, we should hopefully

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be able to, kind of contour or converge with

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the curve better and better and better. In the future we'll talk more about

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how we can test for convergences, and how well are we converging, and all

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that type of thing. But with that said, let's just apply the formula

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, then hopefully we'll have the intuition for in the last video. So the

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taylor series expansion for f(x) is equal to e to the x will be the

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polynomial. So what is f of c. Well if x is equal to 3, we're saying

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that c is 3 in this situation. So if c is 3, f of 3 is e to the third power

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. So that is e to the third power plus, what's f prime of c? Well,

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f prime of x is also going to be e to the x. You take the derivative of

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e to the x to get e to the x. That's one of the supercool things

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about e to the x. So thjis is also f prime of x, gfrankly this is also the sma e as

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f, the nth derivative of x. You can just keep taking the derivative

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of this and you'll get e to the x. So f prime of x is e to the x

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, you evaluate that at 3, you get e to the third power again, times

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x minus 3, c is 3, plus the second derivative of our function, we

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it's still e to the x, evaluate that at 3, you get e to the third power

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over 2 factorial times x minus 3 to the 2nd power, we can keep

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going. The third derivative is still ex, evaluate that at 3, c is three in this

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situation so you get e to the third power over 3 factorial times x minus

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3 to the third power, and we can keep going with this, but I think you get

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the general idea. But what's even more interestin than just kind of going through

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the mechanics of finding the expansion is seeing how it

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how as we add more and more terms it approximates e to the x

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better and better and better, and our approximation gets further away from

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x is equal to 3. A nd to to dthat I used wolfram alpha

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available at wolframalpha.com/.And I think I tpyed in taylor series expansion

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e to the x ad xis equals 3, and it knew what i wanted, and just gave

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me all of this business right over here. And you can see that's its the exact same thing that we have

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over here e to the third and e to third times times x-3 plus 1/2. They actually

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expanded out the factorial, instead of 3 factorial they wrote 6 over here,

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and they did a bunch of terms over here. But whats even more interesting is

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that they actually graph each of these polynomials qwith more and more terms

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So in orange we have e to the x, we have fi s equal to e to the x, and then they

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#tell us, order and approximation shown wuth n dots, so the

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order 1 approximation, so that should be the suituatioin where we have

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a first degree polynomial, so literaally a first degree polynomial would be these

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three temrs right over here. Cause this is a zeroth degree,, this is a first degree

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we just hvae x to the first power involved her. That will be, if we just

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qwere to plot this, if this was our polynomial that is plotted nwith

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one dot, and that is this one right over here is one dot, and they plotted, they

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plotted right over here and we can

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can see that it's just a tangent line at , x is equal to 3, that

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is x is equal to 3 right over there. So this is the tangent line. If

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we add a term, now we're getting to a second degree polynomial

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cause we're adding an x squared, and if you expand this out you'll

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have a x squared term and you'll have another x term, but the degree

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of the polynomial will now be a second degree. So let's look for 2 dots

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So that's this one right over here. So let's see, 2 dots coming in

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see you'll not notice one but two dots, so you have 2 dots and it comes in

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and this is a parabola, this is a second degree polynomial, and then it

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comes back like this. But notice, it does a better job, especially around x equals 3

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of approximating e to the x. It stays with the curve a little bit longer

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. You add another term, let me do this in another colour, a colour

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that I have not used, you add another term. Now you hvae a third degree polynomial

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. If you have all of these combined, if this is your polynomial, and you

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were to graph that, and so let's look for the 3 dots right over here

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. So 1, 2, 3, so it's this curve, third degree polynomial is this curve

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right over here. And notice it starts contouring e to the x a little bit

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sooner than the second degree version, and it stays with it a little bit

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longer. It stays with it a little bit longer, and so you have it just

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like that. You add another term to it, you add the fourth degree term

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to it, so now we have all of this plus all of this. All this is

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your polynomial now you have this cool curve right over here. Notice, if you add

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a term, it's getting better and better at approximating e to the x

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furht er and further away fromx is equal ro 3. And if you add anbother

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

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satisfies you that we are getting closer and closer the more terms

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we add, so you can imagine it's a pretty darn good approximation