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# Visualizing Taylor polynomial approximations

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.A (LO)
,
LIM‑8.A.1 (EK)
,
LIM‑8.A.2 (EK)
,
LIM‑8.B (LO)
,
LIM‑8.B.1 (EK)

## Video transcript

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 it would look something like that 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 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 if we have a zero degree polynomial approximating it the best we could probably do is have a constant function going straight through e to the 3rd if we do a first if we do a first-order approximation so we have a first degree term then it will be the tangent line if we add 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 veteran veteran in the future will talk a little bit more about how we can test for convergences and how well are we converging and all of that type of thing but with that said let's just apply the formula that hopefully got the intuition for in the last video so the 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 can 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 of 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 can 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 it how as we add more and more terms it starts to approximate e to the X better and better better and our approximation gets good further and further away from X is equal to 3 and to do that I used Wolfram Alpha available at Wolfram Alpha comm and I think I typed in like Taylor series expansion e to the X and x equals 3 and I just knew what I wanted and gave me all of this business right over here and actually calculated the expansion 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 expand it out the factorial so instead of 3 factorial they wrote a 6 over here and they did a bunch of terms up here but it's even more interesting is that they actually graph each of these each of these 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 and approximation shown with n dots so the order 1 approximation so that should be the 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 zeroth degree this is a first degree we just have X to the first power involved here that will be if we just were to plot this if this was our polynomial that is plotted with one dot and that is this one right over here with one dot and they plot it they plot it right over here and we can see that it's just a tangent line at at X is equal to three that is X is equal to three 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 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 so you'll notice one two dots so you have two dots and it comes in and this is a 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 three of approximating e to the X it stays with the curve a little bit longer you add another term you add another term let me do this in a new color 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 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 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 three and then if you add another term you get this term this one up here but hopefully that satisfies you that we are getting closer and closer the more terms we add so you could imagine it's a pretty darn good approximation as we approach adding an infinite an infinite number of terms
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