Finding Taylor polynomial approximations of functions
Visualizing Taylor polynomial approximations
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.
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