Welcome to Calculus. I'm Professor Greist, and we're about to begin lecture two, bonus material. In our main lesson, we saw a definition for e to the x, in terms of a series, and from that and a little bit of help from Euler's formula. We derived expressions for sin of x and for cosine of x. These can seem a little mysterious, difficult to work with at first but if you think of these as long polynomials. Polynomials of unbounded degree. It can make it easier both to work with them and to understand what the series mean. Let's consider the case of e to the x and see what happens when we cut off the series at some finite polynomial term, since polynomials are so easy to work with. If we stop after the first term. Well, it's a, a bit too simple to conclude much. But if we keep going, looking at 1 plus x. Or 1 plus x plus x squared over 2 factorial. Or continuing to include the cubic term, as well. Then we see that we're getting closer and closer. Two, what the graph of e to the x looks like. Especially if we zoom in very very close to what's happening near the origin. And does the same thing work with the other series that we've looked at? Well, let's consider cosine of x. We know that the terms in the series expansion go, like, one. And then 1 minus x squared over 2. Then, 1 minus x squared over 2, plus x to the fourth over 4 factorial. And what we see is that we're getting closer and closer to what cosine is doing the more terms that we add. In this series, the better an approximation to cosine we obtain. It will come as no surprise to you that the same thing is true of sine, that when we look at the linear term and then the cubic term and then the fifth order term. That, although we're not getting exactly sin, we're getting something that is closer and closer. And this will keep on going. The more terms that we add, the better these polynomial approximations become to the true function. At this point, a few remarks are in order. First of all, these finite polynomial approximations work best near zero. If you want to approximate e to the x, when x is very, very large, then you're going to need quite a few terms. In order to get a good approximation. Now that's okay, e to the x really is this infinite series, but it's precisely because you include all of the terms. And second of all, sometimes you will see the term MacLaurin series used to apply to infinite series of this form. I choose to use the notation Taylor series exclusively, and in a few lessons, we'll see exactly what type of Taylor series we're talking about, when we write out things in this form. Lastly, and most importantly, at this point in the course, you may be ready to press the panic button. We have begun with material that is usually reserved for the end of a second semester calculus course. Relax. Don't panic. You'll see in just a few short lessons what the payoff is for beginning a calculus course with infinite series. Stick with it, learn the mechanics and know that a deeper understanding is coming. This is not an easy course and it's a long one. We have a lot of time left together. Don't panic. Work at it and you'll make it to the end with a deeper understanding of calculus.