 In this video, we'll generate several Taylor polynomials for the function e to the x centered at x equals 0. We'll also explore the accuracy of these polynomials, both graphically and numerically. Now, we're calling the general form of an nth degree Taylor polynomial about x equals 0. In order to generate this polynomial, we need to find derivatives of the function of interest and evaluate them at the point of interest. In this example, all the derivatives of our function are the same, since f of x equals e to the x's derivatives are all e to the x. Even further, these derivatives evaluated at x equals 0 are all equal to 1. So, we have the following coefficients of this nth degree Taylor polynomial at x equals 0. And in general, we can write that as 1 over n factorial. So, in general, our nth degree Taylor polynomial for f of x equals e to the x about x equals 0 is 1 plus x plus 1 over 2 factorial x squared plus 1 over 3 factorial x cubed plus so on, and to the general term, 1 over n factorial x to the n. We can write that in summation formation as the sum from k equals 0 to n of x to the k over k factorial. Let's take a look at a few Taylor polynomials compared to the function e to the x near x equals 0 graphically. So, here's a graph of e to the x, and I'm going to turn on the graph of the linear approximation, which was the first degree Taylor polynomial for e to the x generated about x equals 0. And we see that near 0, it's actually a pretty good approximation, but the further we veer outside of that interval, we see that the linear approximation is not a good one for e to the x. So, let's take a look at the quadratic approximation. This is also the second degree Taylor polynomial about x equals 0, and we see it's a bit of a better approximation near x equals 0. Let's take a look at the third degree, a little better, and the fourth even better. And we also notice that the higher that degree polynomial, the further away from x equals 0, we can go and still get a pretty good approximation. Let's look at these built upon each other, the first degree Taylor polynomial, second degree, third, and fourth. So, we notice that the higher the degree polynomial, the closer the graph mimics the graph of e to the x, and we notice that the higher the degree polynomial, the further from x equals 0 we can go and be within a certain degree of accuracy. Well, e to the 0.1 is approximately 1.1051709. Well, the first degree Taylor polynomial, 4e to the x, about x equals 0, evaluated at 0.1 gives us 1.1. The second degree polynomial gives us 1.105. The third degree polynomial gives us 1.10526 and so on. The fourth degree polynomial gives us 1.1051708. We find that the higher the degree the polynomial, the closer our approximation is to the actual value of e to the 0.1.