 In this video, we'll generate several Taylor polynomials for the function natural log of x centered at x equals 2. 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 naught. We see here we can actually write this in summation notation from k equals 0 to n of the kth derivative of f evaluated at x sub 0 divided by k factorial times x minus x sub 0 to the k. And as we know, in order to generate a Taylor polynomial for a function we need to find the derivatives of the function and evaluate them at the point of interest. So the coefficients of our nth degree Taylor polynomial at x equals 2 is, if we find the first derivative, or excuse me, the zeroth derivative of natural log of x is natural log of x. Evaluated at 2 is natural log of 2. First derivative is 1 over x. Evaluated at 2 is 1 over 2. Second derivative is negative 1 over x squared. Evaluated at 2 is negative 1 fourth. Third derivative is 2 over x cubed. Evaluated at 2 is 1 fourth. So the nth degree Taylor polynomial for natural log of x about x equals 2. We'll start with the linear approximation, which is natural log of 2 plus 1 half times x minus 2. The second degree Taylor polynomial is as you see here. Third degree Taylor polynomial is as you see here. We're going to use these to approximate the value of natural log of x near 2 in a moment. But let's take a look at a few Taylor polynomials compared to the function f of x near x equals 2 graphically. So here's the graph of natural log of x. We're focusing here at x equals 2. The linear approximation is a good approximation of the graph of natural log of x near x equals 2. But we see as we veer outside of a small interval around 2 that the approximation is not a good one. Let's take a look at the second degree Taylor polynomial. Quite a bit better. It mimics the concavity as we would expect. The second degree Taylor polynomial generated about x equals 2. And the third degree Taylor polynomial, even better about x equals 2. So let's look at these all at once. The first degree or the linear approximation, second degree Taylor polynomial, third degree. So we notice that the higher the degree polynomial, the closer the graph mimics the graph of natural log of x near x equals 2. We also notice that the higher the degree, the polynomial, the further from x equals 2 we can go and be within a certain degree of accuracy. Well, natural log of 2.1 is approximately 0.741937. The first degree Taylor polynomial at 2.1 is equal to 0.743147. The second degree Taylor polynomial is 0.741897. And the third degree Taylor polynomial is 0.741939. We find that the higher the degree, the polynomial, the closer our approximation is to the actual value of natural log of 2.1.