 In this video, we'll work through the development of the nth degree Taylor polynomial for a function f about x equals zero. Then we'll generalize this. We'll begin with the general form of our polynomial of interest. T sub n of x, meaning that nth degree Taylor polynomial, equals a sub zero plus a sub one x plus a sub two x squared plus a sub three x cubed plus all the way up to our nth degree term, a sub n x to the n, where a sub k are real numbers for any k greater than or equal to zero. Now, we remember that linear approximations of functions near a point were generated by matching both the function value and the slope at a certain point with the line we're generating. These lines gave us approximations of the function of interest, but the approximations were only good very close to the point of interest, as we can see in this following graph. Suppose we take a look at some function f, and I generate my linear approximation off of x sub zero. This linear approximation is good in some neighborhood around x sub zero, but we wouldn't want to use it to predict values or approximate the function far away from x sub zero. What if we could enhance the accuracy of this approximation? How would we do this? Well, what if we, in addition to matching the function value and slope at a certain point with the approximating polynomial, we match the concavity of the function as well? This would mean that we want the second derivative of the polynomial to match that of the function at the specified point. Expanding on this idea, we match the first n derivatives of our function with that of the polynomial we're generating. We find the following. We would match f of zero to that of the nth degree Taylor polynomial at zero, which is a sub zero. We've matched the first derivative of f at zero with the first derivative of the Taylor polynomial at zero, which we see is a sub one. Matching the second derivative of f at zero and that of the nth degree Taylor polynomial at zero, we obtain two a sub two. Now, I've written that also as two factorial a sub two, which will be explained in a moment. Matching the third derivative of f at zero with the third derivative of the nth degree Taylor polynomial at zero, which we find is three times two a sub three and I've written that also as three factorial a sub three. We find that we have some form of a pattern in these terms. The nth derivative of f at zero matched with the nth derivative of the nth degree Taylor polynomial at zero turns out to be n factorial a sub n. We can then solve for the coefficients of t sub n of x and we find that a sub zero is equal to f of zero. A sub one is equal to first derivative of f at zero. A sub two is second derivative of f at zero divided by two factorial. A sub three is the third derivative of f at zero divided by three factorial and so on. And our general term is the nth derivative of f at zero divided by n factorial. So the nth degree Taylor polynomial centered at x equals zero is as follows. And in general form, we see that the sum from zero to n of the kth derivative of f at zero divided by k factorial x to the k is our nth degree Taylor polynomial about x equals zero. To generalize further, we can think about the Taylor polynomial for a function about any point x equals x naught. Using the same process as we did here, we would find that this polynomial would be t sub n of x equals f of x sub zero plus f prime of x sub zero times x minus x naught. And note this would be our linear approximation. Expanding on that, our quadratic term is second derivative of f at x naught divided by two factorial times x minus x naught squared and so on all the way up to the nth term. And we can write this in summation form as the sum from k equals zero to an n of the kth derivative of f at x naught divided by k factorial times x minus x naught to the k.