 Welcome to this quick recap of section 8.5 on Taylor polynomials and Taylor series. This is a high point of chapter 8, and so to make sure that it's as clear as possible, we're going to start with an extended example of what Taylor polynomials and series are, how you might use them, and why we care. To begin, let's look at the function e to the x, a very common function from algebra and calculus. We've plotted the one and only point on this function that we know how to evaluate by hand. So how can we evaluate other values on this function? Well, in calc 1 we learned one way to do that by using a tangent line plotted here. Using the first derivative, we were able to calculate the slope of the function at x equals 0 and come up with what we call the tangent line or local linear approximation for e to the x at near x equals 0. The tangent line is a close approximation for e to the x as long as we stay near x equals 0, although as you can see it gets farther and farther away from the curve as we get farther away from x equals 0. We can use this to approximate the value of e. If we substitute x equals 1 into both sides of this approximate equality, we get e on the left and an estimate for it, 1 plus 1 on the right. We've plotted these two values on the curve. The black dot is the location of the exact value of e. It's y value is e because it's on the function e to the x. The red dot is the approximation that is given to us by this tangent line. You can see that they're not particularly close together because we're not close to the point of tangency. That's the local part of a local linear approximation. So although we can get an estimate for e, it's not particularly good. But we don't need to stop here. We can find not only a tangent line but even a tangent quadratic. Here this quadratic shares the same point, 0, 1, the same slope as e to the x, but it also has the same concavity as e to the x at x equals 0. That gives it a much better match for the shape of the curve e to the x. By substituting x equals 1 into this quadratic, we get a better estimate for the value of e, and you can see that the red and black dots have gotten closer together. We don't have to stop there either. We can find a tangent cubic with the same point, slope, second derivative, and third derivative at x equals 0, making it an even better match for the shape of the curve. And again, we get a better estimate for the value of e. If we keep repeating this over and over, we can get an infinitely long polynomial following this pattern. It has the same derivative as e to the x at x equals 0, as well as the same second, third, fourth, and so on derivatives, all at x equals 0. By substituting x equals 1 into this infinitely long polynomial, we get the exact value of e, written out as an infinite series. So why would we want to do this? The first thing to notice is that, while e to the x is on the left, written in the familiar way, this infinitely long polynomial is exactly the same thing as e to the x. It's just another way to write it. It's okay for functions to have more than one way to be written. The reason that we like this is that polynomials are much easier to work with than mysterious functions like e to the x. In particular, we know how to evaluate polynomials at any x value that we're interested in. In fact, by doing that, we can get the exact value of the number e. We could even evaluate e to the x using this infinite polynomial at any other x value that we want, and we can calculate e to any number at all as an infinite series. In fact, any calculator or computer that allows you to calculate e to any power uses either this method or one very similar to it to actually do the calculations. This infinitely long polynomial we found is actually called a Taylor series for e to the x centered at x equals 0, and we can use it to find any value that we're interested in. The polynomials that we found on the previous screens are called Taylor polynomials, and they are estimates for e to the x in very much the same way that a tangent line is an estimate or approximation. Now that we have that idea, let's take a look at the definition of Taylor polynomial. The nth order Taylor polynomial of a function f centered at x equals a is given by this formula, which can be simplified to the summation below, or rewritten in this way. Well, this looks intimidating. A couple of things you should notice are that the first part of it right here is exactly the same as the formula for a tangent line, and every term after that uses a higher derivative of the function f evaluated at this number a called the center. This is just an extension of the idea of a tangent line, making it so that we can calculate tangent polynomials of any degree. Some things to know about this is that these are tangent polynomials. They act just like tangent lines, but they share more than just the same slope. In fact, they share all of the same derivatives as the function that we are interested in at this point a. And so this polynomial piece of n, the nth order Taylor polynomial, approximates f of x near the center x equals a, just like a tangent line approximates f of x near the point of tangency. And the kth derivatives of p and f at this point x equals a all agree, which gives these polynomials the same shape, the same local shape, as the function that we're approximating, as long as you are close to x equals a. Polynomials are much easier to work with than other functions, but Taylor polynomials can be used to calculate values for functions that we don't otherwise know how to work with. If we let a Taylor polynomial go on forever, then we get the Taylor series of f centered at x equals a. This formula is exactly the same as the one on the previous slide, but it goes on forever. These are infinitely long Taylor polynomials, and as such, they can be perfect matches for the function, not merely approximations. If a equals zero, we have a special name for these. If the center is zero, we call them Maclaurin series, after the person who first worked with them. Here's another visual example to show what we mean by all of these. Here's a Taylor polynomial of order one for cosine of x centered at pi over two. Notice that when we say the center is pi over two, that turns out to be the point where the tangent line is tangent to it. A first order Taylor polynomial is a tangent line. Here's the third order Taylor polynomial, or a tangent cubic. Again centered at pi over two, it's a better match for the function. The fifth order Taylor polynomial is an even better match, as is the seventh and the ninth. If we let the order go off to infinity, we get an infinitely long Taylor polynomial called the Taylor series for cosine of x, and you can see here that it is a perfect match for cosine of x. For each x value, a Taylor series becomes a normal numerical series that converges or diverges. We saw this with the example of e to the x before. When we substituted x equals one, we went from a polynomial to a normal numerical series. All of the x values together where the series converges is called an interval of convergence. This interval turns out to always be centered at x equals a, called the center of the series. This interval goes from a minus some number r to a plus some number r, and this value r is called the radius of convergence. The endpoints of this interval might be open or closed. You'll need to use convergence tests from sections 8.3 and 8.4 to decide. And the interval might even be infinite. Two examples are shown below. Visually, you can see on the left how the Taylor series does not appear to match the function at all once we get outside of the interval from negative one to one, but it's a perfect match as long as we're inside that interval. Similarly, the series that we saw for cosine is shown on the right, and it's a perfect match for cosine of x at all x values, and so we say its radius of convergence is infinite. Finally, one last idea from this section is the Lagrange error bound. This tells us how good of an approximation a Taylor polynomial of order n is. So if we want to approximate f of c for some number c, remember f is the real function that we're interested in, and if we can find this number m, which is a positive number so that the n plus 1th derivative of our function f is always between m and negative m on the interval from a to c. That's the meaning of these absolute values. Then the difference between our polynomial at c and the real function's value at c is less than or equal to this formula given on the right over here. And although this looks complicated, this allows us to calculate an estimate for how close our Taylor polynomial really is to the function. It turns out that we can use this to show that many Taylor series actually converge perfectly to the function that the Taylor polynomials are approximating. Now that we've seen these, let's take a look at many examples of how to calculate Taylor polynomials and series and how to work with them.