 This lesson is on the development of the Taylor or McLaren polynomial centered at zero. This lesson will show you where the Taylor or McLaren polynomial comes from. It will not give you any examples because the examples are very definite as far as determining all the different derivatives. Okay, what is a polynomial? Well, a polynomial is something that reads, as you well know, a sub zero plus a sub 1x plus a sub 2x squared plus a sub 3x cubed plus on and on and on plus a sub nx to the n and if we want to go on infinitely plus dot dot dot. And that is equal to the summation from n is equal to zero to infinity of a sub nx to the nth power. In developing the Taylor polynomial, we will take successive derivatives of our polynomial. This is also equal to some function that we can say is f of x. So let's go on. f of x is equal to a sub zero plus a sub 1x plus a sub 2x squared plus a sub 3x cubed on and on and on until we get a sub nx to the n. If we substitute in a zero for x, we get f of zero is equal to a sub zero because all the other terms will cancel out because they will be multiplied by zero. So keep this in mind. Let's go on. What about f prime? Well, f prime of x will be a sub 1 plus 2a sub 2x plus 3a sub 3x squared plus 4a sub 4x cubed all the way to the n until we get n times a sub nx to the n minus 1. If we substitute a zero in for x, again we get f prime of zero is equal to a sub 1 because all the other terms will cancel out. Well, what happens if we take the second derivative? Well, the second derivative is 2a sub 2 plus 6a sub 3x all the way on. And if we substitute in a zero, all the terms cancel out except the first one and we get 2a sub 2. Let's go on. Let's take the third derivative. So f triple prime of x is equal to 6a sub 3 plus 24a sub 4x plus 60a sub 5x squared plus on and on and on. When we substitute in a zero for x, we get 6a sub 3. On the fourth derivative, we will have 24a sub 4 plus 120a sub 5x plus on and on and on. And when we substitute in the zero, we'll get 24a sub 4. And I think you can begin to see the pattern. If I found the fifth derivative and evaluated at zero, we would get 120a sub 5. Again, keep all these numbers in mind because this is what we're going to use next. As you notice, there was an a sub zero and a sub one and a sub two and a sub three and a sub four and an a sub five and all of this. So this will be the numbers that we will put into a polynomial, our original polynomial. So let's go on. Well, what does a sub zero equal? Well, we found out that a sub zero is equal to f of zero. And a sub one was equal to f prime of zero. And a sub two now is going to be equal to f double prime of zero over two. And a sub three is equal to f triple prime of zero over six. And a sub four is equal to f to the fourth prime of zero over 24. Well, let's go on and look at these. This is actually equal to f double prime of zero over two factorial. That one, the triple prime is f triple prime of zero over three factorial. And f to the fourth prime of zero over four factorial. So what is happening? Each one of these is developing to the prime of the sub number, the nth number, over the factorial of that number. So what does our Taylor polynomial become? Well, our Taylor polynomial becomes f of x is equal to f of zero plus f prime at zero times x plus f double prime at zero over two factorial times x squared plus f triple prime at zero over three factorial x cubed plus on and on and on until you get f to the nth prime of zero over n factorial x to the n. And this is why when you do Taylor polynomials you do take successive derivatives of our function and then substitute in the zero and then put it two factorial times x squared for this particular term just times x for this one and of course the first one is the constant. This is the reasoning why you do Taylor or McLaren polynomials. And again, remember Taylor is the general form of any series that can be developed into a polynomial in this way. And the McLaren one is a particular one centered around zero. This concludes the development of Taylor McLaren polynomials.