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Chapter7

# Chapter7 - Chapter 7 Taylor Polynomials and Taylors Theorem...

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Chapter 7 Taylor Polynomials and Taylor’s Theorem We conclude the course with special objects called Taylor polynomials . Some functions can be approximated very well using these Taylor polynomials, and so limits involving quotients of these functions—such as the one that appears at the end of the last section—are very easy to evaluate. 7.1 Taylor Polynomials Question: We know that if f ( x ) is differentiable at x = a , then f ( x ) L a ( x ) := f ( a )+ f 0 ( a )( x - a ) for x near a . Given an x , how large is the error | f ( x ) - L a ( x ) | ? What factors affect the error? We know that there are at least two factors affecting the error: the magnitude | x - a | ( i.e. , the distance from a to x ) and the “sharpness” of the curvature of the graph near x = a ( i.e. , the size of | f 00 ( x ) | ). L a ( x ) has these two important properties: 1. f ( a ) = L a ( a ), and 2. f 0 ( a ) = L 0 a ( a ). It is easy to see that those two properties define the linear approxima- tion; that is, L a ( x ) = f ( a ) + f 0 ( a )( x - a ) is the only polynomial of degree one or less with those two properties. It does seem strange, however, that we want to approximate generic curves with straight lines when we could possibly do better with parabolas or higher degree polynomials, which, being curves, should approximate other curves better. What properties do we want from such polynomials? We want them to satisfy a set of properties generalized from the two properties that L a ( x ) satisfy. Such a polynomial P n,a ( x ) := 111

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c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + · · · + c n ( x - a ) n should satisfy these properties: P n,a ( a ) = f ( a ) , (7.1) P 0 n,a ( a ) = f 0 ( a ) , P 00 n,a ( a ) = f 00 ( a ) , P 000 n,a ( a ) = f 000 ( a ) , . . . P ( n ) n,a ( a ) = f ( n ) ( a ) . Does such a polynomial exist? For n = 1, the answer is yes : We have seen that L a ( x ) is the desired poly- nomial. For n = 2, we require P 2 ,a ( x ) := c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 to be such that P 2 ,a ( a ) := c 0 = f ( a ), P 0 2 ,a ( a ) = c 1 + 2 c 2 ( a - a ) = c 1 = f 0 ( a ), and P 00 2 ,a ( a ) = 2 c 2 = f 00 ( a ) c 2 = f 00 ( a ) 2 . This means that P 2 ,a ( x ) := f ( a ) + f 0 ( a )( x - a ) + f 00 ( a ) 2 ( x - a ) 2 is the only candidate polynomial of degree two or less to satisfy equations listed in (7.1). On the other hand, one can easily verify that P 2 ,a ( x ) defined above does indeed satisfy (7.1); therefore, P 2 ,a ( x ) is the unique polynomial of degree two or less with the properties in (7.1). In general, P n,a ( x ) := n X k =0 f ( k ) ( a ) k ! ( x - a ) k is the unique polynomial of degree at most n such that the equalities in (7.1) all hold. We make the following definition. Definition 7.1.1. Suppose that f : S R where S R , and that I S is an open interval containing some point a S . Suppose also that f ( x ) is n -times differentiable at x = a . Then the n th degree Taylor polynomial of f ( x ) centered at x = a is the polynomial P n,a : R R defined by P n,a ( x ) = n X k =0 f ( k ) ( a ) k ! ( x - a ) k = f ( a ) + f 0 ( a )( x - a ) + f 00 ( a ) 2! ( x - a ) 2 + f 000 ( a ) 3! ( x - a ) 3 + · · · + f ( n ) ( a ) n ! ( x - a ) n . Note that P 1 ,a ( x ) = L a ( x ) is the linear approximation for f ( x ) centered at x = a . 112
Example 7.1. Let f : R R be defined by f ( x ) = e x . Then f ( x ) is n -times differentiable at x = 0 for any n N . Thus the n th degree Taylor polynomials of f ( x ) centered at x = a for n = 0 , 1 , . . . are P 0 , 0 ( x ) = f (0) = 1 , P 1 , 0 ( x ) = f (0) + f 0 (0)( x - 0) = 1 + x, P 2 , 0 ( x ) = f (0) + f 0 (0)( x - 0) + f 00 (0) 2 ( x - 0) 2 = 1 + x + x 2 2 , .

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Chapter7 - Chapter 7 Taylor Polynomials and Taylors Theorem...

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