This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Estimation of functions using Taylor polynomials Course notes for MATH 138 If a power series a + a 1 ( x p ) + a 2 ( x p ) 2 + + a k ( x p ) k + ... converges to a function f on some interval ( p R,p + R ) centered at p , then the series coefficients a k must be the Taylor coefficients of f . Namely, a k = f ( k )( p ) k ! , where f ( k ) denotes the derivative of f taken k times . This prompts us to pose the converse question. If a function f has derivatives of all orders f ( k ) ( x ) , where x runs through some interval ( p R,p + R ) , must the resulting Taylor series f ( p ) + f ( p )( x p ) + f 00 ( p ) 2 ( x p ) 2 + + f ( k ) ( p ) k ! ( x p ) k + ... converge to f ( x ) for all x in ( p R,p + R ) ? In order to answer our question, we need to examine the error that the partial sums of the Taylor series for f make in estimating f ( x ) . The Taylor polynomial We start with a function f defined on an interval I that contains a number p . Sup pose that the first n derivatives of f , namely f ,f 00 ,f 000 ,...,f ( n ) , are also all de fined. There is no shortage of such functions. For every one of these f , the poly nomial T n ( x ) = f ( p ) + f ( p )( x p ) + f 00 ( p ) 2! ( x p ) 2 + + f ( n ) ( p ) n ! ( x p ) n is called the Taylor polynomial of f , of degree up to n , expanded about p . As we observed at the start, if f is represented by a power series f ( x ) = a + a 1 ( x p )+ a 2 ( x p ) 2 + + a k ( x p ) k + ... on some interval around p, 1 then the n th partial sum of this series is nothing but the Taylor polynomial of f of degree up to n . In sigmanotation, the Taylor polynomial can also be written as T n ( x ) = n X k =0 f ( k ) ( p ) k ! ( x p ) k . The numbers f ( k ) ( p ) k ! that go with ( x p ) k are called the Taylor coefficients of f at the point p . We start counting our Taylor polynomials at n = 0 . When n = 0 , the Taylor polynomial of degree up to is the constant polynomial T ( x ) = f ( p ) . When n = 1 , the Taylor polynomial is the much more interesting linear function T 1 ( x ) = f ( a ) + f ( a )( x a ) . Of course, T 1 ( x ) is the well known tangent line, also known as the linear approxi mation or the linearization , of f at p . When n = 2 , the second Taylor polynomial becomes T 2 ( x ) = f ( p ) + f ( p )( x p ) + f 00 ( p ) 2 ( x p ) 2 . When n = 3 we get T 3 ( x ) = f ( p ) + f ( p )( x p ) + f 00 ( p ) 2 ( x p ) 2 + f 000 ( p ) 6 ( x p ) 3 , and so on. The Taylor polynomial is special for f because it is the one and only polynomial of degree up to n that agrees with f and all of the derivatives f ,f 00 ,...,f ( n ) at the central point p . That is T n ( p ) = f ( p ) T n ( p ) = f ( p ) T 00 n ( p ) = f 00 ( p ) ....
View
Full
Document
 Winter '07
 Anoymous
 Calculus, Polynomials, Power Series

Click to edit the document details