Taylornotes

Taylornotes - Estimation of functions using Taylor...

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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 sigma-notation, 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 ) ....
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Taylornotes - Estimation of functions using Taylor...

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