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Unformatted text preview: Math 16B S06 Supplementary Notes 6 Error Estimate for Approximation by Taylor Polynomials Recall that the n th Taylor polynomial for the function f ( x ) at the point x = a is the polynomial f ( a ) + f ( a )( x- a ) + f 00 ( a ) 2 ( x- a ) 2 + f (3) ( a ) 3! ( x- a ) 3 + + f ( n ) ( a ) n ! ( x- a ) n . It is the unique polynomial of degree at most n that agrees with f at the point a and whose first n derivatives agree with those of f at the point a . With the summation notation it can be rewritten as n X k =0 f ( k ) ( a ) k ! ( x- a ) k , with the conventions f (0) = f and 0! = 1. The n th Taylor polynomial for f at a not only agrees with f at a , also its rate of change at a agrees with that of f , and the same is true for the rates of change of the first n- 1 derivatives. It is thus reasonable to expect that the Taylor polynomial will approximate f closely for x near a . To quantify this expectation one needs an estimate for the error in the approximation. The difference between f and its n th Taylor polynomial at a is given by R n ( x, a ) = f ( x )- f ( a )- f ( a )( x- a )- f 00 ( a ) 2 ( x- a ) 2- f (3) ( a ) 3! ( x- a ) 3- - f ( n ) ( a ) n ! ( x- a ) n . This is the error in the approximation. Often it is referred to as the remainder in the n th Taylor approximation. How can we obtain a useful estimate of the size of R n ( x, a )? The simplest case is the case n = 0, 0 th-order Taylor approximation. In this case we are approximating f by the constant function f ( a ), ridiculous perhaps, but nevertheless indicative of the general case. We have R ( x, a ) = f ( x )- f ( a ) = Z x a f ( t ) dt....
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