121616949-math.283

# 121616949-math.283 - examples EXAMPLE 10.53 Find a...

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10.11 Taylor’s Theorem 269 Now most of the terms in this expression cancel out, leaving just F ( t ) = f ( N +1) ( t ) N ! ( x - t ) N + B ( N + 1)( x - t ) N ( - 1) . At some z , F ( z ) = 0 so 0 = f ( N +1) ( z ) N ! ( x - z ) N + B ( N + 1)( x - z ) N ( - 1) B ( N + 1)( x - z ) N = f ( N +1) ( z ) N ! ( x - z ) N B = f ( N +1) ( z ) ( N + 1)! . Now we can write F ( t ) = N X n =0 f ( n ) ( t ) n ! ( x - t ) n + f ( N +1) ( z ) ( N + 1)! ( x - t ) N +1 . Recalling that F ( a ) = f ( x ) we get f ( x ) = N X n =0 f ( n ) ( a ) n ! ( x - a ) n + f ( N +1) ( z ) ( N + 1)! ( x - a ) N +1 , which is what we wanted to show. It may not be immediately obvious that this is particularly useful; let’s look at some
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Unformatted text preview: examples. EXAMPLE 10.53 Find a polynomial approximation for sin x accurate to ± . 005. From Taylor’s theorem: sin x = N X n =0 f ( n ) ( a ) n ! ( x-a ) n + f ( N +1) ( z ) ( N + 1)! ( x-a ) N +1 . What can we say about the size of the term f ( N +1) ( z ) ( N + 1)! ( x-a ) N +1 ?...
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