3.04.TaylorFormula - Approximation Authors: Bill Davis,...

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Unformatted text preview: Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl 1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.04 Taylor's Formula BASICS B.1) Taylor's formula for the expansion of f @ x D in powers of H x- b L B.1.a) Look at these: Clear @ f, b, x D ; Normal @ Series @ f @ x D , 8 x, b, 3 <DD f @ b D + H- b + x L f @ b D + 1 2 H- b + x L 2 f @ b D + 1 6 H- b + x L 3 f H 3 L @ b D Normal @ Series @ f @ x D , 8 x, b, 8 <DD f @ b D + H- b + x L f @ b D + 1 2 H- b + x L 2 f @ b D + 1 6 H- b + x L 3 f H 3 L @ b D + 1 24 H- b + x L 4 f H 4 L @ b D + 1 120 H- b + x L 5 f H 5 L @ b D + 1 720 H- b + x L 6 f H 6 L @ b D + H- b + x L 7 f H 7 L @ b D 5040 + H- b + x L 8 f H 8 L @ b D 40 320 What's the message? Answer: Look at some more: Clear @ f, b, x D ; Normal @ Series @ f @ x D , 8 x, b, 2 <DD f @ b D + H- b + x L f @ b D + 1 2 H- b + x L 2 f @ b D This has order of contact 2 with f @ x D at x = b. Normal @ Series @ f @ x D , 8 x, b, 6 <DD f @ b D + H- b + x L f @ b D + 1 2 H- b + x L 2 f @ b D + 1 6 H- b + x L 3 f H 3 L @ b D + 1 24 H- b + x L 4 f H 4 L @ b D + 1 120 H- b + x L 5 f H 5 L @ b D + 1 720 H- b + x L 6 f H 6 L @ b D This has order of contact 6 with f @ x D at x = b. The denominators are factorials. The message is that the expansion of f @ x D in powers of H x- b L is f @ b D + f @ b D H x- b L + f @ b D H x- b L 2 2 ! + f @ 3 D @ b D H x- b L 3 3 ! + ... + f @ k D @ b D H x- b L k k ! + ... Most everyone calls this Taylor's formula. This is worth memorizing. B.1.b) Check out Taylor's formula for the coefficients of the expansion of f @ x D = e x in powers of H x- 1 L . Answer: The expansion of e x in powers of H x- 1 L starts out with: n = 6; b = 1; Clear @ x, f D ; f @ x D = E x ; Normal @ Series @ f @ x D , 8 x, b, n <DD + H- 1 + x L + 1 2 H- 1 + x L 2 + 1 6 H- 1 + x L 3 + 1 24 H- 1 + x L 4 + 1 120 H- 1 + x L 5 + 1 720 H- 1 + x L 6 Taylor's formula for the expansion of f @ x D in powers of H x- 1 L is f @ b D + f @ b D H x- b L + f @ b D H x- b L 2 2 ! + f @ 3 D @ b D H x- b L 3 3 ! + ... + f @ k D @ b D H x- b L k k ! + ... So Taylor's formula gives you: Clear @ a, k D ; a @ k _ D : = H D @ f @ x D , 8 x, k <D . x-> b L k ! Sum A a @ k D H x- b L k , 8 k, 0, n <E + H- 1 + x L + 1 2 H- 1 + x L 2 + 1 6 H- 1 + x L 3 + 1 24 H- 1 + x L 4 + 1 120 H- 1 + x L 5 + 1 720 H- 1 + x L 6 Looking good like a calculation should. Do it again: n = 12; Normal @ Series @ f @ x D , 8 x, b, n <DD + H- 1 + x L + 1 2 H- 1 + x L 2 + 1 6 H- 1 + x L 3 + 1 24 H- 1 + x L 4 + 1 120 H- 1 + x L 5 + 1 720 H- 1 + x L 6 + H- 1 + x L 7 5040 + H- 1 + x L 8 40 320 + H- 1 + x L 9 362 880 + H- 1 + x L 10 3 628 800 + H- 1 + x L 11 39 916 800 + H- 1 + x L 12 479 001 600 Sum A a @ k D H x- 1 L k , 8 k, 0, n <E + H- 1 + x L + 1 2 H- 1 + x L 2 + 1 6 H- 1 + x L 3 + 1 24 H- 1 + x...
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This note was uploaded on 10/11/2011 for the course MATH 231 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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3.04.TaylorFormula - Approximation Authors: Bill Davis,...

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