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3.04.TaylorFormula

# 3.04.TaylorFormula - Approximation Authors Bill Davis...

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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 40320 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 ê .x -> b L ê 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 40320 + H - 1 + x L 9 362880 + H - 1 + x L 10 3628800 + H - 1 + x L 11 39916800 + H - 1 + x L 12 479001600 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 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 40320 + H - 1 + x L 9 362880 + H - 1 + x L 10 3628800 + H - 1 + x L 11 39916800 + H - 1 + x L 12 479001600 Play with these by going back to the beginning and changing f @ x D , b, and n. · B.1.c.i) What use is Taylor's formula? · Answer: Many new practitioners of calculus want to jump on this formula and use it to slam out all expansions. But those who have been around for a while know this is not a good idea even if you are as fast as Mathematica . As a matter of fact, this formula is usually the least efficient way to obtain an expansion of a given function. Just think of the misery involved in calculating many derivatives and plugging in.

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3.04.TaylorFormula - Approximation Authors Bill Davis...

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