3.04.4.Literacy

# 3.04.4.Literacy - Approximation Authors Bill Davis Horacio...

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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 LITERACY · L.1) All you know about a function f @ x D is: f @ 0 D = 2, f £ @ 0 D = 6, and f ££ @ 0 D = - 8. Write down the expansion of f @ x D in powers of x through the x 2 term. · L.2) All you know about a function f @ x D is: f @ 2 D = 1, f £ @ 2 D = - 3, and f ££ @ 2 D = 1. Write down the expansion of f @ x D in powers of H x - 2 L through the H x - 2 L 2 term. · L.3) All you know about a pair of functions f @ x D and g @ x D is: f @ 1 D = 0, f £ @ 1 D = 6, and g @ 1 D = 0, and g £ @ 1 D = 2 . Calculate the lim x Ø 1 f @ x D g @ x D . · L.4) All you know about a pair of functions f @ x D and g @ x D is: f @ 1 D = 0, f £ @ 1 D = 0, f ££ @ 1 D = 8, g @ 1 D = 0, g £ @ 1 D = 0, and g ££ @ 1 D = 2. Calculate lim x Ø 1 f @ x D g @ x D . · L.5) Here is the expansion of f @ x D = e Sin @ p x D in powers of x - 1 through the H x - 1 L 4 term: Normal A Series A E Sin @ p x D , 8 x, 1, 4 <EE 1 - p H - 1 + x L + 1 2 p 2 H - 1 + x L 2 - 1 8 p 4 H - 1 + x L 4 Here is Taylor's formula for the expansion of a cleared function f @ x D in powers of H x - 1 L through the H x - 1 L 4 term: Clear @ f D ; Normal @ Series @ f @ x D , 8 x, 1, 4 <DD f @ 1 D + H - 1 + x L f £ @ 1 D + 1 2 H - 1 + x L 2 f ££ @ 1 D + 1 6 H - 1 + x L 3 f H 3 L @ 1 D + 1 24 H - 1 + x L 4 f H 4 L @ 1 D Use what you see above to write down the values of f £ @ 1 D , f ££ @ 1 D , f H 3 L @ 1 D , and f H 4 L @ 1 D in the case that f @ x D = e Sin @ p x D . · L.6) Here is the expansion of f @ x D = Sin @ Tan @ x DD - Tan @ Sin @ x DD in powers of x through the x 9 term: Normal @ Series @ Sin @ Tan @ x DD - Tan @ Sin @ x DD , 8 x, 0, 9 <DD - x 7 30 - 29x 9 756 No, that's not a misprint. How does this reveal that f @ 0 D = f £ @ 0 D = f ! @ 0 D = f H 3 L @ 0 D = f H 4 L @ 0 D = f H 5 L @ 0 D = f H 6 L @ 0 D = 0 but f H 7 L @ 0 D ! 0?

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