University of Florida/Egm6321/f09.team1.gzc/Mtg6

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Mtg 6: Sun, 16 Jan 11

page6-1

Proof of Taylor series continued

Since

(1)


Int. by parts (1)

page6-2

Combine [ + β - α] into a single int. use (2) p.6-1 in (2) p.5-3:

(1)

HW*2.1: 1) Do integration by parts on last term (integration) of (1) to reveal 3 more terms in Taylor series, i.e. ,


plus remainder

2)Use IMVT to expression remainder in terms of f(s)(ξ) for s belong[x0,x]

3) Assume(3)&(4)p.3-3 correct , do intergration by parts once more


page6-3


to verify(3)&(4) p.3-3 for (n+1) expansionwith R(n+2)(x)

4) UseIMVT on(4) p.3-3to show(5)p.3-3

IMVT:

Use "g(x)" instead of "f(x)" to avoid confusion with "f(x)" is a Taylor Series.






HW*2.2: Constrast Taylor Series of f(.) around

for n = 0,1,2,...,10


Plot these series (for each n) Find (estimate) max


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Note:Motivation for pf of Taylor series expansion. (similar technique will be used)


  • higher order analysis of Trap. rule (not in A.)
  • Richardson extrap.
  • clenshaw-Cwetis quadrature
  • chebyshew poly (orthog.) Recent devel. using chebyshew poly to solveL2_ODE_VC (Linear 2nd order ODE with varying coefficient ) combine of symbolics + numericsReference : Trefethen's chebfun.
page6-5

Mtg6.pg5.fig1.svg