Mtg 6: Sun, 16 Jan 11
Proof of Taylor series continued
Int. by parts (1)
Combine [ + β - α] into a single int.
use (2) p.6-1 in (2) p.5-3:
HW*2.1: 1) Do integration by parts on last term (integration) of (1) to reveal 3
more terms in Taylor series, i.e. ,
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
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
Use "g(x)" instead of "f(x)" to avoid confusion with "f(x)" is a Taylor Series.
Constrast Taylor Series of f(.) around
for n = 0,1,2,...,10
Plot these series (for each n) Find (estimate) max
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.