Question

1. In class, we derived the Central Difference formula for f'(x0) as f'(x) = f(xo + h) – f(20 - h) +002). f'(o)= 2h A similar result, where four points are involved and the error is (h"), is given in the textbook, page 176, Equation (4.6). Prove it.

Please answer question #1 and write legibly - I have also equation 4.6 from the textbook. Thanks

Answer 1

about x = No is , We have to prove : = f'(x) = [ 62-21) – 8f66-n) +8f69+h) - f(x + 1)] + 5* 4°() The taylor series formula for a function fla) f(*th) = f(x) + n f (x) + f'()} & {1/2) + to (16) + k + 160(2) (we have neglected term higher E (6-1) = f(x)-n f(a) + Eup? "8) + )* { "() + dengan pena) + ventet

* (20) * () * () * ( ب ) = (24+ بها یا د ) (5) - (تام ها (%0) * ( - ) = Ak) (4) () + ( : مواد (1) و (2) و (954 م2) - (+) 8 (1-1) 8 - (:- پتا CookA | (۰) هم 4 - ( 8 - (2) + (4) 1- (%) = (ه) مرد (ه" - ( f + (بهاگ - (a f ( x - (۰)] + 3[ (4) + ۸/۶) - *) * " *) **)

B - [fla)t on files to the ") + 8? 4 "(2) + e pa lan) 1 sp Am] = f() [2-848-1]+ 1"(») [-2h +8h +8 17:36] • 8h +8 ha = (x) *0 + f "(as) • 12 h + f"(w) xo + 3*(w) x0 + f'(a) long - 18h f"(x) ++48)6? fº() ish {"(3) = ) A fºly = 12h 4"() - 48hf°() . 2x3x4 x5 e 4x30 430

:: 184 '(2) - Apna jºlf.) - +7 -n) - 8 f(x-1) + 8 +(4+h) - (+dh) 4x30 where į is between 2-2h and to t2h . We have neglected term involving ho and greater than ho ... We have to take fo (2) where ę is between No-2h and to + 2h. lem f'(x) = f(x -21)-8 f(7. –h)+8f66+) - f(+24) + 48 68 ( => }(3) --on ] f(20-an) – 8 f(2-h]+8(*o+n) – ffrwten?

( 2) A[(x - A)-3 (a + + f(a+b) - spon) + s (5) 以)