12. Derive the error estimate for Simpsons rule over the subinterval a; a Dx. (Hint: Use...

12. Derive the error estimate for Simpson’s rule over the subinterval ½a; a þ Dx.

(Hint: Use the Taylor approximation:

f ðxÞ ¼ f ðaÞ þ f 0ðaÞðx  aÞ þ

1 2

f ð2ÞðaÞðx  aÞ2

þ

1 3!

f ð3ÞðaÞðx  aÞ3 þ

1 4!

f ð4ÞðyÞðx  aÞ4;

for some y A ½a; a þ Dx. Calculate

Ð aþDx a f ðxÞ dx, using the second MVT for integrals in (10.35), and also evaluate the expression for I S over this interval, and subtract, recalling the intermediate value theorem in (9.1).)