. For the trapezoidal rule, express pi ( x ) as f ( xi ) + ( x − xi ) f ( xi +1) − f ( xi ) xi +1 − xi . Expand f in Taylor series about xi and evaluate this at x = xi +1. Use the resulting expression...

.
For the trapezoidal rule, express
pi(x) as

f
(xi) + (x
−
xi)

f
(xi+1) −
f
(xi)

xi+1 −
xi

.

Expand
f
in Taylor series about
xi
and evaluate this at
x
=
xi+1. Use the resulting expression in order to prove (5.14).

.
Following the approach in (5.8)–(5.11), derive
Aij
for
j
= 0,
1,
2 for Simpson’s rule.