Given three points i −1, i , and i +1 , this problem considers the quadratic interpolation formula                 It is assumed here that i − i −1 =  and i +1 − i = . (a) Show that the above...

Given three points


_i
−1,


_i
, and


_i+1, this problem considers the quadratic interpolation formula

It is assumed here that


_i

−


_i−1
=
 and


_i+1
−


_i

=
.

(a) Show that the above interpolation formula can be rewritten as

(b) Calculate

₂(x) and
’₂
().

(c) Suppose

₂() interpolates() at


_i−1,


_i
, and


_i+1, so


_i−1
=
(

_i−1),


_i

=
(

_i
), and


_i+1
=
(

_i+1). Setting
=


_i

+
, expand
() and

₂() about
 = 0 and show that

where
 =
 −


_i
.

(d) How does the result in part (c) compare to the result in Theorem 5.2 in the case of when
 =


_i

−
 =


_i

+
, and
 = 2?