Occasionally you will see someone try to adjust their data in an attempt to use Simpson’s rule. To explain, the goal is to evaluate the integral                                 Assume that the value...

Occasionally you will see someone try to adjust their data in an attempt to use Simpson’s rule. To explain, the goal is to evaluate the integral

Assume that the value of
() is known at


_i−1
and


_i+1
but not at


_i
. The question is, can you use the data to find an approximation for
(

_i
) that will enable you to use Simpson’s rule, and in the process get a better result than you would get using the trapezoidal rule?

(a) What approximation of the integral is obtained using the trapezoidal rule?

(b) One possibility for approximating
(

_i
) is to use piecewise linear interpolation using the two data points (

_i−1,


_i−1) and (

_i+1,


_i+1). Doing this, and inserting the resulting approximation for fi into Simpson’s rule, what results? How does this differ from your answer in part (a)?

(c) Suppose one just assumes that there are constants
 and
 so that


_i

=


_i−1
+


_i+1. With this, Simpson’s rule reduces to an integration rule of the form

What do

₁
and

₂
have to be to maximize the precision? How does this differ from your answer in part (a)?