The position ( ), velocity ( ), and acceleration ( ) are related through the equations: ( ) = ’( ) and ( ) = ’( ). In this problem it is assumed that (0) = 0 and (0) = 0. In this case,...

The position
(), velocity
(), and acceleration
() are related through the equations:
() =
’() and
() =
’(). In this problem it is assumed that
(0) = 0 and(0) = 0. In this case,

It is also assumed that
() is known, and the objective of this exercise is to compute the velocity and position from this information.

(a) Given a subinterval


_i

≤
 ≤


_i+1, then


_i

=
(

_i
) and


_i+1
=
(

_i+1) are known. Assuming


_i

and


_i

have already been computed, use the trapezoidal rule to obtain the following expressions

(b) Suppose the interval 0 ≤
 ≤ 3 is subdivided into
equally spaced subintervals. So,


_i

= ( − 1), where
 = 1, 2, 3, ··· ,
 + 1 and
 = 3/. Assuming that
() = sin(
⁴), plot
 as a function of
, for
 = 10, 20, 40. The three curves should be on the same axis.

(c) An accurately computed value for the position at
 = 3 is
(3) = 0.72732289075 ..... What is the difference between this value and what you compute for
(3) at
 = 10, 20, 40? How large does
 need to be so that this value and what you compute for
(3) is less than 10⁻⁸
in absolute value?