(4) Derive the composite trapazoidal rule and composite simpson's rule from simple trapazoidal and simple simpson's rule
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Ref Lecture notes p.9-3
Show Simple Trapazoidal rulep.7-1 Composite Trapazoidal rule p.7-1
and
Show Simple Simpson's rule p.7-2 Composite Simpson's rulep.7-2
Composite Trapezoidal rule
From Simple trapezoidal rule we have ,
similarly we have
.
.
.
.
Summation of all of above expression gives
Hence Proved..
Composite Simpson's Rule
From Simple Simpson's rule we obtain ,
where
Similarly
.
.
.
.
After Summation of above terms we obtain
where n = 2k and k = 1,2,3,4.....
Hence Proved
Ref Lecture p.15-2
Prove that
where
Hence Proved ..
Ref: Lecture Notes p.15-1
Show derivation that
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From (4) in p.14-2, we can write down the expression of
Given :
f(x(t)) = F(t)
Let there exists
but
Hence Proved ,