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University of Florida/Egm6341/s10.team3.aks/HW2

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(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

Problem Statement

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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

Solution

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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

(14) Prove

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Ref Lecture p.15-2

Problem Statement

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Prove that

where


Solution

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Hence Proved ..

(12) Show Derivation

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Ref: Lecture Notes p.15-1

Problem Statement

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Show derivation that

Solution

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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 ,