University of Florida/Egm6341/s10.team3.aks

(10) Prove Simple trapezoidal rule

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Ref. Lecture notes [[[media:Egm6341.s10.mtg8.pdf|p.8-2]]]

Problem Statement

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Use (2) in Slide [[[media:Egm6341.s10.mtg8.pdf|8-2]]] to obtain Simple trapezoidal rule.

Solution

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Given

  +  

where

 ,

 ,

 


 

Now   +  

=  

=  

=  

=  

which is same as equation (1) Slide p.7-1

This completes the Proof of trapezoidal rule

(11) Expansion of Lagrange functions

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Ref: Lecture notes p.8-3 [[[media:Egm6341.s10.mtg8.pdf]]|

Problem Statement

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Expand(4) from Slide (8-3 [[[media:Egm6341.s10.mtg8.pdf]]]]) to obtain  


Solution

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where  

but when    

and when i j  

then only surviving terms are given by

 

(3) Plot Functions Sin (x),-cos (x)and Sin(x)+cos(x)

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Ref. Lecture notes p.3-3 [[[media:Egm6341.s10.mtg3.pdf]]|

Problem Statement

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Plot f(x)= sin(x) and g(x) = - cos(x) in the interval of [0,pi]]

and also find  

Solution

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Plot f(x)= sin(x) in interval [0,pi]

Matlab code :

 
x = 0:pi/100:pi;
y = sin(x);
plot(x,y)
xlabel('x = 0:pi');
ylabel('Sine of x');
title('Plot of the Sine Function');

Plot :

f(x)=y= Sin(x)

 

Plot g(x)= - cos(x) in interval [0,pi]

Matlab code :

x = 0:pi/100:pi;
y = -cos(x);
plot(x,y)
xlabel('x = 0:pi');
ylabel('cosine of x');
title('Plot of the cosine Function');

Plot :

g(x)= y = - cos(x)

 

Plot f(x)-g(x)= Sin(x)+cos(x)

Matlab code :

x = 0:pi/100:pi;
y = sin(x)+cos(x);
plot(x,y)
title('Plot of the Sine+Cosine Function');
ylabel('Sine+Cosine of x'); 
xlabel('x = 0:pi');

 


 

 

 

Abhishekksingh 16:25, 27 January 2010 (UTC)