Example 4.  Show that the degree of precision of the Trapezoidal Rule is  [Graphics:Images/NewtonCotesMod_gr_187.gif].  

Solution 4.

It will suffice to apply Trapezoidal Rule over the interval  [0,1]  with the three test functions  [Graphics:../Images/NewtonCotesMod_gr_188.gif].

[Graphics:../Images/NewtonCotesMod_gr_189.gif]
[Graphics:../Images/NewtonCotesMod_gr_190.gif]

The function  [Graphics:../Images/NewtonCotesMod_gr_191.gif]  is the lowest power of  x  for which the trapezoidal rule is not exact.

 

[Graphics:../Images/NewtonCotesMod_gr_192.gif]


[Graphics:../Images/NewtonCotesMod_gr_193.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004