Example 1.  Consider the function  [Graphics:Images/RichardsonExtrapMod_gr_34.gif].   Find the formula for the third derivative [Graphics:Images/RichardsonExtrapMod_gr_35.gif], it will be used in our explorations for the remainder term and the truncation error bound.  Graph  [Graphics:Images/RichardsonExtrapMod_gr_36.gif].  Find the bound  [Graphics:Images/RichardsonExtrapMod_gr_37.gif].  Look at it's graph and estimate the value  [Graphics:Images/RichardsonExtrapMod_gr_38.gif], be sure to take the absolute value if necessary.

Solution 1.

[Graphics:../Images/RichardsonExtrapMod_gr_39.gif]



[Graphics:../Images/RichardsonExtrapMod_gr_40.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_41.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_42.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_43.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_44.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_45.gif]



[Graphics:../Images/RichardsonExtrapMod_gr_46.gif]


[Graphics:../Images/RichardsonExtrapMod_gr_47.gif]

[Graphics:../Images/RichardsonExtrapMod_gr_48.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_49.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_50.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_51.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_52.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_53.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_54.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_55.gif]
[Graphics:../Images/RichardsonExtrapMod_gr_56.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004