Example 1.  Consider the function  [Graphics:Images/NumericalDiffMod_gr_27.gif].   Find the formula for the third derivative [Graphics:Images/NumericalDiffMod_gr_28.gif], it will be used in our explorations for the remainder term and the truncation error bound.  Graph  [Graphics:Images/NumericalDiffMod_gr_29.gif].  Find the bound  [Graphics:Images/NumericalDiffMod_gr_30.gif].  Look at it's graph and estimate the value  [Graphics:Images/NumericalDiffMod_gr_31.gif], be sure to take the absolute value if necessary.

Solution 1.

[Graphics:../Images/NumericalDiffMod_gr_32.gif]


[Graphics:../Images/NumericalDiffMod_gr_33.gif]


[Graphics:../Images/NumericalDiffMod_gr_34.gif]


[Graphics:../Images/NumericalDiffMod_gr_35.gif]

[Graphics:../Images/NumericalDiffMod_gr_36.gif]
[Graphics:../Images/NumericalDiffMod_gr_37.gif]
[Graphics:../Images/NumericalDiffMod_gr_38.gif]
[Graphics:../Images/NumericalDiffMod_gr_39.gif]
[Graphics:../Images/NumericalDiffMod_gr_40.gif]
[Graphics:../Images/NumericalDiffMod_gr_41.gif]
[Graphics:../Images/NumericalDiffMod_gr_42.gif]
[Graphics:../Images/NumericalDiffMod_gr_43.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004