Example 3.  Consider  [Graphics:Images/BigOMod_gr_96.gif]  and the Taylor polynomials of degree  [Graphics:Images/BigOMod_gr_97.gif]  expanded about  [Graphics:Images/BigOMod_gr_98.gif].  

    [Graphics:Images/BigOMod_gr_99.gif]    [Graphics:Images/BigOMod_gr_100.gif]

    [Graphics:Images/BigOMod_gr_101.gif]    [Graphics:Images/BigOMod_gr_102.gif]  

    [Graphics:Images/BigOMod_gr_103.gif]    [Graphics:Images/BigOMod_gr_104.gif]  

    [Graphics:Images/BigOMod_gr_105.gif]    [Graphics:Images/BigOMod_gr_106.gif]  

Solution 3.

(i)  Show that  [Graphics:../Images/BigOMod_gr_107.gif] over  [Graphics:../Images/BigOMod_gr_108.gif].  

[Graphics:../Images/BigOMod_gr_109.gif]


[Graphics:../Images/BigOMod_gr_110.gif]

[Graphics:../Images/BigOMod_gr_111.gif]


(ii)  Show that  [Graphics:../Images/BigOMod_gr_112.gif]   over  [Graphics:../Images/BigOMod_gr_113.gif].  

[Graphics:../Images/BigOMod_gr_114.gif]


[Graphics:../Images/BigOMod_gr_115.gif]

[Graphics:../Images/BigOMod_gr_116.gif]


(iii)  Show that   [Graphics:../Images/BigOMod_gr_117.gif]   over  [Graphics:../Images/BigOMod_gr_118.gif].  

[Graphics:../Images/BigOMod_gr_119.gif]


[Graphics:../Images/BigOMod_gr_120.gif]

[Graphics:../Images/BigOMod_gr_121.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004