Example 3.  Use the function  [Graphics:Images/TangentParabolaMod_gr_207.gif]  and the point  [Graphics:Images/TangentParabolaMod_gr_208.gif]  and draw the graphs of Newton cubic polynomials with [Graphics:Images/TangentParabolaMod_gr_209.gif] and compare them with the Taylor polynomial.  

Solution 3.

[Graphics:../Images/TangentParabolaMod_gr_210.gif]


[Graphics:../Images/TangentParabolaMod_gr_211.gif]

 

 

We can graph the secant polynomials corresponding to [Graphics:../Images/TangentParabolaMod_gr_212.gif] and the Taylor polynomial.

[Graphics:../Images/TangentParabolaMod_gr_213.gif]


[Graphics:../Images/TangentParabolaMod_gr_214.gif]

[Graphics:../Images/TangentParabolaMod_gr_215.gif]



[Graphics:../Images/TangentParabolaMod_gr_216.gif]


[Graphics:../Images/TangentParabolaMod_gr_217.gif]

[Graphics:../Images/TangentParabolaMod_gr_218.gif]

The next figure shows a comparison of the Taylor polynomial T3(x) and f(x):

[Graphics:../Images/TangentParabolaMod_gr_219.gif]


[Graphics:../Images/TangentParabolaMod_gr_220.gif]

[Graphics:../Images/TangentParabolaMod_gr_221.gif]



[Graphics:../Images/TangentParabolaMod_gr_222.gif]


[Graphics:../Images/TangentParabolaMod_gr_223.gif]

[Graphics:../Images/TangentParabolaMod_gr_224.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004