We call this theTaylorPolynomialof fcenteredat a.
For our discussion,we will focuson the quadraticcase.The TaylorPolynomialcorrespondingto is
givenby
Notethat this is just our linearapproximationwith an addedterm.Hencewe can view it as an approximation
of for valuescloseto
Example3:
Find the quadraticapproximationof the function at and comparethemto the linear
approximationsfrom the first example.
Solution:
Recallthat
Hence.
; so
Hence
So. If we updateour tablefromthe first examplewe can see how the
quadraticapproximationcompareswith the linearapproximation.
Actual
5.951.9875 1.9874 1.9874
5.991.9975 1.9974 1.9974
6 2 2 2
6.012.0025 2.0024 2.0024