Module 5 : Linear and Quadratic  Approximations, Error  Estimates,  Taylor's Theorem,  Newton and Picard Methods
Lecture 14 : Quadratic approximations and error estimates [Section 14.2]
14.2.5
 Note (Rate of convergence for the error):
 

Since

                    

clearly, . In fact, the error in the quadratic approximation tends to zero at a rate faster than . To see this, note that

                    

and hence

                    
  CLICK HERE TO SEE AN INTERACTIVE VISUALIZATION : Applet 14.1

  PRACTICE EXERCISES
1. 

Let and for . If is twice differentiable at , show that
  is the quadratic approximation of near if and only if

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