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]

Like for linear approximations, it is natural to ask the question:

 How well does approximate for near ?

An answer to this question is the following:

14.2.3
 Corollary:
 

Let with and be the closed interval with end points and .Let be such that

(i) The functions are all continuous.

(ii) For every between and ,
                       exists and ,
then satisfies the following:
                      
14.2.4
 Example:
 

Consider the function
                      
Then
                     
We saw in example 14.2.1, that the quadratic approximation for near the point is given by
                     .
Let us estimate the error
                    

1
12