Order of accuracy of central divided difference scheme for first derivative of a function of one variable

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A problem using central divided difference error order

This is a problem I asked in the first examination of my Numerical Methods course in Spring 2009.   The question is that if one gives you an approximate value of the derivative of a function at a certain point using the central divided difference formula for two different step sizes, would you be able to find a better estimate of the derivative?

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com, the textbook on Numerical Methods with Applications available from the lulu storefront, and the YouTube video lectures available at http://www.youtube.com/numericalmethodsguy.

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