Problem of the Week

PROBLEM 211

On a recent homework assignment, a calculus student was asked to compute the average rate of change of a function f  between two points, a and b, which (as all calculus students should know) is defined by the expression.  Unfortunately, the student computed the average of the rates of change, which is defined by the expression .  

For which functions are these two expressions equal for every value of a and b?  (You may assume, if you wish, that the function f is infinitely differentiable.)

For the hard-to-please:  Same problem, but only assume that the function is once differentiable.  

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ã2005 Alberto L. Delgado