a student calculates a functions average rate of change on an interval and finds that it is 0.

Question

a student calculates a functions average rate of change on an interval and finds that it is 0. the student concludes that the function is constant on the interval. explain the students error, and give an example to support your explanation.

Answer 1

One possibility is that the function is non-linear, e.g. f(x) = x².

If the two points selected were (-2, 4) and (2, 4) the average rate of change would be (4 - 4) / (-2 -2) or 0.

Answer 2

To compute the average rate of change of a function over an interval, only the first and last points are considered.  This calculation gives no information about the behavior of the function inside the interval.

It is true that a function that is a constant [f(x)=C] has a rate of change of 0.

However, any function that shared the values at the first and last points, but changed otherwise, need not be constant.

The function y= |x| shares the points (-1,1) and (1,1) with the constant function y=1.  Over that interval, they both have a 0 average rate of change.  however, both inside and outside that interval, the lines are very different.