What type of non-constant function has the same average rate of change and instantaneous rate of change over all intervals and for all values of "x?" Explain.
Any linear function
Any function that satisfies this condition is linear.
Obviously, a linear function
Since this is true for any interval, instantaneous rate of change at any point
A little more interesting is to prove that this class of linear functions is the only set of functions defined for all real numbers having a property of the same rate of change on any interval as well as an instantaneous rate of change.
Here is a proof.
Let's fix two points on the X-axis:
Since the average rate of change is the same on any interval,
From the above we can conclude:
As we see,