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.
1 Answer
Any linear function
Any function that satisfies this condition is linear.
Explanation:
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,