# 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 *average rate of change* on any interval

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, **linear** function, which is exactly what we wanted to prove.