Question

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.

Answer 1

Any linear function `f(x) = Ax+B` satisfies this condition.
Any function that satisfies this condition is linear.

Explanation:

Obviously, a linear function `f(x) = Ax+B` satisfies this condition. Its average rate of change on any interval `[x_1, x_2]` is equal to
`R = [f(x_2)-f(x_1)]/(x_2-x_1) = (Ax_2+B-Ax_1-B)/(x_2-x_1)=A`

Since this is true for any interval, instantaneous rate of change at any point `x_1` is also equal to `A` since it is a limit of average rate of change when the right end of an interval `x_2` gets infinitely close to its left end `x_1`.

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: `x_1`, `x_2` and take any other point `x`.
Since the average rate of change is the same on any interval,
`R = [f(x_2)-f(x_1)]/(x_2-x_1) = [f(x)-f(x_1)]/(x-x_1)`

From the above we can conclude:
`R(x-x_1) = f(x)-f(x_1)`
`f(x) = Rx-Rx_1+f(x_1)`
As we see, `f(x)` is a linear function, which is exactly what we wanted to prove.