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
May 31, 2016

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.