How do you find the average rate of change for the function #f(x) = x^2 - 2x # on the indicated intervals [1,3]?

Question

Ezra Adams · Accepted Answer

The average rate of change of function #f# on interval #[a,b]# is

#(f(b) - f(a))/(b-a)# In light of this, we have #f(3) = 9-6 =3# and #f(1) = 1-2=-1#. Moreover, the typical rate of change is #(f(3) - f(1))/(3-1) = (3-(-1))/(3-1) = 4/2 = 2# The average rate of change of function #f# on interval #[a,b]# is #(f(b) - f(a))/(b-a)# It is the ratio of the changes, it may also be written #(Deltaf)/(Deltax)# and it may be thought of as the slope of the line through the endpoints of the graph of #f# on the interval. Algebraically it is one version of the difference quotient. (The quotient of the differences in #f# values and #x# values..)

Isabella Ackerman · Answer

undefined To find the average rate of change of the function $ f(x) = x^2 - 2x $ on the interval $[1,3]$, you need to calculate the difference in the function values at the endpoints of the interval and then divide by the difference in the input values.

$ f(1) = (1)^2 - 2(1) = 1 - 2 = -1 $

$ f(3) = (3)^2 - 2(3) = 9 - 6 = 3 $

The average rate of change is:

$$
\frac{{f(3) - f(1)}}{{3 - 1}} = \frac{{3 - (-1)}}{{3 - 1}} = \frac{4}{2} = 2
$$

So, the average rate of change of $ f(x) = x^2 - 2x $ on the interval $[1,3]$ is $ 2 $.