Joseph D. answered • 09/09/19

Precalculus Tutor with the help you need.

Hi Brenden w.,

The average rate of change is: Δh/Δx = (h_{f} - h_{i})/(x_{f} - x_{i}).

For h(x) = 3x + 4 on [2, 2+h], (where x_{i} equals 2 and x_{f} equals 2+h):

h_{i}(2) = 3(2) + 4 = 10

h_{f}(2+h) = 3(2+h) + 4 = 3h + 10

For Δh/Δx = (h_{f} - h_{i})/(x_{f} - x_{i}):

(3h + 10 - 10)/(2 + h - 2) = 3h/h = **3** (the average rate of change).

The average rate of change is: Δj/Δx = (j_{f} - j_{i})/(x_{f} - x_{i}):

For j(x) = 3x^{3} on [1, 1+h]:

j_{i}(1) = 3(1)^{3} = 3

j_{f}(1+h) = 3(1+h)^{3} = 3h^{3} +9h^{2} + 9h + 3

For Δj/Δx = (j_{f} - j_{i})/(x_{f} - x_{i}):

(3h^{3} +9h^{2} + 9h + 3 -3)/(1+h-1) = (3h^{3} +9h^{2} + 9h)/h = **3h**^{2}** + 9h +9 **(the average rate of change).

I hope this helps, Joe.