Find the average rate of change of the function on the interval specified for real number h, j(x) = 7x^3 on [1, 1 + h]

Answer 1

The rate of change is the slope and slope is defined as (y₂ - y₁)/(x₂ - x₁). Since the interval [1, 1+h] is the domain or the x-values we are considering, let's find the corresponding y values (aka, j(x) values).

j(1) = 7(1)³ = 7(1) = 7 so the point (x₁, y₁) is (1, 7)

j(1 + h) = 7(1 + h)³ = 7h³ + 21h² + 21h + 7 so the point (x2, y2) is (1 + h, 7h³ + 21h² + 21h + 7)

So (y₂ - y₁)/(x₂ - x₁) = [(7h³ + 21h² + 21h + 7) - 7]/(1 + h - 1) = (7h³ + 21h² + 21h)/h = 7h² + 21h + 21