find the average rate of change

Question

Find the average rate of change of the function over the given interval.

f(t) = 6t² − 3,      [1, 1.1]

Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.

left endpoint
right endpoint

Answer 1

Hi Artavius,

When you find the average rate, you do not care how the function progressed from time t₁ to later time t₂. You will treat it as if it progressed linearly and write

Avg Rate = [f(t₂) - f(t₁)] / [t₂ - t₁]

f(t) = 6t² - 3

t₁ = 1

t₂ = 1.1

f(t₂) = f(1.1) = 6(1.1)² - 3 = 4.26

f(t₁) = f(1) = 6(1)² - 3 = 3

Avg Rate = [4.26 - 3]/[1.1 - 1] = 1.26/0.1 = 12.6

Since f(t) is quadratic and not linear, it likely that the instantaneous rate is ever 12.6.

instantaneous rate = (d/dt)[6t² - 3]

instantaneous rate = 12t

instantaneous rate at t = 1 = 12(1) = 12

instantaneous rate at t = 1.1 = 12(1.1) = 13.2