Compare the estimated average rate of change for a cubic function to the estimated...
Check the final answer first, then review the worked steps.
Problem
Compare the estimated average rate of change for a cubic function to the estimated average rate of change for a square root function over a given interval.
Step-by-step solution
Calculate the average rate of change for b(x) = \sqrt[3]{3x+9} over [-12, -3] as (b(-3)-b(-12))/(-3 - (-12)) = (0 - (-3))/9 = 3/9 = 1/3. Calculate the average rate of change for d(x) = \sqrt{-3x+9} over [-12, -3] as (d(-3)-d(-12))/(-3 - (-12)) = (\sqrt{18} - \sqrt{45})/9 = (3\sqrt{2} - 3\sqrt{5})/9 = (\sqrt{2} - \sqrt{5})/3 \approx (1.414 - 2.236)/3 \approx -0.822/3 \approx -0.274. The graph of b(x) is increasing, and the graph of d(x) is decreasing.
Answer
The estimated average rate of change of d(x) is greater than the estimated average rate of change of b(x) because b(x) is increasing over the interval but d(x) is decreasing.