Final answer:
The average rate of change of f(x) = 0.1x² over the interval 1 to 4 is 0.5, while for g(x) = 0.3x² over the same interval is 1.5, which is three times the average rate of change of f(x).
Step-by-step explanation:
To compare the average rates of change for the functions f(x) = 0.1x² and g(x) = 0.3x² over the interval 1≤x≤4, we will calculate the average rate of change for each function over this interval.
For f(x) = 0.1x²:
Average rate of change = ∂f(x)/∂x = [f(4) - f(1)] / [4 - 1]
= [(0.1*4²) - (0.1*1²)] / [4 - 1]
= (1.6 - 0.1) / 3
= 1.5 / 3
= 0.5
For g(x) = 0.3x²:
Average rate of change = ∂g(x)/∂x = [g(4) - g(1)] / [4 - 1]
= [(0.3*4²) - (0.3*1²)] / [4 - 1]
= (4.8 - 0.3) / 3
= 4.5 / 3
= 1.5
The average rate of change of g(x) over the interval 1 to 4 is three times that of f(x), as the coefficients of the quadratic terms differ by a factor of three.