Final answer:
The average rate of change for the function f(x)=0.1x² within the interval 1 ≤ x ≤ 4 is 0.5, while for g(x)=0.3x², it is 1.5. This means that the average rate of change for g(x) is three times larger than that for f(x), reflective of the corresponding coefficients of x² in each function.
Step-by-step explanation:
To compare the average rates of change for the functions f(x)=0.1x² and g(x)=0.3x² within the interval 1 ≤ x ≤ 4, we calculate the change in y over the change in x (or Δy/Δx) for each function over that interval. For f(x), we evaluate the function at the endpoints of the interval: f(4) = 0.1·(4²) = 0.1·(16) = 1.6 and f(1) = 0.1·(1²) = 0.1·(1) = 0.1. So, the average rate of change for f(x) is Δy/Δx = (f(4) - f(1)) / (4 - 1) = (1.6 - 0.1) / (4 - 1) = 1.5 / 3 = 0.5.
For g(x), similarly, we find g(4) = 0.3·(4²) = 0.3·(16) = 4.8 and g(1) = 0.3·(1²) = 0.3·(1) = 0.3. Thus, the average rate of change for g(x) is Δy/Δx = (g(4) - g(1)) / (4 - 1) = (4.8 - 0.3) / (4 - 1) = 4.5 / 3 = 1.5.
Comparing the average rates of change, g(x) has an average rate that is 3 times larger than the average rate of change for f(x) since 1.5 / 0.5 = 3. This result is expected because the coefficient in front of x² in g(x) is three times the coefficient in f(x), indicating that the rate of increase for g(x) is faster relative to f(x) in the given interval.