Final answer:
The rate of change for the constant functions f(x) and F(x) is 0, while the rate of change for the exponential function h(x) over the interval [1, 4] is 215. Therefore, h(x) has the greatest rate of change, and both f(x) and F(x) have the least rate of change.
Step-by-step explanation:
The student has been provided with three functions and is asked to find the rate of change for each over the interval [1, 4], to identify which function has the greatest and least rate of change. The functions given are:
- f(x) = 3 - 15, which simplifies to f(x) = -12, a constant function.
- F(x) = 4, another constant function.
- h(x) = 0.5(6)^x, an exponential function.
For the constant functions f(x) and F(x), the rate of change over any interval is 0 because they are horizontal lines.
For the exponential function h(x), the rate of change is not constant, so it must be evaluated:
- Calculate h(1) = 0.5(6)^1 = 3.
- Calculate h(4) = 0.5(6)^4 = 0.5(1296) = 648.
- Find the average rate of change over the interval [1, 4] by subtracting h(1) from h(4) and dividing by the change in x: (648 - 3) / (4 - 1) = 645 / 3 = 215.
The greatest rate of change is for h(x) and the least rate of change is for both f(x) and F(x) since they are the same at 0.