Final answer:
To find the average rate of change, we subtract the function values at the endpoints of the interval and divide it by the difference in the input values. Let's calculate the average rate of change of f(x) over the given intervals: [-2, 2], [0, 4], and [-3, 3].
Step-by-step explanation:
The average rate of change of a function can be found by taking the difference in the function values divided by the difference in the input values. In this case, we have the function f(x) = x^3 - 5x + 7. To find the average rate of change over a given interval, we subtract the function values at the endpoints of the interval and divide it by the difference in the input values.
For example, if we want to find the average rate of change of f(x) over the interval [a, b], we use the formula:
Average rate of change = (f(b) - f(a)) / (b - a)
Let's calculate the average rate of change of f(x) over the given intervals:
- Interval 1: [-2, 2]
- Interval 2: [0, 4]
- Interval 3: [-3, 3]
First, let's calculate the function values at the endpoints of each interval:
- For Interval 1: f(-2) = (-2)^3 - 5(-2) + 7 = -1
- For Interval 1: f(2) = (2)^3 - 5(2) + 7 = 9
- For Interval 2: f(0) = (0)^3 - 5(0) + 7 = 7
- For Interval 2: f(4) = (4)^3 - 5(4) + 7 = 53
- For Interval 3: f(-3) = (-3)^3 - 5(-3) + 7 = 55
- For Interval 3: f(3) = (3)^3 - 5(3) + 7 = -1
Now, let's calculate the average rate of change for each interval using the formula:
- For Interval 1: Average rate of change = (9 - (-1)) / (2 - (-2)) = 10 / 4 = 2.5
- For Interval 2: Average rate of change = (53 - 7) / (4 - 0) = 46 / 4 = 11.5
- For Interval 3: Average rate of change = (-1 - 55) / (3 - (-3)) = -56 / 6 = -9.333