Average Rate of Change of f(x) on [a,b]
The average rate of change of a function f(x) on the interval [a, b] is a measure of how the function is changing on that interval
The average rate of change of a function f(x) on the interval [a, b] is a measure of how the function is changing on that interval. It provides the average rate at which the function values are changing as the independent variable (x) varies from a to b.
To calculate the average rate of change, you need to find the difference in the function values divided by the difference in the corresponding x-values. Mathematically, it can be represented as:
Average Rate of Change = (f(b) – f(a))/(b – a)
Here, f(b) represents the value of the function at the upper bound of the interval (b), and f(a) represents the value of the function at the lower bound of the interval (a).
For example, let’s say we have the function f(x) = 2x^2 and we want to find the average rate of change on the interval [1, 3].
First, we calculate f(3) and f(1):
f(3) = 2(3)^2 = 2(9) = 18
f(1) = 2(1)^2 = 2(1) = 2
Next, we find the difference in function values and the difference in x-values:
f(b) – f(a) = 18 – 2 = 16
b – a = 3 – 1 = 2
Finally, we divide the difference in function values by the difference in x-values to get the average rate of change:
Average Rate of Change = 16/2 = 8
Therefore, the average rate of change of f(x) = 2x^2 on the interval [1, 3] is 8. This means that on average, the function values are changing by 8 units as x varies from 1 to 3.
Note: The average rate of change provides a summary measure of the function’s behavior on the interval. It gives you an idea of the overall trend rather than the specific instantaneous rate of change at any point.
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