average rate of change
The average rate of change refers to the rate at which a quantity changes over a specific interval
The average rate of change refers to the rate at which a quantity changes over a specific interval. In mathematics, it is often used to describe the average rate at which a function’s output changes as its input varies.
To calculate the average rate of change, you need to determine the change in the output (dependent variable) divided by the change in the input (independent variable) over a specific interval.
Let’s take an example to better understand this concept:
Suppose we have a function f(x) = 3x^2 + 2x – 1, and we want to calculate the average rate of change of this function over the interval [1, 5].
1. We start by evaluating the function at the lower endpoint of the interval:
f(1) = 3(1)^2 + 2(1) – 1 = 3 + 2 – 1 = 4
2. Next, we evaluate the function at the upper endpoint of the interval:
f(5) = 3(5)^2 + 2(5) – 1 = 75 + 10 – 1 = 84
3. Now, we can determine the change in the output:
Change in output = f(5) – f(1) = 84 – 4 = 80
4. Similarly, we calculate the change in the input:
Change in input = 5 – 1 = 4
5. Finally, we divide the change in output by the change in input to find the average rate of change:
Average rate of change = Change in output / Change in input = 80 / 4 = 20
Therefore, the average rate of change of the function f(x) = 3x^2 + 2x – 1 over the interval [1, 5] is 20. This means that, on average, the output of the function increases by 20 units for every 1 unit increase in the input within that interval.
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