Average Rate of Change Calculator
Understanding Average Rate of Change
The average rate of change of a function measures how much the function's output value changes, on average, with respect to its input value over a specific interval. It essentially tells you the slope of the secant line connecting two points on the function's graph.
The Formula
For a function $f(x)$, the average rate of change between two points $x_1$ and $x_2$ is calculated using the formula:
$$ \text{Average Rate of Change} = \frac{f(x_2) – f(x_1)}{x_2 – x_1} $$
In this calculator:
- $x_1$ is the starting x-value of your interval.
- $x_2$ is the ending x-value of your interval.
- $f(x_1)$ is the value of the function when the input is $x_1$.
- $f(x_2)$ is the value of the function when the input is $x_2$.
How to Use the Calculator
To find the average rate of change for a function over a specific interval, follow these steps:
- Identify the two x-values that define your interval ($x_1$ and $x_2$).
- Calculate or determine the corresponding function values for these x-values ($f(x_1)$ and $f(x_2)$).
- Enter these four values into the respective fields in the calculator.
- Click the "Calculate Average Rate of Change" button.
Example Calculation
Let's consider the function $f(x) = x^2$. We want to find the average rate of change between $x_1 = 2$ and $x_2 = 5$.
- $x_1 = 2$
- $x_2 = 5$
- $f(x_1) = f(2) = 2^2 = 4$
- $f(x_2) = f(5) = 5^2 = 25$
Using the formula:
$$ \text{Average Rate of Change} = \frac{25 – 4}{5 – 2} = \frac{21}{3} = 7 $$
So, the average rate of change of the function $f(x) = x^2$ between $x=2$ and $x=5$ is 7. This means that, on average, for every unit increase in $x$ within this interval, the function's output increases by 7 units.