How to Calculate Average Rate of Change of a Function – Cost Calculator

Average Rate of Change Calculator

First x-value (x₁):

First y-value (f(x₁)):

Second x-value (x₂):

Second y-value (f(x₂)):

Result:

Understanding the Average Rate of Change of a Function

In mathematics, the average rate of change of a function measures how much the function's output (y-value) changes with respect to a change in its input (x-value) over a specific interval. It essentially describes the "steepness" of the line segment connecting two points on the function's graph. This concept is fundamental to understanding calculus, as it forms the basis for the instantaneous rate of change (the derivative).

The Formula

The average rate of change of a function $f(x)$ over the interval $[x_1, x_2]$ is calculated using the following formula:

$$ \text{Average Rate of Change} = \frac{f(x_2) – f(x_1)}{x_2 – x_1} $$

Alternatively, if you are given the coordinates of two points on the function, $(x_1, y_1)$ and $(x_2, y_2)$, the formula simplifies to the familiar slope formula for a line:

$$ \text{Average Rate of Change} = \frac{y_2 – y_1}{x_2 – x_1} $$

How to Calculate It

To calculate the average rate of change, you need two points on the function, each defined by an x-value and its corresponding y-value (function output). Follow these steps:

Identify the two x-values, $x_1$ and $x_2$.
Determine the corresponding y-values (function outputs) for these x-values, $f(x_1)$ and $f(x_2)$. If you are given coordinates $(x_1, y_1)$ and $(x_2, y_2)$, these are $y_1$ and $y_2$ respectively.
Subtract the first y-value from the second y-value: $y_2 – y_1$ (or $f(x_2) – f(x_1)$). This is the "change in y".
Subtract the first x-value from the second x-value: $x_2 – x_1$. This is the "change in x".
Divide the "change in y" by the "change in x".

Important Note: The denominator, $x_2 – x_1$, must not be zero. This means the two x-values must be distinct.

Example

Let's calculate the average rate of change for the function $f(x) = x^2 + 1$ over the interval from $x_1 = 2$ to $x_2 = 5$.

1. Identify the x-values: $x_1 = 2$ and $x_2 = 5$.

2. Find the corresponding y-values:

For $x_1 = 2$, $f(2) = (2)^2 + 1 = 4 + 1 = 5$. So, $y_1 = 5$.
For $x_2 = 5$, $f(5) = (5)^2 + 1 = 25 + 1 = 26$. So, $y_2 = 26$.

3. Calculate the change in y: $y_2 – y_1 = 26 – 5 = 21$.

4. Calculate the change in x: $x_2 – x_1 = 5 – 2 = 3$.

5. Divide the change in y by the change in x:

$$ \text{Average Rate of Change} = \frac{21}{3} = 7 $$

Therefore, the average rate of change of the function $f(x) = x^2 + 1$ between $x=2$ and $x=5$ is 7. This means that, on average, for every unit increase in x, the function's output increases by 7 units over this interval.