Average Rate of Change Calculator

Average Rate of Change Calculator

Enter the function f(x), X₁ and X₂ long with values in the Average Rate of Change Calculator to determine f(x₁), f(x₂), f(x₁)-(x₂), (x₁-x₂), and the rate of change.

Use this online average rate of change calculator to determine the average rate at which a function changes over a given interval. It measures how one quantity changes on average with respect to another. Let’s explore how to calculate the average rate of change using a formula.

What is the Average Rate of Change?

The average rate of change measures how one quantity changes as another quantity changes. In other words, it is the total change in the output of a function divided by the total change in the input over a specific interval.

Average Rate of Change Formula:

The standard formula is:

$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

Where:

(a, f(a)) are the coordinates of the first point
(b, f(b)) are the coordinates of the second point

How to Find the Average Rate of Change of a Function

To calculate the average rate of change, follow these steps:

Identify the interval $[a, b]$ over which you want to calculate the rate.
Evaluate the function at both endpoints: $f(a)$ and $f(b)$.
Apply the formula: $\frac{f(b) - f(a)}{b - a}$.

Example:

Find the average rate of change of the function $f(y) = 3y^2 + 5$ over the interval $[-1, 3]$.

Solution:

Step 1 – Identify the endpoints:

a = -1, b = 3

Step 2 – Evaluate the function at both endpoints:

$$ f(a) = f(-1) = 3(-1)^2 + 5 = 3 + 5 = 8 $$

$$ f(b) = f(3) = 3(3)^2 + 5 = 27 + 5 = 32 $$

Step 3 – Substitute into the average rate of change formula:

$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} = \frac{32 - 8}{3 - (-1)} = \frac{24}{4} = 6 $$

How the Average Rate of Change Calculator Works

Input:

Enter the function for which you want to calculate the average rate.
Specify the interval values $a$ and $b$.
Click the “Calculate” button.

Output:

The calculator displays the function and interval.
It provides a step-by-step solution for clarity.
You can recalculate for different intervals as needed.

Reference:

From Wikipedia: Rate of Change, Slope Formula, Slope from a Graph, Horizontal and Vertical Lines.

From Brilliant: Average and Instantaneous Rate of Change.