Average Rate of Change Calculator

The average rate of change calculator can determine the rate of change of a function and simplify the process of finding the change in quantities over time.

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What is the Average Rate of Change Calculator?

The average rate of change calculator is an online tool that helps you find the rate of change of a function. It is used to simplify the process of finding change in quantities over time under certain conditions.

Average Rate of Change Calculator with Steps

Our average rate of change formula calculator is a versatile tool that can easily do complex calculations of change of an object or quantity over a period. It is the best learning source for students, teachers, and professionals, as it provides information about different objects' average rate of change.

What is the Average Rate of Change?

An average rate of change of a function is a process in which the change in a quantity is divided by its input value over a closed interval in calculus. It provides information about the change in given quantity per unit of change over a specific condition that affects on it.

This method is used in many scientific fields to observe the rate of change in the original quantity; in physics, it is used to find the rate of change in velocity. On the other hand in biology, it is used to find the growth and decay rate, etc.

What is the Average Rate of Change Formula?

The average rate of change expresses the slope of a line that passes through the points (a, f(a)) and (b, f(b)) on the graph of the function. It provides a rate of a function f(x) that changes per unit change in x under the given interval as [a, b]. Mathematically, the average rate of change formula is:

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

Where:

f (a) and f (b) are the points a and b of the function, respectively.
a and b are the points over an interval.

How to Find the Average Rate of Change?

We use its formula to calculate average rate of change, which makes our calculations easier and simpler, especially when manually evaluating the given function. Now let us explain the calculation of average rate of change in steps.

First, determine the given values as f(x) and the interval value [x1, x2], in which the first value denotes “a” and the second value is “b.”

Then, add the values of a and b in the given function f(x) to find the values of f(a) and f(b).

After that, find the difference between a and b as b-a.

Put the values of f(a), f(b), and b-a in the average rate formula and simplify it.

After simplification, you get a solution for the average rate of change in the given function.

You can also use our average rate of change over an interval calculator, which offers precise results along with step-by-step instructions in just a few seconds, to calculate average rate of change.

Solved Example of Average Rate of Change:

Let's see an example with the solution of the average rate of change to get better clarity about this concept.

Example:

Calculate the average rate of change of f(x) = x² - 1/x on the interval [2,4].

Solution:

Identify the interval and put its values in the given function f(x) to find the values of f(a) and f(b).

$$ f(2) \;=\; 2^2 - \frac{1}{2} \;=\; 4 - \frac{1}{2} \;=\; \frac{7}{2} $$

$$ f(4) \;=\; 4^2 - \frac{1}{4} \;=\; 16 - \frac{1}{4} \;=\; \frac{63}{4} $$

Apply the average rate formula:

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

Put the values of a, b, f(a), and f(b) in the above formula. Then simplify it to the solution of the average rate of change of the given function.

$$ Average\;rate\;of\;change \;=\; \frac{f(4) - f(2)}{4 - 2} \;=\; \frac{\frac{63}{4} - \frac{7}{2}}{4 - 2} $$

$$ =\; \frac{\frac{49}{4}}{2} \;=\; \frac{49}{8} $$

How to Use the Average Value of a Function Calculator?

The average rate of change calculator has a simple design, so everyone can use it to calculate the function average rate on the given x-value points. Before adding the input value, you must follow some instructions. These instructions are:

Enter the value of the given function f(x) in the input box.
Enter the value of the closed interval in the form of a and b in the next input box.
Review your input value before hitting the calculate button to start the calculation process to find the solution to the function average rate.
Click on the “Calculate” button to get the required result for the average rate of change problem.
If you want to try out our average rate of change over an interval calculator for the first time, then you can use the load example.
Click on the “Recalculate” button to get a new page for solving more average rate of change over an interval [a, b].

Output from Average Rate Change Calculator:

The avg rate of change calculator gives you the solution to a given function problem when you input the value into it. It may be included as:

Result Option:

You can click on the result option, and it will provide you with a solution for the rate of change of function questions.

Possible Step:

The Possible Steps option provides you with a solution in which evaluation steps for an average rate of change over closed interval questions.

Benefits of Using Average Rate of Change Formula Calculator:

The average value of a function calculator has many valuable benefits that help you get the solution of the average rate of change function whenever you use it. These benefits are:

It saves your time and effort from doing lengthy and complex calculations of the average rate of change question in less than a minute.
Our average rate change calculator is a free tool that you can use to find the average rate of change for a question without paying any charge.
The average rate of change calculator is an easy-to-use tool, so you do not need any technical expertise. Just use it to calculate the given function average rate problems easily.
It is a reliable tool that provides you with accurate solutions every time whenever you use it to calculate the average rate for function problems.
The avg rate of change calculator provides you with a solution procedure in a step-by-step method for more clarity.

Related References

What is average rate of change? Explain
Give some examples of average rate of change:
How to evaluate average rate of change?
An overview of average rate of change:

Frequently Ask Questions

How do find the average rate of change between two points?

When you find the average rate of change between two points, it tells how much a function's value changes per unit change in the input value. Here are the calculation steps that you use to find the average rate of change between two points.

Identify the coordinates of the two points which are (x1, y1) and (x2, y2) of an object and a function f(x) that is present at that point.
Apply the formula for the average rate of change between two points (x1, y1) and (x2, y2):

$$ Average\; Rate\; of\; Change \;=\; \frac{y_2 − y_1}{x_2 − x_1} $$

As y = f(x) so y1 = f(x1) and y2 = f(x2) if the given function is in the form of f(x).

Simplify the values by dividing the difference in the y-values by the difference in the x-values to get the solution of the average rate of change.

How to find the average rate of change from a graph?

The average rate of change from any graph is a useful process to understand how a quantity changes over a specific interval, sketch the graph with the help of given values to show a graphical representation. To find the average rate of change from a graph let's see the given steps:

Identify the two points on the graph as (x1, y1) and (x2, y2).
The average rate of change formula is:

$$ Average\; Rate\; of\; Change \;=\; \frac{y_2 − y_1}{x_2 − x_1} $$

Simplify the given values as y = f(x), then take a difference between y = f(x2) - f(x1) for y values and x2 - x1 for x values.
Lastly, divide the difference result of y value by the difference of x-values to get the solution of the average rate of change.

What is the average rate of change of y=2x?

To find the average rate of change of the function y = 2x between two points, follow these steps:

Identify the two points that are present on the line y = 2x. Let's suppose x = a and x = b for x value then y becomes y = 2a and y = 2b as y = f(x) so f(x) = 2x.
The average rate of change formula is:

$$ Average\; Rate\; of\; Change \;=\; \frac{f(b) − f(a)}{b − a} $$

Put the above value in this formula and simplify it to get the solution of average rate of change.

$$ Average\; Rate\; of\; Change \;=\; \frac{2b − 2b}{b − a} $$

$$ Average\; Rate\; of\; Change \;=\; \frac{2(b - a)}{b − a} $$

Because b - a divide b - a the only value 2 is left.

$$ Average\; Rate\; of\; Change \;=\; 2 $$

Is the average rate of change of the slope is same?

The average rate of change and the slope are the same for linear functions, but they have different concepts for non-linear functions. Since the average rate of change shows the slope of the secant line and the instantaneous rate of change shows the slope of the tangent line at a certain point.

How to find average rate of change over an interval?

For the evaluation of the average rate of change of a function over a given interval, the process is given as:

Suppose f(x) = x² from x = 1 to x = 4, then find the average rate of change of the function.

Identify the function f(x) and the Interval [a, b] over which you want to calculate the average rate of change.

$$ f(x) \;=\; x^2,\; x \;=\; 1,\; x \;=\; 4 \;over\; an\; interval\; [1, 4] $$

Then evaluate the value of the function at the starting point a, denoted as f(a) and the ending point is b, as f(b).

$$ for\; f(1): f(1) \;=\; 1^2 \;=\; 1 $$

$$ for\; f(4): f(4) \;=\; 4^2 \;=\; 16 $$

The average rate of change formula is,

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

Lastly, find the difference between f(a) and f(b) and a and b. Then divide the function with the given interval values.

$$ For\; y\; value: f(b) - f(a) \;=\; f(4) − f(1) $$

$$ =\; 16 − 1 \;=\; 15 $$

$$ For\; x\; value: 4 − 1 \;=\; 3 $$

$$ Average\; Rate\; of\; Change \;=\; \frac{15}{3} $$

$$ =\; 5 $$

So, the average rate of change of f(x) = x² from x = 1 to x = 4 is 5.

Amanda Jepson

Last Updated February 21, 2024