# Average Rate of Change

Suppose you are planning to go on a road trip and want to estimate how much fuel your car will need so that it will not stop midway. Driving it everyday, you get a rough estimation over several days that the average distance your car covers in 1 litre of fuel is 15 kms. But how is this helpful? You can stretch this average rate to estimate if your car can go through the road trip in a whole fuel tank. What we have found out is the Average Rate of Change of distance relative to the fuel consumed. This is a fairly simple scenario, but it also has applications elsewhere, like calculating the average rate of change of electric current going through a circuit in unit time to work out the power and resistance in the circuit.

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We shall discuss the mathematical side of the above scenarios which can then be applied to a plethora of real-world problems.

## Average Rate of Change definition and formula

The word ’average’ suggests that the rate of change we are looking for will be over a considerable interval. The reason why the average rate of change is useful is that it can be extended over a large interval to get fairly accurate results.

The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.

In a mathematical sense, the quantities are replaced by ‘functions’ but the narrative remains the same.

Let $y=f\left(x\right)$ be a function that is defined over the closed interval $\left[a,b\right]$ where a and b are real numbers. We want to know the rate of change of the function over that interval, the more rigorous definition of Average Rate of Change goes as follows:

The Average Rate of Change of a function $y=f\left(x\right)$ over an interval $\left[a,b\right]$ is given as the ratio of change in the function over the interval to the change in the value of the endpoints.

Translating the definition to an equation, the former part of the definition is ’the change in the function over the interval’ which becomes $f\left(b\right)-f\left(a\right)$ and the latter part is ‘the change in the value of the endpoints’ which translates to $b-a$. Now the average rate of change of a function is the ratio of these, which gives us:

$\frac{\Delta y}{\Delta x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Which is the equation for finding the average rate of change of y w.r.t (abbreviation of ‘with respect to’) x over the interval $\left[a,b\right]$. We can also find a similar expression for the average rate of change of x w.r.t. y over the same interval, which will be just the reciprocal of what we got earlier:

$\frac{\Delta x}{\Delta y}=\frac{b-a}{f\left(b\right)-f\left(a\right)}$

But how can this average rate of change be interpreted in a more visual or graphical sense? Let’s take a look.

## Average Rate of Change on a Graph

Let’s take the same familiar function, $y=f\left(x\right)$ over the interval $\left[a,b\right]$. Now let us graph the endpoints to get a better grasp of what’s happening.

The Average rate of change over two points, StudySmarter Originals

Let the two endpoints be A and B which have coordinates $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ respectively. The average rate of change between the two points will be the slope of the line passing through A and B. The reason is that the ratio of the change in y to the change in x is nothing but the gradient of that line.

In other words, the average rate of change over two points is the slope of the secant line through them.

Note that the average rate of change of a function may differ over different intervals since the slope will be altered respectively. For a linear function, the average rate of change will always be the same for any interval, which can be justified graphically since the slope will always be the same line as the function itself.

## Average Rate of Change of a Function as Gradient

The average rate of change between two points can also be equivalently understood as the gradient between the two points.

The gradient over two points as a part of a right triangle, StudySmarter Originals

If we draw the change in y coordinates and the change in x coordinates, we get a right-angled triangle as shown above. From the laws of trigonometry, the gradient can be found as:

$\frac{\Delta y}{\Delta x}=m=slope$

Which is just a slightly different way of looking at the average rate of change between two points.

If the two points A and B are close enough, the average rate of change is more accurate over a bigger interval. The closer the points, the more precise the rate of change is. If the points get close enough, the average rate of change becomes an instantaneous rate of change (discussed in the accompanying article. As $\Delta y\to 0$ and $\Delta x\to 0$, we get: $\frac{\Delta y}{\Delta x}\approx \frac{dy}{dx}$).

## Examples of Average Rate of Change

Find the average rate of change of the function $f\left(x\right)=2{x}^{2}-1$ as x varies from 1 to 3.

Solution:

Step 1: Calculate the value of the function at the endpoints, 1 and 3:

$f\left(1\right)=2{\left(1\right)}^{2}-3=-1$

$f\left(3\right)=2{\left(3\right)}^{2}-3=15$

Step 2: Find the change in x: $3-1=2$.

Step 3: Take the ratio of the change in function to the change in x:

$\frac{\Delta y}{\Delta x}=\frac{f\left(3\right)-f\left(1\right)}{3-1}$

$\frac{\Delta y}{\Delta x}=\frac{15+1}{2}=8$

Hence the average rate of change of the function w.r.t. x between the points 3 and 1 is 8.

Find the average rate of change of the function$y={e}^{2t}-\frac{{t}^{3}}{2}$ between the values of t:$0\mathrm{and}\frac{3}{2}$.

Solution:

Step 1: Calculate the value of the function at the points given:

$f\left(0\right)={e}^{2\left(0\right)}-\frac{{\left(0\right)}^{3}}{2}=1-0=1$

$f\left(\frac{3}{2}\right)={e}^{2.\frac{3}{2}}-\frac{1}{2}{\left(\frac{3}{2}\right)}^{3}={e}^{3}-\frac{27}{16}$

Step 2: Calculate the difference in the t coordinates: $\frac{3}{2}-0=\frac{3}{2}$.

Step3: Take the ratio the the change in the function and the x coordiantes:

$\frac{\Delta y}{\Delta t}=\frac{f\left(\frac{3}{2}\right)-f\left(0\right)}{\frac{3}{2}-0}$

$\frac{\Delta y}{\Delta x}=\frac{{e}^{3}-\frac{27}{16}-1}{\frac{3}{2}}$$=\frac{2{e}^{3}}{3}-\frac{27}{24}-\frac{2}{3}$$=\frac{16{e}^{3}-27-16}{24}$

$\frac{\Delta y}{\Delta t}=\frac{16{e}^{3}-43}{24}$

Hence the average rate of change of the function is $\frac{16{e}^{3}-43}{24}$ between the points $0\mathrm{and}\frac{3}{2}$.

A bus covers a distance of 60 km from its first stop to its last stop. It takes the bus about 2 hours for the whole journey. The bus driver says that he had several stops in between and drove at different speeds at different times. The bus driver has to report the average speed he was driving at so that the authorities can be assured that he was driving under the speed limit on average. The speed limit is 35 km/h. What is his average speed under it?

Solution:

Step 1: Convert the given data into mathematical information. Let d be the distance covered by the bus and t be the time it takes to cover the distance.

Step 2: Find the change in net distance covered: $\Delta d=60-0=60$ km. And the time taken to cover that distance: $\Delta t=2hrs$.

Step 3: The average speed of an object is given by the ratio of the net distance covered by the bus to the time taken to cover the distance, hence:

${V}_{av}=\frac{\Delta d}{\Delta t}$$=\frac{60}{2}\frac{km}{hr}$

${V}_{av}=30km/hr$

Hence the average speed at which the driver was driving was 30 km/hr, which is below the average speed limit of 35 km/hr.

## Average Rate of Change - Key takeaways

• The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.
• In a mathematical sense, the Average Rate of Change of a function$y=f\left(x\right)$ over a interval $\left[a,b\right]$ is given as the ratio of change in the function over the interval to the change in the value of the endpoints.
• The rate of change of a function between two points is equal to the gradient of the line formed by joining the two points.
• The formula for the Average rate of change of a function between two points is given by $\frac{\Delta y}{\Delta x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

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What is the meaning of average rate of change?

The Average Rate of Change of a quantity relative to another is the measure of how much the quantity changes in a given interval per unit change of the other quantity.

How do you find average rate of change?

The Average Rate of Change of a function y =  f(x) over an interval [a, b] is given as the ratio of change in the function over the interval to the change in the value of the endpoints.

What is an example of average rate of change?

An example of the average rate of change is the average distance covered in a certain time interval by a car.

What is the formula for calculating the average rate of change?

The formula to find the average rate of change is to take the ratio of the change of the two variables over the interval.

How to graph the average rate of change?

To graph the graph of average rate of change between two points, plot the two endpoints and draw a line joining them. The slope of the line will be the average rate of change.

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