Average Rate of Change Formula

The average rate of change formula or the aroc formula is the rate of change of one quantity with respect to the changing quantity of something else.

For finding the average rate of change, we have the following rate of change formula algebra for function f from p to q:

In this average rate of change of a function formula, f(p) and f(q) are the values of the function at p and q, respectively. Also, p and q are the limits of the function ‘f.’

On this page, we will understand the average rate of change formula calculus, the way of finding the average rate of change, the average rate of change over an interval formula, and the instantaneous rate of change equation.

Finding the Average Rate of Change 

To understand the average rate of change formula calculus, let’s have a look at the following example:

From the above data, we see that the price change per year is the rate of change because it expresses how an output quantity varies in relation to the change in the input quantity. 

Also, in the above table, we observe that the price change of gasoline is not constant or it changes by the same amount each year, i.e the rate of change is variable. 

Here, if we use only the initial and the final data to determine the rate of change, we will be applying the rate of change formula algebra over the specified period of time. So, to find the average rate of change, we divide the change in the output quantity by the change in the input quantity as;

Average rate of change formula = \[\frac{\text{Change in input}}{\text{Change in output}}\] = \[\frac{\Delta y}{\Delta x}\] ……(2)

We call equation (2) the average rate of change over an interval formula because the change per year is not constant, and therefore, we are finding the average rate of change for a specified period of time.

Rewriting equation (2): 

\[\frac{\Delta y}{\Delta x}\] = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\] = \[\frac{\text{Calculate the difference in y}}{\text{Calculate the difference in x}}\]

Since y is the function of x, now writing the average rate of change formula algebra:

AROC Formula  = \[\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\] ……(3)

This equation is also called the average rate of change of a function formula.

The Greek letter Δ (delta) signifies the change in a quantity; we read the ratio as Δy: Δx or “the change in y divided by the change in x.” 

Occasionally, we write Δf instead of Δy, which represents the change in the value of the function’s output as a result of a change to its input value. However, it does not mean we are changing the function into some other function.​

​​In the table drawn above, the price change of gasoline from the year 1991 to 1998 is calculated as: 4.68  -  3.32, i.e., $1.36.

So, the change in output is the price change = $1.36 and the change in input (time) = 7 years (from 1991 to 1998).

Now, to find average rate of change from equation (2):

= \[\frac{\text{Change in input}}{\text{Change in output}}\] = \[\frac{1.36}{7}\] ≈ 0.194 dollars 

Therefore, the average increase in gasoline price is $0.194 every year. 

Average Rate of Change Over an Interval Formula

Now, to find the average rate of change over an interval, we have the following data:

From this table, we will compute the average rate of change formula graph. Let’s suppose that g(t) is the function and plotting the graph on interval [- 1, 2]:

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In the below graph, at t = - 1, g(t) is g(- 1), which is equal to 4. Similarly, at t = 2, g(t) = 1. 

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We can see the horizontal change with a red arrow as Δt  =  3, while the vertical change g(t) =  - 3; this means that the output changes by - 3 and the input by 3. 

Here,

x\[_{1}\] = - 1

y\[_{1}\] =  4

x\[_{2}\] = 2

y\[_{2}\] = 1

So, the average rate of change is:

\[\frac{\Delta y}{\Delta x}\] = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\] 

= \[\frac{1-4}{2-(-1)}\] =  \[\frac{-3}{3}\] = -1

Instantaneous Rate of Change Equation

The instantaneous rate of change formula helps us determine the rate of change at a particular instant, and it is likewise the change in the derivative value at a particular point.

The instantaneous rate of change formula is:

f’(a)   =  \[\lim_{\Delta x \rightarrow 0}\]  \[\frac{\Delta y}{\Delta x}\] = \[\lim_{\Delta x \rightarrow 0}\] \[\frac{t(a+h)-t(a)}{h}\]   

Here, this equation is with respect to ‘x’ when y = f(x).

Let’s suppose that there is a function f(x) = 3x\[^{2}\] + 2x + 4. Now, compute the instantaneous rate of change at x = 3.

Now, differentiating the given function with respect to ‘x’ as:

f’(x) = \[\frac{d(f(x))}{dx}\] \[\frac{d(3x^{2}+2x+4)}{dx}\] 

f’(x) =  6x + 2….(a)

Now, putting x =3 to find average rate of change from equation (a), we have:

f’(3)   =   6 * 3 + 2   =  20

So, the instantaneous rate of change for the function f’(3) (at a specific point x = 3) is 20.

Conclusion

We define the units on a rate of change as the “output units divided by the input unit.” Hence, the average rate of change between two input values/quantities is equal to the total change of the function values (output values) divided by the variation in the input values.