Rate of Change Formula - What Is the Rate of Change Formula? Examples

The rate of change formula is one of the most used mathematical principles across academics, specifically in physics, chemistry and finance.

It’s most frequently used when talking about momentum, though it has multiple applications across different industries. Due to its utility, this formula is a specific concept that learners should learn.

This article will share the rate of change formula and how you should solve them.

Average Rate of Change Formula

In mathematics, the average rate of change formula describes the change of one value in relation to another. In practical terms, it's utilized to evaluate the average speed of a variation over a specific period of time.

To put it simply, the rate of change formula is expressed as:

R = Δy / Δx

This computes the change of y compared to the change of x.

The variation within the numerator and denominator is shown by the greek letter Δ, expressed as delta y and delta x. It is also expressed as the variation within the first point and the second point of the value, or:

Δy = y2 - y1

Δx = x2 - x1

As a result, the average rate of change equation can also be shown as:

R = (y2 - y1) / (x2 - x1)

Average Rate of Change = Slope

Plotting out these values in a X Y graph, is useful when discussing differences in value A in comparison with value B.

The straight line that links these two points is known as secant line, and the slope of this line is the average rate of change.

Here’s the formula for the slope of a line:

y = 2x + 1

In short, in a linear function, the average rate of change among two figures is the same as the slope of the function.

This is mainly why average rate of change of a function is the slope of the secant line passing through two arbitrary endpoints on the graph of the function. In the meantime, the instantaneous rate of change is the slope of the tangent line at any point on the graph.

How to Find Average Rate of Change

Now that we have discussed the slope formula and what the figures mean, finding the average rate of change of the function is achievable.

To make understanding this topic simpler, here are the steps you should obey to find the average rate of change.

Step 1: Determine Your Values

In these sort of equations, math problems usually give you two sets of values, from which you extract x and y values.

For example, let’s assume the values (1, 2) and (3, 4).

In this situation, then you have to find the values via the x and y-axis. Coordinates are typically given in an (x, y) format, like this:

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Find the Δx and Δy values. As you may recall, the formula for the rate of change is:

R = Δy / Δx

Which then translates to:

R = y2 - y1 / x2 - x1

Now that we have obtained all the values of x and y, we can add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values in place, all that is left is to simplify the equation by subtracting all the numbers. Therefore, our equation then becomes the following.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

As shown, by simply replacing all our values and simplifying the equation, we obtain the average rate of change for the two coordinates that we were provided.

Average Rate of Change of a Function

As we’ve shared before, the rate of change is pertinent to numerous different scenarios. The previous examples were applicable to the rate of change of a linear equation, but this formula can also be applied to functions.

The rate of change of function observes an identical rule but with a distinct formula due to the different values that functions have. This formula is:

R = (f(b) - f(a)) / b - a

In this situation, the values given will have one f(x) equation and one Cartesian plane value.

Negative Slope

Previously if you recall, the average rate of change of any two values can be plotted on a graph. The R-value, then is, equivalent to its slope.

Every so often, the equation concludes in a slope that is negative. This indicates that the line is descending from left to right in the X Y axis.

This means that the rate of change is decreasing in value. For example, velocity can be negative, which results in a declining position.

Positive Slope

In contrast, a positive slope indicates that the object’s rate of change is positive. This shows us that the object is increasing in value, and the secant line is trending upward from left to right. With regards to our previous example, if an object has positive velocity and its position is increasing.

Examples of Average Rate of Change

Now, we will discuss the average rate of change formula with some examples.

Example 1

Calculate the rate of change of the values where Δy = 10 and Δx = 2.

In the given example, all we have to do is a plain substitution because the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Calculate the rate of change of the values in points (1,6) and (3,14) of the X Y axis.

For this example, we still have to look for the Δy and Δx values by employing the average rate of change formula.

R = y2 - y1 / x2 - x1

R = (14 - 6) / (3 - 1)

R = 8 / 2

R = 4

As you can see, the average rate of change is the same as the slope of the line linking two points.

Example 3

Calculate the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].

The last example will be extracting the rate of change of a function with the formula:

R = (f(b) - f(a)) / b - a

When extracting the rate of change of a function, determine the values of the functions in the equation. In this situation, we simply substitute the values on the equation using the values given in the problem.

The interval given is [3, 5], which means that a = 3 and b = 5.

The function parts will be solved by inputting the values to the equation given, such as.

f(a) = (3)2 +5(3) - 3

f(a) = 9 + 15 - 3

f(a) = 24 - 3

f(a) = 21

f(b) = (5)2 +5(5) - 3

f(b) = 25 + 10 - 3

f(b) = 35 - 3

f(b) = 32

With all our values, all we need to do is plug in them into our rate of change equation, as follows.

R = (f(b) - f(a)) / b - a

R = 32 - 21 / 5 - 3

R = 11 / 2

R = 11/2 or 5.5

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