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

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

It’s most often applied when discussing momentum, however it has multiple uses across various industries. Because of its usefulness, this formula is a specific concept that learners should grasp.

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

Average Rate of Change Formula

In math, the average rate of change formula describes the variation of one figure in relation to another. In practical terms, it's employed to define the average speed of a variation over a certain period of time.

At its simplest, the rate of change formula is expressed as:

R = Δy / Δx

This calculates the variation of y compared to the variation of x.

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

Δy = y2 - y1

Δx = x2 - x1

Consequently, the average rate of change equation can also be shown as:

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

Average Rate of Change = Slope

Plotting out these numbers in a X Y graph, is useful when discussing dissimilarities in value A versus value B.

The straight line that joins these two points is called the 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

To summarize, in a linear function, the average rate of change among two values is the same as the slope of the function.

This is why the average rate of change of a function is the slope of the secant line going through two arbitrary endpoints on the graph of the function. Simultaneously, 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 understand the slope formula and what the values mean, finding the average rate of change of the function is feasible.

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

Step 1: Understand Your Values

In these sort of equations, math problems typically give you two sets of values, from which you will get x and y values.

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

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

x1 = 1

x2 = 3

y1 = 2

y2 = 4

Step 2: Subtract The Values

Calculate the Δx and Δy values. As you can recollect, the formula for the rate of change is:

R = Δy / Δx

Which then translates to:

R = y2 - y1 / x2 - x1

Now that we have all the values of x and y, we can plug-in the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our figures inputted, all that is left is to simplify the equation by subtracting all the values. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve mentioned before, the rate of change is applicable to numerous diverse scenarios. The aforementioned examples were applicable to the rate of change of a linear equation, but this formula can also be relevant for functions.

The rate of change of function follows an identical rule but with a distinct formula because of the distinct 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 X Y axis value.

Negative Slope

As you might remember, the average rate of change of any two values can be plotted. The R-value, then is, equal to its slope.

Every so often, the equation results in a slope that is negative. This means that the line is trending downward from left to right in the X Y graph.

This translates to the rate of change is diminishing in value. For example, rate of change can be negative, which means a declining position.

Positive Slope

On the other hand, a positive slope shows that the object’s rate of change is positive. This tells us that the object is gaining 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 ascending.

Examples of Average Rate of Change

Next, we will review the average rate of change formula with some examples.

Example 1

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

In this example, all we need to do is a simple substitution since the delta values are already provided.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

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

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

R = y2 - y1 / x2 - x1

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

R = 8 / 2

R = 4

As provided, 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 final 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 instance, we simply substitute the values on the equation with the values specified 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

Now that we have all our values, all we need to do is substitute 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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