July 18, 2022

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

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 concepts throughout academics, specifically in chemistry, physics and finance.

It’s most frequently used when talking about momentum, although it has many uses throughout various industries. Due to its utility, this formula is a specific concept that students should understand.

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 change of one value in relation to another. In every day terms, it's employed to determine the average speed of a change over a certain period of time.

Simply put, the rate of change formula is expressed as:

R = Δy / Δx

This calculates the variation of y in comparison to the change of x.

The variation within the numerator and denominator is portrayed by the greek letter Δ, read as delta y and delta x. It is further denoted 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 expressed as:

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

Average Rate of Change = Slope

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

The straight line that joins these two points is also 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 between two figures is equivalent to 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 random 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 know the slope formula and what the figures mean, finding the average rate of change of the function is possible.

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

Step 1: Determine Your Values

In these equations, math scenarios usually provide you with 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 case, next you have to find the values along the x and y-axis. Coordinates are typically provided 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 remember, the formula for the rate of change is:

R = Δy / Δx

Which then translates to:

R = y2 - y1 / x2 - x1

Now that we have found 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 in place, all that we have to do is to simplify the equation by subtracting all the numbers. So, our equation becomes 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 obtain the average rate of change for the two coordinates that we were given.

Average Rate of Change of a Function

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

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

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

In this case, the values provided will have one f(x) equation and one X Y graph value.

Negative Slope

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

Every so often, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the Cartesian plane.

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

Positive Slope

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

Examples of Average Rate of Change

In this section, we will talk about the average rate of change formula via some examples.

Example 1

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

In this example, all we must do is a simple substitution because the delta values are already given.

R = Δy / Δx

R = 10 / 2

R = 5

Example 2

Extract the rate of change of the values in points (1,6) and (3,14) of the Cartesian plane.

For this example, we still have to search for the Δy and Δx values by utilizing 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 connecting two points.

Example 3

Find 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 finding the rate of change of a function, solve for the values of the functions in the equation. In this case, we simply replace 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

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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