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

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

It’s most frequently applied when discussing thrust, though it has multiple uses across many industries. Because of its usefulness, this formula is a specific concept that learners should learn.

This article will go over the rate of change formula and how you can solve it.

Average Rate of Change Formula

In mathematics, the average rate of change formula describes the variation of one figure when compared to another. In practice, it's utilized to determine the average speed of a variation over a specific period of time.

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

R = Δy / Δx

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

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

Δy = y2 - y1

Δx = x2 - x1

Because of this, 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 Cartesian plane, is helpful when reviewing dissimilarities in value A in comparison with value B.

The straight line that joins 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 between two figures is equal to the slope of the function.

This is the reason why the average rate of change of a function is the slope of the secant line going 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 achievable.

To make grasping this concept less complex, 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 scenarios generally provide you with 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, next you have to search for the values on 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 add the values as follows.

R = 4 - 2 / 3 - 1

Step 3: Simplify

With all of our values plugged in, all that remains is to simplify the equation by deducting all the numbers. Thus, our equation will look something like this.

R = 4 - 2 / 3 - 1

R = 2 / 2

R = 1

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

Average Rate of Change of a Function

As we’ve shared previously, the rate of change is pertinent to many different situations. The previous 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 observes 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 scenario, the values provided will have one f(x) equation and one X Y axis 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, therefore is, identical to its slope.

Sometimes, the equation concludes in a slope that is negative. This means that the line is trending downward from left to right in the X Y axis.

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

Positive Slope

On the contrary, 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 aforementioned example, if an object has positive velocity and its position is ascending.

Examples of Average Rate of Change

Now, we will talk about the average rate of change formula via some examples.

Example 1

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

In this example, all we have to do is a simple 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 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 identical to the slope of the line joining 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 finding 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, calculate the values of the functions in the equation. In this case, we simply substitute the values on the equation using the values provided 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

Once we have 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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