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 math formulas across academics, most notably in chemistry, physics and accounting.
It’s most frequently utilized when talking about velocity, although it has numerous applications across various industries. Because of its value, this formula is a specific concept that learners should learn.
This article will discuss the rate of change formula and how you can work with them.
Average Rate of Change Formula
In math, the average rate of change formula denotes the variation of one value in relation to another. In every day terms, it's used to evaluate the average speed of a variation over a specific period of time.
At its simplest, the rate of change formula is written as:
R = Δy / Δx
This measures the variation of y in comparison to the variation of x.
The variation through the numerator and denominator is shown by the greek letter Δ, read as delta y and delta x. It is further expressed as the variation within 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 expressed as:
R = (y2 - y1) / (x2 - x1)
Average Rate of Change = Slope
Plotting out these figures in a X Y axis, is helpful when reviewing differences in value A versus value B.
The straight line that links 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
In summation, in a linear function, the average rate of change among two values 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 intersecting two arbitrary endpoints on the graph of the function. Meanwhile, 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 values mean, finding the average rate of change of the function is achievable.
To make understanding this principle easier, here are the steps you need to obey to find the average rate of change.
Step 1: Determine Your Values
In these sort of equations, math questions typically give you two sets of values, from which you solve to find x and y values.
For example, let’s take the values (1, 2) and (3, 4).
In this scenario, then you have to search for the values along 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 found all the values of x and y, we can input 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 deducting all the numbers. Thus, 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 achieve 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 different scenarios. The previous examples were more relevant to the rate of change of a linear equation, but this formula can also be used in functions.
The rate of change of function observes a similar principle but with a different formula because of the distinct values that functions have. This formula is:
R = (f(b) - f(a)) / b - a
In this instance, the values provided will have one f(x) equation and one X Y graph value.
Negative Slope
Previously if you recollect, the average rate of change of any two values can be plotted. The R-value, therefore is, equivalent to its slope.
Sometimes, the equation results in a slope that is negative. This denotes that the line is descending 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 declining position.
Positive Slope
On the other hand, a positive slope means that the object’s rate of change is positive. This means 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 increasing.
Examples of Average Rate of Change
Next, we will talk about the average rate of change formula through some examples.
Example 1
Calculate 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 due to the fact that the delta values are already specified.
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 Cartesian plane.
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 equal to the slope of the line connecting two points.
Example 3
Extract the rate of change of function f(x) = x2 + 5x - 3 on the interval [3, 5].
The last example will be calculating 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, calculate the values of the functions in the equation. In this instance, we simply replace 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
Now that we have all our values, all we need to do is replace 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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