The average rate of change can simply be understood as the rate at which one quality changes with respect to the other. The average rate of change of a function is calculated using a certain method which is described further below in the article. More information about functions can be gathered by joining the online course on Rational Functions.
We will go on explaining the topic in detail but first we need to get you rolling by providing some very easy examples of average rate of change.
Average Rate of Change Examples
The easiest example for the average rate of change is speed. Speed is simply distance covered by a body in a particular amount of time. The formula for speed is:
Speed = Distance Covered/Total Time Taken
Let’s say a body movies in a straight line from a point A to a point B. Point A is 100 Kilometers away and Point B is 150 Kilometers away, then the distance that the body has covered comes out to be 150 – 100 = 50 km. Suppose, the body was at Point A at 04:30 PM and reached Point B at 08:30 PM. The time taken by the body to cover the distance comes out to be 4 hours. Hence, if we want to calculate the average rate of change of distance with respect to time, then it simply comes out to be 50/4 i.e. 12.5 km per hour.
Another very good example of average rate of change is when you find the slope of a line. The slope of a line is nothing but the change in Y coordinates with respect to the change is the X coordinates. If a line crosses two points with coordinates (x1, y1) = (2, 2) and (x2, y2) = (3, 0), then the slope of the line is (y2-y1)/(x2-x1) = (0-2) / (3-2) = -2/1. Hence, the average rate of change in the Y coordinates with respect to the X coordinates a.k.a. the slope of the line is -2.
Average rate of change could be a good metric to find the overall change in a particular quantity with respect to the other. Though the average rate of change may not be as accurate as the instantaneous rate of change that is calculated by differential calculus, it is still very useful in calculations involving motion of the object. You can learn more about differential calculus by enrolling for the Calculus Master Course.
Now that we have the wagon rolling it’s time to throttle and explore some more concepts.
What are Functions and Average Rate of Change of Functions?
Functions are equations which allow you to give input and according to that input they give you an output. For example: a simple line equation in the format of Y = X + 2, lets you input innumerable values of X and accordingly get output values for Y. If you wish to learn more about polynomial functions then you should definitely join the advanced lesson on polynomial functions.
A simple symbol of a function can be F(x) = X+2, where F(x) is the function that gives innumerable values of F(x) with varied values of X. As an example, let’s put the values of x in the equation as: 1, 2 and 3. Accordingly, you will get the output F(x) as 3, 4 and 5. So, you can see that with the variations in the value of x accordingly there is a variation in the value of F(x).
Now, let’s move on to find the value of average rate of change of functions. The average rate of change can be found out by putting respective values in the formula:
Average Rate of Change of Function = Change in the Value 0f F(x)/ Respective Change in the Value of x
For example, if the value of x changes from x1 = 1 to x2 = 2. Then the change in the value of F(x) from the above equation is F(x1) = 3 and F(x2) = 4. Therefore, the Average Rate of Change of the Function is 4-3/2-3 = 1.
In other words, the Average Rate of Change of Function = F(x2) – F(x1) / x2 – x1.
Rate of Function Calculated as a Derivative
The rate of change of a function can also be calculated by using derivative. Derivatives can prove to be very useful when an instantaneous change metric is required. For example, a projectile’s instantaneous velocity can be found by simply finding the out the derivate of the distance with respect to time.
More information on differential calculus and derivatives can be found on the web course on differential equations.
Practical Applications of the Average Rate of Change
As a curious one you may always be thinking as to why you keep on bothering about these X and Y’s. We would like to point out to you that nothing is without any good reason. Everything that you study in Math has a practical application and since you are reading about average rate of change, here are some practical applications:
- As described before, average rate of change is used to measure the speed of an object undergoing motion. An advanced level of the formula is used in various astronomical equations pertaining to rocket science and space travel.
- The equation is used to measure the rate of a chemical reaction in chemistry. This application of average rate of change is very useful in Chemical Engineering Calculations.
- The formula is used in various behavioral calculations relating to management and the human psyche.
- A good application in real life can be found in predicting your electricity bill. If you have an idea of the average rate in which electricity is consumed in your household, then you can predict your electricity bill for a month. These fundamentals are used in various devices that help you monitor your electricity bill.
Finally, as the old saying goes, “More you learn, more you grow.” Your learning does not stop here, so feel free to browse more resources on differential calculus and explore more concepts on calculating average rate of change using differential calculus.