In this explainer, we will learn how to find the average rate of change of a function between two -values and use limits to find the instantaneous rate of change.
Rates of change are an important concept that is central to many areas of mathematics, physics, chemistry, and other scientific disciplines. In fact, it is not just limited to science; in everyday life, we often deal with concepts such as speed, acceleration, and interest, which, at their core, are rates of change. Sometimes we are interested in finding rates of change or predicting the future based on the rate of change in the present.
When we say that a car is traveling at 60 miles per hour, what do we really mean? Can we only define such a concept if we travel for an hour? Of course, we could say we travel one mile a minute. However, if we are traveling for less than a minute at this speed, is it still reasonable to talk about traveling at a mile a minute? Certainly, if no time passes, we do not travel any distance. However, we are still happy talking about traveling at 60 miles per hour even if for an instant. When we say we are traveling at a particular speed at any given instant, we are talking about an instantaneous rate of change. In this explainer, we will look at how we can define this mathematically and how we can calculate it.
The rate of change is the change in the quantity described by a function with respect to the change in the input values, or the dependent and independent variables.
Let’s first define the average rate of change of a function over an interval.
Definition: Average Rate of Change
The average rate of change of a function over an interval (i.e., when changes from to ) is defined by
From this definition, we see that the average rate of change of a function over an interval is the average change in the function output when its input increases by one over this interval. That is, over the interval , for every 1 unit change in , the average change in the value of the function is , which is just a number.
You may notice that the average rate of change over an interval has the same formula for a slope of a straight line. In fact, on the graph of the function, the average rate of change can be interpreted as the slope of a straight line passing through (or joining) any two distinct points, and , on the curve.
If we denote the change in by and the change in by , the slope of the line that passes through the points and is
Let’s consider an example where we have to determine the average rate of change of a quadratic function as changes from two given values.
Example 1: Computing the Average Rate of Change of Polynomial Functions between Two Points
Evaluate the average rate of change for the function when changes from 1 to 1.5.
Answer
In this example, we want to evaluate the average rate of change for a quadratic function as changes between two given values.
Recall that the average rate of change when changes from to is defined as
We want to determine the average rate of change with and ; first we will evaluate the given function at these values:
Thus, the average rate of change is
In the next example, we consider a real-world problem where we have to determine the average rate of change of a farm’s production as the amount of insecticide increases between two given points.
Example 2: Finding the Average Rate of Change of a Rational Function in a Real-World Context
A farm’s production in kilograms, , as a function of the kilograms of insecticide, , is given by . Find the average rate of change in when varies from 13 to 17.
Answer
In this example, we want to determine the average rate of change of the farm’s production in kilograms, as the kilograms of insecticide, , change between two given values.
Recall that the average rate of change of a function over an interval is defined by
Since the kilograms of insecticide, , increase from to , the amount it changes by is . The farm’s production will also change from to and the amount it changes by will be given by
Thus, the average rate of change in when varies from 13 to 17 is
This means that increasing the mass of the insecticide used from 13 to 17 kg leads to an average increase of the crop’s yield of for every 1 kg increase in insecticide.
Since the amount by which changes, , is arbitrary, we can use a variable to express this with and, hence, the average rate of change , as a function of where is calculated over the interval .
Definition: Average Rate of Change Function
The average rate of change function when changes from to or equivalently when is
Notice that if , the avarage rate of change is calculated over the interval .
Now, let’s consider an example where we have to determine the average rate of change of a polynomial function at a given point.
Example 3: Finding the Average Rate of Change of Polynomial Functions at a Point
Determine the average rate of change function, , for at .
Answer
In this example, we want to determine the average rate of change function of a polynomial at a given value of .
Recall that the average rate of change function of a function at is
For the given function at , we have
In the next example, we will determine the average rate of change of a cubic function at an arbitrary value of .
Example 4: Finding the Average Rate of Change of Polynomial Functions at a Point
Find the average rate of change function, , of at .
Answer
In this example, we want to determine the average rate of change function of a cubic function at an arbitrary value of .
Recall that the average rate of change function of a function at is
For the given function at , we have
Sometimes, we may be told to determine the average rate of change function when changes between two arbitrary values, and , but this is equivalent to the average rate of change function at an arbitrary value .
Now, let’s consider an example where we determine the average rate of change function for a rational function as changes between two arbitrary points, and .
Example 5: Finding the Expression of the Average Rate of Change of a Rational Function
Determine the average rate of change function, , for when changes from to .
Answer
In this example, we want to determine the average rate of change function of a rational function when changes between two arbitrary values.
Recall that the average rate of change function of a function when changes from to is
For the given function , we have
Thus, the average rate of change function is
Now, let’s consider a real-world example where we have to determine the average rate of growth of a population, described by a function , as the time changes between two arbitrary values. This average rate of change will be given as a function of , the amount by which changes.
Example 6: Finding the Average Rate of Change of a Polynomial Function between Two Points
Suppose a population is as a function of time . What is the average rate of growth of this population when changes from to ?
Answer
In this example, we want to determine the average rate of change of a quadratic function that describes a population at time , as the value of changes between two arbitrary points.
Recall that the average rate of change, , of a function over the interval is given by
Using this with the given function of the population , the average rate of change when changes from to is
For a particular value of , the average rate of change function, , evaluated at this value is the same as the average rate of change, , over an interval , where . In other words, . For example, for a function over an interval or as changes from 1 to 1.5, the average rate of change is the same as the average rate of change function as changes from 1 to , or evaluated at and .
In the next example, we will determine the average rate of change function of a polynomial function at a particular value of and evaluate this at a given point.
Example 7: Finding the Average Rate of Change of Polynomials
For a function , the average rate of change between a fixed point and another point is . Given that , find when .
Answer
In this example, we want to determine the average rate of change function, , of a quadratic function and then evaluate this at a given value of .
From the definition, the average rate of change function of the given function when , or when changes from 4 to , is
Substituting , we find
This is the average rate of change of as changes between 4 and 4.5.
If we examine what happens to the average rate of change function as becomes arbitrarily small or as the interval shrinks, we find the instantaneous rate of change of a function. This is because for the average rate of change function, changes between and , which as is no change at all; this gives the rate of change at an instant rather than between two values. We can give a formal definition of this in terms of a limit.
Definition: Instantaneous Rate of Change
The instantaneous rate of change of a function at a point is when this limit exists.
For many of the functions we are familiar with, such as polynomial, trigonometric, exponential, logarithmic, and rational functions, it is possible to find their instantaneous rate of change for values of in their domain. However, there are many functions where this is not possible; we will introduce examples of these functions when we discuss the derivative in more detail.
In the next example, we will determine the instantaneous rate of change of a quadratic function at a given value of .
Example 8: Finding the Instantaneous Rate of Change of a Polynomial Function at a Point
Evaluate the instantaneous rate of change of at .
Answer
In this example, we want to determine the instantaneous rate of change of a quadratic function at a given point. The instantaneous rate of change of a function at is
For the given function at , the instantaneous rate of change is
Now, let’s consider an example where we find the instantaneous rate of change of a root function at an arbitrary value of .
Example 9: Instantaneous Rates of Change
Find the instantaneous rate of change of at .
Answer
In this example, we want to determine the instantaneous rate of change of a root function at an arbitrary value of .
Recall that the instantaneous rate of change of a function when is
For the given function , we have
Instantaneous rates of change have many real-world applications; for example, the velocity of a moving body at a particular time is the instantaneous rate of change of the displacement at that time.
In the final example, we will consider a real-world problem and determine the instantaneous rate of change at a particular time for a polynomial function representing the biomass of a bacterial culture.
Example 10: Finding the Rate of Change of a Polynomial Function Representing the Biomass of a Bacterial Culture at a Certain Time
The biomass of a bacterial culture in milligrams as a function of time in minutes is given by . What is the instantaneous rate of growth of the culture when ?
Answer
In this example, we want to determine the instantaneous rate of change of a cubic function representing the biomass of a bacterial culture.
Recall that the instantaneous rate of change of a function when is
For the given function when , the instantaneous rate of change is
Since the rate of change is positive, it is equivalent to the rate of growth.
Hence, the rate of growth of the biomass of a bacterial culture when is
The instantaneous rate of change is also related to the derivative, at an arbitrary point . In particular, the instantaneous rate of change at of a function is the derivative of a function evaluated at . However, this is beyond the scope of the explainer and will be covered elsewhere.
Let’s now summarize a few key points from the explainer.
Key Points
- The average rate of change of a function over an interval (i.e., when changes from to ) is defined by
- The average rate of change function, , of a function as changes between and , or equivalently on the interval , is defined by
- When the values of and are given, the average rate can be interpreted graphically as the slope of the line that passes through the points and .
- To define an instantaneous rate of change, we introduce the concept of a limit. In particular, we take the average rate of change over smaller and smaller intervals,
that is, the interval as gets closer and closer to zero.
Formally, the instantaneous rate of change of a function at a point is given by when this limit exists. - We can also interpret the instantaneous rate of change of at as the slope of the tangent line to the graph of at .
- We can take the limit of the slopes of secants of the curve passing through the graph of the function at and as .
- To evaluate the limit and find the instantaneous rate of change, we often require some algebraic manipulation of the expression to eliminate from its denominator.
- Although instantaneous rates exist for many functions that we are familiar with, there are, in fact, many functions for which the limit does not exist and, as a result, we are unable to define the notion of instantaneous rate of change for them. A number of examples of such functions will be introduced as you study calculus further.
