## How To Find Average Rate Of Change

Whenever we wish to describe how quantities change over time is the basic idea for finding the average rate of change and is one of the cornerstone concepts in calculus.

So, what does it mean to find the average rate of change?

The **average rate of change** finds how fast a function is changing with respect to something else changing.

It is simply the process of calculating the rate at which the **output** *(y-values)* changes compared to its **input** *(x-values)*.

How do you find the average rate of change?

We use the **slope formula!**

To find the average rate of change, we divide the **change in y** *(output)* by the **change in x** *(input)*. And visually, all we are doing is calculating the slope of the secant line passing between two points.

Now for a linear function, the average rate of change (slope) is constant, but for a non-linear function, the average rate of change is not constant (i.e., changing).

Let’s practice finding the average rate of a function, f(x), over the specified interval given the table of values as seen below.

**Practice Problem #1**

### Practice Problem #2

*See how easy it is?*

All you have to do is **calculate the slope** to *find the average rate of change!*

## Average Vs Instantaneous Rate Of Change

But now this leads us to a very important question.

What is the difference is between Instantaneous Rate of Change and Average Rate of Change?

While both are used to find the slope, the average rate of change calculates the slope of the secant line using the slope formula from algebra. The instantaneous rate of change calculates the slope of the tangent line using derivatives.

Using the graph above, we can see that the green secant line represents the average rate of change between points P and Q, and the orange tangent line designates the instantaneous rate of change at point P.

So, the other key difference is that the average rate of change finds the slope over an interval, whereas the instantaneous rate of change finds the slope at a particular point.

### How To Find Instantaneous Rate Of Change

All we have to do is take the derivative of our function using our derivative rules and then plug in the given x-value into our derivative to calculate the slope at that exact point.

For example, let’s find the instantaneous rate of change for the following functions at the given point.

### Tips For Word Problems

But how do we know when to find the average rate of change or the instantaneous rate of change?

We will always use the slope formula when we see the word “average” or “mean” or “slope of the secant line.”

Otherwise, we will find the derivative or the instantaneous rate of change. For example, if you see any of the following statements, we will use derivatives:

- Find the velocity of an object at a point.
- Determine the instantaneous rate of change of a function.
- Find the slope of the tangent to the graph of a function.
- Calculate the marginal revenue for a given revenue function.

### Harder Example

Alright, so now it’s time to look at an example where we are asked to find both the average rate of change and the instantaneous rate of change.

Notice that for part (a), we used the slope formula to find the average rate of change over the interval. In contrast, for part (b), we used the power rule to find the derivative and substituted the desired x-value into the derivative to find the instantaneous rate of change.

*Nothing to it!*

## Particle Motion

But why is any of this important?

Here’s why.

Because “slope” helps us to understand real-life situations like linear motion and physics.

The concept of **Particle Motion**, which is the expression of a function where its independent variable is time, t, enables us to make a powerful connection to the **first derivative** *(velocity)*, **second derivative** *(acceleration)*, and the **position function** *(displacement)*.

The following notation is commonly used with particle motion.

### Ex) Position – Velocity – Acceleration

Let’s look at a question where we will use this notation to find either the average or instantaneous rate of change.

## Find the average rate of change of a function

y | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |

C(y) | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |

The price change per year is a **rate of change** because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in the table above did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the **average rate of change** over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.

Other examples of rates of change include:

- A population of rats increasing by 40 rats per week
- A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
- A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)
- The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
- The amount of money in a college account decreasing by $4,000 per quarter

### A GENERAL NOTE: RATE OF CHANGE

A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.”

The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.

### HOW TO: GIVEN THE VALUE OF A FUNCTION AT DIFFERENT POINTS, CALCULATE THE AVERAGE RATE OF CHANGE OF A FUNCTION FOR THE INTERVAL BETWEEN TWO VALUES X1 AND X2

### What is average rate of change?

The average rate of change of function f*f*f over the interval *a*≤*x*≤*b*, is less than or equal to, x, is less than or equal to, b is given by this expression:

It is a measure of how much the function changed per unit, on average, over that interval.It is derived from the slope of the straight line connecting the interval’s endpoints on the function’s graph.

### EXAMPLE 1: COMPUTING AN AVERAGE RATE OF CHANGE

Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.

y | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |

C(y) | 2.31 | 2.62 | 2.84 | 3.30 | 2.41 | 2.84 | 3.58 | 3.68 |

### SOLUTION

In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is

### Analysis of the Solution

Note that a decrease is expressed by a negative change or “negative increase.” A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.

The following video provides another example of how to find the average rate of change between two points from a table of values.

## Finding average rate of change

### Example 1: Average rate of change from graph

### Example 2: Average rate of change from equation

### Solved Examples

**Question 1:** **Calculate the average rate of change of a function, f(x) = 3x + 12 as x changes from 5 to 8 .**

**Solution:**

Given,

f(x) = 3x + 12

a = 5

b = 8

f(5) = 3(5) + 12

f(5) = 15 + 12

f(5) = 27

f(8) = 3(8) + 12

f(8) = 24 + 12

f(8) = 36

The average rate of change is,

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