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**Average Rate of Change Function** Instructor: Dr. Jo Steig

**DEFINITION:** A function is a process by which every input is associated with exactly one output. When create a process (or series of steps) to do a certain task we are often creating a function. If we want to use it over and over again then to make our lives easier we give it a name. It helps us remember the name when it has something to do with the process that is being described.

**The Average Rate of Change function describes the average rate at which one quanity is changing with respect to something else changing.**

You are already familiar with some average rate of change calculations:

**(a)** Miles per gallon – calculated by dividing the number of miles by the number of gallons used

**(b)** Cost per killowatt – calculated by dividing the cost of the electricity by the number of killowatts used

**(c)** Miles per hour – calculated by dividing the numebr of miles traveled by the number of hours it takes to travel them.

In general, an average rate a change function is a process that calculates the the amount of change in one item divided by the corresponding amount of change in another. Using function notation, we can define the Average rate of Change of a function f from a to x as

- A is the name of this average rate of change function
- x – a represents the change in the input of the function f
- f(x) – f(a) represents the change in the function f as the input changes from a to x

You might have noticed that the Average Rate of Change function looks a lot like the formula for the slope of a line. In fact, if you take any two distinct points on a curve, (x_{1} ,y_{1} ) and (x_{2} ,y_{2} ), the slope of the line connecting the points will be the average rate of change from x_{1} to x_{2}

Example 1: Find the slope of the line going through the curve as x changes from 3 to 0.

Step 1: f (3) = -1 and f (0) = -4

Step 2: Use the slope formula to create the ratio

Step 3: Simplify.

Step 4: So the slope of the line going through the curve as x changes from 3 to 0 is 1.

Example 2: Find the average rate of change of from 3 to 0.

Since the average rate of change of a function is the slope of the associated line we have already done the work in the last problem. That is, the average rate of change of from 3 to 0 is 1. That is, over the interval [0,3], for every 1 unit change in x, there is a 1 unit change in the value of the function.

Here is a graph of the function, the two points used, and the line connecting those two points.

Now suppose you needed to find series of slopes of lines that go through the curve and the point (3, f(3)) but the other point keeps moving. We will call the second point (x, f(x)). It will be useful to have a process (function) that will do just that for us. The average rate of change function also deterines slope so that process is what we will use.

Example 3: Find the average rate of change function of from 3 to x.

Step 2: Use the average rate of change formula to define A(x) and simplify.

Step 3: The average rate function of change of from 3 to x is

Example 4: Use the result of Example 3 to find the average rate of change of from 3 to 6.

Solution: The average rate function of change of from 3 to x is

So, the average rate of change of from 3 to 6 is A(6) = 9/3 = 3.

Example 5: Use the result of Example 3 to find the average rate of change of from 3 to 0.

The average rate of change of from 3 to 0 is A(0) = 3/3 = 1.

2009 Jo Steig