How to find average rate of change in AP Calculus AB: an example

The function $f$ is given by $f(x) = x^2 + 3x$. What is the average rate of change of $f$ on the interval from $x = 1$ to $x = 4$?

• A) 5
• B) 11
• C) 24
• D) 8

Solving this Problem

Average rate of change is just the slope between two points (so the change in $f$ or $y$, divided by the change in $x$). Once you recognize this, these problems become easy. There is no calculating derivatives or finding instantaneous rates of change required for this problem.

In calculus, you will need to be able to find average rate of change over intervals, use variations of the average rate of change formula to define the derivative, and use the average rate of change to estimate derivative values. Make sure you understand this now.

Step 1: Find the function's value at each endpoint

The interval goes from $x = 1$ to $x = 4$, so find the $f(x)$ value at each endpoint to use in the numerator of the slope formula:

Step 2: Plug in to the slope formula and solve

So the average rate of change over the interval $[1, 4]$ is 8. This means the correct answer is choice D, 8.

Having an average rate of change of 8 means that on average, for each increase of $x$ by one, the $y$ increases by 8 over the interval $[1, 4]$. This does not mean that it will increase by exactly 8 every time; it just means it will increase by 8 when all increases are averaged out. For more practice on this check the check your understanding quiz on lesson 2.1 Defining Average and Instantaneous Rates of Change at a Point.

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