# The Many Faces of the Mean Value Theorem

Karen Sleno is an expert teacher at Flushing High School where she has taught everything from Algebra 1 through AP Calculus and serves as the department chair. Additionally, she is an adjunct instructor at Mott Community College and at the Center for Talented Youth at Johns Hopkins University. In 2022, she won the Michigan Department of Education Regional Teacher of the Year award in Region 5 for her teaching expertise gained over 30 years in the profession. She is a College Board consultant for AP Calculus and has held various roles (including question/exam leader) at the annual AP reading. Her efforts in education and in the AP program earned her recognition as Educator of the Year for her district in 2015.

The Mean Value Theorem (MVT) is one of the big theorems of AP Calculus, connecting a function's average rate of change over an interval with its instantaneous rate of change, i.e. its derivative. In this article we'll unpack the many ways that the MVT shows up on the AP exam and how this impacts your instruction. But first...

## What does the Mean Value Theorem say?

## In plain English?

If a function is well-behaved over a given interval (no jumps, holes, gaps, sharp turns, etc.) then the instantaneous rate of change of the function must match the average rate of change over that interval at least once during that interval.

## How is the MVT tested on the AP Exam?

Once upon a time, on an AP Calculus test far away, students knew exactly what to expect when the Mean Value Theorem was the topic of a question. It went something like this: *Given the function f(x) = ______________, find the value of c that satisfies the for the Mean Value Theorem on the interval [a, b]*. Even the 2001 College Board “acorn” book only states that students should know “The Mean Value Theorem and its graphical applications”, yet conversations with other veteran AP teachers confirm that typically what actually was assessed was finding that elusive value of *c*.

Fast forward to current AP Calculus expectations which are certainly much more rigorous and broad in their scope of how understanding of the Mean Value Theorem can be assessed. Let’s take a look at some of the most common question types that have emerged:

**1. SECANT/TANGENT LINES**

*For the function f(x) = __________ on the interval [1, 4], find the value of x where the secant line is parallel to the tangent line.*

In this instance, students must understand that the formula stated in the Mean Value Theorem is actually an expression of two slopes, one for the secant line (f(b)-f(a)/(b-a) and one for the tangent line f'(c). The fact that those expressions are equated indicates that those lines are parallel. So, a question presented in the following way would not only require students to recognize a graphical application of the Mean Value Theorem, but also be able to set up and solve the resulting equation that follows.

**2. AVERAGE vs. INSTANTANEOUS VELOCITY**

*Find the time t on the interval [1,4] where the average and instantaneous velocities are equal for the position function s(t) = _________________.*

The use of secant and tangent lines to model average and instantaneous velocity propels the Mean Value Theorem into a real life application that can reference particle motion or a comparison of rates of change of any scenario. This type of question raises the bar even higher by not only requiring a knowledge of the theorem, but also making the connection that instantaneous velocity and average velocity are actually the two expressions that result in f'(c)=(f(b)-f(a))/(b-a).

**3. TABULAR APPROACH**

This type of Mean Value Theorem question requires the highest level of understanding of the concept of how this theorem can be applied. Students will be presented a table of values for a given scenario (sometimes with f(x) and f'(x) values and other times with f(x) values only). The question is typically worded in this way:

In a table with both f(t) and f'(t) values, students often attempt to apply the Intermediate Value Theorem by observing whether f'(t) ever passes the given value. If not, many students may assume the answer is “no”. However, a more appropriate approach may be to find the average rate of change on the given interval (or a subinterval) using f(t) values. If the result matches the given value, the student can conclude that the Mean Value Theorem applies and respond appropriately.

Quite a change from the “old days”, right? By sharing these varied approaches to this single topic with students, we emphasize that math learning isn’t about being able to solve one type of problem, but rather it is about deeper conceptual understanding that gives you access to *many *types of problems…exactly what Calc Medic and EFFL lessons attempt to promote! Furthermore, Connecting Representations, Justification, and Communication embody three of the four Mathematical Practices we strive for in AP Calculus, and all of them find their home in problems such as those detailed above. So how do we get students there?

## Resources for teaching the MVT

Just giving students the theorem and then having them practice is not enough! We want students to *experience* the concept and make sense of it on their own before we add on the formal notation and language.

Check out the Calc Medic EFFL lesson that introduces the MVT with a lively conversation about speeding tickets!