\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \)

Mean Value Theorem

You CAN Ace Calculus

17calculus > derivatives > mean value theorem

Topics You Need To Understand For This Page

Tools, Related Topics and Links

external links you may find helpful

mean value theorem youtube playlist

WikiBooks: Mean Value Theorem (less detail)

Wikipedia: Mean Value Theorem (more detail with advanced discussion)

Rolle's Theorem

Rolle's Theorem is a special case of the Mean Value Theorem where \(f(a)=f(b)\).

The Mean Value Theorem is one of the coolest applications of the derivative. Using it doesn't introduce any new notation or concept. It is just another way to apply the derivative on a continuous function. Here is a formal version of the theorem.

Mean Value Theorem

For any function that is continuous on \([a, b]\) and differentiable on \((a, b)\) there exists some \(c\) in the interval \((a, b)\) such that the secant joining the endpoints of the interval \([a, b]\) is parallel to the tangent at \(c\).

Let's break this down.
What This Theorem Requires
1. First, we are given a closed interval \([a,b]\). Notice that all these intervals and values of \(c\) refer to the independent variable, \(x\).
1. Second, we must have a function that is continuous on the given interval \([a,b]\). We don't care what's going on outside this interval.
2. Next, the function must be differentiable inside the open interval \((a,b)\). This means there must not be any sharp corners inside the interval. Also, we don't need to have the function differentiable at the end points, only inside the interval.

Source: Wikipedia - Mean Value Theorem

Notice that the orange and green 'lines' are drawn as vectors since they have an arrow at one end. In this context, we would usually want to draw lines and not vectors. But check with your instructor to see what they expect.

What This Theorem Gives Us
If we have met the conditions above, we are guaranteed the following.
3. We can draw a line (called the secant line) between the end points.
4. We are guaranteed that there is a tangent line to the curve that is parallel to the secant line.
5. This tangent line occurs at some \(x=c\) where \( a < c < b \).

Graphically, it looks like this.
The idea is that we are given the function (which we must check to see if it is meets conditions 1 and 2 above). Then we draw the orange secant line. We are then guaranteed that the green tangent line exists. Let's see how we can actually find this point \(x=c\) and the equation of the tangent line.

This equation should have some familiar pieces.
\( \displaystyle{ f'(c) = \frac{f(b)-f(a)}{b-a} }\)
The right side is the slope of a line through the points \((a, f(a))\) and \((b, f(b))\). So this gives us the slope of the secant line. The left side is the slope of the tangent line through the point \((c,f(c))\). So you set those equal to each other and solve for \(c\). Once you have \(c\), you can find the equation of the tangent line.

Okay, so there are a lot of equations so far but what does this mean intuitively? This first video clip will help you really understand the mean value theorem, what it is saying and where it comes from (with a proof).

MIT OCW - Lec 14 | MIT 18.01 Single Variable Calculus, Fall 2007

Here is a good video explaining the mean value theorem in some detail with a great example. The example is quite extensive but he takes the time to explain some of the things you may run across when using this theorem.

PatrickJMT - The Mean Value Theorem


China celebrates the Mean Value Theorem. Pretty cool, eh? [ Source: Wikipedia - Mean Value Theorem ]

Okay, time for some practice problems.

Search 17Calculus


let us know


do you prefer worked out solutions over video solutions?

Practice Problems

Instructions - - Unless otherwise instructed, determine all numbers c which satisfy the Mean Value Theorem (or Rolles Theorem) for the given functions and intervals, if possible. If the theorem does not apply, explain why. Give your answers in exact form.

Level A - Basic

Practice A01

\(f(x)=3x^2+6x-5\); \([-2,1]\)


Practice A02

\(f(x)=x^2-2x\); \([0,2]\)


Practice A03

\(f(x)=\sqrt{x}-x/3\); \([0,9]\)


Practice A04

\(f(x)=x^{2/3}\); \([0,1]\)


Practice A05

\(f(x)=x^2-3x+2\); \([1,2]\)


Practice A06

\(f(x)=(x^2-2x)e^x\); \([0,2]\)


Practice A07

\(f(x)=\sin(2x)\); \([\pi/6,\pi/3]\)


Level B - Intermediate

Practice B01

\(f(x)=2\sin(x)\cos(x)\); \([0,\pi]\)


Practice B02

\(f(x)=1-\abs{x}\); \([-1,1]\)


Practice B03

\(\displaystyle{f(x)=\frac{x}{x+1}}\); \([-1/2,2]\)


17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades of any individual. We have worked, to the best of our ability, to ensure accurate and correct information and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

17calculus logo

top   -   search   -   practice problems

page like? 0