Calculus How To

Mean Value Theorem & Rolle’s Theorem

Share on

Contents:

  1. Mean Value Theorem
  2. Rolle’s Theorem

What is the Mean Value Theorem?

For any arc between any two endpoints (a,b), a point c exists where the tangent at c is parallel to the secant line through (a,b).



The Mean Value Theorem (MVT) for derivatives states that if the following two statements are true:

  1. A function is a continuous function on a closed interval [a,b], and
  2. If the function is differentiable on the open interval (a,b),

…then there is a number c in (a,b) such that:
mean-value-theorem-formula-300x79

The Mean Value Theorem is an extension of the Intermediate Value Theorem.

The special case of the MVT, when f(a) = f(b) is called Rolle’s Theorem.

There is also a mean value theorem for integrals.

Watch the video for an overview and a simple example, or read on below:


Can’t see the video? Click here.

The Common Sense Explanation

The “mean” in mean value theorem refers to the average rate of change of the function. It’s basic idea is: given a set of values in a set range, one of those points will equal the average. This is best explained with a specific example.



Let’s say you travel from your house to work, varying your speed between 40 and 50 mph. The speedometer needle will fluctuate between 40 and 50, and let’s say you average 45 mph. As the needle moves from 40 to 50, it has to pass this point at least once. I picked 45 mph arbitrarily, but you could pick any number between 40 and 50 (i.e. the “closed interval”) and the needle would have to pass that point.

Mean Value Theorem Example Problem

Example problem: Find a value of c for f(x) = 1 + 3√(x – 1) on the interval [2,9] that satisfies the mean value theorem.

Note: The following steps will only work if your function is both continuous and differentiable.

Step 1: Find the derivative. This is where knowing your derivative rules come in handy. You can find the derivative for this particular function using the chain rule.
average value of a function



Step 2: Place your answer from Step 1 into the formula in the f′ (c) position:
mean value theorem derivative

Step 3: Plug in the two boundaries (from the question: 2 and 9) into the formula. “b” is the highest value on the number line, and “a” is the smallest value. Make sure you put those values in the numerator and the denominator:
mean-value-theorem-2


Step 4: Work the right side of the equation. For f(2) and f(9), you solve by plugging each value, 2 and 9, into the formula from the question.
In other words, solve f(x) = 1 + 3√(2 – 1) and f(x) = 1 + 3√(9 – 1). I used Google’s calculator, which solves cubed roots if you type in the words (i.e. 1 + the cubed root of (2 – 1)).
mean-value-theorem-3-300x77

Step 5: Solve for x. Solid algebra skills will probably come into play for most questions:
average value function

  1. Multiply by 3 and rewrite as 23 power.
    mean-value-theorem-algebra-2
  2. Multiply by (x – 1)23.
    function average value
  3. Multiply by 73.
    mvt
  4. Raise to 32.
    mean-value-theorem-algebra-5
  5. Add 1.
    mean-value-theorem-algebra-6-150x114

The solution is:
mean-value-theorem-algebra-7

Note: If you have two solutions (e.g. a positive and negative value), check your interval to make sure both values are in the interval. In some cases, only one of the values will be in the interval, giving you just one solution.

Extended Mean Value Theorem

There are several “extended mean value theorems”. Most authors, when referencing the EMVT are usually referring to Cauchy’s (Extended) Mean Value Theorem although in some cases they are referring to the mean value theorem for integrals or even Taylor’s theorem.

The “Classical” Mean Value Theorem states:

For a continuous function g defined on the closed interval [a, b] and differentiable on the open interval (a, b), there is at least one number c in (a, b) for which g′(c) = (g(b) – g(a)) / (b – a).

New extensions of the MVT are being developed all of the time, for a wide variety of applications including for one-sided differentiable functions [1] and holomorphic Functions [2].

Cauchy’s Mean Value Theorem

Cauchy’s (extended) mean value theorem is a generalization of the classical MVT. It is usually stated as follows: If functions f and g are continuous of the closed interval [a, b] and differentiable on the open interval (a, b), then there is at least one point c ∈ (a, b) such that:
cauchy's extended mean value theorem

As long as g′ (x) ≠ 0.

Geometrically, the theorem is telling us that there is a value c in the open interval (a, b) for which the tangent line to the curve at (f(c), g(c)) is parallel to the line that connects the two endpoints:
cauchy extended mean value theorem graphic

Other Extended Mean Value Theorem / Special Cases

  • Rolle’s theorem: A special case of the MVT, when f(a) = f(b)
  • The mean value theorem for integrals: states that somewhere under the curve of a function, there is a rectangle with an area equal to the whole area under a curve.
  • Taylor’s Theorem: Although some authors refer to this as an extension of the MVT [3], it could be viewed more like an application. The MVT can be used to prove the a generalized Taylor’s theorem (with Lagrange form of the remainder term) [4] or deduce Taylor’s theorem in one variable [5].

Extended Mean Value Theorem: References

[1] Slota, D. et al. (2012). Mean Value Theorems for One-Sided Differentiable Functions. PDF: http://www.math.put.poznan.pl/artykuly/FM48(2012)-WitulaR-HetmaniokE-SlotaD.pdf
[2] Cakmak, D. & Tiryaki, A. Mean Value Theorem for Holomorphic Functions. Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 34, pp. 1–6.
ISSN: 1072-6691Retrieved May 6, 2021 from: https://ejde.math.txstate.edu/Volumes/2012/34/cakmak.pdf
[3] Robbin J. (2000). Taylor’s Formula (The Extended Mean Value Theorem). Retrieved May 6, 2021 from: https://people.math.wisc.edu/~robbin/221dir/taylor.pdf
[4] Yunsen, C. Mean value theorem. Retrieved May 6, 2021 from: https://www.academia.edu/23708876/Mean_value_theorem
[5] Pinkham, H. (2014). Analysis, Convexity, and Optimization. Lecture 4: Taylor’s Theorem in One Variable. Retrieved May 6, 2021 from: https://www.math.columbia.edu/~pinkham/Optimizationbook.pdf

Mean Value Theorem for Integrals

The mean value theorem for integrals states that somewhere under the curve of a function, there is a rectangle with an area equal to the whole area under a curve.
mvt integral

Usually, questions concerning the mean value theorem for integrals ask you to find a value for c. To answer that, you need the formal definition of the theorem.

Formal Definition of the Mean Value Theorem for Integrals

A continuous function f on a closed, bounded interval [a, b] has at least one number c in the interval (a, b) for which [1]:
theorem mvt integrals

In English, it’s saying that the definite integral from a to b is going to be equal, at some point, to a rectangle. The area of the rectangle is found with the equation f(c)*(a – b), which is the function value at c multiplied by the interval length.

Let’s look at an example of how you can use the formal definition to find a value for c.

Example: How to Find Point “c”

Example question: Given the function f(x) = x(1 – x), what value of c satisfies the MVT for integrals on [0, 1]?

Step 1: Find the indefinite integral
example step 1

Step 2: Add the bounds of integration to your answer from Step 1. We’re looking for “c” on the interval [0, 1], so:
example step 2

Step 3: Set the formula from Step 3 equal to your original function, replacing all x’s in the function with c’s.
example step 3

We’re doing this because we’re looking for one specific point in the function that’s equal to the definite integral on the given bounds.

Step 4: Solve the right side of the equation (Step 3) for the integral bounds:
example step 4

Step 5: Expand the left side of the equation (Step 3) and then set this equal to Step 4:
c – c2 = 1/6

Step 6: Solve for c:
solution to example mvt
For brevity, I skipped writing down all of the algebra steps of solving for c, but you can find the steps here on Symbolab.

Mean Value Theorem for Integrals: References

[1] Larson, R. & Edwards, B. (2016). Calculus. Cengage Learning.


What is Rolle’s Theorem?

Rolle’s theorem is a special case of the mean value theorem. It states that for any continuous, differentiable function with two equal values at two distinct points, the function must have a point where the first derivative is zero.

Theorem in Graphical Terms

rolle'stheorem-150x150
What this means is:

  1. Take any interval on the x-axis (for example, -10 to 10). Make sure two of your function values are equal.
  2. Draw a line from the beginning of the interval to the end. It doesn’t matter if the line is curved, straight or a squiggle—somewhere along that line you’re going to have a horizontal tangent line where the derivative, (f′) is zero. Try it!

Rolle’s Theorem in Math Terms

The standard version of Rolle’s Theorem goes like this: Let’s say you have a function f with the following characteristics:

Then there is some c, with a ≤ c ≤ b such that f′(c) = 0.

This is illustrated by the diagram below, which shows a real-valued, continuous function on a closed interval. According to our theorem, the function equals itself at the two endpoints of the interval means that there is a point where the tangent line is horizontal.

Rolle's Theorem

Note that Rolle’s lemma tells us that there is a point with a derivative of zero, but it doesn’t tell us where it is. It doesn’t give us a method of finding that point either. Still, this theorem is important in calculus because it is used to prove the mean-value theorem.

How to use Rolle’s Theorem

Example question: Use Rolle’s theorem for the following function:
f(x) = x2 – 5x + 4 for x-values [1, 4]

The function f(x) = x2 – 5x + 4 [1, 4]. Graph generated with the HRW graphing calculator.


Step 1: Find out if the function is continuous. You can only use Rolle’s theorem for continuous functions.

This function f(x) = x2 – 5x + 4 is a polynomial function. Polynomials are continuous for all values of x. (How to check for continuity of a function).

Step 2: Figure out if the function is differentiable. If it isn’t differentiable, you can’t use Rolle’s theorem. the easiest way to figure out if the function is differentiable is to simply take the derivative. If you can take the derivative, then it’s differentiable.
f′(x) = 2x – 5

Step 3: Check that the derivative is continuous, using the same rules you used for Step 1.

f′(x) = 2x – 5 is a continuous function.

If the derivative function isn’t continuous, you can’t use Rolle’s theorem.

Step 4: Plug the given x-values into the given formula to check that the two points are the same height (if they aren’t, then Rolle’s does not apply).

  • f(1) = 12 -5(1) + 4 = 0
  • f(4) = 42 -5(4) + 4 = 0

Both points f(1) and f(4) are the same height, so Rolle’s applies.

Step 5: Set the first derivative formula (from Step 2) to zero in order to find out where the function’s slope is zero.

  • 0 = 2x – 5
  • 5 = 2x
  • x = 2.5

The function’s slope is zero at x = 2.5.

That’s it!

History

Rolle’s theorem has a long history: we have reason to believe it was known by Indian mathematician Bhaskara II, who lived between 1114-1185. It is named after Michel Rolle, who published a proof of the polynomial case in 1691. The name was first used in 1834, by mathematician and philosopher Moritz Wilhelm Drobisch.

References

Ghosh, J. (2004). How to Learn Calculus of One Variable. New Age International.
Hosch, Wiliam L. Rolle’s. Encyclopædia Britannica
Publisher: Encyclopædia Britannica, inc. Date Published: August 05, 2011
Retrieved from https://www.britannica.com/science/Rolles-theorem
on April 04, 2019

CITE THIS AS:
Stephanie Glen. "Mean Value Theorem & Rolle’s Theorem" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/calculus-problem-solving/intermediate-value-theorem/mean-value-theorem/
------------------------------------------------------------------------------

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!