Calculus How To

Mean Value Theorem & Rolle’s Theorem

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) 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.

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 54 mph. As the needle moves from 40 to 50, it has to pass this point at least once. I picked 54 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-21-300x69

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 interval [2,9] is positive, so negative values aren’t part of the solution. The average value is:
mean-value-theorem-algebra-7

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

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