**Mean Value Theorem Calculator** is a useful online tool that you can use to calculate the rate of change of a function using the **Mean Value Theorem** (MVT). This calculator & grapher, as shown below, can conveniently carry out your calculation in a swift and accurate manner.

It can also produce the derivative of the given function in just a second or two.

* *Click on the button above to launch the Mean Value Theorem calculator and grapher. If the button doesn't appear, if you are using an incompatible browser or your browser doesn't have proper Javascript support. Try opening this page in Google Chrome or any other modern browser. Should work well on mobile devices too.

Note: $x_{min}$ and $x_{max}$ are just graph plotting ranges. These don't have anything to do with the Mean Value Theorem. Increase or decrease the values to increase the graph size.

## What is the Mean Value Theorem?

The mean value theorem is one of the most important concepts in calculus, the mathematical study of continuous change. It is widely used to learn about the behavior of a particular function.

According to the mean value theorem:

If **f** is a continuous function, which is closed on the interval **[a,b]** and is differentiable on the open interval **(a,b)**, then** there will be a point** **c** in the open interval (a,b), where the differential of **f** at point **c**, will be defined by f'(c) by this formula:

$$ f'(c) = \frac{f(b)-f(a)}{b-a}$$

**A more technical definition:** If $f(x)$ is defined and continuous on the interval $[a, b]$ and differentiable on $(a, b)$, then there is at least one number $c$ in the interval $(a, b)$ (that is, $a < c < b$ ) such that $$f'(c) = \frac{f(b)-f(a)}{b-a}$$

### How can this calculator be used?

My mean value theorem calculator is very user-friendly and simple to use. You can carry out a calculation by following the steps given in this article.

### What are the Benefits of using the Mean Value Theorem Calculator?

This Mean Value Theorem calculator is free, fast and easy to use.

## How to use the Mean Value Theorem Calculator?

This mean value theorem calculator is very user-friendly and simple to use. You can carry out a calculation by following the steps given below:

- Type in the function and limits in the input field. Click on the “Submit” button when you are done.
- You should be able to see the value displayed on the screen. Soon, a new window will pop up automatically and the rate of change of the function, calculated using the mean value theorem, will be shown there.

## Benefits of Using the Mean Value Theorem Calculator

This Mean Value Theorem calculator has some notable advantages, which have been listed below:

**It is easily available**. You can access it anytime, anywhere, without having to pay anything at all.**It is fast and accurate**. The calculator will carry out the calculation for you in a few seconds, and give you a reliable and accurate result. You will not even see the actual mathematical operation being performed, but you can cross-check it for accuracy and see for yourself.**It saves you a lot of time**. Traditional mathematicians understandably frown upon the use of calculators of any sort. However, it is also true that advanced mathematical calculations can be lengthy, time-consuming, and prone to human errors. In today’s busy lifestyle, we do not always have so much time in hand. In such a situation, a mean value theorem is an excellent tool for getting accurate results in a very short span of time.

For professional mathematicians, students, and teachers alike, the mean value theorem calculator is a valuable tool and highly recommended.

The mean value theorem has a special case known as Rolle's Theorem.

Let's try to understand Rolle's Theorem.

## Rolle's Theorem

As I said earlier that Rolle's theorem is an extension of Mean Value Theorem. Let's state the Mean Value Theorem again:

If $f(x)$ is defined and continuous on the interval $[a, b]$ and differentiable on $(a, b)$, then there is at least one number $c$ in the interval $(a, b)$ (that is, $a < c < b$ ) such that $$f'(c) = \frac{f(b)-f(a)}{b-a}$$

What if $f(a)=f(b)$, with $a < b$ in above example?

Then $$ f'(c) = \frac{f(b)-f(a)}{b-a} = 0$$

This is **Rolle's Theorem.**

A more technical statement of Rolle's theorem is this:

If $f(x)$ is defined and continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ such that $f(a) = f(b)$, then $f'(c) = 0$ for some c with $ a \leq c \leq b$.

**Did you know: **This theorem was already stated by the Indian Mathematician Bhaskara II in 12th century without any formal proof. Michel Rolle (1691) proved it with proper details around 500 years later.

You can use above calculator as **Rolle's theorem calculator** as well. You will need just to enter **the values of a and b for which f(a) = f(b)**. Such values are provided in your textbook questions.

Don't be confused with Intermediate Value Theorem.

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