# Mean Value Theorem Calculator

## Apply the mean value theorem step by step

The calculator will find all numbers $$$c$$$ (with steps shown) that satisfy the conclusions of the mean value theorem for the given function on the given interval. Rolle's theorem is a special case of the mean value theorem (when $$$f(a)=f(b)$$$).

The online Mean Value Theorem Calculator allows you to find values where the instantaneous rate of change of a function is equal to the average rate of change of that function. Our calculator is a powerful tool for applying the MVT. With step-by-step solutions, it helps to deepen the knowledge of this concept.

## How to Use the Mean Value Theorem Calculator?

### Input

Enter or paste a function into the appropriate field. Enter the values of $$$a$$$ and $$$b$$$ to define the closed interval $$$[a, b]$$$ to which you want to apply the theorem.

### Calculation

After inputting all the data, click the "Calculate" button.

### Result

The calculator will find the values of $$$c$$$ from the specified interval that satisfy the conditions of the theorem.

## What is the Mean Value Theorem?

The Mean Value Theorem (MVT) is a fundamental concept in calculus that establishes a relationship between the average rate of change of a function over a certain interval and its instantaneous rate of change at some point from that interval.

**Formulation of MVT**

Formally stated, if a function $$$f(x)$$$ meets the following conditions:

- $$$f(x)$$$ is continuous on the closed interval $$$[a,b]$$$,
- $$$f(x)$$$ is differentiable on the open interval $$$(a,b)$$$.

Then there exists at least one number $$$c$$$ in the open interval $$$(a,b)$$$ such that

$$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$$Here:

- $$$f^{\prime}(c)$$$ is the derivative of the function $$$f(x)$$$ at the point $$$x=c$$$, representing the instantaneous rate of change of the function at that point.
- The fraction on the right-hand side, i.e. $$$\frac{f(b)-f(a)}{b-a}$$$, represents the average rate of change of $$$f(x)$$$ over the interval $$$[a,b]$$$.

The Mean Value Theorem says that if you drive from point A to point B, there is at least one moment when your instantaneous speed (speed at a particular moment) is equal to your average speed for the entire journey.

The importance of the mean value theorem lies in its ability to relate average rates of change to instantaneous rates of change, offering insight into the behavior of functions over intervals.

**Example of Applying MVT**

**Function:** Let's consider the function $$$f(x)=x^2$$$.

**Interval:** We'll applly the theorem on the interval $$$[1,2]$$$.

$$\frac{f(2)-f(1)}{2-1}=\frac{2^2-1^2}{2-1}=3$$**Compute the Average Rate of Change:**Thus, the average rate of change of $$$f(x)=x^2$$$ over the interval $$$[1,2]$$$ is $$$3$$$.

$$f^{\prime}(x)=2x$$**Differentiate the Function:**

$$f^{\prime}(c)=3$$$$2c=3$$$$c=\frac{3}{2}=1.5$$**Find $$$c$$$:**

**Conclusion:** The instantaneous rate of change of $$$f(x)=x^2$$$ at $$$x=1.5$$$ is the same as the average change of $$$f(x)$$$ over the interval $$$[1,2]$$$.

## Why Choose Our Mean Value Theorem Calculator?

### Precision

Our calculator uses advanced algorithms to deliver accurate results every time, ensuring you can trust them.

### User-Friendly Interface

Designed with the users in mind, the interface is intuitive and easy to navigate.

### Fast Results

Our calculator is optimized for speed, delivering results almost instantly.

### FAQ

#### What is the Mean Value Theorem Calculator?

The Mean Value Theorem Calculator is an online tool designed to determine a specific point (or points) in an interval where the instantaneous rate of change of a function equals its average rate of change over that interval.

#### Does the function have to meet any conditions to use the Mean Value Theorem?

Yes, the function must be continuous on the closed interval $$$[a,b]$$$ and differentiable on the open interval $$$(a,b)$$$.

#### What is $$$c$$$?

$$$c$$$ represents a specific point (or points) within your chosen interval where the function's instantaneous rate of change change is equal its average rate of change over the interval.

#### What types of functions can I input?

Although our calculator is optimized for a wide range of functions, extremely complex or long functions may have a slight delay in calculation or may not be processed.