**Mean Value Theorem Calculator + Online Solver With Free Steps **

The** Mean Value Theorem Calculator **is an online calculator that helps to calculate the value which is recognized as the **critical point c**. This critical point c is the instant where the average rate of change of the function becomes equal to the instantaneous rate.

The** Mean Value Theorem Calculator** helps to find the find c in any interval [a, b] for a function f(x), where the secant line becomes parallel to the tangent line. Note that there must only be one value of c within the specified interval a and b.

The **Mean Value Theorem Calculator** is only applicable to solve for those functions f(x) in which f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b).

## What Is the Mean Value Theorem Calculator?

**The Mean Value Theorem Calculator is a free online calculator that helps the user to determine the critical point c where the instantaneous rate of any function f(x) becomes equal to its average rate. **

In other words, this calculator assists the user in figuring out the point where the secant line and tangent line of any function f(x) become** parallel** to each other within a specified interval [a,b]. One essential thing to note is that within each interval, only one critical point c can exist.

The **Mean Value Theorem Calculator **is an effective calculator that provides accurate answers and solutions in a matter of seconds. This type of calculator applies to all kinds of functions and all kinds of intervals.

Although the **Mean Value Theorem Calculator **provides swift answers for all kinds of functions and intervals, due to certain mathematical conditions of the theorem, some limitations are also applied to the use of this calculator. The** Mean Value Theorem Calculator **can only solve for those functions f(x) which adhere to the following conditions:

**f(x) is continuous on the closed interval [a,b].**

**f(x) is differentiable on the open interval (a,b).**

If these two conditions are met by the function f(x), then the Mean Value Theorem can be applied to the function. Similarly, only for such functions, the Mean Value Theorem Calculator can be used.

The Mean Value Theorem Calculator makes use of the following formula for calculating the critical point c:

\[ f’(c) = \frac{f(b) – f(a)} {b – a} \]

## How To Use Mean Value Theorem Calculator?

You can start using the** Mean Value Theorem Calculator** for finding the mean value of a function by entering the derivative of a function and the upper and lower limits of the function. It is fairly easy to use due to its simple and user-friendly interface. The calculator is extremely efficient and reliable as it provides accurate and precise results in just a few seconds.

The interface of the calculator consists of three input boxes. The first input box prompts the user to enter the desired function for which they need to calculate the critical point c.

The second input box prompts the user to enter the starting value of the interval, and similarly, the third input box prompts the user to insert the ending value of the interval. Once these values are inserted, the user simply needs to click the “**Submit”** button to get the solution.

The **Mean Value Theorem Calculator **is the best online tool for calculating the critical points c for any function. A detailed step-by-step guide for using this calculator is given below:

### Step 1

Choose the function for which you wish to calculate the critical point. There are no restrictions in the selection of the function. Also, analyze the interval for the selected function f'(x).

### Step 2

Once you have selected your function f(x) and your interval [a,b], insert the derivative function f'(x) and the values of the interval in the designated input boxes.

### Step 3

Review your function and your interval. Make sure that your function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

### Step 4

Now that you have entered and analyzed all the values, simply click on the **Submit** button. The Submit button will trigger the **Mean Value Theorem Calculator **and in a matter of seconds, you will get the solution for your function f(x).

## How Does the Mean Value Theorem Calculator Work?

The **Mean Value Theorem Calculator** works by calculating the critical point c for any given function f(x) under any specified interval [a,b].

To understand the workings of the **Mean Value Theorem Calculator**, we first need to develop an understanding of the Mean Value Theorem.

### Mean Value Theorem

The Mean Value Theorem is used to determine a single point c in any interval [a, b] for any specified function f(x), provided that the function f(x) is differentiable on the open interval and continuous on the closed interval.

The Mean Value Theorem formula is given below:

\[ f’(c) = \frac{f(b) – f(a)} {b – a} \]

The Mean Value Theorem also sets the basis of the renowned Rolle’s Theorem.

## Solved Examples

The** Mean Value Theorem Calculator** is ideal for providing accurate and quick solutions to any type of function. Given below are a few examples for using this calculator that will help you to develop a better understanding of the **Mean Value Theorem Calculator.**

### Example 1

Find the value of c for the following function in the interval [1, 4]. The function is given below:

\[ f(x) = x^{2} + 1 \]

### Solution

First, we need to analyze the function to evaluate whether the function obeys the conditions for the Mean Value Theorem.

The function is given below:

\[ f(x) = x^{2} + 1 \]

Upon analyzing the function, it is evident that the given function is polynomial. Since the function f(x) is a polynomial function, it follows both the conditions of the Mean Value Theorem under the given interval.

We can now use the Mean Value Theorem Calculator to determine the value of c.

Insert the value of the function f(x) in the input box and the values of the interval [1,4] in their respective input boxes. Now click on Submit.

Upon clicking on Submit, the calculator provides the solution for the value of $c$ for the function f(x). The Mean Value Theorem Calculator performs the solution by following the formula given below:

\[ f’(c) = \frac{f(b) – f(a)} {b – a} \]

The solution for this function f(x) in the interval [1,4] is:

**c = 2.5 **

Thus, the critical point for the function f(x) is 2.5 under the interval [1,4].

### Example 2

For the function given below, determine the value of c for the interval [-2, 2]. The function is:

\[ f(x) = 3x^{2} + 2x – 1 \]

### Solution

Before using the Mean Value Theorem Calculator, determine if the function obeys all the conditions of the Mean Value Theorem. The function is given below:

\[ f(x) = 3x^{2} + 2x – 1\]

Since the function is polynomial, this means that the function is continuous as well as differentiable on the interval [-2, 2]. This satisfies the conditions for the Mean Value Theorem.

Next, simply insert the values of the function f(x) and the values of the interval [2, -2] in their destined input boxes. After you have entered these values, click on the button labelled Submit.

The Mean Value Theorem Calculator will instantly provide you with the solution for the value of c. This calculator makes use of the following formula for determining the value of c:

\[ f’(c) = \frac{f(b) – f(a)} {b – a} \]

The solution for the given function and the given interval turns out to be:

**c = 0.0**

Hence, the critical point for the function f(x) under the interval [-2.2] is 0.0.

### Example 3

Determine the value of c on the interval [-1, 2] for the following function:

\[ f(x) = x^{3} + 2x^{2} – x \]

### Solution

To find the value of the critical point c, first, determine if the function obeys all the conditions of the Mean Value Theorem. Since the function is polynomial, it obeys both the conditions.

Insert the values of the function f(x) and the values of the interval [a,b] in the input boxes of the calculator and click on Submit.

Upon clicking on Submit, the Mean Value Theorem Calculator makes use of the following formula for calculating the critical point c:

\[ f’(c) = \frac{f(b) – f(a)} {b – a} \]

The answer for the given function f(x) turns out to be:

**c = 0.7863**

Hence, the critical point for the function f(x) in the interval [-1,2] is 0.7863.

### Example 4

For the following function, find out the value of c that satisfies the interval [1,4]. The function is given below:

\[ f(x) = x^{2} + 2x \]

### Solution

Before using the calculator, we need to determine if the given function f(x) satisfies the conditions of the Mean Value Theorem.

Upon analyzing the function f(x), it appears that the function is a polynomial. Hence, this means that the function is continuous and differentiable on the given interval [1,4].

Now that the function has been verified, insert the function f(x) and the values of the interval into the calculator and click on Submit.

The calculator makes use of the Mean Value Theorem formula to solve for the value of $c$. The formula is given below:

\[ f’(c) = \frac{f(b) – f(a)} {b – a} \]

The answer turns out to be:

**c= 0.0**

Hence, for the function f(x) under the interval [1,4], the value of c is 0.0.