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# Mean Value Theorem Calculator

**Mean Value Theorem Calculator** calculates the rate of change for the given function. The average rate of change function describes the average rate at which one quantity is changing with respect to another.

## What is Mean Value Theorem Calculator?

Mean Value Theorem Calculator is an online tool that helps to calculate the rate of change for the given function. This online mean value theorem calculator helps you to calculate the rate of change in a few seconds. To use this mean value theorem calculator

### Mean Value Theorem Calculator

**NOTE:** Enter the function in terms of x.

## How to Use Mean Value Theorem Calculator?

Please follow the steps below to find the rate of change using an online mean value theorem calculator:

**Step 1:**Go to Cuemath’s online mean value theorem calculator.**Step 2:**Enter the function in terms of x in the given input box of the mean value theorem calculator.**Step 3:**Enter the values of 'a' and 'b' in the given input box of the mean value theorem calculator.**Step 4:**Click on the**"Calculate"**button to calculate the rate of change for the given function.**Step 5:**Click on the**"Reset"**button to clear the fields and enter the new values.

## How Mean Value Theorem Calculator Works?

The mean value theorem formula tells us about a point c that must exist in a function if it follows the following conditions:

Let f(x) be a function defined on [a, b] such that

- It is continuous on [a, b].
- It is differentiable on (a, b).

Then there exists a real number c∈(a, b). The formula to calculate the rate of change is given by

f′(c) = f(b) - f(a) / (b - a)

Let us understand this with the help of the following example.

**Solved Examples on Mean Value Theorem Calculator**

**Example 1:**

Calculate the rate of change(c) for the function, f(x) = x ^{2} - 5x + 7 for the interval [-1, 3] and verify it using the mean value theorem calculator?

**Solution**:

Given: Function f(x) = x ^{2} - 5x + 7, a = -1 and b = 3

f(a) = f(-1) = (-1)^{2} - 5(-1) + 7 = 13

f(b) = f(3) = (3)^{2} - 5(3) +7 = 1

f '(x) = 2x - 5

f '(c)= [f(b) - f(a)] / (b - a)

2c - 5 = (1 - 13) / (3 + 1)

2c - 5 = -3

2c = 2

c = 1, it lies between (-1,3)

**Example 2:**

Calculate the rate of change(c) for the function, f(x) = 3x^{2} + 8x - 11 for the interval [2, 4] and verify it using the mean value theorem calculator?

**Solution**:

Given: Function f(x) = 3x^{2} + 8x - 11, a = 2 and b = 4

f(a) = f(2) = 3(2)^{2} + 8(2) - 11 = 17

f(b) = f(4) = 3(4)^{2} + 8(4) - 11 = 69

f '(x) = 6x + 8

f '(c) = [f(b) - f(a)] / (b - a)

6c + 8 = (69 - 17) / (4 - 2)

6c + 8 = 26

6c = 18

c = 3

c = 3, it lies between (2,4)

Now, try the mean value theorem and find the rate of change for:

- x
^{3}+ 6x + 5, for the interval [-1,5] - 2x
^{2}+ 2x + 4, for the interval [2, 5]

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