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An online mean value theorem calculator helps you to find the rate of change of the function using the mean value theorem. Also, this Rolle’s Theorem calculator displays the derivation of the intervals of a given function. In this context, you can understand the mean value theorem and its special case which is known as Rolle’s Theorem.

In mathematics, the mean value theorem is used to evaluate the behavior of a function. The mean value theorem asserts that if the f is a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), then there is at least one point c on the open interval (a, b), then the mean value theorem formula is:

$$f’ (c) = [f(b) – f (a)] / b – a$$

The mean value theorem for integral states that the slope of a line consolidates at two different points on a curve (smooth) will be the very same as the slope of the tangent line to the curve at a specific point between the two individual points.

Let f be the function on [a, b]. Then the average f (c) of c is

$$1/ b – a∫_a^b f(x) d(x) = f (c)$$

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However, an Online Integral Calculator helps you to evaluate the integrals of the functions with respect to the variable involved.

**Example:**

Find the value of f (x)=11x^2 – 6x – 3 on the interval [4,8].

**Solution:**

In the given equation (f) is continuous on [4, 8].

$$F (C) = 1/b – a ∫ f(x) dx = 1/ 8 – 4∫_4^8 (11x^2 – 6x – 3) dx$$

$$= 1/4 [x^3 – x^2]^8_4$$

$$= 1/4 [(216 – 36) – (8 – 4)]$$

$$= 1/4 [(180 – 4)]$$

$$= 176/4 = 44$$

Here the value of c is 44 that provides the average value of the given function.

Now put x=16 in the function.

$$f(x)=11x^2 – 6x – 3 = 44$$

$$=11x^2 – 6x – 47$$

$$=(x + 2.32)(x – 2.80)=0$$

Hence 2.80 is the value of c. The online mean value theorem calculator gives the same results when you plug in the similar values and intervals in it.

Cauchy’s mean value theorem is the generalization of the mean value theorem. It states: if the function g and f both are continuous on the end interval [a, b] and differentiable on the start interval (a, b), then there exists c e(a, b), such that

$$(f (b) – f (a)) g’c = (g(b) – g (a)) f’c$$

Here’s g (a) ≠ g (b) and g’ (c) ≠0, so this is equivalent to:

$$f’(c) / g’(c) = f(b) – f(a)/ g(b) – g(a)$$

However, an Online Derivative Calculator helps to find the derivative of the function with respect to a given variable.

**Example:**

Find a value of “C” that is the conclusion of the mean value theorem:

f(x) = -4x^3 + 6x – 2 on the interval [-4 , 2].

**Solution:**

f(x) is a polynomial function and is differentiable for all real numbers.

Let evalute f(x) at x = -4 and x = 2

$$f(-4) = -4(-4)^3 + 6(-4) – 2 = 20$$

$$f(2) = -4(2)^3 + 6(2) – 2 = – 4$$

Now, substitute the values in [f(b) – f(a)] / (b – a)

$$[f(b) – f(a)] / (b – a) = [-6 – 4] / (2 – (-4)) = -2$$

Let us now find f ‘(x)

$$f ‘(x) = – 6x^2 + 6$$

We now create an equation, which is based on f ‘(c) = [f(b) – f(a)] / (b – a)

$$-6c + 6 = -2$$

You can find the value of c by using the mean value theorem calculator:

$$c = 2 \sqrt{(1/3)} and c = – 2 \sqrt{(1/3)}$$

Rolle’s theorem says that if the results of a differentiable function (f) are equal at the endpoint of an interval, then there must be a point c where f ’(c)=0.

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**Example:**

Find all values of point c in the interval [−4,0]such that f′(c)=0.Where f(x)=x^2+2x.

**Solution:**

First of all, check the function f(x) that satisfies all the states of Rolle’s theorem.

- f(x) is continuous function in [−4,0] as the quadratic function;
- It is differentiable over the start interval (−4,0);

$$f(−2)=(−4)2+2⋅(−4)=0$$

$$f(0)=02+2⋅0=0$$

$$f(−4)=f(0)$$

So we can use Rolle’s theorem calculator to find the point c

$$f′(x)=(x2+2x)′=2x+2$$

Now, solve the equation f′(c)=0:

$$f′(c)=2c+2=0$$

$$c=−1$$

Thus,

$$f′(c)=0 for c=−1$$

This free Rolle’s Theorem calculator can be used to compute the rate of change of a function with a theorem by upcoming steps:

- First, enter a function for different variables such as x, y, z.
- Now, enter start and end intervals of the continuous function
- Click on the calculate button to see the results

- The mean value theorem calculator provides the answer
- Displays the derivation of entered functions

A restricted form of the mean value theorem was proved by M Rolle in the year 1691; the outcome was what is now known as Rolle’s theorem, and was proved for polynomials, without the methods of calculus. The mean value theorem in its latest form which was proved by Augustin Cauchy in the year of 1823.

f(b)−f(a) = f′(c)(b−a). This theorem is also known as the First Mean Value Theorem that allows showing the increment of a given function (f) on a specific interval through the value of a derivative at an intermediate point.

Use this handy mean value theorem calculator that allows you to find the rate of change of a function, if f is continuous on the closed interval and differentiable on the open interval, then there exists a point c in the interval. The mean value theorem formula is difficult to remember but you can use our free online rolles’s theorem calculator that gives you 100% accurate results in a fraction of a second.

From the source of Wikipedia: Cauchy’s mean value theorem, Proof of Cauchy’s mean value theorem, Mean value theorem in several variables.

From the source of Pauls Online Notes: The Mean Value Theorem, Rolle’s Theorem, Proofs From Derivative Applications.

From the source of Calc Workshop: Mean Value Theorem for Integrals, Average Value, Mean Value Theorem.