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Mean Value Theorem Calculator

Enter the function f(x), specify the interval [a,b], and click “Calculate” to determine the point c that satisfies the Mean Value Theorem, with step-by-step calculations shown.

Enter a Function:

W.R.T:

Start Interval:

End Interval:

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Mean Value Theorem Calculator

This online mean value theorem calculator finds the point "c" within the interval [a,b] for the given function f(x) when the function satisfies the condition of continuity on the closed interval [a,b] and differentiability over the open interval (a,b).

What Is the Mean Value Theorem?

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). At this point, the instantaneous rate of change of function is equal to the average rate of change of that function.

The mean value theorem formula is:

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

f'(c) represents the derivative of the function at c.
(f(b) - f(a)) / (b - a) is the average rate of change over the interval [a,b].

Mean Value Theorem for Integrals

The mean value theorem for integral states that if a function f(x) is continuous, then a point “c” exists within the interval [a,b] such that:

\[ f(c) = \frac{1}{b-a} \int_a^b f(x) \, dx \]

You can use an online integral calculator to evaluate definite integrals quickly.

Example:

Find the average value of f(x) = 5x² - 4x + 3 on [1,3].

Solution:

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

\[ F(c) = \frac{1}{3-1} \int_1^3 (5x^2 - 4x + 3) \, dx = \frac{1}{2} \left[ \frac{5x^3}{3} - 2x^2 + 3x \right]_1^3 = 11.5 \]

The corresponding c value satisfies:

\[ 5c^2 - 4c + 3 = 11.5 \quad \Rightarrow \quad 5c^2 - 4c - 8.5 = 0 \]

\[ c \approx 1.76 \]

Cauchy's Mean Value Theorem

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 point “c” e(a, b), such that:

\[ (f(b)-f(a)) g'(c) = (g(b)-g(a)) f'(c) \quad \Rightarrow \quad \frac{f'(c)}{g'(c)} = \frac{f(b)-f(a)}{g(b)-g(a)} \]

Use an online derivative calculator to find derivatives easily.

Example:

Find c for f(x) = x³ - 6x + 2 on [-2,1].

\[ f(-2) = 6, \quad f(1) = -3 \]

\[ \frac{f(1)-f(-2)}{1-(-2)} = \frac{-3-6}{3} = -3 \]

\[ f'(x) = 3x^2 - 6 \quad \Rightarrow \quad 3c^2 - 6 = -3 \quad \Rightarrow \quad c = \pm \sqrt{1} = \pm 1 \]

Rolle's Theorem

Rolle’s theorem is the special case of the mean value theorem. In this case, the mean value theorem is applied but with the additional condition which is:

f(a) = f(b)

Example:

Find c in [-3,1] such that f'(c) = 0, where f(x) = x² + 2x.

Solution:

Check conditions:

f(x) is continuous on [-3,1]
f(x) is differentiable on (-3,1)

\[ f'(x) = 2x + 2, \quad f'(c) = 0 \Rightarrow 2c + 2 = 0 \Rightarrow c = -1 \]

How To Use The Mean Value Theorem Calculator?

Enter The Function: Enter the function of a single variable (e.g., f(x))
Choose the Specific Variable: Choose the independent variable (e.g., 'x') from the “W.R.T” drop-down
Define the Interval: Input start and end intervals of the continuous function [a,b]
Click Calculate: Click on the “Calculate” button to see the results. If the mean value theorem applies within the given interval, then you will see point 'c' within the interval [a, b], which satisfies the mean value theorem

FAQs

Who Proved The Mean Value Theorem?

Rolle's Theorem: A restricted form of the mean value theorem was proved by Michel Rolle in the year 1691, the outcome was what is now known as Rolle's theorem, and was applied to polynomials, developed without the use of calculus
Mean Value Theorem (General Form): The latest form of mean value theorem was proved by Augustin Cauchy in 1823

What is the First Mean Value Theorem?

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

References

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