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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 c that satisfies the Mean Value Theorem, with step-by-step calculations shown.

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Mean Value Theorem Calculator

Use this online Mean Value Theorem (MVT) calculator to find the point c in the interval [a,b] for a function f(x) that meets the conditions of continuity on [a,b] and differentiability on (a,b).

What Is the Mean Value Theorem?

The Mean Value Theorem describes the behavior of a function on a closed interval. If a function f is continuous on [a,b] and differentiable on (a,b), there is at least one point c in (a,b) such that the instantaneous rate of change equals the average rate of change:

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

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

If a function f(x) is continuous on [a,b], there exists a point c 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:

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

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

This generalizes the MVT. If f and g are continuous on [a,b] and differentiable on (a,b), there exists a c in (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 a special case of the MVT when:

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: Input a single-variable function, e.g., f(x).
Select The Variable: Choose the independent variable, usually 'x'.
Define The Interval: Enter the start and end points [a,b].
Click Calculate: The calculator identifies the point c that satisfies the Mean Value Theorem.

FAQs

Who developed the Mean Value Theorem?

Rolle's Theorem: Michel Rolle, 1691 (polynomial case)
General MVT: Augustin Cauchy, 1823

What is the First Mean Value Theorem?

It allows evaluating a function’s increment on an interval by relating it to the derivative at some intermediate point.

References

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