Mean Value Theorem (MVT)

Welcome to KnowledgeKnot! In this article, we will learn about the Mean Value Theorem (MVT) in calculus, its statement, conditions, geometric interpretation, and applications with solved examples.

What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental result in calculus that describes the behavior of functions over a closed interval. It states:

If a function $f(x)$ satisfies the following conditions:

The function $f(x)$ is continuous on the closed interval $[a, b]$ .
The function $f(x)$ is differentiable on the open interval $(a, b)$ .

Then, there exists at least one point $c \in (a, b)$ such that:

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

Here, the slope of the tangent at $c$ is equal to the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$ .

Geometric Interpretation

The Mean Value Theorem implies that for a continuous and differentiable curve, there exists at least one point where the tangent to the curve is parallel to the secant line joining the endpoints of the interval.

In other words, there is a point where the instantaneous rate of change (slope of the tangent) matches the average rate of change (slope of the secant).

Conditions for the Mean Value Theorem

For the Mean Value Theorem to hold, the function must satisfy:

Continuity on the closed interval $[a, b]$ .
Differentiability on the open interval $(a, b)$ .

If these conditions are not met, the theorem does not apply.

Example Problems

Example 1: Polynomial Function

Verify the Mean Value Theorem for $f(x) = x^2$ on the interval $[1, 3]$ .

Solution:

Step 1: Check the conditions:

Continuity: The function $f(x) = x^2$ is a polynomial, so it is continuous everywhere.
Differentiability: Polynomials are differentiable everywhere.

Step 2: Calculate the slope of the secant line:

\text{slope} = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4

Step 3: Differentiate $f(x)$ :

f'(x) = 2x

Step 4: Solve $f'(c) = 4$ to find $c$ :

2c = 4 \implies c = 2

Since $c = 2 \in (1, 3)$ , the Mean Value Theorem is verified.

Example 2: Trigonometric Function

Verify the Mean Value Theorem for $f(x) = \sin(x)$ on $[0, \pi]$ .

Solution:

Step 1: Check the conditions:

Continuity: $\sin(x)$ is continuous everywhere.
Differentiability: $\sin(x)$ is differentiable everywhere.

Step 2: Calculate the slope of the secant line:

\text{slope} = \frac{f(\pi) - f(0)}{\pi - 0} = \frac{0 - 0}{\pi} = 0

Step 3: Differentiate $f(x)$ :

f'(x) = \cos(x)

Step 4: Solve $f'(c) = 0$ to find $c$ :

\cos(c) = 0 \implies c = \frac{\pi}{2}

Since $c = \frac{\pi}{2} \in (0, \pi)$ , the Mean Value Theorem is verified.

Applications of the Mean Value Theorem

Helps in analyzing the behavior of functions and curves.
Forms the basis for advanced results in calculus, such as Taylor's theorem.
Validates physical phenomena like velocity and acceleration relations.

Practice Problem

Problem:

Verify the Mean Value Theorem for $f(x) = x^3 - 3x^2 + 4$ on $[1, 4]$ .

Hint:

Follow the steps: check continuity, differentiability, and calculate the secant slope. Then find $f'(x)$ and solve $f'(c) = \text{slope}$ .