The Mean Value Theorem: Explained with Examples and Application.

Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus that relates the average rate of change of a function to the instantaneous rate of change at some point within an interval

The Mean Value Theorem is a fundamental concept in calculus that relates the average rate of change of a function to the instantaneous rate of change at some point within an interval. It states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the instantaneous rate of change (slope) is equal to the average rate of change over the interval.

Mathematically, the Mean Value Theorem can be stated as follows:

If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) such that:

f'(c) = (f(b) – f(a))/(b – a)

In simpler terms, this means that if we connect the endpoints of the interval with a straight line, the instantaneous rate of change (slope) at some point within the interval will be equal to the slope of this line.

To better understand this theorem, let’s consider an example:

Suppose we have a function f(x) = x^2 on the interval [0, 3].

Step 1: Check for continuity and differentiability:
Since the function x^2 is a polynomial, it is continuous for all real values of x. Additionally, it is also differentiable for all real values of x.

Step 2: Calculate the average rate of change:
Using the formula (f(b) – f(a))/(b – a), we can calculate the average rate of change over the interval [0, 3]:
average rate of change = (f(3) – f(0))/(3 – 0) = (9 – 0)/3 = 3

Step 3: Find the point where the instantaneous rate of change equals the average rate of change:
To determine the point where f'(c) = 3, we need to find the derivative of the function f(x).
f'(x) = 2x

Now we solve the equation f'(c) = 3 for x:
2c = 3
c = 3/2 = 1.5

So, according to the Mean Value Theorem, there exists at least one point c in the interval (0, 3) where the instantaneous rate of change of f(x) equals the average rate of change, which in this case is 3. The point c would have a derivative of 3, indicating that the slope of the tangent line at that point is also 3.

This theorem is particularly useful when we want to find specific values or information about a function based on its average rate of change over an interval. It provides a guarantee that such a point exists, although it does not give any information about other possible points where this equality may hold.

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