Understanding the Mean Value Theorem: Calculus’s Key Relation between Average and Instantaneous Rates of Change

Mean Value Theorem

The Mean Value Theorem is an important theorem in calculus that relates the average rate of change of a function to its instantaneous rate of change

The Mean Value Theorem is an important theorem in calculus that relates the average rate of change of a function to its instantaneous rate of change. It states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that the instantaneous rate of change of f(x) at c, which is given by the derivative f'(c), is equal to the average rate of change of f(x) over the interval [a, b], which is given by the difference quotient:

f'(c) = [f(b) – f(a)] / (b – a)

Graphically, the Mean Value Theorem tells us that if we have a function f(x) that is continuous and differentiable on an interval, then there must be a point at which the tangent line to the graph of f(x) is parallel to the secant line connecting the endpoints of the interval.

To illustrate this with an example, let’s consider the function f(x) = x^2 on the interval [0, 4]. First, we check that the function is continuous on this interval, which is true. Next, we check that the function is differentiable on the open interval (0, 4), which is also true. Therefore, we can apply the Mean Value Theorem to this function on the given interval.

The average rate of change of f(x) over [0, 4] is (f(4) – f(0))/(4 – 0) = (16 – 0)/(4 – 0) = 4.

Next, we find the derivative of f(x) using the power rule: f'(x) = 2x.

Now, we want to find a number c in the interval (0, 4) such that f'(c) = 2c is equal to the average rate of change, which is 4. Solving the equation:
2c = 4
c = 2

So, by the Mean Value Theorem, there exists a number c = 2 in the interval (0, 4) such that f'(c) = 2, which is equal to the average rate of change 4.

In summary, the Mean Value Theorem guarantees that at some point c within the interval (a, b), the instantaneous rate of change of a function equals its average rate of change over the interval. This theorem is widely used in calculus to prove other important results and to analyze the behavior of functions.

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