# Mean values

## Mean values

Mean Value Theorem Basics

• The Mean Value Theorem is a fundamental result in calculus that captures the intuitive notion that a function will always take on its mean value at some point in an interval.
• 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 value `c` in the range `(a, b)` such that the derivative `f'(c)` equals the average rate of change of the function over the interval.
• Formally, the theorem is expressed as: `f'(c) = [f(b) - f(a)] / (b - a)`.
• The value `c` is known as the mean value of the function.

Understanding the Mean Value Theorem

• Geometrically, the Mean Value Theorem states that there is a tangent to the curve of `f(x)` at some point `c` in `(a, b)` that is parallel to the secant line from `(a, f(a))` to `(b, f(b))`.
• This theorem is a generalization of the Intermediate Value Theorem, in the sense that it introduces a derivative to the condition.
• The Mean Value Theorem thereby provides a precise mathematical link between the average rate of change and the instantaneous rate of change of a function.

Using the Mean Value Theorem

• The Mean Value Theorem is applied in various areas of mathematics, including calculus and differential equations, to provide guarantees about a function’s behaviour given certain conditions.
• It’s useful in proving other important mathematical principles like the Fundamental Theorem of Calculus.
• It is often used to estimate function values, to prove inequalities, or to show that a function has at least one root.
• If a function does not verify the Mean Value Theorem, this means that it is either not continuous or not differentiable on the considered interval.

Applications of the Mean Value Theorem

• The Mean Value Theorem forms the foundation of calculus-based concepts such as Taylor’s theorem, which is crucial in sciences and engineering for approximating functions.
• It is also used to prove the Contraction Mapping Theorem in real analysis, which is a key methodology in numerical solving of equations.
• Real-world applications include physics (e.g., motion along a smooth curve), economics (e.g., changes in market trends), and other fields where analysing the mean behaviour of a dynamic system is required.