Glossary

14.2 The Mean Value Theorem and its applications

3 min read•july 22, 2024

The is a powerful tool in calculus, linking a function's to its . It states that for a continuous, differentiable function on an interval, there's a point where the derivative equals the average rate of change.

This theorem has wide-ranging applications, from proving function properties to establishing inequalities. It's used to show when functions are constant, relate different functions, and find average values. Understanding it is key to grasping deeper calculus concepts.

The Mean Value Theorem

Mean Value Theorem statement

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  • The Mean Value Theorem asserts the existence of a point cc within the open interval (a,b)(a, b) where the derivative of a function ff at cc equals the average rate of change of ff over the closed interval [a,b][a, b], given that ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b)
    • Mathematically expressed as f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
  • Geometrically interprets the theorem by relating the line at point cc to the line connecting the endpoints of the function on the interval [a,b][a, b]
    • The tangent line at cc is parallel to the secant line through (a,f(a))(a, f(a)) and (b,f(b))(b, f(b))

Mean Value Theorem proof

  • Proves the Mean Value Theorem using as an intermediate step
  • Defines an auxiliary function g(x)g(x) as g(x)=f(x)−f(a)−f(b)−f(a)b−a(x−a)g(x) = f(x) - f(a) - \frac{f(b) - f(a)}{b - a}(x - a)
  • Observes that g(a)=g(b)=0g(a) = g(b) = 0, making gg continuous on [a,b][a, b] and differentiable on (a,b)(a, b), satisfying the conditions for Rolle's Theorem
  • Applies Rolle's Theorem to gg, guaranteeing a point cc in (a,b)(a, b) where g′(c)=0g'(c) = 0
  • Computes g′(x)g'(x) as g′(x)=f′(x)−f(b)−f(a)b−ag'(x) = f'(x) - \frac{f(b) - f(a)}{b - a}
  • Sets g′(c)=0g'(c) = 0 to obtain f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}, proving the Mean Value Theorem

Applications of Mean Value Theorem

Applications of Mean Value Theorem

  • Applies the Mean Value Theorem to show that if ff is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and f′(x)=0f'(x) = 0 for all xx in (a,b)(a, b), then ff is constant on [a,b][a, b]
    • Proves this by using the Mean Value Theorem to show f(b)=f(a)f(b) = f(a) for any aa and bb in the interval
  • Uses the Mean Value Theorem to demonstrate that if ff and gg are continuous on [a,b][a, b], differentiable on (a,b)(a, b), and have equal derivatives on (a,b)(a, b), then f(x)=g(x)+Cf(x) = g(x) + C for some constant CC
    • Proves this by considering the function h(x)=f(x)−g(x)h(x) = f(x) - g(x) and showing that hh is constant on [a,b][a, b]

Inequalities from Mean Value Theorem

  • Establishes inequalities and bounds for functions using the Mean Value Theorem
  • Shows that if ff is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and m≤f′(x)≤Mm \leq f'(x) \leq M for all xx in (a,b)(a, b), then m(b−a)≤f(b)−f(a)≤M(b−a)m(b - a) \leq f(b) - f(a) \leq M(b - a)
    • Proves this by applying the Mean Value Theorem and the given bounds on f′(x)f'(x)

Average value of functions

  • Defines the average value of a function ff over an interval [a,b][a, b] as 1b−a∫abf(x)dx\frac{1}{b - a} \int_a^b f(x) dx
  • Proves that if ff is continuous on [a,b][a, b], then there exists a point cc in (a,b)(a, b) such that f(c)=1b−a∫abf(x)dxf(c) = \frac{1}{b - a} \int_a^b f(x) dx
    • Proves this by defining an antiderivative F(x)=∫axf(t)dtF(x) = \int_a^x f(t) dt and applying the Mean Value Theorem to FF

Key Terms to Review (14)

Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician who made significant contributions to analysis and calculus, particularly in the development of the concept of limits, continuity, and differentiability. His work laid the groundwork for many modern mathematical theories, particularly in understanding inverse functions, the Mean Value Theorem, and the application of L'Hôpital's Rule in solving indeterminate forms.
Average rate of change: The average rate of change of a function over an interval measures how much the function's output value changes relative to the input value changes across that interval. This concept is crucial in understanding how functions behave between two points and plays a significant role in motion analysis and the application of the Mean Value Theorem, which bridges the gap between average and instantaneous rates of change.
C in (a, b): In the context of the Mean Value Theorem, 'c in (a, b)' refers to a specific point within the open interval (a, b) where the instantaneous rate of change of a function equals the average rate of change over that interval. This concept is crucial because it establishes that if a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one point 'c' where the tangent line to the curve is parallel to the secant line connecting points a and b. It connects the behavior of a function's derivative with its overall change across an interval.
Continuous on [a, b]: A function is said to be continuous on the interval [a, b] if it is continuous at every point within that closed interval. This means that for any point in [a, b], the limit of the function as it approaches that point from either side equals the value of the function at that point. Understanding this concept is crucial because continuity plays a key role in many important theorems and applications in calculus, including ensuring that a function behaves predictably over an interval.
Differentiable on (a, b): A function is said to be differentiable on the open interval (a, b) if it has a derivative at every point within that interval. This means that the function is smooth and continuous without any sharp corners or vertical tangents in the interval, allowing for the application of the Mean Value Theorem, which relies on the concept of differentiability to establish relationships between the values of the function and its slopes.
Example with a Polynomial Function: An example with a polynomial function involves a specific instance where a polynomial is analyzed or applied, showcasing its properties such as continuity, differentiability, and behavior on intervals. These examples help illustrate important concepts such as the Mean Value Theorem, which asserts that for any continuous function over a closed interval, there exists at least one point where the instantaneous rate of change (the derivative) equals the average rate of change over that interval. By using polynomial functions in these examples, one can see clear and predictable behavior, making them ideal for understanding core principles.
Example with Trigonometric Functions: An example with trigonometric functions typically refers to a problem or scenario that incorporates the properties and applications of trigonometric functions, such as sine, cosine, and tangent. These examples often illustrate the use of these functions in real-world contexts, such as modeling periodic behavior or calculating angles and distances in various fields like physics, engineering, and astronomy.
F'(c) = (f(b) - f(a)) / (b - a): This formula represents the average rate of change of a function between two points, a and b, and establishes a key concept known as the Mean Value Theorem. It asserts that there exists at least one point c within the interval [a, b] where the instantaneous rate of change (the derivative f'(c)) equals this average rate of change. This connection shows how the behavior of functions can be analyzed through their derivatives and emphasizes the importance of continuity and differentiability.
Instantaneous Rate of Change: The instantaneous rate of change refers to the rate at which a function is changing at any specific point, which can be understood as the slope of the tangent line to the graph of the function at that point. This concept is essential for understanding how functions behave at particular values and is closely related to the derivative, which formalizes this idea mathematically. In practical terms, it helps in analyzing motion, understanding changes in variables, and applying important theorems related to functions.
Isaac Newton: Isaac Newton was a mathematician, physicist, and astronomer, widely recognized as one of the most influential scientists of all time. He made groundbreaking contributions to the fields of calculus and mechanics, particularly through his formulation of the laws of motion and universal gravitation, which laid the foundation for classical physics. His work is crucial for understanding concepts like rates of change and the behavior of objects under forces, which relate directly to various mathematical principles.
Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change of the function over that interval. This theorem provides a bridge between the behavior of a function and its derivatives, showing how slopes relate to overall changes.
Rolle's Theorem: Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval between the endpoints, and if the function takes the same value at both endpoints, then there exists at least one point in the open interval where the derivative of the function is zero. This theorem highlights a crucial relationship between differentiability, continuity, and the behavior of functions on intervals.
Slope of the secant: The slope of the secant line between two points on a curve represents the average rate of change of the function over that interval. It is calculated by taking the difference in the y-values and dividing it by the difference in the x-values of those two points, which provides insight into how the function behaves between them. This concept is foundational for understanding more complex ideas such as instantaneous rate of change and derivatives.
Slope of the tangent: The slope of the tangent refers to the instantaneous rate of change of a function at a particular point, which is represented by the slope of the tangent line to the graph of the function at that point. This concept is crucial for understanding how a function behaves locally, allowing us to approximate values and analyze trends using linear functions. The slope of the tangent is foundational in calculus and connects deeply with concepts such as differentiation and linear approximation.
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