14.2 The Mean Value Theorem and its applications

The Mean Value Theorem is a powerful tool in calculus, linking a function's average rate of change to its instantaneous rate of change. 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

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

Mean Value Theorem proof

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

Applications of Mean Value Theorem

Applications of Mean Value Theorem

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

Inequalities from Mean Value Theorem

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

Average value of functions

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