∬Differential Calculus Unit 14 – The Mean Value Theorem

Key Concepts and Definitions

• The Mean Value Theorem (MVT) is a fundamental theorem in differential calculus that relates the average rate of change of a function over an interval to the instantaneous rate of change at a point within that interval
• Assumes the function $f(x)$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$
• States that there exists at least one point $c$ in the open interval $(a, b)$ such that the derivative of the function at $c$ equals the average rate of change of the function over the interval $[a, b]$
  • Mathematically, this is expressed as: $f'(c) = \frac{f(b) - f(a)}{b - a}$
• The point $c$ is called the "mean value" or the "average value" of the function on the interval
• The MVT is a generalization of Rolle's Theorem, which is a special case when $f(a) = f(b)$
• The MVT is crucial for understanding the behavior of functions and their derivatives, and it has numerous applications in calculus and analysis

Historical Context and Development

• The Mean Value Theorem was first formulated and proved by French mathematician Joseph-Louis Lagrange in the late 18th century
  • Lagrange's proof relied on the concept of infinitesimals and was not entirely rigorous by modern standards
• Augustin-Louis Cauchy later provided a more rigorous proof of the MVT in the early 19th century, using the concept of limits and the properties of continuous functions
• The MVT is a consequence of the Intermediate Value Theorem, which states that if a function is continuous on a closed interval, it takes on all values between its minimum and maximum values
• The development of the MVT was a significant milestone in the history of calculus, as it provided a deeper understanding of the relationship between a function and its derivative
• The MVT has since become a fundamental tool in mathematical analysis and has been generalized to higher dimensions and abstract spaces (e.g., Banach spaces)

Theorem Statement and Proof

• Statement: Let $f(x)$ be a function that is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$. Then there exists at least one point $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Proof:
  1. Define a new function $g(x)$ as follows: $g(x) = f(x) - \frac{f(b) - f(a)}{b - a}(x - a) - f(a)$
  2. Observe that $g(a) = g(b) = 0$, since $g(a) = f(a) - \frac{f(b) - f(a)}{b - a}(a - a) - f(a) = 0$ and $g(b) = f(b) - \frac{f(b) - f(a)}{b - a}(b - a) - f(a) = 0$
  3. Apply Rolle's Theorem to the function $g(x)$ on the interval $[a, b]$. Since $g(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $g(a) = g(b)$, there exists a point $c$ in $(a, b)$ such that $g'(c) = 0$
  4. Calculate $g'(c)$ using the chain rule: $g'(c) = f'(c) - \frac{f(b) - f(a)}{b - a}$
  5. Set $g'(c) = 0$ and solve for $f'(c)$: $f'(c) = \frac{f(b) - f(a)}{b - a}$, which proves the Mean Value Theorem

Geometric Interpretation

• The Mean Value Theorem has a clear geometric interpretation when considering the graph of a function $f(x)$ on an interval $[a, b]$
• The theorem states that there exists a point $c$ in the open interval $(a, b)$ where the tangent line to the graph of $f(x)$ is parallel to the secant line connecting the points $(a, f(a))$ and $(b, f(b))$
  • The slope of the secant line is the average rate of change of the function over the interval, given by $\frac{f(b) - f(a)}{b - a}$
  • The slope of the tangent line at the point $(c, f(c))$ is the instantaneous rate of change of the function, given by $f'(c)$
• Geometrically, the MVT guarantees the existence of a point where the instantaneous rate of change equals the average rate of change
• This interpretation helps visualize the relationship between the function and its derivative, and it provides insight into the behavior of the function on the interval
• The geometric interpretation also highlights the connection between the MVT and the Intermediate Value Theorem, as the point $c$ can be seen as an intermediate value of the derivative function $f'(x)$ on the interval $(a, b)$

Applications and Examples

• The Mean Value Theorem has numerous applications in calculus, analysis, and other areas of mathematics
• One common application is in proving the Fundamental Theorem of Calculus, which relates the definite integral of a function to its antiderivative
  • The MVT is used to show that if two functions have the same derivative on an interval, they differ by a constant on that interval
• The MVT is also used to establish inequalities involving functions and their derivatives, such as the following:
  • If $f'(x) \geq 0$ on an interval $[a, b]$, then $f(x)$ is increasing on $[a, b]$
  • If $|f'(x)| \leq M$ on an interval $[a, b]$, then $|f(x) - f(y)| \leq M|x - y|$ for all $x, y$ in $[a, b]$ (Lipschitz condition)
• Example: Consider the function $f(x) = x^3$ on the interval $[0, 2]$. By the MVT, there exists a point $c$ in $(0, 2)$ such that $f'(c) = \frac{f(2) - f(0)}{2 - 0} = \frac{8 - 0}{2} = 4$. Solving for $c$, we find that $c = \sqrt[3]{4} \approx 1.59$
• The MVT is also used in physics and engineering to analyze the behavior of physical systems, such as the motion of objects under the influence of forces

Related Theorems and Concepts

• The Mean Value Theorem is closely related to several other important theorems and concepts in calculus and analysis
• Rolle's Theorem is a special case of the MVT when $f(a) = f(b)$. It states that if a function is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists a point $c$ in $(a, b)$ such that $f'(c) = 0$
• The Intermediate Value Theorem (IVT) is a prerequisite for the MVT and states that if a function is continuous on $[a, b]$ and $k$ is a value between $f(a)$ and $f(b)$, then there exists a point $c$ in $[a, b]$ such that $f(c) = k$
• The Extreme Value Theorem is another consequence of the IVT and states that a continuous function on a closed interval attains its maximum and minimum values on that interval
• The Generalized Mean Value Theorem (Cauchy's Mean Value Theorem) extends the MVT to two functions, stating that if $f(x)$ and $g(x)$ are continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a point $c$ in $(a, b)$ such that $[f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c)$
• The MVT is a key tool in proving the Fundamental Theorem of Calculus, which relates the concept of integration to that of differentiation

Common Misconceptions and Pitfalls

• One common misconception about the Mean Value Theorem is that it guarantees the existence of a unique point $c$ satisfying the theorem. In fact, there may be multiple points or even infinitely many points that satisfy the condition $f'(c) = \frac{f(b) - f(a)}{b - a}$
• Another misconception is that the MVT applies to any function on any interval. The theorem requires the function to be continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$. If these conditions are not met, the MVT may not hold
• Students sometimes confuse the Mean Value Theorem with the Intermediate Value Theorem, which deals with the existence of a point $c$ such that $f(c) = k$ for some value $k$ between $f(a)$ and $f(b)$. While the IVT is a prerequisite for the MVT, they are distinct theorems with different implications
• When applying the MVT, it is essential to verify that the function satisfies the continuity and differentiability conditions on the appropriate intervals. Failing to do so may lead to incorrect conclusions
• In some cases, the MVT may not provide enough information to fully characterize the behavior of a function or its derivative. It is important to use the MVT in conjunction with other theorems and techniques to gain a comprehensive understanding of the function's properties

Practice Problems and Solutions

1. Verify that the function $f(x) = x^2$ satisfies the Mean Value Theorem on the interval $[1, 4]$, and find a point $c$ that satisfies the theorem. Solution:

   • $f(x)$ is continuous on $[1, 4]$ and differentiable on $(1, 4)$, so the MVT applies
   • $f(1) = 1$, $f(4) = 16$, and $f'(x) = 2x$
   • By the MVT, there exists a point $c$ in $(1, 4)$ such that $f'(c) = \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5$
   • Solving for $c$, we find that $2c = 5$, so $c = \frac{5}{2}$

2. Prove that if $f(x)$ is differentiable on $[a, b]$ and $f'(x) = 0$ for all $x$ in $[a, b]$, then $f(x)$ is constant on $[a, b]$. Solution:

   • Let $x_1$ and $x_2$ be any two points in $[a, b]$
   • By the MVT, there exists a point $c$ between $x_1$ and $x_2$ such that $f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$
   • Since $f'(x) = 0$ for all $x$ in $[a, b]$, we have $\frac{f(x_2) - f(x_1)}{x_2 - x_1} = 0$
   • This implies that $f(x_2) - f(x_1) = 0$, or $f(x_2) = f(x_1)$
   • Since $x_1$ and $x_2$ were arbitrary, $f(x)$ is constant on $[a, b]$

3. Use the Mean Value Theorem to prove that $|\sin x - \sin y| \leq |x - y|$ for all $x, y$ in $\mathbb{R}$. Solution:

   • Let $f(x) = \sin x$. Then $f'(x) = \cos x$, and $|\cos x| \leq 1$ for all $x$ in $\mathbb{R}$
   • For any $x, y$ in $\mathbb{R}$, apply the MVT to $f(x)$ on the interval $[x, y]$ (or $[y, x]$ if $y < x$)
   • There exists a point $c$ between $x$ and $y$ such that $f'(c) = \frac{f(y) - f(x)}{y - x}$, or $\cos c = \frac{\sin y - \sin x}{y - x}$
   • Taking the absolute value of both sides and using the fact that $|\cos c| \leq 1$, we have $\left|\frac{\sin y - \sin x}{y - x}\right| \leq 1$
   • This implies that $|\sin y - \sin x| \leq |y - x|$, which is the desired inequality