Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), and if \(f(a)=f(b)\), then there is at least one number \(c\) in \((a, b)\) such that \(f^{\prime}(c)=0 .\) Show that Rolle's Theorem is just a special case of the Mean Value Theorem. (Michel Rolle (1652-1719) was a French mathematician.)

Step-by-Step Solution

Answer

Rolle's Theorem is a special case of the Mean Value Theorem where the slope of the secant line is zero.

Step 1: Understand the Mean Value Theorem (MVT)

The Mean Value Theorem states that if a function \( f \) is continuous on \( [a, b] \) and is differentiable on \( (a, b) \), then there exists at least one number \( c \) in \( (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). This means the slope of the tangent at some point \( c \) is equal to the slope of the secant line joining \( (a, f(a)) \) and \( (b, f(b)) \).

Step 2: Set Up Rolle's Theorem Conditions

In Rolle's Theorem, we have the additional condition that \( f(a) = f(b) \). This means the slope of the secant line \( \frac{f(b) - f(a)}{b - a} = 0 \).

Step 3: Apply Mean Value Theorem to Rolle's Conditions

Since \( f(a) = f(b) \), when we apply the Mean Value Theorem, we get \( f'(c) = \frac{f(b) - f(a)}{b - a} = \frac{0}{b - a} = 0 \). This shows that in this specific condition, the derivative \( f'(c) = 0 \) for some \( c \) in \( (a, b) \).

Step 4: Conclusion About The Theorems

Rolle's Theorem is simply a special case of the Mean Value Theorem where the endpoints have the same function value, leading to a secant slope of zero. This forces the instant slope (derivative) at some point \( c \) to also be zero, satisfying the condition \( f'(c) = 0 \).

Key Concepts

Mean Value Theorem

The Mean Value Theorem is an essential concept in calculus. It bridges the gap between derivatives and the average rate of change over an interval. This theorem tells us that if a function \( f \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \(c\) within that interval where the derivative \( f'(c) \) equals the average rate of change over \([a, b]\).
This can be expressed as:\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]This equation shows that the slope of the tangent at some point \(c\) is the same as the slope of the secant line connecting the endpoints \((a, f(a))\) and \((b, f(b))\).

• The function must be continuous on \([a, b]\).
• The function must be differentiable on \((a, b)\).
• These conditions ensure that a "tangent" slope exists that matches the "average" slope of the entire interval.

continuity and differentiability

Continuity and differentiability are fundamental prerequisites for applying both the Mean Value Theorem and Rolle's Theorem.
### ContinuityContinuity of a function means there are no interruptions, jumps, or breaks in the graph of the function within the interval in question. A continuous function maintains its pathway smoothly across the interval \([a, b]\).

• In simple terms, you can draw the graph of a continuous function without lifting your pencil from the paper.

### DifferentiabilityDifferentiability implies that a function has a derivative at each interior point within the interval \((a, b)\).

• If a function is differentiable, it must also be continuous.
• The existence of a derivative means that the graph of the function has a tangent line at every point within the interval.

Both conditions are critical, ensuring that the behaviors of the function can be predictably analyzed within the interval, allowing for meaningful application of calculus theorems.

calculus problems

Calculus problems often involve analyzing the behaviors and properties of functions. Tools like the Mean Value Theorem and Rolle's Theorem help us solve such problems efficiently by providing a framework to understand change and slopes within an interval.
The key steps to tackle calculus problems involving these theorems are:

• Identify the given interval \([a, b]\) on which the function \( f \) is defined.
• Verify that \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\).
• Determine any additional conditions, like \( f(a) = f(b) \), that might simplify the problem using Rolle's Theorem.
• Apply the appropriate theorem to find the required derivative or slope equivalence.

Through these steps, calculus problems become manageable by breaking them down logically, based on the theorem's conditions.

derivatives

Derivatives are foundational in calculus, representing the rate of change of a function. In simpler terms, the derivative at a point provides the slope of the tangent to the graph of a function at that precise location.
Understanding derivatives is crucial for exploring the behavior of complex calculus problems.
### Tangent Lines and SlopesA tangent line just skims the graph of a function at a given point, mirroring its immediate direction without crossing it:

• The slope of this tangent line, \( f'(c) \), reveals the immediate rate of change at \( c \).
• If \( f'(c) = 0 \), the tangent line is flat, indicating a local maximum, minimum, or a constant region.

### Calculating DerivativesTo calculate the derivative of a function, we use techniques such as:

• Power Rule: For \( f(x) = x^n \), \( f'(x) = nx^{n-1} \).
• Product and Quotient Rules for products or divisions of functions.
• Chain Rule for compositions of functions.

Grasping these basics allows us to utilize derivatives to understand and solve deeper calculus problems, linking seamlessly with theorems like the Mean Value and Rolle's Theorem.