Problem 40 - Calculus Early Transcendentals

Question

Use the Mean-Value Theorem to prove the following result: Let \(f\) be continuous at \(x_{0}\) and suppose that \(\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) exists. Then \(f\) is differentiable at \(x_{0},\) and $$ f^{\prime}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} f^{\prime}(x) $$ [Hint: The derivative \(f^{\prime}\left(x_{0}\right)\) is given by $$ f^{\prime}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}} $$ provided this limit exists.

Step-by-Step Solution

Answer

\( f \) is differentiable at \( x_0 \) and \( f'(x_0) = \lim_{x \rightarrow x_0} f'(x) \).

Step 1: Establish the Mean-Value Theorem

The Mean Value Theorem (MVT) states that if a function \( f \) is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\), then there exists some \( c \) in \((a,b)\) such that \[ f'(c) = \frac{f(b) - f(a)}{b - a}. \] This theorem will guide us in relating the function \( f \) and its derivative.

Step 2: Apply the Mean-Value Theorem Near \( x_0 \)

Consider applying the Mean Value Theorem on the interval \([x_0, x]\), where \( x \) is close to \( x_0 \). On this interval, there exists some \( c \) between \( x_0 \) and \( x \) such that \[ f'(c) = \frac{f(x) - f(x_0)}{x - x_0}. \]

Step 3: Use the Limit of the Derivative

Given that \( \lim_{x \rightarrow x_0} f'(x) \) exists, as \( x \rightarrow x_0 \), it follows that \( f'(c) \rightarrow \lim_{x \rightarrow x_0} f'(x) \). Thus, as \( c \rightarrow x_0 \), the expression \( \frac{f(x) - f(x_0)}{x - x_0} \) approaches \( \lim_{x \rightarrow x_0} f'(x). \)

Step 4: Determine Differentiability at \( x_0 \)

By definition, if the limit \( \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0} \) exists, it represents the derivative \( f'(x_0) \). Therefore, \( f \) is differentiable at \( x_0 \) and \( f'(x_0) = \lim_{x \rightarrow x_0} f'(x). \)

Key Concepts

Continuity

Continuity plays a crucial role in calculus and mathematical analysis. A function is said to be continuous at a point if, intuitively, you can draw the function at that point without lifting your pencil from the paper. More formally, a function \( f \) is continuous at a point \( x_0 \) if the following condition is met:\[ \lim_{{x \to x_0}} f(x) = f(x_0). \] This means that as the values of \( x \) get closer to \( x_0 \), their corresponding function values \( f(x) \) get closer to \( f(x_0) \).
Continuity ensures that functions behave in a predictable and smooth manner. If a function is not continuous at a point, it may have a jump or a gap, which means the mechanics of calculus, like differentiability, might break down. In the context of the Mean Value Theorem, continuity on a closed interval is a crucial precondition to ensure that the theorem holds.

• Continuous functions are essential in discussing limits, as they provide stability in the value of the function.
• Without continuity at \( x_0 \), the limit \( \lim_{{x \to x_0}} f'(x) \) might not be meaningful or applicable.

Differentiability

Differentiability is the property of a function that allows it to have a derivative at a particular point. For a function \( f \) to be differentiable at a point \( x_0 \), it must be smooth around that point, meaning you should be able to draw a tangent line uninterrupted. Mathematically, \( f \) is differentiable at \( x_0 \) if the following limit exists:\[f'(x_0) = \lim_{{x \to x_0}} \frac{f(x) - f(x_0)}{x - x_0}.\]Differentiability implies continuity, but continuity does not necessarily imply differentiability. For instance, the function may have a sharp corner or cusp at \( x_0 \), making it continuous but not differentiable.
Differentiability is a vital concept in calculus because it provides the rate of change or the slope of the function at that point. In the context of the exercise, showing differentiability at \( x_0 \) not only satisfies the conditions of the Mean Value Theorem but also assures us that we can determine a specific value for the derivative at that point.

• Differentiability requires smoothness; a break, cusp, or vertical tangent means a function isn't differentiable.
• Even if \( \lim_{{x \to x_0}} f'(x) \) exists, the differentiability ensures that this is the actual slope at \( x_0 \).

Limit of Derivative

The limit of a derivative is a fascinating concept that combines the ideas of limits and rates of change. When we say that \( \lim_{{x \to x_0}} f'(x) \) exists, it suggests that as \( x \) approaches \( x_0 \), the values of the derivative \( f'(x) \) converge to a single number. This means that the behavior of the function's slope becomes predictable close to \( x_0 \).
This is important for proving differentiability at \( x_0 \) because if the derivative itself approaches a limit, it aligns with the criteria for the function to be differentiable at that point. In the exercise, by establishing \( \lim_{{x \to x_0}} f'(x) \) and proving the corresponding limit for the difference quotient, we confirm differentiability at \( x_0 \).

• The existence of \( \lim_{{x \to x_0}} f'(x) \) ensures smooth transition in slope values of the function.
• Convergence of \( f'(x) \) as \( x \to x_0 \) supports the theorem's requirement that \( f'(x_0) = \lim_{{x \to x_0}} f'(x) \).