Problem 84 - Essential Calculus

Question

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b],\) then \(f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t\).

Step-by-Step Solution

Answer

The given generalization of the Mean Value Theorem is proved by applying the method of integration by parts to the integral in the provided equation, simplifying the expression and rearranging the equation to match the original format. Therefore, the provided generalization is correct.

Step 1: Understand the Components

The function f is twice differentiable on the interval [a, b]. This means it has a first and second derivative on this interval. The generalization of the Mean Value Theorem involves these derivatives, expressed as \(f'\) and \(f''\), as well as an integral involving \(f''\).

Step 2: Integration by Parts

Apply the method of integration by parts to the integral component of the theorem. The formula for integration by parts is \(\int f*g = f*G - \int g*F\), where \(F\) and \(G\) are the antiderivatives of \(f\) and \(g\), respectively. Consider \(f''(t)\) as \(f\) and \((t-b)\) as \(g\). Start by finding the antiderivatives of \(f\) and \(g\). This will transform the equation into: \(f(b)-f(a) = f'(a)(b-a) - [f'(t)(t-b)]_a^b + \int_a^b f'(t)dt \)

Step 3: Simplify the Expression

Firstly, calculate and subtract the values of \(f'(t)(t-b)\) at \(b\) and \(a\). Then observe that \(\int_a^b f'(t)dt = f(b)-f(a)\). Substitute this into the equation, yielding: \(2f(b)-2f(a) = f'(a)(b-a) + [f'(t)(t-b)]_a^b \)

Step 4: Final Calculations

Rearrange the equation to the original form: \(f(b)-f(a) = f'(a)(b-a) - [f''(t)(t-b)]_a^b\)

Key Concepts

Twice Differentiable Functions

When we talk about twice differentiable functions, we are referring to a mathematical concept where a function, denoted as \( f(x) \), is not only differentiable once, but its derivative is also differentiable. This implies that there are two derivatives for the function: the first derivative \( f'(x) \), which provides the rate of change of the function at any point, and the second derivative \( f''(x) \), which gives us information about the curvature or acceleration of the function.

In the context of the Mean Value Theorem generalization, a twice differentiable function on a closed interval \( [a, b] \) ensures that both the first and second derivatives exist and are continuous within that range. This continuous differentiability is crucial as it implies the function is smooth without any sharp turns or cusps, allowing us to apply calculus tools such as integration and the Mean Value Theorem itself.

• A twice differentiable function can reveal the function’s concavity on \( [a, b] \).
• Understanding the behavior of the function and its derivatives is essential for proving the generalization of the Mean Value Theorem.
• The second derivative plays a significant role when integrated, as it is linked to the change in the slope (first derivative) of the function over the interval.

Integration by Parts

Integration by parts is a technique used in calculus that comes in handy when an integral is a product of two functions that would be difficult to integrate directly. It’s based on the product rule for differentiation and is formally stated as \( \int u dv = uv - \int v du \), where \( u \) and \( dv \) are functions of a variable, say \( x \).

To apply integration by parts to the integral in our generalization of the Mean Value Theorem, we would select \( f''(t) \) as our \( dv \) and \( t-b \) as our \( u \). The process involves:

• Identifying the correct choice of \( u \) and \( dv \) to simplify the integral.
• Finding the antiderivatives of each function.
• Applying the integration by parts formula to solve the integral.

The resulting integration helps simplify complex integrals within theorem proofs, as we've shown in the step-by-step solution. It is a fundamental tool in calculus for tackling integrals involving products of functions.

Derivatives

Derivatives are one of the foundational building blocks of calculus, representing the slope of a tangent to a function’s graph at any given point—essentially describing the rate at which the function's value changes. For a function \( y = f(x) \), the derivative at any point \( x \) is denoted by \( f'(x) \) or \( \frac{dy}{dx} \).

Critical to this theorem is the function's first derivative \( f'(x) \), representing the instantaneous rate of change, and the second derivative \( f''(x) \), providing insight into the acceleration of the function's rate of change.

• \( f'(a) \) is crucial in establishing the relationship between the function’s value and its rate of change at the point \( a \).
• The derivative connects tightly to the integral in the theorem through the Fundamental Theorem of Calculus, which links the antiderivative to the definite integral.
• In the proof of the generalization of the Mean Value Theorem, derivatives illustrate how a function's average rate of change over an interval can be equated to the instantaneous rate of change at some point within the interval.

Understanding derivatives not only helps in solving calculus problems but also in understanding the behavior of physical systems and changes over time.