Problem 8 - A Primer of Real Analysis

Question

Suppose \(I\) is an open interval, \(f: I \rightarrow \mathbb{R}, g: I \rightarrow \mathbb{R},\) and \(f^{\prime}(x)=g^{\prime}(x)\) for all \(x \in I .\) Show that there exists \(\alpha \in \mathbb{R}\) such that $$ g(x)=f(x)+\alpha $$ for all \(x \in I\).

Step-by-Step Solution

Answer

The functions differ by a constant; i.e., \( g(x) = f(x) + \alpha \) for all \( x \) in \( I \).

Step 1: Understand the Problem

We are given two functions, \( f \) and \( g \), defined on an open interval \( I \) such that their derivatives are equal, i.e., \( f'(x) = g'(x) \) for all \( x \) in \( I \). We need to show that \( g(x) = f(x) + \alpha \) for some constant \( \alpha \).

Step 2: Integrate Both Functions

Since \( f'(x) = g'(x) \), consider their integral forms. By the Fundamental Theorem of Calculus, integrating \( f'(x) \) and \( g'(x) \) over any interval gives us the functions \( f(x) \) and \( g(x) \) up to constants. Thus, \( f(x) = \int f'(x) \, dx + C_1 \) and \( g(x) = \int g'(x) \, dx + C_2 \).

Step 3: Set Up Equation from Equal Derivatives

Given \( f'(x) = g'(x) \), the integrals of the derivatives are the same: \( \int f'(x) \ dx = \int g'(x) \, dx \). This implies \( f(x) + C_1 = g(x) + C_2 \), where \( C_1 \) and \( C_2 \) are constants of integration.

Step 4: Determine Constant Relationship

From the equation \( f(x) + C_1 = g(x) + C_2 \), we can rearrange it to find a constant \( \alpha \): \( g(x) - f(x) = C_1 - C_2 = \alpha \). This means \( g(x) = f(x) + \alpha \) for all \( x \) in \( I \), where \( \alpha = C_2 - C_1 \).

Step 5: Conclusion

We have shown that \( g(x) = f(x) + \alpha \), where \( \alpha = C_2 - C_1 \). Hence, \( g(x) \) and \( f(x) \) differ by a constant \( \alpha \) for every point in the interval \( I \).

Key Concepts

Integration

Integration is the process of finding an antiderivative. When we integrate a function, we're essentially finding a function whose derivative is the given function. This process is crucial in solving various mathematical problems, especially when dealing with differential equations and physics.

In the exercise, we integrate the derivatives of functions \( f'(x) \) and \( g'(x) \). This is possible because, according to the Fundamental Theorem of Calculus, the integral of a derivative results in the original function up to a constant.

• \( \int f'(x) \, dx = f(x) + C_1 \)
• \( \int g'(x) \, dx = g(x) + C_2 \)

Here, \( C_1 \) and \( C_2 \) are constants arising from the integration. These constants represent any potential vertical shifts in the antiderivative function.

Understanding integration and the role it plays in describing functions and their behavior is fundamental in various fields, from engineering to economics.

Derivative Equality

Derivative equality is a concept where two functions have identical derivatives. In simple terms, if the derivative of function \( f \) is equal to the derivative of function \( g \), it means both functions change in the same way on their domain.

In our exercise, the condition \( f'(x) = g'(x) \) implies both functions are moving the same way across the interval \( I \). This condition tells us something critical: despite any differences in their values, their rate of change at any point is exactly the same. Hence, they are consistently parallel over this interval.

This draws upon a crucial implication about integration and differentiation — given the same rate of change, the original functions differ at most by a constant, which connects directly to the constant of integration concept we explore further.

Constant of Integration

The constant of integration arises from the process of finding antiderivatives or integrating functions. Whenever we find an antiderivative of a function, we must add a constant \( C \) to account for any vertical translations that could result from different initial conditions.

For instance, if we integrate a function like \( f'(x) \), we get \( f(x) + C_1 \) rather than just \( f(x) \). This shows that there is an entire family of functions that can have the same derivative, differing only by a constant.

• From \( f'(x) = g'(x) \), integrating gives \( f(x) + C_1 = g(x) + C_2 \)
• Simplifying, we find \( g(x) = f(x) + \alpha \)
• Here, \( \alpha = C_2 - C_1 \)

This constant \( \alpha \) captures the idea that while two functions can have identical derivatives, they might be vertically shifted versions of each other. The constant of integration is fundamental in calculus, ensuring solutions to differential equations are complete and reflective of all possible scenarios.