Problem 47 - Calculus

Question

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(g(x)\) is even, then \(f(x)\) is odd.

Step-by-Step Solution

Answer

False. The statement is not necessarily true.

Step 1: Understanding the Problem

We have a differential equation \( \frac{dy}{dx} = g(x) \). The exercise states that if \( g(x) \) is even, then \( f(x) \) is odd. We need to determine if this implication is true or false.

Step 2: Define Even and Odd Functions

A function \( g(x) \) is even if \( g(-x) = g(x) \) for all \( x \). A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \). We need to check if the evenness of \( g(x) \) implies the oddness of \( f(x) \).

Step 3: Consider the Derivative of Even Functions

If \( g(x) \) is even, \( g(x) = \frac{dy}{dx} \) implies that the derivative of \( f(x) \) is even. We need to explore whether this necessarily implies that \( f(x) \) is odd.

Step 4: Counterexample to Test the Statement

Let's consider an example: \( g(x) = x^2 \), which is even. Finding \( f(x) \) such that \( \frac{dy}{dx} = x^2 \), we integrate: \( f(x) = \frac{x^3}{3} + C \). \( f(x) \) is not odd since it does not satisfy the property \( f(-x) = -f(x) \).

Step 5: Conclusion Based on the Counterexample

Since we can find cases where \( g(x) \) is even but \( f(x) \) is not odd, the statement is false. The evenness of \( g(x) \) does not necessarily imply the oddness of \( f(x) \).

Key Concepts

Even Functions

In mathematics, an even function exhibits symmetry when reflected across the y-axis. This symmetry means that for every positive input, the function returns the same output as it does for the corresponding negative input. Consider the definition:

• A function \( g(x) \) is even if \( g(-x) = g(x) \) for all \( x \).
• Examples include \( g(x) = x^2 \), \( g(x) = \cos(x) \).

Even functions play a crucial role in differential equations. They are often contrasted with odd functions in exercises to test your understanding of function properties. When reviewing differential equations, identifying even functions can guide you to find other potential characteristics of solutions.

Odd Functions

An odd function is defined by a different type of symmetry, one that revolves around the origin of the function's graph. This symmetry means that for every positive input, the function returns the opposite output for the corresponding negative input. Key details include:

• A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).
• Examples include \( f(x) = x^3 \), \( f(x) = \sin(x) \).

These functions are important in differential equation exercises, especially when trying to discern the implications of derivatives or integrals. When considering function properties, highlighting whether a given function is odd might aid in determining the characteristics of its derivative or integral.

Counterexample

A counterexample is a powerful tool used to disprove statements or hypotheses in mathematics. It involves an example that contrasts the statement, showing that the statement does not hold in all cases. In solving the problem posed: 'If \( g(x) \) is even, then \( f(x) \) is odd', we utilize a counterexample:

• Consider \( g(x) = x^2 \), an even function.
• Integrating gives \( f(x) = \frac{x^3}{3} + C \), which is not odd.

The presence of this counterexample, where \( g(x) \) is indeed even but \( f(x) \) fails to be odd, disproves the initial statement. Crafting such counterexamples requires a depth of understanding but is invaluable in testing mathematical conjectures.

Function Derivatives

Derivatives are fundamental in calculus for understanding the rate of change of functions. In differential equations, the derivative takes center stage in expressing the relationship between functions and their rates of change. Notably:

• The derivative of an even function \( g(x) \) may not result in an odd function \( f(x) \).
• Derivatives can highlight peculiar properties, such as symmetry or asymmetry of functions.

In our exercise, the derivative relation \( \frac{dy}{dx} = g(x) \) was explored, and while \( g(x) \) was even, \( f(x) \) was not necessarily odd. Understanding basic derivative rules and exploring their implications in differential equations can expand problem-solving strategies significantly.