Problem 55 - Calculus Early Transcendentals

Question

Recall that a function \(f\) is called even if \(f(-x)=f(x)\) for all \(x\) in its domain and odd if \(f(-x)=-f(x)\) for all such \(x\) . Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.

Step-by-Step Solution

Answer

The derivative of an even function is odd, and the derivative of an odd function is even.

Step 1: Define the Properties of Even and Odd Functions

Recall that for a function to be even, it must satisfy \( f(-x) = f(x) \) for all \( x \) in its domain. Similarly, a function is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.

Step 2: Differentiate an Even Function

Let \( f(x) \) be an even function, meaning \( f(-x) = f(x) \). Find the derivative: \( f'(x) = \frac{d}{dx}f(x) \). Using the definition of a derivative, we can write:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Step 3: Verify the Odd Property for the Derivative of Even Function

For the function's derivative to be odd, it must satisfy \( f'(-x) = -f'(x) \). Compute \( f'(-x) \):\[ f'(-x) = \lim_{h \to 0} \frac{f(-x+h) - f(-x)}{h} \]Since \( f(-x) = f(x) \), we substitute to get:\[ f'(-x) = \lim_{h \to 0} \frac{f(-(x-h)) - f(x)}{h} = \lim_{h \to 0} \frac{-f(x-h) + f(x)}{h} = -f'(x)\]This confirms the derivative \( f'(x) \) is odd.

Step 4: Differentiate an Odd Function

Let \( g(x) \) be an odd function, so \( g(-x) = -g(x) \). The derivative \( g'(x) = \frac{d}{dx}g(x) \), is given as:\[ g'(x) = \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} \]

Step 5: Verify the Even Property for the Derivative of Odd Function

To show \( g'(x) \) is even, we need \( g'(-x) = g'(x) \). Calculate \( g'(-x) \):\[ g'(-x) = \lim_{h \to 0} \frac{g(-x+h) - g(-x)}{h} \]Because \( g(-x) = -g(x) \), substituting gives:\[ g'(-x) = \lim_{h \to 0} \frac{-g(x-h) + g(x)}{h} = g'(x) \]Thus, \( g'(x) \) is even.

Key Concepts

Even Functions

Even functions are an important concept in mathematics and are defined by their symmetrical property. Specifically, a function \( f(x) \) is termed even if it satisfies the condition \( f(-x) = f(x) \) for every value of \( x \) in its domain. This symmetry means that the graph of an even function is a mirror image over the y-axis.
Common examples of even functions include:\

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• quadratic functions like \( f(x) = x^2 \)
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• cosine functions such as \( f(x) = \cos(x) \)
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• absolute value functions represented as \( f(x) = |x| \)
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Understanding even functions is crucial as they possess unique properties that influence their derivatives, resulting in specific behavioral patterns that are utilized in mathematical proofs and calculus problems.

Odd Functions

Odd functions have a unique set of characteristics distinct from even functions. A function \( g(x) \) is considered odd if it satisfies the condition \( g(-x) = -g(x) \) for every \( x \) within its domain. The graph of an odd function has rotational symmetry around the origin.
Some prime examples of odd functions are:

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• linear functions like \( g(x) = x \)
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• cubic functions such as \( g(x) = x^3 \)
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• sine functions like \( g(x) = \sin(x) \)
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The property of rotational symmetry facilitates the understanding of the derivatives of odd functions, particularly, how they transform into even functions. Knowing these features helps in calculus when working on proofs or understanding the evolution of function behavior upon differentiation.

Properties of Derivatives

Derivatives reveal much about the behavior of functions. Specifically, they can transform the nature of even and odd functions upon differentiation. When taking a derivative of an even function, the result is an odd function. Conversely, the derivative of an odd function yields an even function. This pattern emerges from the formal derivative definition.
For example, given an even function \( f(x) \), its derivative \( f'(x) \) demonstrates rotational symmetry such that \( f'(-x) = -f'(x) \).
Similarly, for an odd function \( g(x) \), its derivative \( g'(x) \) is even such that \( g'(-x) = g'(x) \). These properties are intrinsic to calculus and are critical when proving theorems or solving problems involving complex functions. They help streamline calculations and enhance understanding of function behaviors.

Calculus Proofs

Understanding calculus proofs involves combining several key concepts and applying them to derive certain properties or conclusions about functions. In the context of even and odd functions, calculus proofs illustrate the transformation properties of derivatives.
The proof begins by recalling the symmetric properties of even (\( f(-x) = f(x) \)) and odd (\( g(-x) = -g(x) \)) functions. Using the definition of a derivative - the limit of the difference quotient - we can show how derivatives transform function types:

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• For an even function \( f \), its derivative \( f'(x) \) is proven odd by showing \( f'(-x) = -f'(x) \).
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• For an odd function \( g \), the derivative \( g'(x) \) is shown to be even by demonstrating \( g'(-x) = g'(x) \).
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These proofs reinforce the symmetrical relationship between the functions and their derivatives, solidifying the understanding of how function behaviors inherently change through differentiation.