Problem 52 - Calculus Single Variable

Question

If \(f(-x)=f(x)\) for every \(x,\) then \(f\) is said to be an even function. If \(f(-x)=-f(x),\) for every \(x,\) then \(f\) is said to be an odd function. Assume that \(f\) is differentiable. Show that if \(f\) is even, then \(f^{\prime}\) is odd. Show that if \(f\) is odd, then \(f^{\prime}\) is even.

Step-by-Step Solution

Answer

If \( f \) is even, \( f' \) is odd; if \( f \) is odd, \( f' \) is even.

Step 1: Define the Problem for Even Function

We need to show that if \( f(x) \) is even, then its derivative \( f'(x) \) is odd. Recall, \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).

Step 2: Differentiate the Even Function

Take the derivative of both sides of the equation for an even function. By using the differentiation rules, the derivative of \( f(-x) = f(x) \) with respect to \( x \) is: \( f'(-x) \cdot (-1) = f'(x) \). This simplifies to \( -f'(-x) = f'(x) \).

Step 3: Conclude the Odd Nature of \(f'(x)\)

The equation \( -f'(-x) = f'(x) \) means that \( f'(-x) = -f'(x) \), which fits the definition of an odd function, thus proving \( f'(x) \) is odd when \( f(x) \) is even.

Step 4: Define the Problem for Odd Function

Next, we show if \( f(x) \) is odd, then \( f'(x) \) is even. Recall, \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).

Step 5: Differentiate the Odd Function

Differentiate both sides of the equation for an odd function. The derivative of \( f(-x) = -f(x) \) with respect to \( x \) gives: \( f'(-x) \cdot (-1) = -f'(x) \). This can be rewritten as \( f'(-x) = f'(x) \).

Step 6: Conclude the Even Nature of \(f'(x)\)

The result \( f'(-x) = f'(x) \) shows that \( f'(x) \) is even, proving that if \( f(x) \) is odd, then \( f'(x) \) is even.

Key Concepts

Even Function

In mathematics, an even function is one that maintains its value symmetry around the vertical axis. This means that for a function to be classified as even, it must satisfy the equation \(f(-x) = f(x)\) for every real number \(x\). This condition makes the graph of an even function reflect across the y-axis, making both sides identical.

A great example of an even function is \(f(x) = x^2\), a simple parabola opening upwards. If you replace \(x\) with \(-x\), the equation remains the same: \((-x)^2 = x^2\). When dealing with even functions in calculus, a common goal is identifying the properties of their derivatives. Specifically, when a function is even, its derivative is not. Through differentiation, it's proven that the derivative of an even function is actually an odd function, illustrating a beautiful interplay between symmetry and calculus operations.

Odd Function

An odd function contains a different type of symmetry—rotational symmetry about the origin. For a function \(f\) to be labeled as odd, it must meet the condition \(f(-x) = -f(x)\) for all \(x\). Essentially, this means that if you reflect the graph of the function over both the x-axis and y-axis, the graph remains unchanged.

A classic example of an odd function is \(f(x) = x^3\). Substituting \(-x\) for \(x\) results in \((-x)^3 = -x^3\), perfectly satisfying the condition for oddness. Just like even functions have their corresponding odd derivatives, odd functions have even derivatives. The derivative of an odd function is even, a fascinating trait that underscores the dual nature of these geometric transformations within calculus.

Derivative

Differentiation is a fundamental concept in calculus that deals with finding the derivative of a function. The derivative measures how a function's output value changes as its input changes—the rate of change or slope of the function.

To differentiate a given function \(f(x)\), you produce a new function \(f'(x)\), which provides the slope of \(f(x)\) at any given point. For instance, the process of differentiating a polynomial function like \(f(x) = x^2\) results in \(f'(x) = 2x\). Here, \(2x\) represents how quickly \(f(x) = x^2\) is changing at any specific point along the x-axis.

This concept extends beautifully when linked to the nature of functions (even or odd). As discussed, the derivative of an even function turns out to be odd, and vice versa. This reflects how differentiation not only provides insight into the behavior of functions but also transforms the structural symmetries in intriguing and predictable ways.