Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f\) be differentiable and let \(f^{\prime}\left(x_{0}\right)=m\). Find \(f^{\prime}\left(-x_{0}\right)\) if (a) \(f\) is an odd function. (b) f is an even function.

Step-by-Step Solution

Answer

(a) For an odd function, \( f'(-x_0) = -m \); (b) For an even function, \( f'(-x_0) = m \).

Step 1: Understand Derivatives of Odd and Even Functions

An odd function, by definition, satisfies the condition that \( f(-x) = -f(x) \). An even function satisfies \( f(-x) = f(x) \). Since \( f \) is differentiable, we can use these properties to find the derivative at \( -x_0 \).

Step 2: Find the Derivative for Odd Function

For a differentiable odd function, the relationship \( f(-x) = -f(x) \) holds. Differentiating both sides with respect to \( x \) gives \( -f'(-x) = -f'(x) \). Hence, \( f'(-x) = f'(x) \). Therefore, if \( f'(x_0) = m \), \( f'(-x_0) = -m \).

Step 3: Find the Derivative for Even Function

For a differentiable even function, the relationship \( f(-x) = f(x) \) holds. Differentiating both sides with respect to \( x \) gives \( -f'(-x) = f'(x) \). Thus, \( f'(-x) = -f'(x) \). Therefore, if \( f'(x_0) = m \), \( f'(-x_0) = m \).

Key Concepts

Odd Functions

Odd functions have a unique symmetry about the origin. This means if you rotate the graph 180 degrees around the origin, it remains unchanged. Mathematically, this translates to the condition: \[ f(-x) = -f(x) \].

This property tells us that when you input the negative of any number into the function, the output is the negative of what you would get with the positive number.

Key Characteristics of Odd Functions:

• Symmetric with respect to the origin.
• The graph reflects about the line y = -x.

Now, let's translate this into derivatives. For an odd function, if you differentiate both sides with respect to \(x\): \[ -f'(-x) = -f'(x) \].

From this, we deduce:\[ f'(-x) = -f'(x) \].

Thus, if the derivative at some point \(x_0\) is \(m\), then the derivative at the negative of that point, \(-x_0\), equals \(-m\). This mirrors the function's original property of reflecting its values across the origin.

Even Functions

Even functions are defined by their symmetry about the y-axis. This means if you fold the graph along the y-axis, it matches perfectly on both sides. The defining equation is: \[ f(-x) = f(x) \].

In simple terms, no matter which positive or negative value of \(x\) you choose, if the values are the same, their outputs will be the same as well.

Key Characteristics of Even Functions:

• Symmetric with respect to the y-axis.
• The graph reflects about the vertical line x = 0.

Using this property with derivatives means: when you differentiate both sides: \[ -f'(-x) = f'(x) \].

Simplifying, we have:\[ f'(-x) = -f'(x) \].

Thus, for the derivative of an even function, if at one point \(x_0\) the derivative is \(m\), the derivative at \(-x_0\) remains \(m\), showing that the slopes are consistent on either side of the y-axis.

Differentiability

Differentiability is the concept that a function has a derivative at each point in its domain. For a function to be differentiable at a point, it must not only be continuous at that point but also have a defined slope there. This means you should be able to draw a tangent line at each point of the function's graph.

Key Conditions for Differentiability:

• The function is continuous at the point.
• There are no sharp corners or cusps at the point.
• The tangent at the point is well-defined.

Differentiability ensures that the function changes smoothly, without jumps or interruptions. When dealing with odd and even functions, differentiability allows us to confidently apply the rules of calculus, such as taking derivatives, and understand how the function behaves at points like \(-x_0\) and \(x_0\). As shown in the odd and even functions, correctly applying derivative properties based on differentiability gives us insight into the function's slope behavior depending on its symmetry properties.