Problem 28 - Calculus

Question

(a) If \(f\) is even and \(f^{\prime}(10)=6,\) what is \(f^{\prime}(-10) ?\) (b) If \(f\) is any even function and \(f^{\prime}(0)\) exists, what is \(f^{\prime}(0) ?\)

Step-by-Step Solution

Answer

(a) 6; (b) 0.

Step 1: Introduction to Even Function

A function \( f \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). This property will help us solve both parts of the problem.

Step 1: Understanding Derivatives of Even Functions

The derivative of an even function, \( f'(x) \), should satisfy \( f'(-x) = -f'(x) \), as the derivative of \( f(x) \) is interpreted as the slope of the tangent at any point \( x \). However, for an even function, the slope does not change sign.

Step 2: Solve for f'(−10)

Given that \( f \) is even and \( f'(10) = 6 \), we need to find \( f'(-10) \). For even functions, the derivative's property at symmetric points states \( f'(-x) = -f'(x) \). However, since the function is even, it implies symmetry in the derivative, not negation. Thus, \( f'(-10) = f'(10) = 6 \).

Step 3: Solve for f'(0)

For any even function \( f \), if \( f'(0) \) exists, its derivative at \( x = 0 \) must satisfy \( f'(0) = -f'(0) \). The only real number that equals its own negation is zero. Hence, \( f'(0) = 0 \).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, representing the rate of change or the slope of a function at any given point. When we talk about the derivative of a function, we are often interested in how quickly the function is changing at that particular spot. The derivative, noted as \( f'(x) \), gives us this rate.

• The derivative conveys the slope of the tangent line to the function at any point \( x \).
• This slope indicates how steeply the function is rising or falling at that point.
• If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), the function is decreasing.

By understanding derivatives, you can analyze the behavior of functions more deeply and predict their movements precisely.

Exploring Function Symmetry

Function symmetry is an intriguing characteristic that simplifies many calculus problems. A function \( f(x) \) is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain. This means it is mirror-symmetrical about the y-axis. Let's explore this further:

• For even functions, \( f(-x) \) gives the same value as \( f(x) \), reflecting symmetry about the y-axis.
• This symmetry helps in predicting values at negative and positive points equally.
• In calculus, even function symmetry influences the behavior of its derivatives, which leads to certain predictable properties.

This symmetry property is crucial in solving problems where functions need to be analyzed over a symmetric interval or point.

Solving Calculus Problems with Symmetry

Calculus problem solving often involves leveraging symmetry properties like those found in even functions. When dealing with an even function, such properties can considerably simplify the process of finding derivatives at symmetric points. Here's how you can tackle these problems:

• Recognize the symmetry: Identify whether the function is even using \( f(-x) = f(x) \).
• Apply symmetry to derivatives: For an even function, the symmetry does not imply the derivative changes sign at symmetric points, as it would with an odd function.
• Use symmetry to solve: Use the property \( f'(-x) = f'(x) \) when dealing with symmetry in derivatives, as seen in the original exercise part \( f'(-10) = f'(10) \).

Understanding and utilizing the symmetry of functions allows you to efficiently solve problems and understand the behavior of derivatives better.

Tangent Slopes of Even Functions

The tangent slope of a function at a particular point tells us how steeply the function is climbing or falling there. For even functions, analyzing the slope of the tangent has unique considerations due to symmetry.

• At symmetric points \( +x \) and \( -x \), the slopes are equal because the function is even.
• This means for a point like \( x = 0 \), the slope or \( f'(0) \) must be zero, ensuring the slope of the tangent is "level" to maintain symmetry.
• For the original exercise, because of symmetry, \( f'(10) = 6 \) implies \( f'(-10) = 6 \) as well.

The insight into tangent slopes facilitates the understanding of how evenly an even function changes over its domain, making this concept vital for correctly analyzing and predicting function behavior.