Problem 29 - Calculus

Question

If \(g\) is an odd function and \(g^{\prime}(4)=5,\) what is \(g^{\prime}(-4) ?\)

Step-by-Step Solution

Answer

\( g'(-4) = 5 \)

Step 1: Understanding Odd Functions

An odd function is one where for every input \( x \), the function satisfies \( g(-x) = -g(x) \). This symmetry means the negative input gives a negative of the output of the positive input.

Step 2: Analyzing the Derivative of an Odd Function

If \( g(x) \) is an odd function, its derivative \( g'(x) \) is an even function. An even function satisfies \( g'(-x) = g'(x) \) for every \( x \). This is because differentiation of an odd function results in an even function.

Step 3: Applying Even Function Property to the Derivative

Since \( g'(x) \) is even, we apply the property: \( g'(-x) = g'(x) \). Given \( g'(4) = 5 \), it follows that \( g'(-4) = g'(4) = 5 \).

Key Concepts

Odd Function

An odd function is a unique type of function in calculus, characterized by a specific symmetry. For an odd function, if you have a point at the input value of x, the corresponding output is at -x with the opposite sign. Practically, this means such functions have rotational symmetry about the origin. If you fold the graph along the line y = x, the ends will coincide due to this symmetry.

Mathematically, an odd function satisfies the condition:

• \( g(-x) = -g(x) \)

Example functions like \( f(x) = x^3 \) or \( f(x) = \sin(x) \) often appear in calculus due to their symmetry and application. Understanding this symmetry helps in determining the behavior of the function especially during differentiation and integration.

Even Function

An even function is another important symmetry in mathematics where the function's graph is symmetrical about the y-axis. If you reflect the graph over the y-axis, it lays perfectly on top of itself. For every positive input x, the function gives the same output for -x.

The mathematical expression for an even function is:

• \( f(-x) = f(x) \)

Common examples include functions like \( f(x) = x^2 \) or \( f(x) = \cos(x) \).
In relation to the topic, when differentiating an odd function, its derivative turns out to be an even function. This plays a critical role in calculus problem solving as it affects how we calculate derivatives and predict function behavior.

Derivative Symmetry

The concept of derivative symmetry is vital when dealing with odd and even functions. Derivative symmetry can be used to determine the behavior of derivatives of functions based on their symmetries. If you have an odd function, such as \( g(x) \), its derivative \( g'(x) \) becomes an even function.

The derivative of an even function or odd function behaves in a predictable way:

• The derivative of an odd function is even: \( g'(-x) = g'(x) \).
• The derivative of an even function is odd.

Understanding this symmetry is crucial, especially in calculus for analyzing the functions' rates of change and predicting how the function behaves beyond given values.

Calculus Problem Solving

Solving calculus problems often involves understanding the properties and symmetries of functions. By leveraging the symmetry of odd and even functions in derivatives, you can solve problems more efficiently.

When you know a function is odd and you find its derivative, problems often become simpler because its derivative is even, allowing you to make direct comparisons like \( g'(-x) = g'(x) \). For example, if you understand that \( g'(4) = 5 \) for an even function, it follows easily that \( g'(-4) = 5 \), simplifying complex calculus problems.

By breaking down the properties of functions and understanding the logical steps in problem-solving, practicing these will lead to better comprehension and ease in working through other intricate calculus tasks.