Question

In Exercises,Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The second derivative represents the rate of change of the first derivative.

Short answer

The statement is true. The second derivative represents the rate of change of the first derivative.

Step-by-step solution

Understanding Derivatives

The derivation operation in calculus gives the rate of change. When a function \( f(x) \) is derived, the result, \( f'(x) \), is called the first derivative which represents the slope of \( f(x) \) at a given point, or the rate of change of \( f(x) \). If the function \( f(x) \) is derived again, the result, \( f''(x) \), is called the second derivative.

Interpret Second Derivative

The second derivative, \( f''(x) \), is the derivative of the first derivative \( f'(x) \). Therefore, it shows how the slope or the rate of change in \( f(x) \) (i.e., \( f'(x) \)) changes for different values of \( x \).

Evaluate the Statement

'The second derivative represents the rate of change of the first derivative.' Given the interpretation of the second derivative, the statement is in fact true. The second derivative does represent the rate of change of the first derivative, because it indicates how the slope or the rate of change in the function is itself changing.

Key concepts

Rate of Change

The rate of change is a fundamental concept in calculus that tells us how one quantity changes with respect to another. Imagine you are driving a car; your speedometer shows your speed at any given moment, which is a direct measure of the rate at which you are changing position in relation to time.
The rate of change can be applied to various contexts, such as population growth over a certain period, the increase or decrease in temperature, or any situation where there is a change over time.

• In mathematical terms, this is often represented by a derivative, specifically the first derivative, which expresses the rate of change of a function.
• When you calculate a derivative, you are essentially finding out how much a function's output value changes per unit change in input.

Understanding this concept helps in analyzing how functions behave and their responsiveness to change.

First Derivative

The first derivative of a function, usually denoted as \( f'(x) \), is a critical part of understanding changes in the behavior of the function. It provides insights into the slope of the function at any given point.
Think of the first derivative as a tool that helps to measure how steeply a function is rising or falling.

• For example, in a graph representing distance over time, the first derivative gives the velocity, showing how fast the distance is changing at any moment.
• The first derivative is not only about speed; it is also about understanding the pattern of change - whether the function is increasing, decreasing, or remaining constant.

To calculate the first derivative, you differentiate the original function, \( f(x) \), which often involves applying basic differentiation rules. Thus, the first derivative serves as a foundation for further analysis, including determining the nature of curves.

Slope Analysis

Slope analysis involves examining the slope or gradient of a function, primarily using derivatives. This analysis helps to understand how a function behaves across different intervals.
In any function, the slope at a particular point indicates the pace at which the function's value is changing.

• The first derivative, \( f'(x) \), is directly related to the slope. When the slope is positive, the function is increasing; when negative, it is decreasing.
• Slope analysis is enriched by the second derivative, \( f''(x) \), which reveals how the slope itself is changing, offering insights into the concavity and convexity of the function.

Conducting a slope analysis can help identify turning points, or points where the function changes from increasing to decreasing or vice versa, and points of inflection, where the concavity changes.