Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

A function \(f\) is said to be nondecreasing on an interval \(I\) if \(x_{1}

Step-by-Step Solution

Answer

Sketch horizontal lines for both, such as \(f(x) = 2\). This is constant, meeting the criteria for the requested functions.

Step 1: Understand the Problem

We need to sketch two graphs: one for a function that is nondecreasing but not increasing, and another for a function that is nonincreasing but not decreasing. Remember, a nondecreasing function satisfies \(x_1

Step 2: Sketch Nondecreasing but Not Increasing Function

To sketch a nondecreasing but not increasing function, consider a step function. For example, a function where \(f(x)\) is constant for all \(x\), such as \(f(x) = 2\) for all \(x\) in the interval. This satisfies the condition because no matter \(x_1\) or \(x_2\), \(f(x_1) \leq f(x_2)\) holds true, but it's not strictly increasing because \(f(x_1) = f(x_2)\) for all \(x_1 < x_2\).

Step 3: Sketch Nonincreasing but Not Decreasing Function

To sketch a nonincreasing but not decreasing function, also consider a step function. For example, a function where \(f(x)\) is a constant, such as \(f(x) = 2\) for all \(x\) in the interval. This satisfies the condition because \(f(x_1) \geq f(x_2)\) holds true for each point, but it's not strictly decreasing because \(f(x_1) = f(x_2)\) for all \(x_1 < x_2\).

Step 4: Visual Confirmation

Draw the graphs and ensure that they exhibit horizontal lines, representing constant values over the interval. In both cases, the functions remain constant, showing that they meet the criteria for nondecreasing (not necessarily increasing) and nonincreasing (not necessarily decreasing) functions.

Key Concepts

Nonincreasing Function

A nonincreasing function is a mathematical concept where a function, denoted as \( f \), does not increase as its input \( x \) moves along the interval \( I \). This means for any two points \( x_1 \) and \( x_2 \) within the interval, where \( x_1 < x_2 \), the function must satisfy \( f(x_1) \ge f(x_2) \). This often means that as \( x \) gets larger, \( f \) either stays the same or decreases in value.

Characteristics of nonincreasing functions include:

• Can be constant over an interval: \( f(x) = 3 \) for all \( x \) in \( I \).
• Can have portions where they decrease.
• Not allowed to go up in value as \( x \) increases.

With these properties, step functions commonly serve as examples since they remain constant or only decrease at certain points along the interval.

Step Function

A step function jumps between different constant values over different intervals. This is often visualized as a series of steps in a graph rather than a smooth line or curve. Step functions are piecewise constant functions, meaning that each piece of their domain is a constant value.

Key features of step functions include:

• Consist of horizontal segments, creating "steps" in the graph.
• Remain constant over each interval, only changing value at certain points.
• Useful for modeling situations where changes occur at discrete intervals, such as pricing brackets or tax rates.

In the context of nondecreasing or nonincreasing functions, step functions like \( f(x) = c \) are nonincreasing since they maintain their value or drop at the jump points.

Graphical Representation

Graphically representing functions allows us to interpret their behavior visually. For step functions, the graph is characterized by flat, horizontal segments that represent constant values over intervals.

When drawing such a graph:

• The graph does not ascend; instead, it can either remain flat or descend.
• The horizontal lines show the regions where the function doesn't change its value.
• Vertical jumps occur at specific points, highlighting where the function changes from one level to another.

This form of representation clearly demonstrates the properties of the function, such as being nonincreasing, or nondecreasing, without the confusion of gradual slopes or curves.

Interval Analysis

Interval analysis is a process used in determining the behavior of a function over specific parts of its domain. This approach is essential for understanding functions like nonincreasing or nondecreasing functions. It helps in identifying the sections where a function remains constant, increases, or decreases.

Steps in conducting interval analysis:

• Identify the intervals where the function maintains a specific behavior (constant, increasing, or decreasing).
• Examine the endpoints of intervals where any change might occur, such as where a step function "jumps" to another value.
• Confirm the function's behavior across these points to ensure compliance with its definition (e.g., nonincreasing means no interval should show an increase).

This analysis is integral in visualizing and understanding functions over the range of their domain, providing insight into their "step-like" behavior, as seen with step functions and piecewise functions.

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Key Concepts

Nonincreasing Function

A nonincreasing function is a mathematical concept where a function, denoted as \( f \), does not increase as its input \( x \) moves along the interval \( I \). This means for any two points \( x_1 \) and \( x_2 \) within the interval, where \( x_1 < x_2 \), the function must satisfy \( f(x_1) \ge f(x_2) \). This often means that as \( x \) gets larger, \( f \) either stays the same or decreases in value.

Characteristics of nonincreasing functions include:

• Can be constant over an interval: \( f(x) = 3 \) for all \( x \) in \( I \).
• Can have portions where they decrease.
• Not allowed to go up in value as \( x \) increases.

With these properties, step functions commonly serve as examples since they remain constant or only decrease at certain points along the interval.

Step Function

A step function jumps between different constant values over different intervals. This is often visualized as a series of steps in a graph rather than a smooth line or curve. Step functions are piecewise constant functions, meaning that each piece of their domain is a constant value.

Key features of step functions include:

• Consist of horizontal segments, creating "steps" in the graph.
• Remain constant over each interval, only changing value at certain points.
• Useful for modeling situations where changes occur at discrete intervals, such as pricing brackets or tax rates.

In the context of nondecreasing or nonincreasing functions, step functions like \( f(x) = c \) are nonincreasing since they maintain their value or drop at the jump points.

Graphical Representation

Graphically representing functions allows us to interpret their behavior visually. For step functions, the graph is characterized by flat, horizontal segments that represent constant values over intervals.

When drawing such a graph:

• The graph does not ascend; instead, it can either remain flat or descend.
• The horizontal lines show the regions where the function doesn't change its value.
• Vertical jumps occur at specific points, highlighting where the function changes from one level to another.

This form of representation clearly demonstrates the properties of the function, such as being nonincreasing, or nondecreasing, without the confusion of gradual slopes or curves.

Interval Analysis

Interval analysis is a process used in determining the behavior of a function over specific parts of its domain. This approach is essential for understanding functions like nonincreasing or nondecreasing functions. It helps in identifying the sections where a function remains constant, increases, or decreases.

Steps in conducting interval analysis:

• Identify the intervals where the function maintains a specific behavior (constant, increasing, or decreasing).
• Examine the endpoints of intervals where any change might occur, such as where a step function "jumps" to another value.
• Confirm the function's behavior across these points to ensure compliance with its definition (e.g., nonincreasing means no interval should show an increase).

This analysis is integral in visualizing and understanding functions over the range of their domain, providing insight into their "step-like" behavior, as seen with step functions and piecewise functions.