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

(a) A step function that staircases upwards and plateaus. (b) A step function that staircases downwards and plateaus.

Step 1: Understand Nondecreasing Functions

A function is nondecreasing on an interval if, whenever you have two points in that interval, the function value at the larger point is at least as large as the function value at the smaller point. Importantly, this allows the function to stay constant in some portions and still be considered nondecreasing.

Step 2: Create a Nondecreasing but Not Increasing Graph

To sketch a nondecreasing function that is not strictly increasing, choose a basic step function. For instance, consider a graph that remains constant for some interval and then steps up or increases in a manner such as plateauing after each increase. An example is: a horizontal line at a certain y-value, then steps up to another horizontal line at a higher y-value.

Step 3: Understand Nonincreasing Functions

A function is nonincreasing on an interval if each function value on the right is at most as large as the function value on the left. Essentially, this allows the function to stay constant in some regions while being nonincreasing overall.

Step 4: Create a Nonincreasing but Not Decreasing Graph

Sketch a function that decreases or stays constant but never strictly decreases. Think of a step-like graph that remains constant for a while, then steps down or decreases. For example, imagine a graph that consists of a horizontal segment, stepping down to another horizontal segment at a lower y-value.

Key Concepts

Nondecreasing Functions

A fundamental concept in calculus is that of nondecreasing functions. These are functions where, within a specified interval, the value of the function does not decrease as the input increases. Mathematically, this is expressed as: if you pick two points, say \(x_1\) and \(x_2\), where \(x_1 < x_2\), the function \(f\) at these points will satisfy \(f(x_1) \leq f(x_2)\).

Such a function can stay constant over some intervals, which means it does not necessarily have to increase continuously. A simple way to visualize this is to think of a step function. These functions can maintain a constant height (or y-value) over certain sections and then jump up to a higher constant. This stepping behavior ensures that the function is nondecreasing overall without being strictly increasing throughout.

In real-world terms, imagine the altitude of a hiking trail where you sometimes walk flat but always end the day at a higher elevation than you started. Understanding this aspect of nondecreasing functions is key to correctly sketching and interpreting these types of graphs.

Nonincreasing Functions

Nonincreasing functions are quite the opposite when compared to nondecreasing functions. In such functions, as we move along the x-axis from a smaller value to a larger value within an interval, the function value does not increase. This relationship is captured by the condition: given two points \(x_1 < x_2\), then \(f(x_1) \geq f(x_2)\).

Interestingly, these functions can maintain the same y-value for some intervals or segments, implying that they can remain constant and still be considered nonincreasing. They are allowed to decrease or remain flat without strictly decreasing throughout. A classic representation is a step-like graph which follows this rule: it has horizontal or flat segments that occasionally step down to lower values.

If you were to connect this idea to everyday scenarios, think of descending a staircase that sometimes has flat landings. You never go up but occasionally go down. This understanding is vital when sketching these functions, ensuring that even though a function is not decreasing continuously, it embodies the essence of nonincreasing behavior within its defined interval.

Function Graph Sketching

Function graph sketching is a critical skill in calculus that involves translating mathematical properties into visual representations. When sketching a graph, such as one for nondecreasing or nonincreasing functions, it is essential to account for the fundamental properties that define these functions.

For a nondecreasing function, envision drawing horizontal lines that occasionally jump to higher levels. These jumps or steps reflect an increase after periods of constant function values, like a staircase ascending with each step. Conversely, when sketching a nonincreasing function, imagine horizontal stretches that take steps downward.

Each step in the graph represents constant values, followed by a jump up or down. This mimics how nonincreasing and nondecreasing functions naturally behave over intervals. Sketching these graphs requires attention to their respective properties, ensuring visual accuracy and a meaningful interpretation of the function's behavior.