Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose your are traveling an interstate highway and that every 10 miles there is an emergency telephone. Let \(D\) be the function defined by $$ D(x) $$ is the distance to the nearest emergency telephone where \(x\) is the mileage position on the highway. a. Draw a graph of \(D\). b. Find the period and amplitude of \(D\).

Step-by-Step Solution

Answer

The graph of \(D(x)\) is a zigzag; the period is 10 miles and the amplitude is 5 miles.

Step 1: Understanding the Problem

We need to determine the distance to the nearest emergency telephone located every 10 miles on the highway. These telephones create a repeating pattern of distances to vehicles traveling the highway.

Step 2: Graphing the Function

The function \(D(x)\) represents the distance to the nearest emergency telephone at any given position \(x\) along the highway. This function is periodic, repeating every 10 miles. The graph of this function will form a series of 'V' shapes: at mile 0, 10, 20, etc. the distance \(D(x) = 0\) since you are exactly at a telephone, and it increases to 5 miles at mile 5 or 15, decreases back to 0 at mile 10 or 20, and so on. This creates a zigzag graph.

Step 3: Identifying the Period of \(D\)

Since the emergency telephones are placed every 10 miles, the pattern repeats every 10 miles. Thus, the period of the function \(D(x)\) is 10 miles.

Step 4: Identifying the Amplitude of \(D\)

The amplitude is the maximum distance from the central axis (which, in this case, is 0) to the maximum point in one period. As the distance \(D(x)\) reaches 5 miles halfway between telephones, the amplitude of \(D(x)\) is 5 miles.

Key Concepts

Distance Function

The term "distance function" in this context refers to how far you are from the nearest emergency telephone as you travel along the highway. If you imagine traveling in a straight line along the road, every 10 miles, you'll encounter an emergency phone.
At any point on this journey, the value of the distance function, denoted as \(D(x)\), gives you the shortest distance back to the nearest phone. For example, if you're 3 miles past one, you're 3 miles away, but if you are 7 miles past one, the closest one is 3 miles ahead of you, so the distance remains the same: 3 miles. This function doesn't consider which direction you go; it only gives the shortest distance possible, creating a step-pattern every 5 miles on either side within a 10-mile stretch.
Understanding this pattern is crucial as it influences how the rest of the function behaves on a graph and particularly when interpreting periodicity.

Amplitude

Amplitude in periodic functions describes how far the graph of the function extends vertically from its central axis to its peak. Here, the function \(D(x)\) peaks at 5 miles since that is the farthest point from a phone before a new phone appears along your path.
This measurement is crucial because it tells you the maximum distance you'd ever have to travel to reach an emergency phone. Imagine yourself at mile 0; the phone is right there. Move 5 miles, and you are at the farthest point from either the last or the next phone. This means the 'wave' formed by the distance graph will rise to this highest point before returning to its lowest point (0 miles) back where you find another phone.
Therefore, understanding the amplitude helps you grasp the function's full range of variations within one cycle of its repeating pattern.

Graph Interpretation

Interpreting the graph of a periodic function like \(D(x)\) can help you visualize and understand the behavior of the distance function as you move along the highway. The graph shows a series of 'V' shapes, representing the varying distances to the nearest telephone every 10 miles.
The key points of interpretation are:

• The "V" points touch the baseline at distances corresponding to miles 0, 10, 20, etc. where the distance \(D(x) = 0\).
• The peak of each "V" is at mile 5, 15, 25, etc., where the distance reaches its amplitude of 5 miles.
• This pattern repeats every 10 miles, indicating a period of 10 miles.

By grasping how to interpret these elements on the graph, you gain insights into the traveling conditions, especially in scenarios where knowing the distance to the nearest phone is crucial.

Patterns in Functions

Patterns are fundamental to understanding periodic functions, as they consistently repeat over regular intervals. In the case of \(D(x)\), the pattern emerges as a symmetric zigzag that repeats every 10 miles.
This pattern makes the function predictable: after every 10 miles, you'll encounter the same structural rhythm. This repetition is known as the function's period. Recognizing this allows one to anticipate and plan based on expected distances without constantly computing new values.
Such predictability is helpful in real-world applications beyond the highway scenario, such as anticipating tidal movements, managing power supply cycles, and much more. Recognizing, interpreting, and analyzing these patterns can thus enhance your ability to make reasoned decisions based on periodicity.