Problem 2 - Applied Calculus

Question

Use the description of the function to sketch a possible graph. Put a label on each axis and state whether the function is increasing or decreasing. The height of a sand dune is a function of time, and the wind erodes away the sand dune over time.

Step-by-Step Solution

Answer

The graph is a decreasing function with time on the x-axis and height on the y-axis.

Step 1: Understand the Function Description

The problem states that the height of the sand dune decreases over time due to wind erosion. We understand this to mean that as time passes, the height gets smaller, indicating a decreasing function.

Step 2: Identify the Axis Labels

In this scenario, time is typically represented on the horizontal axis (x-axis) and height on the vertical axis (y-axis). Label the x-axis as "Time" and the y-axis as "Height of the Sand Dune".

Step 3: Determine the Behavior of the Function

Since the function is decreasing, the graph should show a downward trend as time increases. Starting from a certain height on the y-axis, draw a line that slopes downwards to the right, indicating the decrease in height over time.

Step 4: Sketch the Graph

Now, sketch the graph according to the previous steps: draw the axes with labels, plot a starting point at a positive height value on the y-axis, and draw a line or curve that trends downward towards the x-axis, indicating a constant erosion over time.

Step 5: Conclude the Nature of Function

Since the height decreases with time due to constant wind erosion, we confirm the function is indeed decreasing.

Key Concepts

Graph Sketching

When you're tasked with sketching a graph from a verbal description, it's important to gather what key features the graph should have. In this case, we're focusing on the erosion of a sand dune over time. This process can be visualized as a graph where time is on one axis and height is on the other. This task involves understanding the relationship between these two quantities, which will help in sketching an accurate representation.
Graph sketching typically starts with identifying the axes:

• X-axis (Horizontal): This usually represents the independent variable, which in our example is time.
• Y-axis (Vertical): This represents the dependent variable, or the height of the sand dune in this scenario.

With the axes labeled, consider the function's description: the sand dune's height decreasing over time due to wind. This will be represented by a line or curve sloping downwards as it moves from left to right. By sketching the line this way, we show how the height decreases progressively. Remember, the exact shape and steepness of the line depend on how quickly the dune is eroding.

Time and Height Relationship

The relationship between time and height, as given in this problem, is a perfect example of a real-life scenario translating into a mathematical function. As time moves forward, we see changes in the sand dune's height due to constant weathering by wind.
In this context, let's simplify our understanding:

• Time (X-axis): As time progresses, it moves towards the positive direction on the x-axis.
• Height (Y-axis): The height starts from an initial value and decreases over time due to erosion, indicating it moves downward on the y-axis.

Here, the function shows a "cause and effect" kind of relationship: time causes the height to decrease. This relationship is essential to understanding how different factors in the real world can create a predictable mathematical pattern that we can graph and analyze.

Function Behavior Analysis

Understanding the behavior of the function is crucial when interpreting its real-world implications. In our example, the sand dune height's decrease over time due to erosion depicts a clear trend.
Key aspects to consider in the behavior of this function include:

• Decreasing Nature: Since the height consistently lowers as time advances, this function is categorized as decreasing.
• Visual Representation: A decreasing function graph is commonly shown with a downward slope from left to right.

Analyzing function behavior helps to predict future values and understand the relationship between the variables better. In practical terms, it allows for anticipating how the sand dune might look after a certain period, aiding in environmental and scientific planning. Therefore, recognizing that this function decreases is not just about drawing it correctly but also about grasping the broader implications of such a process in real-world applications.