Problem 19 - Applied Calculus

Question

The number of species of lizards, \(N,\) found on an island off Baja California is proportional to the fourth root of the area, \(A,\) of the island. \(^{85}\) Write a formula for \(N\) as a function of \(A .\) Graph this function. Is it increasing or decreasing? Is the graph concave up or concave down? What does this tell you about lizards and island area?

Step-by-Step Solution

Answer

The function \( N(A) = k \cdot A^{1/4} \) is increasing and concave down, indicating larger islands support more lizard species, but with diminishing returns as area increases.

Step 1: Identifying the Relationship

Understand that the number of lizard species \( N \) is linked to the island area \( A \) by a proportional relationship, specifically with the fourth root of \( A \). This means \( N = k \cdot \sqrt[4]{A} \), where \( k \) is a proportionality constant.

Step 2: Writing the Function

Based on the relationship, we write the formula for the number of lizard species as a function of the island's area: \( N(A) = k \cdot A^{1/4} \). This expression uses the fourth root, expressed as an exponent of \( 1/4 \).

Step 3: Graphing the Function

Graph \( N(A) = k \cdot A^{1/4} \) to visualize the relationship. Choose some values for \( k \) (such as \( k=1 \) for simplicity) and plot several points by selecting different values for \( A \). The graph will show that as \( A \) increases, \( N \) also gently increases.

Step 4: Analyzing the Graph's Behavior

Because the function is of the form \( A^{1/4} \), it is an increasing function, meaning \( N(A) \) increases as \( A \) increases. Additionally, the graph is concave down, since the rate of increase decreases as \( A \) becomes larger. This indicates the species count grows slower with increasing area.

Step 5: Interpreting the Biological Implication

The relationship shows that as islands get larger, the number of lizard species increases at a decreasing rate. This means that larger areas support more species, but each addition in area leads to smaller gains in species count.

Key Concepts

Proportional Relationship

A proportional relationship is a mathematical connection between two quantities. Here, the number of lizard species, represented by \( N \), is proportional to the fourth root of the island's area, \( A \). This can be expressed with the formula \( N = k \cdot \sqrt[4]{A} \), where \( k \) is the constant of proportionality.

In simpler terms, as the area of the island increases, the number of lizard species also increases but not linearly. Instead, it follows a power function where the exponent is \( \frac{1}{4} \). This relationship helps us understand how biological factors such as habitat size can influence the diversity of species in an ecosystem.

In practical scenarios, determining \( k \) would usually require empirical data relating to the specific island being studied. However, the theoretical foundation remains the same, showing the nature of the relationship between size and biodiversity in this context.

Graphing Functions

Graphing functions allows us to visually interpret mathematical relationships between variables. For this exercise, we are looking at the function \( N(A) = k \cdot A^{1/4} \). To graph it, we first consider selecting different values for \( A \), calculating \( N(A) \) using our chosen \( k \) value, and plotting these points.

Once plotted, the graph of this function will exhibit an upward curve due to the positive exponent. This is because as \( A \) increases, \( N \) also increases. This visual representation can be vital for understanding how the lizard species number changes with the island area.

• Start by picking a value for \( k \). If not given, \( k = 1 \) is a sensible starting point.
• Calculate \( N \) for several values of \( A \); for instance, \( A = 1, 10, 100, 1000 \).
• Plot these points on a graph with \( A \) on the x-axis and \( N \) on the y-axis.
• Connect these points with a smooth curve to see the trend.

The graph ultimately shows us not just whether the function increases, but indicates the specific non-linear way in which this happens, providing a deeper understanding of changes in biodiversity across island sizes.

Concavity in Graphs

Concavity in a graph describes how a function curves and can tell us much about the rate of change of a relationship. When examining \( N(A) = k \cdot A^{1/4} \), we observe that its graph is concave down. This means that as the island's area grows, the increase in the number of lizard species slows down.

The key to understanding concavity here lies in the function's shape:

• Concave down graphs "bend" downward, like an upside-down bowl, indicating that while the function is increasing, the rate of increase is slowing.
• In mathematical terms, the second derivative of the function \( N(A) \), which examines how the rate of change itself changes, is negative.

This understanding is crucial from a biological perspective, as it indicates that larger habitats do indeed support greater biodiversity, but the benefits decrease with size. This could imply that conservation efforts might focus on increasing habitat areas until diminishing returns suggest other approaches to biodiversity conservation.