Problem 66 - College Algebra and Calculus: An Applied Approach

Question

The gestation period of rabbits is about 29 to 35 days. Therefore, the population of a form (rabbits' home) can increase dramatically in a short period of time. The table gives the population of a form, where \(t\) is the time in months and \(N\) is the rabbit population. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline N & 2 & 8 & 10 & 14 & 10 & 15 & 12 \\ \hline \end{array} $$ (a) Use a graphing utility to graph the population as a function of time. (b) Find any points of discontinuity in the function. Explain your reasoning.

Step-by-Step Solution

Answer

After graphing, one can visually inspect the graph for any points of discontinuity. This would be points where there are sudden leaps or drops in the population not explained by the surrounding values.

Step 1: Plotting the Data

Use a graphing tool to plot the given data points. On the x-axis, specify time \(t\) in months, and on the y-axis, specify the rabbit population \(N\). Each pair of (time, population) — (0, 2), (1, 8), (2, 10), (3, 14), (4, 10), (5, 15), and (6, 12) - represents a point on your graph.

Step 2: Creating the Graph

Once the points are plotted, link them with a curve or straight lines to illustrate the rabbit population trend over time. The choice between curve and straight lines will depend on understanding the nature of the data. Since in reality the population growth rate may not be constant, using curve linkage could provide a more accurate depiction of the population trend.

Step 3: Identifying Points of Discontinuity

By observing the graph, identify any sharp changes or deviations in the growth trend. In the context of this problem, a point of discontinuity would imply a sudden, unnatural fluctuation in the rabbit population which might not be explained by natural breeding.

Key Concepts

Discontinuity

Discontinuity in a function occurs when there is an abrupt change or 'break' in the graph of the function. This could mean a sudden jump in the values, an undefined point, or a gap where the function isn't continuous.

In the context of rabbit population growth, a point of discontinuity might suggest an unexpected increase or decrease in the population numbers that doesn't follow the trend. Looking at the given data, note any sharp spikes or dips. For example, if the population changed from 14 to 10, there might be a discontinuity if such a change is not expected naturally.

To find discontinuities, examine the plotted graph and identify where the line connecting the points shows a discontinuity, such as sharp angles or any significant departure from the expected smooth curve of population growth. Remember, naturally smooth changes should create a more predictable curve.

Graphing Functions

Graphing functions involves plotting points on a coordinate grid based on given data and using those points to visualize the relationship between variables. For rabbit population growth, each entry in the data table gives coordinates with time on the x-axis and population on the y-axis.

Here's how you can graph the function:

• Plot each point from the table: (0, 2), (1, 8), (2, 10), etc.
• After plotting, connect the points to see the trend. A curve typically represents a natural growth model better than straight lines.

Using a curve can emphasize natural fluctuations in the population due to births or environmental factors over time, providing insight beyond mere numbers. Observing the graph allows us to see trends such as peaks and drops more clearly, helping to better understand and predict future population behavior.

Data Plotting

Data plotting is a fundamental process in graphing where data points are marked on a graph to represent numerical information visually. This can provide a clear picture of trends and is especially useful in situations like population studies.

To plot the data correctly, follow these steps:

• Select an appropriate scale for both the x-axis (time in months) and the y-axis (rabbit population).
• Mark the given data pairs on the graph: for example, (0, 2), (1, 8).
• These points help form a visual trend, making patterns or anomalies - such as those suggesting discontinuity - more apparent.

Accurate data plotting is crucial because it sets the foundation for analyzing trends and making predictions based on the visual information.
Ensure each plotted point accurately reflects the corresponding data to allow for proper interpretation and further mathematical exploration.