Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Suppose an insect population density in thousands per acre during year \(n\) can be modeled by the following recursively defined sequence. $$\begin{aligned} &a_{1}=8\\\ &a_{n}=2.9 a_{n-1}-0.2 a_{n-1}^{2}, \text { for } n>1 \end{aligned}$$ (a) Find the population for \(n=1,2,3\) (b) Graph the sequence for \(n=1,2,3, \ldots, 20 .\) Use the window \([0,21]\) by \([0,14] .\) Interpret the graph.

Step-by-Step Solution

Answer

(a) For \( n=1 \), \( a_1=8 \); for \( n=2 \), \( a_2=10.4 \); for \( n=3 \), \( a_3=8.528 \).

Step 1: Calculate for n=1

From the given information, we know that \( a_1 = 8 \). There is no calculation required for \( n = 1 \) as it is provided in the problem.

Step 2: Calculate for n=2

To find \( a_2 \), use the given recursive formula: \( a_n = 2.9 a_{n-1} - 0.2 a_{n-1}^2 \). Substitute \( a_1 = 8 \) into the formula: \[a_2 = 2.9 \times 8 - 0.2 \times 8^2 \]This simplifies to: \( a_2 = 23.2 - 12.8 = 10.4 \).

Step 3: Calculate for n=3

To find \( a_3 \), use the recursive formula with \( a_2 = 10.4 \):\[a_3 = 2.9 \times 10.4 - 0.2 \times 10.4^2 \]This simplifies to: \( a_3 = 30.16 - 21.632 = 8.528 \).

Step 4: Graph the Sequence

To graph the sequence, calculate additional values for \( n \) from 4 to 20 using the recursive formula, continuing similar calculations as above for each subsequent \( n \). Plot the calculated \( a_n \) on the graph with \( n \) on the x-axis and \( a_n \) on the y-axis. The window range is set as \([0,21]\) by \([0,14]\).

Step 5: Interpret the Graph

Upon examining the graph, the pattern of the population denisty displays interesting behavior. Initially, there's a rise, followed by some fluctuations, which could indicate an oscillation or stabilization period. The graph highlights how population might approach an equilibrium due to non-linear interactions in the recursive formula.

Key Concepts

Population Modeling

Population modeling refers to using mathematical equations to represent how a population changes over time. In this exercise, we are introduced to a recursively defined sequence to model the density of an insect population. A recursive sequence allows us to calculate future population sizes based on previous sizes. This particular model starts with an initial population and uses the relation:

• \( a_1 = 8 \) sets the initial population density in thousands per acre.
• \( a_n = 2.9 a_{n-1} - 0.2 a_{n-1}^{2} \) determines subsequent sizes, factoring in previous sizes.

By plugging previous terms back into this equation, we can predict population changes over years. It's important to note that the quadratic term \(-0.2 a_{n-1}^2\) indicates that the population cannot grow indefinitely—it accounts for negative feedback effects like limited resources. Thus, this model helps in understanding how an insect population might stabilize over time or experience fluctuations.

Graph Interpretation

Graph interpretation plays a key role in understanding recursively defined sequences in population modeling. When we plot the insect population over multiple years, we get a visual snapshot of its trend. Initially, this sequence demonstrates growth. For example, moving from \(n=1\) to \(n=2\), we see an increase in the population. However, when plotted over the window \[0, 21\] by \[0, 14\], the graph begins to show oscillations and eventually stabilizes around a specific population density. These fluctuations are indicative of the balance achieved by recursive relationships where linear and quadratic terms interact. By examining the graph:

• We can observe how the insect population growth initially accelerates but gradually diminishes as the negative feedback effect becomes more pronounced.
• The graph may suggest a possible equilibrium point where the population stabilizes.
• Patterns like periodicity might also emerge depending on equation specifics.

Graph interpretation in such models helps in anticipating future population behavior under similar circumstances.

Precalculus Applications

Precalculus provides the necessary mathematical foundation to handle recursive sequences and functions like this population model. This exercise uses precalculus to bridge understanding between algebraic manipulation and real-world applications. Mastering recursive sequences deepens one's comprehension of dynamic systems found in fields like ecology and economics. The recursive formula \[ a_n = 2.9 a_{n-1} - 0.2 a_{n-1}^2 \] illustrates precalculus principles by

• requiring manipulation of terms to predict the next term in a sequence.
• introducing concepts of iterative calculations, which are prevalent in programming and analysis.

This exercise also demonstrates how recursions are not just abstract concepts but apply to modeling realistic problems, such as tracking animal or insect populations. By gaining proficiency in these precalculus tools, students acquire useful skills for interpreting graphs, analyzing patterns, and applying them to complex systems.