Problem 479 - Precalculus

Question

For the following exercises, refer to Table 4.31. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {2} & {4} & {5} & {7} & {8} & {10} & {11} & {15} & {17} \\ \hline f(x) & {12} & {28.6} & {52.8} & {70.3} & {99.9} & {112.5} & {125.8} & {127.9} & {135.1} & {135.9} \\\ \hline\end{array}$$ To the nearest whole number, what is the predicted carrying capacity of the model?

Step-by-Step Solution

Answer

The predicted carrying capacity is 136.

Step 1: Understanding the Problem

We are given a table with values for a function \(f(x)\) at different points \(x\). We need to predict the carrying capacity of this model, which is typically the maximum value \(f(x)\) can reach as \(x\) increases indefinitely.

Step 2: Identify Highest Given Value

Look at the values of \(f(x)\) in the table: 12, 28.6, 52.8, 70.3, 99.9, 112.5, 125.8, 127.9, 135.1, and 135.9. The highest value in the table is 135.9 at \(x = 17\).

Step 3: Predicting the Carrying Capacity

The carrying capacity is often the greatest value that \(f(x)\) approaches or reaches, considering that the table might not extend infinitely. Given the values, especially the highest one at \(x = 17\), and noting if \(f(x)\) appears to be leveling out or slowly increasing, 135.9 seems to be the carrying capacity.

Step 4: Round the Carrying Capacity

To find the carrying capacity to the nearest whole number, round 135.9. This rounds up to 136.

Key Concepts

Function Analysis

When tasked with analyzing a function, it's crucial to first understand what the function represents and how it behaves at different points. Function analysis involves scrutinizing the values of a function, like the ones provided in our table, to determine trends and patterns.

Here, we observe the function values at specific values of the variable, denoted by the table. Through this examination, we can identify where the function might reach its plateau or carrying capacity. This particular function analysis helps in understanding whether the function is continuously increasing, decreasing, or leveling off.

Key points when performing function analysis:

• Identify any patterns or trends in the data. Is it steadily increasing, decreasing, or leveling out?
• Determine if the changes between points are becoming smaller, indicating a possible approach to a limit.
• Pinpoint any anomalies or outliers that do not fit the trend.

This simplification is crucial when you need to predict future values or identify maximum potential values, as in the exercise's case of determining the carrying capacity.

Predicting Values

Predicting values is an essential part of understanding functions, especially when you're given a dataset like in our problem. By analyzing the trend or pattern that the data follows, you can infer future behavior. In our exercise, predicting involves observing values of the function as x increases.

The values assure us that as x increases, the values of f(x) also mostly increase or stabilize. From 135.1 at x = 15 to 135.9 at x = 17, the increase is minimal, hinting that the function is leveling off. Such stabilization often suggests a natural limit or carrying capacity.

When predicting values, consider these aspects:

• Are the increases between values decreasing, hinting at a plateau?
• Could there be logical reasons for the eventual flattening, like physical or theoretical limits?
• Is there consistency in how the values change over the range provided?

These observations are pivotal in estimating the carrying capacity for the function, helping to turn raw data points into meaningful insights.

Maximum Value

In the context of our exercise, the maximum value refers to the greatest value that the function f(x) can achieve given the data, which can also be called the carrying capacity in this context.

Identifying the maximum value from a set of data involves locating the highest number in the dataset, like 135.9 in our table. Understanding that, for the purposes of this exercise, the maximum value of f(x) might be approached as values continue to rise but may not actually exceed the maximum provided due to plateauing.

When determining the maximum value:

• Look for the highest number in your dataset.
• Consider whether the function could realistically exceed this value based on its trend, or if it will level off near this point.
• Ensure understanding of the context: in growth models, the maximum value or carrying capacity suggests a theoretical limit that the function values cannot surpass.

Rounding this value to the nearest whole number, as we've done with the value of 135.9 to get 136, provides a clearer and more usable result for practical purposes. This whole number approximation is often more applicable in real-world scenarios.