Problem 54
Question
For the following exercises, refer to Table 12 . $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} x & 0 & 2 & 4 & 5 & 7 & 8 & 10 & 11 & 15 & 17 \\ \hline f(x) & 12 & 28.6 & 52.8 & 70.3 & 99.9 & 112.5 & 125.8 & 127.9 & 135.1 & 135.9 \end{array} $$ To the nearest whole number, what is the predicted carrying capacity of the model?
Step-by-Step Solution
Verified Answer
The predicted carrying capacity is 136.
1Step 1: Understand the Problem
Recognize that we need to find the carrying capacity of the model given the table of values for \(x\) and \(f(x)\). The carrying capacity is the horizontal asymptote where \(f(x)\) levels off and does not increase significantly as \(x\) continues to increase.
2Step 2: Analyze the Table Values
Examine the \(f(x)\) values as \(x\) increases. Notably, observe that \(f(x)\) rises steeply until \(x=8\) and then increases very little from \(x=10\) to \(x=17\). This suggests that \(f(x)\) is approaching a horizontal asymptote, which would be the carrying capacity.
3Step 3: Look for the Asymptotic Value
To find the asymptotic value, examine the \(f(x)\) values for larger \(x\) values. At \(x=15\) and \(x=17\), \(f(x)\) is 135.1 and 135.9, respectively. The minor increase suggests these values are close to the carrying capacity.
4Step 4: Determine the Closest Whole Number
Given that 135.9 at \(x=17\) is the largest \(f(x)\) value and chance of it leveling, the carrying capacity is approximately 136, rounding to the nearest whole number because 135.9 is closer to 136 than 135.
Key Concepts
Horizontal AsymptoteRegression AnalysisFunction Behavior Analysis
Horizontal Asymptote
The concept of a horizontal asymptote is central to understanding the carrying capacity in this exercise. A horizontal asymptote represents a value that a function approaches but never quite reaches as the input grows infinitely large. It's like a flat line the curve cannot cross. In our context, the function \(f(x)\) represents a model with values getting closer to a certain number as \(x\) increases. Observing the table, notice the steep rise in \(f(x)\) up until \(x = 8\). Then from \(x = 10\) to \(x = 17\), \(f(x)\) barely increases, hinting towards the asymptote.
Think of this asymptote as the theoretical limit or maximum value the model can achieve. This value is identified as the carrying capacity. In our table, the final values show \(f(x)\) reaching 135.1 and 135.9. These values help us conclude that the carrying capacity is approximately 136, serving as the horizontal asymptote.
Think of this asymptote as the theoretical limit or maximum value the model can achieve. This value is identified as the carrying capacity. In our table, the final values show \(f(x)\) reaching 135.1 and 135.9. These values help us conclude that the carrying capacity is approximately 136, serving as the horizontal asymptote.
Regression Analysis
Regression analysis is a statistical process for estimating the relationships among variables. It's particularly useful in determining trends and forecasts. In this exercise, if we were to apply regression analysis, it would help reinforce the determination of the carrying capacity by fitting a curve model to the given data.
With the given data plot, you would first establish the trend line using a software or statistical tool. This line helps in identifying the nature of the change as \(x\) grows. More sophisticated regression techniques such as logistic regression could point out how \(f(x)\) approaches a limit, thus supporting the asymptote conclusion.
In practical scenarios, regression helps confirm not just the overall trend, but also the adequacy and accuracy of the model in prediction.
With the given data plot, you would first establish the trend line using a software or statistical tool. This line helps in identifying the nature of the change as \(x\) grows. More sophisticated regression techniques such as logistic regression could point out how \(f(x)\) approaches a limit, thus supporting the asymptote conclusion.
In practical scenarios, regression helps confirm not just the overall trend, but also the adequacy and accuracy of the model in prediction.
Function Behavior Analysis
When studying a function, analyzing its behavior is essential to understand trends, limits, and capacities. The term 'function behavior' generally refers to how the function values change as the input values change. In this case, tracking \(f(x)\) offers insights into the function's increasing, decreasing, and leveling off phases.
Initially, \(f(x)\) values increase sharply as seen up until \(x = 8\). Then, as the input continues to increase, the rate at which \(f(x)\) grows begins to shrink. This leveling off signifies that a maximum threshold -- or carrying capacity -- is being reached. Analysts look for these behaviors to determine important characteristics such as limits or bounds of a function.
Analyzing a function's behavior involves looking for both quantitative changes and patterns in its growth. It's about assessing whether the function will plateau, how soon it might stop growing, and ultimately what value it is approaching, as seen in our table where the function behavior helped deduce a carrying capacity of 136.
Initially, \(f(x)\) values increase sharply as seen up until \(x = 8\). Then, as the input continues to increase, the rate at which \(f(x)\) grows begins to shrink. This leveling off signifies that a maximum threshold -- or carrying capacity -- is being reached. Analysts look for these behaviors to determine important characteristics such as limits or bounds of a function.
Analyzing a function's behavior involves looking for both quantitative changes and patterns in its growth. It's about assessing whether the function will plateau, how soon it might stop growing, and ultimately what value it is approaching, as seen in our table where the function behavior helped deduce a carrying capacity of 136.
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